Mapping children

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Mapping childrens understanding of mathematical
equivalenceRoger S. Taylor, Bethany
Rittle-Johnson, Percival G. Matthews, Katherine
L. McEldoon
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Algebraic Reasoning

• 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)

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Algebra Equivalence

• 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)

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Two views of Equivalence

• 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

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Three Components of Equivalence

• 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)
• Meaning of the Equal Sign
• Knowledge of equivalence and the equal sign
• Recognizing Valid Equations
• Knowledge of allowable equation formats
• Solving Equations
• Problem solving ability with different equation
  formats

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Focus of Study

• 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)

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Overview of the Construct
Difficulty
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Setting 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 Equal Sign
Difficulty
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Equal Sign Wright Map
Respondents Items
Higher Difficulty
Higher Ability
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Equal 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.
• 5

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Equal 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 Levels
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Construct Component Recognizing Valid Equations
Difficulty
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Recognizing Equations Wright Map
Respondents Items
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Recognizing Equations Wright Map
Respondents Items

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

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

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Recognizing Equations Levels
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Construct Component Solving Equations
Difficulty
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Solving Equations Wright Map
Respondents Items
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Solving Equations Wright Map
Respondents Items
SE10 (L3) 3 6 8 __
SE5 (L2) 8 6 __
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Solving Equations Levels
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Three Construct Components
Solving Equations
Recognizing Valid Equations
Meaning of the Equal Sign
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Summary

• 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

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Future Research

• Test-retest reliability
• Convergent and divergent validity
• Design of instructional interventions
• Utilization of assessment in other research
  projects