Using Visualization to Extend Students

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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

2
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).
• 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.
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A picture is worth a thousand words.

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Do you see what I see?

An old mans face or two lovers kissing?
Not everyone sees what you may see.
Cat or mouse?
A face or an Eskimo?
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What do you see?

Everyone does not necessarily hear/see/interpret
experiences the way you do.
www.couriermail.com.au/lifestyle/left-brain-v-righ
t-brain-test/story-e6frer4f-1111114604318
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ManipulativesHands-On ConcreteVisual
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T Is four-eighths greater than or less than
four- fourths?J (thinking to himself)
Now thats 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. Thats
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 doesnt quite make sense, because the pieces
of one bar are much bigger than the pieces of the
other one. So, whats wrong with Ls drawing?

Ball, D. L. (1992).  Magical hopes 
Manipulatives and the reform of mathematics
education (Adobe PDF). American Educator, 16(2),
14-18, 46-47.
10
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? Hes 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 3 2 1
10 20 30 40
Adults 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 learners 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
• Visual Cuing
• Highlighting (talking about, pointing out)
  significant ideas in students work.

?
48 36 ?
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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 visualizin
  g 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?
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A Students 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 ProblemsA Common Approach for
Learners

• Randomly combining numbers without
• trying to make sense of the problem.

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20

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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? 

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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 visualizin
  g 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.

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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?

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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)

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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
weight of little dog
9
9
18
18
18
18
18
weight of big dog
5 x 18 90 pounds
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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
9
9
9
x
2/3 x weight of little dog
9
9
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
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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
Kevin
Sara
Natalie
Katie
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Mapping one visual depiction of solution for the
Cookie Problem to algebraic solution
Sara
?(½x)
Sol 1
¼(?(½x))
Kevin
Natalie
Katie
1
½x
2
x
Sol 4
?(½x)
¼(?(½x))
x
½x
2
1
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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
• 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
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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?
• Singapore Math, Primary Mathematics 5A

Betty
63
?
184 63 ?
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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?
• 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
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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.
• Why?

4
3412
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).
tomatoes
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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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VideoStudents Using Bakers Tray (430)

• 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
Skip counted by twos
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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
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Area/Array ModelProgression
Area model using grid paper
Open array
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 (502) (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.

48

• Take a minute and write down two things you are
  thinking about from this mornings session.
• Share with a neighbor.