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

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The Traditional Multiplication Hierarchy 357 x 43 357 x 3 35 x 3 5 x 3

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It Looks Like Multiplication 357 x 43 1071 1071 + 1428.. 15351 15351 How many steps? 2 2 1 2

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The Symbols Scale Tips Heavily Toward Procedures 2201 -345 589 x 73 43 589 5789 + 3577 3 2727 + 7 9 10 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

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Better Theories of Learning Conceptual Demonstrations Visual 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 100101 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 + 2 100 1 1 Hundreds Tens Ones

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3 102 100 1 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 10 Hundreds Tens Ones 100

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3 102 1 1 10 Hundreds Tens Ones

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3 102 1 1 10 Hundreds Tens Ones 10

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3 102 3 1 1 10 Hundreds Tens Ones

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3 102 3 9 1 1 10 Hundreds Tens Ones

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3 102 1 1 10 3 -9 1 10 Hundreds Tens Ones

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3 102 1 1 10 3 -9 1 Hundreds Tens Ones

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3 102 3 -9 1 1 1 1 1 1 1 1 1 1 1 11 10 Hundreds Tens Ones 10

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2 3 102 3 -9 1 2 1 1 1 1 1 1 1 1 1 1 11 10 Hundreds Tens Ones

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2 3 1 0 2 4 3 4 -9 1 2 10 1 1 1 1 1 1 1 1 1 1 1 1 Hundreds Tens Ones

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2 3 1 0 2 4 3 4 -9 1 2 -1 2 -1 2 Hundreds Tens Ones 10 1 1 1 1 1 1 1 1 1 1 1 1

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2 3 1 0 2 4 3 4 -9 1 2 -1 2 -1 2 0 Hundreds Tens Ones 10 1 1 1 1 1 1 1 1 1 1 1 1

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3 1 0 2 3 4 -9 1 2 -1 2 0 10 1 1 1 1 1 1 1 1 1 1 1 1 Hundreds Tens Ones

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The Case of Fractions 2525 3535 + 2525 3737 + 2525 3737 - 2525 3737 2525 3737 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 1 4 1 3 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 dont line up 0 1 1 4 0 1

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Common Fair Share Parts Solves the Problem 1 3 4 12 3 12 1 4

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Work around Common Units Solves the Problem 4 12 7 12 3 12 Now we can see how common units are combined

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The Same Issue Applies to Subtraction 1 3 1 4 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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Lets 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 x 4 4 units Begin with an area representation

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An Area Model of Multiplication x 4 4 units Begin with an area representation 1/2 units

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An Area Model of Multiplication x 4 4 Begin with an area representation 1/2

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An Area Model of Multiplication x 4 4 1/2 4 red units

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An Area Model of Multiplication x 4 4 1/2 of the 4 red shown in stripes

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An Area Model of Multiplication x 4 = 4/2 or 2 units 4 1/2 2 units =

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Multiplication of Proper Fractions 1212 1313 x=

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Multiplication of Fractions Begin with an area representation 1212 1313 x 1 1

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Multiplication of Fractions 1212 1313 x= halves 1

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Multiplication of Fractions Show 1/2 1212 1313 x= halves 1

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Multiplication of Fractions Break into 1/3s 1212 1313 x= halves 1

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Multiplication of Fractions Show 1/3 of 1/2 1212 1313 x= halves thirds

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Multiplication of Fractions 1212 1313 x= halves The product is where the areas of 1/3 and 1/2 intersect thirds 1616

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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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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

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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

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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. 3/4 1/2 1

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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

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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

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Division of Decimals 0.60.2 = or.20.6 0.2 0.6

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Division of Decimals 0.60.2 = or.20.6 0.2 0.6 1 time

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Division of Decimals 0.60.2 = or.20.6 0.2 0.6 2 times

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Division of Decimals 0.60.2 = or 2 6.0 0.2 0.6 3 times 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 Learning Conceptual Demonstrations Visual Representations Discussions Controlled and Distributed Practice Return to Periodic Conceptual Demonstrations