Developing Concepts and Generalizations to Build Algebraic Thinking

Abstract

Many students with learning disabilities (LD) in mathematics receive their mathematics education in general education inclusive classes; therefore, these students must be capable of learning algebraic concepts, including developing algebraic thinking abilities, that are part of the general education curriculum. To help students develop algebraic thinking, teachers should ask questions in different ways to promote the ability to think algebraically. This article describes three types of questions—reversibility, flexibility, and generalizations—which support the acquisition of broader concepts leading to algebraic thinking. Examples of the question types within the contexts of rational numbers and integers are provided to assist teachers in creating similar questions for teaching mathematics to students with LD.

Helping Students With Mathematics Difficulties Understand Ratios and Proportions

“…Students with persistent mathematics difficulties benefit from learning mathematics more deeply through instruction that includes three types of questions—reversibility, flexibility, and generalization—that are important for intensive instruction (Dougherty, Bryant, Bryant, Darrough, & Pfannenstiel; see Table 1 for examples). Reversibility questions give students the answer and then they create the question.…”

Exploring the Potential Role of Reversible Reasoning: Cognitive Research on Inverse Function Problems in Mathematics

“…Currently, fewer literature has been found related to reversible reasoning investigations in conceptual relationships for inverse function problems. The focus of previous researchers are more on the operational aspects, for example by identifying error reversals that students make for the problem of "students and professors" (González-Calero, Arnau, & Laserna-Belenguer, 2015;Soneira, González-Calero, & Arnau, 2018;Tunç-Pekkan, 2015), reversible multiplication relationships (Hackenberg, 2010), cognitive conflict and insufficient mental processes to reverse problem situations (Ramful, 2014), the type of task that causes reversible reasoning (B. Dougherty, Bryant, Bryant, & Shin, 2017;B. J. Dougherty, Bryant, Bryant, Darrough, & Pfannenstiel, 2015;Sangwin & Jones, 2017;Simon, Kara, et al, 2016;Vilkomir & O'Donoghue, 2009).…”

Technical Adequacy of Procedural and Conceptual Algebra Screening Measures in High School Algebra

“…These areas were chosen because of their foundational importance to algebra and because they were not well represented in the existing procedural measures. Within each domain, items were created using three processes identified by Krutetskii (1976), a Russian psychologist, and adapted by the Curriculum Research & Development Group (Rachlin, 1998) and others (Dougherty, Bryant, Bryant, Darrough, & Pfannenstielf, 2015).…”

“…We drew upon these question types as we developed items for the conceptual measures (c.f., Dougherty et al, 2015). We classified items as representing generalization, flexibility, and reversibility.…”