Using Strategic Interruptions to Effectively Integrate Whole Class and Small Group Instruction in Mathematics

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

Teachers' views about multiple strategies in middle and high school mathematics: Perceived advantages, disadvantages, and reported instructional practices

Despite extensive scholarship about the importance of teaching mathematics with multiple strategies in the elementary grades, there has been relatively little discussion of this practice in the middle and high school levels or in the context of introductory algebra. This paper begins our exploration of this practice by addressing the following questions: (1) What do middle and high school Algebra I teachers describe as the advantages of instruction that includes a focus on multiple strategies?; and (2) What disadvantages to this practice do these teachers describe?. Our analysis, based on the data from interviews (N=13) and surveys (N=79) conducted with experienced middle and secondary mathematics teachers, indicates that middle and secondary math teachers’ reported views surrounding multiple strategies appear to differ in important ways from those typically associated with teaching with multiple strategies in the elementary grades

Views of struggling students on instruction incorporating multiple strategies in Algebra I: An exploratory study

Although policy documents promote teaching students multiple strategies for solving mathematics problems, some practitioners and researchers argue that struggling learners will be confused and overwhelmed by this instructional practice. In the current exploratory study, we explore how six struggling students viewed the practice of learning multiple strategies at the end of a yearlong algebra course that emphasized this practice. Interviews with these students indicated that they preferred instruction with multiple strategies to their regular instruction, often noting that it reduced their confusion. We discuss directions for future research that emerged from this work

It pays to compare: An experimental study on computational estimation

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

Compared to what? The effects of different comparisons on conceptual knowledge and procedural flexibility for equation solving

Researchers in both cognitive science and mathematics education emphasize the importance of comparison for learning and transfer. However, surprisingly little is known about the advantages and disadvantages of what types of things are being compared. In this experimental study, 162 7th- and 8th-grade students learned to solve equations by comparing equivalent problems solved with the same solution method, by comparing different problem types solved with the same solution method, or by comparing different solution methods to the same problem. Students' conceptual knowledge and procedural flexibility were best supported by comparing solution methods, and to a lesser extent by comparing problem types. The benefits of comparison are augmented when examples differ on relevant features, and contrasting methods may be particularly useful in mathematics learning

Meeting the Needs of Students with Learning Disabilities in Inclusive Mathematics Classrooms: The Role of Schema-Based Instruction on Mathematical Problem-Solving

In this article, we discuss schema-based instruction (SBI) as an alternative to traditional instruction for enhancing the mathematical problem solving performance of students with learning disabilities (LD). In our most recent research and developmental efforts, we designed SBI to meet the needs of middle school students with LD in inclusive mathematics classrooms by addressing the research literatures in special education, cognitive psychology, and mathematics education. This innovative instructional approach encourages students to look beyond surface features of word problems to grasp the underlying mathematical structure of ratio and proportion problems. In addition, SBI introduces students to multiple strategies for solving ratio and proportion problems and encourages the selection of appropriate strategies

Procedural and Conceptual Knowledge: Exploring the Gap Between Knowledge Type and Knowledge Quality

Following Star (2005, 2007) we continue to problematize the entangling of type and quality in the use of conceptual knowledge and procedural knowledge. Although those whose work is guided by types of knowledge and those whose work is guided by qualities of knowledge seem to be referring to the same phenomena, actually they are not. This lack of mutual understanding of both the nature of the questions being asked and the results being generated causes difficulties for the continued exploration of questions of interest in mathematics teaching and learning, such as issues of teachers’ knowledge

The nature and development of experts' strategy flexibility for solving equations

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