GSE Scholarly Articles

Background, context, and purpose of study: During the middle school years, students frequently show significant declines in motivation toward school in general and mathematics in particular. One way in which researchers have sought to spark students’ interests and build their sense of competence in mathematics and in STEM more generally is through the use of technology. Yet evidence regarding the motivational effectiveness of this approach is mixed. Here we evaluate the impact of three brief technology-based activities on students’ short-term motivation in math. 16,789 5th to 8th grade students and their teachers in one large school district were randomly assigned to three different technology-based activities, each representing a different framework for motivation and engagement and all designed around an exemplary lesson related to algebraic reasoning. We investigated the relationship between specific technology-based activities that embody various motivational constructs and students’ engagement in mathematics and perceived competence in pursuing STEM careers.

Discussions where teachers engage students in the comparison of multiple solution strategies to a single problem have been recommended in curriculum policy documents, yet integrating these discussions into teachers' normative routines is not widespread. In this paper, we begin to explore variations in teachers' implementation of Algebra I curriculum materials specifically focused on comparison. We explore (via case studies) implementation of the curriculum materials by two teachers with similar teaching backgrounds. The case studies suggest that these two teachers’ implementation of the comparison materials differed markedly, raising questions about possible factors which may have contributed to implementation differences.

Mastery of algebra is an important yet difficult milestone for students, suggesting the need for more effective teaching strategies in the algebra classroom. Learning by comparing worked-out examples of algebra problems may be one such strategy. Comparison is a powerful learning tool from cognitive science that has shown promising results in prior small-scale studies in mathematics classrooms. This study reports on a yearlong randomized controlled trial testing the effect of an Algebra I supplemental comparison curriculum on students’ mathematical knowledge. 141 Algebra I teachers were randomly assigned to either implement the comparison curriculum as a supplement to their regular curriculum or to be a ‘business as usual’ control. Use of the supplemental curriculum was much less frequent than requested for many teachers, and there was no main effect of condition on student achievement. However, greater use of the supplemental curriculum was associated with greater procedural student knowledge. These findings suggest a role for comparison in the algebra classroom but also the challenges of supporting teacher integration of new materials into the curriculum.

Current curriculum initiatives (e.g. National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010) advocate that models be used in the mathematics classroom. However, despite their apparent promise, there comes a point when models break, a point in the mathematical problem space where the model cannot, or arguable should not, be used. In this work, we explore the breaking point of the chip model for integer subtraction and the area model for fraction addition. Breaking is inevitable – either because no one model is robust enough to be applicable in a very large problem space and/or because the modifications required to keep the model functioning weaken or even eliminate its benefits. While models are often intended to serve as visual illustrations or embodiments of a concept, adaptions at the model breaking point can turn model use into nothing more than executing the graphical analogue to a not-well-understood procedure. The act of identifying model breaking points illuminates the affordances and constraints of the model. This provides students a unique opportunity to discriminate across mathematical models to develop a meta-level understanding of the relationship between models and the mathematics those models are intended to support.

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.

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.

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.

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.

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.