Kyle, a kindergartner from a low-income family, is shown a set of three black dots on a white mat. His teacher then hides the dots with a small box lid and lays down an additional set of two dots. She pushes the two dots under the cover, one at a time. Kyle must now choose the number of dots "hiding" under the box from a set of four…

While understanding rational numbers forms an essential part of mathematical literacy, research has repeatedly shown that they form a stumbling block in education. A big source of difficulty is the inappropriate application of natural number knowledge. The research literature points at three main aspects where natural number knowledge is…

The natural number bias is known to explain many difficulties learners have with understanding rational numbers. The research field distinguishes three aspects where natural number properties are sometimes inappropriately applied in rational number tasks: density, size, and operations. The overall goal of this study was to characterize the…

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…

Usually a student learns to solve a system of linear equations in two ways: "substitution" and "elimination." While the two methods will of course lead to the same answer they are considered different because the thinking process is different. In this paper the author solves a system in these two ways to demonstrate the…

Despite convincing research and strong public support, education budget cuts have forced schools and districts to limit or even eliminate arts instruction. Author Robin Ward argues that educators need to address the critical need to preserve and, in those cases, re-integrate the arts back into the school curriculum--mostly because the arts are…

Rational number interpretations can include part-whole, measure, ratio, quotient, and operator. These are all subconstructs of partitioning (Barnett-Clarke et al. 2010; Behr et al. 1980; Clarke, Roche, and Mitchell 2008; Flores, Samson, and Yanik 2006). Each of these subconstructs uses different cognitive skills (Driscoll 1984), so it is important…

The process of prime factor splicing to generate home primes raises opportunity for conjecture and exploration. The notion of "home primes" is relatively new in the chronicle of mathematics. Heleen (1996-97) first described a procedure called "prime factor splicing" (PFS). The exploration of home primes is interesting and…

Our research focuses on the learning of series as a consequence of institutional choices for their teaching. Our analyses of textbooks and teaching practices led us to conjecture the existence of some implicit contract rules in the teaching of series: in particular, the teaching of series is made almost exclusively in the algebraic setting, with…

The main goal of this paper is to show how Vasily Davydov's powerful ideas about the nature of mathematical thinking and learning can transform the teaching and learning of additive word problem solving. The name Vasily Davydov is well known in the field of mathematics education in Russia. However, the transformative value of Davydov's theoretical…

Multiplicative thinking has been widely accepted as a critically important "big idea" of mathematics and one which underpins much mathematical understanding beyond the primary years of schooling. It is therefore of importance to consider the capacity of children to think multiplicatively but also to consider the capacity of their…

This paper explores how a young child (56 m) builds an understanding of the cardinality principle through communicative, touchscreen-based activities involving talk, gesture and body engagement working via multimodal, touchscreen interface using contemporary mobile technology. Drawing upon Nemirovsky's perceptuomotor integration theoretical lens…

In contrast to previous studies on Spontaneous Focusing on Quantitative Relations (SFOR), the present study investigated not only the "extent" to which children focus on (multiplicative) quantitative relations, but also the "nature" of children's quantitative focus (i.e., the types of quantitative relations that children focus…

This paper describes and analyses the implementation of a "flipped classroom" approach, in an undergraduate mathematics course on numerical methods. The approach replaced all the lecture contents by instructor-made videos and was implemented in the consecutive years 2014 and 2015. The sequential case study presented here begins with an…

This research explores the interplay between students' understandings of proportional and functional relationships. Approximately 90 students participated in an early algebra intervention in Grades 3- 5. Before the intervention and after each year of the intervention, we evaluated their understandings of proportional and functional relationships.…