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This is Session 106 on mathematical problem solving.

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- 1. NCTM Regional Conference & Exposition Denver Colorado<br />8 – 9 October 2010<br />Session 106<br />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. <br />Colorado Convention Center, Korbel 3B<br />
- 2. Problem Solving in the Singapore Classrooms<br />Session 106<br />DrYeap Ban Har<br />Marshall Cavendish Institute<br />Singapore<br />banhar@sg.marshallcavendish.com<br />www.banhar.blogspot.com<br />
- 3. Beliefs<br />Interest<br />Appreciation<br />Confidence<br />Perseverance<br />Monitoring of one’s own thinking<br />Self-regulation of learning<br />Attitudes<br />Metacognition<br />Numerical calculation<br />Algebraic manipulation<br />Spatial visualization<br />Data analysis<br />Measurement<br />Use of mathematical tools<br />Estimation<br />Mathematical Problem Solving<br />Reasoning, communication & connections<br />Thinking skills & heuristics<br />Application & modelling<br />Skills<br />Processes<br />Concepts<br />Numerical<br />Algebraic<br />Geometrical<br />Statistical<br />Probabilistic<br />Analytical<br />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).<br />
- 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. <br />intellectual competence<br />
- 5. example one<br />Source<br />Earlybird Kindergarten Mathematics – Kindergarten Textbook in Singapore<br />
- 6. PCF Kindergarten TelokBlangah, Singapore<br />
- 7.
- 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?<br />example two<br />Source<br />Primary Three Examination Item from Henry Park Primary School, Singapore<br />
- 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?<br />example two<br />Angela<br />Bala<br />872<br />Cheng Hao<br />50<br />Source<br />Primary Three Examination Item from Henry Park Primary School, Singapore<br />
- 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?<br />1 unit = 822 ÷ 6<br />1 unit = 137<br />Bala has 137 marbles.<br />example two<br />Angela<br />Bala<br />822<br />Cheng Hao<br />600<br />42<br />180<br />Source<br />Primary Three Examination Item from Henry Park Primary School, Singapore<br />
- 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?<br />4y + y + (y + 50) = 872<br />6y + 50 = 872<br />6y = 872 – 50 <br />y = 822 ÷ 6 <br />example two<br />y = 137<br />Angela<br />y<br />y<br />y<br />y<br />Bala<br />y<br />872<br />Cheng Hao<br />50<br />y<br />Source<br />Primary Three Examination Item from Henry Park Primary School, Singapore<br />
- 12. Using a box to represent known quantities.<br />Using a box to represent unknown quantities.<br />Angela<br />Using a letter to represent unknown quantities.<br />4y + y + (y + 50) = 872<br />
- 13. example three<br />Source<br />Grade 4 Lesson on Long Division in Keys Grade School, Manila<br />
- 14. Source<br />Grade 4 Lesson on Long Division in Keys Grade School, Manila<br />
- 15. 1<br />2<br />0 <br />4<br />4816<br />4<br />4816<br />4<br />81<br />4000<br />16<br />8<br />800<br />16<br />16<br />0<br />
- 16. Hugo has ? toy cars.<br />His friend has ? toy cars.<br />They have ? toy cars altogether.<br />example four<br />Source<br />2oBasico Lesson on Model Drawing in EscuelaRepublica de Guatemala, Chile<br />
- 17. Hugo has 10 toy cars.<br />His friend has 12 toy cars.<br />They have ? toy cars altogether.<br />Source<br />2oBasico Lesson on Model Drawing in EscuelaRepublica de Guatemala, Chile<br />
- 18. Hugo has 10 toy cars.<br />His friend has 12 toy cars.<br />They have ? toy cars altogether.<br />Source<br />2oBasico Lesson on Model Drawing in EscuelaRepublica de Guatemala, Chile<br />
- 19. Hugo has 121 toy cars.<br />His friend has 74 toy cars.<br />They have ? toy cars altogether.<br />?<br />121<br />74<br />Source<br />2oBasico Lesson on Model Drawing in EscuelaRepublica de Guatemala, Chile<br />
- 20. Hugo has 20 toy cars.<br />His friend has 74 toy cars.<br />They have ? toy cars altogether.<br />Source<br />2oBasico Lesson on Model Drawing in EscuelaRepublica de Guatemala, Chile<br />
- 21. Hugo has 100 toy cars.<br />His friend has 74 toy cars.<br />They have ? toy cars altogether.<br />Source<br />2oBasico Lesson on Model Drawing in EscuelaRepublica de Guatemala, Chile<br />
- 22. Hugo has ? toy cars.<br />His friend has 74 toy cars.<br />They have 100 toy cars altogether.<br />Source<br />2oBasico Lesson on Model Drawing in EscuelaRepublica de Guatemala, Chile<br />
- 23. example five<br />Source<br />My Pals Are Here! Maths – Grade 1 Textbook in Singapore<br />
- 24. Use five consecutive whole numbers so that the vertical sum is equal to the horizontal sum. Start with 1 to 5.<br />example six<br />Source<br />Research Lesson at a Professional Development Course in Singapore<br />
- 25. Use five consecutive whole numbers so that the vertical sum is equal to the horizontal sum. Start with 1 to 5.<br />1<br />3<br />2<br />5<br />3<br />2<br />5<br />3<br />1<br />5<br />2<br />1<br />4<br />4<br />4<br />3<br />2<br />3<br />2<br />4<br />5<br />5<br />6<br />2<br />5<br />3<br />4<br />6<br />6<br />4<br />
- 26. 2<br />4<br />3<br />5<br />6<br />Princess Elizabeth Primary School, Singapore<br />
- 27. Princess Elizabeth Primary School, Singapore<br />
- 28. Use the digits 0 to 9 not more than once to make a correct multiplication sentence.<br />x<br />example seven<br />Source<br />Teacher Professional Development Course in Singapore<br />
- 29. 1 2<br />x 5<br />6 0<br />4 3<br />x 2<br />8 6<br />2 9<br />x 3<br />87<br />15<br />x 4<br />60<br />1 2<br />x 7<br />8 4<br />1 5<br />x 2<br />3 0<br />18<br />x 2<br />3 6<br />1 8<br />x 3<br />5 4<br />1 8<br />x 4<br />7 2<br />1 2<br />x 8<br />9 6<br />responses<br />1 4<br />x 7<br />9 8<br />
- 30. problem solving for various instructional goals<br />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.<br />
- 31. PK – K <br />G 1 <br />problem solving to teach a new concept or skill<br />G 2 <br />G 3<br />G 4 <br />
- 32. PK – K <br />visualization<br />metacognition<br />G 1 <br />G 2 <br />visualization<br />G 3<br />G 4 <br />number sense<br />
- 33. generalization<br />problem solving to consolidate a skill or concept<br />x<br />reasoning<br />
- 34. problem solving for students to apply their knowledge<br />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?<br />
- 35. Problem Solving in the Singapore Classrooms<br />Slides are available at<br />www.banhar.blogspot.com<br />Da Qiao Primary School, Singapore<br />

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