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Paul Drijvers
Freudenthal Institute
Utrecht University
www.uu.nl/Staff/PHMDrijvers
2017-8-16
Mathematical Thinking
2017
Example 1: Circular reasoning
Please prove:
RED area =
= BLUE area
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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
?
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
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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
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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
www.uu.nl/Staff/PHMDrijvers
2017-8-16
Thank you for your attention!
2017
http://www.mathematikum.de/
Example 1: Finger arithmetic