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We used a construct modeling approach to unify diverse item types into a single instrument designed to measure mathematical equivalence knowledge. Analyses showed that the assessment cohered well, measuring a unidimensional construct. Data gathered from the new assessment 1) help expand our abilities to model the variability of student’s knowledge of equivalence, 2) explicitly demonstrate how problem difficulty varies with equation structure, and 3) provide strong empirical evidence that some typically algebraic items load heavily on the equivalence construct.

Mathematical Equivalence Construct Map

Results – The Rasch Model

The data were analyzed using Rash modeling (Bond & Fox, 2007). The Rasch model estimates both respondent ability and item difficulty simultaneously, yielding the probability that a particular respondent will answer a particular item correctly. A test of unidimensionality indicated that the items on the assessment measured a single construct, offering evidence for construct validity.

Understanding the Equals Sign as a Gateway to Algebraic ThinkingPercival G. Matthews, Bethany Rittle-Johnson, Roger S. Taylor, Katherine L. McEldoon

Level Description

Level 4: Comparative Relational. Children successfully solve and evaluate equations by comparing the expressions on the two sides of the equals sign, including using compensatory strategies. They consistently generate a relational definition of equals sign.

Level 3: Relational. Children successfully solve equation structures with operations on both sides of the equals sign. They also rate relational definitions of the equals sign as “a good definition.”

Level 2: Flexible operational. Children successfully solve atypical equation structures that remain compatible with an operational view of the equals sign.

Level 1: Rigid operational. Children are only successful with equations in operations-equals-answer structure. Their definitions of the equals sign are strictly operational.

- Core Equation Structures
- Operations on both sides with multi-digit numbers or multiple instances of an unknown:
- m + m + m = m + 12
- Judge “67 + 86 = 68 + 85” as True or False
- without computing
- Operations on both sides:
- 7 + 6 + 4 = 7 + __ __ + 2 = 6 + 4
- Operations on right:
- 8 = 6 + __
- No operations:
- Judge “8 = 8” as true or false
- Operations on left:
- 6 + 2 = __ __ + 5 = 9

Charting Variability

Table 1 lists the probabilities that students at various ability levels will generate correct answers for selected items. Focusing across rows within a given column allows us to compare the differences in the probabilities of success on select items for students of a given ability level. The assessment ultimately models equivalence knowledge as a continuous construct.

The table helps highlight an interesting point about how equation structure affects difficulty in a similar fashion across classes – it appears that open-equation and equation structure items were of similar difficulty when they were in the same equation formats (e.g., 8 = 6 + __ vis-à-vis “Judge 4 = 4 + 0 as true or false”).

Differences between Level 3 and Level 4 understandings can go undetected if solution method is unmonitored. Measuring use of compensatory strategies reveals differences in equivalence knowledge.

Increasing Sophistication

- Background
- Knowledge of mathematical equivalence is a foundational concept of mathematics that serves as a key link between arithmetic and algebra. Unfortunately, measurement issues have limited our abilities to chart the variability in children’s developing conceptions of mathematical equivalence. In particular, the diversity of items used by different researchers has made it difficult to compare results across studies. Typically, children’s knowledge of equivalence is measured using three classes of items:
- Open equation-solving items, such as 8 + 4 = _ + 5
- Equation-structure items, such as deciding if
- 3 + 5 = 5 + 3 is true or false
- Equal-sign-definition items, such as asking children to provide explicit verbal interpretations of the equals sign
- We used a construct modeling approach (Wilson, 2005) to develop an assessment that could integrate these diverse types of measurement items on the same scale.

Prototypical Level 3 Performances

Student use of solve-and-compare strategy is relatively inefficient and does not capture comparative relational thinking.

Prototypical Level 4 Performances

Students use a compensation strategy, recognizing the fact that 87 is two less than 89, which means that the addend coupled with the 87 must be two more than the one coupled with 89 in order to preserve equality.

Students use the fact that performing the same operations on both sides of an equation preserves equivalence.

Summary

Several decades of research have catalogued the difficulties that American elementary school children have with understanding the concept of mathematical equivalence. The current study adds resolution to that picture by combining diverse measurement items onto a single scale. Specifically, it allows us to compare the difficulty of different classes of items, demonstrating that equation structure is the key factor in determining difficulty. Additionally, this is the first instrument that explicitly maps typical algebraic items onto the same scale as more basic items measuring mathematical equivalence.

- The Assessment
- Most of the items on our assessment were taken directly from previously published works or created based on items present in those works (e.g., Baroody & Ginsburg, 1983; Behr, 1980; Carpenter et al., 2003; Knuth et al., 2006). Items were classified as Level 1, 2, 3, or 4 as outlined in the our construct map (Rittle-Johnson et. al., 2011).
- We also included two new classes of items that require more advanced algebraic thinking:
- Equations involving letter variables (or literals) such as
- n + n + n + 2 = 17 (Jacobs et. al., 2007)
- Items requiring children to reason about how performing the same operation on each side of an equation preserves equivalence such as,
- If we know that 76 + 45 = 121, can we tell without adding whether or not 76 + 45 – 9 = 121 – 9 ?
- The assessment was administered on a whole class basis to 224 students in Grades 2 – 6 at 2 suburban schools in Tennessee.

Table 1: Probability of Success on Selected Items by Student Ability Estimate

References

Baroody, A. J., & Ginsburg, H. P. (1983). The effects of instruction on children's understanding of the" equals" sign. The

Elementary School Journal, 199-212.

Behr, M. (1980). How Children View the Equals Sign. Mathematics Teaching, 92, 13-15.

Bond, T. G., & Fox, C. M. (2007). Applying the Rasch model: Fundamental measurement in the human sciences.

Lawrence Erlbaum.

Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking Mathematically: Integrating Arithmetic and Algebra in

Elementary School. Heinemann, 361 Hanover Street, Portsmouth, NH.

Jacobs, V. R., Franke, M. L., Carpenter, T. P., Levi, L., & Battey, D. (2007). Professional development focused on

children's algebraic reasoning in elementary school. Journal for research in mathematics education, 38(3), 258.

Knuth, E. J., Stephens, A. C., McNeil, N. M., & Alibali, M. W. (2006). Does understanding the equal sign matter?

Evidence from solving equations. Journal for Research in Mathematics Education, 37(4), 297.

Rittle-Johnson, B., Matthews, P. G., Taylor, R. S., & McEldoon, K. L. (2011). Assessing Knowledge of Mathematical

Equivalence: A Construct-Modeling Approach. Journal of Educational Psychology, 103(1), 85-104.

Wilson, M. (2005). Constructing measures: An item response modeling approach. Lawrence Erlbaum.

Contact Information

Percival G. Matthews, Department of Psychology, University of Notre Dame, Notre Dame, IN, 46628. E-mail: pmatthew@nd.edu

This research was supported with funding from the National Science Foundation grantDRL-0746565 to the first author, as well as Institute of Education Sciences, U.S. Department of Education, training grants R305B040110 and R305B080008 to Vanderbilt University. The opinions expressed are those of the authors and do not represent views of the U.S. Department of Education.

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