When middle school students learn science content withgraphs, the graphing and science knowledge may be mutuallyreinforcing: understanding the science content may help stu-dents interpret a related graph, and information from a graphmay illustrate a scientific concept. We examine this relation-ship between graphing and science by studying how studentslearn from interactive computer models with accompanyingdata graphs. The computer models provide an animated simu-lation that illustrates an unobservable phenomenon, while thedata graph tracks one or more quantities over time. This or-dering study, on middle school students learning about photo-synthesis, indicates that engaging with novel graph conceptshelped students interpret their data as they experimented withthe computer model. The study also provided some supportfor the opposite direction: experimenting with the model firsthelped students make sense of the graphs.

Do students activate conceptual and procedural knowledgesimultaneously when learning fraction addition? In groundedfeedback, student actions on a target, to-be-learnedrepresentation are reflected in a more familiar feedbackrepresentation to promote conceptual learning withinprocedural practice. An experiment with 163 4th and 5thgraders shows improved learning with a grounded feedbacktutor over a symbols-only control with step-level right/wrongfeedback. Learning with grounding also transferred tosymbols-only assessment items, providing some support forthe simultaneous activation view.

What is the role of fraction magnitude knowledge in learningfraction addition? An experiment with 71 6th and 7 th gradestudents compared fraction addition instruction and practicewith a magnitude representation to a tightly controlled non-magnitude condition. In the magnitude condition, studentswith better fraction magnitude estimation skills benefittedmore from the conceptual instruction and this relationshipwas moderated by students’ knowledge of how magnituderelates to fraction addition and equivalence. However,students with better fraction magnitude estimation skillsbenefitted less from the practice problems with magnitude. Inthe non-magnitude condition, fraction magnitude estimationwas not predictive of learning. This study indicates thatstudents with magnitude knowledge can leverage it to learnfraction addition concepts from magnitude representations,but, for those students, magnitude representations may be adistraction from practicing the procedure.

Empirical results from a fraction addition task reveal a surprising gap in prior knowledge: difficulty applying the transitive property of equality in a symbolic context. 13 out of the 182 4th and 5th graders (7%) correctly applied the transitive property of equality to identify the sum of two fractions in a step-by-step worked example. This difficulty was robust to brief instruction on transitivity (after which performance rose to 11%). Students’ demonstrated difficulty with transitivity is surprising, especially because common instructional techniques, such as worked examples, assume that the learner understands this concept and where it applies.