Cross-Disciplinary Thematic Special Series

Working Group Background

In 2012, mathematics education and special education scholars have organized, for the first time in PME’s 30 years of activity, a Working Group within the organization’s international annual conference. As noted above, the purpose of this Working Group has been to explore issues at the intersection of mathematics education and special education. The Working Group members recognized that substantial research has been conducted on mathematical cognition, development, and reasoning of students in general education. However, much less is known about the development of mathematical thinking/reasoning in students with disabilities, let alone how to support their learning. The lacuna of research addressing this subset of students may be due in part to the incompatibility of theoretical perspectives that drive research, and practice, in each discipline.

The Working Group’s first meeting took place during the PME-NA 34th Annual Meeting in Kalamazoo, Michigan (see Thouless et al., 2012). It consisted of 15 researchers (faculty and graduate students) and two practitioners. That meeting was devoted to better understand mathematical learning disabilities in light of mathematics education research on students’ multiplicative and fractional reasoning. Since then, while capitalizing on previous and ongoing studies of the Working Group members, more collaborative research projects that integrated research-based practices from both fields have emerged. For instance, one such project focused on nurturing multiplicative reasoning of elementary students with LDM (Xin, Tzur, & Si, 2008–2015). Another such project documented learning trajectories of elementary school children with LDM as they come to understand fractions as quantities (Hunt, Tzur, & Westenskow, 2016; Tzur & Hunt, 2015). These preliminary collaborations have demonstrated promising learning outcomes of students with LDM, and led to two of the four papers in this Special Series. This Special Series was conceived of a way to expand such collaboration, deepen the conversation and understanding among researchers and professionals in both fields, and promote further dialogue and cross-disciplinary collaboration.

Article Foci

In this section, we provide a brief summary of each of the four papers that constitute this Special Series on the intersection of mathematics and special education. In the first article, Lewis explores the nature of and patterns in errors made by two adult students (at the age of 18 and 19) with mathematical learning disabilities who were engaged in solving fraction comparison problems. To this end, Lewis designed a study on the basis of a sociocultural framework, which focuses on “mediational” tools that support the child’s development of mathematics or language abilities. Lewis selected the participants based on a stringent criterion, namely, not only demonstrating low performance but also not responding to the mathematics instruction—four 1-hr long tutoring sessions she provided previously for fostering conceptual understanding of fraction equivalence as well as fraction addition and subtraction. Lewis collected rich data from this case study, which she analyzed both quantitatively and qualitatively. Her analyses on unique errors made by the two participants on fraction comparison problems shed light on atypical ways in which the adults with mathematical disabilities in her study understood and represented fractional quantities. Consequently, we gain potential insights into the error patterns that affected their performance, which can in turn provide direction for adapting the intervention for future research involving students who may understand fractions similarly.

Next, Hunt, Welch-Ptak, and Silva compare and contrast how students with LD and students diagnosed as at risk (receiving Tier II Response to Intervention program) for LD comprehend fractional quantities. Like Lewis, Hunt used qualitative and quantitative methodologies, but used a constructivist framework that utilizes problematic situations to solicit student activities/solutions and to document their understanding of targeted concepts. Hunt conducted individual, clinical interviews with 43 students in second, third, fourth, and fifth grades, and conducted conceptually focused analysis of each child’s solution to challenging situations. The results indicated students with LD and those with mathematics difficulties share similar, underdeveloped mathematical conceptions that are potentially malleable. Specifically, participants in her study viewed fractions as “parts within whole,” rather than as a multiplicative relation of “parts-to-whole” (i.e., a multiple of the unit fraction, n × 1/n). Hunt’s study makes an important contribution in that it focused not on what students do not know (e.g., errors, or “misconceptions”), but rather on what they do know. Her findings provide a potential evidence for the similarities in what different student populations within the larger category of struggling students (LDM, at risk) seem to experience and understand about fractions.

In the third article, Liu and Xin use a single-case design to study the effect of an intervention program that teaches mathematical reasoning and communication skills to three students with LD. Building on the framework of Conversational Repair (CR; attempt to address communicative breakdowns or inaccuracy by way of repeating what have been said or putting them in another way) borrowed from the field of linguistics, the intervention focused on eliciting not just or mainly correct answers, but rather self-explanations to solutions for mathematics word problems. The CR framework emphasizes the importance of expressing one’s own reasoning process in communication while using an implicit to explicit prompting system. Thus, it integrated a heuristic approach from mathematics education and an explicit strategy instruction, one of the evidence-based practices, from special education. The qualitative analyses of students’ reasoning and the quantitative analyses of their problem-solving performance pointed out to the substantial gains participants in this exploratory study made. Findings from this exploratory study have implications to including students with LDM in reformed-based mathematics instructions that engage students in explaining, clarifying, and justifying their thinking or problem-solving process. This study addresses an understudied problem, namely, the potential of how teaching students with LDM to reason and communicate about their solution strategies can enhance both understanding and word problem-solving outcomes.

The fourth article, Xin, Tzur, Hord, Liu, Park, and Si, reports on a study that used a randomized control trial design to test the effect of a math problem-solving intelligent tutor software program, PGBM-COMPS program (Xin et al., 2008–2015), on multiplicative reasoning and problem solving of third- and fourth-grade students with LDM. This software draws on (a) a constructivist view of learning from mathematics education—for example, the “Please go and bring for me. . . .” (PGBM; Tzur et al., 2013) games engage students in building equal-sized towers to nurture fundamental mathematical ideas such as number as a composite unit, and (b) an explicit strategy teaching of conceptual model-based problem solving (COMPS; Xin, 2012; Xin et al., 2012) from special education. Results of comparisons between a typical, school-based, teacher-taught group (control) and a tutor-taught group (experimental) of students with LDM indicated statistically significant gains in favor of the experimental group on both a researcher-developed multiplicative reasoning criterion test and a commercially published standardized test. The fact that students who learned through the PGBM-COMPS tutor substantially surpassed the control group students on the transfer measure is a vital indication of the importance of conceptual learning for supporting potential knowledge transfer.

Taken together, the four articles in this Special Series provide a window into the ongoing, dialogic work carried out by members of the PME Working Group. The intent is that these articles will spark further, substantial collaboration among scholars, as well as teachers, in both the field of mathematics and special education. Indeed, joining the Working Group is open to and warmly encouraged for all who are interested in such collaboration (http://www.igpme.org/). Finally, stay tuned to the Final Commentary to the Special Series (authored by John Woodward and Ron Tzur), which will appear in the next issue.