Elementary Mathematics in US: How can “more” be “less”?

Liping Ma

The Carnegie Foundation for the Advancement of Teaching

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

W–

W+

W÷

W×

Foundation type 1 Foundation type 2

A loose vs. solid foundation

F +

F –

W +

W –

W ×

W ÷ F ÷

F ×

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

W +

W –

W ×

W ÷ F ÷

F ×F +

F –

W–

W+

W÷

W×

Foundation type 1 Foundation type 2

A loose vs. solid foundation: the consequence

F +

F –

W–

W+

W÷

W×

Foundation type 1 Foundation type 2

Building a Solid Foundation

W +

W –

W ×

W ÷ F ÷

F ×

How can more be less?

1. More vs. less

2. How can less be more: an example

3. The “tightest” chain

“Unit (one)”, a simple but powerful concept -- the following quotations are from Sheldon’s Complete Arithmetic (1886)

Quotation 1A unit is a single thing or one; as one apple, one dollar, one hour, one.

Quotation 2Like 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 3Can 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 ?

With multiplication and division, the concept of “unit” is expanded:

Quotation 1A unit is a single thing or one.

Quotation 2A 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”)?

With fractions, the concept of “unit” is expanded one more time:

Quotation 1A unit is a single thing or one.

Quotation 2A 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 3In order to distinguish between these two kinds of units, the first is called an integral unit, and the second a fractional unit.

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.

How can more be less?

1. More vs. less

2. How can less be more: an example

3. The “tightest” chain

ssRatio and proportion

Organizing the topics (the tightest chain and breakups)

Numbers 0 to 10 , addition and subtractionNumbers 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 subtractionMultiplication 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 numberDivision with divisor as a two-digit number

Multiplication with multiplier as a three-digit numberDivision with divisor as a three-digit number

Fractions – the basic conceptsDecimals – meaning and features Decimals – addition and subtraction

Decimals – multiplication and divisionDivisibility

Fractions – meaning and featuresFractions – addition and subtraction

Fractions – multiplicationFractions – division

ssPercentages

MoneyMultiplication and division with multiplication tables

TimeWeight

Area of rectanglesAngles & lines

Length

LengthWeight

Perimeter of rectangles

Circle (perimeter & area); cylinder & cone (area and volume)

Area of triangles & trapezoids;Prism and cubic(volume)

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)

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

Where did the “more” come from?

A Metaphor

(1)

(2) (3)

(4)

If the above metaphor makes sense, who will take the responsibility to make the change?

Thank you !