The present study was designed to explore how learning magnitude knowledge and learning procedural knowledge, with respect to both whole numbers and fractions, might be causally related. Neither magnitude knowledge nor procedural knowledge is necessary or sufficient for learning the other, and yet, correlations between the two are ubiquitous (e.g., Siegler & Pyke, 2013). Using correlational data (Structural Equation Models) and accuracy data (Knowledge Space Theory), potential causal models to describe the data were tested. Structural equation models did not differentiate between learning magnitude knowledge helping to learn procedural knowledge or vice versa. However, knowledge space model testing models of accuracy data provides support for the notion that learning procedural knowledge helps learning magnitude knowledge, and evidence against the reverse notion that learning magnitude knowledge helps learning procedural knowledge. Key Words: Magnitude; Procedure; Structural Equation Models; Knowledge Space Theory

In practice, mathematics education is blocked (i.e., teachingone topic at a time; CCSS, 2010), but research generallypromotes interleaving (i.e., teaching multiple topics together;Rohrer & Taylor, 2007). For example, fraction arithmetic isblocked with students being taught fraction addition beforefraction multiplication. Since students often confuse fractionoperations to produce arithmetic errors, interleaved fractionarithmetic instruction might be more productive than blockedinstruction to teach students to discriminate between theoperations. Additionally, a cognitive task analysis suggeststhat fraction multiplication may be a prerequisite to fractionaddition and thus reversing the blocking order may enhancelearning. Two experiments with fraction addition and fractionmultiplication were run. Experiments 1 and 2 show thatinterleaved instruction is generally better than the currentblocked instruction. Experiment 2 provides evidence thatblocking that reverses the standard order -- providing practiceon fraction multiplication before fraction addition -- producesbetter learning.

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.