using visualization to extend students’ number sense and problem solving skills in grades 4-6...

Using Visualization to ExtendStudents’ 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 2012 2

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.

A picture is worth a thousand words….


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.


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 6

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

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

Manipulatives…Hands-On… Concrete…Visual


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 8

http://www-personal.umich.edu/~dball/articles/BallMagicalHopes.pdf


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, 46-47.

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?




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, 46-47.

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.”



Base Ten Pieces and Number

10 20 30 40

4 3 2 1

Adult’s perspective: 31


What quantity does this “show”?

Is it 4?

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

?


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?

.


The Doubting Teacher

Do they “see” what I “see”?How do I know?


Visualization strategies to make significant ideas explicit

Color Coding

Visual Cuing

Highlighting (talking about, pointing out) significant ideas in students’ work.

48 + 36 70 +14 84

48 + 36 = ?


Perimeter

Area All Over⅓

Teaching Number Sense through Problem Solving and

Visualization

Contextual (Word) ProblemsEmphasis 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 senseDevelops students’ reasoning and understandingGreat formative assessment tool

16Lovin 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 SolvingRule 1

If at all possible, avoid reading the problem. Reading the problem only consumes time and causes confusion.

Rule 2 Extract the numbers from the problem in the order they appear. Watch for numbers written as words.

Rule 3 If there are three or more numbers, add them.

Rule 4 If there are only 2 numbers about the same size, subtract them.

Rule 5 If there are only two numbers and one is much smaller than the other, divide them if it comes out even -- otherwise multiply.

Rule 6 If the problem seems to require a formula, choose one with enough letters to use all the numbers.

Rule 7 If 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.


Randomly combining numbers without

trying to make sense of the problem.

Solving Word Problems:A Common “Approach” for Learners


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?

Key Words


Real problems do not have key words!


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 senseDevelops students’ reasoning and understandingGreat formative assessment toolAVOIDs the sole reliance on key words.


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 = 9x = 27 pounds

2/3 x = 18 pounds (little dog)5(2/3 x) = 5(18) = 90 pounds (big dog)


weight of medium dog

weight of little dog

weight of big dog

9

99

18 18 18 18 18

99

5 x 18 = 90 pounds


x = weight of medium dog

2/3 x = weight of little dog

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

9

99

18 18 18 18 18

99

x

2/3 x

5 (2/3 x)

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

The Cookie ProblemKevin 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?


Different visual depictions of problem

solutions for the Cookie Problem:

Kevin

Sara

Natalie Katie

KevinSaraNatalieKatie2

Sol 1

Sol 2

Sol 3


Mapping one visual depiction of solution for

the Cookie Problem to algebraic solution:

Kevin

Sara

Natalie KatieSol 1


Sol 4

1

⅓(½x)

¼(⅔(½x))

2½x

x

+ ⅓(½x) + ¼(⅔(½x)) + 1 + 2 = 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)


Supporting StudentsDiscuss the differences between pictures and diagrams.

Ask students toExplain 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


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

Singapore Math, Primary Mathematics 5A

$184Meilin

$63?

Betty


Singapore Math

184 – 63 = ?


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


girls

boys

24

121212

12

4 x 12 = 48 children

x = # of girls3x = x + 242x = 24x = 12

Contextual (Word) ProblemsUse 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.


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

Why?


3

4 3×4=12

Use Real Contexts – Grocery Store (Multiplication)


MultiplicationContext – Grocery Store

How many plums does the grocer have on display?

plums38Lovin NESA Spring 2012

Multiplication - Context – Grocery Store

apples lemons

tomatoes

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


How many muffins does the baker have?


Other questionsHow 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?


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)


Students’ Work


JackieEdward

Counted by ones Skip counted by twos

Wendy Students’ Work


Sam

Amanda

Decomposed larger amounts and doubled: 8 + 8 = 16;16 + 16 + 4 = 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 ModelProgression

45

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

Area model using grid paperOpen arra

y

Lovin NESA Spring 2012

4 x 39How could you solve this? (Can you find a couple of ways?)

Video (5:02) (1.1.2) Multiplication mini-lessons


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.



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

Share with a neighbor.

using visualization to extend students’ number sense and problem solving skills in grades 4-6...

Documents

magical hopes

developing number sense

students number sense

hallmarks of number

louann lovin

number of pieces th

number sense good intuition

j lines