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

Strategy flexibility in university mathematics

We present results and ideas on flexibility and its teaching for university mathematicians based on research and our experience in professional development programs. We believe this could provide food for thought for instructors of calculus courses and beyond.Comment: An earlier version of the paper was titled "Strategic flexibility in university mathematics

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

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

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

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