SISB Heuristics


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SISB Heuristics

  1. 1. singapore international school of bangkok problem-solving heuristics for primary mathematics Yeap Ban Har | | Learning Outcomes Participants are expected to learn about problem-solving heuristics suitable for primary school and how to teach problem-solving type lessons. The model method will be given some attention. Presenter Dr. Yeap Ban Har is an established name in mathematics education and teacher professional development. He spent ten years at Singapore's National Institute of Education training preservice and in-service educators. A leading educator, speaker and trainer, Ban Har has authored dozens of textbooks, math readers and assorted titles for teachers. He has been a keynote speaker at international conferences, and is currently the Principal of the Marshall Cavendish Institute. He is also Director of Curriculum and Professional Development at Pathlight School, a K-10 school for students with autism. Ban Har’s interest is in early numeracy, problem solving and teacher professional development. In Thailand, he has given mathematics education seminars at Kasetsart University, Srinakharinwirot University, Chiangmai University, Rajabhat University Phuket, Rajabhat University Mahasarakham, and Khon Kaen University. He is teaching graduate courses for masters and doctoral students as a visiting professor this coming academic year. He will also conduct a seminar in Bangkok jointly organized by Shinglee Publishers on 26 November 2013. coursebook 1|Problem Solving
  2. 2. introduction The Singapore approach to teaching and learning mathematics was the result of trying to find a way to help Singapore students who were mostly not performing well in the 1970’s. In 1980, only 58 percent of first-grade students completed secondary school. By 2000, 93 percent of first-grade students completed secondary school. Achievement, as measured by national examination, also improved. In 1981, 40 percent of the first-graders would graduate 10th grade with passes in three subjects. In 1991, the proportion increased to 65 percent. In 2010, the proportion was 88 percent. Providing students with differentiated curriculum and differentiated examination has arguably resulted in “a very low attrition rate and a very high average achievement.” | Yeap 2012 Improving the Education for All: Curriculum Development and Implementation in Singapore American Institutes for Research In the first national examination conducted in 1960, a total of 30,615 students sat for the Primary School Leaving Examination (PSLE) and 45 percent of them passed. In 2011, a total of 45,261 students sat for the high-stakes examination and 97 percent passed. Only 2.6 percent did not meet the proficiency required for secondary schooling. Since the 1990s, Singapore has always done well in various international studies in literacy, science, and mathematics. It is always among the top-performing countries in Progress in International Reading Literacy Study (PIRLS), Trends in International Mathematics and Science Study (TIMSS), and Program for International Student Assessment (PISA). More important than the ranking is that about 12 percent of 15-year-old students in Singapore were performing at a high level in reading, mathematics, and science in PISA 2009. This compares well with the Organization of Economic Cooperation and Development (OECD) average, which was 4 percent. High-performing OECD economies such as New Zealand, Finland, Japan, and Australia had between 8 percent and 10 percent of students reaching the same level. In Shanghai, it was almost 15 percent. The proportion of students who reach the highest levels was about 15 percent in reading (almost 20 percent in Shanghai with the OECD average at 8 percent); about 35 percent in mathematics (50 percent in Shanghai with the OECD average at 13 percent); and 20 percent in science (almost 25 percent in Shanghai with the OECD at being 9 percent). Similarly, in TIMSS, the proportion of students who reached the advanced international benchmark has been consistently high. In TIMSS 2007, 36 percent of Grade 4 and 32 percent of Grade 8 students reached this level in science (international median were 7 percent and 3 percent, respectively), and 41 percent of Grade 4 and 40 percent of Grade 8 students reached this level in mathematics (international median was 5 percent and 2 percent, respectively). These results were in great contrast to Singapore’s rank of 16th out of 26 participating countries in the Second International Science Study in 1982. | Yeap 2012 Improving the Education for All: Curriculum Development and Implementation in Singapore American Institutes for Research 2|Problem Solving
  3. 3. The CPA Approach as well as the Spiral Approach are fundamental to teaching mathematics in Singapore schools. The national standards, called syllabus in Singapore, is designed based on Bruner’s idea of spiral curriculum. Textbooks are written based on and teachers are trained to use the CPA Approach, based on Bruner’s ideas of representations. “A curriculum as it develops should revisit this basic ideas repeatedly, building upon them until the student has grasped the full formal apparatus that goes with them”. | Bruner 1960 “I was struck by the fact that successful efforts to teach highly structured bodies of knowledge like mathematics, physical sciences, and even the field of history often took the form of metaphoric spiral in which at some simple level a set of ideas or operations were introduced in a rather intuitive way and, once mastered in that spirit, were then revisited and reconstrued in a more formal or operational way, then being connected with other knowledge, the mastery at this stage then being carried one step higher to a new level of formal or operational rigour and to a broader level of abstraction and comprehensiveness. The end stage of this process was eventual mastery of the connexity and structure of a large body of knowledge.” | Bruner 1975 spiral approach concrete-pictorial-abstract Bruner's constructivist theory suggests it is effective when faced with new material to follow a progression from enactive to iconic to symbolic representation; this holds true even for adult learners. | Bruner 1966 3|Problem Solving
  4. 4. problem-solving approach | Singapore 2013 …able to pose and solve problems … | Australia 2012 Make sense of problems and preserve in solving them. | Common Core State Standards USA 2010 .. can solve problems by applying their mathematics to a variety of problems with increasing sophistication, including in unfamiliar contexts and to model real-life scenarios … | United Kingdom 2012 4|Problem Solving
  5. 5. |Example 1 Find the value of 6 + 5 + 7. 5|Problem Solving
  6. 6. |Example 2 Make 4 equal parts. 6|Problem Solving
  7. 7. |Example 3 Use one set of digit tiles 1 to 9 to make two 3-digit numbers. Make the difference the least possible. 7|Problem Solving
  8. 8. |Example 4 There are 1111 ducks in a farm. There are 299 fewer ducks than chickens. How many chickens are there? How many chickens will be left if 418 chickens are sold? Source | New Syllabus Primary Mathematics |Example 5 At first, Mario had 8 coins more than Nina. Then, Nina gave Mario 3 coins. Who had more coins after Nina gave Mario the coins? How many more? Source | Kong Hwa School (Singapore) Lesson Study 8|Problem Solving
  9. 9. |What if …? At first, Nina had 8 coins more than Mario. Then, Nina gave Mario 3 coins. Who had more coins after Nina gave Mario the coins? How many more? |Example 6 There were 94 children in a group. 2 3 of the boys and of the girls in a group chose swimming as their after-school activity. 5 7 In all, 39 children chose swimming as their after-school activity. 9|Problem Solving
  10. 10. |Example 7 10 coins are either 2 baht or 5 baht. Their total value is 41 baht. How many of the coins are 2-baht coins? |Example 8 Draw a polygon with four dots on its sides. Investigate its area. 10 | P r o b l e m S o l v i n g
  11. 11. 11 | P r o b l e m S o l v i n g
  12. 12. Additional Examples |Example 9 37 + 29 |Example 10 1 3 3  2 4 |Example 11 Multiplication Facts     18 + 6 =  Use 3 × 6 = 18 to get 4 × 6.        12 | P r o b l e m S o l v i n g
  13. 13. |Example 12 Cindy has 6 more candies than David. Cindy has 15 candies. How many candies do they both have altogether? |Example 13 Feliz had twice as much money as Ginny. After Feliz gave Ginny $12, they both had the same amount of money. How much did they both have altogether? |Example 14 Elvi made some paper cranes on Monday. On Tuesday she made 3 more paper cranes than on Monday. Subsequently, she made 3 more paper cranes than the day before. Elvi said that she had made 50 paper cranes by Saturday the same week. Is that possible? |Example 15 Han used ¼ of his savings to buy a gift for his father and ½ of the remainder on a book which cost $12. What did the gift cost? bar model method visualization |Example 16 The number of boys to the number of girls in a group was in the ratio 2 : 3. After ¼ of the boys and 69 girls left the group, there were 51 more girls than boys in the group. 13 | P r o b l e m S o l v i n g
  14. 14. |Example 17 The figure, not drawn to scale, shows an isosceles triangle XYZ, and a parallelogram UVWX. Given that UY and WZ are straight lines, XY = XZ and the sum of  YZX and  XUV is 162o, find  UVW. Source|Keming Primary School, Singapore |What if …? The figure, not drawn to scale, shows an equilateral triangle XYZ, and a parallelogram UVWX. Given that UY and WZ are straight lines, find  UVW. 14 | P r o b l e m S o l v i n g
  15. 15. |Example 18 A figure is formed using a length of wire. The figure, shown below, consists of 4 squares. X Y The length of the straight line XY is 15 cm. Find the length of the wire used. |Example 19 A group of boys share some sweets in a box. When they tried to take 11 sweets each, one boy got 6 sweets. When they took 8 sweets each, there were 25 sweets leftover. Find the number of sweets in the box. singapore textbooks Singapore textbooks are designed for a problem-solving approach. They are also designed for differentiated instruction. A typical mathematics lesson is in a three-part format – anchor task, guided practice and independent practice. 15 | P r o b l e m S o l v i n g