Using Visualization to Extend Students’ Number Sense and Problem Solving Skills in Grades 4-6 Mathematics (Part 1) LouAnn Lovin, Ph.D. Mathematics Education James Madison University

Number Sense What is number sense? Turn to a neighbor and share your thoughts. Lovin NESA Spring

Number Sense “…good intuition about numbers and their relationships.” It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms” (Howden, 1989). “Two hallmarks of number sense are flexible strategy use and the ability to look at a computation problem and play with the numbers to solve with an efficient strategy” (Cameron, Hersch, Fosnot, 2004, p. 5). Flexibility in thinking about numbers and their relationships. Lovin NESA Spring Developing number sense through problem solving and visualization.

A picture is worth a thousand words…. Lovin NESA Spring

Do you see what I see? Cat or mouse? A face or an Eskimo? An old man’s face or two lovers kissing? Not everyone sees what you may see. Lovin NESA Spring

What do you see? Everyone does not necessarily hear/see/interpret experiences the way you do. e6frer4f Lovin NESA Spring

Manipulatives…Hands-On… Concrete…Visual Lovin NESA Spring

T: Is four-eighths greater than or less than four- fourths? J: (thinking to himself) Now that’s a silly question. Four-eighths has to be more because eight is more than four. (He looks at the student, L, next to him who has drawn the following picture.) Yup. That’s what I was thinking. Ball, D. L. (1992). Magical hopes: Manipulatives and the reform of mathematics education (Adobe PDF). American Educator, 16(2), 14-18, Magical hopes: Manipulatives and the reform of mathematics education (Adobe PDF). Lovin NESA Spring

But because he knows he was supposed to show his answer in terms of fraction bars, J lines up two fraction bars and is surprised by the result: Ball, D. L. (1992). Magical hopes: Manipulatives and the reform of mathematics education (Adobe PDF). American Educator, 16(2), 14-18, Magical hopes: Manipulatives and the reform of mathematics education (Adobe PDF). J: (He wonders) Four fourths is more? T: Four fourths means the whole thing is shaded in. J: (Thinks) This is what I have in front of me. But it doesn’t quite make sense, because the pieces of one bar are much bigger than the pieces of the other one. So, what’s wrong with L’s drawing? Lovin NESA Spring

T: Which is more – three thirds or five fifths? J: (Moves two fraction bars in front of him and sees that both have all the pieces shaded.) J: (Thinks) Five fifths is more, though, because there are more pieces. Ball, D. L. (1992). Magical hopes: Manipulatives and the reform of mathematics education (Adobe PDF). American Educator, 16(2), 14-18, Magical hopes: Manipulatives and the reform of mathematics education (Adobe PDF). This student is struggling to figure out what he should pay attention to about the fraction models: is it the number of pieces that are shaded? The size of the pieces that are shaded? How much of the bar is shaded? The length of the bar itself? He’s not “seeing” what the teacher wants him to “see.” Lovin NESA Spring

Base Ten Pieces and Number Adult’s perspective: 31 Lovin NESA Spring

What quantity does this “show”? Is it 4? Could it be 2/3? (set model for fractions) ? Lovin NESA Spring

Manipulatives are Thinker Toys, Communicators Hands-on AND minds-on The math is not “in” the manipulative. The math is constructed in the learner’s head and imposed on the manipulative/model. What do you see? What do your students see?. Lovin NESA Spring

The Doubting Teacher Do they “see” what I “see”? How do I know? Lovin NESA Spring

Visualization strategies to make significant ideas explicit Color Coding Visual Cuing Highlighting (talking about, pointing out) significant ideas in students’ work = ? Lovin NESA Spring Perimeter Area All Over ⅓

Teaching Number Sense through Problem Solving and Visualization Contextual (Word) Problems Emphasis on modeling the quantities and their relationships (quantitative analysis) Helps students to get past the words by visualizing and illustra ting word problems with simple diagrams. Emphasizes that mathematics can make sense Develops students’ reasoning and understanding Great formative assessment tool 16 Lovin NESA Spring 2012 What are the purposes of word problems? Why do we have students work on word problems? and Visualization

A Student’s Guide to Problem Solving Rule 1If at all possible, avoid reading the problem. Reading the problem only consumes time and causes confusion. Rule 2Extract the numbers from the problem in the order they appear. Watch for numbers written as words. Rule 3If there are three or more numbers, add them. Rule 4If there are only 2 numbers about the same size, subtract them. Rule 5If there are only two numbers and one is much smaller than the other, divide them if it comes out even -- otherwise multiply. Rule 6If the problem seems to require a formula, choose one with enough letters to use all the numbers. Rule 7If rules 1-6 don't work, make one last desperate attempt. Take the numbers and perform about two pages of random operations. Circle several answers just in case one happens to be right. You might get some partial credit for trying hard. 17 Lovin NESA Spring 2012

Randomly combining numbers without trying to make sense of the problem. Solving Word Problems: A Common “Approach” for Learners 18 Lovin NESA Spring 2012

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This strategy is useful as a rough guide but limited because key words don't help students understa nd the problem situation (i.e. what is happening in the probl em). Key words can also be misleading because the same word may mean different things in different situations. Wendy has 3 cards. Her friend gives her 8 more cards. How many cards does Wendy have now? There are 7 boys and 21 girls in a class. How many more girls than boys are there? Key Words 21 Lovin NESA Spring 2012

Real problems do not have key words! 22 Lovin NESA Spring 2012

Teaching Number Sense through Problem Solving and Visualization Contextual (Word) Problems and Visualization Emphasis on modeling the quantities and their relationships (quantitative analysis) Helps students to get past the words by visualizing and illustra ting word problems with simple diagrams. Emphasizes that mathematics can make sense Develops students’ reasoning and understanding Great formative assessment tool AVOIDs the sole reliance on key words. 23 Lovin NESA Spring 2012

The Dog Problem A big dog weighs five times as much as a little dog. The little dog weighs 2/3 as much as a medium-sized dog. The medium-sized dog weighs 9 pounds more than the little dog. How much does the big dog weigh?

Let x = weight of medium dog. Then weight of little dog = 2/3 x And weight of big dog = 5(2/3 x) x = 9 + 2/3 x (med = 9 + little) 1/3 x = 9 x = 27 pounds 2/3 x = 18 pounds (little dog) 5(2/3 x) = 5(18) = 90 pounds (big dog)

A big dog weighs five times as much as a little dog. The little dog weighs 2/3 as much as a medium-sized dog. The medium-sized dog weighs 9 pounds more than the little dog. How much does the big dog weigh? weight of medium dog weight of little dog weight of big dog x 18 = 90 pounds

A big dog weighs five times as much as a little dog. The little dog weighs 2/3 as much as a medium-sized dog. The medium-sized dog weighs 9 pounds more than the little dog. How much does the big dog weigh? x = weight of medium dog 2/3 x = weight of little dog 5(2/3 x) = weight of big dog x 2/3 x 5 (2/3 x) So….how do you solve this problem from here?

The Cookie Problem Kevin ate half a bunch of cookies. Sara ate one-third of what was left. Then Natalie ate one-fourth of what was left. Then Katie ate one cookie. Two cookies were left. How many cookies were there to begin with? 28 Lovin NESA Spring 2012

Different visual depictions of problem solutions for the Cookie Problem : Kevin Sara NatalieKatie KevinSara Natalie Katie 2 Sol 1 Sol 2 Sol 3 29 Lovin NESA Spring 2012

Mapping one visual depiction of solution for the Cookie Problem to algebraic solution : Kevin Sara NatalieKatie Sol 1 30 Lovin NESA Spring 2012 Sol 4 1 ⅓(½x) ¼(⅔(½x)) 2 ½x x + ⅓(½x) + ¼(⅔(½x)) = x ½x

Visual and Graphic Depictions of Problems Research suggests….. It is not whether teachers use visual/graphic depictions, it is how they are using them that makes a difference in students’ understanding. Students using their own graphic depictions and receiving feedback/guidance from the teacher (during class and on mathematical write ups) Graphic depictions of multiple problems and multiple solutions. Discussions about why particular representations might be more beneficial to help think through a given problem or communicate ideas. (Gersten & Clarke, NCTM Research Brief) 31 Lovin NESA Spring 2012

Supporting Students Discuss the differences between pictures and diagrams. Ask students to Explain how the diagram represents various components of the problem. Emphasize the the importance of precision in the diagram (labeling, proportionality) Discuss their diagrams with one another to highlight the similarities and differences in various diagrams that may represent the same problem. Discuss which diagrams are most appropriate for particular kinds of problems. 32 Lovin NESA Spring 2012 little medium big

Visual and Graphic Depictions of Problems Meilin saved $184. She saved $63 more than Betty. How much did Betty save? Singapore Math, Primary Mathematics 5A $184 Meilin $63 ? Betty 33 Lovin NESA Spring 2012 Singapore Math 184 – 63 = ?

Visual and Graphic Depictions of Problems There are 3 times as many boys as girls on the bus. If there are 24 more boys than girls, how many children are there altogether? Singapore Math, Primary Mathematics 5A Lovin NESA Spring girls boys x 12 = 48 children x = # of girls 3x = x x = 24 x = 12

Contextual (Word) Problems Use to introduce procedures and concepts (e.g., multiplication, division). Makes learning more concrete by presenting abstract ideas in a familiar context. Emphasizes that mathematics can make sense. Great formative assessment tool. 35 Lovin NESA Spring 2012

Multiplication A typical approach is to use arrays or the area model to represent multiplication. Why? 36 Lovin NESA Spring ×4=12

Use Real Contexts – Grocery Store (Multiplication) 37 Lovin NESA Spring 2012

Multiplication Context – Grocery Store How many plums does the grocer have on display? plums 38 Lovin NESA Spring 2012

Multiplication - Context – Grocery Store apples lemons tomatoes Groups of 5 or less subtly suggest skip counting (subitizing). 39 Lovin NESA Spring 2012

How many muffins does the baker have? 40 Lovin NESA Spring 2012

Other questions How many muffins did the baker have when all the trays were filled? How many muffins has the baker sold? What relationships can you see between the different trays? 41 Lovin NESA Spring 2012

Video: Students Using Baker’s Tray (4:30) What are the strategies and big ideas they are using and/or developing How does the context and visual support the students’ mathematical work? How does the teacher highlight students’ significant ideas? Video from Landscape of Learning Multiplication mini-lessons (grades 3-5) 42 Lovin NESA Spring 2012

Students’ Work 43 Lovin NESA Spring 2012 Jackie Edward Counted by onesSkip counted by twos

Wendy Students’ Work 44 Lovin NESA Spring 2012 Sam Amanda Decomposed larger amounts and doubled: = 16; = 36 Used relationships between the trays. Saw the right hand tray has 20, so the middle tray has 4 less or 16. Skip counted by 4. Used relationships between the trays. Saw the middle and last tray were the same as the first.

Area/Array Model Progression 45 Context (muffin tray, sheet of stamps, fruit tray) Area model using grid paper Open array Lovin NESA Spring 2012

4 x 39 How could you solve this? (Can you find a couple of ways?) Video (5:02) (1.1.2) Multiplication mini-lessons 46 Lovin NESA Spring 2012

Number Sense “…good intuition about numbers and their relationships.” It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms” (Howden, 1989). “Two hallmarks of number sense are flexible strategy use and the ability to look at a computation problem and play with the numbers to solve with an efficient strategy” (Cameron, Hersch, Fosnot, 2004, p. 5). Flexibility in thinking about numbers and their relationships. Lovin NESA Spring

Lovin NESA Spring Take a minute and write down two things you are thinking about from this morning’s session. Share with a neighbor.