Using Visualization to Develop
Children's Number Sense and
Problem Solving Skills
in Grades K-3 Mathematics (Part 2)
LouAnn Lovin, Ph.D.
Mathematics Education
James Madison University
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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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)
 Discussions about why particular representations might
be more beneficial to help think through a given problem
or communicate ideas.
 Graphic depictions of multiple problems and multiple
solutions.
(Gersten & Clarke, NCTM Research Brief)
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Supporting Students
 Discuss the differences between pictures and
diagrams.
 Ask students to
Katie
Natalie
Sara Kevin
 Explain how the diagram represents various components
of the problem.
 Emphasize the the importance of precision in the
diagram.
 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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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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Summary of A Common “Approach” for Learners to
Solve Word Problems
Randomly combine 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.
 There are 7 boys and 21 girls in a class. How many
more girls than boys are there?
 Wendy has 3 cards. Her friend gives her 8 more cards. How
many cards does Wendy have now?
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Real problems do not have key words!
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Teaching Mathematical Concepts and
Skills through Word Problems
Contextual (Word) Problems
 Introduce procedures and concepts using contextual problems
(e.g., subtraction; multiplication).
 Makes learning more concrete by presenting abstract ideas in
a familiar context.
 AVOIDs the sole reliance on key words.
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Visual and Graphic
Depictions of Word Problems
Quantitative Analysis
Visual models (like Singapore Math, VandeWalle)
 Helps children to get past the words by visualizing and illus
trating word problems with simple diagrams.
 Emphasis is on modeling the quantities and their
relationships.
 Difference between pictures and diagrams.
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Visual and Graphic
Depictions of Problems
Ben has 5 cats and his cousin, Jerry, has 3 cats. How many
cats do they have together?
5
How would you write
this computation as
an equation?
Ben
Jerry
3
Jerry has 3 cats. Ben has 5 more cats than his cousin
Jerry. How many cats does Ben have?
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Visual and Graphic
Depictions of Problems
Jerry has 3 cats. Ben has 5 more cats than his cousin
Jerry. How many cats does Ben have?
8
Ben
5
Jerry
3
How would you write
this computation as
an equation?
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Visual and Graphic
Depictions of Problems
Meilin saved $184. She saved $63 more than Betty.
How much did Betty save?
$184
Meilin
Betty
?
$63
How would you write this computation?
(Primary Mathematics volume 3A, page 21, problem 7.)
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Solve these problems:
 Jacob had 8 cookies. He ate 3 of them. How
many cookies does he have now?
 Jacob has 3 dollars to buy cookies. How many
more dollars does he need to earn to have 8
dollars?
 Nathan has 3 dollars. Jacob has 8 dollars. How
many more dollars does Jacob have than
Nathan?
Perspectives
Most adults think 8 – 3 = 5, because it’s the
most efficient way to solve these tasks.
Young children see these as 3 different
problems and use the action or situation in the
problem to solve it – so they solve each of these
using different strategies.
(Unfortunately, too often children are told to
subtract – because that’s how we interpret the
problem.)
A first grader…
 Jacob has 8 cookies. He ate 3 of them. How many
cookies does he have now?
X
X
X
1
2
3
4
5
 Jacob has 3 dollars. How many more dollars does
he need to earn to have 8 dollars?
1
2
3
4
1
5
2
6
3
47
58
 Nathan has 3 dollars. Jacob has 8 dollars. How
many more dollars does Jacob have than Nathan?
1
2
3
4
5
Rekenrek
While students can use the rekrenrek to generate different
strategies for solving basic facts, they can also use it to solve
story problems such as the ones below. Visualization is key to
helping find a solution.
Together, Claudia and Robert have 7 apples. Claudia has one
more apple than Robert. How many apples do Claudia and
Robert have?
Claudia had 4 apples. Robert gave her some more. Now she has
7 apples. How many did Robert give her?
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Town Sports ordered 99 scooters. They have received 45
scooters. How many scooters is Town Sports waiting on?
45
?
99
99 – 45 = ______
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OR 45 + ____ = 99
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Joining
Taiwan’s
Cookie
Problem
Physical
Action
Separate
Part-part Whole
No
Physical
Action
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Comparing
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Start Unknowns
Bear Dog had some cookies. Taiwan gave him 8 more
cookies. Then he had 13 cookies. How many cookies did
Bear Dog have before Taiwan gave him any?
?
8
13
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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
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Edward
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Students’ Work
Sam
Wendy
Amanda
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Area Model
Grid Paper
 Show a 2 x 3 rectangle
 Show a 4 x 5 rectangle
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Open Array
12
5
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Area/Array Model
Progression
Context (muffin tray, sheet of stamps, fruit tray)
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2 x 30
How do you think about determining what 2 x 30 is?
What do we mean by “adding a zero”?
Video 1 (:19) (1.1.5) and Video 2 (3:59) (1.1.6)
Multiplication mini-lessons
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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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Take Aways
 Help children create diagrams to represent the
quantities and their relationships in problems.
 Children can solve the same problem using different
operations.
 Take advantage of children’s tendencies to subitize
(rekenreks and arrays)
 Use real world contexts to introduce arrays
(multiplication)
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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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References

Carpenter, Fennema, Franke, Levi, Empson. (1999). Children’s Mathematics: Cognitively Guided Instruction. Heinemann:
Portsmouth, NH.

Diesmann, C., & English, L. (2001). Promoting the use of diagrams as tools for thinking. In A. Cuoco & F. Curcio (Eds.), The Roles
of Representation in School Mathematics, pp. 77-89. Reston, VA: NCTM.

Dolk, M., Liu, N., & Fosnot, C. (2008). The Double-Decker Bus: Early Addition and Subtraction. Portsmouth, NH: Heinemann.

Fosnot, C. & Dolk, M. (2001). Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction.
Portsmouth, NH: Heinneman.

Fosnot, C. (2008). Bunk Beds and Apple Boxes: Early Number Sense. Portsmouth, NH: Heinemann.

Fostnot, C. & Cameron, A. (2007). Games for Early Number Sense. Portsmouth, NH: Heinneman.

Gersten, R. & Clarke, B. (2007). Research Brief: Effective Strategies for Teaching Students with Difficulties in Mathematics.
NCTM: Reston, VA.

Ministry of Education Singapore. (2009). The Singapore Model Method. Panpac Education: Singapore.

NCTM (2000). Principles and Standards of School Mathematics. NCTM: Reston, VA.

Parrish, S. (2010). Number Talks: Helping Children Build Mental Math and Computation Strategies. Math Solutions: Sausalito, CA.

Storeygard, J. (2009). My Kids Can: Making Math Accessible to All Learners. Heinemann: Portsmouth, NH.

Wright, R., Martland. J, Stafford, A., & Stanger, G. (2006). Teaching Number: Advancing Children’s Skills and Strategies. London:
Sage.

Using the Rekenrek as a Visual Model for Strategic Reasoning in Mathematics by Barbara Blanke
(www.mathlearningcenter.org/media/Rekenrek_0308.pdf)

VandeWalle, J. & Lovin, L. (2005). Teaching Student-Centered Mathematics: Grades K-3. Boston: Pearson.
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Cognitively Guided Instruction
Strategies
 Direct Modeling Strategies
 Counting Strategies
 Derived Number Facts
 Known Number Facts (as in recall)
return
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