Children’s Equal Sign Definition Endorsement Predicts Performance on Test of Equivalence Knowledge

• In Event: Poster Session 13
  In Poster Session: PS 13 Topic - Cognitive Processes

Abstract

Understanding of mathematical equivalence is foundational to success in middle and high school mathematics (Knuth et al., 2006; Baroody & Ginsburg, 1983). However, many students struggle to develop a conceptual understanding of the equal sign as a relational symbol, which may hinder their understanding of algebraic concepts (Behr et al., 1980; McNeil & Alibali, 2005). The current experiment investigated whether instruction about equivalence that included conceptually-based instruction plus analogy (i.e., solving an equation is like balancing a teeter-totter) is more beneficial than conceptually based instruction without analogy or than procedural instruction.

We randomly assigned 3rd (N=114) and 6th grade students (N=103) to the conceptual, conceptual with analogy, or procedural instruction condition and examined pretest-to-posttest gains in understanding. The intervention was a five-minute one-on-one session with an experimenter in which students were asked to think about equivalence items such as 5 + 4 + 3 = 5 + ___ and were given a lesson on solving such items consistent with their assigned condition. In all conditions, the lesson emphasized that the equal sign indicates that the amounts on either side of an equation are the same. There were three post-test sessions: immediately, 3 weeks, and 4 months post-intervention. Assessments included equation encoding, true/false, open equations, and defining the equal sign (Matthews et al., 2012).

Contrary to our hypothesis, conceptually-based equivalence instruction did not improve children’s ability to solve equivalence problems more than procedural instruction for 3rd graders (repeated measures ANCOVA, condition*time interaction F(6,429) = .254, p =.96) nor 6th graders (condition*time interaction F(6,386) = .198, p =.98) at any post-test session. Given these unexpected findings, we explored relations among assessment items to investigate how students’ conceptions of equivalence relate to their solving of equivalence items.

Third graders who only endorsed an incorrect operational definition of the equal sign (“the answer” or “the total”) as a “good” definition performed significantly worse on the equivalence assessment across all sessions, compared to children who also endorsed a correct relational definition (“two amounts are the same” or “the same as”) (linear mixed effects model, B = -2.48, t = -4.82, p <.001). This finding aligns with the idea that failure to understand the equal sign as a relational symbol impairs early math performance (Matthews et al., 2012). Among 6th graders, only endorsing an operational definition was uncommon, however, a majority of 6th graders endorsed both an operational and relational definition. Interestingly, there was no significant difference between endorsing both definitions and only endorsing a relational definition on assessment accuracy (linear mixed effects model, B = .305, t = .982, p =.33).

Although our findings suggest that it is difficult to enhance understanding of equivalence is difficult in a single, brief intervention session, they confirm that this understanding is an important predictor of children’s ability to solve equivalence problems in 3rd grade. A better understanding of how conceptions of the equal sign develop may inform the design of curricular materials for teaching equivalence.

Authors

• Emily Szkudlarek, University of Wisconsin - Madison
  Presenting Author

• Andrea Marquardt Donovan, University of Wisconsin - Madison
  Non-Presenting Author

• Ana Stephens, University of Wisconsin - Madison
  Non-Presenting Author

• Burcu Alapala, UW Madison
  Non-Presenting Author

• Allison Monday, University of Wisconsin-Madison
  Non-Presenting Author

• Martha Wagner Alibali, University of Wisconsin - Madison
  Non-Presenting Author

• Percival G Matthews, University of Wisconsin - Madison
  Non-Presenting Author