Students often struggle with concepts from abstract algebra. Typical classes incorporate few ways to make the concepts concrete. Using a set of woven paper artifacts, this paper proposes a way to visualize and explore concepts (symmetries, groups, permutations, subgroups, etc.). The set of artifacts used to illustrate these concepts is derived…

Making sense of mathematical concepts and solving mathematical problems may demand different forms of reasoning. These could be either domain-based, such as algebraic, geometric or statistical reasoning, while others are more general such as inductive/deductive reasoning. This article aims at giving visibility to a particular form of reasoning…

Mr. Buxton is a perplexed elementary mathematics teacher. He co-teaches a second-grade classroom, with Ms. Snyder. In their classroom they have 25 students; five are identified as academically at risk, and three receive special education services. In the past Mr. Buxton successfully used an instructional approach consisting of (a) modeling, (b)…

This article explores some of the common mathematical difficulties that 11- to 16-year-old students experience with respect to their learning of physics. The definition of "understanding" expressed in the article is in the sense of transferability of mathematical skills from topic to topic within physics as well as between the separate…

The mathematical proficiencies in the "Australian Curriculum: Mathematics" of understanding, problem solving, reasoning, and fluency are intended to be entwined actions that work together to build generalised understandings of mathematical concepts. A content analysis identifying the incidence of key proficiency terms (KPTs) embedded in…

Although there is no consensus in regard to a unique meaning for abstraction, there is a recognition of the existence of several theories of abstraction, and that the ability to abstract is imperative to learning and doing meaningful mathematics. The theory of "reducing abstraction" maps the abstract nature of mathematics to the nature…

The purpose of this paper is twofold: On the one hand, this work frames a variety of considerations on cognitive processes underlying mathematical concept construction in two research strands, namely an actions-first strand and an objects-first strand, that mainly shapes past and current approaches on abstraction in learning mathematics. This…

Conventions play an important communicative role in mathematics. Likely due to the complex relationship between conventions and school mathematics, few education researchers have questioned or investigated the consequences of instruction and curricula that primarily, if not unquestionably, maintain conventions. Drawing on Piagetian notions of…

This article explores one segment of an extended research and development project that was conducted to better understand the ways online teacher professional development can support teachers' development of deep and connected mathematical understandings. In particular, this article discusses teachers' understandings of the concept of…

This paper uses a comparative methodology to examine the teaching of abstraction in two mathematics lessons, in Denmark and England. In doing so it aims to extend previous work by the authors, examining the effect of local, cultural issues on the form of teaching in order to understand how these also affect the subject content too. The analysis…

In this study, we follow students' modeling and covariational reasoning in the context of learning about big data. A three-week unit was designed to allow 12th grade students in a mathematics course to explore big and mid-size data using concepts such as trend and scatter to describe the relationships between variables in multivariate settings.…

This article explores the potential for aspects of abstract algebra to be influential for the teaching of school algebra (and early algebra). Using national standards for analysis, four primary areas common in school mathematics--and their progression across elementary, middle, and secondary mathematics--where teaching may be transformed by…

Ratio, rate, and proportion are central ideas in the Common Core State Standards (CCSS) for middle-grades mathematics (CCSSI 2010). These ideas closely connect to themes in earlier grades (pattern building, multiplicative reasoning, rational number concepts) and are the foundation for understanding linear functions as well as many high school…

The presented study used a problem-solving experience in engineering design with LEGO robotics materials as the real-world mathematics-learning context. The goals of the study were (a) to determine if a short but intensive extracurricular learning experience would lead to significant student learning of a particular academic topic and (b) to…

In this paper, we discuss the process of identifying and validating students' abilities to think proportionally. More specifically, we describe the methodology we used to identify these proportional reasoning attributes, beginning with the selection and review of relevant literature on proportional reasoning. We then continue with the…