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- 1. Singapore Mathematics:An IntroductionWhat is Singapore Mathematics?There is no such thing as Singapore Mathematics in Singapore. Whathas come to be known as Singapore Mathematics is the way studentslearn Mathematics and the way teachers learn to teach Mathematicsin Singapore. This includes the curriculum, the textbooks, and thecorresponding teacher professional development.The Singapore Mathematics curriculum is derived from an educationsystem that focuses on thinking and places a strong emphasis onconceptual understanding and mathematical problem-solving. Thescope and sequence of the curriculum is well articulated and follows aspiral progression. A pedagogy that is based on students progressingfrom concrete to pictorial and then to abstract representations,helps the majority of students acquire conceptual understanding ofmathematical concepts. Visuals are used extensively in textbooks.Mathematics as a Vehicle toDevelop Thinking SkillsSince the late 1990s, the Singapore education system has emphasizedon thinking skills as one of its pillars. Schools are encouraged to useschool subjects to help students to acquire good thinking skills anddevelop good thinking habits. In this vein, the ‘Thinking Schools,Learning Nation’ philosophy was introduced in 1997.The latest version of the Singapore Mathematics (Primary) syllabusstates that Mathematics is “an excellent vehicle for the developmentand improvement of a person’s intellectual competence in logicalreasoning, spatial visualisation, analysis and abstract thought” (Ministryof Education of Singapore, 2006, p. 5).The following Lesson 1 helps students develop visualization ability.Lesson 2 helps students develop the ability to see patterns andgeneralize the patterns. Visualization and generalization are examples ofintellectual competence that can be developed through Mathematics. 1
- 2. Lesson 1 In the Kitchen Allan puts some brown sugar on a dish. The total weight of the brown sugar and dish is 110 g. Bella puts thrice the amount of brown sugar that Allan puts on an identical dish, and the total weight of the brown sugar and dish is 290 g. Find the weight of the brown sugar that Bella puts on the dish.2
- 3. Lesson 2 Name Patterns Cheryl and David tried counting the letters in their names in a certain way. For example, Cheryl counted the letters in CHERYL back and forth such that the letter C is the 1st, H is the 2nd, E is the 3rd, R is the 4th, Y is the 5th, and L is the 6th. Then she counts backwards such that Y is the 7th, R is the 8th, E is the 9th, H is the 10th, C is the 11th. Using this way of counting, Cheryl’s 19th letter is E. When David counts his name DAVID, the 19th letter is V. Find Cheryl’s and David’s 99th letter using this way of counting. 3
- 4. Mathematical Problem Solving as the Focus of Learning Mathematics Based on research from around the world, Singapore developed a Mathematics curriculum in the late 1980s to enable students to develop mathematical problem-solving ability. This was introduced to Primary 1 students in 1992. Beliefs Interest Appreciation Monitoring of one’s own thinking Confidence M et Self-regulation of learning Perseverance es ac d og tu nit A tti ion Mathematical s Problem esse Numerical calculation Reasoning, communication Skill Algebraic manipulation Solving and connections Proc Thinking skills and heuristics s Spatial visualization Application and modelling Data analysis Measurement Concepts Use of mathematical tools Estimation Numerical Algebraic Geometrical Statistical Probabilistic Analytic In Lessons 3, 4, and 5, students do not have rules that they can follow to solve the problems. Thus, students have to think of ways to solve the problems and in this way, develop problem-solving strategies. Lesson 3 helps to teach students a new skill. In Lesson 4, the problem is used to consolidate a skill, by allowing students to practice. In Lesson 5, students have to apply a previously learned skill to solve a problem involving the formula for a figure that is different from the ones they have been taught, as students are not taught the formula to find the area of a trapezium (also known as trapezoid) in the Singapore primary curriculum.4
- 5. Lesson 3 Sharing Three-Quarters Share three-quarters of a cake equally among 4 persons. What fraction of the cake does each person get? 5
- 6. Lesson 4 Make a Multiplication Sentence Use one set of digit tiles to make correct multiplication sentences. × Resources: Digit Tiles (See Appendix)6
- 7. Lesson 5 What is the Area? Find the area of the figure. 5 cm 4 cm 5 cm 8 cm 7
- 8. Learning Theories A strong foundation is necessary for the students to do well in Mathematics. In the Singapore textbooks, such a strong foundation is achieved through the application of a few learning principles or learning theories. Jerome Bruner The Concrete Pictorial Abstract Approach — the progression from concrete objects to pictures to abstract symbols is recommended for concept development. This is based on the work of Jerome Bruner on enactive, iconic, and symbolic representations. Students learn a new concept or skill by using concrete materials. Bruner referred to this as the enactive representations of the concept or skill. Later, pictorial representations are used before the introduction of symbols (abstract representations). Reinforcement is achieved by going back and forth between the representations. For example, students learn the concept of division by sharing 12 cookies among 4 persons as well as by putting 12 eggs in groups of 4 before progressing to using drawings to solve division problems. Later, they learn to write the division sentence 12 ÷ 4 = 3. This is referred to as the CPA Approach. Let ’s Learn! 15 Division How To Divide Sharing Let ’s Learn! 1 Sharing Equally Googol has 6 mangosteens. 1 There are 12 cookies. He wants to divide the mangosteens into 2 equal groups. Googol has 4 friends. How many mangosteens are there in each group? He gives each friend the same number of cookies in a bag. ÷ stands for division. 6÷2 =3 Indu Huiling Amil Weiming There are 3 mangosteens in each group. I try putting 2 cookies Now he wants to divide them into equal 3 groups. in each bag. Then I have 4 cookies left. 6÷3 =2 There are 2 mangosteens in each group. How do I read 6 ÷ 2 = 3 and 6 ÷ 3 = 2 are division sentences. 6 ÷ 3 = 2? We read 6 ÷ 2 = 3 as six divided by two Now I put 1 more cookie is equal to three. in each bag. I have no Each friend gets 3 cookies. cookies left. 79 81 079-083 MthsP1B U15.indd 79 6/27/06 6:14:37 PM MPH!Mths 2A U04.indd 81 7/7/06 8:25:15 PM Pupil’s Book 1B p. 79 Pupil’s Book 2A p. 818
- 9. Grouping Let ’s Learn! 3 Googol has 12 snap cards. He divides the cards equally among his friends. Finding The Number Of Groups First put 4 eggs Each friend gets 4 cards. How many friends are there? 1 There are 12 eggs. into 1 bowl. Put 4 eggs into each bowl. How many bowls do you need? Do this until all the eggs × 4 = 12 are put into the bowls. 12 ÷ 4 = There are friends. 4 Sulin has 18 cards. She gives the cards to some friends. I need 3 bowls. If each friend gets 3 cards, how many friends are there? 2 Crystal has 15 toy cats. She puts 3 toy cats on each sofa. How many sofas are needed for all the toy cats? × 3 = 18 18 ÷ 3 = sofas are needed for all the toy cats. There are friends. 81 83079-083 MthsP1B U15.indd 81 6/27/06 6:15:11 PMPupil’s Book 1B p. 81 Pupil’s Book 2A p. 83 MPH!Mths 2A U04.indd 83 1/25/07 9:30:34 AMSource: My Pals are Here! Maths (2nd Edition)The Spiral Approach — students revisit core ideas as they deepen theirunderstanding of those ideas. This is also one of Jerome Bruner’s theories.For example, students learn to divide discrete quantities without theneed to write division sentences in Primary 1. I put 15 apples equally on 3 plates. I put 3 apples in a group. Divide 15 apples into 3 equal groups. Divide 15 apples into groups of 3. There are apples in each group. There are groups. 62 61 PFP_1BTB_Chpt15.indd 62 2/2/07 4:41:06 PMTextbook 1B p. 61 Textbook 1B p. 62 PFP_1BTB_Chpt15.indd 61 2/2/07 4:41:01 PMIn Primary 2, they revisit this idea and use division sentences torepresent the word problems. 4. 2 Division 1. Divide 8 mangoes into 2 equal groups. Divide 15 children into groups of 5. whole There are 4 mangoes in each group. There are 3 groups. We write: Divide 8 by 2. We write: The answer is 4. 8 ÷ 2=4 15 ÷ 5 = 3 Divide 15 by 5. The answer is 3. part part part We also divide to find This is division. the number of groups. Divide 12 balloons into groups of 4. We divide to find the number in each group. Divide 12 balloons into 3 equal groups. 97 Exercise 5, pages 100-102 94 95 Textbook 2A p.97 PFP_2A_TB_Chap5.indd 97 2/2/07 3:25:51 PMTextbook 2A p.94 Textbook 2A p.95 PFP_2A_TB_Chap5.indd 94 2/2/07 3:25:40 PM PFP_2A_TB_Chap5.indd 95 2/2/07 3:25:43 PMSource: Primary Mathematics (Standards Edition) 9
- 10. In Primary 3, the idea is extended to include the idea of a remainder. They also learn to regroup before dividing 2-digit and 3-digit numbers. 2 4 furries shared 11 seashells equally among themselves. Let ’s Learn! Division With Regrouping In Tens And Ones 1 Fandi and Farley went ﬁshing and caught some ﬁshes and crabs. They shared the 52 ﬁshes equally between themselves. How many ﬁshes did each boy get? a How many seashells did each furry receive? b How many seashells were left? 52 ÷ 2 = ? First, divide the tens by 2. 2 Tens Ones 5 tens ÷ 2 2 ͤෆෆ 5 2 a 11 ÷ 4 = ? = 2 tens with remainder 1 ten 4 Divide the 1 11 seashells into 4 equal groups. 4×2=8 8 is less than 11. 4 × 3 = 12 12 is more than 11. Regroup the remainder ten: 2 Tens Ones Choose 2. 1 ten = 10 ones 2 ͤෆෆ 5 2 Add the ones: 4 10 ones + 2 ones = 12 ones 1 2 11 ones ÷ 4 = 2 ones with remainder 3 ones = 2 R3 Quotient = 2 ones 2 R3 Then, divide the ones by 2. 2 6 Remainder = 3 ones 4 ͤෆෆ 1 1 Tens Ones 12 ones ÷ 2 = 6 ones 2 ͤෆෆ 5 2 8 4 Each furry received 2 seashells. 3 1 2 So, 52 ÷ 2 = 26. 1 2 b 3 seashells were left. 0 Each boy got 26 ﬁshes. 94 101 MthsP3A_U07(1 July) 94 7/5/06 2:39:28 PM MthsP3A_U07(1 July) 101 7/5/06 2:40:54 PM Pupil’s Book 3A p. 81 Pupil’s Book 3A p. 101 In Primary 4, 4-digit numbers are used and in Primary 5, division of continuous quantities are dealt with where 168 ÷ 16 = 10.5 rather than 10 remainder 8. Division Let’s Learn! 7 a Divide 4572 by 36. Press Display C 0 Division By A 1-Digit Number 4572 4572 1 6381 sweets were given to the children at a fun fair. Each child received ÷ 36 36 3 sweets. How many children were there at the fun fair? = 127 Th H T O The answer is 127. Step 1 2 Divide 6 thousands by 3. 3 6 3 8 1 b 6 2؋3 What is 168 � divided by 16? Press Display 6 thousands ، 3 = 2 thousands C 0 = 2000 Step 2 168 168 2 1 3 6 3 8 1 ÷ 16 16 Divide 3 hundreds by 3. 6 = 10.5 3 hundreds ، 3 = 1 hundred 3 168 � divided by 16 is 10.5 �. = 100 3 1؋3 Step 3 2 1 2 3 6 3 8 1 Divide 8 tens by 3. 6 8 Carry out this activity. 3 Remember to press 8 tens ، 3 = 2 tens with remainder 2 tens 3 C before you start = 20 with remainder 20 Work in pairs to do these: working on each sum. 8 6 2؋3 a 1065 ؋ 97 b 13 674 ؋ 7 c 1075 ، 25 2 d 10 840 ، 40 e 25 m ؋ 48 m f 406 g ، 28 Step 4 2 1 2 7 Divide 21 ones by 3. 3 6 3 8 1 Think of one multiplication and one division sentence. Get your 6 partner to work them out using a calculator. Check that your 21 ones ، 3 = 7 ones 3 partner’s answers are correct using your calculator. 3 =7 8 6 WB 5A, p 23 When 6381 is divided by 3, the quotient Practice 1 2 1 is 2127 and the remainder is 0. 2 1 7؋3 There were 2127 children at the fun fair. 0 57 32 Chapter 2: Whole Numbers (2) J43 4A CB U03 (57-70) 28Jul 57 7/28/06 2:56:07 PM Pupil’s Book 4A p. 57 Pupil’s Book 5A p. 32 CB5A_U02(29-42).indd 32 9/7/07 10:26:28 AM Source: My Pals are Here! Maths (2nd Edition) Jerome Bruner proposed the idea of spiral curriculum in 1960s. A curriculum as it develops should revisit [the] basic ideas repeatedly, building upon them until the student has grasped the full formal apparatus that goes with them. Bruner, 1960 Bruner recommended that when students first learn an idea, the emphasis should be on grasping the idea intuitively. After that, the curriculum should revisit the basic idea repeatedly, each time adding to what the students already know until they understand the idea fully. Bruner emphasized that ideas are not merely repeated but… revisited later with greater precision and power until students achieve the reward of mastery. Bruner, 197910
- 11. Bruner explained that ideas that have been introduced in an intuitivemanner were then revisited and reconstrued in a more formal oroperational way, then being connected with other knowledge, themastery at this stage then being carried one step higher to a new levelof formal or operational rigour and to a broader level of abstractionand comprehensiveness. The end stage of this process was eventualmastery of the connexity and structure of a large body of knowledge. Bruner, 1960Bruner gave an example of how students learn the idea of primenumbers and factoring.The concept of prime numbers appears to be more readily graspedwhen the child, through construction, discovers that certain handfulsof beans cannot be laid out in completed rows and columns. Suchquantities have either to be laid out in a single file or in an incompleterow-column design in which there is always one extra or one too fewto fill the pattern. These patterns, the child learns, happen to be calledprime. Bruner, 1973Zoltan DienesSystematic Variation – Students are presented with a variety of tasksin a systematic way. This is based on the ideas of Zoltan Dienes(Dienes, 1960). Let ’s Learn! 2 Addition And Subtraction Within 1000 Addition With Regrouping In Ones 1 347 + 129 = ? Let’s Learn! First, add the ones. Hundreds Tens Ones 3 14 7 Simple Addition Within 1000 + 1 29 1 Add using base ten blocks. First, add the ones. 347 6 Use the place value chart to help you. 1 23 7 ones + 9 ones a 123 + 5 = ? + 5 = 16 ones 8 Regroup the ones. Hundreds Tens Ones 16 ones = 1 ten 6 ones 3 ones + 5 ones 129 = 8 ones Then, add the tens. Then, add the tens. 3 14 7 123 + 1 29 1 23 + 5 Hundreds Tens Ones 7 6 4 tens + 2 tens + 1 ten 28 5 = 7 tens 2 tens + 0 tens = 2 tens Lastly, add the hundreds. Lastly, add the 476 So, 123 + 5 = 128. 3 14 7 hundreds. + 1 29 1 23 + 5 47 6 3 hundreds + 1 hundred 1 28 So, 347 + 129 = 476. = 4 hundreds 1 hundred + 0 hundreds = 1 hundred 27 35 MPH!Mths 2A U02.indd 27 7/7/06 7:43:55 PM MPH!Mths 2A U02.indd 35 7/7/06 7:46:27 PM Pupil’s Book 2A p. 27 Pupil’s Book 2A p. 35Source: My Pals are Here! Maths (2nd Edition)The above example shows mathematical variability. The variation is inthe Mathematics — addition without regrouping and with regrouping. 11
- 12. Multiplying by a Les 1-Digit Number The next example shows perceptual variability — the mathematical concept is the same but students are presented with different ways to perceive a 2-digit number. Lesson Objective • Use different methods to multiply up to 4-digit numbers by 1-digit numbers, with or without regrouping. arn Represent numbers using place-value charts. Le 213 can be represented in these ways. nTextbook 1B p. 30 o Less Multiplying by a Source: Primary Mathematics (Standards Edition) Hundreds multiple embodiment is to use different ways to Tens Ones The idea of 1-Digit Numbera represent the same concept. In the above example, the concept of 2-digit number Lesson Objective such as 34 is represented in multiple ways — using sticks, coins and base ten blocks. Is the representation more abstract • Use different methods to multiply up to 4-digit numbers than another? by 1-digit numbers, with or without regrouping. In the next example, 3-digit numbers are represented using base ten arn Represent numbers using place-value charts. Le blocks, number discs and digits. 213 can be represented in these ways. Multiplying by a n o Less Lesson Objective 1-Digit Number Hundreds • Use different methods to multiply up to 4-digit numbers Tens Ones by 1-digit numbers, with or without regrouping. arn Represent numbers using place-value charts. Le 213 can be represented in these ways. Hundreds Tens Ones Hundreds Tens Ones Hundreds Tens Ones Hundreds Tens Ones 2 1 3 Lesson 3.1 Multiplying by a 1-Digit Number 77 Student Book 4A Source: Math in Focus: The Singapore Approach Hundreds Tens Ones The base ten blocks are shown proportionately using concrete materials. For example, looks ten times as large as . However, the number discs are non-proportionate. Hundreds For example, Hundreds Tens does not look ten times as large as . Tens OnesOnes 2 1 3 2 This is another example of representing the same mathematical concept in different ways, some are more abstract than others. It is im- 1 3 portant to provide students with these variations in a systematic way. Additional reading: Read an article by Post (1988) which has a section on Dienes’ ideas of variability (http://www.cehd. umn.edu/rationalnumberproject/88_9.html), as well as the six-stage theory of learning mathematics 3.1 Lesson Multiplying by a 1-Digit Nu (http://www.zoltandienes.com/?page_id=226)12
- 13. Richard SkempRichard Skemp (Skemp, 1976) provides Mathematics teachers with away to think about what constitutes understanding in Mathematics.Skemp distinguished between the ability to perform a procedure,(for example, long division), and the ability to explain the procedure,(for example, explaining the rationale for ‘invert-and-multiply’ whendividing a proper fraction by a proper fraction). He refers to the formeras instrumental understanding (or procedural understanding) and thelatter as relational understanding (conceptual understanding).Singapore Mathematics curriculum expects instrumental understandingto be accompanied by relational understanding. It is pointless to learna procedure without having a conceptual understanding.“Although students should become competent in the variousmathematical skills, over-emphasising procedural skills withoutunderstanding the underlying mathematical principles should beavoided” (Ministry of Education of Singapore, 2006, p. 7).Conventional understanding involves the ability to understand the useof conventions. For example, it is a convention to use + as the symbolfor addition. Some conventions are not universal. For example, ÷ isused as the symbol for division in some countries, but : is used as thesymbol for division in others. Conventions that are universal includethe order of operations. There are some facts, names, notations, andusage which are universally agreed upon, and there are no particularreasons for using those conventions. 12. Chelsea has 5 apple tarts. She cuts each tart into 1 . Find the number 2 of half-tarts Chelsea has. 5 ÷ 1 = 10 When you cut a whole into 2 halves, you get 2 halves. So, in 5 wholes there are Chelsea has half-tarts. 5 × 2 halves. 13. 3 cakes are shared equally among some children. Each child gets 3 of a cake. How many children got 3 of a cake? 4 4 When you cut a whole into three-quarters, you get 1 1 three-quarters. So, in 3 wholes 3 3÷ 3 =4 there are 3 × 1 1 three-quarters. 4 3 children shared the cakes. 66 Textbook 6A 6ATB_Unit 3.indd 66 4/24/09 5:10:13 PMSource: Primary Mathematics (Standards Edition)Additional reading:Read the classic article originally published in Mathematics Teaching (1976) at http://www.grahamtall.co.uk/skemp/pdfs/instrumental-relational.pdf 13
- 14. Lesson 6 Long Division Find the value of 51 ÷ 3. In some countries, this can also be written as 51 : 3. 5114
- 15. Lesson 7 Bar Model 1 1 Marcus gave of his coin collection to his sister and of the remainder 4 2 to his brother. As a result, Marcus had 18 coins. Find the number of coins in his collection at first. In Lessons 6 and 7, we are able to explain the long division algorithms as well as the procedure to multiply fractions. We are said to possess relational understanding of these procedures. 15
- 16. References 1. Bruner, J. (1960). The Process of Education. Cambridge, MA: Harvard University Press. 2. Bruner, J (1966). On Knowing: Essays for the Left Hand. Cambridge, MA: Harvard University Press. 3. Bruner, J. S. (1973). Beyond the information given: Studies in the psychology of knowing, pp. 218-238. New York: W. W. Norton & Co Inc. 4. Dienes, Z. P. (1960). Building up mathematics. London: Hutchinson Educational Ltd. 5. Ministry of Education of Singapore. (2006). Mathematics Syllabus (Primary). Singapore: Curriculum Planning and Development Division. from http://www.moe.gov.sg/education/syllabuses/sciences/files/ maths-primary-2007.pdf 6. Post, T. (1988). Some notes on the nature of mathematics learning. Teaching Mathematics in Grades K-8: Research Based Methods , pp. 1-19. Boston: Allyn & Bacon. http://www.cehd.umn.edu/rationalnumberproject/88_9.html 7. Skemp, R. R. (1976). Relational and instrumental understanding. Mathematics Teaching, 77, pp. 20-26. http://www.grahamtall.co.uk/skemp/pdfs/instrumental-relational.pdf16
- 17. Appendix 0 1 0 1 2 3 2 3 4 5 4 5 6 7 6 7 8 9 8 9 17
- 18. Notes18

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