This document summarizes a presentation on developing thinking through high school mathematics. It discusses how student achievement declines from elementary to high school and calls for changing mindsets. It provides examples of engaging students in high school math concepts like algebra, geometry, and problem solving. These hands-on examples from schools in Singapore, Indonesia, and elsewhere encourage students to make sense of formulas and collaborate on problems. The conclusion is that math can develop thinking beyond just content if taught in this way. International test scores are also presented showing Singapore performs among the highest in math achievement and affect.

1. LEAP Educators’ Conference 2011 Developing Thinking Through High School Mathematics Yeap Ban Har Marshall Cavendish Institute Singapore banhar@sg.marshallcavendish.com Presentation is available on www.banhar.blogspot.com

2. Link to Keynote Lectures First lecture raised the question: “Why is the achievement level eroded when students progress from elementary to high school?”. The second lecture calls for changing mindset. This lecture attempts to answer the question and suggest an alternate way of looking at school mathematics.

3. Introduction Mathematics is an excellent vehicle for the development and improvement of a person’s intellectual competence… Ministry of Education 2006

5. Problem Solving National Test

6. National Test Grade 10 (Secondary 4) A fly, F, starts at a point with position vector (i + 12j) cm and crawls across the surface with a velocity of (3i + 2j) cm s-1. At the instant the fly starts crawling, a spider, S, at the point with position vector (85i + 5j) cm, sets off across the surface with a velocity of (-5i + kj) cm s-1, where k is a constant. Given that the spider catches the fly, calculate the value of k.

7. Problem Solving Textbook

8. Draw a 4-sided polygon. Cut it into triangles. How many triangles can you get?

10. Example: Algebra Factoring a quadratic expression.

11. Algebra Tiles

17. Example: Algebra Completing the square.

25. Example: Algebra Solve .

26. Solve .

27. Example: Coordinate Geometry Find the distance between two points.

28. Problem: Find the distance between points A (2, 3) and B (10, 9). Students working collaboratively on a problem to learn the coordinate geometry topic on distance between points. Traditionally, students were given a formulae to do this. Instead students had to conduct a mini-inquiry to use what they had previously learned. BinaBangsa School, Indonesia

29. Every group was able to use their previous learning (Pythagoras Theorem) to solve the main problem. BinaBangsa School, Indonesia

30. BinaBangsa School, Indonesia Later they were also asked to find points where the distance is a whole number.

31. Example: Circles Learning definition and making sense of formulae.

32. Example To engage pupils in the learning of mathematics. Fuchun Primary School

33. Task 1 Find the longest distance between two points on a circle. Fuchun Primary School, Singapore

34. Fuchun Primary School, Singapore

35. Fuchun Primary School, Singapore

36. Task 2 Measure circumference and diameter of four circles. Fuchun Primary School, Singapore

37. Task 3 Draw a circle given that its circumference is 6 metres. Fuchun Primary School, Singapore

38. Fuchun Primary School, Singapore

39. Fuchun Primary School, Singapore

41. Example: Circles Learning definition and making sense of formulae.

45. Area > 2r2

46. Area < 4r2

47. 2r2 < Area < 4r2

53. Conclusion Mathematics is not just for its content but also for the opportunities to develop generic thinking skills.

54. Example: Trapezium a c b h d

55. Example: Trapezium a d c b h d a

56. International Benchmarks

57. TIMSS 2007 Achievement Singapore International USA Grade 8 Advanced 40 6 2 High 70 31 15 Intermediate 88 67 46 Low 97 92 55 Photo: Catholic High School (Primary)

58. TIMSS 2007 Affect PATM SVM SCM Grade 8 High 60 41 77 Medium 20 34 19 Low 20 25 4 Photo: Catholic High School (Primary)