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.
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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).
Developing number sense
 “Two hallmarks of number
sense are flexible strategy
through
use and the ability to look at a computation problem
problem solving and visualization.
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.
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A picture is worth a thousand
words….
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Do you see what I see?
An old man’s
face or two lovers
kissing?
Cat or mouse?
Not
everyone sees what you may see.
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A face or an Eskimo?
5
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/storyLovin NESA Spring 2012
e6frer4f-1111114604318
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Manipulatives…Hands-On…
Concrete…Visual
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T: Is four-eighths greater than or less than fourfourths?
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.
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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.
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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.
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Base Ten Pieces and Number
4
10
3
2
20
30
1
40
Adult’s perspective: 31
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What quantity does this
“show”?
?
 Is it 4?
 Could it be 2/3? (set model for fractions)
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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?

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The Doubting Teacher
Do they “see” what I “see”?
How do I know?
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Visualization strategies to make
significant ideas explicit
 Color Coding
⅓
Area
All Over
 Visual Cuing
Perimeter
 Highlighting (talking about, pointing out) significant
ideas in students’ work.
48 + 36 = ?
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48
+ 36
70
+14
84
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Teaching Number Sense
through Problem Solving and Visualization
Contextual (Word) Problems and Visualization
 Emphasis
quantities
and their relationships
What areon
themodeling
purposes the
of word
problems?




Why do we analysis)
have students work on word problems?
(quantitative
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
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A Student’s Guide to Problem Solving
Rule 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.
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Solving Word Problems:
A Common “Approach” for Learners
Randomly combining numbers without
trying to make sense of the problem.
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Key Words
 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?
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Real problems do not have key words!
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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.
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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?
9
9
9
weight of medium dog
9
9
18
weight of little dog
18
18
18
weight of big dog
5 x 18 = 90 pounds
18
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?
9
9
x = weight of medium dog
9
x
9
9
2/3 x
18
2/3 x = weight of little dog
So….how do you solve this problem from here?
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?
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Different visual depictions of problem
solutions for the Cookie Problem:
Sara
Sol 1
Kevin
Natalie
Katie
Sol 2
Sol 3
2
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Katie
Natalie
Sara
Kevin
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Mapping one visual depiction of solution for the
Cookie Problem to algebraic solution:
Sara
Sol 1
Kevin
Natalie
½x
⅓(½x)
Katie 1
¼(⅔(½x))
2
x
Sol 4 ½x + ⅓(½x) + ¼(⅔(½x)) + 1 + 2 = x
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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)
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Supporting Students
 Discuss the differences between pictures and
diagrams.
 Ask students to
little
medium
big
 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.
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Visual and Graphic
Depictions of Problems
Singapore
Math
Meilin saved $184. She saved $63 more than Betty.
How much did Betty save?
$184
Meilin
Betty
?
$63
184 – 63 = ?
Singapore Math, Primary Mathematics 5A
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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?
girls
12
24
boys
12
12
12
4 x 12 = 48 children
Singapore Math, Primary Mathematics 5A
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x = # of girls
3x = x + 24
2x = 24
x = 12
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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.
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Multiplication
A typical approach is to use arrays or the area model to
represent multiplication.
4
3×4=12
Why?
3
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Use Real Contexts – Grocery Store
(Multiplication)
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Multiplication
Context – Grocery Store
How many plums does the
grocer have on display?
plums
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Multiplication - Context –
Grocery Store
apples
lemons
Groups of 5 or less subtly suggest
skip counting (subitizing).
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How many muffins does the
baker have?
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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?
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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)
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Students’ Work
Jackie
Edward
Counted by ones
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Skip counted by twos
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Wendy
Students’ Work
Sam
Skip counted by 4. Used
relationships between the
trays. Saw the middle and last
tray were the same as the first.
Amanda
Decomposed larger
amounts and
doubled: 8 + 8 = 16;
16 + 16 + 4 = 36
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Used relationships
between the trays.
Saw the right hand
tray has 20, so the
middle tray has 4
less or 16.
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Area/Array Model
Progression
Context (muffin tray, sheet of stamps, fruit tray)
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4 x 39
How could you solve this? (Can you find a couple of
ways?)
Video (5:02) (1.1.2) Multiplication mini-lessons
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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
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 Take a minute and write down two things you are
thinking about from this morning’s session.
 Share with a neighbor.
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