The Role of Visual Representations in Learning

Mathematics

John Woodward

Dean, School of EducationUniversity of Puget Sound

Summer Assessment InstituteAugust 3, 2012

Information Processing Psychology

• How Do We Store Information?

• How Do We Manipulate It?

• What Mechanisms Enhance Thinking/ Problem Solving?

Information Processing Psychology

t e x ti m a g e s

Monitoring or Metacognition

The Traditional Multiplication Hierarchy

357x 43

357x 3

35x 3

5x 3

It Looks Like Multiplication

How many steps?

2 2 1 2

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

Old Theories of Learning

Show the concept or procedure

Practice

Practice

Practice

Practice

Practice

Practice

Better Theories of Learning

Conceptual DemonstrationsVisual Representations

Discussions

Controlled and Distributed

Practice

Return to Periodic Conceptual

Demonstrations

The Common Core Calls for Understanding as Well as Procedures

Tools

Manipulatives

Place Value or Number Coins 100 10 1

Number Lines

Tools

Fraction Bars

Integer Cards

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 =

Long Division

3 10 2

How would you explain the problem conceptually to students?

1 0 2 Hundreds Tens Ones

100 + 0 + 2

1001

1

Hundreds Tens Ones

3 102

1001

1

Hundreds Tens Ones

3 102 Hundreds Tens Ones

100

1

1

3 102 Hundreds Tens Ones

100

1

1

3 102

1

1

1010

10

1010

1010

10

1010

Hundreds Tens Ones

100

3 102

1

1

1010

10

1010

1010

10

1010

Hundreds Tens Ones

3 1023

1

110

10

10

10

10

10

10

10

10

10

Hundreds Tens Ones

3 102 3

-9 1

11

11

11

11

11

11

10

10

10

10

10

10

10

10

10

Hundreds Tens Ones

10

3 1023

-9 1 2

11

11

11

11

11

11

10

10

10

10

10

10

10

10

10

Hundreds Tens Ones

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

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

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

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

The Case of Fractions25

35

+ 25

37

+

25

37

-

25

37

÷

25

37

x

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 =

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

Counting with Whole Numbers

Counting with Whole Numbers is Familiar and Predictable

0 1 2 3 98 99 100 ... ...

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 ... ...

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.

Counting with Fractions

Counting with Fractional Numbers is not Necessarily Familiar or Predictable

0 1/3 1

?

The Logic of Adding and Subtracting Fractions

1 3

14

13

1+ 4

?

We can combine the quantities, but what do we get?

Students Need to Think about the Part/Wholes

1 3

The parts don’t line up

0 1

1 40 1

0 1

Common Fair Share Parts Solves the Problem

1 3

412

312

14

Work around Common Units Solves the Problem

Now we can see how common units are combined

The Same Issue Applies to Subtraction

1 3

14

What do we call what is left when we find the difference?

-

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

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

Let’s Think about Whole Number Multiplication

3 groups of 4 cubes = 12 cubes

=

3 x 4 = 12

An Area Model of Multiplication

3 x 4

4 units

Begin with an area representation

An Area Model of Multiplication

3 units

3 x 4

4 units

Begin with an area representation

An Area Model of Multiplication

3 x 4 = 12

3 units

4 units

An Area Model of Multiplication

½ x 4

4 units Begin with an area representation

An Area Model of Multiplication

½ x 4

4 units

Begin with an area representation

1/2 units

An Area Model of Multiplication

½ x 4

4 Begin with an area representation

1/2

An Area Model of Multiplication

½ x 4

4

1/2

4 red units

An Area Model of Multiplication

½ x 4

4

1/2

½ of the 4 red shown in stripes

An Area Model of Multiplication

½ x 4 = 4/2 or 2 units

4

1/2

2 units=

Multiplication of Proper Fractions

12

13 x =

Multiplication of Fractions

Begin with an area representation

12

13 x

1

1

Multiplication of Fractions

12

13 x =

halves

1

Multiplication of Fractions

Show 1/2

12

13 x =

halves

1

Multiplication of Fractions

Break into 1/3s

12

13 x =

halves

1

Multiplication of Fractions

Show 1/3 of 1/2

12

13 x =

halves

thirds

Multiplication of Fractions

12

13 x =

halves

The product is where the areas of 1/3 and 1/2

intersect

thirds

16

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

8

0 2 4 6 8 10

The divisor (or unit) of 2 partitions 8 four times.

2

Dividing Whole Numbers

4

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

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

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

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

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

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

Division of Decimals

0.6 0.2 =

or

.2 0.6

0.2 0.6

Division of Decimals

0.6 0.2 =

or

.2 0.6

0.2 0.6

1 time

Division of Decimals

0.6 0.2 =

or

.2 0.6

0.2 0.6

2 times

Division of Decimals

0.6 0.2 =

or

2 6.0

0.2 0.6

3 times

3.0

3.0

Chronic Errors: Operations on Integers

• -1 + -3 = -4

• -1 - -3 = -4

• -1 1 = -1

• -1 -1 = -1

• -1 -1 -1 = 1

Algebra Tiles

Positive integer

Negative integer

Addition: Beginning with A Fundamental Concept

3 + 2 = 5

“Adding Quantities to a Set”

Addition and Subtraction of Integers

3 + 2 = 5 1 1

1

1

1

Addition and Subtraction of Integers

-3 + -2 = -5 1 1

1

1

1

3 – 2 = 1

The fundamental concept of “removal from a set”

Subtraction of Integers: Where the Challenge Begins

3 – 2 = 1 1 1

1

Subtraction of Integers

3 + -2 =

A new dimension of subtraction. Algebraic thinking where a – b = a + -b.

Subtraction of Integers: Where the Challenge Begins

1

3 + -2 = 1 1

1

1

1

Subtraction of Integers

3 – (-2) = 5

This is where understanding breaks down

Subtraction of Integers

3 – (-2) = 5

1 1

1

We add 2 + -2 or a “zero pair”

1

1

1

1

Subtraction of Integers

Better Theories of LearningConceptual Demonstrations

Visual RepresentationsDiscussions

Controlled and Distributed

Practice

Return to Periodic Conceptual

Demonstrations