Effects of Curriculum Variation on Structure in Middle School Mathematics

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Effects of Curriculum Variation on Structure in
Middle School Mathematics

• Robert M. Capraro, Victor L.Willson, Mary
  Margaret Capraro
• Texas AM University
• CEHD Symposium - February 18, 2005

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Presentation Information

• Paper originally presented at the 2004 Annual
  Meeting of the American Educational Research
  Association. San Diego, CA, April 12-16, 2004.
• This research was supported by an NSF-IERI grant,
  2001-2006- Improving Mathematics Teaching and
  Achievement through Professional Development, (J.
  Roseman, G. Kulm, J. Manon, Co-PIs).

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Focus of Study

• Two foundational content strands are the focus of
  this investigation algebra and number
• Middle grades
• AAAS Benchmarks for Science Literacy (2000)
• NCTM Standards (2000)

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Benchmarks

• Algebra - symbolic equations can be used to
  summarize how the quantity of something changes
  over time or in response to other changes
• Number- use, interpret, and compare numbers in
  several equivalent forms such as integers,
  fractions, decimals, and percents

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Algebra Construct

• Field of mathematics that explores relationships
  among different quantities and represents them as
  symbols
• Students in middle school not only study change
  itself, but how fast something is changing, and
  how the rate of change depends on some other
  quantity (relates to science change)

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Arithmetic to Algebra

• Difficulties that middle school students have in
  algebra come from moving from arithmetic
  reasoning to algebraic reasoning manifesting in
  ideas of change (Kilpatrick, Swafford, Findell,
  2001)
• Arithmetic emphasizes numerical expressions while
  algebra focuses on representing the
  generalization of problems through equations
  (Kieran, 1989).

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Algebra Construct

• Fluidity and flexibility with variables is
  essential to being successful with formal algebra
  (Kilpatrick, Swafford, Findell, 2001.
• Understanding how one variable changes as other
  conditions change is essential to developing deep
  understandings (Davis, 1998).
• Understanding ideas of equality are essential to
  know that an equal sign does not mean to do
  something rather what is presented on one side
  has the same value as what is on the other side
  of the sign (Van de Walle, 2004).

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Number Construct

• In middle grades students should deepen their
  understanding of fractions, decimals, percents,
  and integers, and they should become proficient
  in using them to solve problems.
• Researchers note that number sense develops
  gradually, and varies 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).

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Numerals and Number Sense bridges all
Mathematical Concepts

• Ekenstam (1977) stated, The lack of
  understanding of what numerals mean must present
  insuperable barriers to learning mathematics (p.
  317).
• Much work done
• Behr et al. (1997) stated, Although much
  research has been accomplished on the knowing,
  learning, and teaching of these concepts among
  populations during the last decade, much work
  remains to be done (p. 48).
• Behr, Harel, Post, and Lesh (1992) argued that
  explicit information is lacking among researchers
  as regards the concepts underlying the separate
  subconstructs.

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Research on Number Sense

• As a community of scholars (Behr, Khoury, Harel,
  Post, Lesh, 1997 Behr, Lesh, Post, Silver,
  1983 NCTM, 1991, 2000 Kieren, 1976, 1988
  Rittle-Johnson, Siegler, Alibali, 2001), many
  regard number sense to encompass
• (a) understanding number meanings
• (b) comprehending multiple relationships
  (meanings) among numbers
• (c) recognizing the relative magnitude of
  numbers and
• (d) knowing the relative effect of operating on
  numbers.

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Analytic Justification for Confirmatory Factor
Analysis

• CFA is primarily used to confirm theoretical
  constructs by using measured variables to
  estimate latent constructs.
• Because constructs are unobserved but really are
  the very things many researchers wish to study,
  factor analysis is intimately involved with
  questions of validity. . . and is at the heart
  of the measurement of psychological constructs
  (Nunnally, 1978, pp. 112-113).

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• There are several techniques for examining
  structure of a set of variables or indicators, of
  which probably the most useful is some form of
  factor analysis Pedhazur and Schmelkin (1991).
• Jöreskogs (1969) seminal work established a
  group of maximum likelihood estimation techniques
  that have been loosely termed confirmatory factor
  analysis (CFA).
• CFA tests specific hypotheses regarding the
  nature and relationship of factors. Gorsuch
  (1983)
• CFA allows researchers to directly test the fit
  between theories and the structure of data in
  hand (Kieffer, 1999).

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• In confirmatory factor analysis (a) the theory
  comes first, (b) the model is then derived from
  it, and finally (c) the model is tested for
  consistency with the observed data using a
  SEM-type approach (Raykov Marcoulides, 2000,
  p.95).
• CFA does not provide definitive proof that a
  theoretical model is definitely true but rather
  it provides several fit indices to help the
  researcher to determine which competing
  theoretical models best fits the data (Kieffer,
  1999).

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Statement of the Problem

• This study was designed to determine if test
  structure adequately approximated the theorized
  algebra and number learning models.
• Confirmatory factor analysis was used because a
  theoretical model existed in the Atlas of Science
  by the AAAS.

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Methodology

• Tests co-developed by researchers from
  University of Delaware, Texas AM University, and
  AAAS at Project 2061.
• Tests designed to measure sub-constructs of
  algebra and number as defined in the literature
  (NCTM, 2000 AAAS, 1993).
• Tests consist of 20 tasks across 3 parts ranging
  from multiple-choice, short answer, and extended
  response (super-item) questions - Tables 1 2
• Reliability (data in hand) .62 and .84 on
  pretest .82 and .86 on posttest for algebra and
  number, respectively

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Methodology cont.

• Both tests analyzed - structural equation
  modeling
• Posttest data modeled using factor loadings
  from
• pretest CFA
• Algebra data did not adequately fit model
• Exploratory factor analysis conducted to examine
  the underlying structure, which revealed a
  possible arithmetic factor.
• The test adequately measures the construct but
  the addition of a fourth construct improved the
  model variance fit.

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Participants

• 2002 - pilot testing number and algebra
• Pre and post testing began during 2002-03
  school year
• 6th grade students - number
• 7th/ 8th grade students - algebra
• Table 3 - sample demographics

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CFA for Algebra

• CFA provides support for the construct validity
  of the algebra test
• CFA results (Figure 3) indicated latent
  variables of change, variable and equality
  were highly correlated (understandable in terms
  of literature, benchmark item, NCTM and state
  standards)
• Overall fit indices evaluate how well the model
  explained the data reported in Table 4.

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CFA indices

• Normed Fit Index (NFI)
• Goodness of Fit (GFI)
• Tucker-Lewis Index (TLI)
• Comparative Fit Index (CFI)
• Root Mean Square Error of Approximation (RMSEA)

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Algebra EFA

• Algebra postest structure did not conform to
  pre-test structure either exactly or in model
  form
• Exploratory factor analysis performed to
  examine possible structures
• Principal axis (common factor) analysis with
  both Varimax and Promax rotations performed
• Squared multiple correlations were employed as
  initial communality estimates.

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EFA Interpretation

• 4 factors were interpreted (Table 5) pattern
  matrix
• Factor 1 Change
• Factor 2 Equations
• Factor 3 Variables
• Factor 4 Arithmetic
• 20 separately scored items
• 9 were sufficiently variable and correlated with
  factors to be assigned to a factor.
• 11remaining items - 3 reasonably included with
  a less stringent .30 pattern loading
• measurement structural model (Anderson
  Gerbing, 1984) based on the exploratory factor
  analysis was constructed (Figure 4)

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CFA Number Interpretations

• Table 6
• Normed Fit Index (NFI)
• Goodness of Fit (GFI)
• Tucker-Lewis Index (TLI)
• Comparative Fit Index (CFI)
• Root Mean Square Error of Approximation (RMSEA)

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CFA Number Interpretations

• Information when considered in aggregate
  indicates that the model provides a reasonable
  approximation of the data
• May be infinitely many models may fit this data,
  the combined a priori theoretical construct with
  the fit indices supports the theorized model.

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