1. Mathematical Thinking: How to develop it in the classroom
24 Octber 2015, SEAMEO Qitep in Mathematics
Mathematical Thinking: How to develop it in the classroom
Developing Children who learn mathematics by and for themselves.
Masami Isoda, PhD
Professor, University of Tsukuba, Japan.
Project Overseer, APEC Lesson Study Project
Honorary PhD, Khon Kaen University, Thailand (2011)
Honorary Professor, Universidad San Ignatio de Loyora, Peru (2014)
Best Educational Software, Awarded by Minister of Education, Japan (2005)
Most Beautiful Book of the Year in the Area of Natural Science,
Awarded by Japan Publisher Association (2010)

3. Learning by themselves
This is the real image of mathematical activity almost 500 years ago.

4. Keywords to search e-textbook for this lecture in Spanish: dbook, Isoda, Schooten, APEC
dbook site:

5. What is lesson study? It is reproductive science!
Lesson study is plan, do and see activities with various groups. Through the shared knowledge, PCK has been theorized and being integrated and unified as teachers’ theories for teaching mathematics.
Participating Teachers
Theories of Education
Lesson
study
Teacher
Children
Subject M.
Theory of Mathematics

6. What is the 21st centuries skill. http://skills. oecd

7. The case of USA, not the case of others
The case of USA, not the case of others. However, what is your expectation?

8. 国立教育政策研究所 2012

9. Objective of Math Education
We should develop children who can use what they leaned before with out our support. If they developed, they can reply the question what do you wan to do next.
Human Character Formation
Attitude and Value: Beautifulness, Curiosity, Reasonableness, Appreciation
Skills for Learning: Learning How to Learn
Mathematical Thinking:
Extension, Generalization, Anticipation, Integration, Change the representation for explaining
Knowledge and Skills
Traditional way of calculation
New way of calculation
Pattern on the calculations

10. 37x3= 37x6= =37x3x14 What?! Strange?? By Chilean Professor!!
Shall we continue?
37x 30 =1110
37x 33 =1221
37x =1332
37x =1443
=37x3x10
=37x3x11
=37x3x12
=37x3x___
Then,37x42
=37x3x14 1 5 5 4
Yes, there is another Pattern!
Aha!
37x 3 =111
37x 6 =222
37x =333
37x =999
What do you want to do next? ↓ ↓
It’s the end?
What do you want to do next? ↓ ↓
Interesting however its end.
37x3=
How did you get it?
37x6=
3x9
Then?
Did you find? What shall we do?

11. 111 X 14 444 111_ 1554 37x3= 37x6= =37x3x14 What?! Strange??
By Chilean Professor!!
Aha, how reasonable it is!
Shall we continue?
37x 30 = 1110
37x 33 =1221
37x =1332
37x =1443
=37x3x10
=37x3x11
=37x3x12
=37x3x___
Then,37x42
=37x3x14 1 5 5 4
Yes, there is another Pattern!
111
X 14
444
111_
1554
Aha!
37x 3 =111
37x 6 =222
37x =333
37x =999
Yes, let’s explain!
Until explanation, feeling Bad/Sick. You have the good mind as well as mathematician.
What do you want to do next? ↓ ↓
It’s the end?
What do you want to do next? ↓ ↓
Interesting however its end.
37x3=
How did you get it?
37x6=
3x9
Then?
Did you find? What shall we do?

12. 111 X 14 444 111_ 1554 37x3= 37x6= =37x3x14 What?! Strange??
By Chilean Professor!!
Aha, how reasonable it is!
Shall we continue?
37x 30 = 1110
37x 33 =1221
37x =1332
37x =1443
=37x3x10
=37x3x11
=37x3x12
=37x3x___
Then,37x42
=37x3x14 1 5 5 4
Yes, there is another Pattern!
111
X 14
444
111_
1554
Aha!
37x 3 =111
37x 6 =222
37x =333
37x =999
What do you want to do next? ↓ ↓
It’s the end?
What do you want to do next? ↓ ↓
Interesting however its end.
37x3=
How did you get it?
37x6=
3x9
Until understand and explanation, feeling Bad/Sick
Then?

13. 15873x7= Ability to imagine the future! The objective of the question
‘What do you want to do next?’
Ability to imagine the future!
Generalization?
Let’s Reflect on the Activity!
What do you learn from this experience
We can!
What do you want to teach from this example?
Beautifulness?
Reasonableness?
Mathematical Explanation by
The Procedure,
The Reason (Meaning/Different Representation)
The Aims, Objectives and Values
Pattern of calculation?
Shall I have to say something?
No, please do not support us.
15873x7=

14. Objective of Math Education
We should develop children who can use what they leaned before with out our support. Iif they developed, they can reply the question what do you wan to do next.
Human Character Formation
Attitude and Values: Beautifulness, Curiosity, Reasonableness, Appreciation
Skills for Learning: Learning How to Learn
Mathematical Thinking:
Extension, Generalization, Anticipation, Integration, Change the representation for explaining
Knowledge and Skills
Traditional way of calculation
New way of calculation
Pattern on the calculations

15. 2) Attempting to take logical actions
Lists of Mathematical Thinking Type by Katagiri A) Mathematical Attitudes (Mind Set)
1) Attempting to grasp one’s own problems or objectives or substance clearly, by oneself
(1) Attempting to formulate questions, (2) Attempting to maintain a problem consciousness, (3) Attempting to discover mathematical problems in phenomena
2) Attempting to take logical actions
(1) Attempting to take actions that match the objectives, (2) Attempting to establish a perspective, (3) Attempting to think based on the data that can be used, previously learned items, and assumptions
3) Attempting to express matters clearly and simply
(1) Attempting to record and communicate problems and results clearly and simply, (2) Attempting to sort and organize objects when expressing them
4) Attempting to seek better things
(1) Attempting to raise thinking from the concrete level to the abstract level, (2) Attempting to evaluate thinking both objectively and subjectively, and to refine thinking, (3) Attempting to economize thought and effort

16. Lists of Mathematical Thinking Type by Katagiri B) Mathematical Thinking: Related to Mathematical Methods in General
1) Inductive thinking 2) Analogical thinking 3) Deductive thinking 4) Integrative thinking (including extensional thinking) 5) Developmental thinking 6) Abstract thinking (Abstraction) (thinking that abstracts, concretizes, idealizes, and thinking that clarifies conditions) 7) Thinking that simplifies (Simplifying) 8) Thinking that generalizes (Generalizing) 9) Thinking that specializes (Specializing) 10) Thinking that symbolizes (Symbolizing) 11) Thinking that represents with numbers, quantities, and figures

17. C) Mathematical Thinking Related to Mathematical Content(Mathematical Ideas)
1) Clarifying sets of objects for consideration and objects excluded from sets, and clarifying conditions for inclusion (Idea of Sets) 2) Focusing on constituent elements (units) and their sizes and relationships (Idea of Units) 3) Attempting to think based on the fundamental principles of expressions (Idea of Expression) 4) Clarifying and extending the meaning of things and operations, and attempting to think based on this (Idea of Operation) 5) Attempting to formalize operation methods (Idea of Algorithm) 6) Attempting to grasp the big picture of objects and operations, and to use the result of this understanding (Idea of Approximation) 7) Focusing on basic rules and properties (Idea of Fundamental Properties) 8) Attempting to focus on what is determined by one’s decisions, to find rules of relationships between variables, and to use the same (Functional Thinking) 9) Attempting to express propositions and relationships as formulas, and to read their meaning (Idea of Formulas)

18. List of Questions for Mathematical Thinking <Posing Problem>
Questions Regarding Mathematical Attitudes
A11 What kinds of things (to what extent) are understood and usable? (Clarifying the problem)
A12 What is needed to understand, and can this be stated clearly? (Clarifying the problem)
A13

19. Concluding Discussion
This lecture explained the way to develop mathematical thinking in classroom with reflection and appreciation.
Mathematical Thinking can be taught based on the curriculum which extend the children’s ability based on what they learned.
The list of Mathematical Thinking shows the views for explaining what it is and what shall we teach. However, they are the ravels for teachers and unusual technical terms for children. Depending on the development of children, teacher teach them by the meaningful way. The terms such as ‘for example’ or ‘Aha’ are alternative representation for recognize the thinking itself by children.

20. Reference Let’s Search: dbook, Isoda, Schooten

21. Reference 2

22. Thank you very much!