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Using Visualization to Extend Students ’ Number Sense and Problem Solving Skills in Grades 4-6 Mathematics (Part 1)

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### Using Visualization to ExtendStudents’Number Sense andProblem Solving Skillsin Grades 4-6 Mathematics (Part 1)

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?

LouAnn Lovin, Ph.D.

Mathematics Education

James Madison University

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.

Developing number sense

through

problem solving and visualization.

Lovin NESA Spring 2012

A picture is worth a thousand words….

Lovin NESA Spring 2012

Do you see what I see?

An old man’s face or two lovers kissing?

Not everyone sees what you may see.

Cat or mouse?

A face or an Eskimo?

Lovin NESA Spring 2012

What do you see?

Everyone does not necessarily hear/see/interpret experiences the way you do.

www.couriermail.com.au/lifestyle/left-brain-v-right-brain-test/story-e6frer4f-1111114604318

Lovin NESA Spring 2012

Manipulatives…Hands-On… Concrete…Visual

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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, 46-47.

Lovin NESA Spring 2012

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:

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?

Ball, D. L. (1992). Magical hopes: Manipulatives and the reform of mathematics education (Adobe PDF).American Educator, 16(2), 14-18, 46-47.

Lovin NESA Spring 2012

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.

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.”

Ball, D. L. (1992). Magical hopes: Manipulatives and the reform of mathematics education (Adobe PDF). American Educator, 16(2), 14-18, 46-47.

Lovin NESA Spring 2012

What quantity does this “show”?

- Is it 4?
- Could it be 2/3? (set model for fractions)

?

Lovin NESA Spring 2012

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 2012

Visualization strategies to make significant ideas explicit

- ColorCoding
- Visual Cuing
- Highlighting (talking about, pointing out) significant ideas in students’ work.

48

+ 36

70

+14

84

Area All Over

Perimeter

⅓

48 + 36 = ?

Lovin NESA Spring 2012

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 illustrating word problems with simple diagrams.
- Emphasizes that mathematics can make sense
- Develops students’ reasoning and understanding
- Great formative assessment tool

and Visualization

What are the purposes of word problems?

Why do we have students work on word problems?

Lovin NESA Spring 2012

A Student’s Guide to Problem Solving

Lovin NESA Spring 2012

Solving Word Problems:A Common “Approach” for Learners

Randomly combining numbers without

trying to make sense of the problem.

Lovin NESA Spring 2012

Key Words

- This strategy is useful as a rough guide but limited because key words don't help students understand the problem situation (i.e. what is happening in the problem).
- 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?

Lovin NESA Spring 2012

Real problems do not have key words!

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 illustrating 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.

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?

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

9

9

9

9

9

weight of little dog

18

18

18

18

18

weight of big dog

5 x 18 = 90 pounds

x = weight of medium dog

9

9

9

x

9

9

2/3 x = weight of little dog

So….how do you solve this problem from here?

2/3 x

18

18

18

18

18

5 (2/3 x)

5(2/3 x) = weight of big dog

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?

Lovin NESA Spring 2012

Different visual depictions of problem solutions for the Cookie Problem:

Sara

Sol 1

Kevin

Natalie

Katie

Sol 2

Sol 3

2

Katie

Natalie

Sara

Kevin

Lovin NESA Spring 2012

Mapping one visual depiction of solution for the Cookie Problem to algebraic solution:

Sara

⅓(½x)

Sol 1

¼(⅔(½x))

Kevin

Natalie

Katie

1

½x

2

x

+ ¼(⅔(½x))

= x

½x

Sol 4

+ ⅓(½x)

+ 2

+ 1

Lovin NESA Spring 2012

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)

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.

little

medium

big

Lovin NESA Spring 2012

Visual and Graphic Depictions of Problems

Singapore Math

Meilinsaved $184. She saved $63 more than Betty. How much did Betty save?

Singapore Math, Primary Mathematics 5A

Betty

?

$63

$184

Meilin

184 – 63 = ?

Lovin NESA Spring 2012

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

12

girls

24

x = # of girls

3x = x + 24

2x = 24

x = 12

12

12

12

boys

4 x 12 = 48 children

Lovin NESA Spring 2012

- 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.

Lovin NESA Spring 2012

Multiplication

A typical approach is to use arrays or the area model to represent multiplication.

Why?

4

3×4=12

3

Lovin NESA Spring 2012

Use Real Contexts – Grocery Store (Multiplication)

Lovin NESA Spring 2012

MultiplicationContext – Grocery Store

How many plums does the grocer have on display?

plums

Lovin NESA Spring 2012

Multiplication - Context – Grocery Store

apples

lemons

Groups of 5 or less subtly suggest skip counting (subitizing).

tomatoes

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How many muffins does the baker have?

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?

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 1.1.3 from Landscape of Learning Multiplication mini-lessons (grades 3-5)

Lovin NESA Spring 2012

Students’ Work

Wendy

Sam

Skip counted by 4. Used relationships between the trays. Saw the middle and last tray were the same as the first.

Used relationships between the trays.

Saw the right hand tray has 20, so the middle tray has 4 less or 16.

Amanda

Decomposed larger amounts and doubled: 8 + 8 = 16;

16 + 16 + 4 = 36

Lovin NESA Spring 2012

Area/Array ModelProgression

Area model using grid paper

Open array

Context (muffin tray, sheet of stamps, fruit tray)

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

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 2012

Take a minute and write down two things you are thinking about from this morning’s session.

- Share with a neighbor.

Lovin NESA Spring 2012

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