Paul Drijvers Freudenthal Institute Utrecht … Drijvers Freudenthal Institute Utrecht University ... Mathematics is the science of order, patterns, ......

Paul Drijvers

Freudenthal Institute

Utrecht University

[email protected]/

www.uu.nl/Staff/PHMDrijvers

2017-8-16

Mathematical Thinking

2017

Example 1: Circular reasoning

Please prove:

RED area =

= BLUE area

2

3

Example 2: Driving to Hamburg

U H

110

O

215

B

115

Distance to O

Dis

tan

ce t

o B

Driving to Hamburg: schematizing

?

5

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!

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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

2017

Outline

✓Two examples

▪ What do we mean by Mathematical Thinking?

• Problem solving

• Modeling

• Abstracting

▪ How to foster students’ Mathematical Thinking?

• Tasks

• Teaching

• Assessment

• Professional development

▪ Conclusion

9

2015

What do we mean by Mathematical Thinking?

2017

What is Mathematics?

▪ A (too?) broad, philosophical / epistemological question….

▪ Devlin (2000, p. 74): Mathematics is the science of order, patterns, structure, and logical relationships

▪ […] Mathematics offers distinctive modes of thought which are both versatile and powerful, including modeling, abstraction, optimization, logical analysis, inference from data, and use of symbols (NRC, 1989, p. 31).

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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

Mathematical Thinking: an educational perspective

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▪ Mathematical thinking is a whole way of looking at things, of stripping them down to their numerical, structural, or logical essentials, and of analyzing the underlying patterns (Devlin, 2011, p. 59).

▪ MT is figuring out how to use mathematical tools to solve a (non-routine) problem (Drijvers, 2015)

▪ Mathematical thinking is a way of life (math teacher)

Mathematical Thinking is…

…figuring out how to use mathematical tools tosolve 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 activitieswhich appeal for analytical thinking, for creativity, for strategy development, for flexibility, and forreflection.

… 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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USA: 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

▪ The notion of Mathematical Thinking Activity (MTA) to counterbalance the stress on basic skills / procedural fluency / “symbol pushing”.

▪ The notion of MTA highlights that mathematics is more than procedures, and that MTA is a core goal of mathematics education.

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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 problemsolving

4. Manipulating formulas

5. Abstracting

6. Reasoning and proving

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My reduced list:

▪ Problem solving (Mason 2000, Pólya1945/1962, Schoenfeld 1992, 2007, 2013, 2014, Van Streun, 1989)

▪ Modeling (cf., Blum & Niss 1991, Gravemeijer 1999; Treffers 1987 on horizontal mathematization; Swaak, Van Joolingen, & De Jong 1998 on science education)

▪ Abstracting (cf., Devlin 2000, 2012, Piaget 1936, Skemp 1986, Tall 2013; also Treffers1987 on vertical mathematization)

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Mathematical Thinking Model

Problem Solving

Modeling Abstracting

Math

Thinking

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 intomathematical form and backwards

▪ Modeling cycle:

Blum and Leiß (2006)

▪ Cf horizontal mathematization

▪ See Van Joolingen’s talk

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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 developmentwww.math.wisc.edu/~wilson/Courses/.../ReflectiveAbstraction.pdf

▪ Cf vertical mathematization

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Abstraction

▪ The greatest barrier to doing mathematics (Devlin, 2000, p.11)

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

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Abstraction according to Piaget (1936)

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▪ Empirical abstraction, based on concrete experiences with properties of objects

▪ Pseudo-empirical abstraction: based on actions on objects

▪ Reflective abstraction: mentally, internal, separated from concrete objects and experiences

Reflective abstraction is central in mathematics

Abstraction according to Tall (2013)

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structural,

“sensori-”

operational,

“-motor”

Mathematical Thinking Model

Problem Solving

Modeling Abstracting

How to fosterMathematical Thinking?

2017

Outline

✓Two examples

✓What do we mean by Mathematical Thinking?

• Problem solving

• Modeling

• Abstracting

▪ How to foster students’ Mathematical Thinking?

• Tasks

• Teaching

• Assessment

• Professional development

▪ Conclusion

31

2015

• Open-ended team competitions on modeling, for example, Math-Alympiad / Mathematics Day / Math-B Day (see presentation Monica Wijers)

• Algebra: see lecture by Martin Kindt

Tasks

August 2013 Summerschool - Open ended problems 32

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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Professional development

▪ Designed in collaboration with pilot teachers and the Association of Mathematics Teachers

▪ Quite some enrollment

▪ Positive evaluations

▪ Delivery in other universities

▪ Themes:

• Orientation on MT

• MT in task design

• MT in teaching

• MT in assessment

▪ Homework + individual feedback

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Professional development

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Conclusion

2017

Conclusion

▪ Mathematical thinking is one of the main goals of mathematics education

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

▪ Core MT aspects: problem solving, modeling, abstracting

▪ To address mathematical thinking in teaching, we need appropriate activities and assessments, 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

[email protected]/

www.uu.nl/Staff/PHMDrijvers

2017-8-16

Thank you for your attention!

2017

http://www.mathematikum.de/

Example 1: Finger arithmetic