The study reported on in this paper is an interview study conducted with 20 7th and 8th grade students whose purpose was to understand the generalizations they could make about non-linear meanings of multiplication (NLMM) and non-linear growth (NLG) in the context of solving combinatorics problems. The paper identifies productive challenges for…

Generalization has been identified as a cornerstone of algebraic thinking (e.g., Lee, 1996; Sfard, 1995) and is at the center of a rich conceptualization of K-8 algebra (Kaput, 2008; Smith, 2003). Moreover, mathematics teachers are being encouraged to use figural-pattern generalizing tasks as a basis of student-centered instruction, whereby…

Several types of converging evidence have suggested recently that skilled adults solve very simple addition problems (e.g., 2 + 1, 4 + 2) using a fast, unconscious counting algorithm. These results stand in opposition to the long-held assumption in the cognitive arithmetic literature that such simple addition problems normally are solved by fact…

This study examined the utility of brief academic assessments to identify effective generalization procedures for individual students. Specifically, the study built on the proposal that brief assessments of antecedent and consequence manipulations can identify the most effective generalization strategy for individual students. The design was an…

There is a need to teach the pivotal skill of mathematical problem solving to students with severe disabilities, moving beyond basic skills like computation to higher level thinking skills. Problem solving is emphasized as a Standard for Mathematical Practice in the Common Core State Standards across grade levels. This article describes a…

The effectiveness of a multicomponent intervention to improve the problem-solving performance of students with autism spectrum disorders (ASD) during vocational tasks was examined. A multiple-probe across-students design was used to illustrate the effectiveness of point-of-view video modeling paired with practice sessions and a self-operated cue…

According to Every Student Succeeds Act, schools are accountable for the improvement of all children; including those with disabilities or difficulties, recently arrived English language learners (ELs) as well as long-term ELs, and all students for whom English is not their native language. To meet the accountability requirements, it is essential…

This article presents "The Two Children Problem," published by Martin Gardner, who wrote a famous and widely-read math puzzle column in the magazine "Scientific American," and a problem presented by puzzler Gary Foshee. This paper explains the paradox of Problems 2 and 3 and many other variations of the theme. Then the authors…

This study used a multiple probe across participants design, replicated across two interventions and counterbalanced across participant groups to examine the effects of number sense intervention and extensive practice intervention on strategy transformation when students with mathematics learning disabilities (MLD) solved basic fact problems. In…

Algebra has been explicit in many school curriculum programs from the early years but there are competing views on what content and approaches are appropriate for different levels of schooling. This study investigated 12-13-year-old Australian students' algebraic thinking in a hybrid environment of functional and equation-based approaches to…

The No Child Left Behind Act (NCLB, 2006) set a precedent that established even higher expectations for all students, including those with disabilities. More recently, the National Governors Association Center for Best Practices and the Council of Chief State School Officers developed a common set of state standards for proficiency in English…

This paper reports on the results of a study investigating the potential to embed Informal Statistical Inference in statistical investigations, using TinkerPlots, for assisting 8th grade students' informal inferential reasoning to emerge, particularly their articulations of uncertainty. Data collection included students' written work on a…

A well-known mathematical puzzle regarding a worm crawling along an elastic rope is considered. The resulting generalizations provide examples for use in a teaching context including applications of series summation, the use of the integrating factor for the solution of differential equations, and the evaluation of definite integrals. A number of…

How many rods does it take to brace a square in the plane? Once Martin Gardner's network of readers got their hands on it, it turned out to be fewer than Raphael Robinson, who originally posed the problem, thought. And who could have predicted the stunning solutions found subsequently for various generalizations of the problem?

As an alternative to looking solely at linear functions, a three-lesson learning progression developed for Year 6 students that incorporates triangular numbers to develop children's algebraic thinking is described and evaluated.