Transcript Learning Stuff - Peabody College

Mapping children’s
understanding of mathematical
equivalence
Roger 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)
1.
Meaning of the Equal Sign
•
2.
Recognizing Valid Equations
•
3.
Knowledge of equivalence and the equal sign
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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Difficulty
Overview of the Construct
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Setting & Data Collection
•
School: Urban, parochial, serving working-to middleclass, predominantly Caucasian students
•
Written assessment (single period)
•
Administered to 181 students in 10 2nd-6th grade
classrooms (2 per grade)
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Difficulty
Construct Component:
Equal Sign
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Equal Sign Wright Map
Respondents | Items
Higher
Ability
Higher
Difficulty
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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.
a)+
b)=
c)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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Difficulty
Construct Component:
Recognizing Valid Equations
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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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Difficulty
Construct Component:
Solving Equations
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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
Meaning of the Equal Sign Recognizing Valid Equations
Solving Equations
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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
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