The Role of Visual Representations in Learning Mathematics John Woodward Dean, School of Education University of Puget Sound Summer Assessment Institute August 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 t i m a g e s Monitoring or Metacognition

The Traditional Multiplication Hierarchy 357 x x 3 35 x 3 5 x 3

It Looks Like Multiplication 357 x How many steps?

The Symbols Scale Tips Heavily Toward Procedures x What does all of this mean? 4x + 35 = 72 + x y = 3x

Old Theories of Learning Show the concept or procedure Practice

Better Theories of Learning Conceptual Demonstrations Visual 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 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 ÷ = = =

Long Division How would you explain the problem conceptually to students?

1 0 2 Hundreds Tens Ones

Hundreds Tens Ones

Hundreds Tens Ones

3 102 Hundreds Tens Ones

3 102 Hundreds Tens Ones

Hundreds Tens Ones 100

Hundreds Tens Ones

Hundreds Tens Ones 10

Hundreds Tens Ones

Hundreds Tens Ones

Hundreds Tens Ones

Hundreds Tens Ones

Hundreds Tens Ones 10

Hundreds Tens Ones

Hundreds Tens Ones

Hundreds Tens Ones

Hundreds Tens Ones

Hundreds Tens Ones

The Case of Fractions ÷ 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

Counting with Whole Numbers Even When We Skip Count, the Structure is Predictable and Familiar

The “Logic” Whole Number Addition Whole Numbers as a Point of Contrast = 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 ? 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

Common Fair Share Parts Solves the Problem

Work around Common Units Solves the Problem Now we can see how common units are combined

The Same Issue Applies to Subtraction 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 = 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 x=

Multiplication of Fractions Begin with an area representation x 1 1

Multiplication of Fractions x= halves 1

Multiplication of Fractions Show 1/ x= halves 1

Multiplication of Fractions Break into 1/3s x= halves 1

Multiplication of Fractions Show 1/3 of 1/ x= halves thirds

Multiplication of Fractions x= halves The product is where the areas of 1/3 and 1/2 intersect thirds 1616

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

The divisor (or unit) of 2 partitions 8 four times. 2 Dividing Whole Numbers 4

/2 Dividing a Whole Number by a Fraction The divisor (or unit) of 1/2 partitions 8 sixteen times. 16

1313 ÷ 2323 Division of Proper Fractions The divisor (or unit) of 1/3 partitions 2/3 two times. 0 1/3 2/3 1 2/3 1/3 or 2

1212 ÷ 3434 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 3/4 1/2 or

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. 3/4 1/2 1

Division of Proper Fractions The divisor (or unit) of 1/2 partitions 3/4 one and one half times. 3/4 1/2 1 1 time

Division of Proper Fractions The divisor (or unit) of 1/2 partitions 3/4 one and one half times. 3/4 1/2 1 1 time 1/2 time

Division of Decimals  = or

Division of Decimals  = or time

Division of Decimals  = or times

Division of Decimals  = or times 3.0

Chronic Errors: Operations on Integers = =  1 =  -1 =  -1  -1 = 1

Algebra Tiles Positive integer Negative integer

Addition: Beginning with A Fundamental Concept = 5 “Adding Quantities to a Set”

Addition and Subtraction of Integers =

Addition and Subtraction of Integers =

3 – 2 = 1 The fundamental concept of “removal from a set” Subtraction of Integers: Where the Challenge Begins

3 – 2 = Subtraction of Integers

= A new dimension of subtraction. Algebraic thinking where a – b = a + -b. Subtraction of Integers: Where the Challenge Begins

= Subtraction of Integers

3 – (-2) = 5 This is where understanding breaks down Subtraction of Integers

3 – (-2) = We add or a “zero pair” Subtraction of Integers

Better Theories of Learning Conceptual Demonstrations Visual Representations Discussions Controlled and Distributed Practice Return to Periodic Conceptual Demonstrations