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)

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

Keywords to search e-textbook for this lecture in Spanish: dbook, Isoda, Schooten, APEC dbook site: http://math-info.criced.tsukuba.ac.jp/museum/dbook_site/#d

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

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

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

国立教育政策研究所 2012

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

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?

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?

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?

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=

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

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

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

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)

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

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.

Reference Let’s Search: dbook, Isoda, Schooten

Reference 2

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