1. HUE City, Vietnam Hue University’s College of Education 32 Le Loi St. APEC-Khon Kaen International Symposium August 2007

2. A LESSON THAT MAY DEVELOP MATHEMATICAL THINKING OF PRIMARY STUDENTS IN VIETNAM Dr. Tran Vui Hue University, Vietnam APEC-Khon Kaen International Symposium August 2007

3. FIND TWO NUMBERS THAT THEIR SUM AND A RESTRICTED CONDITION ARE KNOWN APEC-Khon Kaen International Symposium August 2007 12 bottles containing 33 liters

4. Find and such that: + = 12. and  2 +  5 = 33 (Condition) xy xy Teachers’ Mathematics Background

5. Teachers’ Mathematics Background in Solving System of Equations x + y = 12 (1) 2x + 5y = 33 2x + 2y = 24 2x + 5y = 33 3y = 9 Time (1) by 2 means that: If all... are...,... then

6. SOCIAL-CULTURAL CONTEXT MATHEMATICAL THINKING (UNIVERSAL) TMT SMT

7. How Do TEACHER DEVELOP STUDENTS’ Mathematical Thinking In a CLASSROOM Setting? (with Teacher’s Background in Solving System of Equations and Beliefs)

8. SOME BACKGROUNDS ON THE REFORM OF VIETNAMESE EDUCATION BEFORE DISCUSSING THE RESEARCH LESSON

10. 2006 2001 2000 1995 CHANGING CURRICULUM & TEXTBOOKS 1985 Current Cur. Less Academic, Skills, Techniques Old Cur.: Academic, Logic, Proof, Algorithms Reform Cur. Problematic Situation, Mathematical Thinking through PS. 5-year Pilot Study

11. Students’ mathematical thinking can be: - defined, - taught, - observed, - tested, - evaluated, and - reported THROUGH its products: the students’ works, talks and representations when they solve mathematical problems. WE BELIEVE FOLOWING PRINCIPLES

12. Challenging between Teachers and Students Teaching algorithms, procedures, techniques, rules to solve difficult problems (Practicing) Finding answers for structured problematic situations (Solving)

13. Mathematical Thinking Process Content Knowledge Reform Curriculum VIETNAMESE REFORM CURRICULUM 15%-25% Step by Step REFORM

14. COMMUNICATING INVESTIGATING P.SOLVING exercising exploring LOGICAL REASONING MATHEMATICAL thinking St.

15. Passive Active Student HIERARCHY OF MATHEMATICAL THINKING Problem Solving Practicing Skills & Algorithm Stephen Krulik, 1993

16. OUTPUTinputprocessing MATHEMATICAL THINKING Observing Inquiring Recalling Summarizing Symbolizing Exploring Analyzing Applying Logic Reasoning Inducing Deducing Problem Solving Investigating Generalizing Reflecting Evaluating Questioning Synthesizing understanding manipulating generating

17. BACK TO THE RESEARCH LESSON At the end of Grade 4, students know how to solve and express solutions of problems having three operations of natural numbers. Example. A toy train has 3 wagons with the length of 2 cm, and 2 wagons with the length of 4 cm. Find the length of the train? Answer. 3  2 + 2  4 = 14 (cm). PRACTICING

18. SETTING THE PROBLEM IN A REVERSE WAY A toy train has two types of wagon: 2 cm- wagons and 4 cm-wagons. This train has the length of 14 cm including 5 wagons. Find the numbers of 2 cm-wagons and 4 cm-wagons of the train. Find and such that: + = 5. and  2 +  4 = 14 (Condition)

19. ANALYSIS OF INTRODUCTORY TASK Open-ended Task Use 2 cm-cards and 4 cm-cards to make a toy train of 5 wagons? PLAYING AROUND AND OBSERVING Pupils can arrange the cards to make a train, use the strategy "guess and check" to get many answers

20. MAKE A SYSTEMATIC LIST N. of reds012345 N. of blues543210 The length201816141210

21. THE RELATIONSHIP BETWEEN THE LENGTH AND THE NUMBERS OF REDS AND BLUES If the number of red wagons increases one, then the length of the train decreases 2 cm. If the length of the train is given then we can find exactly the N. of reds wagons and N. of blues. The length of the train is understood as a restricted condition

22. T: How many red wagons and blue wagons in your train? S: 3 and 2. We have 3  2 + 2  4 = 14 cm.

23. ANALYSIS OF TASK 1 Open-ended Task Make a train with the length of 16 cm. MANIPULATING AND OBSERVING Pupils can arrange the cards to make a train, use the strategy "guess and check" to get many answers

24. MAKE A SYSTEMATIC LIST N. OF REDS N. OF BLUES TOTAL Students analysed number 16 as follows: 16 = 8  2 + 0  4 16 = 2  2 + 3  4 16 = 6  2 + 1  4 16 = 0  2 + 4  4 16 = 4  2 + 2  4

25. THE RELATIONSHIP BETWEEN THE LENGTH AND THE NUMBERS OF REDS AND BLUES The number of red wagons is always even. ? If the number of wagons of the train is given then we can find exactly the N. of reds wagons and N. of blues. The number of wagons of the train is understood as “a restricted condition”.

26. LOGICAL REASONING T: If the train has 6 wagons, how many red wagons and blue wagons in this train? S: From the table I saw that this train has 4 red wagons and 2 blue wagons. T: If we do not make the table, can you explain your solution? S: If all 6 wagons are red, the train's length decreases 4 cm. So I got 2 blue wagons. T: Who can express the answer by using mathematical operations? S: (16 - 6  2) ÷ 2 = 4 ÷ 2 = 2 (blue wagons).

27. ANALYSIS OF TASK 2 Inducing A train with the length of 50 cm including 20 wagons, how many red wagons and blue wagons are there? The teacher guided students to induce a procedure by using the temporary assumption to solve the problem. T: If 20 wagons are red, what is the length of the train? S: 40 cm. T: Why does the length decrease? S: Because we replaced blue wagons by red wagons? T: How many blue wagons did we replace? S: 5 blue wagons. T: How did you get 5? S: (50 - 40) ÷ 2 = 5.

28. ANALYSIS OF TASK 3 Generalization A train with the length of 100 cm including 36 wagons, how many red wagons and blue wagons are there? The teacher guided students to generalize the procedure by using the temporary assumption to solve the problem. Students applied the procedure to solve Task 3. N. of blue wagons: (100 - 36  2) ÷ 2 = 14 (wagons). The number of red wagons: 36 - 14 = 22 (wagons).

29. OBSERVATIONS GENERALIZATION INDUCTION

30. Creative thinking Inductive, generalizing, conjecturing... GOAL 1 GOAL 2 START Divergent

31. ANALYSIS OF QUIZ Application There are 33 liters of fish sauce contained in 2-liter bottles and 5-liter bottles. The number of bottles used is 12. Find the number of 2-liter bottles and 5-liter bottles used. Known that all bottles are full of fish sauce.

32. ANALYSIS OF QUIZ Application 2 5 3 : Difference The number of 5-liter bottles: (33 - 12  2) ÷ 3 = 3.

33. With this kind of teaching, teacher helps students dig deeply into a textbook problem and build up a habit of unsatisfying with achieved results; Encourage students to be interested in investigating and seeking for another solutions, and creative in learning mathematics. Teacher helps students develop their mathematical thinking.

34. INVESTIGATING TASKS SOLVING “PROBLEMS” PRACTICING EXERCISES, SKILLS EXPLORING & RECALLING FACTS, PRINCIPLES, PROCEDURES MATHEMATICAL THINKING DEFINED IN VN CURRICULUM

35. FOUR MAIN ACTIVITIES IN A LESSON THAT TEACHERS SHOULD FOLLOW TO DEVELOP STUDENTS’ MATHEMATICAL THINKING (MOET 2006): Activity 1. Examine and Consolidate the previous knowledge involved with new lesson; Activity 2. Teacher facilitates students explore mathematical knowledge and construct new knowledge by themselves. Activity 3. Students practice the new knowledge by solving exercises and problems in the textbook and exercise book. Activity 4. Teacher concludes what students have learnt from new lesson and assigns the homework.

36. ENGAGING TO THE LESSON, THE PUPILS WILL HAVE OPPORTUNITIES TO SHOW THEIR MATHEMATICAL THINKING THROUGH: The ability of observing, predicting, rational reasoning and logical reasoning; Knowing how to express procedures, properties by language at specific levels of generalization (by words, word formulas); Knowing how to investigate facts, situations, relationships in the process of learning and practicing mathematics; Developing ability on analyzing, synthesis, generalization, specifying; and starting to think critically and creatively.

37. The level of difficulty and complexity of a problem is defined by the achievement objectives in the standard curriculum for each strand of mathematics. The exercises in the practice lessons are ranked: - From easy to difficult, - From simple to complicated, - From direct practice to flexible and combined applications.

38. It is possible and desirable to call upon pupils’ Mathematical thinking powers by offering challenging and meaningful Questions Exercises Problems to work on