Comparison is a powerful and important way that we learn. To support teachers in the use of comparison in their instruction, the authors developed an instructional routine called compare and discuss multiple strategies (CDMS). Similar to other instructional routines, CDMS is a structured, specific, repeatable minilesson that teachers can use to…

Background: The format of a mathematics problem often influences students' problem-solving performance. For example, providing diagrams in conjunction with story problems can benefit students' understanding, choice of strategy, and accuracy on story problems. However, it remains unclear whether providing diagrams in conjunction with symbolic…

Background: Students, parents, teachers, and theorists often advocate for direct instruction on both concepts and procedures, but some theorists suggest that including instruction on procedures in combination with concepts may limit learning opportunities and student understanding. Aims: This study evaluated the effect of instruction on a math…

Feedback is a ubiquitous learning tool that is theorized to help learners detect and correct their errors. The goal of this study was to examine the effects of feedback in a classroom context for children solving math equivalence problems (problems with operations on both sides of the equal sign). The authors worked with children in 7 second-grade…

Engaging learners in exploratory problem-solving activities prior to receiving instruction (i.e., explore-instruct approach) has been endorsed as an effective learning approach. However, it remains unclear whether this approach is feasible for elementary-school children in a classroom context. In two experiments, second-graders solved mathematical…

Background: A key learning outcome in problem-solving domains is the development of procedural flexibility, where learners know multiple procedures and use them appropriately to solve a range of problems (e.g., Verschaffel, Luwel, Torbeyns, & Van Dooren, 2009). However, students often fail to become flexible problem solvers in mathematics. To…

An emerging consensus suggests that guided discovery, which combines discovery and instruction, is a more effective educational approach than either one in isolation. The goal of this study was to examine two specific forms of guided discovery, testing whether conceptual instruction should precede or follow exploratory problem solving. In both…

Comparing and contrasting examples is a core cognitive process that supports learning in children and adults across a variety of topics. In this experimental study, we evaluated the benefits of supporting comparison in a classroom context for children learning about computational estimation. Fifth- and sixth-grade students (N = 157) learned about…

Providing exploratory activities prior to explicit instruction can facilitate learning. However, the level of guidance provided during the exploration has largely gone unstudied. In this study, we examined the effects of 1 form of guidance, feedback, during exploratory mathematics problem solving for children with varying levels of prior domain…

People remember information better if they generate the information while studying rather than read the information. However, prior research has not investigated whether this generation effect extends to related but unstudied items and has not been conducted in classroom settings. We compared third graders' success on studied and unstudied…

Encouraging students to share and compare solution methods is a key component of reform efforts in mathematics, and comparison is emerging as a fundamental learning mechanism. To experimentally evaluate the effects of comparison for mathematics learning, the authors randomly assigned 70 seventh-grade students to learn about algebra equation…

We present a methodology for designing better learning environments. In Phase 1, 6th-grade students' (n = 223) prior knowledge was assessed using a difficulty factors assessment (DFA). The assessment revealed that scaffolds designed to elicit contextual, conceptual, or procedural knowledge each improved students' ability to add and subtract…