Many students fail to develop adequate understanding of mathematical equivalence in early grades, which impacts later algebra learning. Work from McNeil and colleagues proposes that this failure is partly due to the format of traditional instruction and practice with highly similar problems, which encourages students to develop ineffective representations of problem types (McNeil, 2014, McNeil & Alibali, 2005). In the current study, we explore students’ learning trajectories in two matched equivalence interventions. We show that, relative to an active control, the principle-based treatment intervention gives rise to a greater number of successful learners, a designation that, in turn, leads to improved performance on distal transfer assessments. We further demonstrate a predictive relationship between students’ engagement with the intervention, via workbook completion, and likelihood of becoming a successful learner. Our findings have implications for early detection of learning and subsequent scaffolding for low-performing students.

This study examined how changes in children's problem encoding influenced their strategy selection. Fourth-grade students {N=51} solved six mathematical equivalence problems (e.g., 3+4+5=_+5) in a pretest. Children's problem encoding was then manipulated in one of two ways, or was not manipulated in a Control group. In the Subtle group, children solved four additional problems with the equal sign highlighted in red. In the Direct group, children solved the same four problems, and were directed to notice the equal sign in each problem. Children then solved six problems in a posttest, and did so again four weeks later in a follow-up test. The strategies children conveyed in their spoken and gestured explanations were assessed. Children in the Direct group considered multiple strategies for the posttest problems more often than children in the other groups, as reflected in their spoken and gestured explanations. Children in the Direct group were also most likely to generate gestured strategies and to abandon verbal strategies over the course of the study. These findings suggest that changes in problem encoding lead to changes in strategy selection.

Elementary math instruction traditionally has emphasizedprocedures rather than concepts. Thus, students tend to lack astrong understanding of foundational concepts likemathematical equivalence. Cognitive scientists andmathematics educators have found small yet effective ways tomodify traditional arithmetic instruction to promote students’conceptual understanding of math equivalence. Educationalstandards also now reflect this academic research. However, itis unclear whether classroom practices have caught up withresearch and policy. In the current study, we observed teachers’practices during arithmetic instruction. The goal was todetermine if teachers are using research-based practices thatpromote understanding of math equivalence and if variation inuse of research-based practices is associated with students’growth in understanding of math equivalence across the schoolyear. Eight second and third grade classrooms (M students perclassroom = 23) were observed twice during math instruction.Students completed a math test both before and after theobservation period. Research-based practices were rarelyobserved in any classrooms, so there was not much variation inclassroom use of research-based practices to predict studentgrowth. Students improved their performance on all problemtypes tested, but performance on math equivalence problemswas significantly lower than on other problem types. Resultssuggest that policies and practices designed to improvestudents’ understanding of math equivalence may not havefiltered down to affect instructional practices in classrooms.

Deep understanding of mathematical equivalence is criticalfor later mathematical understandings. However, researchstudies and national test results have repeatedly demonstratedthat many students fail to develop adequate understanding ofequivalence. Recent work from McNeil and colleaguesproposes that this failure is partly due to the format oftraditional instruction and practice with highly similarproblems. Specifically, the change-resistance account(McNeil & Alibali, 2005) proposes that students struggle withequivalence because they have developed overgeneralized“rules” that affect how they process and approach mathproblems, (e.g., the operators are always on the left side, theequal sign means to “do something” or “give the answer”)and fail to see equations having two separate sides that arebeing related to one another. Extensive practice withproblems in a similar format (e.g., those that present allarithmetic operations on the left side of the equal sign)encourages students to develop ineffective mental models ofproblem types. We replicate and extend prior work that bringscognitive science research to the classroom. Our findingsindicate that applying research-based design principles toarithmetic practice improves student understanding ofmathematical equivalence enough to support transfer to novelproblem types.

Based on prior work, we predicted that traditional, 2-D counting books would be better than tactile counting books at promoting young children's numeracy. However, the first author suspected that the effect of tactility on learning would differ for Black versus non-Black children. To test this, we examined data from an existing project on preschoolers' learning from shared counting book reading. Participants included 325 preschoolers, ages 2 to 6, 41% of whom were Black. Findings suggest that race moderates the effect of tactility on numeracy. Non-Black children conformed to the original hypothesis that non-tactile counting books would be best for promoting children's numeracy, but Black children did not. This finding is important because much of the research on children's early numeracy is conducted with homogeneous, convenience samples, so theories and predictions are being built on incomplete data. Without studying diverse samples, the field risks making inaccurate conclusions about how children learn.