WERA2019, Tokyo, Aug. 6th 2019

A Strategy for STEM Learning in a Changing Society:
Focusing on the Undergraduate Program

Mathematics Education
Using Real-World Problems
Mitsuru Kawazoe
Osaka Prefecture UniversityJapanWERA2019, Tokyo, Aug. 6th 2019

Background• It is increasingly important for all students at all educational levels toacquire mathematical skills for use in real-world situations.
Mathematical Literacy• An important issue in Japan: mathematical literacy education for ‘Bunkei’

students (=humanities & social sciences students)• Aims of mathematical literacy educations for Bunkei students:• to foster students’ mathematical skill for use in real-world situations• to improve students’ attitude towards math
• However, it is challenging because many of those students have• math anxiety and difficulty in learning math,• little knowledge about how mathematics is used in the real world.
• Question: How should we design mathematical literacy
education for Bunkei students?

WERA2019, Tokyo, Aug. 6th 2019

New Math Courses (2012-)• Math courses for the humanities & social sciences students of the firstyear at Osaka Prefecture University.
• Basic Math I (spring semester)• Basic Math II (Autumn semester)• 90 min/week for 14 weeks followed by an examination period• 4 classes taught by 4 teachers (60-80 students in each class)

WERA2019, Tokyo, Aug. 6th 2019

Design Principles(Kawazoe et al., 2013; Kawazoe & Okamoto, 2017)

1. Design lessons according to mathematical modelling processes.
2. Choose topics & contexts by considering which mathematicalknowledge students are likely to encounter in real life and in whichsituation they will encounter it.
3. Present problems in different contexts associated with the samemathematical knowledge.
4. Connect different mathematical knowledge together by using different
mathematizations of the same problem or mutually related contexts.
5. Explain mathematical concepts & operations using both mathematicallanguage & everyday language.
6. Engage students in group activities rather than individual ones.
7. Design worksheets based on analysis of students’ understanding process
and use them as tools for formative assessment.

WERA2019, Tokyo, Aug. 6th 2019

Design Principles(Kawazoe et al., 2013; Kawazoe & Okamoto, 2017)

1. Design lessons according to mathematical modelling processes.
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tceeflRWERA2019, Tokyo, Aug. 6th 2019

Design Principles(Kawazoe et al., 2013; Kawazoe & Okamoto, 2017)

1. Design lessons according to mathematical modelling processes.
2. Choose topics & contexts by considering which mathematicalknowledge students are likely to encounter in real life and in whichsituation they will encounter it.
3. Present problems in different contexts associated with the samemathematical knowledge.
4. Connect different mathematical knowledge together by using different
mathematizations of the same problem or mutually related contexts.
5. Explain mathematical concepts & operations using both mathematicallanguage & everyday language.
6. Engage students in group activities rather than individual ones.
7. Design worksheets based on analysis of students’ understanding process
and use them as tools for formative assessment.

WERA2019, Tokyo, Aug. 6th 2019

Design Principles
Basic Math I (Spring semester)(Kawazoe et al., 2013; Kawazoe & Okamoto, 2017)

1. Design lessons according to mathematical modelling processes.
• systems of linear equations/inequalities (linear programming)
2.numberChoosetopics(savings,& contexts
by considering
which mathematicalsequences
loanpayment,
pharmacokinetics,of addiction, etc.)• knowledge students are likely to encounter in realmodel
life and in which& vectors (spreadsheets, social networks analysis)• matrices
situation they will encounter it.
(mental rotation, pharmacokinetics, bacterial growth, pandemic, etc.)•3.functions
Present
problems
indifferent
contexts
associatedwith
thesame(lottery, disease examination, birthday paradox, Bayesian estimation)• probability
mathematical knowledge.
4. Connect different mathematical knowledge together by using different
Basic Math II (Autumn semester)mathematizations of the same problem or mutually related contexts.
(population dynamics, optimal distribution, etc.)•5.eigenvalues/vectors
Explain mathematical concepts & operations using both mathematical(cyclical movement of electric demand, sound composition, etc.)• functions
language & everyday language.
(innovation diffusion, population growth, logistic function, marginal profit)•6.derivatives
Engage students in group activities rather than individual ones.
• integrals (speed & distance, accumulated radiation level, standard normal distribution)
7.
Designworksheets
basedonanalysisofstudents’
understanding
process
multivariable
functions(loansimulator,formulas
forestimating
vital
capacity,•and use them as tools for formative assessment.
optimization problem, etc.)

WERA2019, Tokyo, Aug. 6th 2019

Design Principles(Kawazoe et al., 2013; Kawazoe & Okamoto, 2017)

1. Design lessons according to mathematical modelling processes.
2. Choose topics & contexts by considering which mathematicalknowledge students are likely to encounter in real life and in whichsituation they will encounter it.
3. Present problems in different contexts associated with the samemathematical knowledge.
4. Connect different mathematical knowledge together by using different
mathematizations of the same problem or mutually related contexts.
5. Explain mathematical concepts & operations using both mathematicallanguage & everyday language.
6. Engage students in group activities rather than individual ones.
7. Design worksheets based on analysis of students’ understanding process
and use them as tools for formative assessment.

WERA2019, Tokyo, Aug. 6th 2019

Design Principles(Kawazoe et al., 2013; Kawazoe & Okamoto, 2017)Classroom(Examples)1. Designmaterials
lessons according
to mathematical modelling processes.
Context2.
Choose topics& contexts byconsidering whichmathematicalContext
Context
Context
Context
Blood
Analyzing the spreading
Loan payment
Bloodcaffeinelevelknowledge
students
arelikely
toencounterin
reallife
andin which
Salmonella
fexofenadine level
pattern of disease.
after drinking a cup ofgrowth(Pharmacy)
situation they
will encountercoffee
it.
(FMD outbreak, H1N1)
3. Present problems in different contexts associated with the samemathematical knowledge.
GeometricSemiRecurrence
Exponential
progression
Log Plot
4.NumberConnect different
knowledge
together by using
different
relation mathematical
Function
Functionsan+1same=c an problem or mutually related contexts.
an+1=c an+d of thesequences
mathematizations
5. Explain mathematical concepts & operations using both mathematicalContext
language & everyday language.
Items in Final ExamConsidering a newspaper article with aContext
6. Rivo
Engagestudents in group
activities rather thanindividualones. cesiumsemi-log
graph of radioactive(Examples)
payment ofconcentration in soil after Fukushima
shopping
7. Design worksheets based on analysis of students’nuclear
understanding
process
disasterand use them as tools for formative assessment.

WERA2019, Tokyo, Aug. 6th 2019

Design Principles(Kawazoe et al., 2013; Kawazoe & Okamoto, 2017)

1. Design lessons according to mathematical modelling processes.
2. Choose topics & contexts by considering which mathematicalknowledge students are likely to encounter in real life and in whichsituation they will encounter it.
3. Present problems in different contexts associated with the samemathematical knowledge.
4. Connect different mathematical knowledge together by using different
mathematizations of the same problem or mutually related contexts.
5. Explain mathematical concepts & operations using both mathematicallanguage & everyday language.
6. Engage students in group activities rather than individual ones.
7. Design worksheets based on analysis of students’ understanding process
and use them as tools for formative assessment.

WERA2019, Tokyo, Aug. 6th 2019

Example: Eigenvalues/vectors (2-weeks lesson)
Rental Bicycle Service Problem

Lecture BuildingB- 2 Parkings (A, B)- Bicycles can be dropped off at anyof the 2 parkings- Monitoring investigation (weeklydata)- The service will start with morethan 100 bicycles.

A- How to divide bicycles between 2parkings?

Students already learned the transition
matrix, but have not learned eigenvectors/values yet.

From \ ToA

BAB

70

30

2080

Main Gate(%)

WERA2019, Tokyo, Aug. 6th 2019•

1st week: discovering eigenvectors
How to divide bicycles between the 2parkings? (Group discussion)
1. A>B,

2. A=B,

3. A<BSimulation! What occurs?

• Choose an initial data and calculate
the distribution of the following 5weeks.
(Groupactivity)✓◆ ✓ ◆ ✓ ◆0.7 0.2
0.3 0.8

n200100=???
???

on· ·the= 1, 2, 3,·• Plot the data of allngroups
graph paper. (Group activity)

What can be found?

• Eigenvectors/values are introduced.

From \ ToA

BA
B

7030

2080(%)

✓◆✓ ◆
2311

WERA2019, Tokyo, Aug. 6th 2019

2nd week: explain with eigenvectors• Re-interpretation:✓◆✓ ◆✓ ◆200
21=??
+??
10031✓◆✓ ◆✓
◆80
n 200
n 120
nP=P+P100
18080
✓
◆✓
◆
12080
n=+ 0.518080• What is an optimal solution?
✓ Optimal solution•=stable point=eigenvector with eigenvalue 1Homework:✓ a similar problem, but in another context

WERA2019, Tokyo, Aug. 6th 2019

Evaluation of Teaching Practices• The average of mean scores of the final exam (2012-2018):• 72.3 (BMI) & 74.2 (BMII) (full mark=100)• Results of a self-report questionnaire (Kawazoe et al., 2013):• The rates of students who answered that their interest in mathincreased during the semester were 60.4% (BMI) & 60.6% (BMII).

WERA2019, Tokyo, Aug. 6th 2019

Evaluation of Teaching Practices• The average of mean scores of the final exam (2012-2018):• 72.3 (BMI) & 74.2 (BMII) (full mark=100)• Results of a self-report questionnaire (Kawazoe et al., 2013):• The rates of students who answered that their interest in mathincreased during the semester were 60.4% (BMI) & 60.6% (BMII).
• The rates of students who answered that their mathematical thinkingskills improved during the semester were 74.0% (BMI) & 70.8%(BMII).

WERA2019, Tokyo, Aug. 6th 2019

Evaluation of Teaching Practices• The average of mean scores of the final exam (2012-2018):• 72.3 (BMI) & 74.2 (BMII) (full mark=100)• Results of a self-report questionnaire (Kawazoe et al., 2013):• The rates of students who answered that their interest in mathincreased during the semester were 60.4% (BMI) & 60.6% (BMII).
• The rates of students who answered that their mathematical thinking
skills improved during the semester were 74.0% (BMI) & 70.8%(BMII).
• Analysis of students’ free comments in worksheets (BMI in 2018):• The frequency of comments containing descriptions showing metacognitive reflection per student was greater for students who did not likemathematics than for students who liked mathematics at the beginningof the course. (Kawazoe, 2019)

WERA2019, Tokyo, Aug. 6th 2019

Evaluation of Teaching Practices• The average of mean scores of the final exam (2012-2018):• 72.3 (BMI) & 74.2 (BMII) (full mark=100)• Results of a self-report questionnaire (Kawazoe et al., 2013):• The rates of students who answered that their interest in mathincreased during the semester were 60.4% (BMI) & 60.6% (BMII).
• The rates of students who answered that their mathematical thinking
skills improved during the semester were 74.0% (BMI) & 70.8%(BMII).
• Analysis of students’ free comments in worksheets (BMI in 2018):• The frequency of comments containing descriptions showing metacognitive reflection per student was greater for students who did not likemathematics than for students who liked mathematics at the beginningof the course. (Kawazoe, 2019)

WERA2019, Tokyo, Aug. 6th 2019

Thank you.
kawazoe@las.osakafu-u.ac.jp