Algebra

In this paper, we share the design and impact of a set of two-hour online mathematics professional development modules adapted from face-to-face video-based materials.

We investigate the algebraic discourse of secondary mathematics teachers with respect to the topic of equation solving by analyzing five teachers’ responses to open-ended items on a questionnaire that asks respondents to analyze hypothetical student work related to equation solving and explain related concepts.

The ReLaTe-SA project investigates the research question: what language do teachers use to describe and explain routines in algebra classes? The goal of this article is to inform readers about some ways we have learned to describe the discourse that teachers use when solving linear equations.

The researchers conducted a qualitative analysis of the perceptions of school personnel and pre-service teachers about an Algebra I tutoring program for students with learning disabilities. The researchers surveyed and interviewed the participants about the effectiveness of the program for the mathematics learning of the students with LD at the school and as a learning experience for the pre-service teachers.

This article addresses the nature of student-generated representations that support students’ early algebraic reasoning in the realm of generalized arithmetic.

This book aims to advance ongoing debates in the field of mathematics and mathematics education regarding conceptions of argumentation, justification, and proof and the consequences for research and practice when applying particular conceptions of each construct. Through analyses of classroom practice across grade levels using different lenses - particular conceptions of argumentation, justification, and proof - researchers consider the implications of how each conception shapes empirical outcomes. In each section, organized by grade band, authors adopt particular conceptions of argumentation, justification, and proof, and they analyse one data set from each perspective. In addition, each section includes a synthesis chapter from an expert in the field to bring to the fore potential implications, as well as new questions, raised by the analyses. Finally, a culminating section considers the use of each conception across grade bands and data sets.

Several recent studies have focused on helping students understand the limitations of empirical arguments (e.g., Stylianides, G. J. & Stylianides, A. J., 2009, Brown, 2014). One view is that students use empirical argumentation because they hold empirical proof schemes—they are convinced a general claim is true by checking a few cases (Harel & Sowder, 1998). Some researchers have sought to unseat students’ empirical proof schemes by developing students’ skepticism, their uncertainty about the truth of a general claim in the face of confirming (but not exhaustive) evidence (e.g., Brown, 2014; Stylianides, G. J. & Stylianides, A. J., 2009). With sufficient skepticism, students would seek more secure, non-empirical arguments to convince themselves that a general claim is true. We take a different perspective, seeking to develop students’ awareness of domain appropriateness (DA), whether the argument type is appropriate to the domain of the claim. In particular, DA entails understanding that an empirical check of a proper subset of cases in a claim’s domain does not (i) guarantee the claim is true and does not (ii) provide an argument that is acceptable in the mathematical or classroom community, although checking all cases does both (i) and (ii). DA is distinct from skepticism; it is not concerned with students’ confidence about the truth of a general claim. We studied how ten 8th graders developed DA through classroom experiences that were part of a broader project focused on developing viable argumentation. 

Students’ difficulties with argumentation, proving, and the role of counterexamples in proving are well documented. Students in this study experienced an intervention for improving their argumentation and proving practices. The intervention included the eliminating counterexamples (ECE) framework as a means of constructing and critiquing viable arguments for a general claim. This framework involves constructing descriptions of all possible counterexamples to a conditional claim and determining whether or not a direct argument eliminates the possibility of counterexamples. This case study investigates U.S. eighth-grade (age 13) mathematics students’ conceptions about the validity of a direct argument after the students received instruction on the ECE framework. We describe student activities in response to the intervention, and we identify students’ conceptions that are inconsistent with canonical notions of mathematical proving and appear to be barriers to using the ECE framework.

Students’ difficulties with contrapositive reasoning are well documented. Lack of intuition about contrapositive reasoning and lack of a meta-argument for the logical equivalence between a conditional claim and its contrapositive may contribute to students’ struggles. This case study investigated the effectiveness of the eliminating counterexamples intervention in improving students’ ability to construct, critique, and validate contrapositive arguments in a U.S. eighth-grade mathematics classroom. The intervention involved constructing descriptions of all possible counterexamples to a conditional claim and its contrapositive, comparing the two descriptions, noting that the descriptions are the same barring the order of phrases, and finding a counterexample to show the claim is false or viably arguing that no counterexample exists.

This presntation addreses 4 research cquestions