22/06/2022

Mathematics Education to Develop Students Agency
Part III: Shape and Figure
Difference of Shape and Figure

Overview

Student Agency:
Learned how to learn

 Constructivism is the theory forknowing on Epistemology(Philosophy).
 On this philosophicalperspective, any learning can be
explained by constructivism
from the perspective ofstudents: Even though theylearned mathematics byInjection Approach.
 Thus, it does not mean teachingapproach itself. However, somemath-educators try to explain
their own developed teachingapproaches based on
constructivism: It is possible
however it produced
misunderstanding that theapproaches themselves isconstructivism:
 Why I do not think so?
Crucial issue is what content
students learned in relation tohow to learn. Depending on theobjective, learning content
2
itself changes.

ISODA Masami, Prof/PhD
Director of CRICED
University of Tsukuba, Japan

1

Review of Part III, Lesson 4

Appreciation
Reflection

StudentAgency inthis lecture:
Developstudents
who learn
mathematics
forthemselves

Acquisition

All these are the content of learning!!

Long term task sequence

Student Agency:
Learned how to learn

Crucial issue is whatcontent students learnedin relation to how to learn.
Depending on theobjective, learning content
itself changes.
Curriculum and Literacy
frameworks are the source ofobjectives:
 The frameworks just show
us the necessary ways ofwriting objective only.
 Concrete objectives
themselves are usually
written based on thecurriculum standards.

On this lecture, weuse SEA-BES: CCRLS
3
to explain objectives

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22/06/2022

Mathematization (Freudenthal, 1973) and van Hiele Levels (van Hiele,1986)

Curriculum Sequence under Mathematization Principle (Freudenthal, 1973)

Freudenthal defined Mathematization by the re-organization of (mathematical) experience by mathematical means.

Freudenthal defined Mathematization by the re-organization of (mathematical) experience via mathematical means .

van Hiele Levels (Level of Thinking by Freudenthal): Teaching is the activity for students to be able to think on upper level.

Van Hiele, P. M. (1986). Structure and Insight, Accademic Press. p39.

Le Object
velOperation/
Means

Nature of language:
ways of thinking

L1

Shape (形)Concrete object istreated by thehidden attribute ofeach shape

L2

ConcreteObjectFigure(図形)attributevan Hiele Levels (Level of Thinking by Freudenthal): Teaching is the activity for students to be able to think on upper level.

Level

ObjectOperation/Means Nature of language: ways of thinking

L1

Concrete O.

Shape (形)Concrete object is treated by the hidden attribute of each shape

L2

Figure(図形)attribute

Attribute belonging each shape is treated by the properties of figure

L3

Properties

Proposition

Figures are recognized/operated by properties.

L4

Proposition

Proof

Proposition are explained by proof (local theory)

L5

ProofLogic

Each proposition become a part of Axiomatic system (general5 theory)

L3

Properties Proposition(平面図形:plane figure)

Attribute belonging
each figure is treatedas the properties offigure
Figures arerecognized byproperties.

Contradiction between levels
Round shape is not a circle
Round shape is not a circle(asfigure).
(as figure).

Fold
shapesandandfind
Fold roundround shapes
finda circle
wasa circle
whenwhenit wasitfolded
foldedjust overlap.
just overlap.
Circle isis drawn
drawn byCircle
bycompasswith center
andcenterradius.andcompass
withradius.
How can we provide our students forthe opportunity to think by and forthemselves, and learning how to?

Contradiction is theopportunity to think anddiscuss mathematically
Square,rhombusand
andSquare, rhombus
rectangle
are different
figure(shape).
rectangle
aredifferentfigure
PropertiesPropertiesof figure are(shape).
of figureattributeof figure(shape).
areattributeof figure(shape).
Square is rhombus.

Squareisrectangle.
rhombus.
Square is
Squareisisrectangle.
Rhombussquare if ... .
Rectangle is... .
Rhombusis rhombus
square ifif ...
Rectangle is rhombus if ... .
By Isoda such as 2015 6and 2018

Mathematization from L1 to L2
Why this activity
is necessary?
 Japanese had alreadydeveloped the task sequenceof it from 1934 textbook up to1950s curriculum reform
because they had been
distinguished shape, figure andplane figure in Japanese at thebeginning of school system in1870s.
 If we use appropriate
terminology to distinguishconceptual differences, we canproduce appropriate task
sequence for students’ activity.
 What terminology is the key
on those two pages’ activity?
Shape, Attribute via
collecting, touching,stacking and making

Let’s explain byusing terminology.
Did you use?
Shape, Attribute via tracing, cutting and drawing782

22/06/2022

Those pages are NOT
independently existed.
It has task sequence
beyond chapters andgrades as well as othercontent-strands.
https://support.gakuto.co.jp/
r2s_sansu_translation/

910

Which level is it?

11123

22/06/2022

Mathematization (Freudenthal, 1973) and van Hiele Levels (van Hiele,1986)
Freudenthal defined Mathematization by the re-organization of (mathematical) experience by mathematical means.
van Hiele Levels (Level of Thinking by Freudenthal): Teaching is the activity for students to be able to think on upper level.

Le Object
velOperation/
Means

Nature of language:
ways of thinking

L1

Shape (形)Concrete object istreated by thehidden attribute ofeach shape

L2

L3

Concrete
ObjectFigure(図形)attribute

Properties Proposition(平面図形:plane figure)

Attribute belonging
each figure is treatedas the properties offigure
Figures arerecognized byproperties.

Contradiction between levels
Round shape is not a circle
Round shape is not a circle(asfigure).
(as figure).

Foldshapesandandfind
Fold roundround shapesfinda circle
wasa circle
whenwhenit wasitfolded
foldedjust overlap.
just overlap.
Circle isis drawn
drawn byCircle
bycompasswith center
andcenterradius.andcompass
withradius.
How can we provide our students forthe opportunity to think by and forthemselves, and learning how to?

Contradiction is theopportunity to think anddiscuss mathematically
Square,rhombusand
andSquare, rhombus
rectangle
rectangle
arefiguredifferent
figureare different(shape).
(shape).
of figure
PropertiesPropertiesof figure areareattributeof figureattributeof figure(shape).
(shape).
Square is rhombus.

Square is rhombus.
Square is rectangle.
Squareis isrectangle.
Rhombussquare if ... .
Rhombussquare ifif ...
Rectangle isis rhombus... ..
Rectangle is rhombus if ... .
By Isoda such as 201513
and 2018

4