The Role of Visual Representations in Learning Mathematics
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The Role of Visual Representations in Learning
Mathematics
John Woodward
Dean, School of EducationUniversity of Puget Sound
Summer Assessment InstituteAugust 3, 2012
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Information Processing Psychology
• How Do We Store Information?
• How Do We Manipulate It?
• What Mechanisms Enhance Thinking/ Problem Solving?
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Information Processing Psychology
t e x ti m a g e s
Monitoring or Metacognition
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The Traditional Multiplication Hierarchy
357x 43
357x 3
35x 3
5x 3
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It Looks Like Multiplication
How many steps?
2 2 1 2
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The Symbols Scale Tips Heavily Toward Procedures
2201-345589
x 73
43 589
5789+ 3577
3 27 +
7 910
What does all of this
mean? 4x + 35 = 72 + x
y = 3x + 1
.0009823
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Old Theories of Learning
Show the concept or procedure
Practice
Practice
Practice
Practice
Practice
Practice
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Better Theories of Learning
Conceptual DemonstrationsVisual Representations
Discussions
Controlled and Distributed
Practice
Return to Periodic Conceptual
Demonstrations
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The Common Core Calls for Understanding as Well as Procedures
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Tools
Manipulatives
Place Value or Number Coins 100 10 1
Number Lines
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Tools
Fraction Bars
Integer Cards
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The Tasks
3 ) 102
1/3 + 1/4
1/3 - 1/4
1/3 x 1/2
2/3 ÷ 1/2 3/4 = 9/12 as equivalent fractions
.60 ÷ .20
4 + -3 =
4 - -3 =
4 - 3 =
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Long Division
3 10 2
How would you explain the problem conceptually to students?
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1 0 2 Hundreds Tens Ones
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100 + 0 + 2
1001
1
Hundreds Tens Ones
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3 102
1001
1
Hundreds Tens Ones
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3 102 Hundreds Tens Ones
100
1
1
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3 102 Hundreds Tens Ones
100
1
1
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3 102
1
1
1010
10
1010
1010
10
1010
Hundreds Tens Ones
100
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3 102
1
1
1010
10
1010
1010
10
1010
Hundreds Tens Ones
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3 1023
1
110
10
10
10
10
10
10
10
10
10
Hundreds Tens Ones
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3 102 3
-9 1
11
11
11
11
11
11
10
10
10
10
10
10
10
10
10
Hundreds Tens Ones
10
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3 1023
-9 1 2
11
11
11
11
11
11
10
10
10
10
10
10
10
10
10
Hundreds Tens Ones
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3 1 0 23 4
-9 1 2
1010
10
10
1010
10
1010
1
1
1
1
1
1
1
1
1
11
1
Hundreds Tens Ones
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3 1 0 23 4
-9 1 2 -1 2
Hundreds Tens Ones
1010
10
10
1010
10
1010
1
1
1
1
1
1
1
1
1
11
1
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3 1 0 23 4
-9 1 2 -1 2 0
Hundreds Tens Ones
1010
10
10
1010
10
1010
1
1
1
1
1
1
1
1
1
11
1
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3 1 0 23 4
-9 1 2 -1 2 0
1010
10
10
1010
10
1010
1
1
1
1
1
1
1
1
1
11
1
Hundreds Tens Ones
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The Case of Fractions25
35
+ 25
37
+
25
37
-
25
37
÷
25
37
x
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Give Lots of Practice to Those who Struggle
2/3 + 3/5 =
3/4 - 1/2 =
3/5 x 1/6 =
4/9 ÷ 1/2 =
2/3 + 3/5 =
3/4 - 1/2 =
3/5 x 1/6 =
4/9 ÷ 1/2 =
3/5 x 1/6 =
4/9 ÷ 1/2 =
2/3 + 3/5 =
3/4 - 1/2 =
3/5 x 1/6 =
4/9 ÷ 1/2 =
2/3 + 3/5 =
3/4 - 1/2 =
3/5 x 1/6 =
4/9 ÷ 1/2 =
3/5 x 1/6 =
4/9 ÷ 1/2 =
2/3 + 3/5 =
3/4 - 1/2 =
3/5 x 1/6 =
4/9 ÷ 1/2 =
2/3 + 3/5 =
3/4 - 1/2 =
3/5 x 1/6 =
4/9 ÷ 1/2 =
3/5 x 1/6 =
4/9 ÷ 1/2 =
2/3 + 3/5 =
3/4 - 1/2 =
3/5 x 1/6 =
4/9 ÷ 1/2 =
2/3 + 3/5 =
3/4 - 1/2 =
3/5 x 1/6 =
4/9 ÷ 1/2 =
3/5 x 1/6 =
4/9 ÷ 1/2 =
2/3 + 3/5 =
3/4 - 1/2 =
3/5 x 1/6 =
4/9 ÷ 1/2 =
2/3 + 3/5 =
3/4 - 1/2 =
3/5 x 1/6 =
4/9 ÷ 1/2 =
3/5 x 1/6 =
4/9 ÷ 1/2 =
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Why Operations on Fractions Are So Difficult
• Students are used to the logic of whole number counting– Fractional numbers are a big change
• Operations on fractions require students to think differently– Addition and subtraction of fractions require one kind of
thinking– Multiplication and division require another kind of thinking– Contrasting operations on whole numbers with operations on
fractions can help students see the difference
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Counting with Whole Numbers
Counting with Whole Numbers is Familiar and Predictable
0 1 2 3 98 99 100 ... ...
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Counting with Whole Numbers
Even When We Skip Count, the Structure is Predictable and Familiar
0 1 2 3 4 5 6 98 99 100 101 102 ... ...
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The “Logic” Whole Number Addition
Whole Numbers as a Point of Contrast
3 + 4 = 7
0 1 2 3 4 5 6 7 8 9
Students just assume the unit of 1 when they think addition.
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Counting with Fractions
Counting with Fractional Numbers is not Necessarily Familiar or Predictable
0 1/3 1
?
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The Logic of Adding and Subtracting Fractions
1 3
14
13
1+ 4
?
We can combine the quantities, but what do we get?
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Students Need to Think about the Part/Wholes
1 3
The parts don’t line up
0 1
1 40 1
0 1
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Common Fair Share Parts Solves the Problem
1 3
412
312
14
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Work around Common Units Solves the Problem
Now we can see how common units are combined
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The Same Issue Applies to Subtraction
1 3
14
What do we call what is left when we find the difference?
-
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Start with Subtraction of Fractions
We Need Those Fair Shares in Order to be Exact
4 12
3 - 12
1 12
=
Now it is easier to see that we are removing 3/12s
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Multiplication of Fractions
Multiplication of Fractions: A Guiding Question
When you multiply two numbers, the product is usually larger than either of the two factors.
When you multiply two proper fractions, the product is usually smaller. Why?
3 x 4 = 12
1/3 x 1/2 = 1/6
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Let’s Think about Whole Number Multiplication
3 groups of 4 cubes = 12 cubes
=
3 x 4 = 12
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An Area Model of Multiplication
3 x 4
4 units
Begin with an area representation
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An Area Model of Multiplication
3 units
3 x 4
4 units
Begin with an area representation
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An Area Model of Multiplication
3 x 4 = 12
3 units
4 units
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An Area Model of Multiplication
½ x 4
4 units Begin with an area representation
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An Area Model of Multiplication
½ x 4
4 units
Begin with an area representation
1/2 units
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An Area Model of Multiplication
½ x 4
4 Begin with an area representation
1/2
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An Area Model of Multiplication
½ x 4
4
1/2
4 red units
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An Area Model of Multiplication
½ x 4
4
1/2
½ of the 4 red shown in stripes
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An Area Model of Multiplication
½ x 4 = 4/2 or 2 units
4
1/2
2 units=
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Multiplication of Proper Fractions
12
13 x =
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Multiplication of Fractions
Begin with an area representation
12
13 x
1
1
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Multiplication of Fractions
12
13 x =
halves
1
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Multiplication of Fractions
Show 1/2
12
13 x =
halves
1
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Multiplication of Fractions
Break into 1/3s
12
13 x =
halves
1
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Multiplication of Fractions
Show 1/3 of 1/2
12
13 x =
halves
thirds
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Multiplication of Fractions
12
13 x =
halves
The product is where the areas of 1/3 and 1/2
intersect
thirds
16
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Division of Fractions
When you divide two whole numbers, the quotient is usually smaller than the dividend.
When you divide two proper fractions, the quotient is usually larger than the dividend. Why?
12 ÷ 4 = 3
2/3 ÷ 1/2 = 4/3
A Guiding Observation
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8
0 2 4 6 8 10
The divisor (or unit) of 2 partitions 8 four times.
2
Dividing Whole Numbers
4
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8
0 1 2 3 4 5 6 7 8
1/2
Dividing a Whole Number by a Fraction
The divisor (or unit) of 1/2 partitions 8 sixteen times.
16
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13÷2
3
Division of Proper Fractions
The divisor (or unit) of 1/3 partitions 2/3 two times.
0 1/3 2/3 1
2/31/3or2
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12÷3
4
Another Example: Division of Proper Fractions
Begin with the dividend 3/4 and the divisor 1/2
0 1/4 1/2 3/4 1
3/41/2or
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Division of Proper Fractions
0 1/4 1/2 3/4 1
The divisor (or unit) of 1/2 partitions 3/4 one and one half times.
3/41/2
11/2
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Division of Proper Fractions
The divisor (or unit) of 1/2 partitions 3/4 one and one half times.
3/41/2
11/2
1 time
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Division of Proper Fractions
The divisor (or unit) of 1/2 partitions 3/4 one and one half times.
3/41/2
11/2
1 time1/2 time
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Division of Decimals
0.6 0.2 =
or
.2 0.6
0.2 0.6
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Division of Decimals
0.6 0.2 =
or
.2 0.6
0.2 0.6
1 time
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Division of Decimals
0.6 0.2 =
or
.2 0.6
0.2 0.6
2 times
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Division of Decimals
0.6 0.2 =
or
2 6.0
0.2 0.6
3 times
3.0
3.0
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Chronic Errors: Operations on Integers
• -1 + -3 = -4
• -1 - -3 = -4
• -1 1 = -1
• -1 -1 = -1
• -1 -1 -1 = 1
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Algebra Tiles
Positive integer
Negative integer
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Addition: Beginning with A Fundamental Concept
3 + 2 = 5
“Adding Quantities to a Set”
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Addition and Subtraction of Integers
3 + 2 = 5 1 1
1
1
1
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Addition and Subtraction of Integers
-3 + -2 = -5 1 1
1
1
1
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3 – 2 = 1
The fundamental concept of “removal from a set”
Subtraction of Integers: Where the Challenge Begins
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3 – 2 = 1 1 1
1
Subtraction of Integers
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3 + -2 =
A new dimension of subtraction. Algebraic thinking where a – b = a + -b.
Subtraction of Integers: Where the Challenge Begins
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1
3 + -2 = 1 1
1
1
1
Subtraction of Integers
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3 – (-2) = 5
This is where understanding breaks down
Subtraction of Integers
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3 – (-2) = 5
1 1
1
We add 2 + -2 or a “zero pair”
1
1
1
1
Subtraction of Integers
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Better Theories of LearningConceptual Demonstrations
Visual RepresentationsDiscussions
Controlled and Distributed
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