Osaka Central Advanced Mathematical Institute (OCAMI)
Osaka Metropolitan University
MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics

OCAMI Reports Vol. 6 (2023)
doi: 10.24544/omu.20231219-001

The 21th International Conference
on the Teaching of Mathematical Modellingand Applications
Organized by
Toshikazu Ikeda
Akihiko Saeki
Keiichi Nishimura
Akio Matsuzaki
Takashi Kawakami

September 10–15, 2023

© 2023 OCAMI.

OCAMI. The 21th International Conference on the Teaching of Mathematical Modelling and Applications. OCAMI Reports. Vol. 6, Osaka Central Advanced Mathematical Institute, Osaka Metropolitan
University. 2023, 160 pp. doi: 10.24544/omu.20231219-001

Organizers
Toshikazu Ikeda
Yokohama National University
Akihiko Saeki
Kanazawa Institute of Technology
Keiichi Nishimura
Tokyo Gakugei University
Akio Matsuzaki
Saitama University
Takashi Kawakami
Utsunomiya University

The 21th International Conference
on the Teaching of Mathematical
Modelling and Applications

Conference Booklet

10-15 September 2023
Awaji Island, Japan

Table of Contents

Welcome to ICTMA21 ..................................................... 1
Local Organizing Committee ............................................ 2
ICTMA21 Schedule ......................................................... 5
Keynote, Plenary, Panel, & Lectures ................................. 6
Scientific Programme .................................................... 10
Floor Map .................................................................... 13
Programme ................................................................. 15
Abstract ...................................................................... 34
Keynote .............................................................. 34
Plenary Lecture .................................................... 36
Plenary Panel ....................................................... 40
Special Lecture ..................................................... 42
Long & Short Presentation ..................................... 44
Poster Presentation .............................................. 137
Conference Information ............................................... 144
Supporting Organizations & Supporters .......................... 153
List of Participants ....................................................... 154

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ICTMA21 Schedule
SundayMondayTuesday

WednesdayThursdayFridaySeptember 10th

September 11th

September 12th

September 13th

September 14th

September 15th

8:45

8:45
Registration
8:45-10:00

9:30

Plenary lecture1
Opening Ceremony

10:00

10:00-10:30
10:30

Coffee break

10:45

10:30-11:00

9:30

Plenary lecture2(Greefrath)(Czocher)

9:30-10:30

9:30-10:30

Session6

Plenary panel

9:30-11:00

9:30-11:00
10:30
10:45

Excursion

11:00
11:15

Keynote speech(Goos & Carreira)Session3
10:45-12:15

11:00(Option)Pollak Award

10:30Lunch (Take away
lunch boxes)

11:00-12:30

12:00

Session7
11:15-12:45

Lunch

12:15-13:30

12:30-13:45

Lunch

12:00

12:45
13:00

12:45-13:45

13:30

13:30

13:45
Early
CareerResearch

Registration

13:00-16:30
Executive

15:45

MeetingBoard
15:0017:00

13:45

Session4
Session1

13:30-15:00

13:45-15:15

Day

13:00-17:00

15:15
15:30

Session8
13:45-15:15
15:00

Coffee break
Coffee break

15:00-15:30

Session5
Session2

Coffee break
15:15-15:45

15:15-15:45

15:30-17:00

15:45-17:15

Plenary lecture3

15:15
15:30
15:45

(Shimizu)
15:45-16:45

16:45

16:30
16:45

General meeting

17:00
17:15

Special Lecture
Welcome reception
17:00-(Galluzzo& Cheung)

Poster session &16:45-17:45

Happy hour

17:00
17:15
17:45
18:00

17:00-18:30

17:30-18:30

18:30

18:30
Happy hour

19:00

11:15-12:00

12:30

Lunch

12:45
13:00

17:45
18:00

Closing ceremony
12:15

12:30

16:30

11:15

ceremony &

12:15

15:00

10:00

Conference dinner

18:30-19:30

18:00-

19:00

19:30

19:30

20:00

20:00

5

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Plenary Lecture 1
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Plenary Panel
RELATIONS AMONG PROBLEM SOLVING/POSING, CREATIVITY AND
MATHEMATICAL MODELLING
Chair: Jonas Bergman Ärlebäck
Linköping University, Sweden

Jinfa Cai
University of Delaware, USA

Gabriele Kaiser
University of Hamburg, Germany& Nord University, Norway

Roza Leikin
University of Haifa, Israel

Special Lecture
IMMC: CELEBRATING 10 YEARS OF INFLUENCING EDUCATIONALCHANGE
Alfred Cheung
Benjamin Galluzzo
NeoUnion ESC Organization,
Clarkson University, USA
Hong Kong, China

8

Early Career Researcher Day
EARLY CAREER RESEARCH DAY: REFLECTIONS ON PUBLISHING ON
MODELLING RESEARCH
Gabriele Kaiser
University of Hamburg, Germany& Nord University, NorwayŞerife Sevinç
Middle East Technical University,
TurkeyBEGINNING WALKING IN TWO WORLDS AS A TEACHER AND
RESEARCHER OF MATHEMATICAL MODELLING IN THE CLASSROOM
Gloria Stillman
Retired, Australia

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12

Floor Map
Sep.11th
12th
14th

Rooms 401 and 402 are the LOC office rooms.
Please come here if you need anything.
CORPORATE EXHIBITIONS

Session room

Sep.
10th
(Sun)

EXECUTIVE BOARD MEETING

Break area
(lunch & coffee)

401

Session room

Session room

Feel free to use
the lounge.

Break area
(lunch & coffee)

Session room

Lounge
308

309

310

307

306
303
304

305

From the Hotel
Sep.
14th
(Thu)

Connecting Corridor to the Hotel

CONFERENCE DINNER

201

202

13

Conference Center Entrance

101
102
103

CAREER

The hotel entrance is near the bus
Hotel Entrance
Bus Stop

stop. If you enter via the hotel

Conference Center Entrance (1F)

entrance, you can access thesecond floor of the conference
center via the connecting corridor
on the second floor of the hotel.

14

ProgrammeSunday, September 10, 2023
Time/Location

Activity

13:00-17:00/BF1 Lobby

REGISTRATIONEARLY CAREER RESEARCH DAY
E1 EARLY CAREER RESEARCH DAY: REFLECTIONS ON
PUBLISHING ON MODELLING RESEARCH
Gabriele Kaiser , University of Hamburg,

Germany & Nord University, NorwayŞerife Sevinç , Middle East Technical University, Turkey

13:00-16:30/B1F Event Hall

E2 BEGINNING WALKING IN TWO WORLDS AS A TEACHER AND
RESEARCHER OF MATHEMATICAL MODELLING IN THE
CLASSROOM
Gloria Stillman , Retired, Australia
Chair
Takashi Kawakami , Utsunomiya University, Japan

15:00-17:00/Room402

EXECUTIVE BOARD MEETING

17:00-19:00/B1F Lobby

WELCOME RECEPTION

15

㻹㼛㼚㼐㼍㼥㻘㻌㻿㼑㼜㼠㼑㼙㼎㼑㼞㻌㻝㻝㻘㻌㻞㻜㻞㻟Time/Location

Activity㻤㻦㻠㻡㻙㻝㻜㻦㻜㻜㻛㻮㻝㻲㻌㻸㼛㼎㼎㼥㻾㻱㻳㻵㻿㼀㻾㻭㼀㻵㻻㻺㻝㻜㻦㻜㻜㻙㻝㻜㻦㻟㻜㻛㻮㻝㻲㻌㻱㼢㼑㼚㼠㻌㻴㼍㼘㼘㻻㻼㻱㻺㻵㻺㻳㻌㻯㻱㻾㻱㻹㻻㻺㼅㻝㻜㻦㻟㻜㻙㻝㻝㻦㻜㻜㻛㻮㻝㻲㻌㻸㼛㼎㼎㼥㻯㻻㻲㻲㻱㻱㻌㻮㻾㻱㻭㻷㻌㻔㻯㼛㼒㼒㼑㼑㻌㻮㼞㼑㼍㼗㻌㻭㼞㼑㼍㻌㼛㼚㻌㻮㻝㻲㻌㻸㼛㼎㼎㼥㻕㻷㻱㼅㻺㻻㼀㻱㻌㻿㻼㻱㻱㻯㻴㻌㻔㻶㻻㻵㻺㼀㻌㻼㻾㻱㻿㻱㻺㼀㻭㼀㻵㻻㻺㻕㻷㻝䚷㻯㻻㻺㻯㻱㻼㼀㼁㻭㻸㻵㻿㻵㻺㻳㻌㼀㻴㻱㻌㻾㻱㻸㻭㼀㻵㻻㻺㻿㻴㻵㻼㻮㻱㼀㼃㻱㻱㻺㻌㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻭㻸㻌㻹㻻㻰㻱㻸㻸㻵㻺㻳㻌㻭㻺㻰㻵㻺㼀㻱㻾㻰㻵㻿㻯㻵㻼㻸㻵㻺㻭㻾㼅㻌㻿㼀㻱㻹㻌㻱㻰㼁㻯㻭㼀㻵㻻㻺㻝㻝㻦㻜㻜㻙㻝㻞㻦㻟㻜㻛㻱㼢㼑㼚㼠㻌㻴㼍㼘㼘㻹㼑㼞㼞㼕㼘㼥㼚㻌㻳㼛㼛㼟 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㼠㼔㼑㻌㻿㼡㼚㼟㼔㼕㼚㼑㻌㻯㼛㼍㼟㼠㻘㻌㻭㼡㼟㼠㼞㼍㼘㼕㼍㻿㼡㼟㼍㼚㼍㻌㻯㼍㼞㼞㼑㼕㼞㼍 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㼠㼔㼑㻌㻭㼘㼓㼍㼞㼢㼑㻌㼍㼚㼐㻌㻵㼚㼟㼠㼕㼠㼡㼠㼑㻌㼛㼒㻱㼐㼡㼏㼍㼠㼕㼛㼚㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻸㼕㼟㼎㼛㼚㻘㻌㻼㼛㼞㼠㼡㼓㼍㼘㻯㼔㼍㼕㼞㻹㼕㼘㼠㼛㼚㻌㻾㼛㼟㼍 㻘㻌㻲㼑㼐㼑㼞㼍㼘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻻㼡㼞㼛㻌㻼㼞㼑㼠㼛㻘㻌㻮㼞㼍㼦㼕㼘㻝㻞㻦㻟㻜㻙㻝㻟㻦㻠㻡㻛㻟㻲㻌㻒㻌㻠㻲㻌㻸㼁㻺㻯㻴㻌㻔㼀㼍㼗㼑㻌㼍㼣㼍㼥㻌㼘㼡㼚㼏㼔㻌㼎㼛㼤㼑㼟㻌㼕㼚㻌㻯㼛㼒㼒㼑㼑㻌㻮㼞㼑㼍㼗㻌㻭㼞㼑㼍㼟㻌㼛㼚㻌㻟㻲㼍㼚㼐㻌㻠㻲㻕

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㻔㼀㼑㼍㼏㼔㼑㼞㻌㻱㼐㼡㼏㼍㼠㼕㼛㼚㻕㻯㼔㼍㼕㼞㻾㼕㼠㼍㻌㻮㼛㼞㼞㼛㼙㼑㼛㻌㻲㼑㼞㼞㼕㼅㼛㼚㼓㼖㼕㼍㼚㻌㻿㼡㼚㼅㼛㼚㼓㼖㼕㼍㼚㻌㻿㼡㼚 㻘㻌㻱㼍㼟㼠㻌㻯㼔㼕㼚㼍㻌㻺㼛㼞㼙㼍㼘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㻯㼔㼕㼚㼍㻶㼕㼚㼓㻌㻯㼔㼑㼚㼓 㻘㻌㻱㼍㼟㼠㻌㻯㼔㼕㼚㼍㻌㻺㼛㼞㼙㼍㼘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㻯㼔㼕㼚㼍㻻㻭㻞㻌㻔㻿㼔㼛㼞㼠㻦㻌㻟㻜㼙㼕㼚㻚㻕䚷㻭㻌㼀㻱㻿㼀㻌㻵㻺㻿㼀㻾㼁㻹㻱㻺㼀㻌㻲㻻㻾㻌㻼㻾㻻㻲㻱㻿㻿㻵㻻㻺㻭㻸㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻭㻸㻌㻷㻺㻻㼃㻸㻱㻰㻳㻱㻌㻻㻺㻌㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻭㻸㻌㻹㻻㻰㻱㻸㻸㻵㻺㻳㻌㻙㻲㻵㻾㻿㼀㻌㻾㻱㻿㼁㻸㼀㻿㻾㼕㼠㼍㻌㻮㼛㼞㼞㼛㼙㼑㼛㻌㻲㼑㼞㼞㼕 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻷㼍㼟㼟㼑㼘㻘㻌㻳㼑㼞㼙㼍㼚㼥㻳㼕㼘㼎㼑㼞㼠㻌㻳㼞㼑㼑㼒㼞㼍㼠㼔 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻹㾇㼚㼟㼠㼑㼞㻘㻌㻳㼑㼞㼙㼍㼚㼥㻴㼍㼚㼟㻙㻿㼠㼑㼒㼍㼚㻌㻿㼕㼘㼘㼑㼞 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㼃㾇㼞㼦㼎㼡㼞㼓㻘㻌㻳㼑㼞㼙㼍㼚㼥㻻㻭㻟䠄㻸㼛㼚㼓㻦㻌㻠㻡㼙㼕㼚㻚䠅䚷㻿㼅㻿㼀㻱㻹㻌㻰㼅㻺㻭㻹㻵㻯㻿㻌㻿㻻㻲㼀㼃㻭㻾㻱㻌㻭㻺㻰㻿㻯㻱㻺㻭㻾㻵㻻㻿㻌㻲㻻㻾㻌㻯㻻㻺㼀㻵㻺㼁㻻㼁㻿㻌㻼㻾㻻㻲㻱㻿㻿㻵㻻㻺㻭㻸㻌㻰㻱㼂㻱㻸㻻㻼㻹㻱㻺㼀㻾㼛㼛㼙㻟㻜㻝㻮㻌㻔㻸㼛㼚㼓㻛㻿㼔㼛㼞㼠㻦㻌㻝㻟㻦㻠㻡㻙㻝㻡㻦㻜㻜㻕
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Carolina Guerrero-Ortiz㼅㼡㼗㼕㼗㼛㻌㻭㼟㼍㼙㼕㻙㻶㼛㼔㼍㼚㼟㼟㼛㼚㻻㻭㻞㻟䚷㼃㻴㻱㻺㻌㻰㻵㻳㻵㼀㻭㻸㻌㼀㻻㻻㻸㻿㻌㻮㻱㻯㻻㻹㻱㻌㻭㻌㻿㼁㻼㻼㻻㻾㼀㻌㻲㻻㻾㻹㻻㻰㻱㻸㻸㻵㻺㻳㻌㻭㻯㼀㻵㼂㻵㼀㻵㻱㻿㻦㻌㻼㻾㻱㻙㻿㻱㻾㼂㻵㻯㻱㻌㼀㻱㻭㻯㻴㻱㻾㻿䇻㻵㻺㻿㼀㻾㼁㻯㼀㻵㻻㻺㻭㻸㻌㼀㻭㻿㻷㻿㻯㼍㼞㼛㼘㼕㼚㼍㻌㻳㼡㼑㼞㼞㼑㼞㼛㻙㻻㼞㼠㼕㼦 㻘㻌㻼㼛㼚㼠㼕㼒㼕㼏㼕㼍㻌㼁㼚㼕㼢㼑㼞㼟㼕㼐㼍㼐㻌㻯㼍㼠㽾㼘㼕㼏㼍㻌㼐㼑㻌㼂㼍㼘㼜㼍㼞㼍㽸㼟㼛㻘㻌㻯㼔㼕㼘㼑㻻㻭㻞㻠䚷㻭㻌㼀㻴㻱㻻㻾㻱㼀㻵㻯㻭㻸㻌㻹㻻㻰㻱㻸㻌㻲㻻㻾㻌㻰㻱㻿㻯㻾㻵㻮㻵㻺㻳㼀㻱㻯㻴㻺㻻㻸㻻㻳㻵㻯㻭㻸㻘㻌㻼㻱㻰㻭㻳㻻㻳㻵㻯㻭㻸㻘㻌㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻭㻸㻌㻭㻺㻰㻌㻱㼄㼀㻾㻭㻙㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻭㻸㻌㻷㻺㻻㼃㻸㻱㻰㻳㻱㻌㻲㻻㻾㻌㼀㻱㻭㻯㻴㻵㻺㻳㻌㻹㻻㻰㻱㻸㻸㻵㻺㻳㻶㼛㼟㼑㼒㼍㻌㻯㼍㼟㼠㼕㼘㼘㼛㻙㻲㼡㼚㼑㼟 㻘㻌㻼㼛㼚㼠㼕㼒㼕㼏㼕㼍㻌㼁㼚㼕㼢㼑㼞㼟㼕㼐㼍㼐㻌㻯㼍㼠㽾㼘㼕㼏㼍㻌㼐㼑㻌㼂㼍㼘㼜㼍㼞㼍㽸㼟㼛㻘㻌㻯㼔㼕㼘㼑㻯㼍㼞㼛㼘㼕㼚㼍㻌㻳㼡㼑㼞㼞㼑㼞㼛㻙㻻㼞㼠㼕㼦 㻘㻌㻼㼛㼚㼠㼕㼒㼕㼏㼕㼍㻌㼁㼚㼕㼢㼑㼞㼟㼕㼐㼍㼐㻌㻯㼍㼠㽾㼘㼕㼏㼍㻌㼐㼑㻌㼂㼍㼘㼜㼍㼞㼍㽸㼟㼛㻘㻌㻯㼔㼕㼘㼑18

㻿㻼㻱㻯㻵㻭㻸㻌㻸㻱㻯㼀㼁㻾㻱㻿㻝䚷㻵㻹㻹㻯㻦㻌㻯㻱㻸㻱㻮㻾㻭㼀㻵㻺㻳㻌㻝㻜㻌㼅㻱㻭㻾㻿㻌㻻㻲㻌㻵㻺㻲㻸㼁㻱㻺㻯㻵㻺㻳㻱㻰㼁㻯㻭㼀㻵㻻㻺㻭㻸㻌㻯㻴㻭㻺㻳㻱㻝㻣㻦㻟㻜㻙㻝㻤㻦㻟㻜㻛㻮㻝㻲㻌㻱㼢㼑㼚㼠㻌㻴㼍㼘㼘㻮㼑㼚㼖㼍㼙㼕㼚㻌㻳㼍㼘㼘㼡㼦㼦㼛 㻘㻌㻯㼘㼍㼞㼗㼟㼛㼚㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㼁㻿㻭㻭㼘㼒㼞㼑㼐㻌㻯㼔㼑㼡㼚㼓 㻘㻌㻺㼑㼛㼁㼚㼕㼛㼚㻌㻱㻿㻯㻌㻻㼞㼓㼍㼚㼕㼦㼍㼠㼕㼛㼚㻘㻌㻴㼛㼚㼓㻌㻷㼛㼚㼓㻘㻌㻯㼔㼕㼚㼍㻯㼔㼍㼕㼞㻵㼞㼑㼚㼑㻌㻲㼑㼞㼞㼍㼚㼐㼛 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼍㼠㻌㼐㼑㻌㼂㼍㼘㽳㼚㼏㼕㼍㻘㻌㻿㼜㼍㼕㼚㻝㻤㻦㻟㻜㻙㻝㻥㻦㻟㻜㻛㻮㻝㻲㻌㻸㼛㼎㼎㼥㻴㻭㻼㻼㼅㻌㻴㻻㼁㻾㻝㻜㻦㻟㻜㻙㻝㻣㻦㻟㻜㻛㻾㼛㼛㼙㻠㻜㻠㻯㻻㻾㻼㻻㻾㻭㼀㻱㻌㻱㼄㻴㻵㻮㻵㼀㻵㻻㻺㻿19㼀㼡㼑㼟㼐㼍㼥㻘㻌㻿㼑㼜㼠㼑㼙㼎㼑㼞㻌㻝㻞㻘㻌㻞㻜㻞㻟Time/Location

Activity㻼㻸㻱㻺㻭㻾㼅㻌㻸㻱㻯㼀㼁㻾㻱㻌㻝㻼㻝䚷㼀㻱㻭㻯㻴㻱㻾㻌㻱㻰㼁㻯㻭㼀㻵㻻㻺㻌㻭㻺㻰㻌㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻭㻸㻹㻻㻰㻱㻸㻸㻵㻺㻳㻌䇵㻌㻼㻾㻱㻙㻿㻱㻾㼂㻵㻯㻱㻌㼀㻱㻭㻯㻴㻱㻾㻿䇻㻼㻾㻻㻲㻱㻿㻿㻵㻻㻺㻭㻸㻌㻯㻻㻹㻼㻱㼀㻱㻺㻯㻱㻌㻲㻻㻾㻌㼀㻴㻱㻌㼀㻱㻭㻯㻴㻵㻺㻳㻻㻲㻌㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻭㻸㻌㻹㻻㻰㻱㻸㻸㻵㻺㻳㻥㻦㻟㻜㻙㻝㻜㻦㻟㻜㻛㻮㻝㻲㻌㻱㼢㼑㼚㼠㻌㻴㼍㼘㼘㻳㼕㼘㼎㼑㼞㼠㻌㻳㼞㼑㼑㼒㼞㼍㼠㼔 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻹㾇㼚㼟㼠㼑㼞㻘㻌㻳㼑㼞㼙㼍㼚㼥㻯㼔㼍㼕㼞㻶㼛㼚㼍㼟㻌㻮㼑㼞㼓㼙㼍㼚㻌㽏㼞㼘㼑㼎㽯㼏㼗 㻘㻌㻸㼕㼚㼗㾁㼜㼕㼚㼓㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㻿㼣㼑㼐㼑㼚㻝㻜㻦㻠㻡㻙㻝㻞㻦㻝㻡䚷䚷㻿㻿㻱㻿㻿㻵㻻㻺㻌㻟㻻㻮㻝䚷㼁㻿㻵㻺㻳㻌㻭㻺㻌㻵㻺㻿㼀㻾㼁㻯㼀㻵㻻㻺㻭㻸㻌㼂㻵㻰㻱㻻㻌㼀㻻㻌㻿㼁㻼㻼㻻㻾㼀㻌㼁㻼㻼㻱㻾㻿㻱㻯㻻㻺㻰㻭㻾㼅㻌㻿㼀㼁㻰㻱㻺㼀㻿㻌㻵㻺㻌㻯㻾㻱㻭㼀㻵㻺㻳㻌㻭㻌㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻭㻸㻌㻹㻻㻰㻱㻸㻸㼍㼡㼞㼍㻌㼃㼕㼞㼠㼔 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻹㾇㼚㼟㼠㼑㼞㻘㻌㻳㼑㼞㼙㼍㼚㼥㻳㼕㼘㼎㼑㼞㼠㻌㻳㼞㼑㼑㼒㼞㼍㼠㼔 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻹㾇㼚㼟㼠㼑㼞㻘㻌㻳㼑㼞㼙㼍㼚㼥㻾㼛㼛㼙㻟㻜㻝㻭㻌㻔㻸㼛㼚㼓㻕㻔㻵㻯㼀㻌㻒㻌㻹㼛㼐㼑㼘㼘㼕㼚㼓㻕㻯㼔㼍㼕㼞㻹㼡㼟㼠㼍㼒㼍㻌㻯㼑㼢㼕㼗㼎㼍㼟㻸㼍㼡㼞㼍㻌㼃㼕㼞㼠㼔㻻㻮㻞䚷㻲㻸㻵㻼㻼㻵㻺㻳㻌㼁㻺㻵㼂㻱㻾㻿㻵㼀㼅㻙㻮㻭㻿㻱㻰㻌㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻭㻸㻹㻻㻰㻱㻸㻸㻵㻺㻳㻌㻿㻱㻹㻵㻺㻭㻾㻿㻌㻫㻌㻵㻺㻿㻵㻳㻴㼀㻿㻌㻲㻾㻻㻹㻌㻼㻾㻱㻙㻿㻱㻾㼂㻵㻯㻱㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻿㻌㼀㻱㻭㻯㻴㻱㻾㻿㻹㼡㼟㼠㼍㼒㼍㻌㻯㼑㼢㼕㼗㼎㼍㼟 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻴㼍㼙㼎㼡㼞㼓㻘㻌㻳㼑㼞㼙㼍㼚㼥㻳㼍㼎㼞㼕㼑㼘㼑㻌㻷㼍㼕㼟㼑㼞 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻴㼍㼙㼎㼡㼞㼓㻘㻌㻳㼑㼞㼙㼍㼚㼥㻘㻌㻺㼛㼞㼐㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻺㼛㼞㼣㼍㼥
㻰㼑㼚㼕㼟㼑㻌㻹㼕㼑㽪㼑㼘㼑㼞 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻴㼍㼙㼎㼡㼞㼓㻘㻌㻳㼑㼞㼙㼍㼚㼥㻻㻮㻟䚷㻹㻻㻰㻱㻸㻸㻵㻺㻳㻌㼁㻺㻯㻱㻾㼀㻭㻵㻺㻌㻲㼁㼀㼁㻾㻱㻿㻦㻌㻌㻼㻾㻻㻮㻸㻱㻹㻿㻘㻌㻹㻱㼀㻴㻻㻰㻿㻘㼀㻻㻻㻸㻿㻾㼛㼛㼙㻟㻜㻝㻮㻌㻔㻸㼛㼚㼓㻕㻔㼀㼑㼏㼔㼚㼛㼘㼛㼓㼥㻌㻒㻌㻹㼛㼐㼑㼘㼘㼕㼚㼓㻼㼑㼐㼍㼓㼛㼓㼥㻕㻯㼔㼍㼕㼞㻮㼑㼚㼖㼍㼙㼕㼚㻌㻳㼍㼘㼘㼡㼦㼦㼛㻰㼕㼍㼚㼍㻌㻹㻚㻌㻲㼕㼟㼔㼑㼞㻼㼑㼠㼑㼞㻌㻳㼍㼘㼎㼞㼍㼕㼠㼔 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻽㼡㼑㼑㼚㼟㼘㼍㼚㼐㻘㻌㻭㼡㼟㼠㼞㼍㼘㼕㼍㻰㼕㼍㼚㼍㻌㻲㼕㼟㼔㼑㼞 㻘㻌㻼㼛㼞㼠㼘㼍㼚㼐㻌㻿㼠㼍㼠㼑㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㼁㻿㻭㻻㻮㻠䚷㻿㼀㻭㻾㼀㻵㻺㻳㻌㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻭㻸㻌㻹㻻㻰㻱㻸㻸㻵㻺㻳㻌㻲㻾㻻㻹㻌㻱㼄㻵㻿㼀㻵㻺㻳㻹㻻㻰㻱㻸㻿
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㻷㼍㼞㼕㼚㼍㻌㻭㼘㼑㼟㼟㼍㼚㼐㼞㼍㻌㻼㼑㼟㼟㼛㼍㻌㼐㼍㻌㻿㼕㼘㼢㼍 㻘㻌㻲㼑㼐㼑㼞㼍㼘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㼀㼑㼏㼔㼚㼛㼘㼛㼓㼥㻌㻙㻌㻼㼍㼞㼍㼚㼍㻘㻌㻮㼞㼍㼦㼕㼘㻻㻮㻝㻜㻌㼀㻾㻭㻺㻿㻵㻱㻺㼀㻌㻼㻴㻱㻺㻻㻹㻱㻺㻭㻌㻭㻿㻌㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻭㻸㻌㻹㻻㻰㻱㻸㻸㻵㻺㻳㻹㻭㼀㻱㻾㻵㻭㻸㻿㻌㻵㻺㻌㻿㻯㻵㻱㻺㻯㻱㻌㻱㻰㼁㻯㻭㼀㻵㻻㻺㻾㼛㼛㼙㻠㻜㻟㻌㻔㻿㼔㼛㼞㼠㻦㻌㻝㻜㻦㻠㻡㻙㻝㻝㻦㻠㻡㻕㻔㻿㼀㻱㻹㻌㻒㻌㻹㼛㼐㼑㼘㼘㼕㼚㼓㻕㻯㼔㼍㼕㼞㼅㼛㼟㼔㼕㼠㼍㼗㼍㻌㻭㼎㼑㻻㼟㼍㼙㼡㻌㻵㼚㼛㼙㼛㼠㼛㻻㼟㼍㼙㼡㻌㻵㼚㼛㼙㼛㼠㼛 㻘㻌㻯㼑㼚㼠㼑㼞㻌㼒㼛㼞㻌㻲㼡㼚㼐㼍㼙㼑㼚㼠㼍㼘㻌㻱㼐㼡㼏㼍㼠㼕㼛㼚㻘㻌㻷㼥㼡㼟㼔㼡㻌㻿㼍㼚㼓㼥㼛㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㻶㼍㼜㼍㼚㻻㻮㻝㻝㻌㻭㻺㻌㻭㻺㻭㻸㼅㻿㻵㻿㻌㻻㻲㻌㼀㻴㻱㻌㼀㻴㻱㻻㻾㻱㼀㻵㻯㻭㻸㻌㻲㻾㻭㻹㻱㼃㻻㻾㻷㻌㻲㻻㻾㻺㼁㻹㻱㻾㻵㻯㻭㻸㻌㻯㻻㻳㻺㻵㼀㻵㻻㻺㻌㻵㻺㻌㼆㻭㻹㻮㻵㻭㼅㼛㼟㼔㼕㼠㼍㼗㼍㻌㻭㼎㼑 㻘㻌㻴㼕㼞㼛㼟㼔㼕㼙㼍㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㻳㼞㼍㼐㼡㼍㼠㼑㻌㻿㼏㼔㼛㼛㼘㻌㼛㼒㻌㻴㼡㼙㼍㼚㼕㼠㼕㼑㼟㼍㼚㼐㻌㻿㼛㼏㼕㼍㼘㻌㻿㼏㼕㼑㼚㼏㼑㼟㻘㻌㻶㼍㼜㼍㼚21㻻㻮㻝㻞䚷㻯㻴㻭㻾㻭㻯㼀㻱㻾㻵㻿㼀㻵㻯㻿㻌㻻㻲㻌㼀㻴㻱㻌㻹㻻㻰㻱㻸㻵㻺㻳㻌㻼㻾㻻㻮㻸㻱㻹㻿㻼㻻㻿㻱㻰㻌㻮㼅㻌㻼㻾㻻㻿㻼㻱㻯㼀㻵㼂㻱㻌㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻿㻌㼀㻱㻭㻯㻴㻱㻾㻿㻿㼑㼞㼕㼒㼑㻌㻿㼑㼢㼕㼚㼏 㻘㻌㻹㼕㼐㼐㼘㼑㻌㻱㼍㼟㼠㻌㼀㼑㼏㼔㼚㼕㼏㼍㼘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㼀㼡㼞㼗㼑㼥㻾㼛㼛㼙㻠㻜㻡㻌㻔㻿㼔㼛㼞㼠㻕㻔㻼㼞㼛㼎㼘㼑㼙㻌㻼㼛㼟㼕㼚㼓㻌㻒㻌㻹㼛㼐㼑㼘㼘㼕㼚㼓㻕㻯㼔㼍㼕㼞㻿㼛㼚㼓㼏㼔㼍㼕㻌㼁㼓㼟㼛㼚㼗㼕㼐㻿㼑㼞㼕㼒㼑㻌㻿㼑㼢㼕㼚㼏㻻㻮㻝㻟䚷㻲㻭㻯㻵㻸㻵㼀㻭㼀㻵㻺㻳㻌㻼㻾㻻㻮㻸㻱㻹㻌㻼㻻㻿㻵㻺㻳㻌㻵㻺㻌㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻭㻸㻹㻻㻰㻱㻸㻸㻵㻺㻳㻦㻌㼀㻴㻱㻌㻯㻭㻿㻱㻌㻻㻲㻌㻝㻞㼀㻴㻙㻳㻾㻭㻰㻱㻌㻿㼀㼁㻰㻱㻺㼀㻿㻷㼛㼟㼡㼗㼑㻌㼅㼍㼠㼍 㻘㻌㻶㼛㼠㼛㻌㻴㼕㼓㼔㻌㻿㼏㼔㼛㼛㼘㻘㻌㻶㼍㼜㼍㼚㻭㼗㼕㼔㼕㼗㼛㻌㻿㼍㼑㼗㼕 㻘㻌㻷㼍㼚㼍㼦㼍㼣㼍㻌㻵㼚㼟㼠㼕㼠㼡㼠㼑㻌㼛㼒㻌㼀㼑㼏㼔㼚㼛㼘㼛㼓㼥㻘㻌㻶㼍㼜㼍㼚㻺㼍㼛㼔㼕㼞㼛㻌㻹㼕㼚㼍㼓㼍㼣㼍 㻘㻌㻺㼍㼞㼡㼠㼛㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻱㼐㼡㼏㼍㼠㼕㼛㼚㻘㻌㻶㼍㼜㼍㼚㼀㼍㼗㼍㼟㼔㼕㻌㻷㼍㼣㼍㼗㼍㼙㼕 㻘㻌㼁㼠㼟㼡㼚㼛㼙㼕㼥㼍㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㻶㼍㼜㼍㼚㻻㻮㻝㻠䚷㻱㻺㻴㻭㻺㻯㻵㻺㻳㻌㻼㻾㻱㻿㻱㻾㼂㻵㻯㻱㻌㼀㻱㻭㻯㻴㻱㻾㻿㻌㻾㻱㻭㻸㻙㼃㻻㻾㻸㻰㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻭㻸㻌㻼㻾㻻㻮㻸㻱㻹㻙㻼㻻㻿㻵㻺㻳㻌㻭㻮㻵㻸㻵㼀㻵㻱㻿㻦㻌㻰㻱㻿㻵㻳㻺㻵㻺㻳㻸㻱㻭㻾㻺㻵㻺㻳㻌㼀㻾㻭㻶㻱㻯㼀㻻㻾㼅㻿㼛㼚㼓㼏㼔㼍㼕㻌㼁㼓㼟㼛㼚㼗㼕㼐 㻘㻌㻷㼍㼟㼑㼠㼟㼍㼞㼠㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㼀㼔㼍㼕㼘㼍㼚㼐㻿㼍㼗㼛㼚㻌㼀㼍㼚㼓㼗㼍㼣㼟㼍㼗㼡㼘 㻘㻌㻷㼍㼟㼑㼠㼟㼍㼞㼠㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㼀㼔㼍㼕㼘㼍㼚㼐㼃㼑㼑㼞㼍㼣㼍㼠㻌㼀㼔㼍㼕㼗㼍㼙 㻘㻌㻺㼍㼗㼔㼛㼚㼟㼍㼣㼍㼚㻌㻾㼍㼖㼍㼎㼔㼍㼠㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㼀㼔㼍㼕㼘㼍㼚㼐㻝㻞㻦㻝㻡㻙㻝㻟㻦㻟㻜㻛㻟㻲㻌㻒㻌㻠㻲㻸㼁㻺㻯㻴㻌㻔㼀㼍㼗㼑㻌㼍㼣㼍㼥㻌㼘㼡㼚㼏㼔㻌㼎㼛㼤㼑㼟㻌㼕㼚㻌㻯㼛㼒㼒㼑㼑㻌㻮㼞㼑㼍㼗㻌㻭㼞㼑㼍㼟㻌㼛㼚㻌㻟㻲㼍㼚㼐㻌㻠㻲㻕

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㻿㼔㼍㼘㼕㼚㼥㻌㻷㼍㼚㼚㼍㼜㼕㼞㼍㼚 㻘㻌㻿㼑㼗㼛㼘㼍㼔㻌㻷㼑㼎㼍㼚㼓㼟㼍㼍㼚㻌㻿㼑㼞㼕㻌㼀㼍㼟㼕㼗㻌㻷㼡㼍㼘㼍㻌㻸㼡㼙㼜㼡㼞㻘㻹㼍㼘㼍㼥㼟㼕㼍
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㼀㼡㼑㼟㼐㼍㼥㻘㻌㻿㼑㼜㼠㼑㼙㼎㼑㼞㻌㻝㻞㻘㻌㻞㻜㻞㻟Time/Location

Activity㻼㻻㻿㼀㻱㻾㻌㻿㻱㻿㻿㻵㻻㻺㻼㻻㻝䚷㻱㼄㻭㻹㻵㻺㻵㻺㻳㻌㻱㻭㻾㻸㼅㻙㻳㻾㻭㻰㻱㻌㻯㻴㻵㻸㻰㻾㻱㻺䇻㻿㻌㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻭㻸㻹㻻㻰㻱㻸㻿㻌㻵㻺㻌㻼㻾㻻㻼㻻㻾㼀㻵㻻㻺㻭㻸㻌㻿㻵㼀㼁㻭㼀㻵㻻㻺㻿㻦㻌㻭㻌㻳㻸㻵㻹㻼㻿㻱㻌㻲㻾㻻㻹㼀㻭㻿㻷㻙㻮㻭㻿㻱㻰㻌㻵㻺㼀㻱㻾㼂㻵㻱㼃㻿㻷㼑㼕㼗㼛㻌㻴㼕㼚㼛 㻘㻌㼁㼠㼟㼡㼚㼛㼙㼕㼥㼍㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㻶㼍㼜㼍㼚㻴㼕㼞㼍㼗㼡㻌㻵㼏㼔㼕㼗㼍㼣㼍 㻘㻌㻹㼕㼥㼍㼓㼕㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻱㼐㼡㼏㼍㼠㼕㼛㼚㻘㻌㻶㼍㼜㼍㼚㻴㼕㼟㼍㼑㻌㻷㼍㼠㼛 㻘㻌㻴㼥㼛㼓㼛㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㼀㼑㼍㼏㼔㼑㼞㻌㻱㼐㼡㼏㼍㼠㼕㼛㼚㻘㻌㻶㼍㼜㼍㼚㻼㻻㻞䚷㻴㻻㼃㻌㻰㻻㻌㻿㼀㼁㻰㻱㻺㼀㻿㻌㻯㻾㻵㼀㻵㻯㻭㻸㻸㼅㻌㻵㻺㼀㻱㻾㻼㻾㻱㼀㻌㻼㻸㼁㻾㻭㻸㼃㻭㼅㻿㻌㼀㻻㻌㻰㻱㼀㻱㻾㻹㻵㻺㻱㻌㼀㻴㻱㻌㻹㻻㻿㼀㻌㻼㻻㻼㼁㻸㻭㻾㻌㻻㻺㻵㻳㻵㻾㻵㻫㻙㻭㼀㻻㻹㻵㻿㼀㻵㻯㻌㻭㻼㻼㻾㻻㻭㻯㻴㻌㻲㻻㻯㼁㻿㻱㻰㻌㻻㻺㻌㻵㻺㼀㻱㻾㻼㻾㻱㼀㻭㼀㻵㻻㻺㻘㼂㻭㻸㻵㻰㻭㼀㻵㻻㻺㻌㻭㻺㻰㻌㻹㻻㻰㻵㻲㻵㻯㻭㼀㻵㻻㻺㻙㻹㼕㼚㼍㼙㼕㻌㻭㼟㼍㼔㼍㼞㼍 㻘㻌㼅㼛㼗㼛㼔㼍㼙㼍㻌㻺㼍㼠㼕㼛㼚㼍㼘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㻔㻹㼍㼟㼠㼑㼞䇻㼟㻌㼟㼠㼡㼐㼑㼚㼠㻕㻘㻶㼍㼜㼍㼚㼀㼍㼗㼍㼟㼔㼕㻌㻿㼡㼙㼕㼛㼗㼍 㻘㻌㼅㼛㼗㼛㼔㼍㼙㼍㻌㻺㼍㼠㼕㼛㼚㼍㼘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㻔㻹㼍㼟㼠㼑㼞䇻㼟㻌㼟㼠㼡㼐㼑㼚㼠㻕㻘㻶㼍㼜㼍㼚㻹㼍㼗㼛㼠㼛㻌㻯㼔㼛㼚㼍㼚 㻘㻌㼅㼛㼗㼛㼔㼍㼙㼍㻌㻺㼍㼠㼕㼛㼚㼍㼘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㻔㻹㼍㼟㼠㼑㼞䇻㼟㻌㼟㼠㼡㼐㼑㼚㼠㻕㻘㻶㼍㼜㼍㼚㻷㼍㼦㼡㼗㼕㻌㻷㼛㼕㼗㼑 㻘㻌㼅㼛㼗㼛㼔㼍㼙㼍㻌㻺㼍㼠㼕㼛㼚㼍㼘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㻔㻹㼍㼟㼠㼑㼞䇻㼟㻌㼟㼠㼡㼐㼑㼚㼠㻕㻘㻶㼍㼜㼍㼚㻼㻻㻟䚷㻴㻻㼃㻌㼀㻻㻌㼁㻿㻱㻌㻱㻸㻱㻹㻱㻺㼀㻭㻾㼅㻌㻹㻭㼀㻴㻌㼀㻱㼄㼀㻮㻻㻻㻷㻿㻌㼀㻻㻼㻾㻻㻹㻻㼀㻱㻌㻹㻭㼀㻴㻱㻹㻭㼀㻵㼆㻭㼀㻵㻻㻺㻴㼍㼠㼟㼡㼐㼍㻌㻴㼕㼞㼛㼗㼕 㻘㻌㼅㼛㼗㼛㼔㼍㼙㼍㻌㻺㼍㼠㼕㼛㼚㼍㼘㻌㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻔㻹㼍㼟㼠㼑㼞㻓㼟㻌㼟㼠㼡㼐㼑㼚㼠㼟㻕㻘㻝㻣㻦㻜㻜㻙㻝㻤㻦㻟㻜㻛㻮㻝㻲㻌㻸㼛㼎㼎㼥㻶㼍㼜㼍㼚㻼㻻㻠䚷㻲㻻㻿㼀㻱㻾㻵㻺㻳㻌㻿㼀㼁㻰㻱㻺㼀㻿䇻㻌㻭㻮㻵㻸㻵㼀㼅㻌㼀㻻㻌㻯㻻㻹㻼㻭㻾㻱㻌㻭㻺㻰㻯㻾㻵㼀㻵㻯㻭㻸㻸㼅㻌㻱㼄㻭㻹㻵㻺㻱㻌㻼㻸㼁㻾㻭㻸㻌㻹㻻㻰㻱㻸㻿㻌㼀㻻㻌㻿㻻㻸㼂㻱㻌㻾㻱㻭㻸㻌㼃㻻㻾㻸㻰㻼㻾㻻㻮㻸㻱㻹㻿㻙㼀㻱㻭㻯㻴㻵㻺㻳㻌㻱㼄㻼㻱㻾㻵㻹㻱㻺㼀㻌㻮㼅㻌㼁㻿㻵㻺㻳㻌㻭㻌㼀㻱㻺㼀㻼㻾㻻㻮㻸㻱㻹㻙
㻿㼔㼛㼠㼍㻌㻷㼕㼠㼍 㻘㻌㼅㼛㼗㼛㼔㼍㼙㼍㻌㻺㼍㼠㼕㼛㼚㼍㼘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻔㻹㼍㼟㼠㼑㼞䇻㼟㻌㼟㼠㼡㼐㼑㼚㼠㻕㻘㻌㻶㼍㼜㼍㼚㻹㼍㼟㼍㼟㼔㼕㻌㻵㼗㼑㼙㼡㼞㼍 㻘㻌㻌㼅㼛㼗㼛㼔㼍㼙㼍㻌㻺㼍㼠㼕㼛㼚㼍㼘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻔㻹㼍㼟㼠㼑㼞䇻㼟㻌㼟㼠㼡㼐㼑㼚㼠㻕㻘㻶㼍㼜㼍㼚㻷㼛㼐㼍㼕㻌㻭㼞㼍㼠㼍㼚㼕 㻘㻌㻌㼅㼛㼗㼛㼔㼍㼙㼍㻌㻺㼍㼠㼕㼛㼚㼍㼘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻔㻹㼍㼟㼠㼑㼞䇻㼟㻌㼟㼠㼡㼐㼑㼚㼠㻕㻘㻶㼍㼜㼍㼚㼀㼛㼟㼔㼕㼗㼍㼦㼡㻌㻵㼗㼑㼐㼍 㻘㻌㼅㼛㼗㼛㼍㼔㼍㼙㻌㻺㼍㼠㼕㼛㼚㼍㼘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㻶㼍㼜㼍㼚㻼㻻㻡䚷㼃㻴㻭㼀㻌㻹㻱㼀㻭㻼㻴㻻㻾㻿㻌㻰㻻㻌㻯㻴㻵㻸㻰㻾㻱㻺㻌㼁㼀㻵㻸㻵㼆㻱㻌㼀㻻㻌㻹㻭㻷㻱㻯㻻㻺㻺㻱㻯㼀㻵㻻㻺㻿㻌㻮㻱㼀㼃㻱㻱㻺㻌㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻿㻌㻭㻺㻰㻌㼀㻴㻱㻌㻾㻱㻭㻸㼃㻻㻾㻸㻰㻫
㻷㼑㼚㼟㼡㼗㼑㻌㻷㼛㼕㼦㼡㼙㼕 㻘 㻌㻳㼡㼚㼙㼍㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㻶㼍㼜㼍㼚㻾㼥㼡㼠㼍㻌㼀㼍㼚㼕 㻘㻌㼀㼍㼚㼍㼗㼍㻌㻳㼍㼗㼡㼑㼚㻌㻾㼕㼠㼟㼡㼙㼑㼕㼗㼍㼚㻌㻷㼑㼕㼟㼥㼛㻌㻼㼞㼕㼙㼍㼞㼥㻌㻿㼏㼔㼛㼛㼘㻘㻶㼍㼜㼍㼚㻾㼥㼛㻌㻴㼍㼚㼦㼍㼣㼍 㻘㻌㻿㼑㼥㼍㻌㻼㼞㼕㼙㼍㼞㼥㻌㻿㼏㼔㼛㼛㼘㻘㻌㻶㼍㼜㼍㼚㻼㻻㻢䚷㻴㻻㼃㻌㻰㻻㻱㻿㻌㻲㻵㻾㻿㼀㻙㼅㻱㻭㻾㻌㻿㻯㻵㼀㻱㻯㻴㻌㻱㼄㻼㻱㻾㻵㻱㻺㻯㻱㻌㻻㻲㻼㻾㻻㻲㻱㻿㻿㻵㻻㻺㻭㻸㻌㻰㻱㼂㻱㻸㻻㻼㻹㻱㻺㼀㻌㻱㻲㻲㻱㻯㼀㻌㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻱㻰㼁㻯㻭㼀㻵㻻㻺㻫㻿㼍㼠㼛㼞㼡㻌㻲㼡㼖㼕㼠㼍㼚㼕 㻘㻌㼀㼍㼙㼍㼓㼍㼣㼍㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㻶㼍㼜㼍㼚㻹㼛㼠㼛㼗㼛㻌㻲㼡㼖㼕㼠㼍㼚㼕 㻘㻌㻶㼛㼑㼠㼟㼡㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻱㼐㼡㼏㼍㼠㼕㼛㼚㻘㻌㻶㼍㼜㼍㼚26

Wednesday, September 13, 2023
Time/Location

Activity
PLENARY LECTURE 2

P2 IN THEIR OWN WORDS: EXPLANATIONS OF STEMSTUDENTS’ REASONING DURING MATHEMATICAL
MODELLING
9:30-10:30/Event Hall

Jennifer A. Czocher , Texas State University, USA
Chair
Cyril Julie , University of the Western Cape, South Africa

10:30-

EXCURSION (OPTION)LUNCH (Take away lunch boxes at B1F Lobby)

27㼀㼔㼡㼞㼟㼐㼍㼥㻘㻌㻿㼑㼜㼠㼑㼙㼎㼑㼞㻌㻝㻠㻘㻌㻞㻜㻞㻟Time/Location

Activity㻥㻦㻟㻜㻙㻝㻝㻦㻜㻜䚷䚷㻿㻿㻱㻿㻿㻵㻻㻺㻌㻢㻾㼛㼛㼙㻟㻜㻝㻭㻌㻔㻸㼛㼚㼓㻕㻔㻹㼛㼐㼑㼘㼘㼕㼚㼓㻌㻼㼞㼛㼏㼑㼟㼟㻕㻯㼔㼍㼕㼞㻭㼗㼕㼛㻌㻹㼍㼠㼟㼡㼦㼍㼗㼕㻱㼘㼑㼚㼍㻌㻺㼍㼒㼠㼍㼘㼕㼑㼢㻻㻯㻝䚷㻯㻻㻺㻿㼀㻾㼁㻯㼀㻵㻺㻳㻌㻹㻱㻭㻺㻵㻺㻳㻌㻲㻻㻾㻌㻵㻺㼀㻱㻾㻭㻯㼀㻵㼂㻱㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻭㻸㻌㻹㻻㻰㻱㻸㻿㻌㼃㻵㼀㻴㻵㻺㻌㻭㻌㻰㻵㻭㻸㻱㻯㼀㻵㻯㻭㻸㻌㻸㻱㻭㻾㻺㻵㻺㻳㻱㻺㼂㻵㻾㻻㻺㻹㻱㻺㼀㻱㼘㼑㼚㼍㻌㻺㼍㼒㼠㼍㼘㼕㼑㼢 㻘㻌㻭㼏㼔㼢㼍㻌㻭㼏㼍㼐㼑㼙㼕㼏㻌㻯㼛㼘㼘㼑㼓㼑㻘㻌㻵㼟㼞㼍㼑㼘㻻㻯㻞䚷㻭㻰㼂㻭㻺㻯㻵㻺㻳㻌㼀㻴㻱㻌㻿㼀㼁㻰㼅㻌㻻㻲㻌㻭㻌㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻵㻭㻺䇻㻿㻹㻻㻰㻱㻸㻸㻵㻺㻳㻌㻼㻾㻻㻳㻾㻱㻿㻿㻌㻰㻵㻿㻼㻸㻭㼅㻱㻰㻌㼃㻵㼀㻴㻌㻭㻼㻼㻸㻵㻱㻰㻌㻾㻱㻿㻼㻻㻺㻿㻱㻭㻺㻭㻸㼅㻿㻵㻿㻌㻹㻭㻼㻼㻵㻺㻳㻭㼗㼕㼛㻌㻹㼍㼠㼟㼡㼦㼍㼗㼕 㻘㻌㻿㼍㼕㼠㼍㼙㼍㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㻶㼍㼜㼍㼚㻾㼛㼛㼙㻟㻜㻝㻮㻌㻔㻸㼛㼚㼓㻕㻔㼀㼑㼍㼏㼔㼑㼞㻌㻱㼐㼡㼏㼍㼠㼕㼛㼚㻕㻯㼔㼍㼕㼞㼄㼕㼍㼛㼘㼕㻌㻸㼡㻳㼑㼛㼞㼓㼑㻌㻱㼗㼛㼘㻻㻯㻟䚷㻭㻺㻭㻸㼅㻿㻵㻺㻳㻌䇺㻿㼀㼁㻰㻱㻺㼀㻙㼀㻻㻙㻿㼀㼁㻰㻱㻺㼀㻙㼀㻻㻙㼀㻱㻭㻯㻴㻱㻾䇻㻯㻻㻸㻸㻭㻮㻻㻾㻭㼀㻵㼂㻱㻌㻸㻱㻭㻾㻺㻵㻺㻳㻌㻼㻾㻭㻯㼀㻵㻯㻱㻿㻌㻵㻺㻌㻭㻌㻼㻾㻱㻙㻿㻱㻾㼂㻵㻯㻱㻭㻼㻼㻸㻵㻯㻭㼀㻵㻻㻺㻿㻌㻭㻺㻰㻌㻹㻻㻰㻱㻸㻸㻵㻺㻳㻌㻯㻻㼁㻾㻿㻱㻚㻳㼑㼛㼞㼓㼑㻌㻱㼗㼛㼘 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㼃㼕㼠㼣㼍㼠㼑㼞㼟㼞㼍㼚㼐㻘㻌㻿㼛㼡㼠㼔㻌㻭㼒㼞㼕㼏㼍㻻㻯㻠䚷㻼㻾㻱㻙㻿㻱㻾㼂㻵㻯㻱㻌㼀㻱㻭㻯㻴㻱㻾㻿䇻㻌㻼㻱㻾㻯㻱㻼㼀㻵㻻㻺㻌㻻㻺㻌㼀㻴㻱㻵㻾㻌㻾㻻㻸㻱㻿㻵㻺㻌㼀㻴㻱㻌㼀㻱㻭㻯㻴㻵㻺㻳㻌㻻㻲㻌㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻭㻸㻌㻹㻻㻰㻱㻸㻸㻵㻺㻳㼄㼕㼍㼛㼘㼕㻌㻸㼡 㻘㻌㻱㼍㼟㼠㻌㻯㼔㼕㼚㼍㻌㻺㼛㼞㼙㼍㼘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㻯㼔㼕㼚㼍㼅㼍㼚㻌㼆㼔㼡 㻘㻌㻱㼍㼟㼠㻌㻯㼔㼕㼚㼍㻌㻺㼛㼞㼙㼍㼘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㻯㼔㼕㼚㼍㻻㻯㻡䚷㻯㻴㻭㻾㻭㻯㼀㻱㻾㻵㼆㻵㻺㻳㻌㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻭㻸㻌㻹㻻㻰㻱㻸㻸㻵㻺㻳㻌㼀㻭㻿㻷㻿㻌㻵㻺㻱㻹㻼㻵㻾㻵㻯㻭㻸㻌㻸㻵㼀㻱㻾㻭㼀㼁㻾㻱㻾㼛㼛㼙㻟㻝㻝㻭㻌㻔㻸㼛㼚㼓㻕㻔㻸㼕㼠㼑㼞㼍㼠㼡㼞㼑㻌㻾㼑㼢㼕㼑㼣㻕㻯㼔㼍㼕㼞㻺㼕㼚㼍㻌㼁㼚㼟㼔㼑㼘㼙㻾㼛㼟㼑㻌㻹㼍㼞㼥㻌㼆㼎㼕㼑㼗㻾㼛㼟㼑㻌㻹㼍㼞㼥㻌㼆㼎㼕㼑㼗 㻘㻌㻼㼑㼚㼚㼟㼥㼘㼢㼍㼚㼕㼍㻌㻿㼠㼍㼠㼑㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㼁㻿㻭㼄㼕㼍㼚㼓㼝㼡㼍㼚㻌㼅㼍㼛 㻘㻌㻼㼑㼚㼚㼟㼥㼘㼢㼍㼚㼕㼍㻌㻿㼠㼍㼠㼑㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㼁㻿㻭㻹㻚㻌㻷㼍㼠㼔㼘㼑㼑㼚㻌㻴㼑㼕㼐 㻘㻌㻼㼑㼚㼚㼟㼥㼘㼢㼍㼚㼕㼍㻌㻿㼠㼍㼠㼑㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㼁㻿㻭㻹㼍㼠㼠㼔㼑㼣㻌㼂㼕㼏㼠㼛㼞㻌㻮㼘㼍㼏㼗 㻘㻌㻼㼑㼚㼚㼟㼥㼘㼢㼍㼚㼕㼍㻌㻿㼠㼍㼠㼑㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㼁㻿㻭㻻㻯㻢䚷㻭㻌㻾㻱㼂㻵㻱㼃㻌㻻㻲㻌㻾㻱㻿㻱㻭㻾㻯㻴㻌㻻㻺㻌㻹㻻㻰㻱㻸㻸㻵㻺㻳㻘㻌㻮㻵㻳㻌㻰㻭㼀㻭㻌㻭㻺㻰㻱㼂㻵㻰㻱㻺㼀㻵㻭㻾㼅㻌㻼㻾㻭㻯㼀㻵㻯㻱㻿㻺㼕㼚㼍㻌㼁㼚㼟㼔㼑㼘㼙 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㼃㼡㼞㼦㼎㼡㼞㼓㻘㻌㻳㼑㼞㼙㼍㼚㼥㼂㼕㼚㼏㼑㻌㻳㼑㼕㼓㼑㼞 㻘㻌㻭㼡㼟㼠㼞㼍㼘㼕㼍㼚㻌㻯㼍㼠㼔㼛㼘㼕㼏㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㻭㼡㼟㼠㼞㼍㼘㼕㼍㻴㼍㼚㼟㻙㻿㼠㼑㼒㼍㼚㻌㻿㼕㼘㼘㼑㼞 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㼃㼡㼞㼦㼎㼡㼞㼓㻘㻌㻳㼑㼞㼙㼍㼚㼥㻿㼍㼞㼍㼔㻌㻰㼕㼓㼍㼚 㻘㻌㻭㼡㼟㼠㼞㼍㼘㼕㼍㼚㻌㻯㼍㼠㼔㼛㼘㼕㼏㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㻭㼡㼟㼠㼞㼍㼘㼕㼍㻭㼚㼐㼞㼑㻌㻳㼞㼑㼡㼎㼑㼘 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㼃㼡㼞㼦㼎㼡㼞㼓㻘㻌㻳㼑㼞㼙㼍㼚㼥㻻㻯㻣䚷㼃㻴㻭㼀㻌㻵㻿㻌㼀㻴㻱㻌㻾㻻㻸㻱㻌㻻㻲㻌㻹㻻㻰㻱㻸㻌㻵㻺㻌㼀㻴㻱㻌㻹㻻㻰㻱㻸㻵㻺㻺㻳㻼㻾㻻㻯㻱㻿㻿
㻷㼛㼕㼏㼔㼕㻌㻺㼍㼗㼍㼙㼡㼞㼍 㻘㻌㼀㼛㼗㼥㼛㻌㻳㼍㼗㼡㼓㼑㼕㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㻶㼍㼜㼍㼚㻾㼛㼛㼙㻟㻝㻝㻮㻌㻔㻿㼔㼛㼞㼠㻕㻔㻹㼛㼐㼑㼘㼘㼕㼚㼓㻌㻱㼜㼕㼟㼠㼑㼙㼛㼘㼛㼓㼥㻕㻯㼔㼍㼕㼞
Blanca Cecilia Fulano-Vargas㻷㼛㼕㼏㼔㼕㻌㻺㼍㼗㼍㼙㼡㼞㼍㻻㻯㻤䚷㻯㻻㻺㻯㻱㻼㼀㻵㻻㻺㻿㻘㻌㼁㻿㻱㻿㻌㻭㻺㻰㻌㻻㻮㻿㼀㻭㻯㻸㻱㻿㻌㻵㻺㻌㼀㻴㻱㻌㼀㻱㻭㻯㻴㻵㻺㻳㻭㻺㻰㻌㻸㻱㻭㻾㻺㻵㻺㻳㻌㻻㻲㻌㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻭㻸㻌㻹㻻㻰㻱㻸㻸㻵㻺㻳㻮㼘㼍㼚㼏㼍㻌㻯㼑㼏㼕㼘㼕㼍㻌㻲㼡㼘㼍㼚㼛㻙㼂㼍㼞㼓㼍㼟 㻘㻌㻿㼑㼏㼞㼑㼠㼍㼞㽸㼍㻌㼐㼑㻌㻱㼐㼡㼏㼍㼏㼕㽾㼚㻌㻰㼑㻌㻮㼛㼓㼛㼠㽬㻙㻯㼛㼘㼛㼙㼎㼕㼍㻘㻌㻯㼛㼘㼛㼙㼎㼕㼍㻺㼑㼘㼟㼛㼚㻌㻱㼚㼞㼕㼝㼡㼑㻌㻮㼍㼞㼞㼕㼛㼟㻌㻶㼍㼞㼍 㻘㻌㻿㼑㼏㼞㼑㼠㼍㼞㽸㼍㻌㼐㼑㻌㻱㼐㼡㼏㼍㼏㼕㽾㼚㻌㻰㼑㻌㻮㼛㼓㼛㼠㽬㻙㻯㼛㼘㼛㼙㼎㼕㼍㻘㻌㻯㼛㼘㼛㼙㼎㼕㼍㻻㻯㻥䚷㻵㻰㻱㻺㼀㻵㻲㼅㻵㻺㻳㻌㻭㼁㼀㻴㻱㻺㼀㻵㻯㻵㼀㼅㻌㻭㻺㻰㻌㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻭㻸㻹㻻㻰㻱㻸㻸㻵㻺㻳㻌㻯㻻㻹㻼㻱㼀㻱㻺㻯㻵㻱㻿㻌㻭㻿㻌㻿㼀㻱㻼㻼㻵㻺㻳㻌㻿㼀㻻㻺㻱㻿㻌㻲㻻㻾㻮㻸㻱㻺㻰㻵㻺㻳㻌㻰㻵㻲㻲㻱㻾㻱㻺㼀㻌㻼㻱㻾㻿㻼㻱㻯㼀㻵㼂㻱㻿㻰㼍㼕㼗㼕㻌㼁㼞㼍㼥㼍㼙㼍 㻘㻌㻵㼚㼠㼑㼞㼚㼍㼠㼕㼛㼚㼍㼘㻌㻼㼍㼏㼕㼒㼕㼏㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㻶㼍㼜㼍㼚28

㻻㻯㻝㻜䚷㼀㻴㻱㻌㻾㻻㻸㻱㻌㻻㻲㻌㻿㼀㼁㻰㼅㻌㻭㻺㻰㻌㼀㻱㻯㻴㻺㻻㻸㻻㻳㼅㻌㻵㻺㻹㻭㼀㻴㻱㻭㻹㼀㻵㻯㻭㻸㻌㻹㻻㻰㻱㻸㻸㻵㻺㻳㻾㼛㼛㼙㻠㻜㻡㻌㻔㻿㼔㼛㼞㼠㻦㻌㻥㻦㻟㻜㻙㻝㻜㻦㻟㻜㻕㻔㼀㼑㼏㼔㼚㼛㼘㼛㼓㼥㻕㻯㼔㼍㼕㼞㻭㼚㼐㼞㼑㼍㼟㻌㻮㼍㼏㼗㻮㼞㼕㼠㼠㼍㻌㻱㼥㼞㼕㼏㼔㻌㻶㼑㼟㼟㼑㼚㻮㼞㼕㼠㼠㼍㻌㻱㼥㼞㼕㼏㼔㻌㻶㼑㼟㼟㼑㼚 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻯㼛㼜㼑㼚㼔㼍㼓㼑㼚㻘㻌㻰㼑㼚㼙㼍㼞㼗㻻㻯㻝㻝䚷㻸㻱㻭㻾㻺㻵㻺㻳㻌㻻㻲㻌㻰㻵㻳㻵㼀㻭㻸㻌㼀㻻㻻㻸㻌㻯㻻㻹㻼㻱㼀㻱㻺㻯㻵㻱㻿㻌㻵㻺㻌㼀㻴㻱㻯㻻㻺㼀㻱㼄㼀㻌㻻㻲㻌㻹㻻㻰㻱㻸㻸㻵㻺㻳㻭㼚㼐㼞㼑㼍㼟㻌㻮㼍㼏㼗 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻹㾇㼚㼟㼠㼑㼞㻘㻌㻳㼑㼞㼙㼍㼚㼥㻳㼕㼘㼎㼑㼞㼠㻌㻳㼞㼑㼑㼒㼞㼍㼠㼔 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻹㾇㼚㼟㼠㼑㼞㻘㻌㻳㼑㼞㼙㼍㼚㼥㻿㼠㼍㼚㼕㼟㼘㼍㼣㻌㻿㼏㼔㼡㼗㼍㼖㼘㼛㼣 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻹㾇㼚㼟㼠㼑㼞㻘㻌㻳㼑㼞㼙㼍㼚㼥㻝㻝㻦㻝㻡㻙㻝㻞㻦㻠㻡㻌㻌㻌㻌㻿㻿㻱㻿㻿㻵㻻㻺㻌㻣㻻㻯㻝㻞䚷㼀㻱㻭㻯㻴㻵㻺㻳㻌㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻭㻸㻌㼀㻴㻵㻺㻷㻵㻺㻳㻦㻌㻾㻱㻭㻿㻻㻺㻵㻺㻳㻘㻹㻻㻰㻱㻸㻸㻵㻺㻳㻘㻌㻼㻾㻻㻮㻸㻱㻹㻌㻿㻻㻸㼂㻵㻺㻳㻾㼛㼛㼙㻟㻜㻝㻭㻌㻔㻸㼛㼚㼓㻕㻔㻹㼛㼐㼑㼘㼘㼕㼚㼓㻌㻯㼛㼙㼜㼑㼠㼑㼚㼏㼥㻕㻯㼔㼍㼕㼞㻶㼍㼚㼕㼚㼍㻌㻷㼞㼍㼣㼕㼠㼦㻰㼍㼓㻌㼃㼑㼐㼑㼘㼕㼚㻰㼍㼓㻌㼃㼑㼐㼑㼘㼕㼚 㻘㻌㻯㼔㼍㼘㼙㼑㼞㼟㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㼀㼑㼏㼔㼚㼛㼘㼛㼓㼥㻘㻌㻿㼣㼑㼐㼑㼚㻻㻯㻝㻟䚷㼀㻭㻷㻵㻺㻳㻌㼁㻼㻌㻻㼃㻺㻱㻾㻿㻴㻵㻼㻦㻌㼃㻴㻭㼀㻌㻯㻭㻺㻌㻮㻱㻌㻸㻱㻭㻾㻺㻱㻰㻌㻲㻾㻻㻹㻼㻻㻿㻵㻺㻳㻌㻹㻻㻰㻱㻸㻸㻵㻺㻳㻌㻼㻾㻻㻮㻸㻱㻹㻿㻫㻌㻶㼍㼚㼕㼚㼍㻌㻷㼞㼍㼣㼕㼠㼦 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻹㾇㼚㼟㼠㼑㼞㻘㻌㻳㼑㼞㼙㼍㼚㼥㻸㼡㼕㼟㼍㻌㻴㼍㼞㼠㼙㼍㼚㼚 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻹㾇㼚㼟㼠㼑㼞㻘㻌㻳㼑㼞㼙㼍㼚㼥㻿㼠㼍㼚㼕㼟㼘㼍㼣㻌㻿㼏㼔㼡㼗㼍㼖㼘㼛㼣 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻹㾇㼚㼟㼠㼑㼞㻘㻌㻳㼑㼞㼙㼍㼚㼥㼃㼑㼞㼚㼑㼞㻌㻮㼘㼡㼙 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻷㼍㼟㼟㼑㼘㻘㻌㻳㼑㼞㼙㼍㼚㼥㻻㻯㻝㻠䚷㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻭㻸㻌㻹㻻㻰㻱㻸㻸㻵㻺㻳㻌㻭㼀㻌㻭㻸㻸㻌㼅㻱㻭㻾㻌㻸㻱㼂㻱㻸㻿㻌㻵㻺㻭㼁㻿㼀㻾㻭㻸㻵㻭㻾㼛㼛㼙㻟㻜㻝㻮㻌㻔㻸㼛㼚㼓㻕㻔㻼㼞㼕㼙㼍㼞㼥㻛㻿㼑㼏㼛㼚㼐㼍㼞㼥㻌㻱㼐㼡㼏㼍㼠㼕㼛㼚㻕㻯㼔㼍㼕㼞㼅㼡㼍㼚㻌㼅㼍㼚
㻼㼍㼡㼘㻌㻮㼞㼛㼣㼚㻼㼍㼡㼘㻌㻮㼞㼛㼣㼚 㻘㻌㻯㼡㼞㼠㼕㼚㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㻭㼡㼟㼠㼞㼍㼘㼕㼍㻻㻯㻝㻡䚷㼀㻴㻱㻌㻵㻺㼂㻱㻿㼀㻵㻳㻭㼀㻵㻻㻺㻌㻻㻲㻌㼀㻴㻱㻌㻹㻻㻰㻱㻸㻸㻵㻺㻳㻌㻯㻻㻹㻼㻱㼀㻱㻺㻯㻵㻱㻿㻻㻲㻌㼀㼃㻻㻌㻳㻾㻻㼁㻼㻿㻌㻻㻲㻌㻳㻾㻭㻰㻱㻌㻱㻵㻳㻴㼀㻌㻿㼀㼁㻰㻱㻺㼀㻿㻌㼃㻴㻻㻌㻾㻱㻯㻱㻵㼂㻱㻰㻰㻵㻲㻲㻱㻾㻱㻺㼀㻌㻹㻻㻰㻱㻸㻸㻵㻺㻳㻌㻵㻺㻿㼀㻾㼁㻯㼀㻵㻻㻺㻿㼅㼡㼍㼚㻌㼅㼍㼚 㻘㻌㻱㼍㼟㼠㻌㻯㼔㼕㼚㼍㻌㻺㼛㼞㼙㼍㼘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㻯㼔㼕㼚㼍㼅㼕㻌㻯㼔㼑㼚 㻘㻌㻿㼔㼍㼚㼓㼔㼍㼕㻌㼃㼑㼚㼝㼕㻌㻹㼕㼐㼐㼘㼑㻌㻿㼏㼔㼛㼛㼘㻘㻌㻯㼔㼕㼚㼍㻻㻯㻝㻢䚷㼂㻭㻸㻵㻰㻭㼀㻵㻻㻺㻌㻵㻺㻌㻿㻵㼄㼀㻴㻙㻳㻾㻭㻰㻱㻌㻰㻭㼀㻭㻙㻰㻾㻵㼂㻱㻺㻌㻹㻻㻰㻱㻸㻸㻵㻺㻳㼀㼍㼗㼍㼟㼔㼕㻌㻷㼍㼣㼍㼗㼍㼙㼕 㻘㻌㼁㼠㼟㼡㼚㼛㼙㼕㼥㼍㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㻶㼍㼜㼍㼚㻷㼍㼠㼟㼡㼗㼕㻌㻭㼗㼕㼦㼍㼣㼍 㻘㻌㻱㼘㼑㼙㼑㼚㼠㼍㼞㼥㻌㻿㼏㼔㼛㼛㼘㻌㻭㼠㼠㼍㼏㼔㼑㼐㻌㼠㼛㻌㼁㼠㼟㼡㼚㼛㼙㼕㼥㼍㻾㼛㼛㼙㻟㻝㻝㻭㻌㻔㻸㼛㼚㼓㻕㻔㻰㼍㼠㼍㻌㻒㻌㻹㼛㼐㼑㼘㼘㼕㼚㼓㻕㻯㼔㼍㼕㼞㼂㼕㼚㼏㼑㻌㻳㼑㼕㼓㼑㼞㼀㼍㼗㼍㼟㼔㼕㻌㻷㼍㼣㼍㼗㼍㼙㼕㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㻶㼍㼜㼍㼚㻻㻯㻝㻣䚷㻿㼀㻾㻱㻺㻳㼀㻴㻱㻺㻵㻺㻳㻌㼀㻱㻭㻯㻴㻱㻾㻿䇻㻌㻯㻭㻼㻭㻮㻵㻸㻵㼀㻵㻱㻿㻌㼃㻵㼀㻴㻌㻮㻵㻳㻰㻭㼀㻭㼂㼕㼚㼏㼑㻌㻳㼑㼕㼓㼑㼞 㻘㻌㻭㼡㼟㼠㼞㼍㼘㼕㼍㼚㻌㻯㼍㼠㼔㼛㼘㼕㼏㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㻭㼡㼟㼠㼞㼍㼘㼕㼍㻴㼍㼚㼟㻙㻿㼠㼑㼒㼍㼚㻌㻿㼕㼘㼘㼑㼞 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㼃㼡㼞㼦㼎㼡㼞㼓㻘㻌㻳㼑㼞㼙㼍㼚㼥㻿㼍㼞㼍㼔㻌㻰㼕㼓㼍㼚 㻘㻌㻭㼡㼟㼠㼞㼍㼘㼕㼍㼚㻌㻯㼍㼠㼔㼛㼘㼕㼏㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㻭㼡㼟㼠㼞㼍㼘㼕㼍㻭㼚㼐㼞㼑㻌㻳㼞㼑㼡㼎㼑㼘 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㼃㼡㼞㼦㼎㼡㼞㼓㻘㻌㻳㼑㼞㼙㼍㼚㼥㻺㼕㼚㼍㻌㼁㼚㼟㼔㼑㼘㼙 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㼃㼡㼞㼦㼎㼡㼞㼓㻘㻌㻳㼑㼞㼙㼍㼚㼥29㻻㻯㻝㻤䚷㻵㻹㻼㻾㻻㼂㻵㻺㻳㻌㼁㻺㻰㻱㻾㻳㻾㻭㻰㼁㻭㼀㻱㻌㻿㼀㻭㼀㻵㻿㼀㻵㻯㻿㻌㻱㻰㼁㻯㻭㼀㻵㻻㻺㻦㻱㻰㼁㻯㻭㼀㻵㻻㻺㻭㻸㻌㻸㻱㻿㻿㻻㻺㻿㻌㻲㻾㻻㻹㻌㻼㻱㻰㻭㻳㻻㻳㻵㻯㻭㻸㻌㻱㼄㻼㻱㻾㻵㻹㻱㻺㼀㻿㻾㼛㼛㼙㻟㻝㻝㻮㻌㻔㻿㼔㼛㼞㼠㻦㻌㻝㻝㻦㻝㻡㻙㻝㻞㻦㻝㻡㻕㻔㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㻒㻌㼀㼑㼍㼏㼔㼑㼞㻌㻱㼐㼡㼏㼍㼠㼕㼛㼚㻕㻯㼔㼍㼕㼞㻭㼘㼕㼚㼍㻌㻭㼘㼣㼍㼟㼠㼄㼕㼚㻌㻹㼍

㻭㼚㼡㼟㼔㼗㼍㻌㻷㼍㼞㼗㼑㼘㼍㼚㼛㼢㼍 㻘㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻷㼑㼚㼠㼡㼏㼗㼥㻘㻌㼁㻿㻭㼄㼕㼚㻌㻹㼍 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻷㼑㼚㼠㼡㼏㼗㼥㻘㻌㼁㻿㻭㼃㼕㼘㼘㼕㼍㼙㻌㻾㼍㼥㼑㼚㼟 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻷㼑㼚㼠㼡㼏㼗㼥㻘㻌㼁㻿㻭㻻㻯㻝㻥䚷㻼㻾㻱㻙㻿㻱㻾㼂㻵㻯㻱㻌㼀㻱㻭㻯㻴㻱㻾㻿䇻㻌㻺㻻㼀㻵㻯㻵㻺㻳㻌㻻㻲㻌㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻭㻸㻹㻻㻰㻱㻸㻸㻵㻺㻳㻌㻼㻾㻻㻯㻱㻿㻿㻱㻿㻭㼘㼕㼚㼍㻌㻭㼘㼣㼍㼟㼠 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻴㼍㼙㼎㼡㼞㼓㻘㻌㻳㼑㼞㼙㼍㼚㼥㻻㻯㻞㻜䚷㻯㻻㻹㻮㻵㻺㻵㻺㻳㻌㻮㻾㼁㼀㻱㻌㻲㻻㻾㻯㻱㻌㻭㻺㻰㻌㻵㼀㻌㼀㻻㻌㻿㻻㻸㼂㻱㻌㻭㻯㻸㻭㻿㻿㻵㻯㻭㻸㻌㻿㻴㻻㻼㻼㻵㻺㻳㻌㻼㻾㻻㻮㻸㻱㻹㻸㼡㼐㼣㼕㼓㻌㻼㼍㼐㼕㼠㼦 㻘㻌㻰㼞㼑㼟㼐㼑㼚㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻭㼜㼜㼘㼕㼑㼐㻌㻿㼏㼕㼑㼚㼏㼑㼟㻘㻌㻲㼍㼏㼡㼘㼠㼥㻌㼛㼒㻵㼚㼒㼛㼞㼙㼍㼠㼕㼏㼟㻌㼍㼚㼐㻌㻹㼍㼠㼔㼑㼙㼍㼠㼕㼏㼟㻘㻌㻳㼑㼞㼙㼍㼚㼥㻾㼛㼛㼙㻠㻜㻡㻌㻔㻿㼔㼛㼞㼠㻕㻔㼀㼑㼏㼔㼚㼛㼘㼛㼓㼥㻕㻯㼔㼍㼕㼞㻿㼛㼚㼖㼍㻌㻮㼘㼑㼥㼙㼑㼔㼘㻸㼡㼐㼣㼕㼓㻌㻼㼍㼐㼕㼠㼦㻻㻯㻞㻝䚷㻭㻌㻿㼀㼁㻰㼅㻌㻻㻺㻌㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻭㻸㻌㻹㻻㻰㻱㻸㻵㻺㻳㻌㼁㼀㻵㻸㻵㼆㻵㻺㻳㻭㻼㻻㻸㻸㻻㻺㻵㼁㻿㻓㻌㻯㻵㻾㻯㻸㻱㼀㼍㼗㼡㼙㼍㻌㼀㼍㼗㼍㼥㼍㼙㼍 㻘㻌㻷㼛㼚㼍㼚㻌㻶㼡㼚㼕㼛㼞㻌㻴㼕㼓㼔㻌㻿㼏㼔㼛㼛㼘㻘㻌㻶㼍㼜㼍㼚㻻㻯㻞㻞䚷㻱㼄㻼㻸㻻㻾㻭㼀㻻㻾㼅㻌㻿㼀㼁㻰㼅㻌㻻㻺㻌㻿㼀㼁㻰㻱㻺㼀㻿㻓㻌㼁㻺㻰㻱㻾㻿㼀㻭㻺㻰㻵㻺㻳㻻㻲㻌㻿㻵㻹㼁㻸㻭㼀㻵㻻㻺㻿㻌㻙㻌㻴㻻㼃㻌㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻭㻸㻌㻹㻻㻰㻱㻸㻸㻵㻺㻳㻯㻻㻹㻼㻱㼀㻱㻺㻯㼅㻌㻺㻱㻱㻰㻿㻌㼀㻻㻌㻮㻱㻌㻯㻻㻹㻼㻸㻱㻹㻱㻺㼀㻱㻰㻌㼀㻻㻌㻼㻾㻻㻹㻻㼀㻱㼁㻺㻰㻱㻾㻿㼀㻭㻺㻰㻵㻺㻳㻌㻻㻲㻌㻿㻵㻹㼁㻸㻭㼀㻵㻻㻺㻿㻿㼛㼚㼖㼍㻌㻮㼘㼑㼥㼙㼑㼔㼘 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻱㼐㼡㼏㼍㼠㼕㼛㼚㻌㻸㼡㼐㼣㼕㼓㼟㼎㼡㼞㼓㻘㻌㻳㼑㼞㼙㼍㼚㼥㻯㼔㼞㼕㼟㼠㼕㼚㼑㻌㻮㼑㼟㼏㼔㼑㼞㼑㼞 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻱㼐㼡㼏㼍㼠㼕㼛㼚㻌㻸㼡㼐㼣㼕㼓㼟㼎㼡㼞㼓㻘㻌㻳㼑㼞㼙㼍㼚㼥㻝㻞㻦㻠㻡㻙㻝㻟㻦㻠㻡㻛㻟㻲㻌㻒㻌㻠㻲㻸㼁㻺㻯㻴㻌㻔㼀㼍㼗㼑㻌㼍㼣㼍㼥㻌㼘㼡㼚㼏㼔㻌㼎㼛㼤㼑㼟㻌㼕㼚㻌㻯㼛㼒㼒㼑㼑㻌㻮㼞㼑㼍㼗㻌㻭㼞㼑㼍㼟㻌㼛㼚㻌㻟㻲㼍㼚㼐㻌㻠㻲㻕

㻿㻱㻿㻿㻵㻻㻺㻌㻤㻝㻟㻦㻠㻡㻙㻝㻡㻦㻝㻡㻌㻌㻌㻌㻿㻻㻯㻞㻟䚷㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻭㻸㻌㻹㻻㻰㻱㻸㻵㻺㻳㻌㻼㻾㻻㻶㻱㻯㼀㻿㻌㻵㻺㻌㻭㻰㼂㻭㻺㻯㻱㻰㻻㻾㻰㻵㻺㻭㻾㼅㻌㻰㻵㻲㻲㻱㻾㻱㻺㼀㻵㻭㻸㻌㻱㻽㼁㻭㼀㻵㻻㻺㻿㻌㻯㻻㼁㻾㻿㻱㻿㻌㻲㻻㻾㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻿㻌㻭㻺㻰㻌㻱㻺㻳㻵㻺㻱㻱㻾㻵㻺㻳㻌㻿㼀㼁㻰㻱㻺㼀㻿㻾㼛㼛㼙㻟㻜㻝㻭㻌㻔㻸㼛㼚㼓㻕㻔㻿㼀㻱㻹㻌㻒㻌㻹㼛㼐㼑㼘㼘㼕㼚㼓㻕㻯㼔㼍㼕㼞㻹㼕㼠㼟㼡㼞㼡㻌㻷㼍㼣㼍㼦㼛㼑㼅㼡㼞㼕㼥㻌㻾㼛㼓㼛㼢㼏㼔㼑㼚㼗㼛㼅㼡㼞㼕㼥㻌㻾㼛㼓㼛㼢㼏㼔㼑㼚㼗㼛 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻭㼓㼐㼑㼞㻘㻌㻷㼞㼕㼟㼠㼕㼍㼚㼟㼍㼚㼐㻘㻌㻺㼛㼞㼣㼍㼥㻿㼢㼕㼠㼘㼍㼚㼍㻌㻾㼛㼓㼛㼢㼏㼔㼑㼚㼗㼛 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻭㼓㼐㼑㼞㻘㻌㻳㼞㼕㼙㼟㼠㼍㼐㻘㻌㻺㼛㼞㼣㼍㼥㻻㻯㻞㻠䚷㻹㻻㻰㻱㻸㻸㻵㻺㻳㻌㼀㻭㻿㻷㻿㻌㻲㻻㻾㻌㻺㻻㻺㻙㻿㼀㻱㻹㻌㼁㻺㻵㼂㻱㻾㻿㻵㼀㼅㻿㼀㼁㻰㻱㻺㼀㻿㻌㻭㻺㻰㻌㼀㻴㻱㻌㻯㻴㻭㻾㻭㻯㼀㻱㻾㻵㻿㼀㻵㻯㻿㻌㻻㻲㻌㼀㻴㻱㻵㻾㻌㻹㻻㻰㻱㻸㻸㻵㻺㻳㻼㻾㻻㻯㻱㻿㻿
㻹㼕㼠㼟㼡㼞㼡㻌㻷㼍㼣㼍㼦㼛㼑 㻘㻌㻻㼟㼍㼗㼍㻌㻹㼑㼠㼞㼛㼜㼛㼘㼕㼠㼍㼚㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻘㻌㻶㼍㼜㼍㼚㻷㼛㼖㼕㻌㻻㼠㼍㼗㼕 㻘㻌㻴㼛㼗㼗㼍㼕㼐㼛㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻱㼐㼡㼏㼍㼠㼕㼛㼚㻘㻌㻶㼍㼜㼍㼚30㻻㻯㻞㻡䚷㻯㻻㻹㻼㻭㻾㻵㻺㻳㻌㻭㻺㻰㻌㻯㻻㻺㼀㻾㻭㻿㼀㻵㻺㻳㻌㻿㼀㻭㼀㻵㻿㼀㻵㻯㻭㻸㻌㻭㻺㻰㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻭㻸㻌㻹㻻㻰㻱㻸㻸㻵㻺㻳㻙㻌㻭㻌㻿㼀㼁㻰㼅㻌㻻㻲㻌㼀㻴㻱㻌㻾㻻㻸㻱㻌㻻㻲㻌㻰㻭㼀㻭㻯㼍㼙㼕㼘㼘㼍㻌㻴㼑㼘㼘㼟㼠㼑㼚㻌㽣㼟㼠㼑㼞㼓㼍㼍㼞㼐 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㻯㼛㼘㼘㼑㼓㼑㻌㻯㼛㼜㼑㼚㼔㼍㼓㼑㼚㻘㻰㼑㼚㼙㼍㼞㼗
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31㻼㻸㻱㻺㻭㻾㼅㻌㻸㻱㻯㼀㼁㻾㻱㻌㻟㻼㻟䚷㻸㻱㻿㻿㻻㻺㻌㻿㼀㼁㻰㼅㻌㻭㻺㻰㻌㻵㼀㻿㻌㻾㻱㻸㻭㼀㻵㻻㻺㻿㻌㼀㻻㻹㻭㼀㻴㻱㻹㻭㼀㻵㻯㻭㻸㻌㻹㻻㻰㻱㻸㻸㻵㻺㻳㻝㻡㻦㻠㻡㻙㻝㻢㻦㻠㻡㻛㻮㻝㻲㻌㻱㼢㼑㼚㼠㻌㻴㼍㼘㼘㼅㼛㼟㼔㼕㼚㼛㼞㼕㻌㻿㼔㼕㼙㼕㼦㼡 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㼀㼟㼡㼗㼡㼎㼍㻘㻌㻶㼍㼜㼍㼚㻯㼔㼍㼕㼞㻿㼠㼍㼚㼕㼟㼘㼍㼣㻌㻿㼏㼔㼡㼗㼍㼖㼘㼛㼣 㻘㻌㼁㼚㼕㼢㼑㼞㼟㼕㼠㼥㻌㼛㼒㻌㻌㻹㾇㼚㼟㼠㼑㼞㻘㻌㻳㼑㼞㼙㼍㼚㼥㻝㻢㻦㻠㻡㻙㻝㻣㻦㻠㻡㻛㻮㻝㻲㻌㻱㼢㼑㼚㼠㻌㻴㼍㼘㼘㻳㻱㻺㻱㻾㻭㻸㻌㻹㻱㻱㼀㻵㻺㻳㻝㻤㻦㻜㻜㻙㻌㻛㻞㻲㻌㻾㼑㼏㼑㼜㼠㼕㼛㼚㻌㻴㼍㼘㼘㻌㻮㻯㻻㻺㻲㻱㻾㻱㻺㻯㻱㻌㻰㻵㻺㻺㻱㻾㻝㻜㻦㻟㻜㻙㻝㻣㻦㻟㻜㻛㻾㼛㼛㼙㻠㻜㻠㻯㻻㻾㻼㻻㻾㻭㼀㻱㻌㻱㼄㻴㻵㻮㻵㼀㻵㻻㻺㻿

32

Friday, September 15, 2023
Time/Location

Activity
PLENARY PANEL

P4 RELATIONS AMONG PROBLEM SOLVING/POSING,
CREATIVITY AND MATHEMATICAL MODELLING
9:30-11:00/B1F Event Hall

Jinfa Cai, University of Delaware, USA
Gabriele Kaiser , University of Hamburg, Germany & Nord

University, Norway
Roza Leikin , University of Haifa, Israel
Chair
Jonas Bergman Ärlebäck , Linköping University, Sweden

11:15-12:00/B1F Event HallPOLLAK AWARD CEREMONY & CLOSING CEREMONY

33

Abstract

Keynote34

CONCEPTUALISING THE RELATIONSHIP BETWEEN MATHEMATICAL
MODELLING AND INTERDISCIPLINARY STEM EDUCATION
Merrilyn Goos1 and Susana Carreira22

1

University of the Sunshine Coast, Australia; 2University of the Algarve and Institute of Education,
University of Lisbon, Portugal

Around the world, STEM education is promoted by governments as a means of addressing social andeconomic challenges and creating a scientifically, mathematically, and technologically literate
citizenry. In many countries, policies and reports by governments and business groups aim toincorporate STEM into the school curriculum, encourage young people to engage in STEM education,
and advocate for STEM careers. Yet, STEM education research is still in an embryonic state and thefield lacks a scientific evidence base that can inform theory, policy, and practice (Maass et al., 2019).
It is also unclear how mathematical concepts and practices contribute to a better understanding of theother STEM disciplines; nor do we understand well enough how STEM education enhance students’learning of mathematics (English, 2016).
One way of exploring the role of mathematics in interdisciplinary STEM education is to examinesynergies between STEM and mathematical modelling. For example, both approaches may involve
making connections between mathematics and other disciplines (Goos, 2020). In STEM education,
however, the disciplines connected to mathematics are science, technology, and engineering, while
many real world situations to be modelled need not be related to such disciplines, despite the argumentthat mathematical modelling is inherently interdisciplinary (Stillman et al., 2023). Some perspectives
see the engineering design process as foundational in STEM tasks (English, 2016), while others haveproposed to articulate the design and the modelling cycles (Baioa & Carreira, 2021). In thispresentation, we will consider the impact of the STEM education movement on interdisciplinary
curriculum development, ask whether good interdisciplinary “M in STEM” tasks are also goodmodelling tasks (and vice versa), and compare the influence of STEM and mathematical modellingon school mathematics education.
References
Baioa, A. M., & Carreira, S. (2021). Mathematical thinking about systems – students modeling a biometrics
identity
verification
system.
MathematicalThinking
andLearning,
https://doi.org/
10.1080/10986065.2021.2012736
English, L. (2016). STEM education K-12: Perspectives on integration. International Journal of STEM
Education, 3(3).
Goos, M. (2020). Mathematics crossing borders: A comparative analysis of models for integrating
mathematics with other disciplines in pre-service teacher education, In N. Radakovic & L. Jao (Eds.),
Borders in mathematics pre-service teacher education (pp. 91–115). Springer.
Maass, K., Geiger, V., Romero Ariza, M., & Goos, M. (2019). The role of mathematics in interdisciplinary
STEM education. ZDM Mathematics Education, 51(6), 869–884.
Stillman, G. A., Ikeda, T., Schukajlow, S., Araújo, J. L., & Ärlebäck, J. B. (2023). Survey of interdisciplinary
aspects of the teaching and learning of mathematical modelling in mathematics education. In G. Greefrath,
S. Carreira, & G. A. Stillman (Eds.), Advancing and consolidating mathematical modelling (pp. 21–41).
Springer.

35

Plenary Lecture

36

TEACHER EDUCATION AND MATHEMATICAL MODELLING –PRE-SERVICE TEACHERS’ PROFESSIONAL COMPETENCEFOR THE TEACHING OF MATHEMATICAL MODELLING
Gilbert GreefrathUniversity of Münster, Germany

Teaching mathematical modelling is considered challenging (Blum, 2011). However, it also offers
special opportunities for mathematics teaching due to the openness and self-differentiating propertiesof modelling tasks. We know that pre-service teachers prefer tasks with low modelling content andhave little knowledge about mathematical modelling. Thus, an important goal of teacher education isto enable pre-service teachers to acquire this specific professional competence for the teaching ofmathematical modelling.
Professional competence is agreed to include affective-value-oriented aspects in addition tocognitive-oriented knowledge dimensions (Blömeke et al., 2015; Kunter et al., 2013). In this context,
pedagogical content knowledge represents a central factor in determining the cognitive activation
potential of teaching. These characteristics of professional competence can be transferred to the arenaof mathematical modelling. In particular, the area of pedagogical content knowledge appears to be
especially significant.
When designing modelling seminars for pre-service teachers with the participation of students inpractical phases for the acquisition of professional competence, the question arises in particular as towhich special facets of pedagogical content knowledge specific to modelling should be emphasised.
The results show that, overall, the pedagogical content knowledge of pre-service teachers for teachingmathematical modelling can be significantly improved and that it can be very helpful for theacquisition of competence in some facets if the pre-service teachers develop modelling tasks for thepractical phases themselves (Greefrath et al., 2022). Furthermore, we also take a look at thedevelopment of affective-value-oriented aspects. Going further, the question arises to what extent theuse of technology can be considered in the acquisition of professional competences for teachingmathematical modelling and what new possibilities arise from this. Initial results will be reported on
this.
References
Blömeke, S., Gustafsson, J.-E., & Shavelson, R. J. (2015). Beyond Dichotomies: Competence Viewed as aContinuum. Zeitschrift für Psychologie, 223(1), 3–13.
Blum, W. (2011). Can Modelling Be Taught and Learnt? Some Answers from Empirical Research. In G.
Kaiser, W. Blum, R. Borromeo Ferri, & G. A. Stillman (Eds.), Trends in Teaching and Learning ofMathematical Modelling (pp. 15–30). Springer.
Greefrath, G., Siller, H.-S., Klock, H., & Wess, R. (2022). Pre-service secondary teachers’ pedagogical content
knowledge for the teaching of mathematical modelling. Educational Studies in Mathematics, 109(2), 383–
407.
Kunter, M., Baumert, J., Blum, W., Klusmann, U., Krauss, S., & Neubrand, M. (Eds.). (2013). Cognitive
activation in the mathematics classroom and professional competence of teachers. Results from the coactiv
project. Springer.

37

IN THEIR OWN WORDS: EXPLANATIONS OF STEM STUDENTS’
REASONING DURING MATHEMATICAL MODELLING
Jennifer A. Czocher
Texas State University, USA

Teaching mathematics through mathematical modelling means moving beyond dressing up
mathematical procedures with real-world terminology. It involves interacting with students’representational systems and ways of thinking about and with those systems – and those ways ofthinking are often non-normative, incomplete, and self-contradictory to an objective observer, like ateacher or a researcher. Thus, one question the field has posed is How should the observer respondto students’ work? There is a strong temptation to attend to accuracy and to correct students’modelling work because it does not match the intended solution. However, we know that consistent
negative feedback dismissing students’ real-world experiences teaches them to ignore their own
intuitions and hinders them in learning to use mathematics to solve realistic problems.
In this talk, I start from the position that it is not possible to answer the question How should theobserver respond? without understanding Why did the students model the way they did? This second
question is partially answered. The field already has a number of high-level, descriptive answers tothis second question. For example, interpretations of students’ work might point towards
metacognitive fault (the student was “not paying attention” to or “did not notice” some feature theobserver thought was important), implicate obstacles to knowledge transfer (the student did not
activate target mathematics content), or even indicate blockages to phases of a modelling cycle (thestudent did not include/ignore variables). I argue that descriptive framings, even when couched insuitable theories, fall short of explaining what the student might have been thinking that led them totheir modelling decisions which limits our capacity to design contingent responses to students’modelling work. To enhance the field’s ability to address students’ patterns of reasoning, I argue weought to be asking What were they thinking? A simple analogy is that a doctor should seek theunderlying cause of symptoms to prescribe a proper treatment.
Awareness of the typical ways students think about mathematical content while modelling and thekinds of explanations that are sensible to students is an essential step towards equipping instructors
with the pedagogical skills they need to contingently interpret student thinking and respondconstructively to students’ reasoning patterns. Using STEM students’ own words, I will share somearchetypal ways students think about mathematical content while modelling including: their desire
for precision, accuracy and authenticity, the mathematical structures they anticipate, and therationales they use to justify the mathematics selected for their models. My goal is to increase thesophistication we attribute to students’ modelling decisions by deepening the field’s understandingof students’ mathematics – as they use it in modelling – as an alternative to focusing exclusively on
accuracy of solutions to modelling problems.

38

LESSON STUDY AND ITS RELATIONS TO MATHEMATICAL
MODELLING
Yoshinori Shimizu
University of Tsukuba, Japan

Lesson study that originated in Japan is an approach to improve teaching and learning mathematics
through a particular form of activity in which a group of teachers works collaboratively to design,
implement, observe, and reflect on the proposed “research lessons”. Research and practice on lesson
study has spread internationally with a focus on its capacity as a vehicle for the professional
development of teachers. While recent advocacy of lesson study is focused on its function of
professional development, key impacts of lesson study also include development of mathematics
curriculum with the design of tasks and instructional sequences (Lewis, 2016).
If mathematical modelling is being introduced as a new product into the complex system ofmathematics education in many countries, it has to fit with the existing parts and interfaces in thissystem (Pollak, 2015). For more than the decades, various approaches were taken for exploring thenature of teaching and learning of mathematical modelling in the classroom. Such approaches include
teaching experiment, design research (Lesh et al., 2010), and case studies (Schukajlow et al, 2018).
By examining the phases of lesson study and the process of teaching and learning mathematicalmodelling in the classroom, this talk illustrates how a cyclic nature of lesson study can contribute todesigning tasks and activities for the students and to develop an instructional sequence for teaching
and learning mathematical modelling embedded in a teaching unit of mathematical topics.
Focusing on the process of mathematical modelling (e.g. “model-elicit-activities” or “being aware of
the prerequisite of a model”), lesson study provide the community of teachers illuminating examples
with the “evidences” of students’ work, tasks to be appeared in the textbooks, and a shared vision on
the new curriculum. The case of “Be Careful with the Blind Corner from the Car” (Nishimura, 2012),
appeared in an “Open House” at a university-affiliated high school, is used for describing how task
design and its implementation through lesson study procced. The example with other cases illustrates
how cycles of lesson study can contribute to the task design and then to the curriculum development.
References
Lesh, R., Young, R. & Fennewald, T. (2010). Modeling in K-16 mathematics classrooms – and beyond. In R.
Lesh (Eds.) Modeling Students’ Mathematical Modeling Competencies (pp. 275–283). New York: Springer.
Lewis, C. (2016). How does lesson study improve mathematics instruction? ZDM Mathematics Education 48,
571–580.
Nishimura, K. (2012). Task Design and Implementation for Developing Students’ Mathematical Modelling
Competencies. Tokyo: Toyokan (in Japanese).
Pollak, H. O. (2015). The place of mathematical modelling in the system of mathematics education. In G. A.
Stillman et al. (Eds.), International Perspectives on Teaching and Learning Mathematical Modelling. (pp.
265–276). New York: Springer.
Schukajlow, S., Kaiser, G. & Stillman, G. (2018) Empirical research on teaching and learning of mathematicalmodelling: a survey on the current state-of-the-art. ZDM Mathematics Education 50, 5–18.

39

Plenary Panel

40

RELATIONS AMONG PROBLEM SOLVING/POSING, CREATIVITY AND
MATHEMATICAL MODELLING
Jonas Bergman Arleback1 (Chair), Jinfa Cai2, Gabriele Kaiser3, and Roza Leikin4
1

Linköping University, Sweden; 2University of Delaware, USA; 3University of Hamburg, Germany& Nord University, Norway; 4University of Haifa, Israel

Mathematical problem solving and mathematical modelling have several commonalities as has been
pointed out on various occasions (Blum & Niss, 1991; Niss & Højgaard, 2019). In the current
discourse on mathematics education and mathematics education research, problem solving/posing,creativity and modelling play an important role and their fostering is emphasized in many curricular
standards all over the world, being quite often strongly connected within the curricula.
However, although mathematical modelling and mathematical problem solving/posing share many
commonalities in their emphasis on non-routine and open-ended solution processes, where creativity
is concerned, both approaches have also many differences in their orientation and foci. Especially
their relation to the real world as a context for the problems tackled and their requirements concerning
authenticity and proximity to reality are quite different. The creativity-directed approach involves
students in creative mathematical processing. The fostering of creativity through challenging
mathematical problems and tasks play an important role within mathematics education and have beenconnected in the past especially to problem solving and complex problems in mathematicalcompetitions. However, in the recent years the scope of fostering mathematical creativity has been
widened, focusing on mathematical problems with different stages of complexity and a great variety
of solution processes, also within real world contexts.
In all three approaches scaffolding activities play an important role as the teacher should function asa mentor supporting students to find their own solutions on their own pace, as independent as possible.
The panel will discuss the complex relations between these three important approaches concerning
the following topics:- What kind of tasks are important in the three approaches, how can they be characterised?
- How can the solution strategies be characterised?
- How can problem solving/posing competency and modelling competency be conceptualised,
respectively? How can they mutually support each other for further advances?
- Which scaffolding activities have proven to be successful for mentoring students in theirsolution processes?
- Which kind of theoretical framework is appropriate, for example concerning the relation toreal-world contexts or to metacognition?
- Under which conditions foster problem solving/posing and modelling tasks creative
mathematical processing?
The panel will close with prospects for further research and effective practices.
References
Blum, W., & Niss, M. (1991). Applied mathematical problem solving, modelling, applications, and links toother subjects: state, trends and issues in mathematics instruction. Educational Studies in Mathematics, 22,
37–68.
Niss, M., & Højgaard, T. (2019). Mathematical competencies revisited. Educational Studies in Mathematics,
102, 9–28.

41

Special Lecture

42

IMMC: CELEBRATING 10 YEARS OF INFLUENCINGEDUCATIONAL CHANGE
Benjamin Galluzzo1and Alfred Cheung2
1

Clarkson University, USA; 2NeoUnion ESC Organization, Hong Kong, China

The Consortium for Mathematics and Its Applications (COMAP) has had over 35 years of experiencein running contests in mathematical modelling – at the undergraduate level with the MathematicalContest in Modeling / the Interdisciplinary Contest in Modeling (MCM/ICM), at the secondary levelwith the High School Mathematical Contest in Modeling (HiMCM) and now at the middle schoollevel with the Middle Mathematical Contest in Modeling (MidMCM). While these are international
contests, we have never attracted more than 20 countries to participate, and entrants are primarily
from China and the U.S. In an attempt to level the playing field, COMAP joined with NeoUnion ESC
Organization (NeoUnion) of Hong Kong to create a modelling challenge based on COMAP’s practiceand the Olympiad model as well. And so, in the International Mathematical Modeling Challenge(IMMC or styled as IM2C) each country or region is permitted to submit no more than two papers to
the international judging round. This model has proved to be quite successful and pre-Covid we had
48 countries registering teams. Indeed, during the last ten years, the IMMC has played an increasingrole in the education scene around the world. It has empowered students, teachers, and policy makers
to place mathematical modelling at the center of curricula reform and educational change.
Our aim in this talk is to invite you to join us, not simply as a participating country, but also as a
research partner with the shared goal to promote the value and importance of mathematical modellingas a critical skill all students should have the opportunity to learn. Of course, we are only at the
beginning, but we now have an extensive database that we could (and should) make use of together.
Currently we have access to student work from the first ten years of the IMMC, representing morethan 400 teams from over 50 countries. We are excited to utilize this data to address questions suchas: To what extent and in what sense does cultural context influence model development? What is theimpact of developing a pre-selection competition in a country? What makes a math modelling task
meaningful for an international audience? These are just a few examples; we look forward to sharing
our data and hearing your ideas.
The second part of the talk will highlight one country’s story. We will focus on what China does in
IMMC and how attitudes towards modelling, including curricula, have dramatically changed over the
past ten years. An emphasis on policy changes and influence, made possible through the IMMC,
showing what a focus on mathematical modelling contests has made possible.

References
Boaler, J., Cordero, M., & Dieckmann, J. (2019). Pursuing gender equity in mathematics competitions: A caseof mathematical freedom. Focus, 39(1), 18-20.
Garfunkel, S., Niss, M., & Brown, J. P. (2021). Opportunities for modelling: An extra-curricular challenge.
In Mathematical Modelling Education in East and West (pp. 363-375). Cham: Springer International
Publishing.

43

Long & Short
Presentation

44

PRE-SERVICE TEACHERS’ INTERVENTIONS IN MATHEMATICAL
MODELING ACTIVITIESYongjian Sun1 and Jing Cheng2
1,2

East China Normal University, China

The importance of teacher pedagogical interventions for student learning has been well recognizedby substantial literature, however there has been little research on pedagogical interventions inmathematical modeling. Employing scaffolding theory (Stender & Kaiser, 2015) and the theory aboutteacher interventions in cooperative learning (Ferguson-Patrick, 2016), this paper presented theinvestigation of the intervention strategies used by pre-service mathematics teachers when they were
offering assistance for students in doing mathematical modelling.
This study presented a modelling activity which included a total number of 18 pre-service
mathematics teachers and 32 grade eight students who were divided into 6 groups, one group with 56 students. When the students were doing a modelling task in groups, the 18 pre-service mathematics
teachers were playing different roles: usually a lead teacher who progressed the process of modelling,
six teachers who sit in each student group to offer instant interventions and the other pre-service
teachers who sit aside to observe the interventions. Since the students did three modelling tasks in theactivity, all the 18 pre-service teachers had the chance to conduct interventions for the students to dodifferent tasks. The entire modelling activity of 240 minutes was both audio and video taped.
In this paper, we analyzed the types of teacher interventions by adopting Stender and Kaiser’s (2015)
classification of teacher interventions (including the categories of motivative, feedback, generalstrategic, content-oriented strategic, and content) and Ferguson-Patrick’s focus on teachers’ socialrole and organizational role when providing interventions. It is expected that different types of teacher
interventions will be recognized in the modelling activities. Moreover, the characteristics of thedifferent types of teacher interventions will be in-detailed analyzed; and the classification of teacher
interventions on mathematical modelling might be influenced by the types of modelling tasks.
Through the analytic results and discussions based on the results, this study might provide insights
for the learning and teaching of mathematical modelling, especially in the context of China.
References
Ferguson-Patrick K. (2016). The importance of teacher role in cooperative learning : The effects of high-stakes
testing on pedagogical approaches of early career teachers in primary schools. Education3-13, 46 (1), 89101.
Stender, P., Kaiser, G. (2015). Scaffolding in complex modelling situations. ZDM Mathematics Education, 47,
1255–1267.

45

A TEST INSTRUMENT FOR PROFESSIONAL MATHEMATICALKNOWLEDGE ON MATHEMATICAL MODELLING - FIRST RESULTS
Rita Borromeo Ferri1, Gilbert Greefrath2 and Hans-Stefan Siller3
1

University of Kassel, Germany; 2University of Münster, Germany; 3University of Würzburg,
Germany

The professional knowledge of mathematics teachers provides a rational basis for decisions inmathematics teaching. It also provides arranging the contents, necessary simplifications and the useof application possibilities (Vollrath, 1988). However, specific professional knowledge onmathematical modelling is required especially for the fostering of mathematical modellingcompetence of students at all ages. This can generally be considered as consisting of different areasof knowledge (Shulman, 1986). Although studies on mathematical modelling knowledge alreadyexist, less attention has been paid to the aspect of professional mathematical modelling knowledge ofpre-service teachers. Previous test instruments on the modelling-specific mathematical knowledge ofpre-service teachers use either school-level mathematical knowledge of modelling (Yang et al., 2022),
or knowledge that goes well beyond secondary school (Czocher et al., 2021; Haines et al., 2001).
First test items on content knowledge at the level of in-depth school mathematical knowledge in thearea of calculus were developed (Greefrath et al., 2022). In our test construction, an atomistic
approach is chosen; the items address individual sub-competencies. In this study, we present the first
results of piloting the test items. 101 pre-service teachers from three German universities participatedin the pilot study. A scale for modelling-specific content knowledge for pre-service teachersconsisting of a total of five multiple-choice test items for different sub-competencies was created. We
present the test instrument and solution rates are reported and discussed against the background of
the selected population - with surprising results.
References
Czocher, J. A., Melhuish, K., Kandasamy, S. S., & Roan, E. (2021). Dual measures of mathematical modelingfor engineering and other stem undergraduates. International Journal of Research in Undergraduate
Mathematics Education, 7(2), 328–350.
Greefrath, G., Borromeo Ferri, R., & Siller, H.-S. (2022). Professional content knowledge of mathematicalmodelling of preservice teachers: First considerations for a new test instrument. In J. Hodgen, E. Geraniou,
G. Bolondi, & F. Ferretti (Eds..), Proceedings of the Twelfth Congress of the European Society for Researchin Mathematics Education (CERME12) (pp. 1067–1074). ERME / Free University of Bozen-Bolzano.
Haines, C. R., Crouch, R., & Davis, J. (2001). Understanding students’ modelling skills. In J. F. Matos, W.
Blum, K. Houston, & S. P. Carreira (Eds..), Modelling and Mathematics Education (pp. 366–380).
Horwood.
Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2),Article 2.
Yang, X., Schwarz, B., & Leung, I. K. C. (2022). Pre-service mathematics teachers’ professional modelingcompetencies: A comparative study between Germany, Mainland China, and Hong Kong. Educational
Studies in Mathematics, 109(2), 409–429.
Vollrath, J. (1988). The role of mathematical background theories in mathematics education. In H.-G. Steiner,& A. Vermandel (Eds.), Foundations and methodology of the discipline mathematics education.
Proceedings of the 2nd TME conference (pp. 120–137). Bielefeld: IDM.

46

SYSTEM DYNAMICS SOFTWARE AND SCENARIOS FOR CONTINUOUS
PROFESSIONAL DEVELOPMENT
Cyril Julie
University of the Western Cape, South Africa

Software to assist with the investigation of behaviour over time of natural and social phenomena havebeen used for some time. Galbraith & Fisher (2021) present a concise but fairly comprehensive
exposition on how the availability of free software opened opportunities to explore complex
phenomena from a systems dynamic perspective. They suggest that “it is necessary to invest someeffort into understanding the software…before their potential can be realised.” (p. 620). This
presentation is about this investment of effort to come to grips with a particular free version of system
dynamics
modelling
software,VensimPLE(Ventana
Systems,
https://www.ventanasystems.com/software/), to develop scenarios of continuous professional
development (CPD) offerings for mathematics teachers.
Continuous professional of mathematics teachers is a complex social issue. An umbrella review (areview of reviews of the evidence); systematic reviews and single research studies on effective CPD
reveal that at least 25 issues are at stake for the successful delivery of CPD. CPD providers in South
Africa were surveyed to rate the 25 issues. This was done since qualitative data “allows decision
makers and social actors to be more aware of the directions their decisions could lead the system; andwhat the key variables are for the implementation of public policies to achieve the desired (or agreed)
scenario”. (Corral-Quintana, et al., 2015, p. 83). Using the ratings of the CPD providers in South
Africa, my own experience with CPD provision and some other literature scenarios for effective CPD
were developed. I report, in a quasi-autoethnographic manner, on two different scenarios for the
implementation of CPD that were developed.
The concluding argument revolves around the need for more personal-like accounts of laying bear
the blind alleys we encounter when coming to grips with using software for mathematical modelling.
References
Corral-Quintana, S., Nuez, D. L., Verna, C. L., Jesús Hernández Hernández, J. H. & David Romero-Manriquede Lara, D. R. (2015). How to improve strategic decision-making in complex systems when only qualitative
information is available. Land Use Policy, January 2016, 83-101.
Galbraith, P., & Fisher, D. M. (2021). System Dynamics: Adding a String to the Modelling Bow Mathematicalmodel-making in problem solving: Japanese pupils’ performance and awareness of assumptions. In F. K.
S. Leung, G. A. Stillman, G. Kaiser, & K. L. Wong (Eds.), Mathematical Modelling Education in East andWest (pp. 619 – 629). Springer.

47

FACTORS AFFECTING CHINESE IN-SERVICE HIGH SCHOOL
MATHEMATICS TEACHERS’ IMPLEMENTATION OF MATHEMATICAL
MODELLING IN CLASSROOM: AN INTERVIEW STUDY
Xinrong YangSouthwest University, China

Mathematical modelling has been widely accepted as a core component of mathematical literacy forstudents internationally nowadays. Due to such importance, mathematical modelling has been
integrated into school mathematics curricula worldwide to promote connections between schoolmathematics and real-world situations (Kaiser, 2017). Similarly, the idea of mathematical modelling
was started to be emphasized in China in 2003 to change high school students’ mathematics learning
experience, and it is now listed as one of the six key mathematical competences in the high schoolmathematics curriculum (MOE, 2017). However, after almost twenty years, like in many other places,
there is still few mathematical modelling in everyday mathematics teaching in China at the moment(Blum, 2015). Without any doubt, teachers play the most decisive role in the successful incorporationof mathematical modelling in their everyday teaching (Stillman, 2010). In view of this, the study
investigated what factors influence teachers to implement mathematical modelling in classroom.
Thirty in-service high school mathematics teachers with various teaching experience (1-32 years)
from various places such as Shanghai and Chongqing in China attended this study. Each of them wasinterviewed on the topics of their experiences and difficulties or obstacles of implementation
mathematical modelling in everyday teaching. Constant comparative analysis approach (Strauss &Corbin, 1990) was applied to analyse the transcribed documents.
Generally, five factors/obstacles were identified: 1) the confusion of teachers with respect to whatmathematical modelling is, and the lack of teacher knowledge; 2) shortage of professional
development opportunities at both pre-service and in-service stages; 3) shortage of recourses such as
no enough or suitable modelling examples; 4) examination culture since mathematical modelling is
not included in university-entrance examination and shortage of teaching time; 5) the lack ofknowledge for students because students did not have plenty previous experience. These
factors/obstacles will be further discussed with referring to societal and cultural factors such as
traditions of teacher education within the Chinese context.
References
Blum, W. (2015). Quality teaching of mathematical modelling: What do we know, what can we do? In S. J.
Cho (Ed.), Proceedings of the 12th international congress on mathematical education (pp. 73–96). Springer.
Kaiser, G. (2017). The teaching and learning of mathematical modeling. In J. Cai (Ed.), Compendium for
Research in Mathematics Education (pp. 267–291). National Council of Teachers of Mathematics.
Stillman, G. (2010). Implementing applications and modelling in secondary school: issues for teaching andlearning. In B. Kaur & J. Dindyal (Eds.), Mathematical applications and modelling (pp. 300–322).
Singapore: World Scientific.
Strauss, A., & Corbin, J. (1990). Basics of qualitative research: Grounded theory procedures and techniques.
Newbury Park, CA: Sage.

48

DEVELOPING LOCAL REPRESENTATIVE SOLUTIONS TO FERMI
PROBLEMS
Albarracín, Lluís 1, Jonas Bergman Ärlebäck2, Serife Sevinc3, Irene Palomares4, Daniel Orey5 and
Milton Rosa5
Universitat Autònoma de Barcelona, Spain; 2Linköpings Universitet, Sweden; 3Orta Doğu TeknikÜniversitesi, Turkey; 4Universitat de València, Spain; 5Universidade Federal de Ouro Preto, Brazil
1

According to Niss and Blum (2020), mathematical modelling refers to the process of constructing
mathematical representations or abstractions that capture essential features of real-world phenomena,situations, or problems. It involves translating a non-mathematical context into a mathematicalframework that enables analysis, exploration, prediction, or decision making related to the originalcontext. The modelling process begins by considering a real-world context and problem that
motivates and engages the problem solver(s). Sriraman and Lesh (2006) suggested that Fermi
problems (FPs) could serve as an effective starting point for engaging students in modelling becauseof their accessibility and ability to emphasise the connection between the real world and themathematical domain. FPs often involve situations from everyday life, such as the number of cars
travelling through a busy street in a day. Solving FPs requires strategies and estimations based on
personal experiences (Ärlebäck, 2009), and provide an opportunity to explore how cultural aspects
come into play during the modelling process. As FPs often involve estimating quantities that mayvary significantly across different contexts or cultures, they provide a tool for investigating theinfluence of intercultural awareness (Baker 2011) in the solving of modelling tasks (Albarracin et al.,2022).
In this presentation we explain an empirical study of the solutions of a Fermi problem by prospective
teachers from 4 universities in 3 different countries. A first phase of the research is to investigate thedifferences in the solutions from different countries, both in terms of mathematical methods used andin terms of the cultural elements that come to the fore. In the second stage, to enable students
effectively to compare productions from all countries, it was necessary to develop what we call"cultural representative solutions". The aim of this presentation is to provide a methodological andtheoretical basis for developing such 'representative solutions' as a research tool in understanding thecultural aspects of the models developed in the solutions to a FPs, detail the process of how data isanalysed to define, design, and validate these "cultural representative solutions", as well as discuss
the potential contribution of using such representative solutions in teaching and learning modelling.
References
Albarracin, L., Ärlebäck, J., Ferrando, I., Orey, D., Rosa, M. & Sevinç, S. (2022, Setember, 24-27). Eliciting
Cultural Aspects in Modeling Processes Using Fermi Problems. ICTMA 20.
Ärlebäck, J. B. (2009). On the use of realistic Fermi problems for introducing mathematical modelling inschool. The Mathematics Enthusiast, 6(3), 331–364.
Baker, W. (2011). From cultural awareness to intercultural awareness: Culture in ELT. ETL Journal, 66(1),62-70.
Niss, M., & Blum, W. (2020). The learning and teaching of mathematical modelling. Routledge.
Sriraman, B., & Lesh, R. A. (2006). Modeling conceptions revisited. ZDM, 38(3), 247–254.

49

DEVELOPMENT OF SECONDARY MATHEMATICS INSTRUCTIONALSTRATEGY FOR ACQUIRING CONCEPTUAL KNOWLEDGE
Shota Higuchi1 and Yuki Watanabe1
1Tokyo University of Science, Japan

Japan’s Ministry of Education, Culture, Sports, Science, and Technology (MEXT) (2018) has
emphasized the significance of acquiring conceptual knowledge through mathematical activity. In
Japan, mathematical modelling is a part of mathematical activity. Fujimura (2011) has pointed outthat junior and senior high school students in Japan need more conceptual knowledge. Problemsolving prior to instruction (PS-I) (Loibl & Rummel 2014) is an instructional strategy for acquiring
conceptual knowledge. Sinha and Kapur (2019) suggested that PS-I may not be effective for somelearner profiles. MEXT (2018) pointed out that inadequate motivation to learn mathematics is one of
the characteristics of Japanese students. As a result, they may stop trying to solve problems.
In this study, taking into account the characteristics of Japanese students, we designed an instructional
strategy for acquiring conceptual knowledge based on PS-I.
The instructional strategy consists of six stages. The first stage is problem solving (part 1). Students
work on a real-world problem that incorporates a fresh concept. The second stage involves evaluating
the generated solutions. In the third stage, students organize the elements of existing knowledge. The
fourth stage is problem solving (part 2). Based on the activities from stages 1–3, students work on thesame problem. In the fifth step, they are informed about the correct solution. In the sixth step, students
compare the correct solution with the generated solution and incorporate their understanding of thecharacteristics of each solution.
In December 2022, we conducted a mock session on quadratic functions with 22 pre-service
mathematics teachers. We investigated whether the instructional strategy developed in this study
would be effective in helping students acquire conceptual knowledge when compared withconventional teaching. According to the results, pre-service mathematics teachers believed that thislesson was sufficient to help students gain conceptual knowledge when compared with conventional
teaching, thus signifying the value of this lesson. However, the mathematics domain may ascertain
how useful However, how useful the teaching strategy from this study is may depend on mathematics
domain. Going forward, we intend to conduct this class with high school students to collect moredata. We plan to present our practice plan at the conference.
Reference
Fujimura, N. (2011). Change processes of studentsʼ mathematical concepts through teaching-learning
activities. Japanese Psychological Review, 54(3), 296-311. (in Japanese)
Loibl, K., & Rummel, N. (2014). The impact of guidance during problem-solving prior to instruction on
students’ inventions and learning outcomes. Instructional Science, 42, 305-326.
MEXT (2018). Courses of Study for High School Mathematics (2018 Edition). (in Japanese)
Sinha, T., & Kapur, M. (2019). When productive failure fails. Proceedings of the Annual Meeting of the
Cognitive Science Society, 41, 2811-2817.

50

DESIGNING MATHEMATICAL MODELLING TASKS INVOLVING A
SIMULATION WITH PHYSICAL EXPERIMENTS: ON THE TASK OF
MARKER RECUPTURE METHODS
Kosuke Mineno,1 Akihiko Saeki,2 and Takashi Kawakami3
1

Toin University of Yokohama, Japan; 2Kanazawa Institute of Technology, Japan;
3

Utsunomiya University, Japan

In order to properly utilize reasoning of sampling in the real world, which includes a variety ofvariables, it is necessary to focus on the factors of variation in the real world and to have a deep
understanding of their methods. When designing a task for this purpose, the requirements are that thecontext be one of modelling from an authentic problem, that the student be able to trace a situation
as close as possible to an actual real-world situation, and that the validity of the model be verifiable.
To fulfill these requirements, we propose modelling involving simulation with physical experiments.
The simulation with physical experiments has a role in verifying how the model works in reality (e.g.,
Carreira & Baioa, 2018). The purpose of this study is to provide a case study of 9th graders in Japanto see how the design of a mathematical modelling task that encompasses simulation with physicalexperiments contributed to a focus on factors of variation in the real world and a deeper understandingof the methods of reasoning of sampling.
Designing a modelling task is not easy, so in this paper, we focus on the sign recapture method, which
is often treated in Japanese textbooks, and redesign it into a modelling task. The original task was toestimate the total number of fish in a lake, that is, the population, based on the ratio of colored touncolored fish when some fish were captured, colored, returned to the lake, and recaptured a few days
later. This is the basis of a model that is frequently used in biological population studies. This task
was redesigned to incorporate a simulation with physical experiments using a fish tank, aiming tocreate a more realistic and authentic situation setting. Then, it revealed that the task of estimating thetotal number of fish lacked authenticity for the students and discarded many variables. In theredesigned task, a situation requiring estimation was first set up as an exotic fish problem. And basedon the simulation using a fish tank, an activity was set up to understand the statistical estimation
method and to critically interpret and validate the method in relation to the real world. Then, a modelwith a better estimation method was re-planned and implemented again through simulation.
In the practical lesson, the first problem with the existing model was that in reality, random sampling
is not possible due to natural variation and migration by herds. Several methods based on multiple
sampling were proposed as improvement measures. Two of the methods that students consideredwere suggested as characteristic examples of critical reflection: setting the number of colors to be
applied and changing the number of colors to be applied. In both considerations, the students
compared the mathematical model with reality while experimenting and refining their plans. They
modified their models from a realistic and statistical perspective. This case study suggests thatsimulation with physical experiments of a realistic and authentic situation setting can help to betterinterpretation and validation of mathematical models in statistics.
References
Carreira, S., & Baioa, A. M. (2018). Mathematical modelling with hands-on experimental tasks: on thestudent’s sense of credibility. ZDM, 50, 201–215.

51

CAN MATHEMATICAL MODELLING WORK AS A
CRITICAL THINKING‑DEMANDING ACTIVITY?
Liezle Boshoff
Cape Peninsula University of Technology, South Africa
This research stems from a PhD research project that investigates how engineering undergraduates
experience critical thinking (CT) – a study motivated by educators’ struggle to understandengineering undergraduates’ ways of engaging with CT. Part of the project called for an activity thatprovides an opportunity for students to think critically and this paper illustrates how the inherent
structure of model-eliciting activities provides a suitable framework to design such an activity. Lu
and Kaiser (2022) illustrated that mathematical modelling has the potential for being a creativitydemanding activity. This project further-develops this framing as the same question is being asked,
this time about CT.
What is CT?
CT is a complex phenomenon which is difficult to define. Ahern et al.’s (2012) conception of CT
does however provide a framework for designing activities that afford the development ofengineering students’ CT skills:
CT is a movement from the concrete, from the factual to the abstract and back again – an ability
not only to use knowledge and facts to create ideas, concepts and solve problems but also to use
these developed concepts, theories, and ideas in the real world.
What is mathematical modelling?
Geiger et al. (2022) maintain:
Mathematical modelling consists of identifying a problem within a real-world context, developinga relevant mathematical representation, determining a subsequent mathematical solution,interpreting the solution within the original context, and evaluating the solution’s validity for
resolving the problem.
Lu and Kaiser (2022) affirm mathematical modelling’s potential as a creativity-demanding activity
via its impact on the modelling cycle. Drawing on the definition of mathematical modelling, thispaper argues that there is the potential for mathematical modelling to be reframed as CT-demanding.
References
Ahern, A., O'Connor, T., McRuairc, G., McNamara, M., & O'Donnell, D. (2012). Critical thinking in theuniversity curriculum – the impact on engineering education. European Journal of Engineering Education,
37(2), 125–132.
Geiger, V., Galbraith, P., Niss, M., & Delzoppo, C. (2022). Developing a task design and implementation
framework for fostering mathematical modelling competencies. Educational Studies in Mathematics,
109(2), 313–336.
Lu, X., & Kaiser, G. (2022). Can mathematical modelling work as a creativity-demanding activity? An
empirical study in China. ZDM: The International Journal on Mathematics Education, 54(1), 67–81.

52

THE CO-PRODUCTIVITY CHAIN OF MODELLING –ON MODELS USED BY CITIZENS, OPERATORS AND CONSTRUCTORSPeter Frejd 1 and Pauline Vos 2
1

Linköping University, Sweden; 2 University of Agder, Norway

In this theoretical paper, we describe, analyse, and theorize how mathematical models play a role invocational work and in daily life. Whilst many people do not engage in the modelling in itself, they
productively engage with mathematical models in several ways.
We present several cases of public and vocational use of line and bar graphs (Kemp & Kissane, 2010).
Figure 1 shows one such a line graph, namely of electricity prices fluctuating by the hour (in green).
The hourly prices are published by a power company one day ahead. Reading the graph as prescriptivemodel (Niss, 2015), customers can plan to use less electricity when prices are high (blue bars). Their
power consumption then offers descriptive feedback so the company can re-model their pricing. This
interactivity is productive in two ways: citizens can try to reduce costs and companies can try to
reduce power peaks.

Building on Skovsmose (2005), we distinguish across groups of people who are, in different ways,
involved in the co-productivity chain of modelling, which involves data feedback and re-modelling.
Civil or industrial institutions hire professional constructors of models (Frejd & Bergsten, 2016).
These models are often hidden in technology but can materialize in spreadsheets or graphs. Operators(vocational workers) apply the models for safety, quality control, decisions, etc. Citizens benefit (ornot) from processes regulated through mathematical models, and their behaviour is monitored andfed back into new models.
For the fostering of mathematical modelling in schools, we highlight that not all students will be
constructors of mathematical models. We will describe competencies needed by operators using
mathematical models in vocational work, and customers using models in daily life.
References
Frejd, P., & Bergsten, C. (2016). Mathematical modelling as a professional task. Educational
Studies in Mathematics, 91(1), 11–35.
Kemp, M., & Kissane, B. (2010). A five-step framework for interpreting tables and graphs in their context. In
C. Reading (Ed.), Data and context in statistics education: Towards an evidence-based society. ICOTS.
Niss, M. (2015). Prescriptive modelling–challenges and opportunities. In Mathematical modelling in education,
research and practice (pp. 67–79). Springer.
Skovsmose, O. (2005). Travelling through education: Uncertainty, mathematics, responsibility. Sense
Publishers.

53

COLLABORATIVE DESIGN OF MATHEMATICAL MODELLING TASKSFOR MIDDLE SCHOOL STUDENTS
Sakon Tangkawsakul1, Weerawat Thaikam2 and Nuttapat Mookda3
1

Kasetsart University, Thailand; 2Nakhonsawan Rajabhat University, Thailand;3
Independent Scholar, Thailand

One of the crucial learning outcomes in mathematics, both in the Thailand curriculum and worldwide,
is for students to develop mathematical literacy and the ability to apply mathematical knowledge tomodel real-world situations in various contexts outside the classroom. (Stacey & Turner, 2014;
Klainin, 2015) So, they should be encouraged to apply their mathematical knowledge and experiences
as tools to hone their mathematical thinking and competency (Blum & Borromeo Ferri, 2009; NCTM,
2014). To achieve the point, the objective of this study was to develop a prototype of the modelling
tasks and establish principles to guide design tasks for middle school students. The researchmethodology involved three phases of design research: preparation and design, experimentation, anda retrospective study as described by Bakker (2018). The participants consisted of educators, teachers,
and students. The instruments were open-ended tasks, which posed questions that encouraged thestudents to formulate, employ, interpret, and evaluate data to solve real-world problems. Semistructured interviews were conducted and the results were assessed using descriptive statistics andcontent analysis. Results revealed that 1) The prototype consisted of three tasks, with each exhibiting
three important characteristics as (i) real-world situations relevant to the students, (ii) tasks thatencouraged the students to formulate, employ, interpret, and evaluate in order to solve real-world
problems, and (iii) open-ended and challenging problems. 2) The principles for designing the tasks
consisted of three principles as (P1) having a shared understanding with designers regarding
mathematical literacy, modelling and how designing frameworks were valuable for modifying tasks(P2) selecting real-world situations relevant to students and posing questions that encouraged thestudents to apply their mathematical knowledge and experience, and (P3) design collaboration toensure that the tasks were valid, reliable, and appropriate. 3) The result of tasks experimentation
showed that most students were able to effectively use mathematics to solve problems. However,some faced obstacles and challenges when formulating the situations in mathematical contexts, andin interpreting and evaluating solutions in real-world contexts. The study findings can be used todevelop the tasks that assess students’ mathematical literacy in a contextually relevant manner.
References
Bakker, A. (2018). Design research in education: A practical guide for early career researchers. Routledge.
Blum, W., & Borromeo Ferri, R. (2009). Mathematical modelling: Can it be taught and learnt? Journal of
Mathematical Modelling and Application, 1(1), 45-58.
Klainin, S. (2015). Mathematics education at school level in Thailand: The development—The impact—The
dilemmas. Retrieved from https://pisathailand.ipst.ac.th/ipst-959/
National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical successfor all. Reston, VA: National Council of Teachers of Mathematics.
Stacey, K. & Turner, R. (2014). Assessing mathematical literacy: The PISA experience. New York, NY:
Springer International Publishing.

54

THE ROLE OF CARE IN MATHEMATICAL SCIENCES EDUCATIONINCLUDING STUDENTS’ VALUES
Rintaro Ueda1 and Keiichi Nishimura2
1

Tokyo Metropolitan Oizumi Sakura High School, Japan; 2Tokyo Gakugei University, Japan

Our research group focused on decision-making and consensus building using mathematical sciences(e.g., Yamaguchi et al., 2020). Mathematical sciences include mathematics, statistics, and appliedmathematics at their core; however, they are also closely related to other disciplines such as scienceand engineering, depending on the authentic problem being dealt with. In mathematical sciences
education, this decision-making process formulates and addresses real-world problems that requiredecision-making. Regarding the multiple options created by different formulations, it is necessary to
clarify the premise, form a consensus, and make decisions (Yamaguchi et al., 2020). When building
consensus, it is also necessary to pay attention to the values embedded in the mathematical model,
which becomes the focal point of the dialogue.
Previously, the authors have clarified that the lack of “care for others” by students is one of therequirements for consensus building difficulty (Matsushima et al., 2023). “The care for others” is arelational involvement entailing the attention to, and recognition of, others and their material andspiritual need (Radford, 2021). The research question in this paper is the role of care in the practiceof mathematical sciences education, especially in consensus building. The method is to analyse theaspects of consensus building through the lens of the concept of care in mathematical sciences in thepractice targeting upper secondary school.
The practice under analysis is based on “Nurmi counterexample” in social choice theory. “Nurmi
counterexample” is an example that yields different results depending on the models, such as majority
rule, majority rule with runoff. The practice lesson was given for 11 hours from November 2022 toMarch 2023 to eight fourth-year students in a night crafts course at a public upper secondary schoolin Tokyo. In recent years, unreasonable school rules have become an issue in Japanese school
education, and students are expected to participate in revising school rules. Using school rules
revision as the subject topic, the question through the unit is, “What voting method and voteaggregation approach should be adopted to reflect the will of all?”. The students critically examined
the model of majority rule and developed an alternative model of scoring rules with various
weightings. Then, among those models, they discussed what should be adopted and reached aconsensus.
References
Matsushima, M., Ueda, R .and Nishimura, K. (2023). Aspects of Students’ "Consensus Building" inMathematical Sciences Education: From the Perspective of Ethics. Proceedings of the 11th Spring Research
Conference of the Japanese Society for Mathematical Education, 394. (In Japanese)
Radford, L. (2021). The theory of objectification: A Vygotskian perspective on knowing and becoming inmathematics teaching and learning. Brill Sense.
Yamaguchi, T., Nishimura, K., Shimada, I., Matsushima, M. and Matsuzaki, A. (2020). Development ofMathematical Sciences Education in School Education: A Foundational Framework for Curriculum Designin Mathematical Sciences Education. Journal of Science Education in Japan, 44(2), 104–122. (In Japanese)

55

POSSIBILITIES OF STUDYING THE GOLDEN RATIO PROPORTION IN
THE CHRISTS SCULPTURES OF MASTER ALEIJADINHO IN THE TOWN
OF CONGONHAS, MINAS GERAIS, BRAZIL
Daniel Clark Orey1, Milton Rosa2 and Kelly Cristina Santos Rocha3
1

Universidade Federal de Ouro Preto, Brazil; 2Universidade Federal de Ouro Preto, Brazil,
3
Universidade Federal de Ouro Preto, Brazil

This qualitative research was conducted by using data gleaned from 7 participants: 1 museum
mediator, 1 tour guide, 1 sculptor, 1 plastic artist, 1 photographer, 1 mathematics teacher, and 1
tourist. The general objective of this study was to understand how the dialogical approach of
ethnomodelling (Rosa & Orey, 2010) contributed to the study of the golden ratio that were present inthese sculptures and which contributed to the development of the teaching and learning process inmathematics. The specific objectives were related to a) analyze the sculptures of 18th Century
Aleijadinho’s Christs to identify possible use of proportionality in his creation and b) understanding
how a dialogical approach of ethnomodelling can help us in understanding the possibilities of
proportional study in these sculptures.
According to these objectives, this study sought to answer the following research question: How can
the dialogical approach of ethnomodelling contribute to the study of the golden ratio that may bepresent in Aleijadinho Christ Sculptures, in the chapels of the town of Congonhas, in Minas Gerais?
In order to collect data we used: 2 questionnaires (initial and final), 3 semi-structured interviews, 1
focus group with two stages, and the researcher’s field diary. For the presentation and analysis of thedata, and for the interpretation of the results obtained in this study, we use an adapted methodologicaldesign of Grounded Theory (Glaser & Strauss, 1967) to conduct the open and axial codifications thatmade possible, respectively, the identification of the preliminary codes and 4 conceptual categories.
The results obtained in this study favored a new awareness and appreciation of these sculptures, aswell showed a respect for aspects of local culture, through the dialogic approach of ethnomodelling.
This approach helped the participants of this study in understanding the possibilities of the study of
proportionality in these sculptures. This approach contributed to the improvement of cultural
sensitivity of these participants, highlighting the importance of respecting and valuing Aleijadinho's
mathematical knowledge and skills, as well as showing the presence of the golden ratio in some of his
sculptures.
Finally, the results showed the possibility of inserting mathematical knowledge with the knowledge
and actions of the participants through the development of a pedagogical action related to local
history and culture with a theoretical foundation in the golden ratio, in the story of Aleijadinho, in
ethnomathematics, in the sociocultural perspective of mathematical modelling, and in the dialogic
approach of ethnomodelling.
References
Glaser, B., & Strauss, A. (1967). The discovery of grounded theory: strategies for qualitative research.
Sociology Press.
Rosa, M., & Orey, D. C. (2010). Ethnomodelling: a pedagogical action for uncovering ethnomathematicalpractices. Journal of Mathematical Modelling and Application, 1(3), 58-67.

56

MATHEMATICAL MODEL OF YEAST GROWTH: BIOLOGY STUDENTS
USE MATHEMATICS TO UNDERSTAND BIOLOGICAL PROCESSES
Svitlana Rogovchenko1 and Yuriy Rogovchenko2
1

University of Agder, Grimstad, Norway; 2University of Agder, Kristiansand, Norway

In this paper, we analyse how biology undergraduates work on a mathematical modelling (MM) task
guiding them to a discrete model for the yeast growth. This is only one of many tasks addressed by
students in a collaborative research project where biologically meaningful MM tasks were offered tobiology undergraduates to illustrate the use of mathematics and enhance students’ mathematical skills
and confidence. The study involved twelve first-year biology students who engaged in MM activities
during four sessions alongside their standard mathematics course. Led by the second author, an
experienced mathematician, sessions introduced students to different aspects of MM through smallgroup activities aimed at the promotion and integration of mathematical thinking and problem-solving
skills into biology and deepening of students' understanding of mathematical principles in biology.
Logistic growth models are commonly used both in mathematics and biology to study and understandthe growth patterns of microbial populations, including mathematical models of yeast growth.
Mathematicians treat the logistic model of yeast growth as a mathematical abstraction, emphasizing
the construction and analysis of continuous and discrete-time models which should capture thegrowth patterns observed in experimental data. They may explore the properties and behaviour ofmathematical models including existence of equilibrium points, stability properties of solutions,bifurcations, and numerical simulations (Lewis & Powell, 2017).
In biology, the logistic model of yeast growth is considered in the context of understanding theunderlying biological processes that drive population dynamics. Biologists are interested in studying
how environmental factors, nutrient availability, competition, and other biological interactions affectthe yeast growth. They may perform experiments, gather empirical data, and make observations tovalidate or refine their mathematical model. Biologists also investigate the mechanisms andmolecular pathways involved in the yeast growth and analyse how they align with the predictions
made by the mathematical model (Slonczewski et al., 2017; Stewart & Day, 2021).
We explore students’ work on the task by comparing mathematical and biological approaches. The
analysis of students’ discourse and its projections on professional discourses in mathematics andbiology is conducted in terms of the anthropological theory of the didactic (Chevallard, 2019).
References
Chevallard, Y. (2019). Introducing the anthropological theory of the didactic: an attempt at a principledapproach. Hiroshima Journal of Mathematics Education, 12, 71–114.
Lewis, M. & Powell, J. (2017). Model competition to describe data yeast for mathematicians: A ferment of
discovery and model competition to describe data. Bulletin of Mathematical Biology, 79, 356–382.
Slonczewski, J. L., Foster, J. W., & Zinsler, E. R. (2021). Microbiology: An evolving science. W. W. Norton& Company.
Stewart, J. & Day, T. (2015). Biocalculus: Calculus, probability, and statistics for the life sciences, Cengage
Learning.

57

ASSESSING AUTHENTICITY IN MODELLING TEST ITEMS:DEVELOPING AND PILOTING OF A THEORETICAL MODEL
Dominik Schlüter1, Maik Rolfs1 and Michael Besser1
1

Leuphana University Lueneburg, Germany

Authenticity figures prominently in the discussions of the modelling community (Galbraith, 2013)
and is often demanded for both learning tasks and test items. It is not only demanded on a theoreticalbasis normatively, but also empirical studies indicate that the use of authentic resources inmathematical tasks can increase motivation and performance of students (e. g. Vos et al., 2007).
Central to the discourse around authenticity in mathematics tasks is the definition of Niss (1992) aswell as the considerations of Palm (2009), who presents an operationalisation approach in the formof a didactical and pedagogical framework. In recent years, Vos’ (2018) theoretical considerations
have had an additional significant influence on the current international discourse. Her atomistic
approach on authenticity is already being used for the analysis of learning tasks, but is not yet foundin a theoretically and empirically robust model for assessing authenticity in modelling test items. Thisis the more astonishing, since many empirical studies (even PISA) claim at implementing “authentic
modelling tasks” into standardised test situations and since it is therefore of special interest to
empirically understand those studies frameworks on authenticity. This desideratum is taken up by the
research project "AutMod" which aims at developing and piloting such a model and thus aims atinvestigating how to assess authenticity in modelling test items more closely.
In order to being able to set up the intended model on authenticity of modelling items in standardised
test situations, (a) the current state of theory was reappraised, (b) standardised as well as informal
discussions with experts were held and finally (c) an adapted theory-based model on authenticity wasderived. For piloting this model, authenticity of overall 30 mathematical modelling test items fromVERA (standardises performance test for evaluating the quality of math education in Germany) has
been assessed by two independent raters based on a structured coding manual. In our presentation wewill present the qualitative as well as quantitative analysis of the results of the piloting study.
References
Galbraith, P. L. (2013). From conference to community: An ICTMA journey – the Ken Houston inaugurallecture. In G. A. Stillman, G. Kaiser, W. Blum, & J. P. Brown (Eds.), Teaching mathematical modelling:
Connecting to research and practice (pp. 349–360). Dordrecht: Springer.
Niss, M. (1992). Applications and modeling in school mathematics – directions for future development. In I.
Wirzup & R. Streit (Eds.), Developments in school mathematics education around the world V.3 (pp. 346–
361). National Council of Teachers of Mathematics.
Palm, T. (2009) Theory of authentic task situations. In L. Verschaffel, B. Greer, W. Van Dooren & S.
Mukhopadhyay (Eds.), Words and worlds: Modelling verbal descriptions of situations (S. 3–19).
Rotterdam: Sense Publishers.
Vos, P., Devesse, T. G. & Pinto, A. A. R. (2007). Designing mathematics lessons in Mozambique: startingfrom authentic resources. African Journal of Research in Mathematics, Science and Technology Education,
11, 51–66.
Vos, P. (2018). “How real people really need mathematics in the real world” - Authenticity in mathematics
education. Education Sciences, 8(4).

58TASKS DESIGN TO ENCOURAGE AND FACILITATE MODELLINGSKILLS AND ENGAGEMENT IN MATHEMATICS LEARNING
Ruth Rodriguez-Gallegos1 and Rafael Ernesto Bourguet-Díaz2
1

Tecnologico de Monterrey, Mexico; 2Tecnologico de Monterrey, Mexico

This proposal focuses on the design of modelling tasks through two cases aimed at future Engineers.
These cases aim to support the learning of mathematical notions such as Differential Equations (DE)in an undergraduate course of Engineering in a private university in Mexico, in such a way that itallows the student to know more closely approach this object of knowledge. This approximation
allows the student to build meanings of the structure itself and the use of the DE notion in problemsof social contexts. In previous research (Rodriguez & Bourguet, 2015), a high failure rate is reported
on this topic but also an educational proposal to incorporate modelling tasks through system dynamics
approach, this work is a continuation of this one. The intention is to complement the approach deeply
developed by Fisher (2020, 2021). It is also intended to contribute toward this direction in thescientific training of engineers and citizens in general. The design of these tasks in specific contexts
aims to improve the subject-object relationship in the construction of mathematical knowledge. In
this research, we imagine with the students decision-making contexts of near and possible futures inpost-pandemic times and in a digitalized society that offer hope and encourage entrepreneurship for
economic development focused on solutions to social problems.
The purposes of context design are to encourage subject engagement and to facilitate harmonious
closeness with the mathematical object (DE), in addition to develop modelling skills. The cases
present different degree of difficulty and complexity, for example, it is asked to decide in a structured
problem, about a decision to switch (or not) to a new type of employment, but also promote theinclusion of disabled people in the workplace. We also present a structure that we have envisioned
for the design of tasks (cases) inspired by the case study methodology proposed by Harvard. The
research will have a qualitative approach with a high emphasis on the part of the study of the evolutionof critical and systemic thinking when students face this type of scenario during a Mathematics class.
The tasks designed provide conditions to individually reflect on the meaning of the parameters, thestructure of the differential equation and its solution within the framework of a socio-technologicalproblem of a certain complexity.
References
Fisher, D.M. (2020). Algebra Students Build Stock/Flow Models to Study Non-linear, Dynamic Feedback
System Problems. In Stillman, G.A., Kaiser, G., Lampen, C.E. (eds) Mathematical Modelling Educationand Sense-making. International Perspectives on the Teaching and Learning of Mathematical Modelling.
Springer, Cham. https://doi.org/10.1007/978-3-030-37673-4_43
Fisher, D.M. (2021). Global Understanding of Complex Systems Problems Can Start in Pre-college Education.
In Leung, F.K.S., Stillman, G.A., Kaiser, G., Wong, K.L. (eds) Mathematical Modelling Education in Eastand West. International Perspectives on the Teaching and Learning of Mathematical Modelling. Springer,
Cham. https://doi.org/10.1007/978-3-030-66996-6_3
Rodriguez, R. & Bourguet, R. (2015). Building bridges between Engineering and Mathematics: Identifying
modelling practices through differential equations and simulation. Proceedings of the 122nd SEE Annual
Conference-Exposition. Seattle, United States.

59

MATHEMATICAL MODELLING AS A WAY OF SEEING THE WORLD
Bárbara Nivalda Palharini Alvim Sousa1 and Emerson Tortola2
1

State University of Northern Paraná, Brazil; 2Federal University of Technology – Paraná, Brazil

Modelling has been a way of promoting mathematics learning under the assumption of developing
abilities and competences that, on the one hand, are shown to be powerful sources of understanding
and insertion of students in the use of symbolic and formal mathematical systems, through theconstruction of mathematical models (Swan et al., 2007) and, on the other hand, enable a reading,even though partial and idiosyncratic, of reality through mathematics (Sousa & Tortola, 2021). In thiscontext, mathematical models play an important role in assigning meaning to the facts of the world
and to the mathematical concepts, and the research community has sought to clarify and connect
frameworks, roles and uses of mathematical models (Doerr & English, 2003, Carreira & Baioa, 2018).
In this paper, we discuss modelling and mathematical models from a philosophical perspective thatconsider the concept of use, language games, and ways of seeing from Wittgenstein (2009), and a
pragmatic perspective of knowledge that deals with the epistemic activity of constituting meaning
based on the construction of meaning rules (Moreno, 1995, 2018). Concerned about mathematicalmodels use in modelling tasks, we investigate the question “how the construction of models inmodelling activities enable the constitution of a way of seeing the world?”.
To support our reflections, we bring to the debate four modelling activities performed by different
students group in Mathematics Degree. Data analysis were carried out from students’ reports andteacher’s class notes, and our methodology is based on qualitative research. Results point out to areflection on how mathematical models uses promote and format ways of seeing the world throughmodelling. Through different practices, it is possible to reflect on different levels of schooling andthe epistemic activity resulting from modelling process based on the articulations promoted between
mathematics and reality, or sub-realities, when the students investigated topics of their interest, suchas the Covid-19 pandemic, organizing a suitcase to stay within the allowed weight limit, launching a
rocket, or even, locating a portal within a game.
References
Sousa, B. N. P, & Tortola, E. (2021). Mathematical Models in Mathematical Modelling Activities:
philosophical considerations. Rencima, 12(2), 1-25.
Carreira, S., & Baioa, A. M. (2018). Mathematical modelling with hands-on experimental tasks: on thestudent’s sense of credibility. ZDM, 50, 201-215.
Doerr, H. M., & English. L. D. A (2003). Modeling Perspective on Students’ Mathematical Reasoning About
Data. Journal of Research in Mathematics Education, 34(2), 110-136.
Moreno, A. R. (1995). Wittgenstein através das imagens (2. ed.). Editora da Unicamp.
Moreno, A. R. (2018). For an epistemology of use one aspect of the wittgensteinian concept of use:
construction of the sign and constitution of meaning. In R. L. Azize (Org.). Wittgenstein nas américas:
legado e convergências (pp. 27-52). EDUFBA.
Swan, M. et al. (2007). The roles of modelling in learning mathematics. In W. Blum et al. (Eds.). Modellingand Applications in Mathematics Education: New ICMI Study Series (v. 10, pp. 275-284). Springer.
Wittgenstein, L. (2009). Philosophical Investigations (4. ed.). Wiley-Blackwell.

60

PRE-SERVICE UPPER SECONDARY TEACHERS’ SPONTANEOUS VIEWS
ON MATHEMATICAL MODELS AND MODELLING
Jonas Bergman Ärlebäck
Department of Mathematics, Linköping University, Sweden

How teachers conceptualise and understand mathematical models and modelling can be a barrier(Burkhardt, 2006) or a facilitator (Blum, 2015) for the implementation of modelling in the classrooms.
In other words, it is important to provide pre-service teachers with a nuanced and rich understandingof mathematical models and modelling, and their potential for mathematics teaching and learning. In
order to support pre-service-teachers in developing such an understanding in an effective way, thispaper investigates the spontaneous views on mathematical models and modelling held by 69
pre-service upper secondary teachers who were about to start a course focusing on the educational
aspects of models and modelling in their third year of teacher training.
The analysis is based on a phenomenographic quantitative research paradigm (Marton, 1981; 1986)
and uses open- and axial coding (cf. Strauss & Corbin, 1990) to analyse the 69 pre-service upper
secondary students’ written responses to six questions, with the aim of identifying and describing
qualitatively different attitudes and beliefs about the notions, uses, functions, and roles ofmathematical models and modelling. The six questions answered by the students were: (1) What is amodel?; (2) What is a mathematical model?; (3) What does it mean to engage in (mathematical)
modelling?; (4) What are the functions and roles of (mathematical) models in different contexts andsituations?; (5) Are (mathematical) models and modelling important content to know about and learn
how to use? Why/why not?; and (6) How much emphasis and what role should modelling to have inthe teaching and learning of mathematics in schools? The results of the analysis will be discussed inrelation to the existing literature in the field, and in particular in relation to and in contrast to the
cognitive- and affective shaped beliefs and the four types of ideal modellers discussed by Maaß(2015).
References
Blum, W. (2015). Quality teaching of mathematical modelling: What do we know, what can we do? In S. J.
Cho (Ed.), The Proceedings of the 12th International Congress on Mathematical Education: Intellectual
andattitudinal
challenges(pp.
73–96).
SpringerInternational
Publishing.
https://doi.org/10.1007/978-3-319-12688-3
Burkhardt, H. (2006). Modelling in Mathematics Classrooms: reflections on past developments and the future.
Zentralblatt für Didaktik der Mathematik, 38(2), 178–195. https://doi.org/10.1007/BF02655888
Marton, F. (1981). Phenomenography - describing conceptions of the world around us. Instructional Science,10(2), 177-200.
Marton, F. (1986). Phenomenography - A research approach investigating different understandings of reality.
Journal of Thought, 21(2), 28-49.
Maaß, K. (2005). Barriers and opportunities for the integration of modelling in mathematics classes: results ofan empirical study. Teaching Mathematics and Its Applications: International Journal of the IMA, 24(2-3),
61-74. http://doi.org/10.1093/teamat/hri019
Strauss, A. L., & Corbin, J. M. (1990). Basics of qualitative research: Techniques and procedures fordeveloping grounded theory. SAGE.

61

TEACHING ACCOMODATIONS: FACILITATING CLASSROOMINTERACTIONS AROUND FERMI PROBLEMS
Azita Manouchehri1 and Eduan Soto-Martinez1
1

The Ohio State University, USA

In recent years, the use of Fermi problems in mathematics education has gained popularity due totheir perceived capacity for engaging students in reasoning and logical thinking (Barahmeh, Hamad& Barahmeh, 2017). Arlebäck (2009) positioned Fermi problems as a viable venue for bridging
students’ work on fermi tasks and mathematical modeling process, acknowledging that research isneeded to better understand how students’ work around Fermi tasks might be organized so modelingcompetencies are fostered in the course of their implementation. Relying on a teaching experiment
methodology and using interactional analytical techniques (Schütte, Friesen & Jung , 2019) we
examined, over two instructional sessions, teacher-student interactions in a pre-calculus course as
students worked on a Fermi problem. The task asked: How many steps would it take to burn off anentire bag of potato chips?
Dirks and Edge (1983, cited in Ärlebäck 2009) argued that successful work on Fermi tasks requires
the individuals to possess “sufficient understanding of the problem to decide what data might beuseful in solving it, insight to conceive of useful simplifying assumptions, an ability to estimate
relevant physical quantities, and some specific scientific knowledge” (p. 602). In reporting the resultsof using Fermi problems in secondary mathematics classrooms in Sweden, Ärlebäck highlighted thecritical role that students’ personal knowledge played in not only the assumptions they made onceencountering Fermi problems but also in how they validated their answers. Ärlebäck (2009) cautionedthat group composition and participants’ preferences can strongly influence the variables they
consider and assumptions they make, thus influencing the accuracy of models produced. In thesessions we examined, teacher interventions became central in focusing learners’ work along someof the cognitive and social dimensions identified by Ärlebäck (2009). While in some cases theinterventions were motivated by the teacher’s perceived need to equalize the social capital of theparticipants in small groups, a majority of his comments aimed to provoke reflection, encouraging
students to make connections among various answers obtained, addressing their implicit assumptions
by introducing new constructs (age, gender, metabolism, height and weight of individuals), andconfronting some false scientific assumptions they made (i.e sweating means more calories are burnt).
As such the implicit assumptions that the learners constructed became a source for group deliberation.
ReferencesÄrlebäck, J. B. (2009). Mathematical modelling in upper secondary mathematics education in Sweden: A
curricula and design study. Linköping University, Linköping, Sweden Retrieved from http://liu.divaportal.org/smash/get/diva2:302720/FULLTEXT02
Barahmeh, H. M., Hamad, A. M. B., & Barahmeh, N. M. (2017). The Effect of Fermi Questions in theDevelopment of Science Processes Skills in Physics among Jordanian Ninth Graders. Journal of Education
and Practice, 8(3), 186-194.
Schütte, M., Friesen, R. A., & Jung, J. (2019). Interactional analysis: A method for analysing mathematicallearning processes in interactions. Compendium for early career researchers in mathematics education,
101-129.

62

IMMC-SPAIN: AN OPPORTUNITY TO INTRODUCE MATHEMATICAL
MODELLING IN THE CLASSROOMS
Irene Ferrando1, Carlos Segura1 and Lluís Albarracín2
1

Universitat de València, Spain; 2Universtat Autónoma de Barcelona, Spain

In recent decades, the growing interest in the modelling approach to mathematics education has
increasingly been transferred to the design of educational programmes. In the case of Spain,
modelling appears as one of the mathematical competences to be developed by students at secondary
level (grades 7-12) in the new education law (Ministry of Education and Vocational Training, 2022).
As Garfunkel et al. (2021) point out, mathematical modelling can be integrated into teaching inseveral ways: in the "normal" classroom setting, or in the form of extracurricular activities with orwithout direct teacher support.
In this contribution we would like to describe the development of one of these extracurricular
activities, present the design and analyse the results of the IMMC-Spain competition. This national
competition for teams of students, which allows to choose the two teams participating in theInternational Mathematical Modelling Challenge organised by the Consortium for Mathematics andits Application, has as its central objective to promote the teaching of mathematical modelling andapplications in secondary education. To date, four editions of IMMC-Spain have been held and morethan 500 students have been involved in the competition. It is also worth highlighting the good results
achieved in the international competition by some of the selected teams.
The key aspects of both the design of the competition and the modelling activity encouraged will behighlighted in the presentation. On the one hand, we will describe the process and criteria used forthe design of the problems proposed in the IMMC. On the other hand, we will show the elements thatcharacterise the solutions proposed by the students. We will also analyse the participation data andthe performance of the teams. In this analysis, the gender perspective will be considered as a variable
since, as mentioned in Boaler et al. (2019), a higher proportion of girls participate in this type ofmathematical activity than in other individual competitions such as the Maths Olympiads.
Funding:
Authors are grateful to the support provided by the grant PID2021-126707NB-I00 funded by
MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”.
References
Boaler, J., Cordero, M., & Dieckmann, J. (2019). Pursuing gender equity in mathematics competitions: A caseof mathematical freedom. Focus, 39(1), 18-20.
Garfunkel, S., Niss, M., & Brown, J. P. (2021). Opportunities for modelling: An extra-curricular challenge.
In Mathematical Modelling Education in East and West (pp. 363-375). Cham: Springer International
Publishing.
Spanish Ministry of Education and Vocational Training (2022) Real Decreto 217/2022, de 29 de marzo, por
el que se establece la ordenación y las enseñanzas mínimas de la Educación Secundaria Obligatoria. Boletín
Oficial del Estado, 76, 31 March, 2022. https://www.boe.es/eli/es/rd/2022/03/29/217/con

63

STUDENT EXPERIENCES WITH CONNECTING CONTEXT AND
MATHEMATICS WHILE MATHEMATICALLY MODELLING
Kerri Spooner
Auckland University of Technology, New ZealandConnecting mathematics and contexts is a fundamental requirement when mathematically modelling.
However, connecting mathematics and contexts appears to be an unfamiliar experience formathematics students. This study looked at the research question: What are the student experiences
for connecting mathematics and contexts while participating in mathematical modelling? To gain
insight into the student experience, tertiary student mathematical modelling experiences withmodelling activities from three different New Zealand university modelling courses were looked at.
The modelling activities across the three courses used different contexts specific to each course. Data
was collected through twenty individual semi-structured interviews with the students. Reflective
thematic analysis was used to identify themes in students’ experiences with connecting mathematics
and contexts within the mathematical modelling activities. The results of the study showed:
connecting mathematics and contexts was new for students; students had different experiences
depending on their mathematics background, first language, and previous experience with appliedmathematics subjects; and for some students connecting mathematics with contexts mademathematics more relatable. The results of the study will be used to inform developing teachingpractices that enable students to successful connect mathematics with contexts.

64

TRAINING PROSPECTIVE TEACHERS’ DIDACTICAL PERSPECTIVESOF USING GEOGEBRA FOR MATHEMATICAL MODELLING
Yukiko Asami-Johansson1, Mikael Cronhjort1 and Mirko Radic1
1

University of Gävle, Sweden

Dynamic Geometry Software (DGS) can be used in teacher education regarding mathematicalmodelling with two purposes, to help prospective teachers (PTs) construct interactivity connected tomathematical representations (Arzarello, et al., 2012), and to let PTs analyse the benefits of DGS in
the modelling activities (Hernández, et al., 2020; Kokol-Voljc, 2007). However, most empiricalstudies reporting the impact of DGS in teacher education focus on PTs’ mathematical learning ofmodelling, and not on PTs’ didactical analysis of the use of DGS in mathematics classrooms.
We report a case study where prospective secondary school teachers consider and discussmathematical and didactical aspects of applying GeoGebra within modelling activities. The context
is a course at the end of the first year of our teacher education programme. The course objectives
include presenting current contents in secondary school mathematics courses, evaluating PTs’ own
teaching and their pupils’ learning, and using educational tools to illustrate mathematical concepts.
The study reports from the first of two seminars conducted by Zoom with nine PTs in two hours. The
seminar was recorded. The seminar started with an overview of how the Swedish national curriculum
describes the use of GeoGebra in upper/lower secondary school and discussed the underlying reasonsfor the use of DGS. Then the PTs presented their own experiences using GeoGebra and discussed
their perceptions about possibilities and disadvantages using GeoGebra in mathematics lessons(“pupils can even explore a new concept−not only solve a problem by using GeoGebra”. “Comparing
to analogical work by drawing on paper, pupils can be confused when they put misleading data andget erroneous graphs”, etc.). The PTs then discussed the task of creating maximum and minimum
area of a rectangular pasture with a 100 m rope stretched against the shore of a lake. This task was
constructed in such a way that it could be solved without using derivatives. First, the PTs intuitively
guessed the maximum area, which was correct, and drew a graph with GeoGebra using a file available
on the course website. They showed the maximum point on the curve of the quadratic graph andconcluded that the minimum area is indeterminate. When asked to solve the same problem for an
arbitrary numerical value, e.g., 103 m, they cannot guess the solution intuitively and realised thatalgebraic methods are needed to find the exact solution. In this way, the PTs received factors toconsider the dynamics of DGS as a didactical tool, while the importance of giving pupils theopportunity to solve modelling tasks in a mathematical way was also highlighted.
References
Arzarello, F., Ferrara, F., & Robutti, O. (2012). Mathematical modelling with technology: the role of dynamic
representations. Teaching mathematics and its applications, 31(1), 20–30.
Hernández, A., Perdomo-Díaz, J. & Camacho-Machín, M. (2020). Mathematical understanding in problem
solving with GeoGebra: a case study in initial teacher education. International Journal of Mathematical
Education in Science and Technology, 51(2), 208-223. DOI: 10.1080/0020739X.2019.1587022
Kokol-Voljc, V. (2007). Use of mathematical software in pre-service teacher training: the case of DGS. In D.
Küchemann (Ed.) Proceedings of the British Society for Research into Learning Mathematics, 27(3), 55–
60. The British Society for Research into Learning Mathematics.

65

WHEN DIGITAL TOOLS BECOME A SUPPORT FOR
MODELLING ACTIVITIES: PRE-SERVICE TEACHERS’INSTRUCTIONAL TASKS
Carolina Guerrero-Ortiz
Pontificia Universidad Católica de Valparaíso, Chile

In recent years, digital tools have had an increasingly preponderant and diverse role in teaching andlearning of mathematics, particularly in modelling. Exploring the implications of technology in the
study of phenomena is relevant because not only the most of our activities are permeated by different
digital tools, but also because the frequent complexity or inaccessibility to phenomena requires themediation of technology to get closer to them. In the case of modelling, there are at least two waysof conceptualizing the role of technology. On the one hand, as internal support for the transit between
the phases of the modelling process and on the other hand as an additional domain to the world ofmathematics (Greefrath, Hertleif, Siller, 2018).
From the experience of analyzing the design of tasks and their implementation in the classroom(Guerrero-Ortiz, 2021; Guerrero-Ortiz & Camacho-Machín, 2022), some uses of technology havebeen identified, these suggest an alternative perspective to the schematic representation of known
modelling processes. This work presents results of analyzing the implementation of modelling tasks
in the classroom. The tasks were developed by secondary mathematics preservice teachers. In thetasks, access to the situation was facilitated through technological tools. Digital tools were also usedas scaffolding the modelling process. In the students’ work, the reality perceived by them wasmediated by the characteristics of the technology; therefore, processes such as the interpretation andmental representation of reality can be different.
It is argued that the mathematical modelling in the classroom is not the same as that developed byexpert modelers, just as the modelling process that teachers develop in the design of tasks is not thesame as the one that students are encouraged to do as a result of working with the designed task
(Guerrero-Ortiz & Camacho-Machín, 2022). Therefore, the approach to reality that each one
experience is different. In a teaching context, students' experiences, the classroom environment, andthe tools used shape how they perceive task contexts. Therefore, the characteristics of the tasks thatinvolve the use of digital tools shape how individuals approach the proposed situation and how themodelling processes are developed.
References
Greefrath, G.; Hertleif, C.; Siller, H. (2018). Mathematical modelling with digital tools -a quantitative study
on mathematising with dynamic geometry software. ZDM, 50, 233–244.
Guerrero-Ortiz, C. (2021). Pre-service Mathematics Teachers’ Technological Pedagogical Content
Knowledge: The Case of Modelling. In F.K.S. Leung, G.A. Stillman, G. Kaiser, Wong, K.L. (Eds.),
Mathematical Modelling Education in East and West (pp. 141–151). Springer Nature: Cham, Switzerland.
Guerrero-Ortiz, C. & Camacho-Machín, M. (2022). Characterizing Tasks for Teaching Mathematics inDynamic Geometry System and Modelling Environments. Mathematics, 10 (8), 1-20.

66

A THEORETICAL MODEL FOR DESCRIBING TECHNOLOGICAL,PEDAGOGICAL, MATHEMATICAL AND EXTRA-MATHEMATICALKNOWLEDGE FOR TEACHING MODELLING
Josefa Castillo-Funes1 and Carolina Guerrero-Ortiz 2
Pontificia Universidad Católica de Valparaíso, Chile 1,2
A theoretical proposal that considers elements of the TPACK model such as the technology,
pedagogy, and content knowledge and the proposal on teacher knowledge for teaching mathematicalmodelling (Wess et al., 2021) is discussed here. The TPACK model from Mishra and Koehler (2006)
is constituted from three basic sources of knowledge: technological knowledge (TK), pedagogicalknowledge (PK) and content knowledge (CK). While Greefrath (2021) suggests that the teaching ofmathematical modelling requires specific pedagogical knowledge. He and his colleagues structure amodel involving four dimensions of knowledge: interventions, modelling processes, modelling tasks
and, goals and perspectives. The TPACK model considers pedagogical and technological content
knowledge, without special attention to mathematics and not attention to mathematical modelling,
while the proposal by Greefrath et al. (2021) considers pedagogical aspects for teaching modelling,
without emphasis on the use of technology. To explain the knowledge that the teacher must have toteach modelling with technology, a theoretical model (TPMMK) is proposed here. This model
highlights the interactions between mathematical knowledge, didactic knowledge, technologicalknowledge and extra-mathematical knowledge (Figure 1). The model can represent the knowledge
for the teaching of interdisciplinary work with the use of technology. In area 1 we find mathematicalmodelling as an element linking mathematical and extra-mathematical knowledge, which may or may
not require the use of technology (2.1 and 2.2). And by including knowledge for teaching mathematics
and about modelling, we structure what we call an integrated knowledge TPMMK (3). This
theoretical proposal is based on the literature and now is being validated with empirical data andthrough expert judgement (Escobar and Cuervo, 2008).

Figure 1. Dimensions of the TPMMK Conceptual Model
References
Mishra, P., & Koehler, M. J. (2006). Technological Pedagogical Content Knowledge: A Framework for
Teacher Knowledge. Teachers College Record, 108(6), 1017-1054.
Greefrath, G., Siller, H.-S., Klock, H., & Wess, R. (2021). Pre-service secondary teachers' pedagogical content
knowledge for the teaching of mathematical modelling. Mathematics Education Research Journal, 1-24.
Escobar, J. and Cuervo, A. (2008). Content validity and expert judgement. Advances in Measurement Journal,
6, 27-36.
Wess, R., Klock, H., Siller, H.-S., & Greefrath, G. (2021). Measuring professional competence for the teachingof mathematical modelling. En Leung, F. K. S., Stillman, G. A., Kaiser, G., & Wong, K. L. (Eds.),
Mathematical modelling education in east and west: International perspectives on the teaching andlearning of mathematical modelling (pp. 249-260). Springer.

67

USING AN INSTRUCTIONAL VIDEO TO SUPPORT UPPER SECONDARYSTUDENTS IN CREATING A MATHEMATICAL MODEL
Laura Wirth1 and Gilbert Greefrath1
1

University of Münster, Germany

Mathematical modeling is a challenge for many students since each step of the modeling process is apotential barrier (Galbraith & Stillman, 2006). To support students’ modeling processes, various approaches have been developed, including the use of heuristic worked examples which display an
expert’s realistic solution process to a modeling problem. Based on research that text-based heuristic
worked examples have been found to be beneficial in supporting students’ modeling processes (Zöttl
et al., 2010), we implement video-based heuristic worked examples because videos offer various
advantages including that the application of strategies can be presented dynamically. Moreover, using
videos to display the modeling problem offers the potential of helping students to make predictions(Cevikbas et al., 2023). Because understanding the problem is a major starting challenge and mathematizing the structured real-world problem is often one of the hardest parts for students (Galbraith &Stillman, 2006), we are focusing on the following research question: What changes in upper secondary students’ creation of a mathematical model can be observed after they have worked with an instructional video on a modeling problem?
In order to gain insights into students’ modeling processes, we videotaped four pairs of upper secondary students. Each pair collaboratively worked on a modeling problem, an instructional video on
a second modeling problem, and a third modeling problem. The data from the solution processes anda subsequent interview focusing on how the video supported the working processes during the thirdmodeling problem were analyzed using qualitative content analysis (Kuckartz, 2014). The results
indicate that the instructional video offered students confidence in dealing with missing values andbroadened their strategy repertoire in making assumptions, thereby supporting the students in theprocess of understanding and simplifying. We also found that the process of mathematizing was improved because some students chose a more adequate mathematical model than in the first modelingproblem. Our findings suggest that instructional videos are a promising format for providing heuristic
worked examples and may offer an innovative approach for supporting students in creating a mathematical model.
References
Cevikbas, M., Greefrath, G., & Siller, H.-S. (2023). Advantages and challenges of using digital technologies
in mathematical modelling education: A descriptive systematic literature review. Frontiers in Education,
8. https://doi.org/10.3389/feduc.2023.1142556
Galbraith, P., & Stillman, G. A. (2006). A framework for identifying student blockages during transitions inthemodellingprocess.
ZDM—Mathematics
Education,
38(2),
143–162.
https://doi.org/10.1007/BF02655886
Kuckartz, U. (2014). Qualitative text analysis: A guide to methods, practice & using software. SAGE.
https://doi.org/10.4135/9781446288719
Zöttl, L., Ufer, S., & Reiss, K. (2010). Modelling with heuristic worked examples in the KOMMA learning
environment. Journal für Mathematik-Didaktik, 31(1), 143–165. https://doi.org/10.1007/s13138-0100008-9

68

FLIPPING UNIVERSITY-BASED MATHEMATICAL MODELLINGSEMINARS – INSIGHTS FROM PRE-SERVICE MATHEMATICSTEACHERS
Mustafa Cevikbas1, Gabriele Kaiser1,2, Denise Mießeler1
1

University of Hamburg, Germany; 2Nord University, Norway

Mathematical modelling empowers learners’ critical thinking, prepares them for responsible
citizenship in societal developments and promotes them to be effective problem solver (Blum, 2011).
Mathematical modelling takes an important place in various national curricula (Borromeo Ferri,2018). However, it is a complex subject that can be challenging to teach and learn and employing
innovative approaches in modelling is necessary to enhance mathematical modelling competence(Cevikbas et al., 2022, 2023). The flipped classroom (FC) approach including the use of explanatory
videos can be a promising pedagogy to enhance mathematical modelling education maximizing classhours for modelling activities and promoting student engagement, learning motivation, andachievement (Bergmann & Sams, 2012). In this study, we explore the value of using this innovative
approach to teach and learn mathematical modelling from the perspective of pre-service mathematics
teachers, including its benefits and challenges for learners and instructors. We flipped twomathematical modelling seminars at a large German university and gathered data through an online
questionnaire including open-ended questions and semi-structured interviews from 42 pre-service
teachers at master’s level. The results of this qualitative study showed that the use of explanatory
videos and the FC model can improve the understanding of mathematical modelling and promote
engagement, collaboration, and modelling competence. However, there are also pitfalls associatedwith FC, including technological constraints, difficulties with self-directed learning, and inadequate
preparation for in-class modelling activities. Pre-service mathematics teachers expressed positive
attitudes towards flipping modelling instruction, but they highlighted that especially creating newcontent such as explanatory videos was a time-consuming activity for them. The results underscore
the need to incorporate the FC pedagogy in teacher education programs to prepare pre-service
mathematics teachers to effectively teach mathematical modelling in their professional life using
technology-supported emerging approaches.
References
Bergmann, J. & Sams, A. (2012). Flip your classroom: Reach every student in every class every day. ISTE.
Blum, W. (2011). Can modelling be taught and learnt? Some answers from empirical research. In G. Kaiser,
W. Blum, R. Borromeo Ferri, & G. Stillman (Eds.), Trends in Teaching and Learning of Mathematical
Modelling, ICTMA 14, (pp. 15–30). Springer.
Borromeo Ferri, R. (2018). Learning how to teach mathematical modeling in school and teacher education.
Springer.
Cevikbas, M., Kaiser, G. & Schukajlow, S. (2022). A systematic literature review of the current discussion onmathematical modelling competencies: state-of-the-art developments in conceptualizing, measuring, andfostering. Educational Studies in Mathematics, 109(2), 205-236.
Cevikbas, M., Greefrath, G., & Siller, H. S. (2023). Advantages and challenges of using digital technologies
in mathematical modelling education – A descriptive systematic literature review. Frontiers in Education,
8, 1142556.

69

MODELLING UNCERTAIN FUTURES: PROBLEMS, METHODS, TOOLSPeter Galbraith1 and Diana Fisher2
1

University of Queensland, Australia; 2Portland State University, USA

A previous ICTMA presentation addressed a gap in student knowledge of modelling identified
through a problem set in the International Mathematical Modeling challenge 2019 (Garfunkel, et.al. ,(2021). It involved estimating the future carrying capacity of planet Earth, and no student teamentered in this world-wide competition had the resources to approach the problem adequately. Suchproblems contain non-linearities whereby solutions can only be approached through simulation. The
earlier presentation illustrated the type of approach needed using System Dynamics. This presentation
extends its predecessor by adopting the goal of pursuing possible solution(s) to the problem as set.
In doing so ancillary educational implications are considered within a methodological framework.
These include, on the one hand, implications for teacher education in terms of an approach integrating
mathematics with software. Additionally there is information available from classroom experience of
the introduction of such material in secondary school classes, Fisher (2018).
Professional estimates of carrying capacity in terms of population have typically considered thedepletion of interacting and depleting resources. This different approach engages issues that are ofincreasing concern to human society, involving adults and young people alike e.g., global warming.
Predictions of the future population of Earth have been made by several organisations e.g., the United
Nations. These are based heavily on assumptions about future values of birth-rates treated seemingly
in isolation from other factors. Articles and research reports continue to conjecture about the impact
of increasing global temperatures on human welfare and industrial production. But the potential
impact of interactions of these two quantities with each other and with population are not evident in
the predictions. The approach here is to develop such scenarios, both in terms of assumptions madefor models, and in the meaning, interpretation, and implications of outcomes.
The presentation will engage with material drawn from the following content: Modelling problems
where simulation is essential; purpose and structure of system dynamics models; development of abasic model built around interactions between population, industrial growth, and global temperature;interpretation and implications of alternative scenarios; classroom experience with SD modelling.
Output from the model will be discussed relating to areas such as the following: potential impact ofdifferent levels of industrial growth; comparisons between population projections with and without
assumed impacts from changing global temperatures; the relative effectiveness of different policies
suggested for mitigating the effects of global warming.
The significance of such modelling relates to expressions of purpose found in many national
curriculum statements: a goal of mathematics education being to equip students with competence that
extends into the future and beyond the classroom - for the workplace, for individual undertakings,
and as responsible citizens. We would include the word “global” in describing the last of these.
References
Fisher, D. M. (2021). Global understanding of complex systems problems can start in pre-college education.
In F.K.S. Leung, G.A. Stillman, G. Kaiser, & K.L.Wong (Eds.), Mathematical Modelling Education in
East and West (pp. 35-44). Springer, Cham.
Garfunkel, S., Niss, M., & Brown, J. P. (2021). Opportunities for modelling: An extra-curricular challenge. In
F.K.S. Leung, G.A. Stillman, G. Kaiser, & K.L.Wong (Eds.), Mathematical Modelling Education in Eastand West (pp. 363-375). Springer, Cham.

70

STARTING MATHEMATICAL MODELLING FROM EXISTING MODELS
Benjamin Galluzzo1, Adewale Adeolu1, Rose Mary Zbiek2, Amy Brass2 and Jie Chao3
1

Clarkson University, USA; 2Pennsylvania State University, USA; 3Concord Consortium, USA

The mathematical modelling process starts in the “unedited” or “messy” real world, is worked uponmathematically by modelers, and then is transformed back into the real-world context (e.g., Pollak,2012). However, identifying a meaningful “starting point” or developing a quality mathematicalmodelling activity that is cognitively demanding is non-trivial (Borromeo Ferri, 2018, p. 41).
Motivated by the need to provide an accessible, yet holistic math modelling experience for students
new to modelling, we consider an alternative starting point for the mathematical modelling processthat supports students in “translating between the real world and mathematics in both directions”(Blum & Borromeo Ferri, 2009). In this study we present students with an existing model, asking
them to analyse the given model, and then develop a revised model that better represents the realworld context. Specifically, we are interested in the following: what does it mean for a modelling task
to start with an existing model, and how does it impact students' modelling process?
This study is part of the M2Studio project, which aims to develop a new technology for learning
mathematical modelling and a compatible curriculum. Here, we focus on the last activity in the
curriculum, which provided students the opportunity to engage in the modelling process through anexisting model. For this task, we presented a situation that one of our colleagues has faced; whether
it is worthwhile to move to a cheaper apartment that is further away from her workplace. Ourcolleague (the client) shares her existing cost-focused model, but she is unsure of other factors thatshe should (or could) consider. The students are asked to revise her model to help her better
understand her situation and ultimately make a decision. We note that the task, initiated with anexisting model, satisfies characteristics associated with modelling tasks (Maaß, 2007).
In this talk, we present findings from two groups consisting of two students each. Data from video
recordings reveal that students initially analysed the model by interpreting and validating aspects ofthe existing model to identify what needed to be improved. They then continued to move throughout
the modelling process; creating sub questions and making additional choices to develop revised
models. Notably, students in this study repeated this process, constantly interpreting and validating
how their current model compared with the existing model and what further revisions may be needed
to ensure they developed a model that better represents the client’s situation.
References
Blum, W. & Borromeo Ferri, R. (2009). Mathematical modelling: Can it be taught and learnt? Journal of
Mathematical Modelling and Application, 1(1), 45–58.
Borromeo Ferri, R. (2018). Learning How to Teach Mathematical Modeling in School and Teacher Education.
Springer International Publishing AG 2018. https://doi.org/10.1007/978-3-319-68072-9
Maaß, K. (2007). Modelling in class: What do we want the students to learn? In C. Haines, P. Galbraith, W.
Blum, S. Khan, & Mathematical Modelling (Eds.), Education, engineering and economics (pp. 65–78).
Chichester: Horwood Publishing.
Pollak, H., (2012). “What is Mathematical Modeling?” MSRI Publications Volume 60, 2012.

71

DIFFERENCES IN INTERPRETATION OF COMPUTER RESULTS IN THE
MODELLING CYCLE WITH ADDED COMPUTER MODEL: THROUGH
THE MODELLING WORKSHOP ON THE PREMISE OF USINGSCIENTIFIC CALCULATOR
Yuzu Yamamoto1 and Akio Matsuzaki2
1

Graduate School of Education, Saitama University, Japan; 2Saitama University, JapanOne of the Scientific calculator’s function “Regression Calc” displays the results of the regression
calculations based on entered data. We conduct preliminary experiment to measure the distance fromthe projector lens to the projection screen using tape measure, and the brightness of the projection
screen using lux meter. We collect the data obtained from the experiment, and inquiry the relationshipbetween the measured distances and the brightness using “Regression Calc” (Yamamoto et al., 2022).
In the modelling cycle with added computer model (Greefrath, 2011), the experiment is real situation.
A summary of the data on the distance from the projector lens to the screen and the brightness ofprojection screen is real model. The function of scientific calculator “Regression Calc” is computermodel, and the approximation formulas and correlation coefficients obtained using “Regression Calc”
are computer results. The research question is as follow: What are differences in the modeller’sinterpretations of computer results?
We plan and conduct a modelling workshop using scientific calculator and digital tools for 59
undergraduate Japanese students (Yamamoto et al., 2023). The students predicted the relationshipbetween the distance from the projector lens to the screen and the brightness of projection screen.
The experiment was conducted to measure the distance from the projector lens to the projection
screen using tape measure, and the brightness of the projection screen using lux meter. Based on thedate obtained from the experiment, the students explored the relationship using “Regression Calc” ofscientific calculator. The students' descriptions about the relationship can be classified four types asfollows: (ⅰ) proportion (5 students), (ⅱ) inverse proportion (14 students), (ⅲ) inverse proportion to
the square (39 students), (ⅳ) could not find relationship (1 student).
References
Greefrath, G. (2011). Using technologies: New possibilities of teaching and learning modelling Overview. In
G. Kaiser, W. Blum, R. Borromeo Ferri, & G. Stillman. (Eds.), Trends in Teaching and Learning ofMathematical Modelling: ICTMA 14 (pp.301-304). Springer.
Yamamoto, Y., Hara, K., & Matsuzaki, A. (2022, December). Towards the development of teaching materials
for functions of inverse proportion to the square on the premise of using scientific calculator: Through anexperiment by measuring the brightness of the projector light. Oral Presentation at 10th International
Conference of Research on Mathematics and Science Education in 2022. Online.
Yamamoto, Y., Hara, K., & Matsuzaki, A. (2023, March). Towards the lesson plan for functions of inverse
proportion to the square on the premise of using scientific calculator: Practice report of workshop on
problem related to projector for university students. Oral Presentation at the 27th meeting of theMathematical Education Society, Graduate School, etc. in 2023. (written in Japanese)72

CHANGES OF MATHEMATICAL MODELS THAT CAN BE
REPRESENTED BY CHANGING THE SLIDER TOOL IN GEOGEBRA:
FOCUS ON THE TRAJECTORY OF BASKETBALL IN ONE SCENE OF
THE MANGA “THE BASKETBALL WHICH KUROKO PLAYS”
Mikino Tachihara1 and Akio Matsuzaki2
1

Graduate School of Education, Saitama University, Japan; 2Saitama University, JapanThe slider tool in GeoGebra can change variable numbers in functions easily. Namigata (2017)
develops teaching materials on the subject of the manga “The Basketball Which Kuroko Plays” using
GeoGebra. In this teaching materials, he focuses on the trajectory of the ball when a shooter shoots athree-point shot from a certain position, the trajectory of the ball when the shooter shoots a threepoint shot changing the position, and the parabola focused on the ball velocity. Tachihara (2022)
represents the trajectory of the ball using differential equations when the shooter shoots a three-point
shot changing the position. We succeed to change an initial velocity of the ball that is a variable
number of differential equations using GeoGebra and represent a mathematical model based on thethree cases focused by Namigata (2017). We set a research question as follow: How can mathematicalmodels be represented from dynamic graphs displayed on GeoGebra?
In the modelling cycle with added computer model (Greefrath, 2011), one scene of a three-point shotin the manga "The Basketball which Kuroko plays" is real situation, the two trajectories and the one
parabola are real models. The parabolic equation that the shooter shoots a three-point shot changing
𝑔
the position 𝑦 = − 2 2 𝑥 2 + tan𝜃 ∙ 𝑥 + 3 is mathematical model (Tachihara, 2022). The slider
2𝑣0 cos 𝜃tool in GeoGebra that changes variable numbers is computer model. The dynamic graphs displayedin GeoGebra by changing variable numbers are computer results. Therefore, the maximum positionof the range of a three-point shot and the range of the initial velocity of the ball are mathematicalresults. The ball’s angle of incidence to the goal ring is set at 70 degrees (Namigata, 2017). Fixing
70𝜋
9.8
gravitational acceleration g = 9.8 and θ = 180 , a new mathematical model 𝑦 = − 2 270𝜋 𝑥 2 +2𝑣0 cos

70𝜋

180

tan 180 ∙ 𝑥 + 3 is represented. As a result, a range of the initial velocity of the ball is represented.
References
Greefrath, G. (2011). Using technologies: New possibilities of teaching and learning modelling Overview. In
G. Kaiser, W. Blum, R. Borromeo Ferri, & G. Stillman. (Eds.), Trends in Teaching and Learning ofMathematical Modelling: ICTMA 14 (pp.301-304). Springer.
Namigata, M. (2017, March). A Consideration on the trajectory of a projected object using GeoGebra:
Focusing on a scene of three-point shot from the manga “The Basketball which Kuroko plays”, Annual
Spring Conference of the Mathematical Education Society in 2017. (written in Japanese)
Tachihara, M. (2022, December). Towards the development of teaching materials for differential equation
based on the trajectory of basketball: Reconsidering the mathematical model of a three-point shots in the
manga “The Basketball which Kuroko plays”, Oral Presentation at 10th International Conference of
Research on Mathematics and Science Education. Online.

73

IMPROVEMENT TO THE MATHEMATICAL MODELLING LESSON BY
INEXPERIENCED SECONDARY SCHOOL MATHEMATICS TEACHER
Shigekazu Komeda1 and Shogo Ohbayashi1
1

Saga University, JapanThe aim of this study is to identify what is needed for inexperienced secondary school mathematics
teacher who are conducting inadequate mathematics lesson as mathematical modelling to improvetheir own lesson towards the mathematical modelling lesson. All secondary school mathematics
teachers in Japan have experience of conducting mathematics lessons on real-world problems, as
textbooks contain teaching materials that cover real-world problems. However, these are "wordproblems", pointed out by Niss & Blum(2020), and lesson that deal with such problems as they arecannot be called mathematical modelling lesson. How, then, can mathematics teacher who conduct
such lessons improve them towards mathematical modelling lesson?
In this study, “expansive learning" proposed by Engeström, Y.(2001) is used to help mathematics
teacher who are teaching mathematics lesson with "word problems" to improve towards themathematical modelling lesson. In this study, the following methods were implemented based on "thecycle of expansive learning" he proposed.
(1) Interviews were conducted with target teachers to clarify the challenges of dealing with real-world
problems in mathematics lessons. (2) Based on the issues identified in (1), the target teachers were
given experience in mathematical modelling suitable for them and asked to analyse the differences
between the mathematical modelling process and teaching process of their own previous
mathematical lessons of dealing with real-world problems. (3) Have them devise measures toovercome the issues identified in (1) and to compensate for the inadequacies of conventional teaching.
(4) Have the target teachers design and implement the lesson that incorporate the strategies in (3). (5)Have the target teachers analyze and discuss the lessons implemented in (4).
In this study, four secondary school mathematics teachers with less than three years' experience andno previous experience of teaching mathematical modelling were targeted. The reason for selecting
inexperienced mathematics teachers was that, due to their inexperience, we thought they had a strong
desire to improve their lessons.
The results of the study showed that when the teaching process conducted by the four teachers werecompared with their previous teaching process, improvement to the mathematical modelling lesson
were observed for three of the teachers. However, one teacher had not been able to improve. The
requirements of improvement to the mathematical modelling lesson were clarified from the results ofinterviews with the three teachers who showed improvement, their modelling experiences and theirimproved lesson.
References
Engeström, Y.(2010). Expansive Learning at Work: Toward an activity theoretical reconceptualization,
Journal of Education and Work, 14(1), 133-156.
Niss, M. & Blum, W. (2020). The Learning and Teaching of Mathematical Modelling, Oxon & New York:
Routledge.

74

MATHEMATICAL WORK IN A MODELLING TASK: CIRCULATIONS AND
TYPES OF MODELS
Jaime Huincahue1, Paula Verdugo-Hernández2, Carolina Henriquez-Rivas1 and Gonzalo EspinozaVásquez3
1

Universidad Católica del Maule; 2Universidad de Talca; 3Universidad Alberto Hurtado, Chile

The identification of routes for solving mathematical modeling tasks from an individualized approachallows understanding the learning processes in accordance with the characteristics of the subjects.
There are successful examples in the literature that directly nurture the valorization of modeling tasks.
Considering the task "The Assignment" by Reilly (2017), the present study performs an adaptation
aimed at secondary level (15-17 years) and higher students of mathematics teacher training in Chile,with the purpose of characterizing the types of mathematical models and the circulations ofepistemological and cognitive planes that are developed from a modeling task. The task is:
You are building a house plan for your family, with the bedrooms and bathrooms you want. You
have to worry about the economic cost of the interior of the house, specifically the floor and walls,
how much would it cost? Next, you must design the floor plan of two types of houses (one small
and one large), where one of them is the house you have initially designed. How many large houses
could you build to spend the same, or less, on floor and walls? or less on floors and walls than you
would spend on 50 small houses?
To analyze the task and its development, we consider the theoretical framework of Mathematical
Working Spaces (MWS) (Kuzniak et al., 2022), which allows, from the articulation between the
epistemological and cognitive levels, to characterize the existing circulations. With a qualitative
methodology and interpretative approach, the task has been adjusted from the validation by fiveexpert researchers in didactics of mathematics, and subsequently piloted in secondary school students.
Then, a case study has been defined in students of secondary education and Initial Teacher Training.
The preliminary results show that the task has been understood by all the individuals to whom it has
been applied. Particularly, in the case presented, the activation of the semiotic genesis is observed,
where the student visualizes and designs a plan, then provides the values required for the constructionof the house. Finally, he poses an equation, resorting to his referential and estimates a final value asa solution. The data analyzed lack rigor, and there are slight similarities with the a priori analysis
performed. On the other hand, the mathematical models used by the students are identified in a
figurative semiotics, being of interest to perform a more exhaustive data collection, in order to obtainthe multiplicity of alternatives foreseen in the a priori analysis and thus complement the preliminary
results in the final version of this report.
References
Kuzniak, A., Montoya-Delgadillo, E., Richard, P. (2022). Mathematical work in educational context - The
perspective of the Theory of Mathematical Working Spaces. Springer.
Reilly, E. (2017). Developing a mathematical modelling task for all students. In G. Stillman, W. Blum, G.
Kaiser (Eds.), Mathematical modelling and applications. Crossing and researching boundaries inMathematics Education (pp. 443–453). Springer.

75

IMPLICATIONS OF LEARNING THROUGH FOR TEACHING USING
MATHEMATICAL MODELLING
Emerson Tortola1, Jader Otavio Dalto1 and Karina Alessandra Pessoa da Silva1
1

Federal University of Technology - Parana, Brazil

Education in mathematical modelling has been designed mainly on pragmatic and theoreticalepistemological aspects, which aim to structure cycles to guide the modelling practice (Boromeo Ferri,2018). Based on these aspects, we have been investigating and configuring education designs
organised around three axes: learning about modelling, learning through modelling and teaching
using modelling. Teacher education based on these axes has been undertaken to prepare them to know
aspects related to modelling and how to do it and how to define appropriate methods to teach using
modelling (Borromeo Ferri, 2018).
In 2022, nine in-service teachers participated in a course in which they developed a modelling activity.
The discussions about that activity fostered an approach directed towards the educational context.
When we were analysing the discussions, we evidenced characteristics such as those pointed out by
Barbosa (2010), who, based on modelling routes, categorised the students’ mathematical, technicaland reflective discussions in the construction of mathematical models. Notably, in a transversal way,
we evidenced the pedagogical character in the discussions so that the teachers, when learning throughmodelling, related aspects that could be implemented to teach using modelling.
Inspired by these discussions, we investigated the questions: How do in-service teachers becomeaware of actions to teach using modelling when they develop an activity as modellers? What does
this process of awareness reveal? The activity development, which was recorded in audio and video,
together with the activity development reports and a collaborative planning prepared by the teachers
made up the case study we analysed, according to the Content Analysis (Bardin, 2013).
The results indicate that the awareness to teach using modelling is present in what we call pedagogicaldiscussions that were transversal to the other discussions pointed out by Barbosa (2010) and they
were configured in the following categories: clarifying the definition of the problem, structuring datacollection, considering a relevant mathematical approach within reach of students; and enabling thevalidation of the results. In addition, they indicate the need for the three education axes to be worked
on in an articulated way so that learning about modelling is fundamental for teachers to recognise
their actions through the cycles, and to be able to carry them out intentionally, pedagogically speaking,while developing modelling activities and, in this context, envisage actions that can provide subsidies
for the constitution of a didactics to teach using modelling.
References
Barbosa, J. C. (2010). The students’ discussions in the modeling environment. In R. Lesh, P. L. Galbraith, C.
R. Haines, & A. Hurford. (Org.). Modeling students’ mathematical modeling competencies. New York:
Springer, 365-372.
Bardin, L. (2013). L’analyse de contenu. Paris: Presses Universitaires de France.
Borromeo Ferri, R. (2018). Learning How to Teach Mathematical Modeling in School and Teacher Education.
New York: Springer International Publishing.

76

TRANSIENT PHENOMENA AS MATHEMATICAL MODELLING
MATERIALS IN SCIENCE EDUCATION
Osamu Inomoto
Center for Fundamental Education, Kyushu Sangyo University, Japan

In science and physics education, it is essential to formulate phenomena mathematically in order toquantitatively understand natural phenomena and their properties and regularities. Understanding
through experimentation and observation requires a process of "discovering" laws based on
experimental results. In this respect, the methodology of mathematical modeling is effective, andunderstanding can be deepened by repeating the process of formulating a hypothesis, planning andconducting experiments, formulating and interpreting the results, and discussing the results. The roleof mathematical modeling in science and physics education is to deepen understanding of phenomena
and to enhance observation and discussion (Michelsen, 2015). In mathematics education, on the other
hand, mathematical modeling contributes to understanding the relationship between natural andsocial phenomena, for example, visualizing and interpreting equations and their solutions as
phenomena (Burghes & Borrie, 1982). In addition, the importance of modeling physical phenomena
represented by exponential functions is noted (Pisano & Bussotti, 2012).
This paper focuses on applications of the integral method in high school mathematics. In this unit,
students will study initial value problems for the simplest differential equations 𝑑𝑦⁄𝑑𝑥 = 𝑘𝑦, 𝑦(0) =𝑦0 for the variable 𝑦(𝑥), where 𝑘 is a positive or negative constant. Differential equations of this typeare widely found in natural and social phenomena and, depending on the sign of the coefficients, caneither relax to equilibrium values or explode to infinity. A slight change in the right-hand side of thisdifferential equation can also be used to describe biological phenomena, such as allosteric effects, forexample. This is expected to enrich the content of mathematics education by connecting it withnatural phenomena. As for physics education, it provides effective clues for mathematicalunderstanding of transient phenomena such as thermal equilibrium, and enables students to"discover" the laws of phenomena and to deepen their physical interpretation through modelling
procedures.
In this study, we propose a mathematical modeling method of transient phenomena from theviewpoint of physics education and its application to teaching materials. In particular, the author
shows that experimentally "discovering" Newton's cooling law in thermal equilibrium bymeasurement, formulating it as a differential equation, and physically interpreting it can lead to a
deeper understanding of transient phenomena compared to conventional learning. The experiments
and discussions are effective in mathematics education and serves to utilize the research content.
References
Burghes, D.N. & Borrie, M.S. (1982). Modelling with differential equations, Horwood.
Michelsen, C. (2015). Mathematical modeling is also physics - interdisciplinary teaching between mathematics
and physics in Danish upper secondary education. Physics Education, 50, 489.
Pisano, R. & Bussotti, P. (2012). Open Problems in Mathematical Modelling and Physical Experiments.
Exploring Exponential Function. Problems of Education in the 21st Century, 56-69.

77

AN ANALYSIS OF THE THEORETICAL FRAMEWORK FOR NUMERICAL
COGNITION IN ZAMBIA
Yoshitaka Abe
Hiroshima University Graduate School of Humanities and Social Sciences, Japan

It highlights the importance of basic education in African countries, particularly due to their low
academic performance. To deal with the challenges, the need for accurate assessment tools tounderstand the conditions of developing countries include Zambia and enhance instruction isemphasized (World Bank, 2018; Baba et al., 2021). While children in lower grades show partial
understanding and varying mastery up to two digits, expanding the learning units to include three
digits and decimals is crucial for further improvement (Uchida, 2012; Nakawa et.al., 2019). However,the literature on decimals or fractions is limited, with insufficient discussion regarding the knowledge
children usage to comprehend them (Alcock et.al., 2016).
The objective of this study is to evaluate the reliability of the theoretical framework of numericalcognition in developing assessment tools specific to Zambia.
The integration of conceptual and procedural knowledge is identified as a potential challenge that
may impact students' understanding of decimals (Hiebert & Lefever, 1986; Carpenter, 1986). The
significance of structural knowledge for mastering decimals is also emphasised (Annette, 1998). The
relationships among three types of knowledge are defined, and corresponding tasks that are frequently
used in surveys of math competencies and correspond to concepts. (including T1; number
identification, T2; object matching, T3; mental number lines, T4;pattern recognition, T5;
composition/decomposition, and T6; calculations ) (Fig.1).
Structural
knowledge

Conceptual
knowledge
T1

T2

T3

T4

Proceduralknowledge
T5

Fig. 1 Hypothesis theoretical framework

T6

Fig. 2 Correlation among revised theoretical framework in natural number only

To evaluate this hypothetical theory by mathematical modelling. According to Ikeda (1999),
mathematical modelling is an activity in which actual problems are mathematized to create amathematical model, which is then interpreted and checked to then find suitable mathematical model.
I analysed the test results of approximately 150 Zambian elementary school students in grades 6 and7 using the structural equation modelling. As a result, the Fig1 did not fit well. Alternatively, when
focusing on natural numbers only and applying conceptual and procedural knowledge (Fig.2), a
better fit (p-value = 0.050). Consequently, it is concluded that the theoretical framework developedin this study is applicable only to natural numbers. I expect that limitations in adapting the framework
to decimals are attributed to students' insufficient understanding and poor performance in specifictasks. To address these limitations, it is proposed to increase the sample size, modify tasks, andinvolve multiple schools. Furthermore, specific difficulties and typical errors were identified, leading
to the need for redefining the theoretical framework based on these findings.
References
World Bank. (2018). World Development Report: Learning to Realize Education’s Promise.

78

CHARACTERISTICS OF THE MODELING PROBLEMS POSED BYPROSPECTIVE MATHEMATICS TEACHERS
Serife Sevinc
Middle East Technical University, Turkey

Mathematical modeling perspective is based on the premise of connecting mathematical world withthe real-world (Blum & Niss, 1991; Lesh & Doerr, 2003). Hence, modeling process is based oncontextual problems that do not require straightforward answers but construction of a model.
Although modeling problems may differ regarding the modeling perspective they entail, they arecontextual, ill-defined, and open-ended problems. For instance, models-and-modeling perspective
suggests a genre of problems called MEAs that are often presented as realistic customer-driven
situations and open to construction of multiple reasonable models (Lesh & Doerr, 2003). Anothertype of problems that modeling researchers studied are Fermi Problems given in estimation-based
open-ended real-life situations (Albarracín, & Gorgorió, 2014; Ärlebäck, 2009). Furthermore, themodeling potentials of closed versus open real-world problems also took researchers’ attention(Schukajlow et al., 2022). This body of research indicated that modeling problems shared somecommon aspects but also varied in some characteristics. In this study, 34 prospective mathematics
teachers at a large public university in Turkiye worked in pairs to pose modeling problems addressing
geometry topics. They were also asked to write at least one solution to the problem and reflect on thecharacteristics of the problems. Their modeling problems, solutions, and reflections were analyzed
based on the cognitive demand, open-endedness, and contextual aspect of the problem. The study isstill in the stage of on-going analysis; however preliminary results indicated that modeling problems
posed by the prospective teachers involved multiple mathematical ideas while placing geometry as acentral idea, and prospective teachers used relations between mathematics ideas to increase the levelof cognitive demand. Furthermore, the open-endedness of the problem – being open to multiple
correct solutions – helped them to pose problems requiring higher-level mathematical thinking. I
believe that this analysis will expand our understanding of what potentials of modeling problems are
considered by future teachers in posing modeling problems.
References
Albarracín, L., & Gorgorió, N. (2014). Devising a plan to solve Fermi problems involving large numbers.
Educational Studies in Mathematics, 86(1), 79-96.
Ärlebäck, J. B. (2009). On the use of realistic Fermi problems for introducing mathematical modelling inschool. The Montana Mathematics Enthusiast, 6(3), 331–364.
Blum, W., & Niss, M. (1991). Applied mathematical problem solving, modelling, applications, and links to othersubjects: state, trends and issues in mathematics instruction. Educational Studies in Mathematics, 22, 37–68.
Lesh, R., & Doerr, H. (2003). Foundations of a models and modeling perspective on mathematics teaching,learning, and problem solving. In R. Lesh & H. Doerr (Eds.), Beyond constructivism: A models andmodeling perspective (pp. 3–33). Lawrence Erlbaum.
Schukajlow, S., Krawitz, J., Kanefke, J., & Rakoczy, K. (2022). Interest and performance in solving openmodelling problems and closed real-world problems. In C. Fernández, S. Llinares, A. Gutiérrez, & N.
Planas (Eds.). Proceedings of the 45th Conference of the International Group for the Psychology ofMathematics Education (Vol. 3, pp. 403-410). PME.

79

FACILITATING PROBLEM POSING IN MATHEMATICAL MODELLING:
THE CASE OF 12TH-GRADE STUDENTS
Kosuke Yata1, Akihiko Saeki2 Naohiro Minagawa3 and Takashi Kawakami4
1

Joto High School, Japan; 2Kanazawa Institute of Technology, Japan; 3Naruto University of
Education, Japan; 4Utsunomiya University, Japan

Problem posing is crucial in mathematical modelling, but it has been found that it is not easy forstudents. For example, Hartmann et al. (2021) showed that approximately half of the problems posedby students were problems that had artificial connection to reality. We believe that this is influenced
by the difficulty that students face in posing problems where they feel ownership of the problem.
Therefore, the aim of the present study was to propose a strategy to reduce students' difficulty inproblem posing in mathematical modelling, and to find out the effectiveness of problem posing using
this strategy. We have prepared the following two strategies to pose problems in which students canhave authentic connections to reality. The first strategy is to provide information about real-world
situations that are close to students' lives and that facilitates the recall of specific situations such as
personal, educational/occupational, and public situations (OECD, 2018). The second one is to provide
tasks that allow students to pose authentic questions in real-world situations. This applies because ‘aquestion is authentic if it is one that people within the context would ask’ (Vos, 2018, p. 3).
We addressed the following question: How do the two strategies support 12th-grade students’problem poising in mathematical modelling? For this purpose, we designed and conducted a teachingexperiment in which 37 12th-grade students repeatedly reformulated and solved textbook problems
with real-world situations (i.e., using logarithmic function) by referring to information on real-world
situations. First, the students reformulated the textbook problems by referring to information on foodpoisoning bacteria. Secondly, they considered authentic questions based on new real-world situations(i.e., food poisoning in curries, plastic bottles, and packed lunches). Thirdly, they posed and solved
the problems again, based on authentic questions that they had considered themselves. Finally, they
modified the problems in new real-world situations presented by other students and advised by their
biology teacher. We analysed the students' statements in the worksheets used in the teachingexperiment.
The analysis showed that all students were able to pose authentic problems and that theirmathematisation was richer in terms of specifying the assumptions necessary for the solution. Many
students were also able to interpret and validate the results. These results indicate the effectivenessof two strategies in reducing students' difficulties in problem posing with mathematical modellingand in improving their mathematical modelling competencies.
References
Hartmann, L-M., Krawitz, J., & Schukajlow, S. (2021). Create your own problems! When given descriptionsof real‑world situations, do students pose and solve modelling problems? ZDM, 53, 919–935.
OECD (2018). PISA2022 mathematics framework (Draft). https://pisa2022-maths.oecd.org
Vos, P. (2018). “How real people really need mathematics in the real world”: Authenticity in mathematics
education. Education Sciences, 8(4), 195; https://doi.org/10.3390/educsci8040195

80

ENHANCING PRESERVICE TEACHERS REAL-WORLD
MATHEMATICAL PROBLEM-POSING ABILITIES:DESIGNING LEARNING TRAJECTORY
Songchai Ugsonkid1, Sakon Tangkawsakul1 and Weerawat Thaikam2
1

Kasetsart University, Thailand; 2Nakhonsawan Rajabhat University, Thailand

Fostering students’ capabilities to utilize their knowledge, skills, and mathematical processes to solve
real-world mathematical problems (RWMP) is a key objective outlined in the Thai Education
Curriculum and also in many other countries (Klainin, 2015). Critical ways of encouraging students
to solve RWMP related to preservice teachers’ (PSTs) abilities to pose appropriate RWMP (Paolucci& Wessels, 2017). This research developed the hypothetical learning trajectory (HLT) described by
Simon (1995) to enhance the RWMP-posing abilities of PSTs. Two research questions wereaddressed. 1) What are the key characteristics of the HLT that contribute to improving problemposing abilities? 2) What are the RWMP posed by PSTs? The research methodology involved threephases of educational design research following Bakker (2018) as preparation and design, teachingexperiments, and a retrospective study. The participants included collaborative educators and twentyone PSTs engaged in problem-solving and problem-posing activities who were instructed on how topose RWMP. Data were collected and analyzed using qualitative techniques such as interviews,
observations, and content analysis. The results concluded that 1) the HLT consisted of two
characteristics as a) implementing gradual problem-posing activities starting with simple strategy
problem-posing, followed by solving worthwhile problems, practicing multiple approaches inproblem-posing, and gradually incorporating free problem-posing with greater relevance to realworld contexts, and b) encouraging productive interaction among PSTs through group and discussion
forums. The actual learning trajectories of most PSTs aligned with the HLT. Results also suggested
that 2) most RWMP posed by PSTs focused on wrapper problems (Stillman, 1998) which had realistic
context but lacked complexity and opportunities for assumption-making and seeking additional
relevant information, both crucial criteria for modelling problems. These findings offer valuable
insights for educators seeking to enhance the RWMP-posing abilities of PSTs by promoting effective
learning regardless of their previous experience in mathematical modelling.
References
Bakker, A. (2018). Design research in education: A practical guide for early career researchers. Routledge.
Klainin, S. (2015). Mathematics education at school level in Thailand: The development—The impact—The
dilemmas. Retrieved from https://pisathailand.ipst.ac.th/ipst-959/
Paolucci, C., & Wessels, H. (2017). An examination of preservice teachers’ capacity to create mathematicalmodeling problems for children. Journal of Teacher Education, 68(3), 330-344.
Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for
research in mathematics education, 26(2), 114-145.
Stillman, G. (1998). The emperor’s new clothes? Teaching and assessment of mathematical applications at thesenior level. In P. Galbraith, W. Blum, G. Booker, & D. Huntley (Eds.), Mathematical modelling: Teaching
and assessment in a technology-rich world (pp. 243–253). Chichester, UK: Horwood.

81

THE RELATIONSHIP BETWEEN TEXT COMPREHENSION AND NOTETAKING WHILE WORKING ON REALITY-BASED TASKS
Lisa-Marie Wienecke1 and Dominik Leiss1
1

Leuphana University Lueneburg, Germany

Numerous empirical studies have demonstrated that the translation process between reality andmathematics while working on reality-based tasks is associated with cognitive barriers (Galbraith &Stillman, 2006) and that understanding real problem situations is a source of concern for students(Leiss et al., 2010). One promising strategy to overcome these difficulties is note-taking Graham etal., 2020; Rogiers et al., 2020). However, differential analyses on the design of notes and the specificinfluence of note-taking on the solution process while working on reality-based mathematical tasks
represent a research desideratum.
The data on note-taking, including personal characteristics, were collected within the VAMPSProject (Variation Tasks Mathematics Physics Language - funded by German Research Foundation).
Tasks on three different topics (contexts) were varied at three levels of language complexity (Heineet al., 2018), resulting in a total of nine mathematical reality-based tasks. 424 students worked oneach of the three contexts, generating 1 272 responses. These were then used to calculate correlations
and multivariate regressions using SPSS and R Statistics.
Overall, unsolicited notes were made on just under one-third of the tasks completed. When counting
any form of marking or formal structuring as a note, the figure is as high as 92.7 %. The initial results
show that linguistically higher demands, increased reading enjoyment, and the use of different
reading strategies are associated with increased note-taking. In contrast, there are only small effects
related to the comprehension of the task text. Taking notes, however, is clearly related to the correct
calculation path and thus to the correct solution. Another finding is that girls tend to take significantly
more notes than boys and a higher socio-economic status also correlate with more notes of any kind.
I am also pleased to be able to say results on the relationship between notes and performance at theconference.
References
Galbraith, P. L., & Stillman, G. A. (2006). A framework for identifying student blockages during transitionsin the modelling process. ZDM, 38(2), 143–162.
Graham, S., Kiuhara, S. A., & MacKay, M. (2020). The effects of writing on learning in science, social studies,and mathematics: A meta-analysis. Review of Educational Research, 90(2), 179–226.
Heine, L., Domenech, M., Otto, L., Neumann, A., Krelle, M., Leiss, D., Höttecke, D., Ehmke, T., & Schwippert,
K. (2018). Modellierung sprachlicher Anforderungen in Testaufgaben verschiedener Unterrichtsfächer:
Theoretische und empirische Grundlagen. Zeitschrift für Angewandte Linguistik, 2018(69), 69–96.
Leiss, D., Schukajlow, S., Blum, W., Messner, R., & Pekrun, R. (2010). The Role of the Situation Model in
Mathematical Modelling—Task Analyses, Student Competencies, and Teacher Interventions. Journal FürMathematik-Didaktik, 31(1), 119–141.
Rogiers, A., Merchie, E., & Van Keer, H. (2020). Opening the Black Box of students’ text-learning processes:
A process mining perspective. Frontline Learning Research, 8(3), 40–62.

82

STUDENTS’ PREFERENCES AND BELIEFS ABOUT THE OPENNESS OF
MODELLING PROBLEMS
Stanislaw Schukajlow1, Janina Krawitz1, Katharina Wiehe2 and Katrin Rakoczy2
1

University of Münster, Germany; 2University of Gießen, Germany

One important characteristic of modelling problems is their openness (Niss & Blum, 2020). In an
open problem, information that is essential for solving the problem is missing. Prior research revealed
students’ significant barriers in solving open modelling problems. Based on theoretical considerations
and empirical findings (Galbraith & Stillman, 2001; Krawitz et al., 2018), we suggest the importanceof the following processes in dealing with open problems: (1) noticing missing information, (2)identifying the unknown quantities needed to solve the problem, (3) making realistic assumptions
about the missing quantities and (4) integrating open quantities in the mathematical model. In the
randomized control trial carried out in the project OModA (Open Modelling Problems in mathematics
teaching that is oriented toward self-regulation) we aimed to analyze the effects of teaching opennessof modelling problems on students’ learning outcomes. In our talk, we will focus on the effects on
students’ preferences for solving open problems and their beliefs about open problems and theirrelation to school grades. Students’ preferences for solving open problems refer to the positive valueof solving this problem type. Students’ beliefs about open problems address the belief that amathematical problem always includes all information needed for its solution. The research questions
were: (1) Does teaching openness to modelling problems affect students’ preferences and beliefs? (2)Are students’ preferences and beliefs about open problems related to their school grades?
Two hundred and ninety-four German ninth graders (137 female) were randomly assigned withineach class to two conditions. In the experimental condition (EC) students were taught to solve openmodelling problems and in the control condition (CC) closed real-world problems. Before and after
the teaching unit (4 lessons á 45 minutes) students ranked their preferences (4 items) and beliefs (1item) on open modelling problems on a 5-point Likert-Scale and reported school grades.
As expected, teaching open modelling problems decreased students’ belief that a mathematicalproblem always includes all information needed for its solution, compared to the CC. Results revealed
a positive relationship between beliefs (but not preferences) and school grades in the pretest. Further,teaching openness to modelling problems tended to affect positively students’ preference for solving
open problems. One important practical implication is that teaching openness to modelling problems
is a powerful approach to improving beliefs and preferences for this problem type. One theoretical
implication of our study is that although open problems are demanding for students and it can be
expected that these students prefer solving these problems, students on different levels ofmathematical achievement have similar preferences to solve these problems.
References
Galbraith, P. L., & Stillman, G. (2001). Assumptions and context: Pursuing their role in modelling activity. In
J. Matos, W. Blum, K. Houston, & S. Carreira (Eds.), Modelling and Mathematics Education, Ictma 9:
Applications in Science and Technology (pp. 300–310). Horwood Publishing.
Krawitz, J., Schukajlow, S., & Van Dooren, W. (2018). Unrealistic responses to realistic problems withmissing information: What are important barriers? Educational Psychology, 38, 1221–1238.
Niss, M., & Blum, W. (2020). The learning and teaching of mathematical modelling. Routledge.

83

DESIGNING COLLABORATIVE MATHEMATICAL MODELLING TASKSFOR LOWER SECONDARY EDUCATION
Sevinç Göksen-Zayim1, Derk Pik1 and Carla van Boxtel1
1

University of Amsterdam, The Netherlands

Collaborative learning is considered a potentially powerful way to learn mathematics (e.g. Dekker &Elshout-Mohr, 1998; Webb, 2009), but we still have limited knowledge about how group work
contributes to mathematical modelling competencies. The research question we address is: How cangroup tasks for lower secondary education enhance collaborative mathematical modelling?
In an educational design research, a series of five modelling tasks have been developed building upon
design principles for (individual) modelling tasks (Galbraith, 2006; Geiger, et al., 2021) andcollaborative learning tasks (Webb, 2009). We used the following principles: (1) the problems are
open-ended and sufficiently complex to provoke students to work collaboratively; (2) there is agenuine link with the real-world and a connection to students’ perceptions; (3) the problems matchthe level of the students; (4) the different mathematical modelling (sub-)competencies are addressedin the tasks, including assumption making; (5) the tasks encourage the use of mathematical concepts,
linking them to everyday language.
The participants were 37 groups of three students, aged 13 and 14, from grade eight coming from tenclasses from five secondary schools. Each group performed one modelling task and then participating
students individually completed a questionnaire about the task. Students' verbal interactions wereanalyzed using a previously developed coding scheme (Göksen-Zayim, et al., 2022).
The analysis showed that students generally worked well together on the modelling tasks andunderstood the problem. However, students hardly discussed the purpose of the task and possible
approaches. Furthermore, in only half of the groups, students made more than one assumption andless than half constructed interpretations and validations, resulting in an incomplete solution of themodelling task. The results of the questionnaire showed that students considered the tasks interesting.
In conclusion, we found that the group tasks based upon our design principles can engage students incollaborative modelling, but they need more support on making assumptions and solving strategies.
References
Dekker, R., & Elshout-Mohr, M. (1998). A process model for interaction and mathematical level raising.
Educational Studies in Mathematics, 35(3), 303–314.
Galbraith, P. (2006). Real world problems: Developing principles of design. In P. Grootenboer, R.
Zevenbergen, & M. Chinnappan (Eds.), Identities, cultures and learning spaces (pp. 229–236). MERGA.
Geiger, V., Galbraith, P., Niss, M., & Delzoppo, C. (2021). Developing a task design and implementation
framework for fostering mathematical modelling competencies. Educational Studies in Mathematics,1–24.
Göksen-Zayim, S., Pik, D., & van Boxtel, C. (2022). An observation instrument to analyze the collaboration
between students while solving mathematical modelling problems. In J. Hodgen, E. Geraniou, G. Bolondi& F. Ferretti. (Eds.), Proceedings of the European Society for Research in Mathematics Education(CERME12) (pp. 1–8). Free University of Bozen-Bolzano and ERME.
Webb, N.M. (2009). The teacher's role in promoting collaborative dialogue in the classroom. British Journalof Educational Psychology, 79(1), 1–28.

84

COMPARISON OF MATHEMATICAL MODELLING IN SOLVING AWNINGWINDOW PROBLEMS: MALAYSIAN CIVIL EXAMINATION PROBLEM
VS OLD JAPANESE TEXTBOOK PROBLEM
Shaliny Kannapiran1 and Akio Matsuzaki2
1

Sekolah Kebangsaan Seri Tasik Kuala Lumpur, Malaysia; 2Saitama University, JapanThe research question is as follow: What are the similarities and differences in solving awning
window problems between Malaysian civil examination problem and Japanese old textbook problem
based on modelling cycle (Blum & Leiß, 2007)?
Malaysian Civil Examination Problem (Ministry of Education, 2020)
Diagram shows a window that has
been installed in a house. The openingof the window is limited to 30° from
fulcrums E and H. It is given the lengthof EF = EK = 55 cm.
a) Name the angle which corresponds
to ∠FEK.
b) Hence, calculate the verticaldistance from point F to the line EK
when the opening of the window ismaximum.

Japanese Old Textbook Problem(The Lower Secondary Textbook, 1943)
There is a window which
Japanese Old Textbook Problem
have a structure as the(The Lower Secondary Textbook, 1943)
figure. The prop GH is
furnished on midpoint of awindow frame. If you push
the stick AB, the frame CD
slide down along thegroove of the window. In
this time, how does thebottomed frame EF move?

There is a window whichhave a structure as the
Diagramshowsawindowthathasfigure. Thethat
prophasGHbeenis
In the case of Malaysian civil examination (left side of Fig.1) an awning window
been installed in a house. The opening
furnished on midpoint of a
installed in an architecture
is window
a real issituation
In this problem mathematicalhasof thelimited to &30’ problem.
fromwindow frame. model
If you pushfulcrums E and H. It is given the length
the stick AB, the frame CD
already been made for thequestions a) name the angle of opening of the window,
and for questionof EF = EK = 55 cm.
slide down along thea) Name theangle thewhich
corresponds
grooveof thethe window.
b), calculate the vertical distancewhen
openingof the window is maximum.
Thus,
processInof
to ∠FEK.
this time, how does themathematical work leadsb) toHence,a mathematicalis mainly focussed. In thebottomedcase offrame
oldEFJapanese
calculate theresult
verticalmove?
distancepoint
to the linewindow
EK
textbook problem (right sideof from
Fig.1)
anFawningthat has been installed in an architecture iswhen the opening of the window isa real situation & problem.
The focus on the structure of the awning window explains the mechanism
maximum.

Figure 1. Awning window problemsof opening and/or closing of the awning window. The mechanism of opening and/or closing of theawning window is a real model & problem and mathematical model & problem. Thus, modeller willsolve the mechanism of opening and/or closing of the awning window mathematically in a series ofprocess in modelling cycle. Hence the similarities are both problems are presented in the figures ofawning window that are located on a real situation & problem. The differences are that in the case of
old Japanese textbook it is needed to solve the problem in a series of process in modelling cycle whilein the case of Malaysian civil examination it is needed to solve the problem through the process ofmathematical work mainly.
References
Blum, W. & Leiß, D. (2007). How do students and teacher deal with modelling problems? In C. Haines, P.
Galbraith, W. Blum, & S. Khan (Eds.), Mathematical Modelling (ICTMA12): Education, Engineering andEconomics (pp.222-231). Chichester: Horwood.
Ministry of Education (2020). Sijil Pelajaran Malaysia (Malaysian Certificate of Education): Mathematics
Paper 2. Malaysia: Examination Board.
The Lower Secondary Textbook (1943). Second Categories in Mathematics 3 (p.3). The Lower Secondary
Textbook.

85

IDENTIFYING AND TRACKING STUDENT CONCEPTIONS OF
MATHEMATICS
Jill P Brown1 and Gloria Ann Stillman2
1

Deakin University, Australia; 2Retired, Australia

A proposed enabler of successful mathematisation when mathematical modelling is believing that avalid use of mathematics is modelling real phenomena (Niss, 2010). We discuss analysis of datacollected in the Victorian component of the Enablers Project3 to answer the research questions: Whatconceptions of mathematics are held by Victorian Year 11 students? and How are these conceptions
changed by experiencing a collaborative modelling intervention as part of their mathematics lessons?
According to Wood et al. (2012, p. 70), by university entry, “students have already developed
conceptions of mathematics”. Houston et al. (2010) investigated undergraduate students’ conceptionsof mathematics categorising them as Number, Components, Modelling and Abstract, and Life.
Modelling and Abstract are at a similar level and eventually combined if a student develops a Life
conception, viewing “mathematics as an approach to life and a way of thinking” (p. 73). Higher levels
subsume the conception(s) of lower levels, becoming broader. This categorisation was used foranalysing developing conceptions of Grade 6 students (Brown & Stillman, 2017). Wood et al.’s shortform of the Conceptions of Mathematics Questionnaire (CMQ) interrogates, via Likert scales, items
related to what mathematics is as well as the role of mathematics in future studies and career. In the
Enablers Project the CMQ was administered pre and post student participation in modelling lessons.
We used selected items to ascertain students’ conceptions of mathematics according to the Houston
et al. categories. To visualise aggregated student responses, radar diagrams using responses toselected items targeting views underpinning a conception were constructed. A more robust
determination of an individual’s conceptions used information from the Metaphor Assessment
Instrument (Cai & Melino, 2011) collected at the same time and responses in post modelling lesson
implementation interviews. We found teachers and their students needed to adjust their view ofstudents’ variation in conceptions of mathematics. Classes were not as homogeneous in conceptions
as teachers expressed in pre and post lesson interviews. The discussion will address: (1) usefulnessof the techniques for studying students’ conceptions of mathematics, (2) students’ modelling
experiences potential for changing in their awareness of conception variation and necessity toprogress towards broader conceptions including modelling, and (3) implications for upper secondary.
ReferencesCai, J., & Merlino, F.J. (2011). Metaphors: A powerful means for assessing students’ mathematicaldispositions. In D.J. Brahier & W.R. Speer (Eds.), Motivation and disposition (pp. 147-156). NCTM.
Houston, K., Maher, G., Wood L., et al. (2010). Is there life after modelling? Student conceptions ofmathematics. Mathematics Education Research Journal, 22(2), 69–80.
Niss, M. (2010). Modeling a crucial aspect of students’ mathematical modeling. In R. Lesh et al. (Eds.),
Modeling students’ mathematical competencies (pp. 43-59). Springer.
Wood, L.N., Petocz, P., & Reid, A. (2012). Becoming a mathematician: An international perspective. Springer.
3

Australian Research Council funded project DP17010555 Chief Investigators: V. Geiger, G. Stillman, J. Brown, & P.
Galbraith. M. Niss was a Partner Investigator. The views expressed herein are those of the authors.

86

A COMPARATIVE CASE STUDY ON TEACHING MODELLING AT THE
SECONDARY LEVEL WITH A UNIT DEVELOPED FOR THE TERTIARYLEVELRina Durandt1, Werner Blum2, Alfred Lindl3 and Rita Borromeo Ferri2
1

University of Johannesburg, South Africa; 2University of Kassel, Germany; 3University ofRegensburg, Germany

This paper reports on results from the CoSTAMM project (“Comparative Studies into Teaching
Approaches for Mathematical Modelling”). An important goal of research is to explore whichteaching methods are effective for teaching mathematical modelling at different levels of education.
In the first three CoSTAMM studies, two teaching designs were investigated, a teacher-directive
design and a method-integrative design, similar to the German DISUM project. A classical research
design with an entrance test, a pre-test, a treatment (modelling unit with five lessons, altogether ten
tasks), and a post-test was implemented at the tertiary level with engineering and chemistry students
in South Africa (for results in the engineering context see Durandt, Blum & Lindl, 2022a). The main
results from these studies show that both teaching approaches led to a significant competency growth,both in mathematics and in mathematical modelling, with advantages for the method-integrative
groups, similar to the results in DISUM.
Since the mathematics required in the five-lesson modelling unit and in the pre-/post-test is essentiallyat the secondary level (grades 9/10; for details see Durandt, Blum & Lindl, 2022b), the idea was toimplement the modelling unit and the two teaching designs at this level and to see whether similar
results would emerge. This study (without an entrance test) was carried out in 2022 as a case study
in four grade 9 classes in Germany (15-year-olds), where two classes followed a teacher-directive
design (with a total of 34 students), and two classes followed a method-integrative design (with a
total of 35 students). Apart from data on students’ progress, data on teaching quality in the four classes
were also collected. Linear mixed regression models were used to evaluate and to compare the results.
The results show interesting similarities with the previous studies, but also some unexpectedpeculiarities. Globally speaking, the teaching unit and the two designs worked also at the lower
secondary level, in a German context, and were effective for fostering modelling competency.
References
Durandt, R., Blum, W., & Lindl, A. (2022a). Fostering mathematical modelling competency of first year South
African engineering students: What Influence does the teaching design have? Educational Studies inMathematics, 109, 361-381.
Durandt, R., Blum, W. & Lindl, A. (2022b). A Mathematical Modelling Unit for First-Year Engineering
Students. Modelling in Science Education and Learning 15(1), 77-92.

87

DESIGN OF STEM TASKS BASED ON MATHEMATICAL REASONING IN
A SEQUENCE OF MODELLING ACTIVITIESKai-Lin Yang
National Taiwan Normal University, Taiwan

The importance of integrated STEM (science, technology, engineering and mathematics) education
has been acknowledged to develop 21st century skills required in the global market. It indicates theurgent need to design integrated STEM tasks for STEM education. In this presentation, I will
introduce how to design STEM tasks based on mathematical reasoning in a sequence of modelling
activities. First, a sequence of modelling activities is introduced. Second, how the model development
sequence can be modified to design integrated STEM tasks based on mathematical reasoning andillustrated with exemplary tasks. Last, implications and suggestions are provided for further
investigation.
Model-eliciting, model-exploration and model-adaptation activities (MEAs, MXAs, MAAs) arerecommended as a sequence of modelling activities for learning mathematics (Lesh, Cramer, Doerr,
Post, & Zawojewski, 2003). Studies with MEAs have shown that students’ thinking can progress
during at least one or two full class periods. To further develop powerful representation systems,
MXAs following MEAs are necessary. To consolidate the developed mathematical concepts andthoughts, MAAs provide students the opportunities to apply and generalize them.
Mathematics often serves as a tool to be applied in other disciplines. However, this technical aspectof mathematics is not enough for integrated STEM education, and multiple relationships between
mathematics and other STEM disciplines should be included. By modifying the model development
sequence based on mathematical reasoning (MR) where mathematical mental acts, such as explaining,
conjecturing, structuring, predicting, problem solving, interpreting and validating, are involved(Harel, 2008), MEAs, MXAs and MAAs are respectively focused on connection of, interaction
between, coordination of mathematics and other STEM disciplines. The modification will beillustrated with a sequence of STEM tasks, named as MR-STEM connection, interaction andcoordination tasks.
The potentials and design guidelines of MR-STEM connection, interaction and coordination tasks
will be discussed. The approach to the design of a sequence of STEM tasks is theory-based andpractice-oriented. Further investigation of how mathematics and mathematical reasoning can beapplied and developed in a sequence of modelling activities are needed to support multiple aspectsof mathematics.
References
Lesh, R., Cramer, K., Doerr, H. M., Post, T., & Zawojewski, J. S. (2003). Model development sequences. In
R. Lesh, & H. M. Doerr (Eds.), Beyond constructivism: Models and modelling perspectives on mathematics
problem solving, learning, and teaching (pp. 3-33). Mahwah, NJ: Lawrence Erlbaum.
Harel, G. (2008). DNR perspective on mathematics curriculum and instruction, Part I: focus onproving. ZDM, 40, 487-500.

88

HOW CHINESE CITIZENS UNDERSTAND THE INFLECTION POINT OF
THE COVID-19 PANDEMIC: INSIGHT INTO FORMAL MATHEMATICS
AND STREET MATHEMATICS
Yi Wang1 and Lidong Wang1
1

Beijing Normal University, China

Since the beginning of 2020, the COVID-19 pandemic has changed everything, bringing enormous
global challenges to the mathematics education community (Bakker & Wagner, 2020). While doctors
and other medical professionals are fighting the disease, people are struggling to remain safe andsurvive the pandemic.
Also noteworthy is what researchers in mathematics education can do, or are responsible for, during
this time of crisis. Regarding the interaction between mathematics and society (Moreno-Aemella andSantos-Trigo, 2008; National Research Council, 2012), we should focus on the impact of mathematics
education on social events and people’s lives, especially focusing on the fact that mathematics
education should aim to raise mathematically informed citizens who are able to understandmathematical or scientific claims as tentative, as deserving a fitting degree of confidence andskepticism – a stance that is neither gullible nor inflexible, but appropriately critical (Muğaloğlu etal., 2022).
This study investigates how people use mathematics to understand the inflection point of the COVID19 pandemic, using data from 1,371 Chinese people, and explore the possible reasons behind theirunderstanding. Questionnaire surveys were used to collect quantitative data and descriptive statistics
were used to reveal people’s interpretation of the inflection point numerically and graphically. Online
interviews were employed to collect qualitative data that were organized and interpreted to provide
more details regarding people’s understanding. The results indicate that only a few people canprecisely explain what the inflection point of the pandemic is, despite the fact that secondary school
education is supposed to equip people with relevant knowledge and literacy and official media
releases on the pandemic are another source for people to do so. In addition, citizens tend to apply
more informal mathematics in the understanding of inflection point than formal mathematics.
References
Bakker, A., & Wagner, D. (2020). Pandemic: Lessons for today and tomorrow? Educational Studies inMathematics, 104(1), 1–4. https://doi.org/10.1007/s10649-020-09946-3
Moreno-Aemella, L., & Santos-Trigo, M. (2008). Democratic access and use of powerful mathematics in an
emerging country. In L. D. English (Ed.), Handbook of international mathematics education (pp. 319–351).
Routledge.
Muğaloğlu, E. Z., Kaymaz, Z., Mısır, M. E., & Laçin-Şimşek, C. (2022). Exploring the role of trust in scientists
to explain health-related behaviors in response to the COVID-19 pandemic. Science & Education.
https://doi.org/10.1007/s11191-022-00323-5
National Research Council. (2012). Education for life and work: Developing transferable knowledge and skillsin the 21st century. The National Academies Press.

89

STATISTICAL PROBLEM SOLVING BASED ON CHILDREN’S VALUES IN
MATHEMATICAL SCIENCES EDUCATION
Mitsuru Matsushima1, Daisuke Ishikawa2 and Keiichi Nishimura3
1

Kagawa University, Japan; 2The Ninth Haketa Elementary School, Japan,
3

Tokyo Gakugei University, Japan

Our research group focused on decision-making and consensus building using mathematical sciences(e.g., Yamaguchi et al., 2020; Ishikawa, 2020). Mathematical sciences include mathematics, statistics,
and applied mathematics at their core; however, they are also closely related to other disciplines suchas science and engineering, depending on the authentic problem being dealt with. In mathematicalsciences education, this decision-making process formulates and addresses real-world problems that
require decision-making. Regarding the multiple options created by different formulations, it isnecessary to clarify the premise, form a consensus, and make decisions (Yamaguchi et al., 2020).
When building consensus, it is also necessary to pay attention to the values embedded in themathematical model, which becomes the focal point of the dialogue. Through such dialogue, children
can become aware of the close relationship between the existence of diverse values and mathematics.
And it can update children’s values (Ishikawa, 2020). This paper introduces one practice ofmathematical sciences education.
This practice was performed using a statistical unit for sixth graders. In Japan, when a new batch offirst graders enter elementary school, a welcome party is often planned and hosted mainly by thecurrent batch of sixth graders, which is the highest grade. This practice is based on the sixth graders’values, “We want to hold a welcome party that can be enjoyed by both first and sixth graders.” In thispractice, the sixth graders planned a ring-toss game. To make it enjoyable for both, the sixth graders
conducted an experiment to determine the distance to the target and collected data. They also asked
participants how they felt about the ring-toss game during data collection. They determined thethrowing distance using both quantitative and qualitative data. However, when they collected data,
they found that the number of successful quoits among first graders varied. Therefore, it is necessary
to discuss the number of times the ring is successfully thrown to clear the game. During the discussion,
opinions were raised that the characteristics of the group should be understood based on data, andthat the values of both first and sixth graders are important. In subsequent learning, based on the dataof the entire class, they learned the concepts of average, median, mode, maximum, and minimum
values, and captured the characteristics of the data.
From this practical example, we can conclude that children’s values become the focus ofmathematical sciences decision-making, as they are effective when examining the validity of theresults of statistical problem solving, discovering new problems, and refining plans. Value-basedmodeling of real-world problems and statistical problem solving based on real data is an excellent
way to master mathematical science methods. This engenders a real sense of usefulness and is usefulfor raising children with diverse values.
References
Ishikawa, D. (2020). A Study on Roles of Teachers in the Lessons Intended to Mathematical-Scientific
Decision-Making Through the Lesson: “Routing of social studies tour”. Journal of Japan Society ofMathematical Education, 102(6), 3–12. (In Japanese)
Yamaguchi, T., Nishimura, K., Shimada, I., Matsushima, M. and Matsuzaki, A. (2020). Development ofMathematical Sciences Education in School Education: A Foundational Framework for Curriculum Designin Mathematical Sciences Education. Journal of Science Education in Japan, 44(2), 104–122. (In Japanese)90

DIFFERENCES AND SIMILATIRIES IN STUDENTS’ PERFORMANCE ON
MODELLING TASKS WITH MUCH OR LITTLE PERSONAL INTEREST IN
THE REAL-WORLD CONTEXT OF THE TASK – A COMPARISON
Melanie Kämmerer1 and Gilbert Greefrath1
1

University of Münster, Germany

When working on a mathematical modelling task, there is always a real-world context that needs tobe considered to solve the task. The real-world context could therefore be one aspect in the selectionof appropriate tasks. While it is already known that interest in mathematics influences modelling
performance, this is an open question for personal interest in the real-world context (Krug &Schukajlow 2014). Therefore, the PiMo project (Personal interest in modelling tasks) investigates theinfluence of the personal interest in the context of modelling tasks on the modelling process.
Here, it is investigated to what extent personal interest in the real-world context of a modelling task
influences the performance and the procedure shown in the solution process of the modelling task.
To answer this question a qualitative study with 10 students of seventh grade is carried out with athree-step design. For this purpose, three modelling tasks with similar characteristics in terms ofmathematical content, authenticity and openness to the real-world contexts: soccer, running and arts
& crafts were developed. First, the students state their personal interest in the given contexts. Then,for every student one task with much personal interest in the real-world context of the task (task I+)
and one task with little interest (task I-) is selected. Second, students work in small groups on an
example task to get familiar with modelling tasks. In the third step, about one week later, students
work individually, while thinking aloud, on their task I+ and their task I- and a buffering item inbetween. This process is video recorded. The order of the tasks is altered over the whole sample.
The transcripts of the videos are analysed using a qualitative content analysis. Categories are the stepsof the modelling cycle after Blum & Leiß (2007). After consensually coding the modelling steps, theperformance on each step is evaluated using an assessment scheme, where the performance is ranked
in four levels. Then, not only the overall score as well as scores in the individual modelling steps are
examined, but also qualitatively the concrete procedure in the steps. The analysis is still in progress,
but first results show that differences for the solution processes with much or little interest in thecontext of the task be observed.
The results of the PiMo project can supply teachers with information about how they can use thepersonal interest in real-world contexts of modelling tasks to support their students’ modellingcompetences and enhance their performance.
References
Blum, W., & Leiß, D. (2007). How do students and teachers deal with mathematical modelling problems? The
example sugarloaf and the DISUM project. In C. Haines, P. L. Galbraith, W. Blum, & S. Khan, (Eds.),
Mathematical modelling (ICTMA 12). Education, engineering and economics (pp.222–231). Springer.
Schukajlow, S., & Krug, A. (2014). Are interest and enjoyment important for students’ performance? In S.
Oesterle, P. Liljedahl, C. Nicol, & D. Allan (Eds.), Proceedings of the Joint Meeting of PME 38 and PMENA 36 (Vol. 5, pp. 129–136).

91

WRITTEN AND SPOKEN RETELLING OF THE TASK SITUATION AS A
STRATEGY FOR UNDERSTANDING OPEN MODELLING TASKS
Anna Surel1 and Gilbert Greefrath1
1

University of Münster, Germany

When working on open mathematical modelling tasks, students have many difficulties, especially inthe first steps of understanding, simplifying and structuring (Blum, 2015). However, these steps,
which include the formation of a situation model for the task situation, are crucial for successful
completion of the tasks (Leiss et al., 2010). One way to support these first steps is to have the students
retell the task situation in their own words in order to engage more intensively with it. This can bedone verbally or in writing. Both forms have their advantages and disadvantages and it is unclear
which strategy is more helpful for the students. For example, a written approach is often more indepth than a spoken approach, but it can also be more challenging, especially for students with weakerlanguage competence, as it requires a certain level of writing skills (Grabowski et al., 2012).
In a qualitative video study, six seventh graders are observed working on two mathematical modelling
tasks after retelling the given task situation verbally or in writing. For this, the students are given
different instructions to retell the task situation verbally or in writing before they have to solve thetask. Using the method of thinking aloud, the students' thought processes are made visible in order toanalyse the different processing procedures. The extent to which the students' processing procedures
and the retelling processes differ when working on open mathematical modelling tasks with writtenor spoken retelling is investigated. In order to be able to put possible differences also in the context
of different levels of language competence, a short language competence test was conducted with thestudents beforehand.
A qualitative content analysis is used to analyse the transcripts of the videos. Since retelling as a
strategy is meant to support the comprehension process, the focus of the analysis is on the first stepsof the modelling process: the understanding, simplifying and structuring. First results on differences
between the processing of open modelling tasks with written and spoken retelling will be presented
at the ICTMA conference and discussed with regard to possibilities for supporting students in theprocessing of open mathematical modelling tasks.
References
Blum, W. (2015). Quality teaching of Mathematical Modelling: What do we know, what can we do? In S. J.
Cho (Ed.), The Proceedings of the 12th International Congress on Mathematical Education: Intellectualand attitudinal challenges (pp. 73–96). Springer International Publishing. https://doi.org/10.1007/978-3319-12688-3
Grabowski, J., Becker-Mrotzek, M., Knopp, M., Nachtwei, N., Weinzierl, C., Jost, J., & Schmitt, M. (2012).
Subcomponents of writing literacy: Diagnosis and didactical support. In M. Torrance, D. Alamargot, M.
Castelló, F. Ganier, O. Kruse, A. Mangen, L. Tolchinsky, & L. van Waes (Eds.), Learning to write
effectively:
current
trendsinEuropeanresearch(pp.
199–201).
BRILL.
https://doi.org/10.1163/9781780529295_045
Leiss, D., Schukajlow, S., Blum, W., Messner, R., & Pekrun, R. (2010). The role of the situation model inmathematical modelling—task analyses, student competencies, and teacher interventions. Journal FürMathematik-Didaktik, 31(1), 119–141.

92

INFLUENCE OF LANGUAGE USED ON THE MATHEMATISATION：ANALYSIS BASED ON COGNITIVE LINGUISTICS
Masaaki Ishikawa
Aichi University of Education, Japan

Students tend to have difficulty in mathematising of real situation. In the study of word-problem, thepossibility that the cause of this difficulty lies in the students' language ability has been pursued. For
example, studies have focused on the relationship between reading comprehension and problemsolving skills (e.g.,Vilenius-Tuohimaa, Aunola, & Nurmi, 2008) and have analysed the effect of thecomplexity of written expressions on students' problem solving (e.g., Plath & Leiss, 2018). Whilethese studies have developed internationally, they have not developed in Japan. In this study, we
focus on the relationship between the Japanese and mathematical language and analyse how theability to use Japanese affects the mathematisation of real situation. This study utilizes "cognitive
linguistics" as a theoretical framework for consideration.
The purpose of this research is to clarify the effect of differences in the structure of Japanese andmathematical language at the stage of "mathematisation" where the translation from the language inthe real world to the language in the mathematical world. To achieve the purpose, three sub-tasks
were set in this presentation;(1) How has the issue of language used been discussed in Japanese wordproblem and modelling research? (2) What implications does a linguistic approach have for wordproblem research? (3) From the perspective of linguistics, how does the differences in structure
between Japanese and mathematical languages affect learners' thinking?
The results corresponding to the sub-tasks are as follows;(1) In Japan, there was a time when the
structural difference between Japanese and mathematical languages became a topic of discussion, butthere were not many of them. The reason for this may be that studies have been case studies, andtherefore a general methodology has not been developed to capture the effects of structuraldifferences between Japanese and mathematical languages. (2) We pointed out that the field of
linguistics, which has a high affinity with mathematics education and research, is cognitive linguistics,
and organized the basic ideas. Furthermore, based on cognitive linguistics, we organized two relativist
perspectives: "relativism of interpretation between individuals (conceptual relativism)" and"relativism of languages (linguistic relativism)". And we also theoretically argued that these
relativism enable new perspective that are different from the existing framework of mathematicalmodelling research. (3) We conducted a survey and empirically verified that the perspectives of"conceptual relativism" and "linguistic relativism" are effective frameworks for capturing theinfluence of language on mathematisation.
References
Plath, J., & Leiss, D. (2018). The impact of linguistic complexity on the solution of mathematical modelling
tasks. ZDM - Mathematics Education, 50(1–2), 159–171. https://doi.org/10.1007/s11858-017-0897-x
Vilenius-Tuohimaa, P. M., Aunola, K., & Nurmi, J. E. (2008). The association between mathematical word
problems
andreadingcomprehension.
Educational
Psychology,
28(4),
409–426.
https://doi.org/10.1080/01443410701708228

93

ACCUMULATION OF DEGREE-DAYS AND CATEGORY OF MODELLING:
A CASE WITH STUDENTS IN CHILE
Claudio Gaete-Peralta1, Jaime Huincahue2 and Jaime Mena3
1

Universidad Bernardo O´Higgins, Chile; 2Universidad Católica del Maule, Chile; 3Pontificia
Universidad Católica de Valparaíso, Chile

The teaching of the definite integral is traditionally centred on a formal and mechanistic approachthat distances this concept from an interpretation linked to processes of change (Contreras andOrdóñez, 2006). In this regard, Marcía (2020) provides theoretical references for the design oflearning situations that allow students, from a development of category of modelling (Cordero et. al,2022), to construct the integral defined as a model of accumulation degree-days in specific situationsof change in Phenology. Based on the above, the aim of this research was to analyse the constructionof the definite integral of university students when they faced an organised and sequenced learning
situation from an argumentation of accumulation of degree-days in a specific situation of change.
To respond to the objective, a case study (Stake, 2007) was used with 10 students from a Chilean
university, who developed a learning situation, designed based on the framework formulated by
Marcía (2020), whose purpose was to determine the accumulation of degree-days of a particular typeof watermelon harvested in the Azapa Valley (northern Chile) during the month of August 2023. For
the development of the situation, students made use of: a) web pages to obtain data on the averagetemperature forecast for that month, and b) GeoGebra to construct a graph of average temperature𝑇(𝑡) (time 𝑡 in days) that models the data obtained and incorporates area sliders to determine theaccumulation of degree-days. Data collection was obtained through written and audio-visual records
and data analysis used interpretive techniques of content analysis (Flick, 2004).
The development of the situation contributed to the emergence of a category of modelling thatallowed students to construct the integral defined as a model of accumulation degree-days. In thissense, the role of GeoGebra was fundamental as it favoured the students' construction of the function
𝑇(𝑡) and the visualisation of the quantities that accumulated.
References
Contreras, Á., & Ordóñez, L. (2006). Complejidad ontosemiótica de un texto sobre la introducción a la integral
definida. Relime, 9 (1), 65-84.
Cordero, F., Mendoza-Higuera, J., Pérez-Oxté, I., Huincahue, J., & Mena-Lorca, J. (2022). A Category ofModelling: The Uses of Mathematical Knowledge in Different Scenarios and the Learning of Mathematics.
In: M. Rosa, F. Cordero, D. Orey & P. Carranza, (Eds.), Mathematical Modelling Programs in Latin
America (pp. 247–267). Springer.
Flick, U. (2004). Introducción a la investigación cualitativa. Madrid: Morata
Marcía, S. (2020). Resignificación de la integral en una comunidad de estudiantes de docencia de lamatemática. una categoría de acumulación y la perspectiva de identidad disciplinar [Unpublished master's
tesis]. Instituto Politécnico Nacional.
Stake, R. (2007). Investigación con estudio de casos. Madrid: Morata.

94

EXPERIMENTS IN TEACHING ADVANCED CALCULUS: AN EARLYINTRODUCTION TO DOUBLE INTEGRALSJohn Gordon1 and Dawnette Blackwood-Rhoomes2
1

The City University of New York-QCC, USA; 2Tech Prep Solutions Inc., USA

Differentiation and integration are the two major divisions of calculus. Traditionally, in the UnitedStates undergraduate education system, single integration theory is introduced through a first course
in calculus, say Calculus I, via the critical topic of Riemann sums, which is then followed by proof
and application of the First and Second Fundamental Theorems of Calculus (FTC) for well-preparedstudents, say those with a solid understanding of calculus as presented in Spivak, M. (2008), or asketch and conceptual layout of the said proofs for students who might not have had such a deep,
theoretical grounding in the essentials of calculus and who might have been educated in the subject
through texts such as Varberg, D., (2007), or Stewart, J. (2003). This introduction is usually followed
by several applications of the theory involving calculating the area under curves, volumes ofrevolution, finding the length of planar curves, numerical integration, and other applicable topics.
After several other topics and perhaps more advanced calculus courses, say Calculus II,
differentiation in two- and three-dimensional spaces is introduced followed by double and triple
integration in two- and three-dimensional spaces, respectively, over rectangular and other regions.
This paper outlines strategies for experimentation with a much earlier introduction of double integrals
over rectangular regions. The authors encourage introducing the techniques of double integration
over rectangular regions immediately after students have been introduced to and practiced extensively
on applications of single integration of functions of one variable in a first introduction to calculus.
The idea is to present this topic mainly using conceptualization, analogies, and graphics, and thenmaking immediate applications to concepts such as volumes of revolution and numerical integrationof well-behaved functions of one variable. There are at least three reasons for this: (1) The audience
for the calculus sequence (Calculus I, II, and III) in the United States (and elsewhere) tends todecrease significantly with each advanced level. Consequently, many students in a general audience
who took an introductory first course miss the opportunity to see and make use of the power andapplication of advanced techniques like double integration. This strategy has the possible advantageof retaining some of these students for more advanced levels (2) Students who experience difficulties
with concepts such as the computation of volumes of revolution might (and have been observed byus) have a less difficult time doing those computations using double integration and (3) “Repetition
deepens impression,” which puts students having an earlier introduction at a distinct advantage when
they see the concepts for a second time. For these reasons, we believe this is an experiment thatmathematics educators should consider conducting.
The applications of double integrals to mathematical modeling are numerous and make an early
introduction even more worthwhile. Areas of application include density and mass, moments andcenters of mass, surface area, and probability. These are only a small fraction of possible applications
which can engage curious students and pique their curiosity and interest in modeling.
References
Spivak, M. (2008). Calculus. Publish or Perish, Inc.
Varberg, D. (2007). Calculus, early transcendentals. Pearson Education, Inc.
Stewart, J. (2003). Calculus. Thomson Learning, Inc.
Stewart, J. (2015). Calculus, 8th Edition. Cengage Learning.

95

VOCATIONAL MATHEMATICS EDUCATION IN DIFFERENT
Karolina Muhrman1 and Peter Frejd1
1Linköping University, Sweden

The members of the International Community of Teachers of Mathematical Modelling andApplications (ICTMA) have during the last 40 years developed a large set of research literature
mainly focusing on mathematical modelling (Geiger & Frejd, 2015). However, less focus has been
paid on the A in ICTMA, namely applications. The dominating use of mathematics in mostoccupations is connected to applications of mathematics. Thus, the use of realistic applications should
be a part of vocational mathematics education. However, the prevalent mathematics teaching practicein vocational education and training in upper secondary school in Sweden is based on textbooks thatmainly focuses on formal academic mathematics where the link to the labour market is weak. Many
students, therefore, do not see the relevance of using mathematics for their future vocation, which
impacts both on students’ motivation to learn mathematics and their mathematical skills (Muhrman,2016). The overall consequence is that students do not get access to the labour market, since they do
not meet the working life's requirements of mathematical knowledge. There is a national andinternational consensus among researchers of a need for research to explore teaching and learning
methods that strengthen the relation between mathematics education and students’ future occupations(e.g. FitzSimons, 2001). In particular, the problem of organising activities suitable for preparing
students to solve and critically reflect on problems arising from realistic situations at the workplace
is a current matter of research (Frejd & Muhrman, 2020).
This paper will report from an ongoing project where we aim to investigate how mathematics invocational education and training can be organised to support students for their future profession. The
project includes ten upper secondary vocational education schools in Sweden that work with teaching
and learning interventions with applications of mathematics in different contexts (in a regularclassroom and in a workplace authentic environment). Our goal is to explore the effects of theseinterventions on the students' mathematical skills and their motivation to learn mathematics. The
study has a longitudinal design using pre- and post- tests and questionnaires and interviews to collectdata before and after the interventions, and 12 and 18 months later, to investigate if the effect ispermanent. During the conference we will present and discuss our first results from our project.
References
FitzSimons, G. E. (2001). Is there a role for mathematical disciplinarity in productive learning for the
workplace? Studies in Continuing Education, 23(2), 261-276.
Frejd, P., & Muhrman, K. (2020). Is the mathematics classroom a suitable learning space for making workplacemathematics visible? – An analysis of a subject integrated team-teaching approach applied in different
learning
spaces.
JournalofVocational
Education&
Training.
https://doi.org/10.1080/13636820.2020.1760337
Geiger, V., & Frejd, P. (2015). A reflection on mathematical modelling and applications as a field of research:
Theoretical orientation and diversity. In G. A. Stillman, W. Blum, & M. S. Biembengut (Eds.),
Mathematical modelling in education research and practice: Cultural, social and cognitive influences (pp.
161–171). New York: Springer.
Muhrman, K. (2016). Inget klöver utan matematik. En studie av matematik i yrkesutbildning och yrkesliv.
(Doktorsavhandling). Linköpings universitet.

96

CONNECTIONS ESTABLISHED BY 9TH GRADERS AFTER SIMULATING
AND MODELLING THE EFFECT OF ATHEROSCLEROSIS
Susana Carreira1, Nélia Amado2 and Marli Teresinha Quartieri3
1

Universidade do Algarve & UIDEF, Instituto de Educação, Universidade de Lisboa, Portugal;
2
Universidade do Algarve, Portugal; 3Universidade do Vale do Taquari, Brasil

In this study, we aim to know the types of connections, and their preponderance, established by 9th
graders after carrying out a simulation of the accumulation of plaque in the arteries, and creating amathematical model describing the reduction of blood with increased plaque thickness. Establishingconnections, namely between real-world situations and the world of mathematics is one of thecognitive processes inherent to mathematical modelling and has been identified as one of thechallenges that students face in modelling problems. In recent years, mathematical connections havebeen investigated (e.g., Amado et al., 2019), in line with an emphasis on their importance as curricularand educational targets (e.g., OECD, 2019). Businskas (2008) sees the construction of mathematicalconnections as a process through which relationships between ideas, definitions, concepts, ortheorems (intra-mathematical connections); or relationships between these and other subjects or realcontexts (extra-mathematical connections) are established. Mathematical connections are key tomathematical modelling, especially if entailing interdisciplinary work and links to other subjects(Borromeo Ferri & Mousoulides, 2018; Blum & Niss, 1991). This qualitative study involved 83
students, from four 9th grade classes. They worked in small groups, in a total of 24 groups. The task,of an interdisciplinary nature, asked: “How would you explain to a relative the effect of fat plaqueson the volume of blood flowing in the arteries?”. To support their answer, the students had materials
to carry out a simulation and collect data: a cylindrical cup (representing an artery), flexible EVA
sheets (representing fat plaques); syringe for measuring volume; coloured water. The task was donein the mathematics class, where both the science and the mathematics teachers participated. Aftercompleting the task, the students received a sheet containing images (alluding to the experience) andwords (mathematical and non-mathematical concepts). The assignment was: “Draw all the links you
find between pairs of items. Number each link and explain what the link means”. Data were analysedusing a deductive content analysis, and different types of mathematical connections were categorized.
Results showed that students established both intra- and extra-mathematical connections, but thelatter occurred more frequently and with more detailed justifications, namely referring to thepathology of atherosclerosis. Some intra-mathematical connections were not justified by the students,suggesting insufficient knowledge of some mathematical concepts, namely the quadratic function.
References
Amado, N., Carreira, S., & Canavarro, A. P. (2019). Sobre o tema conexões matemáticas. Livro de Atas do
EIEM 2019 - Encontro de Investigação em Educação Matemática (pp. 3–6). Loulé, Portugal: SPIEM.
Blum, W., & Niss, M. (1991). Applied mathematical problem solving, modelling, applications, and links toother subjects. State, trends and issues in mathematics instruction. Educational Studies in Mathematics, 22,
37–68. https://doi.org/10.1007/BF00302716
Borromeo Ferri, R., & Mousoulides, N. (2018). Mathematical modelling as a prototype for inter-disciplinary
mathematics education? Theoretical reflections. In T. Dooley & G. Gueudet (Eds.), Proceedings of CERME
10 (pp. 900–907). Dublin, Ireland: DCU Institute of Education and ERME.
Businskas, A. (2008). Conversations about connections: How secondary mathematics teachers conceptualise
and contend with mathematical connections. Unpublished doctoral dissertation. Simon Fraser University,
Burnaby, Canada.
OECD (2019). PISA 2018 Assessment and Analytical Framework. OECD Publishing.
https://doi.org/10.1787/b25efab8-en

97

USING A SCIENTIFIC CALCULATOR FOR MATHEMATICAL
MODELLING AND APPLICATIONS IN SECONDARY SCHOOL
Barry Kissane
Murdoch University, Australia

Attention to applications of mathematics in secondary schools has morphed over recent decades into
attention to mathematical modelling, perhaps motivated by students themselves and perhaps by others
keen to see how mathematics might be useful. By the nature of secondary schooling, the mathematics
involved is generally relatively unsophisticated. In this paper, the appropriateness of modern
scientific calculators as tools for students and teachers will be assessed and exemplified.
Many formulations of the mathematical modelling process use the term ‘solve’ to describe themathematics involved. In this sense, solving is a key part of both illustrative applications ofmathematics and of mathematical modelling itself. (reSolve Project (2023); Galbraith and Holton(2023)) Usiskin (2011) makes clear that early steps in learning about mathematical modelling involve
careful exposure to mathematical applications, which inevitably require mechanisms of dealing withmathematical representations and their elaborations, including sensitivity analyses.
For secondary school students in many countries, modern scientific calculators provide a rich yet
affordable toolkit for most of the mathematical processes required for first steps in modelling andapplications. The toolkit includes representation and evaluation of user-defined functions, tabulationof functional values, numerical solution of equations, numerical calculus, spreadsheets, elementary
statistical analysis and random processes. While other technologies provide more powerful support(such as graphical support through graphics calculators and algebraic support via computer algebra
systems), the real circumstances of many students do not permit everyday access to these in secondary
schools, either because they are too expensive for universal use or because they are effectively
disallowed by curriculum authorities. So the scientific calculator is an ideal technology for getting
started, to be followed at a later stage by more sophisticated technologies.
The modern scientific calculator is a suitable technology to support early steps in mathematicalmodelling and applications of mathematics in the secondary school, because its functionality has been
designed explicitly to support such work, in the wider economic and curriculum contexts in which
many teachers and students are located. Examples to illustrate both the nature and scope of this
support will be provided in the paper.
References
Galbraith, P. and Holton, D. (2023) Mathematical modelling: A guidebook for teachers and teams. Accessedat https://www.immchallenge.org.au/supporting-resources
reSolve Project (2023). ST7: Teachers’ Guide to Mathematical Modelling. Australian Academy of Science.
Accessed at https://www.resolve.edu.au/mathematical-modelling-teachers-guide?special_topic=1491 .
Usiskin, Z. (2011). Mathematical Modeling in the School Curriculum.
http://ucsmp.uchicago.edu/resources/conferences/2011-09-11/

98

MAKING STUDENTS BECOMING RESPONSIBLE CITIZENS IN 21ST
CENTURY BY FOSTERING MATHEMATICAL REASONING ABOUTREAL-WORLD APPLICATIONSMaike Hagena1, Dominik Schlüter2, Michael Besser2 and Dominik Leiss2
1

University of Hamburg, Germany; 2Leuphana University of Lueneburg, Germany

Promoting students becoming responsible citizens in 21st century is accepted being a central aim ofeducation (see e. g. European Commission, 2019) and implemented in political documents forteaching and learning all over the world (see e. g. NCTM, 2000). With respect to teaching and learning
mathematics at school, this special goal is addressed by highlighting the importance of mathematicalreasoning about real world applications for students’ everyday life: “The ability to reason logicallyand present arguments in honest and convincing ways is a skill that is becoming increasingly
important in today’s world” (OECD, 2018, p. 15). Based on these considerations, it is of special
interest fostering students mathematical reasoning about real-world applications – and it is assumed
that building up both competencies on mathematical modelling (CMM) in particular (Kaiser, 2020)
as well as competencies on critical thinking (CCT) in everyday life in general (Lai, 2011) can support
students’ learning. However, empirical evidence about the impact of CMM and CCT on students’mathematical reasoning about real-world applications is rare. The present study harks back to thisdesideratum and analyses empirically, whether a special training on both CMM and CCT can foster
students’ competencies on mathematical reasoning about real-world problems.
In 2015/2016, overall 380 seventh-grade students (average age: 13 years) from 9 German middle
schools took part in the research project FASAF (principal researchers: D. Leiss, A. Neumann, K.
Schwippert). The research project aimed at fostering students’ competencies on mathematicalreasoning about real-world applications by offering to different types of weekly training, each lastingfor 6 months: a training dealing with CMM and CCT separately (experimental group A; EG A; n =82) and a training dealing with CMM and CCT simultaneously (experimental group B; EG B; n =75). Additionally, a waiting control group (CG; n = 223 students) was part of the project. Tests onmathematical reasoning about real-world applications have been administered in a classical pre-postdesign. Comparing EGA, EG B and CG with respect to these performance tests offers insights into
the effect of those different trainings. These results will be discussed at ICTMA; implications forteaching and learning mathematical reasoning about real-world applications as well as about making
students’ becoming responsible citizens will be highlighted.
References
European Commission (2019). The European qualifications framework – Supporting learning, work andcross-border mobility: 10th anniversary. Publications Office. https://data.europa.eu/doi/10.2767/385613
Kaiser, G. (2020). Mathematical modelling and applications in education. In S. Lerman (Ed.), Encyclopediaof Mathematics Education (pp. 553–561). Springer, Cham.
Lai, E. R. (2011). Critical Thinking: A Literature Review. Pearson’s Research Report, 6 (40-41).
NCTM (2000). Principles and standards for school mathematics. Reston: National Council of Teachers ofMathematics.
OECD (2018). PISA 2022 mathematics framework (draft). https://pisa2022-maths.oecd.org/files/
PISA%202022%20Mathematics%20Framework%20Draft.pdf

99HANDS-ON WORKSHOP: PARTICIPANTS BUILD A SMALL HUMANCARRYING CAPACITY FOR EARTH MODEL IN FREE STELLA ONLINE
Diana M. Fisher
Portland State University, USA

Complex systems are ubiquitous in our world, from climate change, to human physiology, to how theteenage mind operates. Having students or professionals build computer models to capture thebehavior of complex systems has been shown to aid them in understanding how complex systems
operate and test policies to attempt to find leverage points that could modify the behavior of thesesystems (Hung, 2008). If we want to provide opportunities for students to develop the skill ofanalyzing complex systems we must win the hearts and minds of the teachers, who ultimately control
what is taught in the classroom (Tondeur et al., 2019). Research shows that providing hands-on
experience with new technologies can be valuable in reaching teachers at a level that can expand theirbelief system to include broader instructional strategies in their teaching (Beaudin et al., 1997).
In the 2019 IMMC competition the question to estimate the human carrying capacity of the earth
secondary school students did not perform well (Garfunkel et al., 2021). A major issue was, inattempting to mathematize this problem, attention to non-linear dependencies, interconnections, andfeedback was needed, suggesting that simulation would be necessary to perform the analysis(Galbraith & Fisher, 2021). Secondary school students have shown that they can build and analyze
complex systems using System Dynamics (SD) software to develop original models (Fisher, 2018).
In order to provide teacher educators with experience building at least one model using the free, web
based SD software that K-12 students could have used to analyze the 2019 IMMC question a shorthands-on workshop has been prepared. Participants will build a World Population-Temperaturemodel and potentially a second model to gain exposure to a tool that is readily available in the K-12
schools using either laptops, tablets, or Chromebooks. Multiple articles have already occurred in the
ICTMA Proceedings publications explaining the SD method and giving numerous examples.
References
Beaudin, M., & Bowers, D. (1997). Logistics for facilitating CAS instruction. In J. Berry, J. Monaghan, M. Kronfellner,& B. Kutzler (Eds.), The state of computer algebra in mathematics education (pp. 126-135). Lancashire, England:
Chartwell-York.
Fisher, D. M. (2018). Reflections on teaching system dynamics modelling to secondary school students for over 20
years. Systems Journal Special Edition: Theory and Practice of System Dynamics Modelling, 6(12). Available
at: http://www.mdpi.com/2079-8954/6/2/12/htm.
Galbraith, P. L. & Fisher, D. M. (2021). Technology and mathematical modeling: addressing challenges, opening
doors. In: S. Carreira, & W. Blum. (Eds.) Mathematical Modelling in the Teaching and Learning of Mathematics:
Part 1. Quadrante. 30, (1): 198-218.
Garfunkel, S., Niss, M., & Brown, J. P. (2021). Opportunities for modelling: An extra-curricular challenge. In F.K.S.
Leung, G.A. Stillman, G. Kaiser, & K.L.Wong (Eds.), Mathematical Modelling Education in East and West (pp.
363-375). Springer, Cham.
Hung, W. (2008). Enhancing systems‐thinking skills with modelling. British Journal of Educational Technology, 39(6),1099-1120.
Tondeur, J., Scherer, R., Baran, E., Siddiq, F., Valtonen, T., & Sointu, E. (2019). Teacher educators as gatekeepers:
Preparing the next generation of teachers for technology integration in education. British Journal of Educational
Technology, 50(3), 1189-1209.

100

STUDENTS’ MODELLING ACTIVITIES IN THE INQUIRY ON THE SLIDERULE
Tatsuya Mizoguchi1, Kokoro Okada1, Berta Barquero2 and Marianna Bosch2
1

Tottori University, Japan; 2Universitat de Barcelona, Spain

There has been a growing focus on inquiry learning in educational contexts internationally (Artigue& Blomhøj, 2013). Although there is also an emphasis on inquiry in the recently revised Japanese
Course of Study, it seems different from the international context, with more attention paid to a better
application of the knowledge and skills to be learnt than to the inquiry process itself. This may be
attributed to the assumed didactic paradigm in Japan, closer to the paradigm of visiting works than tothe one of questioning the world, as claimed by the Anthropological Theory of the Didactic (ATD)(Chevallard, 2019). We use the proposal of study and research paths (SRPs) as a model of inquiry,
which is proposed in the framework of the ATD. SRPs emphasise the dialectic between the generationof questions to be addressed by the inquirers and the search and construction of models to answer
them. This dialectic of questions and answers provides a first description of the structure of the inquiry
process, as well as the milestones for the paths foreseen during the inquiry (Bosch, 2018).
Our contribution considers university students’ modelling activities carried out in an SRP about thefunctioning of “slide rules” for multiplying and dividing decimal numbers. The research question we
focus on is about: how can the university students’ modelling activities in the process of inquiring on
the slide rule be characterised through a question-answer dialectic? The SRP has been implemented
in a teacher education programme in Japan in spring 2023. It consisted of five sessions of group
activities. Session 1 was an exploration about what can be done with a slide rule. In Sessions 2 and3, students inquired into the following questions: How does a slide rule work? Which scales are
combined to carry out what calculations? How is each scale constructed? Session 4 was devoted to
students’ self-analysis of their inquiry process. The didactic tool of the question-answer maps (QAmaps) (Winsløw et al., 2013) was used for the analysis of the inquiry and modelling processes
followed. Finally, Session 5 consisted of a poster session where the groups of students presented their
final results.
Our analysis in this teacher education activity is primarily carried out based on the QA maps initially
proposed by the students. On the one hand, we focus on the models used by students to explain the
different calculations, mostly using logarithms, which is the basic concept of the slide rule. On theother hand, we look at how the question-answer dialectic differed from one group to another,
especially with regard to the modelling paths taken and their productivity.
References
Artigue, M., & Blomhøj, M. (2013). Conceptualizing inquiry-based education in mathematics. ZDMMathematics Education, 45(6), 797–810.
Bosch, M. (2018). Study and research paths: A model for inquiry. Proceedings of International Congress ofMathematicians – 2018 Rio de Janeiro, Vol. 3, 4001-4022.
Chevallard, Y. (2019). Introducing the anthropological theory of the didactic: An attempt at a principledapproach. Hiroshima Journal of Mathematics Education, 12, 71–114.

101

TEACHER EDUCATION IN MODELLING:
MODELS, SIMULATION AND NEW EPISTEMOLOGICAL NEEDS
Berta Barquero1, Marianna Bosch1, Susana Vásquez1
1
Universitat de Barcelona, Spain
Teacher education at all school levels might take into account the epistemological and didactic needsof the teaching profession. These needs are not always well known, as they often correspond toimportant constraints created by the new curricular demands to teach mathematics as a modelling
tool in inquiry approaches. The collaboration between teachers and researchers to elaborate
epistemological and didactic tools to overcome these constraints is an open question in our field. In
our paper, we address this question from the perspective of the Anthropological Theory of theDidactic (ATD) through the proposal of study and research paths (SRPs) and SRPs for teacher
education (Barquero et al., 2018; Bosch, 2018). The experiences carried out with SRPs and SRPs-TE
by our team at Universitat de Barcelona will serve as a basis for a reflection on the role ofepistemological tools based on the notions of systems, models and simulation in helping teachers
manage didactic processes based on modelling.
More concretely, we choose two case studies of SRPs implemented at secondary and university levels,
respectively. On the one hand, an SRP about “How sure are some padlocks? (Vásquez et al., 2021)
will be used to introduce two essential properties of modelling: those of recursivity and ofreversibility. We present examples of students’ modelling processes to illustrate the above mentioned
properties and analyse their role in the construction and interpretation of models. The implementationof this SRP, after 4 consecutive years, brings light to the tools shared by the team of teachers to build
a common infrastructure to talk, manage and analyse the students modelling processes. The second
SRP is generated by the question “What is school segregation and how is it measured?”. It aims toshow the complex relationships between systems and models in a different situation: when one finds
already-made models (like a formula) and aims at studying and developing them. In this case, themodelling process starts with the consideration of an external model that needs to be tested withdifferent systems, some of them artificial or simulated. The particular role of simulations in the
production and development of models thus appears as an important focus of the analysis.
References
Barquero, B., Bosch, M., & Romo, A. (2018). Mathematical modelling in teacher education: dealing withinstitutional constraints. ZDM Mathematics Education, 50(1-2), 31-43.
Bosch, M. (2018). Study and research paths: A model for inquiry. Proceedings of International Congress ofMathematicians – 2018 Rio de Janeiro, Vol. 3, 4001-4022.
Vásquez S., Barquero B., & Bosch, M. (2021). Teaching and learning combinatorics in secondary school: amodelling approach based on the ATD. Quadrante, 30(2), 200-219.

102

HOW CAN ALEXA UNDERSTAND US? SPEECH RECOGNITION AND AI
IN MATHEMATICS EDUCATION
Stephanie Hofmann1 and Martin Frank2
1

Karlsruhe Institute of Technology, Germany; 2Karlsruhe Institute of Technology, Germany

The rapid development in the field of artificial intelligence (AI) has led to amazing advances innatural language processing (NLP) in recent years. From intelligent voice assistants like Siri andAlexa to chatbots like ChatGPT, we are surrounded by NLP technologies on a daily basis. Given
these technological developments, it seems essential to have a basic understanding of the underlying
mathematics in order to comprehend and critique how these systems work (Long & Magerko, 2020).
In this talk, we will use the example of speech recognition to show that the topic of NLP offers many
links to school mathematics and is thus suitable for taking up highly relevant, authentic applications
in mathematics education. It is shown how a suitable didactic reduction of the underlying
mathematical models enables the learning of basic strategies of machine learning - a collection of
methods in the research field of AI - as well as the application of school mathematical concepts inthe field of NLP. We present the didactic reduction and a possible instructional progression thatenables students aged 16 and up to acquire the mathematical foundations of single word recognition.
Four steps were identified as essential in single word recognition: preprocessing of the speech signal,feature extraction, classification and the evaluation of the classification. The first step is primarily
concerned with the processing and visualization of the data. The speech signals are available in time
period and can be converted into an amplitude spectrum using the Fourier transform. This can bedone using suitable digital tools as a black box. Alternatively, however, the Fourier transform canalso be worked out with students using the example of triads as a decomposition of the signal into
individual sinusoidal oscillations (Wohak & Frank, 2021). In feature extraction, typical features inthe spectrum are to be identified. For specified time steps, the local maxima as the typical features
can be extracted and stored in a vector. Subsequently, a first mathematical model for classification isdeveloped. For this purpose, the feature vectors of signals for which the word meaning is known are
combined to classes of the same word meaning. If a signal is to be classified whose word meaning is
still unknown, the Euclidean distance to the feature vectors of every class is calculated. Then, classby class, the mean value of the distance is calculated and the signal is assigned to the class with thesmallest mean value. As a model improvement, the distance between the feature vectors can be
determined using Dynamic Time Warping to account for the temporal variability of the speech signals(Pfister & Kaufmann, 2017). Finally, the classification result can be evaluated with the learners. Here,the assessment tool of the confusion matrix, which is common for machine learning methods, can beused and the reference to four-field tables, which are typically covered in school lessons, can be made.
References
Long, D. & Magerko, B. (2020). What is AI Literacy? Competencies and Design Considerations.
Pfister, B. & Kaufmann, T. (2017). Sprachverarbeitung. In Springer eBooks.
Wohak, K. & Frank, M. (2021). Compressing audio signals with interactive and cloud-based learningmaterial—a workshop for high-school students. Teaching Mathematics and its Applications: An
International Journal of the IMA.

103

DIDATIC REVISION STRATEGY FOR DESIGNING MATHEMATICAL
MODELING TASKS USING AI TECHNOLOGY
Dong-Joong Kim1, Hee-jeong Kim1, Won Kim1, Young-Seok Oh2 and Gima Lee2
1

Korea University, Korea; 2Graduate School of Education, Korea University, Korea

Mathematical modeling, which involves abstracting mathematical concepts from and applying themto real-life contexts, not only enhances students’ mathematical competency but also fosters anappreciation for the usefulness of mathematics (Gravemeijer & Doorman 1999). Thus, it has been
adopted in the national mathematics curriculum of many countries. However, transforming real-life
problems into modeling tasks suitable for secondary school mathematics classes presents challenges
due to complexity, intricate data analysis, and the added teacher burden (Borromeo Ferri, 2018). AI
technology can provide a solution to these challenges, assisting teachers in data-related difficulties
and modeling task formation. To explore how teachers utilize AI technology to design mathematicalmodeling tasks, we developed an expanded Didactic Revision (DR) strategies framework. The
development of this framework was created based on previous studies (Ahn & Yu, 2023; Kim et al.,2021; O'Neil & Schutt, 2013), and then revised by math education experts and AI math experts.
Our framework includes 8 distinct DR strategies: five from previous study (Kim et al., 2021) andthree new ones identified in this study: (1) DR-S strategy: revising complicated computations for
Simpler problems; (2) DR-B strategy: Breaking a task into smaller subtasks; (3) DR-A strategy:
revising Assumptions in the context of the task for articulation; (4) DR-W strategy: using Wording
appropriate for student levels; (5) DR-N strategy: revising Notations, such as symbols that can cause
student misunderstandings or mistakes; (6) DR-D strategy: specifies criteria for Data collection andprocessing so that students can collect and process the Data needed to construct a mathematicalmodel; (7) DR-PM strategy: provides criteria for ProMpts to discuss with a chatbot (e.g., ChatGPT)
allowing students to monitor their task-solving progress; and (8) DR-PG strategy, which encourages
students to use ProGramming tools to explore collected data or to construct and compute amathematical model. In our presentation, we will provide how this applies to data in which preservice
teachers design modeling tasks with AI tools including GhatGPT, Orange3. Our study is significant
because the framework provides pedagogical guidance for teachers designing mathematical modelingtasks using AI technology, particularly crucial in this era emphasizing both mathematical modelingand AI.
References
Ahn, S., & Yu, W. (2023). Introduction to natural language processing using deep learning. Retrieved fromhttps://wikidocs.net/book/2155
Borromeo Ferri, R. (2018). Learning how to teach mathematical modeling: In school and teacher education.
New York: Springer.
Gravemeijer, K., & Doorman, M. (1999). Context problems in realistic mathematics education: A calculus
course as an example. Educational Studies in Mathematics, 39, 111–129.
Kim, D., Choi, S., Lee, Y., & Lim, W. (2021). Secondary teacher candidates’ mathematical modeling task
design and revision. Mathematics, 9(22), 1-15.
O’Neil, C., & Schutt, R. (2013). Doing data science. Cambridge: O’Reilly Media.

104

LEARNING THEORY BELIEFS ABOUT TEACHING SIMULATIONS AND
MATHEMATICAL MODELLING WITH DIGITAL TOOLS
Jascha Quarder1, Sebastian Gerber2, Hans-Stefan Siller2 and Gilbert Greefrath1
1

University of Münster, Germany; 2University of Würzburg, Germany

In simulations and mathematical modelling, as well as in the use of digital tools, a student-centred
approach is accepted as an important teaching method in mathematics education (Schukajlow et al.,2012; Thurm & Barzel, 2022). A substantial influence on instructional design is attributed to beliefs
of mathematics teachers (Voss et al., 2013). Beliefs are part of teachers' professional competence(Baumert & Kunter, 2013) and can be categorised into constructivist and transmissive beliefs from alearning theory perspective. Teachers with stronger constructivist beliefs more often integrate tasks
that need to be actively worked on by the learners. Teachers with stronger transmissive beliefs
understand learners as passive recipients who absorb knowledge from the teacher (Voss et al., 2013).
Transmissive beliefs therefore often don’t support the independent, process-oriented approach insimulations and mathematical modelling with digital tools, while constructivist beliefs favour it.
This chapter describes domain-specific (i.e. in the area of using of digital tools in simulations andmathematical modelling) beliefs by concretising global constructivist and global transmissive beliefs
of mathematics teachers. We present the conception of a mathematics education course offered at the
Universities of Münster and Würzburg. It aims to promote professional competence for teaching
simulations and mathematical modelling with digital tools and therefore also intends to build up
described constructivist beliefs and reduce described transmissive ones. In an intervention study withtwo measurement points, we investigate to what extent participation in the course can initiate changes
in the mentioned beliefs. For this purpose, data from N = 204 pre-service mathematics teachers werecollected and quantitatively evaluated using two self-developed belief scales (statements with a Likert
scale each) and, among other things, t-tests. Initial results show significant effects on the beliefs infavour of the intervention: They indicate that the domain-specific constructivist beliefs were built up
and the transmissive ones were reduced and that this can be attributed to the course participation.
Based on the evaluations, its effectiveness and implications for teacher education are discussed.
References
Baumert, J., & Kunter, M. (2013). The COACTIV model of teachers’ professional competence. In M. Kunter,
J. Baumert, W. Blum, U. Klusmann, S. Krauss, & M. Neubrand (Eds.), Cognitive activation in themathematics classroom and professional competence of teachers: Results from the COACTIV project (pp.
25–48). Springer US.
Schukajlow, S., Leiss, D., Pekrun, R., Blum, W., Müller, M., & Messner, R. (2012). Teaching methods formodelling problems and students’ task-specific enjoyment, value, interest and self-efficacy expectations.
Educational Studies in Mathematics, 79(2), 215–237.
Thurm, D., & Barzel, B. (2022). Teaching mathematics with technology: A multidimensional analysis ofteacher beliefs. Educational Studies in Mathematics, 109(1), 41–63.
Voss, T., Kleickmann, T., Kunter, M., & Hachfeld, A. (2013). Mathematics teachers’ beliefs. In M. Kunter, J.
Baumert, W. Blum, U. Klusmann, S. Krauss, & M. Neubrand (Eds.), Cognitive activation in themathematics classroom and professional competence of teachers: Results from the coactiv project (pp.
249–271). Springer US.

105

CONSTRUCTING MEANING FOR INTERACTIVE MATHEMATICAL
MODELS WITHIN A DIALECTICAL LEARNING ENVIRONMENTElena Naftaliev
Achva Academic College, Israel

Modelling activities in mathematics have changed in the last decades because of technologicaldevelopment. Using technology to design interactive materials provides a captivating, engaging tool
which encourages learners to explore mathematical models and to devise their own models. Students
need experience with mathematical models to understand the point of mathematical modelling, thatis, its “language” (Schwartz, 2007). Once such models exist in the cognitive “baggage” of learners,
they also become a tool for active mathematical modelling (Wilensky, 1999). While students learn
about dynamic processes and about the mathematical models of the processes, it is especially
important that the materials be represented in a similarly dynamic way by real-world models, such asanimations, and interactive mathematical models (Schwartz, 2007; Naftaliev, 2017).
This presentation focuses on the investigation of interactive materials that encourage a dialectic
process of building meaning for complex and unknown mathematical models by learners. The designof the materials focused on generic combinations of accelerated motion. It was intended to support
the development of awareness of kinematic phenomena (such as rate of change and constant andchanging speeds) and the analysis of their mathematical models. The students identified visual andkinematic conflicts and so reflected on their current mathematical understanding and modified itusing different resources and facing the constraints designed in the materials.
The animation served as a tool for students to reflect on their currently dominant spontaneous
structures mainly because it presented a surprising or contradictory situation to students, who did not
yet understand the aspects of motion, and when the models in the diagram were not significant enoughfor them. But the animation served as a tool for modifying the spontaneous structures because it acted
as an exploration and inquiry tool; running the animation was useful for noticing phenomena andverifying conjectures. The dialectic process emerged from the need to resolve the tensions between
the learners' mathematical understandings and the information presented in the interactive materials.
The animation was a tangible real-world model that helped build meaning for the linked mathematicalmodels, and the linked models acted as tools to characterize the different aspects of the motion process.
References
Naftaliev, E., (2017). Interactive Diagrams Used for Collaborative Learning Concerning Mathematical Modelsof Motion. In Stillman, G., Blum, W. & Kaiser, G. (Eds), Mathematical Modelling and Applications:
Crossing and Researching Boundaries in Mathematics Education, 553-563. Springer.
Schwartz, J. (2007). Models, simulations, and exploratory environments: A tentative taxonomy. In R. Lesh, E.
Hamilton, & J. Kaput (Eds.), Foundations for the future in mathematics education, 161–172. Mahwah:
Erlbaum.
Wilensky, U. (1999). GasLab? An extensible modelling toolkit for exploring micro- and macro- views of gases.
In N. Roberts, W. Feurzeig, & B. Hunter (Eds.), Computer modelling and simulation in science education,
151-178. Berlin: Springer-Verlag.

106

ADVANCING THE STUDY OF A MATHEMATICIAN’S MODELLINGPROGRESS DISPLAYED WITH APPLIED RESPONSE ANALYSISMAPPING
Akio Matsuzaki
Saitama University, Japan

Processing in modelling are different from each modeller because their experiences are different, andchanging models are influenced by some situations. As previous studies I have already researcheddifferent subjects and some case studies (for example, graduate students or working adult and so on).
Research question is as follow: How does an expert in mathematics make progress modelling? I
reported a part of the modelling of a mathematician (Matsuzaki, 2019), and continue my study todisplay all of his modelling in this presentation. I have studied to chase modelling progress withapplied response analysis mapping (Matsuzaki, 2011, 2018). Applied response analysis mapping is amethod to display processing in modelling progress. One of characteristics of this method is reflected
situations, which are used for broad meanings that are imaged based on each modeller’s experiences.
Components of the modelling based on prior experiences of modeller are identified as follows:
components based on real experiences (CRE) and components based on mathematical experiences(CME). Links between CRE and CME during modelling have been confirmed by this method.
Problems for subjects are related to electronics: How much brightness is needed to read a book? I
asked for subjects the following method to answer the problems: to decide conditions, pose models
and solve problem. Adding to I asked the mathematician to talk about situations by think-aloudmethods through solving problems. We can see the modelling progress displayed with applied
response analysis mapping reflects some components (CRE & CME) of experiences of themathematician, and that processing is affected by situations.
References
Blum, W., & Leiß, D. (2007). How do students and teachers deal with modelling problems? In C. Haines, P.
Galbraith, W. Blum, & S. Khan (Eds.), Mathematical modelling (ICTMA12) (pp. 222–231). Chichester:
Horwood.
Matsuzaki, A. (2011). Using response analysis mapping to display modellers’ mathematical modelling
progress. In G. Kaiser, W. Blum, R. Borromeo Ferri, and G. Stillman. (Eds.), Trends in Teaching and
Learning of Mathematical Modelling: ICTMA 14 (International Perspectives on the Teaching and
Learning of Mathematical Modelling) (pp. 499-508). Springer.
Matsuzaki, A. (2018). Study of Mathematical Modelling Competencies Focused on Gen-Bamen: Using the
Method of ‘Phänomenologie’ by Hussel, E. in the Middle Later Period and Applied Response Analysis
Mapping. Kyoritsu syuppan.
Matsuzaki, A. (2019, July). A Mathematician’s modelling progress displayed with applied response analysis
mapping, Oral presentation at 19th International Conference on the Teaching of Mathematical Modellingand Applications (ICTMA19).

107

ANALYSING ‘STUDENT-TO-STUDENT-TO-TEACHER’ COLLABORATIVELEARNING PRACTICES IN A PRE-SERVICE APPLICATIONS AND
MODELLING COURSE
George Ekol
University of Witwatersrand, South Africa
In South Africa, as in many other countries, mathematical modelling and its applications in real life
situations are recognised in the secondary school curriculum (DBE, 2011). However, modelling is
not yet uniformly taught in all secondary schools in the country, and it is not assessed in the national
examinations. As universities train more secondary mathematics teachers to teach modelling, it isexpected that more awareness of, and interest in the topic will be created in the schools and in thecommunities outside the school. The current study contributes to building the modellingcompetencies of pre-service teachers (PST) and preparing them for secondary mathematics teaching.
The purpose of the study is to analyse features of ‘student-to-student-to-teacher’ (S-S-T) collaborative
practices that occur frequently when the PSTs solve an open-ended mathematical modelling task, andwhether such practices enhance or limit PSTs modelling solutions. The research question I address
is: What features of collaborative practices dominate PSTs solutions of an open-ended modelling task,
and how do those practices enhance or limit the quality of their solutions?
Participants in the study are 52 PSTs in their final year who have already been introduced to theprinciples of mathematical modelling (Blum and Leiss, 2007) for 10 hours of two-two hours oflectures, and two-three hours of tutorials per week for two weeks. After two weeks, the PSTs wereassigned an open modeling task, the “Giant shoes” problem (see Blum & Ferri, 2009, p.45). They
were to consult among themselves, but submit an individual report of about 1000 words, describing
the collaborations that they enlisted from peers toward solving the task. The teacher/researcher was
also available to the PSTs for consultation; however, apart from providing the minimum guidelines
given to the PSTs at the beginning, and responding to general questions of clarification, theteacher/researcher remained largely in the background.
Collaborative learning has been associated with positive effect on students’ mathematics learning(e.g., Göksen-Zayim et al., 2022). Discussions in small groups can also promote the development ofmodelling competencies (Durandt et al. 2022; Geiger et al., 2022; Göksen-Zayim, et al., 2022; Maaβ,2006). One of the difficulties for teachers/researchers is to observe the learning that occurs when
students work collaboratively in small groups. In this study, I use collaborative learning framework(Göksen-Zayim, et al., 2022) to analyse PSTs written solutions and based on the analysis, to identify
practices that enhance, and those that limit PSTs modelling solutions.
References
Durandt, R., Blum, W., & Lindl, A. (2022). Fostering mathematical modelling competency of South African
engineering students: which influence does the teaching design have? Educational Studies in Mathematics,
109(2), 361-381.
Geiger, V., Galbraith, P., Niss, M., & Delzoppo, C. (2022). Developing a task design and implementation
framework for fostering mathematical modelling competencies. Educational Studies in Mathematics,
109(2), 313-336.
Göksen-Zayim, S., Pik, D., & van Boxtel, C. (2022, February). An observation instrument to analyse thecollaboration between students while solving mathematical modelling problems. In Twelfth Congress of
the European Society for Research in Mathematics Education.

108

PRE-SERVICE TEACHERS’ PERCEPTION ON THEIR ROLES IN THE
TEACHING OF MATHEMATICAL MODELLING
Xiaoli Lu1 and Yan Zhu2
1

East China Normal University, China; 2East China Normal University, China

This study was situated in a programme for mathematics teacher education to prepare the professional
competencies of teaching mathematical modelling for pre-service mathematics teachers at aprestigious university for teacher education in China. In the programme, 36 pre-service mathematics
teachers received the training of how to teach mathematical modelling at secondary schools, which
followed Borromeo Ferri’s (2018) four competence dimensions for teaching modelling including
theoretical dimension, task-related dimension, teaching-related dimension, and diagnostic dimension.
Especially, at the end of programme, the pre-service teachers were involved into a modelling activity,
working with 63 Grade eight students to tackle with three modelling tasks in groups, during which
they were able to transfer the competence they obtained from the programme to practice.
This paper presented the pre-service teachers’ roles played in the modelling activity. According to
the design of the activity, the pre-service teachers should take one of three roles when the students
were working on one modelling task, and the students worked on three tasks in total. The three roles
were: (1) the main lecturer for the whole class who organised the procedure of modelling anddemonstrated related concepts, skills and etc, through following the instruction which mainly
designed by the main lecturer; (2) an inventor for a student group who sit with a group of 5 to 6
students to provide inventions when necessary to offer assistance when the students doing a modellingtask; and (3) an observer for a student group who sit aside the students to observe their work, his/her
peer’s intervention, and the interactions between students and the inventor. Since there were threemodelling tasks, each pre-service teachers could take at least two roles in the whole modelling activity.
As such, the pre-service teachers could be categorised by their roles in the modelling tasks. Six types
were identified: the pre-service teacher who taking all the three roles, the roles of main lecturer andinventor, the roles of main lecturer and observer, the roles of inventor and observer, the role ofinventor only, and the role of observer only. Interviews were delivered to the different types of preservice teachers to gain their perceptions about the different roles they took when working with school
students on mathematical modelling, such as perceptions on the effectiveness of teacher
demonstration and intervention, on the relationship between demonstration and intervention, and on
the categories and functions of teacher intervention etc. The interview data were analysed through aqualitative text analysis approach (Kuckartz, 2014). The results may reveal not only pre-service
teachers’ reflection on their performances on the modelling instruction, but also a systematic
approach to promoting the professional competence of teaching modelling for pre-service
mathematics teachers.
References
Borromeo Ferri, R. (2018). Learning how to teach mathematical modelling in school and teacher education.
Springer.
Kuckartz, U. (2014). Qualitative text analysis: A guide to methods, practice and using software. New York:
Sage.

109

CHARACTERIZING MATHEMATICAL MODELLING TASKS IN
EMPIRICAL LITERATURE
Rose Mary Zbiek1, Xiangquan Yao1, M. Kathleen Heid1 and Matthew Victor Black1
1

Pennsylvania State University, USA

Mathematical modelling is a process that uses mathematics to represent, analyze, make predictions,or otherwise provide insight into real-world phenomena. Researchers have developed various
diagrammatic representations to characterize the process (e.g., Blum & Leiß, 2007). Althoughdifferent in detail, these diagrams share the common perspective that modelling involves simplifying
and structuring, mathematizing, working mathematically, interpreting, and validating. Thesemodeling activities occur in specific task environments and therefore are shaped by the tasks(Zaslavsky, 2005). Tasks as a dimension of teacher competence (Borromeo Ferri, 2018) underscores
the centrality of tasks to modelling and learning to model and how a specific modeling task might
prompt or hinder modelers’ activity. Implicitly if not explicitly, tasks communicate to students how
teachers and curriculum developers conceptualize modelling. As a result, it is important to examine
the features of mathematical modeling tasks used in research and practice.
This study examined mathematical modelling tasks that were used as instruments for data collectionin empirical studies published in major mathematics education journals. We identified 109
mathematical modeling tasks from research articles on modelling that were published from 2011 to2020 in top-tier mathematics education research journals. Noting that many articles included little
information about implementation of the modelling tasks, our analysis focused on the opportunities
to model afforded by the task statements rather than on potential solutions. For each modelling task,
we considered each phase of the modelling process as we analyzed whether the task requires modelers
to make assumptions and identify variables; whether data, formula, or worksheet are provided in thetask statement; whether it specifies mathematical content, representations, or tools to be used; andwhether the task statement requires modelers to refine and validate their solutions.
Results reveal that few tasks demanded modelers to engage in full modeling cycles and few tasks
demand modelers to refine and validate their solutions. Moreover, most task statements did not
specify mathematical content, representation, and tools to be used. In most cases, the striking
differences among modelling tasks were in their demand for modelers to make assumptions andidentify variables. Informed by these results, we developed different profiles of modelling tasks.
Results from this study not only suggest the existence of different opportunities to model afforded bymodelling tasks in the literature but also inform the design of fully open-ended modelling tasks.
References
Blum, W., & Leiß, D. (2007). How do students and teachers deal with modelling problems? In C. Haines, P.
Galbraith, W. Blum, & S. Khan (Eds.), Mathematical modelling: Education, engineering and economics—
ICTMA 12 (pp. 222–231). Chichester: Horwood.
Borromeo Ferri, R. (2018). Learning how to teach mathematical modeling in school and teacher
education. (n.p.): Springer International Publishing.
Zaslavsky, O. (2005). Seizing the opportunity to create uncertainty in learning mathematics. Educational
Studies in Mathematics, 60, 297-321.

110

A REVIEW OF RESEARCH ON MODELLING, BIG DATA AND
EVIDENTIARY PRACTICES
Nina Unshelm2 Vince Geiger1, Hans-Stefan Siller2, Sarah Digan1 and Andre Greubel2
1

Australian Catholic University, Australia; 2University of Wurzburg, Germany

We live in an age of disruption, including the COVID-19 pandemic, climate change, food security,and poverty (Maass et al., 2019). These events have increased focus on sustainability in government
policy, industry, and education internationally (see UN General Assembly, 2015). For example,
energy, resources and environmental change are elements of Australia’s Science and ResearchPriorities and sustainability is a cross-curricular priority in the Australian Curriculum. Similarly, thecurriculum framework for German federal states includes sustainability as a key element, currently
implemented in secondary mathematics level 1 and under development for secondary level II.
Solutions to sustainability problems often rely on the application of Science, Technology,
Engineering and Mathematics (STEM), which often includes the use of modelling or applying models
to very large data-sets in order to understand or make predictions about a phenomena. These problems
and proposed solutions are communicated to the public through mathematics and statistics – typically
via mass and social media. Thus, a capacity to interpret information and and evaluate findings fromthe analysis of data, often communicated as scientific, mathematical and/or statistical evidence, is
key to a STEM literate citizenry and workforce (Gal & Geiger, 2022).
The capabilities associated with STEM literacy and STEM careers include the ability to posequestions, select relevant data-sets, evaluate the quality of data, and analyse,and critically evaluate
findings (e.g., Duncan et al., 2018) – which we term evidentiary practices. These are key to making
prudent decisions for individuals and societies. Little is known, however, about teaching approaches
that can develop students’ evidentiary practices, especially when using very large data-sets.
In this session, we will present the findings of a review aimed at synthesising relevant research
literature concerned with how evidentiary practices are applied in studies that focus on mathematicalmodelling and very large data sets. Our presentation will address:
1. The nature of evidentiary practices
2. The connections between evidentiary practices and mathematical modelling
3. Findings related to the use of models and modelling to understand and/or analyse large data
sets within educational settings.
References
Gal, I., & Geiger, V. (2022). Welcome to the era of vague news. Educational Studies in Mathematics, 111, 528.
Duncan, R. G., Chinn, C. A., & Barzilai, S. (2018). Grasp of evidence: Problematizing and expanding the next
generation science standards’ conceptualization of evidence. Journal of Research in Science Teaching,55(7), 907-937.
Maass, K., Geiger, V., Ariza, M., & Goos, M. (2019). The role of mathematics in interdisciplinary STEM
education. ZDM Mathematics Education, 51(6), 869-884.
UN General Assembly (2015). Transforming our world: the 2030 Agenda for Sustainable Development.

111

WHAT IS THE ROLE OF MODEL IN THE MODELINNG PROCESS
Koichi Nakamura
Tokyo Gakugei University, Japan

Research on the modelling process has examined students' difficulties with modelling tasks andspecific modelling routes (Blum et al. 2009). In this study, the role of the model is discussed using
the example of the Can problem as one perspective from which to examine the modelling process.

S(t)/Smin

Can problem and modelling process The Can problem is "Find the height and radius of the canthat minimises the surface area so that less material is used to make a cylindrical can with a volumeof 300 ml? "If the height of the cylinder is h and the radius is r, then S=2πr2+2πrh, V=πr2h. Taking
the volume as 300ml and the surface area as a function of the radius, S=2πr2+600/r. This is the firstmodel: Minimum surface model. This model gives us to a minimum surface area at h=2r. This result
allows us to observe real cans in everyday life. We find another problem, "Aren’t real cans designed
to minimise the surface area?" We collected data on cans and investigated some relationships,
diameter and height, and surface area and volume. There is no relationship between diameter andheight as h≠2r. We can get the formula S=5.81V0.66 as the relationship between surface area andvolume of real data. We find the second model, S=5.81V0.66: Real can model. To test the adequacyof this model, data from real drum can is used, the height of is 89cm, the diameter is 58cm. At thesame time, the relationship between surface area and volume is investigated under h=2r conditions,where the surface area is minimised. Expressing the surface area as a function of the volume gives
S=3√54π×V2/3=5.54V2/3. This is the third model that Ideal can model. Real can model helps us to2,0
create the third model. We are surprised to see Real can
1,8
1,6
model and Ideal can model, these models are very similar
1,4
to each other, order and coefficient, although the
1,2
1,0
hypotheses are different, h≠2r and h=2r. We meet again
0,80,6
a new problem, "How much does the surface area of a real
0,40,2
can increase compared to the surface area of an ideal can
0,0
0,0
0,5
1,0
1,5
2,0
2,5
3,0
of the same volume?" and "How does the surface area of
t=h/2ra can of constant volume change when its height anddiameter change?". We make hypotheses, h=2tr, in order toFigure 1 Material Efficiency
specify more accurately capture the relationship between height and radius. We obtain the fourthmodel, R(t)=S(t)/Smin=1/3(1+2t)×t-2/3, the Material Efficiency model (Figure 1). The second and thirdmodels give rise to new problems and contribute to the elaboration of hypotheses.
Discussion We see the role of models, they help to collect and examine the data, to find a new
hypotheses, elaborate hypothesis, and new problems. The review of the tentative model provides
opportunities to facilitate the various phases of modeling activities.
References
Blum, W., & Ferri, R. B. (2009). Mathematical modelling: Can it be taught and learnt? Journal of mathematicalmodelling and application, 1 (1), 45-58
Nakamura, K. (2011). A Study of the Mathematical Modeling Process with the Can Problem, Journal of Japansociety of mathematics education, 93(7), 2-11. (in Japanese)

112

3,5

CONCEPTIONS, USES AND OBSTACLES IN THE TEACHING AND
LEARNING OF MATHEMATICAL MODELLING
Blanca Cecilia Fulano-Vargas1 and Nelson Enrique Barrios Jara1
1

Secretaría de Educación De Bogotá-Colombia, Colombia

This paper presents the advances of the research entitled "Conceptions, uses and obstacles in theteaching practices to strengthen the process of mathematical modelling in basic students of Bogotá,
Colombia South America". The objectives are a) determine the conceptions and uses in the teachingof mathematical modelling by teachers in the town of Fontibón, Bogotá, Colombia; and b) recognize
the obstacles in the learning of mathematical modelling from the perspective of teachers.
When constructing the theoretical framework in relation the epistemological analysis of conceptions
theoretical construct on mathematical modelling process (Blomhøj, 2019; Blum and Leiß, 2007; Frejd,
2014 Fulano and Barrios, 2021); the uses the teaching of mathematical modelling (Villa-Ochoa,
2019) and obstacles in the learning (Brosseau, 2007; Piaget, 1970).
The method the present research is qualitative, with a descriptive scope, through a structured
interview applied to seven mathematics teachers from official schools; five women and two men withan average age of 45 years. The categories conception, use and obstacles are found in an internalconsistency matrix validated by experts.
The results on the conceptions category, teachers use mathematical modelling as a process thatinvolves solving problems inside and outside of mathematics. Regarding the category of obstacles,
the ontogenetic ones associated with difficulties related to the immaturity of the student and the useof technological tools are based first; second, in thinking skills there are problems associated withrecognizing and relating variables, generalizing, formulating hypotheses and validating the model;
Third, in the affective domain, both intrinsic and extrinsic motivation difficulties are evident.
The conception of mathematical modelling is associated with problem solving, teachers use it as atool to strengthen thinking skills. In addition, there are obstacles related to ontogenetic aspects,
thinking skills, affective domain and administrative aspects that affect modelling learning.
References
Brosseau, G. (2007). Iniciación al estudio de la teoría de las situaciones didácticas. Buenos Aires: Editorial
ZorzalBlomhøj, M. (2019). Towards integration of modelling in secondary mathematics teaching. In G.A. Stillman& J.P Brown (Eds) Lines of inquirí in mathematical modelling research in education (ICME-13
Monographs). Cham: Springer
Blum, W., and Leiß, D (2007). How do students and teachers deal with modelling: What do we know, what
can we do? In S.J. Cho (Ed). The Proceedings of the 12th international congress on mathematical education(pp. 73-96). Cham: Springer.
Frejd, P. (2014). Modes of Mathematical Modelling An analysis of how modelling is used and interpreted inand out of school settings. Sweden: LiU-Tryck.
Fulano-Vargas, B., and Barrios N. (2021). Characteristic Elements Influencing the Practices of Mathematics
Teachers Developing the Modelling Process in Ninth Grade. In Mathematical Modelling Education in Eastand West. Switzerland: Springer.
Piaget; J. (1970). Genetic Epistemology (E Duckworth, Trans.). New York: W.W. Norton.
Villa-Ochoa, J. and Souza, E. (2019). Un panorama de investigaciones sobre modelación matemática en
Colombia y Brasil Remat. Revista de Educação Matemática, São Paulo, 16(21), 18-37, jan. /abr. 2019.

113

IDENTIFYING AUTHENTICITY AND MATHEMATICAL MODELLINGCOMPETENCIES AS STEPPING STONES FOR BLENDING DIFFERENTPERSPECTIVES
Daiki Urayama
International Pacific University, Japan
The presence of diversity in the field of mathematical modelling within the realm of mathematics
education has generated both advantages and disadvantages. The existence of diverse perspectives onmathematical modelling has resulted in a lack of consensus regarding the specific methods tointegrate it into the curriculum (Ferri, 2013). A wide range of approaches to mathematical modellinghas been identified, and the current landscape encompasses various perspectives such as Realistic,
Educational, Models and modelling, Socio-critical, and Epistemology (e.g., Abassian et at., 2019;
Ferri, 2013; Kaiser & Sriraman, 2006), offering intriguing considerations such as the frequency ofuse of these perspectives across different countries.
It is plausible that the advancement of curriculum development can be achieved through the
networking or integration of fundamental concepts and assumptions from several perspectives.
However, integrating these central ideas should be approached with meticulous consideration, as
advocated by Scheiner (2020). Embracing a singular stance entails endorsing the notion ofincommensurability put forth by Scheiner (2020), which consequently leads to divergent trajectories
in mathematical modelling research. To foster a cohesive framework, it is imperative to consider the
integration of multiple opposing perspectives to facilitate a convergent viewpoint.
Ledezma & Sala (2022) try to perform the task of networking theories, combining the cognitive
perspective in mathematical modelling with onto-semiotic approaches. However, a comprehensive
integration of multiple perspectives in mathematical modelling has not been accomplished.
Addressing this gap, Kaiser & Sriraman (2006) proposes that adopting a Hegelian inquiry system
may alleviate incommensurability resulting from conflicts among diverse perspectives. Moreover,Abassian et al. (2019) delve into several key aspects of mathematical modelling: goals, definition ofa mathematical model, description of the mathematical modelling cycle, design of the task, examples
of key researchers, and research focus, which are discussed in depth. However, when blending
multiple opposing perspectives, it is pivotal to consider the core concepts of authenticity andmathematical modelling competencies (Kaiser & Brand, 2015) within each perspective. This study
concludes that these concepts are vital to blend these perspectives, presenting an analysis across them.
References
Abassian, A., Safi, F., Bush, S., & Bostic, J. (2019). Five different perspectives on mathematical modeling inmathematics education. Investigations in Mathematics Learning, 12(1), 53–65.
Ferri, R. B. (2013). Mathematical modelling in European education. Journal of Mathematics Education at
Teachers College, 4(2), 18–24.
Kaiser, G., & Brand, S. (2015). Modelling Competencies: Past Development and Further Perspectives. In G.
A. Stillman, W. Blum, & M. Salett Biembengut (Eds.), Mathematical Modelling in Education Researchand Practice (pp. 129-149). Springer International Publishing.
Kaiser, G., & Sriraman, B. (2006). A global survey of international perspectives on modelling in mathematics
education. ZDM, 38(3), 302-310.
Scheiner, T. (2020). Dealing with opposing theoretical perspectives: Knowledge in structures or knowledge inpieces? Educational Studies in Mathematics, 104(1), 127–145.

114

THE ROLE OF STUDY AND TECHNOLOGY IN MATHEAMTICAL
MODELLING
Britta Eyrich Jessen
University of Copenhagen, Denmark

In recent years an emphasis has been put on mathematical modelling drawing on technology (Siller
et al., 2022). Their study show that technology is often used to find information and data, exploring
possible solutions, formulating problems, visualisations and calculations. This often links to students’pragmatic uses of technology. However, Jessen & Kjeldsen (2022) argue that the productive use oftechnology requires epistemic uses of technology done by students as well and demonstrate in twocases how such activities have furthered students learning of mathematics through modelling and thedevelopment of modelling competency.
These findings align with the claim of Artigue and Blomhøj (2013) who argue that modelling can beseen as an approach to inquiry based mathematics education, and Chevallard (2008) who argues thatthe process of study is an overlooked and sometimes ignored part of inquiry based approaches to
education. In this study we revisit existing papers on mathematical modelling to analyse the role ofstudy with an emphasis on how technology provide students with resources to be studied. This may
be teaching resources, but also students’ own productions such as calculations, data collections orvisualisations needed to be scrutinised in order to fully grasp and solve a certain modelling problem.
The papers analysed will cover different approaches to mathematical modelling and feed into thediscussion of how theoretical frameworks for mathematical modelling impact task design for theteaching and learning of mathematical modelling (Barquero & Jessen, 2020), though adding a newperspective regarding how to support students’ autonomy in the modelling process and how thisstrengthens the authenticity of the activities.
References
Artigue, M., Blomhøj, M. (2013) Conceptualizing inquiry-based education in mathematics. ZDM Mathematics
Education 45, 797–810. https://doi.org/10.1007/s11858-013-0506-6
Barquero Farràs, B., & Jessen, B. E. (2020). Impact of theoretical perspectives on the design of mathematicalmodelling task. Avances de investigación en educación matemáticz, 17, 98–113.
https://doi.org/10.35763/AIEM.V0I17.317
Chevallard, Y. (2008). Afterthoughts on a seeming didactical paradox. In International conference “Efficacité& Équité en Éducation”, Rennes, France.
http://yves.chevallard.free.fr/spip/spip/IMG/pdf/Yves_Chevallard_-_On_Inquiry-Based_Instruction.pdf
Jessen, B.E., Kjeldsen, T.H. (2022). Mathematical Modelling and Digital Tools—And How a Merger Can
Support Students’ Learning. In: Jankvist, U.T., Geraniou, E. (eds) Mathematical Competencies in the
Digital Era. Mathematics Education in the Digital Era, vol 20, 99–118. Springer, Cham.
https://doi.org/10.1007/978-3-031-10141-0_6
Siller, H. S., Cevikbas, M., Geiger, V., & Greefrath, G. (2022). The role of digital resources in mathematicalmodelling. In C. Fernández, S. Llinares, A. Gutiérrez, & N. Planas (Eds.). Proceedings of the 45th
Conference of the International Group for the Psychology of Mathematics Education, Vol. 1, 152–155.
Spain.

115

LEARNING OF DIGITAL TOOL COMPETENCIES IN THE CONTEXT OF
MODELLING
Andreas Back1, Gilbert Greefrath1 and Stanislaw Schukajlow1
1

University of Münster, Germany

The use of digital tools in teaching can foster the development of competencies in mathematics(Hillmayr et al., 2020) and especially in mathematical modelling (Cevikbas et al., 2023). Furthermore,studies show that digital tools can be useful in most sub-competencies of mathematical modelling.
However, there is still a need for research on how exactly digital tools can be useful for improving
modelling competencies (Greefrath et al., 2018). This leads to the relevant question for teachers how
digital tools should be introduced and how they should be practised in mathematics classes.
The students could improve their digital tool competencies via an intramathematical or a contextual
learning environment. The open question is: Which way of instruction is better for improving digital
tool competences and modelling competencies? We present the research project Modi+, in which,
we analyse the effect of the type of instruction in the dynamic geometry software GeoGebra(intramathematical vs. in context) on the use of the digital tool and mathematical modelling. Before
the instruction in digital tools, secondary school students (N = 150) were tested in their GeoGebra
competencies. After the instruction students were tested in their GeoGebra competencies again andsolve modelling problems. During the instruction students of one group learned to use GeoGebra byworking on intramathematical tasks and students of the other group learned to use it by working on
tasks in the context of geometrical modelling.
In the talk, we will illustrate the development of the digital tool competence test and introduce its
quality criteria. Further, we will give insights in the first results on the effect the type of learning touse a digital tool has on the acquisition of digital tool competencies. We expect that students, whowere in the intramathematical group, increases stronger their digital tool competencies than students,
who were in the context group.
References
Cevikbas, M., Greefrath, G., & Siller, H.-S. (2023). Advantages and challenges of using digital technologies
in mathematical modelling education – a descriptive systematic literature review. Frontiers in Education,8, 1–17. https://doi.org/10.3389/feduc.2023.1142556
Greefrath, G., Hertleif, C., & Siller, H.-S. (2018). Mathematical modelling with digital tools—A quantitative
study on mathematising with dynamic geometry software. ZDM, 50, 233–244.
https://doi.org/10.1007/s11858-018-0924-6
Hillmayr, D., Ziernwald, L., Reinhold, F., Hofer, S. I., & Reiss, K. M. (2020). The potential of digital tools toenhance mathematics and science learning in secondary schools: A context-specific meta-analysis.
Computers & Education, 153, 1–25. https://doi.org/10.1016/j.compedu.2020.103897

116

TEACHING MATHEMATICAL THINKING: REASONING, MODELLING,
PROBLEM SOLVING
Dag Wedelin
Chalmers University of Technology, Sweden

We describe how we teach mathematical thinking to first year engineering students at Chalmers. We
see our more recent focus on mathematical thinking as a development of our earlier teaching where
we have previously for many years taught mathematical modelling and problem solving toengineering students in their second year, as well as to mathematics teacher students (Wedelin &Adawi, 2014; Wedelin & Adawi, 2015). A basic motivation is to complement the ordinary university
courses in mathematics, as well as our students’ mathematical background from secondary school.
For this purpose, we find the notion of mathematical thinking attractive as it starts with everyone’s
natural ability to think mathematically. Mathematical thinking can therefore be applied by anyone
from the beginning, which is a different point of view than seeing mathematics as a body ofknowledge that first needs to be learned before we can do anything mathematical. This enables us tomaintain a link between common sense and mathematics, to capture creativity in mathematics, and is
helpful in understanding how mathematics and other scientific fields have evolved.
Over the years, we can see that in our own development we started out with a desire to teachmathematical modelling, discovering that it was useful to explicitly combine with concepts ofproblem solving as known in the mathematical literature. In this new course, we additionally include
selected concepts and insights of mathematical reasoning which we have found are also useful tocomplement ordinary mathematics education. So in summary, we conceptualize mathematicalthinking in terms of three main components: reasoning, modelling and problem solving.
It is clearly not possible to develop your mathematical thinking without engaging with actual
problems, applied and theoretical. So while we have established a structure and a language to talkabout mathematical thinking, the main focus must be to actually work with different kinds ofproblems. In order to clarify the purpose of mathematical thinking, we have chosen to group our
problems in four main categories: keeping track, investigating the abstract, investigating the world,and designing. We also link to mathematical knowledge by highlighting different kinds of models
and their associated mathematics, e.g. functions and equations, optimization, probability andstatistics, and so on.
We discuss how our teaching is received by our students, and finally provide some generalexperiences and reflections.
References
Wedelin D. & Adawi T. (2014). Teaching mathematical modelling and problem solving - a cognitive
apprenticeship approach to mathematics and engineering education. International Journal of Engineering
Pedagogy, 4(5), 49-55.
Wedelin D. & Adawi T. (2015). Applied mathematical problem solving - principles for designing small
realistic problems. In Stillman, Blum, Biembengut (eds.), Mathematical Modelling in Education Researchand Practice (pp. 417-427). Springer.

117

TAKING UP OWNERSHIP: WHAT CAN BE LEARNED FROM POSINGMODELLING PROBLEMS?
Janina Krawitz1, Luisa Hartmann1, Stanislaw Schukajlow1 & Werner Blum²
1

University of Münster, Germany; 2University of Kassel, Germany

The central ideas of mathematical modelling include formulating conditions and assumptions toestablish models and evaluating their impact on solving real-world problems (Niss & Blum, 2020).
However, these activities are often overlooked, superficially addressed or cause difficulties when
students solve modelling problems (Galbraith & Stillman, 2001). Hence, students require support indeveloping the modelling competencies that are necessary to master these activities. Prompting
students to pose their own modelling problems could be a promising teaching approach fordeveloping modelling competencies. Problem posing comprises the reformulation or generation ofproblems and involves processes such as problem analysis, condition variation, and problem
evaluation (Baumanns & Rott, 2022). These processes may enhance modelling competencies such as
making assumptions and validating the solution. However, research investigating problem posing
approaches for modelling is still limited.
In the MoPro project (Modelling with Problem Posing), we aim at investigating the effects of problem
posing on students’ modelling competencies and motivation. First, we conducted a qualitative study
to determine the processes and the difficulties that occur when students generate and reformulate
modelling problems. Second, we developed and evaluated a test to assess competencies for generating
and reformulating modelling problems. In a next step, we will investigate the effect of a teachingapproach where students pose and solve modelling problems in comparison to an approach wherestudents only solve modelling problems.
In this presentation, we focus on the results of the qualitative study on problem posing processes.
Sixteen ninth- and tenth-graders, working in pairs, were prompted to reformulate and generatemodelling problems. Through qualitative content analysis, we identified different processes that took
place while students reformulated and generated modelling problems, and we evaluated theimportance of these processes for building modelling competencies. For example, we observed
students engaging in setting up and varying situational and numerical assumptions and discussing
how these assumptions influence the authenticity and the difficulty of the problem. The findings
contribute to a better understanding of the processes involved in reformulating and generating
modelling problems. They also provide a basis for developing teaching approaches that encouragelearners to take ownership of their own problems.
References
Baumanns, L., & Rott, B. (2022). The process of problem posing: development of a descriptive phase modelof problem posing. Educational Studies in Mathematics, 110, pages 251–269.
Galbraith, P., & Stillman, G. (2001). Assumptions and Context: Pursuing their Role in Modelling Activity. In
J. F. Matos, W. Blum, K. Houston, & S. P. Carreira (Eds.), Modelling and Mathematics Education(pp. 300–310). Woodhead Publishing.
Niss, M., & Blum, W. (2020). The Learning and Teaching of Mathematical Modelling. Routledge.

118

MATHEMATICAL MODELLING AT ALL YEAR LEVELS IN AUSTRALIAPaul Brown
Curtin University, Australia

In Australia, a new mathematics curriculum (Australian Curriculum, Assessment and Reporting
Authority [ACARA], Version 9.0, 2022) is being introduced. It covers primary and secondary schoolmathematics classes from Year 1 to Year 10 (students aged from five years to about 16 years). At
every year level, the new curriculum specifies mathematical modelling as a mandated topic, although
the extent to which some Australian states will vary this requirement is not yet known. In Australia,
individual states are free to adopt the ACARA curriculum in full or adapt it to their own requirements,
perhaps by providing student work samples which contain cultural content pertinent to the local area.
The ACARA (2022) curriculum includes compulsory topics and examples, together with noncompulsory “elaborations” which may be of value to particular groups such as First Nations peoples
or high-achieving students. The use of technologies is sometimes specified, especially computerbased technologies such as spreadsheets.
The presentation will consider the background of the “proficiency strand” topics which have been
supplanted by the introduction of mathematical modelling as a separate topic. In addition, the declared
theoretical underpinnings of the revised curriculum will be considered.
The presentation will discuss responses to the new curriculum as a case study. It will examine thework of publishers, including Oxford (Garvey, 2023), who produce hardcopy textbooks for Years 7
to 10, and Mathspace (n. d,), an online interactive tutoring system which caters for students fromYear 4 and upwards. These publishers have been amongst the first to respond, supplying resources intime for the first Australian states to implement the revised curriculum in 2024.
The revised curriculum envisages mathematical modelling introduced at Year 1. This is in the context
of “use mathematical modelling to solve practical problems involving equal sharing and grouping;
represent the situations with diagrams, physical and virtual materials, and use calculation strategies
to solve the problem” (ACARA, 2022, AC9M1N06). In Year 2, mathematical modelling is used “tosolve practical additive and multiplicative problems, including money transactions, representing thesituation and choosing calculation strategies” (ACARA, 2022, AC9M2N06). These objectives may
seem ambitious, or as stretching the meaning of mathematical modelling, and it will be interesting tofollow how systems, publishers, schools and teachers implement the curriculum – especially for themost junior classes.
References
Australian Curriculum, Assessment and Reporting Authority [ACARA]. (2022) The Australian
Curriculum: Mathematics, version 9. https://v9.australiancurriculum.edu.au/Garvey, S. (Ed.). (2023). Oxford Maths. Oxford University Press.
Mathspace (n.d.) Mathspace. https://mathspace.co/au

119THE INVESTIGATION OF THE MODELLING COMPETENCIES OF TWOGROUPS OF GRADE EIGHT STUDENTS WHO RECEIVED DIFFERENTMODELLING INSTRUCTIONS
Yuan Yan1 and Yi Chen2
1

East China Normal University, China; 2Shanghai Wenqi Middle School, China

With the promotion of mathematical modelling in the teaching and learning of mathematics in China,
an increasing attention has been paid to which teaching approaches are appropriate to promotestudents’ modelling competencies.
In this study, we measured two groups of grade eight students’ modelling competencies who received
two different ways of modelling instructions. Group A received a kind of modelling instruction which
was consistent with the traditional approach of mathematical teaching that was teacher-centered anddemonstration-oriented; and the instruction was led by an experienced teacher. Group B received themodelling instruction that organized in the way of group cooperative learning, with five to six
students working together on modelling tasks; and there was one pre-service teachers provided
interventions for each student group. Group A, which involved 45 students, received modellingtraining in June 2022; while Group B included 32 students was involved into the training in May
2023. According to their class teachers, the two groups performed equally on the learning of schoolmathematics. After receiving the modelling instructions, the two groups of students were immediately
asked to individually complete a modelling test which includes three modelling tasks.
In this paper, we presented the measurement of the two groups’ performances on the two samemodelling tasks in the modelling test they received after the modelling instructions. A hybrid
approach (Cevikbas et al., 2021) was employed to assess the students’ modelling competencies
through their performances on the two tasks, which includes a holistic approach which concentrated
on students’ full-scale modelling process and an atomistic approach which focused on the phases ofsimplifying situations and making assumptions. The research results will reveal the differences in thestudents’ performances on the modelling tasks in terms of holistic and atomistic aspects of modellingcompetencies, and further imply the influences of different modelling instructions on students’modelling competencies.
References
Cevikbas, M., Kaiser, G., & Schukajlow, S. (2021). A systematic literature review of the current discussion
on mathematical modelling competencies. Educational Studies in Mathematics, 109, 205–236.

120

VALIDATION IN SIXTH-GRADE DATA-DRIVEN MODELLING
Takashi Kawakami1 and Katsuki Akizawa2
1

Utsunomiya University, Japan; 2Elementary School Attached to Utsunomiya University, Japan

Validation in mathematical and statistical modelling is crucial for all citizens to understand andcritique mathematical and statistical models in a data-driven society (e.g., Czocher et al., 2018; Gal& Geiger, 2022). However, the nature of validation and modification attempted by young learners
remains unclear. To address this research gap, our study focused on validation with sixth-gradestudents during the modelling of a data-rich situation, that is, data-driven modelling (DDM). DDM
refers to activities involving the generation, validation, and modification of models (mathematicaland statistical models), based on data/context to make better predictions and estimates (Kawakami,2023). In the framework, mathematical and statistical models reflect the subject’s deterministic andstochastic interpretation of the data/context, respectively. The framework can conceptualise datacantered validation, which compares models for prediction and estimation with the actual results, andsubsequent modification related to models and data.
We addressed the following question: What attempts do sixth-grade students make when comparing
models for prediction and estimation with the actual results during DDM? For this purpose, we
designed and implemented a DDM teaching experiment consisting of 7 lessons for 35 sixth-gradestudents. The students worked in groups on an authentic ring-decorating task, developing models for
estimating the number of origami sheets needed to decorate the gym with ring-decorations for a
thank-you party they were organising. They repeatedly made assumptions, and generated, measured,
structured, and represented the necessary data. After the lessons, they constructed the decorations,completed a task-based questionnaire in which they watched a video of the hanging of the decorations(because of COVID-19 school closures), and compared the models’ results with the actual
decorations, and modified the models if necessary. We analysed the data from the questionnaire interms of what they focused on in their models, what they relied on for validation, and how they tried
to improve their models.
The preliminary results show that the students validated the context in which the decorations were
made, the way the data were collected and measured, and the assumptions of the models. Students
who noticed measurement errors critically rethought a mathematical model that was considered
deterministic and that assumed proportional relationships, and mentioned the need to reinterpret it
stochastically and to modify it to a statistical model, taking into account errors and uncertainty. Thesefindings suggest that validation can play an age-appropriate and innovative role in learning to model
data-filled situations.
References
Czocher, J., Stillman, G., & Brown, J. (2018). Verification and validation: What do we mean? J. Hunter et al.
(Eds.), Proceedings of MERGA41 (pp. 250–257). Auckland: MERGA.
Gal, I., & Geiger, V. (2022). Welcome to the era of vague news: A study of the demands of statistical andmathematical products in the COVID-19 pandemic media. Educational Studies in Mathematics, 111, 5–28.
Kawakami, T. (2023). A triplet of data/context, mathematical model, and statistical model: Conceptualisingdata-driven modelling in school mathematics. [Paper presentation] CERME13, July 10–14, 2023,
Budapest: Hungary. Retierved from https://www.researchgate.net/publication/370553939

121

STRENGTHENING TEACHERS’ CAPABILITIES WITH BIG DATA
Vince Geiger1, Hans-Stefan Siller2, Sarah Digan1, Andre Greubel2 and Nina Unshelm2
1

Australian Catholic University, Australia; 2University of Wurzburg, Germany

We live in a time of disruption – not just digital disruption but also long-standing international
challenges such as those associated with health, the environment, and food and energy security (seeUN General Assembly, 2015). Responses to disruptions invariably rely on the use of mathematics
and statistics (M&S). M&S are also used to update the public about the state of disruptions, makepredictions about their progress, and to justify decisions made by authorities about responses – allcommunicated in public forums (e.g., mass and social media). This means that citizens must be able
to understand, interpret and evaluate M&S information (Gal & Geiger 2022) to remain informed andempowered to contribute to a prosperous, just and equitable society (Maass et al., 2019).
The COVID-19 pandemic highlighted how M&S can be used as evidence, by government agencies,
politicians, commercial entities, experts, and non-experts, to support arguments about how to respondto the crisis. These arguments often involved claims that were contradictory, inconclusive, andmisleading. For citizens to understand how evidence is used in decision making and assess theveracity of claims made during debates, they must possess the capability to critically evaluate M&S
evidence (Gal & Geiger, 2022). This requires processes described in science education as evidentiary
practices (e.g., Duncan et al., 2018) – a foundation for prudent decision-making.
In this presentation, we describe a current study that aims to develop students’ capabilities to evaluate
claims and arguments in the media through the modelling of large data sets related to sustainability
issues. The session will provide background to the study and opportunity to discuss the challenges
associated with the design of tasks appropriate for students in Years 9/10 and relevant data collection
instruments. Tasks are focused on issues such as levels of atmospheric CO2, and international levelsof poverty, among others. Each task is based on two media items that provide contradictory
information about a disruption, aligned with a selection of at least three databases. Data collection
instruments include observation, computer screen capture, surveys, and student interviews.
The outcomes of the research are focused on: 1. The processes used to identify misleading ot
contradictory information within media items; 2. Students’ choice of data base in response to a task;
3. Student modelling of selected data in respect to the issue of concern; and 4. Students’ decisionmaking processes in forming judgements about the veracity of claims and arguments in a media item.
References
Gal, I., & Geiger, V. (2022). Welcome to the era of vague news. Educational Studies in Mathematics, 111, 528.
Duncan, R. G., Chinn, C. A., & Barzilai, S. (2018). Grasp of evidence: Problematizing and expanding the next
generation science standards’ conceptualization of evidence. Journal of Research in Science Teaching,55(7), 907-937.
Maass, K., Geiger, V., Ariza, M., & Goos, M. (2019). The role of mathematics in interdisciplinary STEM
education. ZDM Mathematics Education, 51(6), 869-884.
UN General Assembly (2015). Transforming our world: the 2030 Agenda for Sustainable Development.

122

IMPROVING UNDERGRADUATE STATISTICS EDUCATION:EDUCATIONAL LESSONS FROM PEDAGOGICAL EXPERIMENTS
Anushka Karkelanova1, Xin Ma1 and William Rayens1
1

University of Kentucky, USA

The main goal of this research is to fill in some gaps in the research literature on the teaching ofstatistics. This research includes two independent studies (experiments). The first study examined the
instructional effects of physical versus virtual manipulatives on learning outcomes in introductory
statistics, whereas the second study investigated the impact of different styles in teaching statistics(inverted classroom versus traditional classroom) on learning outcomes in introductory statistics. In
general, this research attempted to join many other reform efforts to explore instructional ways thatengage students in reasoning and modeling statistically. To combat the abstract nature of probability
and statistics, the use of manipulatives may represent one of the most effective strategies in thestatistics classroom. There are fundamental reasons to inherently value the inverted classroom’s
emphasis on activity-based learning and increased responsibility of the students to become active
participants in their own learning.
The results of the first study revealed that, in terms of GPA one year (as the outcome), there were no
statistically significant interaction effects between types of manipulatives (the business-as-usual
group received traditional concrete manipulatives and the experimental group received online virtual
manipulatives) and high school ACT mathematics scores, informing the literature that ability levels
neither intensify nor weaken the effects of types of manipulatives. The results of the study did notshow a statistically significant difference in GPA one year later between the experimental group andthe business-as-usual group.
The results of the second study revealed some significant differences between the business-as-usual
group who received traditional lecture and the experimental group who received inverted instruction.
We compared seven outcomes for the two groups: projects average, tests average, classwork, midterm
attendance average, class final attendance average, midterm grade, and class final grade. After
controlling for student individual background (age, gender, and ethnicity), students in the traditionalclassroom did better than students in the inverted classroom in projects average, overall classwork,and midterm grade. After controlling for high school background (high school GPA and ACT
mathematics scores), students in the traditional classroom did better than students in the invertedclassroom in projects average, overall classwork, and midterm grade. After controlling for university
program background (university cumulative GPA and student major), students in the traditionalclassroom performed similarly to students in the inverted classroom in all seven outcomes. Finally,
when controlling for all (i.e., student individual background, high school background, and university
program background), students in the traditional classroom did better than students in the invertedclassroom in midterm grade only.

123

PRE-SERVICE TEACHERS’ NOTICING OF MATHEMATICAL
MODELLING PROCESSES
Alina Alwast
University of Hamburg, Germany

For teaching mathematical modelling in a classroom setting broad knowledge about many aspects ofmathematical modelling, such as modelling tasks, modelling cycles, multiple solutions and potential
difficulties (as described in the 4-dimensional model by Borromeo Ferry and Blum (2010)) is needed.
Moreover, it is essential that teachers are able to react spontaneously to solution approaches, whichwere not anticipated, and students’ learning obstacles, which can sometimes not be foreseen. In moredetail, teachers need to selectively perceive noteworthy aspects in a situation with an overwhelming
amount of information, interpret the perceived incident by relying on their knowledge and weighdifferent options to react. This competence of noticing is important not only for teaching-andlearning-processes in general, but for mathematical modelling teaching-and-learning-processes inparticular. It also differs among novice and expert teachers (Cai et al. 2021).
The development of noticing competencies should therefore be already supported in teacher
education, especially as it links theoretical knowledge offered at university courses and practicalactivities in school. To bridge this gap, link theory and practice and prepare pre-service teachers fortheir future work at school, a seminar about mathematical modelling was offered, which included
theoretical concepts on the one hand and the analysis of practice-oriented artifacts, such as students’
written solutions, text vignettes and videos on the other hand in order to foster pre-service teachers’
noticing competencies. Thus, the following research questions are posed:
To what extent were pre-service teachers able to notice important events within modeling processes?
How did their ability to notice differ when analyzing classroom sequences displayed by videos fromtheir ability to notice when analyzing text vignettes?
To answer the research questions pre-service teachers noticing competencies were evaluated with avideo-based test and text-vignettes, which included analogous situations, e.g., the same difficulty
occurs although the modelling problem differed. The instruments were used before and after several
master seminars with 55 pre-service teachers. The answers of the participants were qualitatively
analyzed. It can be hypothesized that video-based formats capture perception more precisely, whileespecially interpretation and decision-making can also be assessed sufficiently by text-based formats.
References
Borromeo Ferri, R. & Blum, W. (2010). Mathematical modeling in teacher education: Experiences from amodeling seminar. In V. Durand-Guerrier, S. Soury-Lavergne & F. Arzarello (Eds.), CERME-6:
Proceedings of the Sixth Congress of the European Society for Research in Mathematics Education (pp.
2046–2055). Lyon: INRP.
Cai, J., LaRochelle, R. Hwang, S. & Kaiser, G. (2022). Expert and preservice secondary teachers’
competencies for noticing student thinking about modelling. Educational Studies in Mathematics, 109, 431453.

124

COMBINING BRUTE FORCE AND IT TO SOLVE A CLASSICALSHOPPING PROBLEM
Ludwig Paditz
Dresden University of Applied Sciences, Faculty of Informatics and Mathematics, Germany

Two classic word problems with a similar background are mathematically modeled and a solution isprovided. The first problem, the puzzle of the "Dutchmen's Wives", can be found as early as 1739 ina women's magazine (May (1739). The Ladies Diary). A similar problem (“Find Ada's Surname”,
Workman (1906)) was formulated at the beginning of the last century. The mathematical model leads
to a Diophantine quadratic equation, which can be solved using the third binomial formula and aprime factor analysis. First, a "stupid trial and error method" (brute force) is used to search for asolution. The CASIO-calculator/emulator ClassPad II will be used.
The first word problem: the puzzle of the “Dutchmen's Wives” “There came three Dutchmen of myacquaintance to see me, being lately married. The men’s names were Hendrick, Claas, and Cornelius;
the women’s, Geertruii, Catriin, and Anna. But I forgot the name of each man’s wife. They told me
they had been at market to buy hogs. Each person bought as many hogs as they gave shillings for eachhog. Hendrick bought 23 hogs more than Catriin, and Claas bought 11 more than Geertruii; likewise,
each man laid out three guineas more than his wife. I desire to know the name of each man’s wife.”
May (1739), Dudeney (1917), Hemme (2019).
The second word problem: the puzzle “Find Ada's Surname” Dudeney (1917), Workman(1906,1918)
We want to sensitize the pupils to MINT (mathematics, computer science (informatics), naturalsciences and technology) and to link interest with personal experiences. We show the pupils that
MINT is much more than the cliché of mathematics and computer programming. For this purpose, theinhibition threshold for dealing with MINT should be lowered in the school project on the basis of
playful, project-oriented didactics and at the same time interest in the topic should be encouraged.
Various projects are available. “In conclusions, we point to the fact that there are several ways todemonstrate interrelations between traditional Maths and IT.” Hvorecký, Korenova, Barot (2022).
References
Dudeney, H.E. (1917, 1958). Amusements in Mathematics. Nelson London, pp. 26-27
http://djm.cc/library/Amusements_in_Mathematics_Dudeney_edited02.pdf
Hemme, H.(2019). Die Holländer kaufen Schweine, Spektrum der Wissenschaft,https://www.spektrum.de/raetsel/die-hollaender-kaufen-schweine/1579982
Hvorecký, J., Korenova, L., Barot, T. (2022). Combining Brute Force and IT to Solve Difficult Problems.
Electronic Proceedings of the 27th Asian Technology Conference in Mathematics (ATCM) 2022,
December 9-12, Prague, https://atcm.mathandtech.org/EP2022/regular/21959.pdf
May, J. (1739). Three Dutchmen, The Ladies’ Diary: Puzzle
https://www.cantorsparadise.com/three-dutchmen-f9f08ac19d73
Workman, W.P. (1906, 1918). The Tutorial Arithmetic: with Answers. Univ. Tutorial Press London, p. 482
https://cs.smu.ca/~dawson/workman/workman3.gif

125

A STUDY ON MATHEMATICAL MODELING UTILIZING APOLLONIUS'CIRCLE
Takuma Takayama
Konan Junior High School, Japan

The purpose of this paper is to present a comparison of student reality in a paper-drawn and
GeoGebra-drawn mathematical modelling and problem-solving class in the field of graphics for thirdyear junior high school students, using the everyday and easily interesting subject ”Where can I see
Tokyo Tower and Sky Tree at the same height?".
As a result of 2015, it became clear that the learners used logical thinking in various situations in theprocess of mathematical modelling problem solving in this problem. (2015).
The study suggested that it is possible to realize what Ikeda calls "the flow from intuitive, analogical,
and inductive understanding to deductive understanding in a curriculum sequence, and to teach insuch a way that students can experience the deepening of logic".
2022, school education in Japan is changing drastically with the deployment of a tablet computer foreach student in junior high schools. Therefore, it is now possible for students themselves to open theirtablets and use GeoGebra to draw Apollonius' circle on a map of Tokyo, including Tokyo Tower and
Sky Tree.
A post-class survey showed that students were more interested
in Apollonius circles when using GeoGebra, and were also moreinterested in the use of mathematical modelling. Furthermore,these two results will be used to discuss the difference between
the thinking of drafting regarding mathematical modelling with
GeoGebra and the thinking of drafting with a compass and ruler.
References
Ikeda, T. (2017). Cultivation of generic ability in mathematics education from the viewpoint of eugenic
purpose and its task: On the intention of task research and logical thinking ability. Proceedings of the 5th
Spring Conference of the Japanese Society for Mathematics Education, 251-254.
Matsuzaki, A. (2018). Study of mathematical modelling competencies focused on gen‐bamen : using themethod of 'Phänomenologie' by Husserl, E. in the middle latter period and applied response analysis
mapping. Kyoritsu Shuppan.
Nishimura, K. (2012). A Study on the Development and Practice of Teaching Materials for Developing theAbility to Perform Mathematical Models. Toyokan Shuppansha.
Takayama, T. (2015). Classroom Practice on Modelling in Figure Domain: Where Sky Tree and Tokyo Tower
appear to be the same height. Journal of the Japanese Society for Mathematics Education, 97th Annual
Meeting, 400.
Takayama, T. (2018). Development and Practice of Developmental Teaching Materials for Relating
Knowledge in Figure Domain. Journal of Mathematics Education Society of Waseda University, 36(1), 2234.

126

EXPLORATORY STUDY ON STUDENTS' UNDERSTANDING OFSIMULATIONS - HOW MATHEMATICAL MODELLINGCOMPETENCY NEEDS TO BE COMPLEMENTED TO PROMOTEUNDERSTANDING OF SIMULATIONS
Sonja Bleymehl1 and Christine Bescherer1
1
University of Education Ludwigsburg, Germany
Computer simulations have been a central method in scientific findings for a long time.
Scientists use simulations to test their hypotheses, generate new ones and make predictions. In
addition, simulations are used for communicating scientific results. Experts exchange
information among themselves or communicate their knowledge to politicians and other stakeholders. Therefore, simulations serve as basis for decisions of social relevance. To maintaincredibility the decision-making process of persons in charge is explained by sharing (simplified)simulations results through the media. This is particularly important when political decisions
require actions from the citizens as for example in the case of the COVID-19 pandemic.
Consequently, computer simulations are no longer just the matter of scientists, they are part ofour everyday life and the informed citizen and consumer need to be able to reflect simulation
results critically. Hence, it is the responsibility of schools to impart competencies forunderstanding simulations (OECD, 2018, p. 26).
This exploitative study aims to come up with a theoretical construct of understanding
simulations. What does it take to understand simulations? Which kind of competencies andskills are needed to be able to access all information provided by the simulation and put it into
the right context?
After analysing various definitions of the term computer simulation, two main characteristics
emerge: Computer simulations are based on mathematical models and the results are theconsequences of the system dynamics (e.g. OECD, 2018, p.18). Therefore, we propose thatmodelling competency is a part of understanding simulations, as well as understanding
mathematical models themselves and system thinking.
To discuss these hypotheses critically and identify more partial competencies and skills, semistructured interviews with experts on computer simulation are being conducted. By engaging
experts from various fields such as education, philosophy, engineering, systems theory andcomputer science, we aim to explore understanding simulations from multiple perspectives andgain comprehensive insights. Outcomes of this interview study will be presented at theconference.
This interview study is part of a broader investigation with the overall aim of developing
learning sequences on computer simulations.
References
OECD (2018). PISA 2022 Mathematics Framework (Draft). https://pisa2022-maths.oecd.org

127

MATHEMATICAL MODELING PROJECTS IN ADVANCED ORDINARY
DIFFERENTIAL EQUATIONS COURSES FOR MATHEMATICS AND
ENGINEERING STUDENTS
Yuriy Rogovchenko1 and Svitlana Rogovchenko2
1

University of Agder, Kristiansand, Norway; 2University of Agder, Grimstad, Norway

Differential equations (DEs) are used for modelling a wide spectrum of phenomena in engineering,
physics, natural and social sciences and are taught in most STEM programs. Traditional courses
concentrate attention mainly on different techniques for solving DEs, whereas contemporary
approaches to teaching and learning of the subject tend to emphasize qualitative, graphical, andnumerical methods. Despite the centrality of DEs in the undergraduate curricula, educational researchis scarce and focuses primarily on the understanding of the concept of DE, solutions to first order
DEs and systems of DEs whereas “mathematical modeling and the role that technology can play inthe modeling process are missing” (Rasmussen & Wawro, 2017, p. 555).
Recent research addressed the use of mathematical modelling (MM) as a vehicle for more authentic
learning of DEs (Czocher et al., 2021), how the emphasis on MM principles can be beneficial forstudents in standard DEs courses (Czocher, 2017) and how modelling tasks motivate the developmentof new explorative mathematical routines contributing to the improvement of students’ mathematicaland programming skills (Rogovchenko, 2021).
In this paper, we discuss how MM projects were integrated in two DEs courses at the University ofAgder, Mathematics for Mechatronics for senior engineering students and Nonlinear Differential
Equations and Dynamical Systems for graduate mathematics students. In the former course, MM
projects were included in summative assessments during the semester and included student small
group work, submission of written individual reports and presentation of the projects to the class. In
the latter course, students could choose to work on theory based or MM individual projects, andsubmission of individual written reports and presentation of the projects were compulsory for theadmission to the final oral exam. In both cases MM projects included experimental parts. We analyseand compare students’ work on MM projects and projects’ impact on students’ learning in courses
where there was little room available in the syllabus for addressing MM in more detail.
References
Czocher, J. A. (2017). How can emphasizing mathematical modeling principles benefit students in a
traditionally taught differential equations course? The Journal of Mathematical Behavior, 45, 78–94.
Czocher, J. A., Melhuish, K., Kandasamy, S. S., & Roan, E. (2021). Dual measures of mathematical modelingfor engineering and other STEM undergraduates. International Journal of Research in Undergraduate
Mathematics Education, 7, 328–350.
Rasmussen, C., & Wawro, M. (2017). Post-calculus research in undergraduate mathematics education. In J.
Cai (Ed.), Compendium for research in mathematics education (pp. 551–581). National Council of
Teachers of Mathematics.
Rogovchenko, S. (2021). Mathematical modelling problems in a mathematics course for engineers: A
commognitive perspective. In: F.K.S. Leung, G.A. Stillman, G. Kaiser, & K.L. Wong, (Eds) Mathematical
Modelling Education in East and West (pp. 561–570). Springer.

128

MODELLING TASKS FOR NON-STEM UNIVERSITY STUDENTS AND
THE CHARACTERISTICS OF THEIR MODELLING PROCESS
Mitsuru Kawazoe1 and Koji Otaki2
1

Osaka Metropolitan University, Japan; 2Hokkaido University of Education, Japan

In Japan, the teaching of mathematical modelling (MM) for all university students, including nonSTEM students, is attracting more attention depending on the spread of data sciences in our society.
However, actually, most Japanese university students, especially non-STEM students, do not havethe opportunity to experience MM in their educational history, due to the lack of MM courses inschool and university. Here are our research questions: What tasks can prompt non-STEM students’
MM activities? How is their MM process, and how does it differ from STEM students?
Regarding the questions, we developed a one-semester MM course with the following topics: (1) theJapanese government’s project to halve the number of deer and wild boar in Japan (cf. Wild animal
population control, 2014), (2) the effectiveness of a new AI-based cancer detection (cf. Klein et al,2021), and (3) changes in the market share of smartphone operating systems (cf. Statcounter, 2023).
The MM task for each topic can be approached using a recurrence formula of a sequence, Bayesianestimation, and Markov chains, respectively. The course was conducted in the spring and the autumn
semesters of the 2022 academic year with mixed but unbalanced classes of STEM and non-STEM
students: non-STEM students dominated 78% of the spring class (non-STEM class); STEM students
dominated 89% of the autumn class (STEM class). Students worked in groups of 3-4. Activity logs
and reports on each task produced by the groups were collected and qualitatively analysed.
The results are as follows. In the non-STEM class, students’ awareness of the mathematics used in
MM activities was weaker and the models were less diverse than in the STEM class: the recurrence
formula in Task (1) appeared only after the teacher’s suggestion to express models in mathematicalterms; only the Markov chain model was used in Task (3). Despite these differences, rich MM
activities, making assumptions, simplifying the real situation, and evaluating, modifying the models,
were observed in both classes for Task (1), (3); the differences seem to be eliminable with thecooperation of both students and with the help of the teacher. On the other hand, such activities werepoor in both classes for Task (2). This was probably because the mathematisation from the situation
models was the main activity for Task (1), (3), whereas, in the case of Task (2), it was to understandthe terminology and formulas used in the medical field that is based on Bayes’ theorem, and then tointerpret the real situation within them. That seemed to be difficult for the students of both types.
References
Klein, E. A., Richards, D., Cohn, A., Tummala, M., Lapham, R., Cosgrove, D., Chung, G., Clement, J., Gao,J., Hunkapiller, N., Jamshidi, A., Kurtzman, K. N., Seiden, M. V., Swanton, C., & Liu, M. C. (2021).
Clinical validation of a targeted methylation-based multi-cancer early detection test using an independentvalidation set. Annals of Oncology, 32(9), 1167–1177, https://doi.org/10.1016/j.annonc.2021.05.806
Statcounter. (2023). Mobile Operating System Market Share in Japan - April 2023. [Data set].
https://gs.statcounter.com/os-market-share/mobile/japan
Wild animal population control. [Editorial]. (2014, May 1). The Japan Times. https://www.japantimes.co.jp/
opinion/2014/05/01/editorials/wild-animal-population-control/

129

COMPARING AND CONTRASTING STATISTICAL AND MATHEMATICAL
MODELLING – A STUDY OF THE ROLE OF DATA
Camilla Hellsten Østergaard1 and Britta Eyrich Jessen2
1

University College Copenhagen, Denmark; 2University of Copenhagen, Denmark

Data is depicted as a central part of the modelling cycle as “the epistemological basis for the different
sub-processes” (Blomhøj & Kjeldsen, 2006, p. 166) and as a crucial role as a driver for the inquiry
process and a unique component of the media to be studied (Chevallard, 2019). To question and study
data are vital for statistical inquiries leading to statistical modelling and the development of statistical
literacy and reasoning. Similar aims can be said for mathematical modelling where modelling candrive the learning of mathematics. According to Chevallard (2019) can data both be quantitative andqualitative in nature (possibly gathered as part of the inquiry or modelling process).
We compare and contrast existing research regarding statistical and mathematical modelling with anemphasis put on the role of data and how data may shape the inquiry and modelling processes
conducted by students. Our main focus is two Danish Study and Research Paths (SRP; Chevallard,2006) studying the real-world problems of ‘health and physically activity’ with grade five students(Østergaard & Larsen, submitted) and ‘facial recognition system’ with upper secondary school
students (Jessen, 2022). We found, how the dynamic data visualisation tool, TinkerPlots, and the
dynamic geometry software, GeoGebra, drew students’ learning processes and were crucial milieus;
to generate first hypotheses, to question and explore students’ collected data and answers found in
media and, hence, as a condition for the questioning and media-milieu dialectics.
We further elaborates on the development and implementation of SRP in classrooms, and we question
specific views of interaction between teachers and researchers and investigate the hypothesis that
research could work as an integrated part of paradidactic infrastructure (Miyakawa & Winsløw, 2013),
such as in the format of lesson study and other forms of teacher–researcher collaborations.
References
Blomhøj, M., Kjeldsen, T.H. (2006). Teaching mathematical modelling through project work. ZDMMathematics Education, 38, 163–177. https://doi.org/10.1007/BF02655887
Chevallard, Y. (2006). Steps towards a new epistemology in mathematics education. In M. Bosch (Ed.),
Proceedings of the Fourth Congress of the European Society for Research in Mathematics Education(CERME 4) (pp. 22-30). Sant Feliu de Guíxols, Spain: FUNDEMI IQS – Universitat Ramon Llull and
ERME.
Chevallard, Y. (2019). Introducing the anthropological theory of the Didactic: an attempt at a principledapproach. Hiroshima journal of mathematics education, 12, 71–114.
Jessen, B. E. (2022). Media and milieu dialectics and new resources available for students. In Pre-proceedingsof the 7th International Conference on the Anthropological Theory of the Didactic, p. 232–239.
Østergaard, C. H. & Larsen, D. M. (submitted). Questioning the world – More than inquiry. Statistics
Education Research Journal
Miyakawa, T., Winsløw, C. (2013). Developing mathematics teacher knowledge: the paradidactic
infrastructure of “open lesson” in Japan. Journal of Mathematics Teacher Educucation 16, 185–209.
https://doi.org/10.1007/s10857-013-9236-5

130ASPECTS OF DATA MODELLING AMONG JAPANESE JUNIOR HIGHSCHOOL STUDENTS: FOCUSING ON CRITICAL CONSIDERATION
Yuichiro Hattori1 and Mayu Nomura2
1

Okayama University, Japan; 2Hayama Junior High School, Japan

In recent years, research linking mathematical and statistical modelling has been actively promoted(e.g. Ärlebäck & Kawakami, 2023; Kawakami & Mineno, 2021). In Japan's new Courses of Study,
published in 2017, there is an urgent need to enhance statistics education and develop students'
abilities to critically examine data through lessons. Thus, practical research must be conducted. The
purpose of this study is to describe the actual mathematical activities demonstrated by students in thesecond grade of a Japanese junior high school through data modelling (Pfannkuch et al., 2018), an
activity type of statistical modelling. The study also aims to clarify the level of criticality
demonstrated by the students.
The lessons were based on the cycle of PPDAC (Wild & Pfannkuch, 1999), and a 16-h unit plan wasdeveloped and implemented. In this unit, we asked, "How can we improve our peripheral visionskills?" To identify the characteristics of students with faster times in the "Touch the Numbers" action
game application, which requires quick number pressing, we conducted an analysis to determine thefactors contributing to their speed. The questionnaire (Q1: grade; Q2: time spent using smartphones
and games; Q3: favourite subject; Q4: club activities; Q5: frequency of reading) was cross-tabulatedwith time and the data were analysed based on statistics (number of data points, minimum, maximum,
mean, and median), histograms, frequency tables, and box-and-whisker diagrams.
The following aspects of the students were observed in class situations, in which they exchanged
analysis results through group activities: 1. When summarising the findings, students tended toincorporate the conclusions of other groups’ analyses into their own; 2. In data analysis, it wasdifficult for students to identify "no trend"; 3. Difficulties were observed in critically interpreting
information beyond the data. These findings can serve as guidelines for improving future data
modelling classes. In addition, this study revealed that students constructed new mathematical models
from the given statistical models when problem-solving. These observations provide suggestive
evidence of the relationship between mathematical and statistical modelling.
ReferencesÄrlebäck, J. B., & Kawakami, T. (2023). The Relationships Between Statistics, Statistical Modelling andMathematical Modelling. In: Greefrath, G., Carreira, S., & Stillman, G. A. (eds) Advancing andConsolidating Mathematical Modelling. International Perspectives on the Teaching and Learning ofMathematical Modelling. Springer Nature.
Kawakami, T., Mineno, K. (2021). Data-Based Modelling to Combine Mathematical, Statistical, andContextual Approaches: Focusing on Ninth-Grade Students. In: Leung, F.K.S., Stillman, G.A., Kaiser, G.,
Wong, K.L. (eds) Mathematical Modelling Education in East and West. International Perspectives on the
Teaching and Learning of Mathematical Modelling. Springer, Cham.
Pfannkuch, M., Ben-Zvi, D., & Budgett, S. (2018). Innovations in statistical modeling to connect data, chanceand context. ZDM, 50(7), 1113–1123.
Wild, C. J., & Pfannkuch, M. (1999). Statistical Thinking in Empirical Enquiry. International Statistical
Review, 67, 223-265.

131

WHAT ARE THE POTENTIAL RELATIONS BETWEEN MATHEMATICAL
MODELLING AND COMPUTING EDUCATION – THE CASE OFSIMULATIONSLouise Meier Carlsen1 and Britta Eyrich Jessen2
1

IT University of Copenhagen, Denmark; 2University of Copenhagen, Denmark

Data science and computing education in upper secondary mathematics has sparked debate amongscholars about the traditional view of math being a gateway to STEM (Boaler & Lewitt, 2019; Conrad& Mazzeo, 2022). Mathematical modelling has emerged as a way to connect STEM and math, but
further research is needed to understand technology's role in this integration (Niss & Blum, 2020;
Siller et al, 2022).
This paper explores the use of technology in mathematical modelling through simulations, building
on previous case studies in this area (Greefrath & Siller, 2017). Drawing on data science education
elements (Heinemann et al., 2018), we investigate the role of technology in a Danish upper secondary
math case study where students work on “Are boys better at mathematics than girls?”
Our study analyses audio and screen recordings to assess how students use technology focusing onquestion-answer dynamics and media-milieu interplay (Jessen, 2022). We ask: What are the potential
relations between mathematical modelling and data science when teaching simulations in upper
secondary school? Can task design that facilitates explicit media-milieu dialectics support students
learning of both.
References
Boaler, J. and Levitt, S.D. (2019). Opinion: Modern high school math should be about data science — not
Algebra 2. LA Times. https://www.youcubed.org/wp-content/uploads/2019/10/LA-times-op-ed.pdf.
Conrad, B. & Mazzeo, R. (2022). Editorial: Mathematicians and the high school math curriculum. IMU News,
111. International Mathematics Union.
Heinemann, B., Opel, S., Budde, L., Schulte, C., Frischemeier, D., Biehler, R., Podworny, S., & Wassong, T.
(2018). Drafting a data science curriculum for secondary schools. In Proceedings of the 18th Koli Calling
International Conference on Computing Education Research (Koli Calling ’18). Association for
Computing Machinery, NY. https://doi.org/10.1145/3279720.3279737

Jessen, B. E. (2022). Characterising the roles of digital resources in mathematical modelling. I J.
Hodgen, E. Geraniou, G. Bolondi, & F. Ferretti (Eds.), Proceedings of the Twelfth Congress ofEuropean Research Society in Mathematics Education. https://hal.science/hal-03759024/
Niss, M., & Blum, W. (2020). The learning and teaching of mathematical modelling. Routledge.
Siller, H. S., Cevikbas, M., Geiger, V., & Greefrath, G. (2022). The role of digital resources in mathematicalmodelling. In C. Fernández, S. Llinares, A. Gutiérrez, & N. Planas (Eds.). Proceedings of the 45th
Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 152-155).
Spain.
Greefrath, G., Siller, HS. (2017). Modelling and Simulation with the Help of Digital Tools. In: Stillman, G.,
Blum, W., Kaiser, G. (eds) Mathematical Modelling and Applications. International Perspectives on the
Teaching and Learning of Mathematical Modelling. Springer, Cham.

132

DIAGNOSTIC COMPETENCE OF PRESERVICE TEACHERS IN THE
CONTEXT OF OPEN MODELLING TASKS
Katharina Wiehe1 and Stanislaw Schukajlow1
1

University of Münster, Germany

Open problems take on an important role in mathematics education. One particular type of open tasks
are open modelling tasks, which are characterized by their relation to reality and require demandingtranslation processes between reality and mathematics (Niss & Blum, 2020). Researchers repeatedly
underlined the necessity of teaching students how to solve open modelling tasks in the classroom(Krawitz et al., 2018). For the teaching of modelling tasks, (preservice) teachers should be able toassess students' performance on open modelling tasks (Greefrath et al., 2022). Indeed diagnostic
competence is a competency facet in the models of teachers' professional competence in modelling(Greefrath et al., 2022) and should address also the openness of the modelling tasks. These tasks
require students to (1) notice missing information and (2) make realistic assumptions for this missing
information (Krawitz et al., 2018). However, empirical considerations on the assessment of student
solutions to open modelling tasks have been lacking. To address this research gap, we conducted aninterview study in the framework of the DOModA project (Diagnose von Schülerlösungen zu
Offenen Modellierungsaufgaben, Diagnostic Modelling Competencies of Student Solutions on Open
Modelling Problems). The aim of this study is to gain insights into the diagnostic competencies of(preservice) teachers in the context of open modelling tasks.
Ten preservice teachers at the University of Münster were first asked to solve four open modelling
tasks and then to rank various student solutions to these open modelling tasks from good to worse
and justify their judgements. The interviews take part in a pair of two preservice teachers. The student
solutions were designed following the model by Krawitz et al. (2018). The student solutions differedregarding (1) whether an assumption was made for a missing data, (2) whether the assumption was
realistic, (3) whether the openness was identified in the answer.
One important finding of the study is that the preservice teachers do not make assumptions, when
they solve modelling problems. Furthermore, preservice teachers do not perceive the necessity forassumptions and do not expect those from students. These findings contribute to a better
understanding of how preservice teachers judges student solutions regarding open modelling tasks.
References
Greefrath, G., Siller, H. S., Klock, H. & Wess, R. (2022). Pre-service secondary teachers’ pedagogical content
knowledge for the teaching of mathematical modelling. Educational Studies in Mathematics, 109(2), 383–
407.
Krawitz, J., Schukajlow, S. & Van Dooren, W. (2018). Unrealistic responses to realistic problems with missing
information: What are important barriers? Educational Psychology, 38(10), 1221–1238.
Niss, M. & Blum, W. (2020). The learning and teaching of mathematical modelling. Routledge.

133

DIMENSIONS OF THE PEDAGOGICAL CONTENT KNOWLEDGE IN A
FIRST MATHEMATICAL MODELLING PRACTICE: THE CASE OF LUCY
Jader Otavio Dalto1, Karina Alessandra Pessoa da Silva1 and Adriana Helena Borssoi1
1

Federal University of Technology - Parana, Brazil

Implementing modelling practices in the classroom is still challenging for many teachers (Niss &Blum, 2020). To overcome these challenges, we understand that teacher training in Mathematical
Modelling, in addition to contemplating theoretical knowledge, needs to articulate knowledge
associated with modelling and its use in classroom practices.
Our research focused on teacher knowledge necessary for teaching, specifically the PedagogicalContent Knowledge (PCK) proposed by Shulman (1986). Borromeo Ferri and Blum (2010) showed
four dimensions of teachers' PCK for modelling: a theoretical dimension, a task dimension, an
instructional dimension, and a diagnostic dimension. In general, each of these dimensions
encompasses specific knowledge that the teaching of mathematics, mediated by modelling, and theteaching of modelling itself require from the modeller teacher.
In this article, we seek to reflect on the following questions: what dimensions of the PCK manifest inplanning and implementing a first practice with modelling from an in-service teacher? How are these
dimensions articulated in the planning and implementation of the practice? To address these questions,
we conducted a case study (Yin, 2002) – the case of Lucy. In 2021, Lucy attended a course on
Mathematical Modelling from the Teaching Perspective, where the collaborative planning of practicewith modelling and its implementation occurred.
The qualitative analysis, structured via Trees of Association of Ideas (Spink, 2013), was supported
by data from audio and video recordings of the classes of the subject in which the planning was
carried out and the implementation of the practice by Lucy with her 8th-grade students.
(13 years old). The results revealed that the PCK dimensions were articulated through teachers'interventions during collaborative planning and the report of the implemented practice, as well as inLucy's actions during classroom management. These results reinforce the need to improve new
designs for training courses that promote the articulation of the teachers’ PCK dimensions.
References
Borromeo Ferri, R., & Blum, W. (2010). Mathematical modelling in teacher education–experiences from amodelling seminar. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.), Proceedings of thesixth Congress of the European Society for Research in Mathematics Education (pp. 2046–2055). INRP,
Lyon.
Niss, M. & Blum, W. (2020). Learning and Teaching of Mathematical Modelling. Routledge, England.
Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2),4–14.
Spink, M. J. (2013). Práticas discursivas e produção de sentidos no cotidiano. Rio de Janeiro: Editora Virtual.
Yin, R. (2002). Case study research: Design and methods. Newbury Park: Sage.

134

AN ETHNOMODELLING PERSPECTIVE OF EMIC, ETIC, AND DIALOGICANALYSIS OF ETHNOMODELS IN THE AFRO-DESCENDANTCARIBBEAN DANCE PALO DE MAYO OF COSTA RICA
Steven Eduardo Quesada Segura1, Milton Rosa1 and Daniel Orey Clark1
1

Universidade Federal Ouro Preto, Brazil

This study is the result of a master’s research on the traditional dances of Costa Rica, which was heldat the Universidade Federal Ouro Preto, in Minas Gerais, Brazil. Its intent was to highlight local
knowledge and ethnomathematical practices that are present in these dances. Thus, this presentation
analyses ethnomodels that can be elaborated in relation to the Afro-descendant dance of Palo Mayo,
which can contribute to the development of pedagogical actions in the perspective of ethnomodelling.
This is qualitative research in which some of the results are related to the elaboration of ethnomodels:a) emic (local) based on observations made with the dancers of the traditional dance of Palo de Mayo,b) etic (global) based on the perceptions of a mathematics teacher and the researcher, and c) dialogic(glocal) related to dialogue that happened between the Palo de Mayo dancers and the researcher, who
has both emic and etic views of mathematical knowledge, as he is a member of the dance culture andhe is also a mathematics teacher.
The connections between ethnomathematics and the sociocultural perspective of mathematicalmodelling provide for the development of pedagogical actions in classrooms, which are directed
towards to the awareness of the relevance of the social and cultural aspects of mathematics (Orey &Rosa, 2021).
Figure 1 shows the elaboration of a dialogic ethnomodel that represents the mathematical knowledge
of Palo de Mayo dancers in relation to the school/academic mathematical knowledge that is implicit
during the development of this dance. This ethnomodel generates an appreciation and respect for boththe emic (local) and etic (global) knowledge through a cultural dynamism (dialogic/glocal).
Figure 1: A dialogic ethnomodel. Source: Personal file of the authors.

The partial results of this study, which are related to data collected in participant observation, show
the development of a dialogue between emic (local)mathematical (cultural dynamism) present in thesociocultural traditional Palo de Mayo dance with school/academic mathematical knowledge, which
values and respects this dynamic of the encounter of distinct cultures.
References
Orey, D. C., & Rosa, M. (2021). Ethnomodelling as a glocalization process of mathematical practices throughcultural dynamism practices through cultural dynamism. The Mathematics Enthusiast, 18(3), 438-468.

135

A SUGGESTION OF A FRAMEWORK TO CAPTURE THE STRUCTURE OF
REGIONAL PROBLEMS AND ITS META PROBLEM
Tadashi Misono1 and Yuki Watanabe2
1

Shimane University, Japan; 2Tokyo University of Science, Japan

Mathematical modeling problems for students to solve in lessons have been developed. For example,
the International Mathematical Modeling Challenge (IM2C) in Australia is releasing problems.
“Waste not, want not” is one of problems developed by the IM2C. Attempting to solve the problem
is beneficial for students. However, in many cases, mathematical modeling problems tend to belocalized to original situations faced by specific areas and residents. Therefore, teachers must modify
the variables in the original problems to suit the local situation in which students live. For example,
the original “waste not, want not” problem has variables such as initial value for population inAustralia and amount of waste per person per year in Australia. Certainly, teachers in Japan must
review the variables for students in Japan. However, the modeling process will be fixed between
specific situations.
To overcome this problem, Misono and Takanashi (2022) suggested a framework to capture thestructure of regional problems and its meta problem. Figure 1 shows its translation to English.

Meta Problem
localize

localizeglobalize
Specific Problem for Region A

globalize
Specific Problem for Region A…

Figure 1. A framework to capture the structure of regional problems and its meta problem

In our previous study, we only refer to problems and situations limited to Japan. However, in thisstudy, we apply this framework to worldwide situations.

Acknowledgments
This work was supported by JSPS KAKENHI (Grant number: 22H01065).
References
International Mathematical Modeling Challenge (n. d.). Example problems. https://www.immchallenge.org.au/supporting-resources/example-problems. (Referred on May 31, 2023)
Misono, T. & Takanashi, Y. (2022). A Suggestion of a Framework for Developing Mathematical Modeling
Activities on Regional Problems Which are Available for Different Regions. JSSE Research Report, 36(3),39–44. (in Japanese)136

PosterPresentation

137

EXAMINING EARLY-GRADE CHILDREN’S MATHEMATICAL MODELSIN PROPORTIONAL SITUATIONS: A GLIMPSE FROM TASK-BASEDINTERVIEWS
Keiko Hino1 Hiraku Ichikawa2 and Hisae Kato3
1

Utsunomiya University, Japan; 2Miyagi University of Education, Japan; 3Hyogo University ofTeacher Education, Japan

The purpose of this paper is to explore aspects of mathematical models related to proportional
reasoning in early-grade primary school children. This paper bases “Models and Modeling
Perspectives” (MMP, Lesh & Lehrer, 2003) and examines children’s interpretations and descriptionsof proportional situations in task-based interviews.
Proportionality is a key idea and model used in mathematics and science. However, it has been shownthat students who have studied proportionality apply it without examining the problem situation (e.g.,
De Bock, et al., 2002). In reality, they do not necessarily master the powerful model of proportionality.
By capturing the case of early-grade children from the MMP, it is expected to reveal factors that havenot been fully recognized so far and to provide useful suggestions on how proportionality can belearned. For example, one of the concerns in previous studies of young children is how they overcome
the overreliance to whole-number information and how a progress from a simple sensory perception
to an attention to numerical relations of proportionality is facilitated (e.g., Hurst & Cordes, 2018).
The research question addressed is “What models do children exhibit to find unknown quantities, andwhat are their own ways of perceiving and hypothesizing quantity relationships?” In this paper, weoffer cases of children’s situated problem-solving from our task-based interviews with 18 children,
aged 5-8, by using everyday situations. In the interview, several proportional situations were
presented in which two quantities are related at a certain rate, e.g., “We exchange 2 chocolate bars
and 3 marbles. How many marbles can you get if you have 6 chocolate bars?”
From the results of the interviews, it became apparent that the children’s models of the relationshipbetween the quantities varied, compared with the ones intended by the interviewer. They attended
broader aspects of the quantities and relationship by reflecting on their everyday experiences andvalues. It is also observed that the children revised their interpretations by trying to resolve theirperceived unfitness. By reflecting on these cases, we discuss what implications children’s models
offer for the teaching/learning of the basis of proportional reasoning in early-grade primary school.
References
De Bock, D., Van Dooren, W., Janssens, D., Verschaffel, L. (2002). Improper use of linear reasoning: An indepth study of the nature and the irresistibility of secondary school students’ errors. Educational Studies inMathematics, 50, 311-334.
Hurst, M. A., & Cordes, S. (2018). Attending to relations: Proportional reasoning in 3- to 6-year-old children.
Developmental Psychology, 54 (3), 428-439.
Lesh, R., & Lehrer, R. (2003). Models and modeling perspectives on the development of students and teachers.
Mathematical Thinking and Learning, 5(2-3), 109-129.

138

HOW DO STUDENTS CRITICALLY INTERPRET PLURAL WAYS TO
DETERMINE THE MOST POPULAR ONIGIRI?
-ATOMISTIC APPROACH FOCUSED ON INTERPRETATION,
VALIDATION AND MODIFICATIONMinami Asahara1, Takashi Sumioka1, Makoto Chonan1 and Kazuki Koike1
1

Yokohama National University (Master’s student), Japan

In a society where online information and tools, such as Chat GPT, are becoming increasingly
prevalent, it is essential to develop competencies that enable individuals to identify and analyze
hidden or explicit assumptions in the given information and make decisions based on that analysis.
This activity is strongly concerned with prescriptive modelling, which aims to pave the way for taking
action based on decisions resulting from a certain type of mathematical consideration (Niss, 2015).
This study aimed to foster sub-competencies in modelling among sixth-grade students by applying
an atomistic approach (Kaiser & Brand, 2015), focusing on interpretation, validation, andmodification. This study sought to determine how students interpret, compare, and validate the threevoting models in response to the question, "What is the best ingredient for an onigiri?"
The teaching experiment involved teachers presenting six different onigiri ingredients to 28 students
and three voting models to determine the best ingredients. The three voting models were (a) a singleentry method, (b) a deciding vote method, and (c) a point-distribution method. The teachers gave thestudents the opportunity to vote using these three models and then analyzed the results.
This study analyzed the transformation of students' thinking, dividing the analysis into three stages:(1) personal thinking after understanding the problem, (2) personal thinking after voting in class, and(3) personal thinking after group discussions. This study also asked about the significance of this typeof problem in the future.
The results showed that more than 60% of the students' thinking was transformed after the threestages, and about 85% of them had a positive opinion on the effectiveness of this kind of problem
solving. During the teaching experiment, students were able to point out the advantages anddisadvantages of the voting method as their ideas differed from each other. For example, an advantageof the deciding vote method is that it results in fewer dead votes. The advantage of the pointdistribution method is that it accurately reflects feelings, whereas the disadvantage is that it results inan artificially extreme score. At the end of the class, some students suggested changing the modelaccording to their purpose.
While the teaching experiment focused on the atomistic approach in this material, which is one of theteaching methods used to encourage students' interpretation and validation, the study found thatstudents were unable to modify the given voting models or create new ones. Therefore, encouraging
students to pursue better models remains a challenge.
References
Niss, M. (2015). Prescriptive modelling – challenges and opportunities. In G.A. Stillman,W. Blum, & M. S.
Biembengut (Eds.), Mathematical modelling in education research and practice: Cultural, social andcognitive Influences (pp.67-79). Cham: Springer.
Kaiser, G., & Brand, S. (2015). Modelling competencies: Past development and further perspectives. In G.A.
Stillman, W. Blum, & M. S. Biembengut (Eds.), Mathematical modelling in education research andpractice: Cultural, social and cognitive Influences (pp.129-149). Cham: Springer.

139

HOW TO USE ELEMENTARY MATH TEXTBOOKS TO PROMOTE
MATHEMATIZATION
Hatsuda Hiroki
Yokohama National University (Master’s student), Japan
Teacher education in the teaching of mathematical modelling has become a worldwide topic (Blum,2015). In Japan, methods to facilitate training for teachers in using modelling at schools are also being
considered (Saeki et al.,2023). In this study, we will focus on methods for arranging problems in
textbooks that teachers use daily to allow non-specialist teachers of mathematics to gradually
incorporate the teaching of modelling at the elementary school level. The purpose of this study is toconsider how word problems can be arranged to promote mathematization and to clarify theireffectiveness and challenges through practical instruction. The following methods are used to arrange
the textbooks: (1) inserting sentences that serve the purpose of the text, (2) using actual objects tomake realistic situations in the text questions, and (3) asking questions to shake up the students' viewof mathematics according to their actual situation.
In the actual class, we select the task from the textbook, "Divide 72 sheets of colored paper among
three students in equal numbers. How many sheets does one person get?" The first step is to state"folding paper airplanes" as the objective for solving the problem. The reason is that the way toanswer the question will change depending on the objective. The second step is to use colored paper
when thinking about the 72 sheets of paper, to remind the students of the implicit assumption that the
sheets are to be divided regardless of color. The problem we arrange is: "Three students make paper
airplanes. 72 sheets of colored paper are divided equally among three students. " As a third step, after
solving the problem and getting the answer from 24 sheets, the teacher asks "One of the three students
says that the bundle of papers over there is better. Why do you think so?"The practical class is given to 32 fourth-grade elementary school students for two hours (45 minutesx 2). In the introductory stage, three students are asked to divide 72 sheets of origami paper equallyin front of everyone. After the students work out the division equation and give the answer of 24
sheets per person, we make evocative questioning by shuffling the paper. Based on the children's
murmurs such as "We should not use gold or silver because the colors are too bright" or "After sorting
the sheets of origami paper by color, we should divide them into three students," a development leads
to the consideration and revision of the mathematization of setting new assumptions.
From the above study, we find that the implicit assumption becomes apparent by including a purposein the text, making the problem situation realistic, and making evocative questioning according to thestudent's actual situation. However, further study is needed to find out how to modify the model basedon the implicit assumption.
References
Blum, W. (2015). Quality teaching of mathematical modelling: What do we know, what can we do? In S. J.
Cho (Ed.), Proceedings of the 12th International Congress on Mathematical Education(pp.73–96).
Springer.
Saeki, A., Kaneko, M., Kawakami., T & Ikeda., T (2023). In-service teachers’ transformation of amathematised task into modelling tasks. In G. Greefrath, S. Carreira, & G. A. Stillman (Eds.), Advancing
and Consolidating Mathematical Modelling (pp. 241–257). Springer.

140

FOSTERING STUDENTS’ ABILITY TO COMPARE AND CRITICALLYEXAMINE PLURAL MODELS TO SOLVE REAL WORLD PROBLEMS-TEACHING EXPERIMENT BY USING A TENT PROBLEMShota Kita1, Masashi Ikemura1, Kodai Aratani1 and Toshikazu Ikeda2
1

Yokohama National University(Master’s student), 2Yokoaham National University, Japan

This article explored how changes in society, such as the proliferation of Internet information andchatbots, make everyday life’s modelling processes more diverse. People are expected to consider at
least two models and make judgments based on a comparison of their advantages and disadvantages.
This kind of activity is related to the atomistic approach to modelling, which focuses on subcompetencies such as interpretation, validation, and modification (Kaiser & Brand, 2015). Thisapproach can be applied to real-world problem-solving. This study aimed to identify the effectivenessand challenges of the atomistic approach for 50 twelfth-grade students by analyzing their descriptions
during and after the teaching experiment.
The modelling task in the teaching experiment was to make the tent interior comfortable. The teacher
posed the question of how to determine the angle between the two poles set in the tent. Two options
were presented: 1) maximizing the volume inside the tent and 2) maximizing the length of the radiusof a semicircle tangent to the triangle composed of two poles. Two questionnaires were administeredat different stages of the experiment. The first question asked which of the two models was moresuitable for the following three stages: individual thoughts before mathematical analysis, individual
thoughts after mathematical analysis, and individual thoughts after group discussions. The second
question, "Is it meaningful in a society to consider a more suitable model by comparing and validating
plural models like this tent problem?" was asked after the teaching experiment using the Likert scale
method. In addition, students were asked to describe their reasons for selecting their preferred model.
In the experimental teaching, students pointed out through group discussions the advantages of having
a high space within the tent, which was derived from Model 2), for greater freedom of movement,
and the usefulness of having a large floor, which was derived from Model 1), for greater comfort
while sleeping. Furthermore, some have pointed out the importance of adapting tent designs
according to their purposes. The results showed that over 70% of the students’ thoughts weretransformed through the three stages of analysis in the first question. Approximately 95% of students
selected “totally agree” and “agree” in the Likert scale questionnaires regarding the second question.
Furthermore, 70% of students described the usefulness of considering at least two models and making
a judgment by comparing and validating multiple models. However, 30% of the students did not
appreciate its usefulness.
Overall, this article highlighted the importance of fostering students’ abilities to compare andcritically examine multiple models for solving real-world problems. The study results suggested thatthe atomistic approach can be an effective way to achieve this, although further discussion is needed
regarding the teacher's role in providing the two models for comparison.
References
Kaiser, G., & Brand, S. (2015). Modelling competencies: Past development and further perspectives. In G.A.
Stillman, W. Blum, & M. S. Biembengut (Eds.), Mathematical modelling in education research andpractice: Cultural, social and cognitive Influences (pp.129-149). Cham: Springer.

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WHAT METAPHORS DO CHILDREN UTILIZE TO MAKE CONNECTIONSBETWEEN MATHEMATICS AND THE REAL WORLD?
Kensuke Koizumi1, Ryuta Tani2 and Ryo Hanzawa3
1

Gunma University, Japan; 2Tanaka Gakuen Ritsumeikan Keisyo Primary School, Japan; 3Seya
Primary School, JapanWe learn to see time as money from the expression "time is money”. Thus, metaphors serve as a kindof filter through which we try to better understand certain things. The filter expresses a perspectivethat reflects the person's own experience. Carreira (2001) focused on the relationship between theconcept of mathematical models and the concept of metaphors, noting that mathematics and the realworld influence each other to create meaning. This study examines the connection between
mathematics and the real world from the perspective of metaphor. The meaning of “metaphor” in thisstudy relies on the discussion of Lakoff & Johnson (2003) and Sfard (1997).
This study was conducted as a case study on the semantic generation of mathematical symbols suchas □ and △, and was intended to identify the following two research questions.
RQ1. How does the connection between mathematics and the real world work in making sense?
RQ2. What kind of effects can be expected by eliciting the generation of meaning by children(especially through classroom interaction)?
The findings of this case study of 23 children (fifth grade elementary school) revealed the follows.
In an activity that attempted to explain the distinction between a) variables and b) unknowns, themeaning of □ and △ were associated with the real world as follows. (The above is related to RQ1)a) chameleons, days of the week, first and last name relationships, community and town relationships,
attendees and absentees relationships, etc.
b) Omikuji (a paper fortune), capsule toy, lucky bag, etc.
When such metaphors are incorporated into lesson planning, the following effects are expected. First,
understanding with conviction related to concrete experiences. Second, understanding ofmathematical concepts with multifaceted meanings. The activity of interpreting multiple
metaphorical expressions that extracted different elements, could constitute an opportunity toorganize the multifaceted meanings of mathematical concepts, including meanings that were implicit
until then. Third, understanding through the diversity of base selection. Furthermore, in some cases,
as a perspective on the development of learning, it can be a trigger for other learning content (in thiscase, for example, “community and town relationships”). (The above is related to RQ2)
References
Carreira, S. (2001). The mountain is the utility – on the metaphorical nature of mathematical models. In J. F.
Matos, W. Blum, S. K. Houston and S. P. Carreira (Eds.). Modelling and mathematics education: ICTMA9– applications in science and technology (pp.15-29). Chichester, Horwood Publishing.
Lakoff, G., & Johnson, M. (2003). Metaphors we live by. The University of Chicago Press, New York.
Sfard, A. (1997). Commentary: On Metaphorical Roots of Conceptual Growth, in English, L. (Ed.),
Mathematical reasoning: analogies, metaphors, and images (pp.339-371). Lawrence Erlbaum Associates.

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HOW DOES FIRST-YEAR SCITECH EXPERIENCE OF
PROFESSIONAL DEVELOPMENT EFFECT MATHEMATIC EDUCATION?
Satoru Fujitani1 and Motoko Fujitani2
1

Tamagawa University, Japan; 2Joetsu University of Education, Japan

This paper reports professional development practices of first-year college education based on theperspective of science, technology, engineering, and mathematics (STEM) education. Meaningful
science for young children builds on the emotional underpinnings of their curiosity and concerns
about the everyday world, and their pleasure in exploring it (Harlan & Rivkin, 2003). Through theimplementation of the course named 'Introductory Study on Natural Environment and Science inChildhood Education', Fujitani (2022) has engaged in activities for first-year teacher-education
students to cultivate STEM interests and to build a foundation in mathematics for primary education.
We describe an example of mathematical modelling in practice in this course as follows.
During a visit to the garden inside the shrine (Figure 1), students
worked on the estimation of the natural regeneration of forests
after learning forester’s research practice at the time ofafforestation for the shrine. The students attempted to find thetime period of regeneration toward the climax stage of forests.
Students also undertook an activity to build an antenna for digital
television broadcasting (Figure 2). The size of the loop antenna
they made was defined by the frequency band of the radio waves
Figure 1: Visit for nature observation
they planned to receive. Following the instruction, the students
cut out aluminium foil according to the wavelength of the radio
waves to produce their antennas.
In addition, the students also have engaged in the experiences ofmammalian and aquatic animal rearing and cultivation ofvegetables and flowers through this course, for instance. By
introducing students to these scientific experiences early on, thestudents, who will be teachers, are able to build the foundation
Figure 2: Making TV antennas
for developing scientific thinking and inquiry skills that would
be the scaffold for the spark of the student’s interest and excitement in STEM fields. The course
makes teacher-training students foster a love and curiosity for science, and encourages them to pursue
further STEM learning opportunities, and it can work as an access of mathematical modelling forthem. It can also affect the children who are taught by the students as teachers in future likewise.
References
Fujitani, S. (2022). First-year education to raise awareness of nature and science in childhood education.
Human and Education: Yearbook of Mejiro University Research Institute of Higher Education, 16, 104110. (in Japanese)
Harlan, J.D., & Rivkin, M.S. (2003). Science experiences for the early childhood years: an integrative affective
approach. Prentice Hall.

143

&RQIHUHQFH,QIRUPDWLRQ
ClimateAwaji has a Humid subtropical climate (Köppen Cfa) characterized by warm summers.
The average annual temperature in Awaji is 16.3 °C / 61.3 °F. The average annual rainfall
is 1,600 mm, with September as the wettest month. The temperatures are highest on
average in August, at around 26.6 °C, and especially deadly heat in this summer. The
temperatures in September are forecasted hot more than usual.

VISA
Information for foreign nationals about visas, including the application procedures andother related FAQs. Please access to the following web site of Ministry of Foreign Affairsof Japan.
https://www.mofa.go.jp/j_info/visit/visa/index.html

Electricity in Japanx

In Japan the power plug sockets are of type A and B. The standard voltage is 100 V
and the frequency is 50 / 60 Hz.

x

In Japan the power plug sockets are mainly type A. Check out the following pictures;Type A: mainly used in North and Central America, China and Japan.

x

The power plug sockets of Awaji Yumebutai International ConferenceCenter are only Type A.

Registrationx

Please check your name, affiliation, country, and lunch tickets.

xx

Have a name holder on a neck strap during the conference.
Excursion is option, and check your choice (your choice is printed on name holder).

  
 
x

Opt. tour: Kyoto  Opt. tour: None

Opt. tour: Himeiji

Please fill in the form ‘Personal Information Handling Agreement’.

144

Time table of the bus stopsxx

Cash or transportation IC card (ICOCA, SUICA, PASMO) are available.
Please check the following web site if you will use transporatation IC card.
[ICOCA] https://www.westjr.co.jp/global/en/howto/icoca/[Suica] https://www.jreast.co.jp/multi/en/welcomesuica/welcomesuica.html[PASMO] https://www.pasmo.co.jp/visitors/en/about

https://www.yumebutai.org/pdf/time_sin_san_mai.pdf

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Wi-FixxFree Wi-Fi is available in each room. Please check SSID & Password in each room.
Free Wi-Fi is also available outside the rooms and in the corridor (SSID is floor nameand no password).

Welcome Reception (9th Sunday)x
x

Welcome reception is held at the lobby (B1F). It’s free of charge.
Light meals, snacks, and drinks are available.

Coffee Break (It’s included in the registration fee.)x

Coffee, tea, and some drinks are available on 11th (Monday), 12th (Tuesday), and14th (Thursday). The place is at the lobby (B1F) in the morning on the 11th (Monday),and the 3rd and 4th floor break area in the rest.

x

In the afternoon on the 10th (Monday) and on the 15th (Friday), coffee break isavailable at the lobby (B1F).

Happy Hour (10th Monday & 11th Tuesday)xx

Happy hours are held at the lobby (B1F). It’s free of charge.
On the 10th (Monday), the time is 18:30 to 19:30.

x

On the 11th (Tuesday), the time is 17:00 to 18:30, and concurrently with poster
session.

x

Light meals, snacks, and drinks are available.

Lunch (It’s included in the registration fee.)xLunch is available every day from 11th (Monday) to the 14th (Thursday).

x
x
x
x

Coffee, tea, and some drinks are also available except on the 13th (Wednesday).
Please exchange lunch ticket for lunch at break area except the 13th (Wednesday).
On the 13th (Wednesday), exchange lunch ticket for lunch box at the lobby (B1F).
Don’t forget take away lunch box before participating the excursion (optional tour)
please.

xLunch menu will NOT change from your registration request (None, Vegetarian, orOther). Please check the lunch tickets.

x

It is possible to take lunch into the building and outside, except to the public floor ofGrand Nikko Awaji (2F).

146

Conference Dinner (14th Thursday) (It’s included in the registration fee.)x

Conference dinner is held at the Reception Hall B (2F).

x

To enter the Reception Hall B, you have to place a name holder on a neck strap.

Corporate Exhibition
Exhibitions by supported companies (Casio Institute for Educational Development,
Gakkotosho, and Kyoiku-shuppan) is held at 404.

Smoking
No smoking on the premises. Smoking is only permitted outside in designated areas (4F
courtyard).

Other inquiresx

If you have questions, please contact LOC staffs and/or students’
staffs who have a yellow-colored name holder on a neck strap.

x

Rooms 401 and 402 are the LOC office rooms. Please come therooms if you need anything.

Excursion (13th Wednesday) *Option
After the plenary lecture 2, please meet at a designated place in front of the building.
Please don’t forget to take away lunch box.

Hospitals
If you have illness or poor physical condition etc…, please inform the staff. Our staff will
introduce the following hospital.
x Seirei Awaji Hospital[URL] https://www.seirei.or.jp/awaji/index.html
Desk: 8:30-11:30, 13:30-16:30 *10th Sunday is closed.
Address: Yumebutai 1-1, Awaji-City, Hyogo
Tel: (+81) 799-72-3636
x Higashiura Heisei Hospital[URL] https://higashihp.jp/news/id_3278
Consultation time: 9:00-12:00, 15:00-17:00 *10th Sunday is closed.
Address: Kuruma 1867, Awaji City, Hyogo
Tel: (+81) 799-74-0503

147

Evacuation routes

148

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153

6XSSRUWHUVList of Participants
[A]

[F]

Abe Yoshitaka

Ferrando Irene

Akizawa Katsuki

Fisher M. Diana

Albarracin Lluis

Frejd Peter

Alwast Alina

Fujiki Ryo

Aratani Kodai

Fujitani Satoru

Arleback Bergman Jonas

Fujitani Motoko

Asahara Minami

Fujiwara Daiki

Asami-Johansson Yukiko

Fukuda Hiroto[B]

Fukushima Takumi

Back Andreas

Fulano-Vargas Cecilia Blanca

Barquero Berta

Funes Castillo Josefa

Barrios Jara Nelson Enrique[G]

Besser Michael

Gaete Peralta Claudio

Bleymehl Johanna Emilie Sonja

Galluzzo Benjamin

Borromeo Ferri Rita

Geiger Vince

Boshoff Liezle

Gerber Sebastian

Boshoff Beyers Willem

Goksen-Zayim Sevinc

Brown Paul

Goos Merrilyn

Brown P. Jill

Gordon Anthony John

[C]

Goto Ayaka

Cai Jinfa

Greefrath Gilbert

Carlsen Meier Louise

Guerrero Carolina

Carreira Susana

Gwak Do Kyeong

Cevikbas Mustafa[H]

Cheung Alfred

Hachiya Yusuke

Chikusa ToshihikoHatanaka Ryohei

Chonan Makoto

Hatsuda Hiroki

Cornelissen Belinda

Hattori Yuichiro

Czocher A. Jennifer

Helder Joanne Pamela[D]

Higuchi Shota

Dalto Otavio Jader

Hino Keiko

Dominguez Angeles

Ho Yee Hung

Durandt Rina

Hofmann Stephanie[E]

Huincahue Jaime

Ekol L. George

154

[I]

Koike Kazuki

Ichikawa Hiraku

Koike Ako

Ikeda Toshikazu

Koizumi Kensuke

Ikemura Masashi

Komeda Shigekazu

Inomoto Osamu

Krawitz Janina

Ishida Ayumu

Kudo suguru

Ishikawa Masaaki[L]

Ishikawa Daisuke

Lee Kin Sum

Ishikawa Ayane

Lee Chun Ting

Ito Yasuhiro

Lee Gima[J]

Leikin Roza

Jessen Eyrich Britta

Lu Xiaoli

Julie Martin Cyril

[M]

[K]

Ma XinKaiser Gabriele

Manouchehri Azita

Kammerer Melanie

Matsumoto Shinichiro

Kaneko Masafumi

Matsushima Mitsuru

Kannapiran Shaliny

Matsuzaki Akio

Kato Hisae

McCurry Liana

Kawakami Takashi

Mineno Kosuke

Kawasaki Tetsushi

Misono Tadashi

Kawazoe Mitsuru

Mizoguchi Tatsuya

Kim Hee-jeong

Morita Daisuke

Kim Dong Joong

Muhrman Karolina

Kim Somin[N]

Kim WonNaftaliev Elena

Kim Hye In

Nagae Hiromi

Kim Jeong Heon

Nakamura Haruka

Kimura Kent

Nakamura Koichi

Kimura Kumiko

Nisawa Yoshiki

Kissane Barry

Nishimura Keiichi

Kita Shota[O]

Kita Mike Masayuki

Obayashi Masanori

Kita Misuzu

Obayashi Shogo

Kitajima Shigeki

Ogawa Misaki

Kohen Zehavit

Oh Chunyoung

Kohen Hanan

Oh Young-Seok

155

Ohtani Minoru

Tachihara Mikino

Okabe Yasuyuki

Takayama Takuma

Okazaki Masakazu

Tangkawsakul Sakon

Omura Rikuto

Tortola Emerson

Orey C. Daniel[U]

Ostergaard Hellsten Camilla

Ueda Rintaro

Ota Shinji

Uematsu Keita

Otaki Koji

Ugsonkid Songchai[P]

Unshelm Nina

Paditz Ludwig

Uozumi Shohei[R]

Urayama Daiki

Rodriguez-Gallegos Ruth

[W]

Rodriguez-Gallegos Daniel

Wang Lidong

Rogovchenko Yuriy

Watanabe Yuki

Rogovchenko Svitlana

Wedelin Dag

Rosa Milton

Wiehe Maria Katharina[S]

Wienecke Lisa-Marie

Saeki Akihiko

Wirth Laura

Sato Kiyoto[Y]

Schlueter Dominik

Yamamoto Yuzu

Schukajlow Stanislaw

Yamanaka Hitoshi

Segura Carlos

Yan YuanSeino Tatsuhiko

Yanagimoto Akira

Sevinc Serife

Yang Kai-Lin

Shiina Mihoko

Yang Xinrong

Shimizu Yoshinori

Yata Kosuke

Shimizu Kunihiko

Yoshimura Noboru

Shioda Teppei[Z]

Sousa Nivalda Barbara

Zavala Genaro

Spooner Kerri

Zbiek Mary Rose

Stillman Gloria
Sumioka Takashi
Sun Yongjian
Surel Anna
Suto Takamasa[T]
Tachibana Yuki

156