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 2011Developing Thinking Through High School MathematicsYeap Ban HarMarshall Cavendish InstituteSingaporebanhar@sg.marshallcavendish.comPresentation is available on www.banhar.blogspot.com

2. Link to Keynote LecturesFirst 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. IntroductionMathematics is an excellent vehicle for the development and improvement of a person’s intellectual competence…Ministry of Education 2006

5. Problem SolvingNational 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 SolvingTextbook

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

10. Example: AlgebraFactoring a quadratic expression.

11. Algebra Tiles

17. Example: AlgebraCompleting the square.

25. Example: AlgebraSolve .

26. Solve .

27. Example: Coordinate GeometryFind 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, IndonesiaLater they were also asked to find points where the distance is a whole number.

31. Example: CirclesLearning definition and making sense of formulae.

32. ExampleTo 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: CirclesLearning definition and making sense of formulae.

45. Area > 2r2

46. Area < 4r2

47. 2r2 < Area < 4r2

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

54. Example: Trapeziumacbhd

55. Example: Trapeziumadcbhda

56. International Benchmarks

57. TIMSS 2007 AchievementSingaporeInternationalUSAGrade 8Advanced4062High703115Intermediate886746Low979255Photo: Catholic High School (Primary)

58. TIMSS 2007 AffectPATMSVMSCMGrade 8High604177Medium203419Low20254Photo: Catholic High School (Primary)