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

Many studies investigating inquiry learning in science domains have appeared over the years. Throughout this period, inquiry learning has been regularly criticized by scholars who favor direct instruction over inquiry learning. In this vein, Zhang, Kirschner, Cobern, and Sweller (2022) recently asserted that direct instruction is overall superior to inquiry-based instruction and reproached policy makers for ignoring this fact. In the current article we reply to this assertion and the premises on which it is based. We review the evidence and argue that a more complete and correct interpretation of the literature demonstrates that inquiry-based instruction produces better overall results for acquiring conceptual knowledge than does direct instruction. We show that this conclusion holds for controlled, correlational, and program-based studies. We subsequently argue that inquiry-based and direct instruction each have their specific virtues and disadvantages and that the effectiveness of each approach depends on moderating factors such as the learning goal, the domain involved, and students' prior knowledge and other student characteristics. Furthermore, inquiry-based instruction is most effective when supplemented with guidance that can be personalized based on these moderating factors and can even involve providing direct instruction. Therefore, we posit that a combination of inquiry and direct instruction may often be the best approach to support student learning. We conclude that policy makers rightfully advocate inquiry-based instruction, particularly when students’ investigations are supplemented with direct instruction at appropriate junctures.