This is Session 106 on mathematical problem solving.

1. NCTM Regional Conference & Exposition Denver Colorado 8 – 9 October 2010 Session 106 This session includes a selection of problems used in grades K – 3 classrooms in Singapore. Look at how these problems are used to achieve different instructional goals. See how these problems play a role in helping children develop a strong foundation for subsequent learning.      Colorado Convention Center, Korbel 3B

2. Problem Solving in the Singapore Classrooms Session 106 DrYeap Ban Har Marshall Cavendish Institute Singapore banhar@sg.marshallcavendish.com www.banhar.blogspot.com

3. Beliefs Interest Appreciation Confidence Perseverance Monitoring of one’s own thinking Self-regulation of learning Attitudes Metacognition Numerical calculation Algebraic manipulation Spatial visualization Data analysis Measurement Use of mathematical tools Estimation Mathematical Problem Solving Reasoning, communication & connections Thinking skills & heuristics Application & modelling Skills Processes Concepts Numerical Algebraic Geometrical Statistical Probabilistic Analytical The mathematical problem-solving curriculum framework was introduced in Singapore in 1992 in response to recommendations made in An Agenda for Action (in the US) and the Cockcroft Report (in the UK).

4. The Singapore mathematics curriculum states that mathematics is an “excellent vehicle for the development and improvement of a person’s intellectual competence” (Ministry of Education 2006). The next few examples show how problem solving can be used to develop visualization, number sense and ability to see patterns as well as to facilitate language development. intellectual competence

5. example one Source Earlybird Kindergarten Mathematics – Kindergarten Textbook in Singapore

6. PCF Kindergarten TelokBlangah, Singapore

8. Angela, Bala, and Cheng Hao have 872 marbles altogether. Angela has 4 times as many marbles as Bala. Cheng Hao has 50 more marbles than Bala. How many marbles does Bala have? example two Source Primary Three Examination Item from Henry Park Primary School, Singapore

9. Angela, Bala, and Cheng Hao have 872 marbles altogether. Angela has 4 times as many marbles as Bala. Cheng Hao has 50 more marbles than Bala. How many marbles does Bala have? example two Angela Bala 872 Cheng Hao 50 Source Primary Three Examination Item from Henry Park Primary School, Singapore

10. Angela, Bala, and Cheng Hao have 872 marbles altogether. Angela has 4 times as many marbles as Bala. Cheng Hao has 50 more marbles than Bala. How many marbles does Bala have? 1 unit = 822 ÷ 6 1 unit = 137 Bala has 137 marbles. example two Angela Bala 822 Cheng Hao 600 42 180 Source Primary Three Examination Item from Henry Park Primary School, Singapore

11. Angela, Bala, and Cheng Hao have 872 marbles altogether. Angela has 4 times as many marbles as Bala. Cheng Hao has 50 more marbles than Bala. How many marbles does Bala have? 4y + y + (y + 50) = 872 6y + 50 = 872 6y = 872 – 50 y = 822 ÷ 6 example two y = 137 Angela y y y y Bala y 872 Cheng Hao 50 y Source Primary Three Examination Item from Henry Park Primary School, Singapore

12. Using a box to represent known quantities. Using a box to represent unknown quantities. Angela Using a letter to represent unknown quantities. 4y + y + (y + 50) = 872

13. example three Source Grade 4 Lesson on Long Division in Keys Grade School, Manila

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

15. 1 2 0 4 4816 4 4816 4 81 4000 16 8 800 16 16 0

16. Hugo has ? toy cars. His friend has ? toy cars. They have ? toy cars altogether. example four Source 2oBasico Lesson on Model Drawing in EscuelaRepublica de Guatemala, Chile

17. Hugo has 10 toy cars. His friend has 12 toy cars. They have ? toy cars altogether. Source 2oBasico Lesson on Model Drawing in EscuelaRepublica de Guatemala, Chile

18. Hugo has 10 toy cars. His friend has 12 toy cars. They have ? toy cars altogether. Source 2oBasico Lesson on Model Drawing in EscuelaRepublica de Guatemala, Chile

19. Hugo has 121 toy cars. His friend has 74 toy cars. They have ? toy cars altogether. ? 121 74 Source 2oBasico Lesson on Model Drawing in EscuelaRepublica de Guatemala, Chile

20. Hugo has 20 toy cars. His friend has 74 toy cars. They have ? toy cars altogether. Source 2oBasico Lesson on Model Drawing in EscuelaRepublica de Guatemala, Chile

21. Hugo has 100 toy cars. His friend has 74 toy cars. They have ? toy cars altogether. Source 2oBasico Lesson on Model Drawing in EscuelaRepublica de Guatemala, Chile

22. Hugo has ? toy cars. His friend has 74 toy cars. They have 100 toy cars altogether. Source 2oBasico Lesson on Model Drawing in EscuelaRepublica de Guatemala, Chile

23. example five Source My Pals Are Here! Maths – Grade 1 Textbook in Singapore

24. Use five consecutive whole numbers so that the vertical sum is equal to the horizontal sum. Start with 1 to 5. example six Source Research Lesson at a Professional Development Course in Singapore

25. Use five consecutive whole numbers so that the vertical sum is equal to the horizontal sum. Start with 1 to 5. 1 3 2 5 3 2 5 3 1 5 2 1 4 4 4 3 2 3 2 4 5 5 6 2 5 3 4 6 6 4

26. 2 4 3 5 6 Princess Elizabeth Primary School, Singapore

27. Princess Elizabeth Primary School, Singapore

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

29. 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

30. problem solving for various instructional goals We have seen the use of problem solving to teach a basic concept or skill, to consolidate learning as well as to provide opportunities for students to apply what they know.

31. PK – K G 1 problem solving to teach a new concept or skill G 2 G 3 G 4

32. PK – K visualization metacognition G 1 G 2 visualization G 3 G 4 number sense

33. generalization problem solving to consolidate a skill or concept x reasoning

34. problem solving for students to apply their knowledge Angela, Bala, and Cheng Hao have 872 marbles altogether. Angela has 4 times as many marbles as Bala. Cheng Hao has 50 more marbles than Bala. How many marbles does Bala have?

35. Problem Solving in the Singapore Classrooms Slides are available at www.banhar.blogspot.com Da Qiao Primary School, Singapore