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Mapping children’s Mapping children’s understanding of understanding of

mathematical equivalencemathematical equivalence

Roger S. Taylor, Bethany Rittle-Johnson, Roger S. Taylor, Bethany Rittle-Johnson, Percival G. Matthews, Katherine L. Percival G. Matthews, Katherine L.

McEldoonMcEldoon

• Goal of “Algebra for All” motivated National Mathematics Panel (2008) report on how to improve students’ readiness for algebra

• To meet this goal, must re-conceptualize algebra as a continuous strand from elementary through high school (NCTM, 2000)

Algebraic ReasoningAlgebraic Reasoning

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• Previous research has shown that knowledge of the equal sign and equivalence (or equality) is critical for learning algebra (Carpenter, et al., 2003; Knuth, Stephens, McNeil, & Alibali, 2006)

• Equivalence is the principle that two sides of an equation represent the same value

• Provides the foundation for two key algebra proficiencies (Kieran, 1992):

• Understanding the equivalence of expressions

• Competence at performing the same operation on both sides of an equation

• Is accessible to elementary school children (Baroody & Ginsburg, 1983; Jacobs et al., 2007)

Algebra & EquivalenceAlgebra & Equivalence

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• Many children view the equal sign operationally, as a command to carry out arithmetic operations (Baroody & Ginsburg, 1983; Carpenter, et al., 2003; McNeil & Alibali, 2005)

• Define “=“ as meaning “add up the numbers”

• Solve equations incorrectly when operations on the right side of the equal sign, such as: 3 + 5 = __ + 6

• Reject number sentences such as 3 + 5 = 5 + 3

• Children need to develop a relational view of the equal sign

Two views of EquivalenceTwo views of Equivalence

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• Knowledge of equivalence assessed through 3 main components (e.g. Alibali, 1999; Falkner, Levi, & Carpenter, 1999; Li, Ding, Capraro, & Capraro, 2008; McNeil, 2007; Rittle-Johnson & Alibali, 1999; Weaver, 1973)

1. Meaning of the Equal Sign

• Knowledge of equivalence and the equal sign

2. Recognizing Valid Equations

• Knowledge of allowable equation formats

3. Solving Equations

• Problem solving ability with different equation formats

Three Components of Three Components of EquivalenceEquivalence

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• Develop framework for assessing & interpreting students’ level of understanding of mathematical equivalence

• Validate assessment

• Initial phase of research project: Creation & validation of Construct Map (Wilson, 2005)

Focus of StudyFocus of Study

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Overview of the Overview of the ConstructConstruct

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Setting & Data CollectionSetting & Data Collection

• School: Urban, parochial, serving working-to middle-class, predominantly Caucasian students

• Written assessment (single period)

• Administered to 181 students in 10 2nd-6th grade classrooms (2 per grade)

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Construct Component: Construct Component: Equal Sign Equal Sign

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Equal Sign Wright MapEqual Sign Wright Map

Respondents | Items

HigherAbility

HigherDifficulty

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Equal Sign Wright MapEqual Sign Wright Map

Respondents | Items

ES4: (L2) Which answer choice below would you Put in the empty box to show that two nickels are the same amount of money as one dime? Circle your answer.

a)+b)=c)5¢

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Equal Sign Wright MapEqual Sign Wright Map

Respondents | Items

ES1: (L3) What does the equal sign (=) mean?

ES5: (L3) Is this a good definition of the equal sign? Circle good or not good.• The equal sign means the same as.

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Equal Sign LevelsEqual Sign Levels

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Construct Component: Construct Component: Recognizing Valid EquationsRecognizing Valid Equations

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Recognizing Equations Recognizing Equations Wright MapWright Map

Respondents | Items

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Recognizing Equations Recognizing Equations Wright MapWright Map

Respondents | Items

RE2b: (L2) For each example, decide if the number sentence is true. In other words, does it make sense?• 8 = 8

RE2h: (L3) For each example, decide if the number sentence is true. In other words, does it make sense?• 6 + 4 = 5 + 5

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Recognizing Equations Recognizing Equations LevelsLevels

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Construct Component: Construct Component: Solving EquationsSolving Equations

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Solving Equations Wright Solving Equations Wright MapMap

Respondents | Items

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Solving Equations Wright Solving Equations Wright MapMap

Respondents | Items

SE5: (L2) 8 = 6 + __

SE10: (L3) 3 + 6 = 8 + __

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Solving Equations LevelsSolving Equations Levels

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Three Construct Three Construct ComponentsComponents

Solving EquationsRecognizing Valid EquationsMeaning of the Equal Sign

• Developed mathematical equivalence construct that consists of three components that seem to develop at different rates

• Created Construct Map

• Provides framework for assessing & interpreting students’ knowledge of mathematical equivalence

• Helped refine our thinking of what makes items hard

• Provides educational researchers and practitioners with a criterion-referenced interpretation of instructional assessments and student performance

SummarySummary

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• Test-retest reliability

• Convergent and divergent validity

• Design of instructional interventions

• Utilization of assessment in other research projects

Future ResearchFuture Research

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