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Elementary Mathematics in US: How can “more” be “less”? Liping Ma The Carnegie Foundation for the Advancement of Teaching

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How can more be less? 1.More vs. less 2.How can less be more: an example 3.The “tightest” chain More vs. less More vs. less

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W–W– W+W+ W÷W÷ W×W× Foundation type 1 Foundation type 2 A loose vs. solid foundation F + F – W + W – W × W ÷ F ÷ F ×

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Mathematics topics intended at each grade: W. Schmidt, R. Houang, & L. Cogan (2002): A Coherent Curriculum U. S. Countries with high math performance US perspective: Arithmetic as a collection of algorithms Whole numbers Fractions × − + ÷ × − + ÷ Arithmetic as a microcosm of mathematics Concept of a Unit × ÷ + − Fractions Whole numbers

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W + W – W × W ÷ F ÷ F × F + F – W–W– W+W+ W÷W÷ W×W× Foundation type 1 Foundation type 2 A loose vs. solid foundation: the consequence

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F + F – W–W– W+W+ W÷W÷ W×W× Foundation type 1 Foundation type 2 Building a Solid Foundation W + W – W × W ÷ F ÷ F ×

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How can more be less? 1.More vs. less 2.How can less be more: an example 3.The “tightest” chain

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“Unit (one)”, a simple but powerful concept -- the following quotations are from Sheldon’s Complete Arithmetic (1886) Quotation 1 A unit is a single thing or one; as one apple, one dollar, one hour, one. Quotation 2 Like numbers are numbers whose units are the same; as $7 and $9. Unlike numbers are numbers whose units are different; as 8 lb. and 12 cents. Quotation 3 Can you add 8 cents and 7 cents? What kind of numbers are they? Can you add $5 and 5lb.? What kind of numbers are they? Quotation 4 Principle: Only like numbers can be added an subtracted. Why do we need to line numbers up when we do addition ?

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With multiplication and division, the concept of “unit” is expanded: Quotation 1 A unit is a single thing or one. Quotation 2 A group of things if considered as a single thing or one is also a unit; as one class, one dozen, one group of 5 students. Quotation 3 There are 3 plates each with 5 apples in it. How many apples are there in all? What is the unit (the “one”)? Some children are sharing 15 apples among them. Each them gets 5 apples. How many children are there? What is the unit (the “one”)? There are 3 children who want to evenly share 15 apples among them. How many apples will each child get? What is the unit (the “one”)?

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With fractions, the concept of “unit” is expanded one more time: Quotation 1 A unit is a single thing or one. Quotation 2 A unit, however, may be divided into equal parts, and each of these parts becomes a single thing or a unit. What is the fractional unit of 3/4 ? of 2/3? Quotation 3 In order to distinguish between these two kinds of units, the first is called an integral unit, and the second a fractional unit.

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With fractions, the concept of “unit” is expanded one more time: Computing 3/4 + 2/3, Why do we need to turn the fractions into fractions with common denominator? Quotation 1 Principle Only like numbers can be added an subtracted.

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How can more be less? 1.More vs. less 2.How can less be more: an example 3.The “tightest” chain

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ss Ratio and proportion Organizing the topics (the tightest chain and breakups) Numbers 0 to 10, addition and subtraction Numbers 11 to 20, addition and subtraction (with concept of regrouping) Numbers up to 100, addition and subtraction (with concept of regrouping) Numbers up to 10,000, notation, addition and subtraction Multiplication with multiplier as a one-digit number Division with divisor as a one-digit number Many-digit numbers, notation, addition and subtraction Multiplication with multiplier as a two-digit number Division with divisor as a two-digit number Multiplication with multiplier as a three-digit number Division with divisor as a three-digit number Fractions – the basic concepts Decimals – meaning and features Decimals – addition and subtraction Decimals – multiplication and division Divisibility Fractions – meaning and features Fractions – addition and subtraction Fractions – multiplication Fractions – division ss Percentages Money Multiplication and division with multiplication tables Time Weight Area of rectangles Angles & lines Length Weight Perimeter of rectangles Circle (perimeter & area); cylinder & cone (area and volume) Area of triangles & trapezoids; Prism and cubic (volume)

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63 + 3 = 40 + 5 = 30 + 20 = 11 + 6 = 15 + 2 = 6 + 9 = 8 + 4 = 6 + 6 = 7 + 3 = 2 + 6 = 7 − 5 = 10 − 3 = 12 − 6 = 12 − 4 = 15 − 9 = 17 − 15 = 17 − 11 = 50 − 30 = 90 − 5 = 66 − 3 = 38 + 25 = 45 + 18 = 27 + 4 = 52 + 12 = 64 − 22= 63 + 20 = 85 − 20 = 72 − 3 = 85 − 16 = 42 − 18 = How number sense can be developed through well arranged exercises 3 + 2 = 4 − 1 = Within 10 With 10 Within 20 (across 10) Within 100 (without regrouping) Within 100 (with regrouping)

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Five categories of “missing pieces” 1)Basic concepts to form arithmetic as a subject 2)Basic terminology in teaching and learning arithmetic as a subject 3) “Anchoring ideas” for future mathematical learning 4)Computational capacity for future mathematical learning 5)The system of word problems

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Where did the “more” come from?

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A Metaphor (1) (2) (3) (4) If the above metaphor makes sense, who will take the responsibility to make the change?

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Thank you !

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