Developing Mathematical Thinking: Lesson Planning and Problem Posing

1. It is a general question that can not be answered without restriction. APEC-KKU Conference 16.8.2007 How to plan a lesson for developing Mathematical ThinkingKyozai Kenkyu教材研究Research Subject Mater A priori Analysis vs Planning on aims Masami Isoda CRICED, University of Tsukuba Knowing and embedding the aims of education in the lesson plan by the classroom problem is a key for improvement. Where do Mathematics problems come from? Some classroom problems come from the extension in curriculum.

2. The Perspectives of Describing Mathematical thinkingin relation to the future of mathematics learning on the document by MEXT (1999) • Mathematization: Reorganization of experience through the reflection The ways of conceptual development Refutation is acceptable. • The World of Invariant Mathematics: Mathematics is the pattern of science. • Mathematical Ways of Thinking: G. Polya Learning how to develop mathematics

3. Problem Solving Approacha model of the lesson to develop Mathematical Thinking • Teachers begin by presenting students with a mathematics problem employing principles they have not yet learned. • They then work alone or in small groups to devise a solution. • After a few minutes, students are called on to present their answers; the whole class works through the problems and solutions, uncovering the related mathematical concepts and reasoning. • (from Teaching Gap. J.Stingler & J. Hiebert) • Problem Posing • Predict the methods for Solution • Solving • Discussion • Reflection • In the process, children learn mathematics but this process is not aimed to represent it. It focuses on how to guide children’s activity.

4. Confirming and understanding the extended meaning and procedure. Appreciation and sense of achievement Reflection/ Summary Facilitate developmental discussion based on meanings and procedures previously learned, and eliminated gaps. Reproducing and reconsidering procedures How did you do that? Reproducing and reconsidering meanings Why did you think that way? Emortional Aspects What is confusing or troubling you? Aiming to eliminate gaps and conflict “What?” and “Why?” Asking themselves and others again No meaning and procedure type Conflict Exposure of gaps in procedure and meaning Comparison to previously learned knowledge “hmmmm,” “what?” Prioritize meaning without procedure (or confused) type Secure procedure and meaning type Prioritize procedure with confused or ambiguous meaning type Prioritize procedure without meaning type Target (extending) Task Students become aware of the gaps and differences with knowledge previously learned.: Concern, uneasiness and conflict Previously learned Procedures and Meanings Recall, Confirmation and Understanding. It goes well!! Sense of efficacy. Previously learned (Known Task)

5. Problem Solving Approacha model of the lesson to develop Mathemetical Thinking by Child Centered Approach

6. Necessary Activity for developing the lesson Plan the lesson with following problems • Find the following pairs of numbers □×△＝ □－△ X ×Y　＝X－Y What kinds of thinking? What if? What if not? Generalization (0,0) (1,△) (1／2，△) (1,1／2) Speicalzation Ｙ＝ｘ／(ｘ＋１） (3，3／4)

7. Necessary Activity for developing the lesson Mathematical Thinking What If Correct Answer Other Approaches Mathematics Problems Wrong Answers Conditions of selection • Problems are used for children enabling them to engage in rich mathematical activity and lean from the reflection. Developing New Prob. What can children learn from the process? The Aim of Lesson, Mathematical Value Anticipating Children’s Activity Embedding the aim For controlling children’s Activity on the aim Classroom Problems in order / sequence (ways of posing in the process of teaching)

8. (1/2, 1/3) … (1/n, 1/(n+1)) Finding examples (0,0), (1,1/2), ………. Solving generally y= x/(x+1) Finding the aims Find following pairs of numbers □×△＝ □－△ X ×Y　＝X－Y • Number Pattern • Invariant vs Variant • Specialization • Inductive Reasoning • Generalization • Power of Symbol • Sequence, Integra Introduction of Symbol (1/□)×(1/△)＝ (1/□)－(1/△)? Why this problem is interesting for teachers? For children? □×△＝ □－△ Guessingexamples; (0,0), (1,1/2),….Finding general pattern (1/□)×(1/△)＝ (1/□)－(1/△) How to represent general pattern? (1/□)×(1/(□＋1))＝ (1/□)－(1/（□＋1)) How can children recognize it as problematical?

9. →Children →Teacher →Children 3 8 6 2 7 5 In case teacher does not have information of children. For example. Seiyama’s lesson □＋□＋□＝？ How can children recognize this very strange phenomenon?

10. Necessary Activity for developing the Lesson Plan Classroom Problems in order (ways of posing in process) How to be clear of the aim and the value of math. in process. Set Children’s Activity through the questions Plan of Activity, Questioning for Interaction and the Black Board Writing Planning the Black Board writing with consideration of classroom interaction is the way to consider real classroom setting.

11. Where do problems come from? • It is true that some mathematics problems originated from daily situation but the problem for each lesson does not always come from daily situation. • Most of mathematics problems are sited in the textbooks, source books and exercise books. Thus, mathematics problems come from textbooks (or curriculum). • Enabling students to recognize the aims and contexts, teachers should embed the aim of the lesson into their classroom problems on their teaching plan. • Even if problems are described in daily situations, we are not sure that students can use the related mathematical ideas on the daily situation if students cannot recognize aims and contexts to use it.

12. How to use students’ misconceptions or wrong ideasas the key problem in the mathematics classroom? But, a child answer is ….. Wow,Teacher used Children’s idea! How many? How to calculate? ● ●●● ●●●●● ●●●●●●● ●●●●●●●●● ●●●●●●●●●●● What is the teacher’s or your expectation? Oh, NO! If you are the teacher, what can you do? ● ●●● ●●●●● ●●●●●●● Why did teacher ask this question? Important Math.Thinking; Generalization, Application, …… Value of them; Faster, Easier, Reasonable , ……

13. Why do teachers feel that a misconception/misunderstanding is a problem? • If it is not expected, it must be a problem but it is expected… • The teaching approach itself included the pedagogical value/educational aim.. • For teaching; Ways of Math. Thinking, Math. Communication, and • Developing Math. Using other’s idea is basic reasoning How can we use students’ misconceptions in the mathematics classroom?Teacher’s Theory for Problem Solving Approach But, a child answer is ….. Wow,Teacher used Children’s idea! How many? How to calculate? ● ●●● ●●●●● ●●●●●●● ●●●●●●●●● ●●●●●●●●●●● What is the teacher’s expectation? Oh, NO! If you are the teacher, what can you do? ● ●●● ●●●●● ●●●●●●● Why did teacher ask this question? Important Math.thinking; Generalization, Application, …… Value of them; Faster, Easier, Reasonable , …… What is Teacher’s Theory? the theory for developing children, not for observing children like Jean-Henri Fabre (a famous entomologist, famous observer). It is the theory of supporting the development of teachers’ eyes for educating the children. It’s usually developed with teachers who are working on the Lesson Study (Plan Do See). Here, we use the terms: Meaning and Procedure in planning the lesson to develop children’s mathematical knowledge based on the curriculum

14. Based on the nature of mathematics learning Because it is the evidence of Thinking Mathematically • Why should we use misconceptions? • Students use what they already learned. • Students try to use easier procedure. • Explain misconception with meaning and procedure from what students learned before. • Dialectic ways of discussion for teaching mathematical thinking and developing mathematical ideas. • Categorizing students’ ideas from the meaning and procedure. Because it is the nature of Mathematics Curriculum including the extension sequence

15. Explain Misconception with Meaning and Procedure from What Students Learned Before.

16. Way of drawing pattern 1 Sample 1 Sample 2 Using the right diagonal line as the base Way of Drawing A: Even intervals along the edges Way of Drawing B: Even intervals from the line To introduce parallel lines, Mr. Masaki started by drawing a sample lattice pattern. The following process shows how students develop the idea of parallel with no instruction on the definition of parallel. Task 1. Let’s draw the sample 1 lattice pattern Task 2. Let’s draw the sample 2 lattice pattern Dialectic Discussion: “What? ”“Why?” Synthesis: Define the parallel line based on the difference

17. Way of drawing pattern 1 Sample 1 Sample 2 Using the right diagonal line as the base Way of Drawing A: Even intervals along the edges Way of Drawing B: Even intervals from the line To introduce parallel, Mr. Masaki started by drawing a sample lattice pattern. The following process shows how students develop the idea of parallel with no instruction on the definition of parallel. Way of drawing 1: Procedure a →Way of Drawing A; Task 2 If you want to draw the model, draw lines spread evenly apart from the top edge of the paper. Way of drawing 1: Procedure b →Way of Drawing B; Task 2 If you want to draw the model, draw lines spread evenly apart. Even if teacher explained many times, there are diversity of children’s understanding. Because children can not distinguish special ideas and general ideas.

18. (A) (B) (C) Where does it come from?② explains ① ..Schoenfeld, A (1986)Isoda, M.(1991, 1996) Write 23 + 5 Decimal notation system meaning ? (Forgotten) Align to the right and write Originated from Extension from Whole Number to Decimal Number Programmed Emergence of Misconception in Curriculum/Teaching (Anyone cannot avoid: Epistemological Obstacle)

19. Meaning and Procedure (Isoda, 1991) • Meaning (here, Conceptual or declarative knowledge) refers to contents (definitions, properties, places, situations, contexts, reason or foundation) that can be described as “ ~ is…” For example, 2+3 is the manipulation of ○○←○○○. The meaning can also be described as: “2+3 is ○○←○○○,” and as such explains conceptual or declarative knowledge. • Procedure (here, Procedural knowledge) on the other hand refers to the contents described as “if…., then do…” The procedure is used for calculations such as mental arithmetic in which calculations are done sub-consciously. For example, “if it is 2x3, then write 6” or “ if it is 2+3, then write the answer by calculating the problem as ○○←○○○.” • If we understood well, meaning and procedure are easier translatable in our mind automatically. • So many cases, they are not translatable easily.

20. Procedurization of meaning (Isoda, 1991)(Procedurization of Concept) • Procedures can be created based on meaning. • For example, when tackling the problem “how many dl are in 1.5l?” for the first time, a long process of interpreting the meaning is applied and the solution “1.5l is 1l 5dl” is found. Additionally, this can be applied to other problems such as “how many dl are in 3.2l?” with the answer being “3.2l is 3l 2dl.” Not before too long, children discover easier procedures by themselves. Simultaneously, children realize and appreciate the value of acquiring procedures that alternate long sequential reasoning to one routine which does not need to reason. • Many teachers believe that the procedure should explain based on the meanings but it should be a kind of preferred alternation because of the simplicity and earliness. Based on the value of mathematics, simplicity, we finally develop proceedings

21. Proceduralization of meaning (Isoda, 1991) • Preferred alternation from meaning to procedure based on faster and easier

22. Meaning entailed by procedureConceptualization of procedure (Isoda, 1991) • Meaning can be created based on procedure • In the first grade, like in the operation activity where “○○○←○○” means 3+2, children learn the meaning of addition from concise operations and then become proficient at mental arithmetic procedures (the procedurization of meaning). At that point, calculations such as 4+2+3 and 2+2+2 are done more quickly than counting, which is seen as a procedure. • Further, in the second grade, comparing with several additional situation, only repeated addition problems lead to the meaning of multiplication. It is here where the specific addition procedure known as “repeated addition (cumulation)” is added as part of the meaning (meaning entailed by procedure). The reason such situations become possible is that children become both proficient at calculations and familiar enough with the procedure to do it instantly as well as the meaning of situation. • Children who are not familiar with the procedure resort to learning addition and multiplication at the same time, which in turn makes it more difficult for children to recognize that multiplication can be regarded as a special circumstance of addition.

23. Procedurization of ConceptConceptualization of Procedureon the Extending Curriculum Sequence (Isoda,1996) More General Case Procedure of C Meaning of C General Case Procedure of B Meaning of B Special Case Meaning of A Procedure of A

24. Dialectic Ways of Discussion for Teaching Mathematical Thinking and Developing Mathematical Ideas.-Beyond the Contradiction-

25. If you are correct, then what will happen? If 1/2+1/3=2/5, then 1/2+1/2 =? Dialectic Discussion1/2 + 1/3=? 1/2 + 1/3=2/5 1/2 + 1/3=5/6 contradiction Prioritize procedure without meaning type 1+1=2, 2+3=5 then 2/5 Secure procedure and meaning type = + Parallel Deadlock Prioritize procedure with confused or ambiguous meaning type

26. Why Dialectic Discussion?Because it is the nature of Math and it is important for Human communication. • Ancient Greek; If your saying or result is given, then • Ways of communication; If your saying it true, … • Socrates Method (in German Pedagogy) on Plato’s School • Indirect proof of Pythagorean school • Analysis and Synthesis of Euclid and Pappus • Renaissance • Rene Descartes • Analysis on Geometry; If we have conclusion (construction), …. • Analysis on Algebra; If we have the answer x, …. • Fermat & Bernoulli, • Analysis on Calculus; If we have the limit, …. • Mathematical logic of discovery; focused on the function of counter examples • Hegelians; Karl Popper, Imre Lakatos

27. Dialectic discussion that eliminates gaps in diverse ideas Children who are aware of the meaning of the procedure and children who are not aware of it contradict each other. Here, the discussion develops based on the ideas and concerns of children who have an ambiguous understanding of meaning or procedure. The reactions of children who are no longer aware of the meaning of the procedure The reactions of children who remain aware of the meaning of the procedure In order to eliminate contradictions and gaps, it is necessary for children to persuade to revise other’s ideas. Contradiction • Two strategies against the parallel discussion. • What if A’s idea is correct? If 1/2+1/3=2/5 is correct, then 1/2+1/2=2/4=1/2. • Facilitating awareness through application of tasks in different situations and examples

28. Categorizing Students’ Ideas from the Meaning and Procedure.

29. 4 0 Previously Learned Task:The problem to confirm the previously learned procedure and the meaning that forms the base of today’s target task • When children who have knowledge of basic division work out the equation, 1600÷400 is done, the following is reviewed: • Take away 00 and calculate: procedure • Explain A as a unit of 100 (bundle): meaning • Substitute A for a 100 yen coin and explain: meaning

30. Target Task (Unknown Task):Unknown problem to press for application of the previously learned meaning and procedure • The target problem presented is 1900÷400, which presents a problem for some children and not for others as to how to deal with the remainder. • Answer to the equation using a procedure in which the meaning is lost. Apply A (Take away 00) and make the remainder 3. Because the meaning is detracted, the children do not question the remainder of 3: Half of the class • Answer to the question when procedures have ambiguous meanings. Using A (Take away 00) and B (Unit 100), the remainder was revised to 300. However, because the meaning was ambiguous, it was changed to 400: Several students. • Answer to the question when the procedure is ambiguous. A (Take away 00) was used, but here a different procedure was selected by mistake. No students question the quotient 400: Very few students • Answer to a question that confirms procedural meanings. Using A (Take away 00) , an explanation of the quotient and remainders from the meaning of B (Unit 100) and C (100 yen).

31. 4 0 Target Task (Unknown Task):Categorizing ideas by Meaning and Procedure. • What does each child know and how does he/she apply it? • Answer to the equation using a procedure in which the meaning is lost. Apply A (Take away 00) and make the remainder 3. Because the meaning is detracted, the children do not question the remainder of 3: Half of the class • Answer to the question when procedures have ambiguous meanings. Using A (Take away 00) and B (Unit 100), the remainder was revised to 300. However, because the meaning was ambiguous, it was changed to 400: Several students. • Answer to the question when the procedure is ambiguous. A (Take away 00) was used, but here a different procedure was selected by mistake. No students question the quotient 400: Very few students • Answer to a question that confirms procedural meanings. Using A (Take away 00) , an explanation of the quotient and remainders from the meaning of B (Unit 100) and C (100 yen). Type 1. Solutions reached through the use of procedures without meaning: Prioritize procedure without meaning type Type 2. Solution reached through the use of procedures with meaning: Prioritize procedure with confused or ambiguous meaning type Type 2. Solution reached through the use of procedures with meaning: Prioritize procedure with confused or ambiguous meaning type Type 3. Solution reached through the use of procedures backed by meaning: Secure procedure and meaning type

32. Confirming and understanding the extended meaning and procedure. Appreciation and sense of achievement Reflection/ Summary Facilitate developmental discussion based on meanings and procedures previously learned, and eliminated gaps. Reproducing and reconsidering procedures How did you do that? Reproducing and reconsidering meanings Why did you think that way? Emortional Aspects What is confusing or troubling you? Aiming to eliminate gaps and conflict “What?” and “Why?” Asking themselves and others again No meaning and procedure type Conflict Exposure of gaps in procedure and meaning Comparison to previously learned knowledge “hmmmm,” “what?” Prioritize meaning without procedure (or confused) type Secure procedure and meaning type Prioritize procedure with confused or ambiguous meaning type Prioritize procedure without meaning type Target (extending) Task Students become aware of the gaps and differences with knowledge previously learned.: Concern, uneasiness and conflict Previously learned Procedures and Meanings Recall, Confirmation and Understanding. It goes well!! Sense of efficacy. Previously learned (Known Task)

33. Way of drawing pattern 1 Sample 1 Sample 2 Using the right diagonal line as the base Way of Drawing A: Even intervals along the edges Way of Drawing B: Even intervals from the line To introduce parallel, Mr. Masaki started by drawing a sample lattice pattern. The following process shows how students develop the idea of parallel in case of the no instruction of the definition of parallel. Way of drawing 1: Procedure a →Way of Drawing A; Task 2 If you want to draw the model, draw lines spread evenly apart from the top edge of the paper. Way of drawing 1: Procedure b →Way of Drawing B; Task 2 If you want to draw the model, draw lines spread evenly apart. Even if teacher explained many times, there are diversity of children’s understanding. Because children can not distinguish special ideas and general ideas.