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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.

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Using Visualization to Develop Children's Number Sense andProblem Solving Skills in Grades K-3 Mathematics (Part 2)

LouAnn Lovin, Ph.D.

Mathematics Education

James Madison University


The cookie problem
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
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


Visual and graphic depictions of problems
Visual and Graphic Cookie ProblemDepictions 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)

Lovin NESA Spring 2012


Supporting students
Supporting Students Cookie Problem

  • 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.

    • 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.

Katie Natalie Sara Kevin

Lovin NESA Spring 2012


A student s guide to problem solving
A Student Cookie Problem’s Guide to Problem Solving

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Summary of a common approach for learners to solve word problems
Summary of A Common Cookie Problem“Approach” for Learners to Solve Word Problems

Randomly combine numbers without trying to make sense of the problem.

Lovin NESA Spring 2012


Lovin NESA Spring 2012 Cookie Problem


Lovin NESA Spring 2012 Cookie Problem


Key words
Key Words Cookie Problem

  • 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. 

    • 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?

Lovin NESA Spring 2012


Real problems Cookie Problemdo not have key words!

Lovin NESA Spring 2012


Teaching mathematical concepts and skills through word problems
Teaching Mathematical Concepts and Skills Cookie Problemthrough 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.

Lovin NESA Spring 2012


Visual and graphic depictions of word problems
Visual and Graphic Cookie ProblemDepictions of Word Problems

Quantitative Analysis

Visual models (like Singapore Math, VandeWalle)

  • Helps children to get past the words by visualizing and  illustrating word problems with simple diagrams.

  • Emphasis is on modeling the quantities and their relationships.

  • Difference between pictures and diagrams.

Lovin NESA Spring 2012


Visual and graphic depictions of problems1
Visual and Graphic Cookie ProblemDepictions of Problems

Ben has 5 cats and his cousin, Jerry, has 3 cats. How many cats do they have together?

How would you write this computation as an equation?

Jerry

3

5

Ben

Jerry has 3 cats. Ben has 5 more cats than his cousin Jerry. How many cats does Ben have?

Lovin NESA Spring 2012


Visual and graphic depictions of problems2
Visual and Graphic Cookie ProblemDepictions 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?

Lovin NESA Spring 2012


Visual and graphic depictions of problems3
Visual and Graphic Cookie ProblemDepictions of Problems

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

How would you write this computation?

(Primary Mathematics volume 3A, page 21, problem 7.)

$184

Meilin

Betty

?

$63

Lovin NESA Spring 2012


Solve these problems
Solve these problems: Cookie Problem

  • 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?

How did you find your answer?


Perspectives
Perspectives Cookie Problem

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
A first grader… Cookie Problem

  • Jacob has 8 cookies. He ate 3 of them. How many cookies does he have now?

  • Jacob has 3 dollars. 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?

X

X

X

1

2

3

4

5

1

2

3

4

1

2

5

6

3

4

7

5

8

1

2

3

4

5


Rekenrek
Rekenrek Cookie Problem

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?

Lovin NESA Spring 2012


Town Sports ordered 99 scooters. They have received 45 scooters. How many scooters is Town Sports waiting on?

45

?

99

99 – 45 = ______

OR 45 + ____ = 99

Lovin NESA Spring 2012


Taiwan s cookie problem
Taiwan scooters. How many scooters is Town Sports waiting on?’s Cookie Problem

Joining

Physical Action

Separate

Part-part Whole

No

Physical Action

Comparing

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Start unknowns
Start Unknowns scooters. How many scooters is Town Sports waiting on?

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

Lovin NESA Spring 2012


Multiplication
Multiplication scooters. How many scooters is Town Sports waiting on?

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
Use Real Contexts – Grocery Store (Multiplication) scooters. How many scooters is Town Sports waiting on?

Lovin NESA Spring 2012


Multiplication context grocery store
Multiplication scooters. How many scooters is Town Sports waiting on?Context – Grocery Store

How many plums does the grocer have on display?

plums

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Multiplication context grocery store1
Multiplication - scooters. How many scooters is Town Sports waiting on?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
How many muffins does the baker have? scooters. How many scooters is Town Sports waiting on?

Lovin NESA Spring 2012


Other questions
Other questions scooters. How many scooters is Town Sports waiting on?

  • 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
Video: scooters. How many scooters is Town Sports waiting on?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
Students scooters. How many scooters is Town Sports waiting on?’ Work

Jackie

Edward

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Students work1
Students scooters. How many scooters is Town Sports waiting on?’ Work

Sam

Wendy

Amanda

Lovin NESA Spring 2012


Area model grid paper
Area Model scooters. How many scooters is Town Sports waiting on?Grid Paper

  • Show a 2 x 3 rectangle

  • Show a 4 x 5 rectangle

Lovin NESA Spring 2012


Open array
Open Array scooters. How many scooters is Town Sports waiting on?

12

5

Lovin NESA Spring 2012


Area array model progression
Area/Array Model scooters. How many scooters is Town Sports waiting on?Progression

Area model using grid paper

Open array

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

Lovin NESA Spring 2012


2 x 30
2 x 30 scooters. How many scooters is Town Sports waiting on?

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

Lovin NESA Spring 2012


Lovin NESA Spring 2012


Take aways
Take about from this morningAways

  • 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)

Lovin NESA Spring 2012


Do you see what i see
Do you see what I see? about from this morning

An old man’s face or two lovers kissing?

Cat or mouse?

Not everyone sees what you may see.

Lovin NESA Spring 2012


References
References about from this morning

  • 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
Cognitively Guided Instruction about from this morningStrategies

  • Direct Modeling Strategies

  • Counting Strategies

  • Derived Number Facts

  • Known Number Facts (as in recall)

return

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