Paul Drijvers

Freudenthal Institute

Utrecht University

p.drijvers@uu.nl www.fisme.science.uu.nl/

www.uu.nl/Staff/PHMDrijvers

2016-8-18

Mathematical Thinking

2016

http://www.mathematikum.de/

Example 1: Finger arithmetic

Example 2: Circular reasoning

Please prove:

RED area =

= BLUE area

3

4

Example 3: Driving to Hamburg

U H

110

O

215

B

115

Distance to O

Dis

tan

ce t

o B

Driving to Hamburg: schematizing

?

6

Driving to Hamburg: animation

H H

110

O

215

B

115

Problem orientation:

Starting point in U: (O, B ) = (215, 330)

End point in H: (O, B ) = (225, 110)

Model: u = distance to Utrecht (independent variable)

O(u) = distance to Osnabruck = | u – 215 | (dependent)

B(u) = distance to Bremen = | u – 330| (dependent)

P(u) = (O(u), B(u))

-> So we have a parametric curve!

7

Driving to Hamburg: model

H H

110

O

215

B

115

? Do these tasks invite Mathematical Thinking?

Why? Which task features are decisive?

What would we mean by Mathematical Thinking?

Outline

2016

Outline

Three examples

What do we mean by Mathematical Thinking?

• Problem solving

• Modeling

• Abstracting

How to foster students’ Mathematical Thinking?

• Tasks

• Teaching

• Assessment

Conclusion

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2015

What do we mean by Mathematical Thinking?

2016

Mathematical thinking as a central goal of mathematics education

George Pólya (1887 – 1985): … first and foremost, it should teach those young people to THINK.

http://www.fisme.science.uu.nl/publicaties/literatuur/Oratie_Paul_Drijvers_facsimile_20150521.pdf

And it still is:

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https://www.coursera.org/course/maththink

https://www.amazon.com/Introduction-Mathematical-Thinking-Keith-Devlin/dp/0615653634

Common Core State Standards for mathematical practice

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the reasoning of others

4. Model with mathematics

5. Use appropriate tools strategically

6. Attend to precision

7. Look for and make use of structure

8. Look for and express regularity in repeated reasoning

(http://www.corestandards.org/Math/Practice)

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Mathematical Thinking in the Dutch 2015 curriculum reform

To go beyond reproduction and symbol pushing, the notion of ‘mathematical thinking activity’ is stressed:

1. Modeling and algebrizing

2. Ordening and structuring

3. Analytical thinking and problem solving

4. Manipulating formulas

5. Abstracting

6. Reasoning and proving

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Mathematical Thinking is…

…figuring out how to use mathematical tools to solve a problem.

How: Which tools, which order, which restrictions?

Tools: Specific (e.g., quadratic formula) and general (problem solving, modeling, abstracting, …)

Problem: Non-routine tasks for the students

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Mathematical Thinking…

… involves engagement in non-routine activities which appeal for analytical thinking, for creativity, for strategy development, for flexibility, and for reflection.

… is relative, depends on preliminary knowlegde, level, age, educational context...

Solve (x - 2)2 + 7 = 16: routine task in grade 9

but requires Mathematical Thinking of a grade 7 student who encounters this type of equation for the first time

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Mathematical Thinking Model

Problem Solving

Modeling Abstracting

Problem solving

Problem solving concerns solving non-routine tasks, for which students don’t have a ready-made strategy available (Schoenfeld, 2007).

Solving a problem means finding a way out of a difficulty, a way around an obstacle, attaining an aim which was not immediately attainable. Solving problems is the specific achievement of intelligence, and intelligence is the specific gift of mankind: solving problems can be regarded as the most characteristically human activity. (Pólya, Mathematical Discovery, 1962, p. v.)

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Problem solving: Pólya’s phases

1. Understanding the problem

2. Devising a plan

3. Carrying out the plan

4. Looking back

Not a linear process!

Education often focuses on phase 3, whereas 1-2-4 are crucial (and the most difficult?)

See presentation Rogier Bos

Modeling

Translating realistic problems into mathematical form and backwards

Modeling cycle:

Blum and Leiß (2006)

Cf horizontal mathematization

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Abstracting

Abstracting is an activity by which we become aware of similarities […] among our experiences. (Skemp, 1986, p. 21)

Tall (1988, p. 2): Abstraction is the isolation of specific attributes of a concept so that they can be considered separately from the other attributes

Piaget (1985) in Dubinsky: Reflective abstraction as the construction of logico-mathematical structures by an individual during the course of cognitive development www.math.wisc.edu/~wilson/Courses/.../ReflectiveAbstraction.pdf

Cf vertical mathematization

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Mathematical Thinking Model

Problem Solving

Modeling Abstracting

How to foster Mathematical Thinking?

2016

• Open-ended team competitions on modeling, for example, Math-Alympiad / Mathematics Day / Math-B Day (see presentations De Haan – Wijers - Tak)

• Algebra: see Tuesday’s lecture by Martin Kindt

Tasks to evoke mathematical thinking

August 2013 Summerschool - Open ended problems 25

Adapt regular tasks and activities

Regular text book: Adapted version:

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Differentiate:

a. f(x) = 2x

b. f(x) = 3x

c. f(x) = (1/2)x

d. f(x) = (1/3)x

Differentiate:

a. f(x) = 2x

b. f(x) = 3x

c. f(x) = x3

d. f(x) = (1/3)x

Teaching mathematical thinking

Don’t answer questions but raise questions Give students the time to think Ask for students’ explanation and reasoning Anticipate student reactions and prepare

activitating follow-up Pay attention to the problem solving process Have whole-class reflections on strategies Gradually fade feedback Assess mathematical thinking Be alert to opportunities to invite math thinking. Enjoy math your self and show it!

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Assess Mathematical Thinking

The PISA assessment piramid:

Active Thinking fits in the

highest category of production

rather than reproduction.

cf Lectures by Mieke Abels and

Harrie Eijkelhof:

• Reproduction

• Connections

• Reflection

51,8 ↔ 𝟑𝟗, 𝟖

Trend in national examinations

Master Thesis Hanneke Buitenhuis:

Decreasing proportion of “Thinking items”

Pilot students slightly better than regular classes

http://dspace.library.uu.nl/bitstream/handle/1874/318236/Onderzoek%20naar%20WDA%20in%20pilotexamens_Hanneke%20Kodde_juli%202015.pdf?sequence=2

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Conclusion

2016

Conclusion

Mathematical thinking is one of the main goals of mathematics education

Core aspects: problem solving, modeling, abstracting

This may get out of sight due to a focus on reproductive skill mastery

To address mathematical thinking in teaching, we need appropriate activities, and teachers who are able to capitalize on the opportunities such tasks offer

Let us all try to be such teachers…..

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Paul Drijvers

Freudenthal Institute

Utrecht University

p.drijvers@uu.nl www.fisme.science.uu.nl/

www.uu.nl/Staff/PHMDrijvers

2016-8-18

Thank you for your attention!

2016