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Star, Jon

This study investigated the effectiveness of an instructional program (schema-based instruction, SBI) designed to teach 7th graders how to comprehend and solve proportion problems involving ratios/rates, scale drawings, and percents. The SBI program emphasized the underlying mathematical structure of problems via schematic diagrams, focused on a 4-step procedure to support and monitor problem solving, and addressed the flexible use of alternative solution strategies based on the problem situation. Blocking by teacher at three middle schools, the authors randomly assigned the 21 classrooms to one of two conditions: SBI and control. Classroom teachers provided the instruction. Results of multilevel modeling used to test for treatment effects after accounting for pretests and other characteristics (gender, ethnicity) revealed the direct effects of SBI on mathematical problem solving at posttest. However, the improved problem solving skills were not maintained a month later when SBI was no longer in effect nor did the skills transfer to solving problems in new domain-level content.

In this paper we explore a new way to think about the use of group work in mathematics instruction through what we refer to as strategic interruptions. Strategic interruptions involve frequent and often rapid transitions between whole class and small group instruction. Through analyses of video of Algebra I teaching, we identify patterns in the frequency, timing, rationale, and instructional practices related to the use of and switching between whole class and small group instructional formats. We postulate that use of strategic interruptions has the potential to be a powerful and easily implementable form of group work that may be especially appropriate in secondary classrooms.

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 estimation either by comparing alternative solution strategies or by reflecting on the strategies one at a time. At posttest and retention test, students who compared were more flexible problem solvers on a variety of measures. Comparison also supported greater conceptual knowledge, but only for students who already knew some estimation strategies. These findings indicate that comparison is an effective learning and instructional practice in a domain with multiple acceptable answers.

Largely absent from the emerging literature on flexibility is a consideration of experts' flexibility. Do experts exhibit strategy flexibility, as one might assume? If so, how do experts perceive that this capacity developed in themselves? Do experts feel that flexibility is an important instructional outcome in school mathematics? In this paper, we describe results from several interviews with experts to explore strategy flexibility for solving equations. We conducted interviews with eight content experts, where we asked a number of questions about flexibility and also engaged the experts in problem solving. Our analysis indicates that the experts that were interviewed did exhibit strategy flexibility in the domain of linear equation solving, but they did not consistently select the most efficient method for solving a given equation. However, regardless of whether these experts used the best method on a given problem, they nevertheless showed an awareness of and an appreciation of efficient and elegant problem solutions. The experts that we spoke to were capable of making subtle judgments about the most appropriate strategy for a given problem, based on factors including mental and rapid testing of strategies, the problem solver's goals (e.g., efficiency, error-free execution, elegance) and familiarity with a given problem type. Implications for future research on flexibility and on mathematics instruction are discussed.

Taking early action may be key to helping students struggling with mathematics. The eight recommendations in this guide are designed to help teachers, principals, and administrators use Response to Intervention for the early detection, prevention, and support of students struggling with mathematics.

The ability to estimate is a fundamental real-world skill; it allows students to check the reasonableness of answers found through other means, and it can help students develop a better understanding of place value, mathematical operations, and general number sense. Flexibility in the use of strategies is particularly critical in computational estimation. The ability to perform complex calculations mentally is cognitively challenging for many students; thus it is important to have a broad repertoire of estimation strategies and to select the most appropriate strategy for a given problem. In this paper, we consider the role of students' prior knowledge of estimation strategies in the effectiveness of interventions designed to promote strategy flexibility across two recent studies. In the first, 65 fifth graders began the study as fluent users of one strategy for computing mental estimates to multi-digit multiplication problems such as 17 x 41. In the second, 157 fifth and sixth graders began the study with moderate to low prior knowledge of strategies for computing mental estimates. Results indicated that students' fluency with estimation strategies had an impact on which strategies they adopted. Students who exhibited high fluency at pretest were more likely to increase use of estimation strategies that led to more accurate estimates, while students with less fluency adopted strategies that were easiest to implement. Our results suggest that both the ease and accuracy of strategies as well as students’ fluency with strategies are all important factors in the development of strategy flexibility.

This study evaluated whether schema-based instruction (SBI), a promising method for teaching students to represent and solve mathematical word problems, impacted the learning of percent word problems. Of particular interest was the extent that SBI improved high- and low-achieving students' learning and to a lesser degree on the indirect effect of SBI on transfer to novel problems, as compared to a business as usual control condition. Seventy 7th grade students in four classrooms (one high- and one low-achieving class in both the SBI and control conditions) participated in the study. Results indicate a significant treatment by achievement level interaction, such that SBI had a greater impact on high-achieving students' problem solving scores. However, findings did not support transfer effects of SBI for high-achieving students. Implications for improving the problem-solving performance of low achievers are discussed.