Transcript Elementary Mathematics in U.S.: How could "more" be "less"?

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 canMore
less be vs.
more:less
an example
3. The “tightest” chain
A loose vs. solid foundation
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F–
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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
Arithmetic as
a microcosm of
mathematics
US perspective:
Arithmetic as a collection of
algorithms
Whole
numbers
−
×
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+
Fractions
−
×
÷
Fractions
Whole
numbers
U. S.
÷
Countries
×
with high
math
+
performance
−
+
Concept of a Unit
A loose vs. solid foundation:
the consequence
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Foundation type 1
Foundation type 2
Building a Solid Foundation
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Foundation type 1
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Foundation type 2
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 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 ?
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”)?
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?
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?
How can more be less?
1. More vs. less
2. How can less be more: an example
3. The “tightest” chain
Organizing the topics (the tightest chain and breakups)
Circle (perimeter & area);
ss
Ratio and proportion
cylinder & cone (area and
ss
Percentages
volume)
Fractions – division
Fractions – multiplication
Area of triangles
Fractions – addition and subtraction
& trapezoids;
Fractions – meaning and features
Prism and cubic
Divisibility
(volume)
Decimals – multiplication and division
Decimals – addition and subtraction
Area of
Decimals – meaning and features
rectangles
Fractions – the basic concepts
Angles & lines
Division with divisor as a three-digit number
Perimeter of
Multiplication with multiplier as a three-digit number
rectangles
Division with divisor as a two-digit number
Length
Multiplication with multiplier as a two-digit number
Weight
Many-digit numbers, notation, addition and subtraction
Division with divisor as a one-digit number
Length
Multiplication with multiplier as a one-digit number
Time
Numbers up to 10,000 , notation, addition and subtraction
Weight
Multiplication and division with multiplication tables
Money
Numbers up to 100 , addition and subtraction (with concept of regrouping)
Numbers 11 to 20 , addition and subtraction (with concept of regrouping)
Numbers 0 to 10 , addition and subtraction
How number sense can
be developed through
well arranged exercises
45 + 18 = 42 − 18 =
38 + 25 = 85 − 16 =
27 + 4 = 72 − 3 =
Within 100
(with
regrouping)
52 + 12 = 64 − 22=
63 + 20 = 85 − 20 =
Within 100
(without
regrouping)
63 + 3 = 66 − 3 =
40 + 5 = 90 − 5 =
30 + 20 = 50 − 30 =
11 + 6 = 17 − 11 =
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=
3+2= 4−1=
Within 10
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)
(4)
(2)
(3)
If the above metaphor makes sense,
who will take the responsibility to make
the change?
Thank you !