Elementary Mathematics in US: How can

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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
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A loose vs. solid foundation
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Foundation type 1
Foundation type 2
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Mathematics topics intended at each grade
W. Schmidt, R. Houang, L. Cogan (2002) A
Coherent Curriculum
Fractions
Whole numbers
Countries with high math performance
U. S.
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A loose vs. solid foundation the consequence
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Foundation type 1
Foundation type 2
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Building a Solid Foundation
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Foundation type 1
Foundation type 2
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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 Sheldons
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.
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.
What is the fractional unit of 3/4 ? of 2/3?
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With fractions, the concept of unit is expanded
one more time
Quotation 1 Principle Only like numbers can be
added an subtracted.
Computing 3/4 2/3, Why do we need to turn the
fractions into fractions with common denominator?
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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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Organizing the topics (the tightest chain and
breakups)
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How number sense can be developed through well
arranged exercises
45 18
42 - 18
Within 100 (with regrouping)
38 25
85 - 16
72 - 3
27 4
64 - 22
52 12
63 20
85 - 20
Within 100 (without regrouping)
63 3
66 - 3
40 5
90 - 5
30 20
50 - 30
17 - 11
11 6
15 2
17 - 15
6 9
15 - 9
Within 20 (across 10)
8 4
12 - 4
6 6
12 - 6
7 3
10 - 3
With 10
2 6
7 - 5
Within 10
3 2
4 - 1
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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)
(4)
(2)
(3)
If the above metaphor makes sense, who will take
the responsibility to make the change?
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Thank you !