This plenary lecture on elementary school mathematics was given at the conference which marks the end of the three-year Leaders and Educators in Asia Programme.

1. LEAP Educators’ Conference 2011 Promoting Higher-Order Thinking through Drill-and-Practice in Elementary School Mathematics Yeap Ban Har Marshall Cavendish Institute Singapore banhar@sg.marshallcavendish.com Presentation is available on www.banhar.blogspot.com

2. This lecture is dedicated to the fifteen students in the Elementary Mathematics Content and Pedagogy Course held at NIE (Sep – Dec 2010)

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

5. Example Arrange consecutive numbers 1 to 5 in the given arrangement so that the vertical sum is equal to the horizontal sum.

6. 3 2 2 1 5 2 3 5 1 5 4 1 4 4 3 Total = 10 Total = 8 Total = 9

7. Example Arrange consecutive numbers 2 to 6 in the given arrangement so that the vertical sum is equal to the horizontal sum.

8. 5 3 3 2 6 3 6 5 2 4 6 2 4 4 5 Total = 12 Total = 11 Total = 13

9. 3 2 2 1 5 2 3 5 1 5 4 1 4 4 3 1 8 Total = 10 Total = 8 Total = 9 3 9 5 10 2 11 4 12 5 3 3 6 13 2 6 3 6 5 2 4 6 2 4 4 5 Total = 12 Total = 11 Total = 13

10. 1 8 1, 2, 3, 4, 5 3 9 5 10 2 11 4 12 2, 3, 4, 5, 6 6 13 3 14 5 3, 4, 5, 6, 7 15 7 16 20 ? 22 20, 21, 22, 23, 24 ? 24 ? m m + 2 m + 4 ? 3(m + 2) ? m, …

11. 2 4 3 5 6 Princess Elizabeth Primary School, Singapore

12. Princess Elizabeth Primary School, Singapore

13. Example Students get to practise addition of three numbers while developing higher-order thinking such as looking for patterns and relations, as well as their number sense.

14. Use the digits 0 to 9 not more than once to make a correct multiplication sentence. x Source Teacher Professional Development Course in Singapore

15. 1 2 x 5 6 0 4 3 x 2 8 6 2 9 x 3 87 15 x 4 60 1 2 x 7 8 4 1 5 x 2 3 0 18 x 2 3 6 1 8 x 3 5 4 1 8 x 4 7 2 1 2 x 8 9 6 responses 1 4 x 7 9 8

16. The Anchor Problem – to make a correct division sentence using one set of digit tiles 0 to 9. Source: Springhurst Elementary School, New York

21. Example How about the learning of multiplication facts or the multiplication table? Can it involve the development of higher-order thinking?

22. Source:Pensar sin Limites (Chile)

23. Source:Pensar sin Limites (Chile)

24. Source: Maths Champion (Indonesia)

25. 2 x 7 = 14 5 x 7 = 35 7 x 7 = ? 3 x 8 = 24 6 x 8 = ? 12 x 8 = ? 10 x 6 = 60 9 x 6 = ? 18 x 6 = ? 17 x 6 = ? 2 x 8 = 13 3 x 8 = ? Source: Primary Mathematics (Singapore)

26. Example Students have a chance to develop the ability to make connections and also to develop visualization ability which is fundamental in the learning of mathematics.

27. 1 2 0 4 4816 4 4 8 8 1 0 16 16 0

28. 1 2 0 4 4816 4 4816 4 8 4000 16 8 800 16 16 0

29. Source Grade 4 Lesson on Long Division in Keys Grade School, Manila

30. Source Grade 4 Lesson on Long Division in Keys Grade School, Manila

32. Conclusion Drill-and-practice is mot mutually exclusive from higher-order thinking. This lecture gives examples of how drills can be used to develop a range of higher-order thinking.