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.