Incremental rehearsal (IR) has consistently been shown to improve students' math fact retention and fluency (Maki et al., Journal of Behavioral Education 30:534-558, 2021). However, less is known about how intervention modifications may support longer-term skill maintenance. The purpose of this study was to compare traditional IR with a modified…

The exploration of children's drawings as mathematical representations is a current focus in early years mathematics education research. This paper presents a qualitative analysis of 72 kindergarten to Grade 3 (5 to 8 years old) children's drawings produced during problem-solving tasks centred on multiplicative strategies. Existing frameworks for…

Developing and supporting understanding of the meaning of multiplication and multiplicative relationships in students with mathematics learning disabilities requires carefully designed instruction that emphasizes strategic representation use. This article discusses three ways in which teachers can incorporate multiple representations within…

This paper reports findings related to the pursuit of describing a lived knowing for teaching mathematics. Specifically, we used an enactivist perspective to describe the knowing exhibited by three experienced high school teachers in their instructional actions while leading whole-class instruction on the topic of the equations of exponential…

Over the last years, there is a growing interest in studying students' difficulties with rational numbers from a cognitive/developmental perspective, focusing on the role of prior knowledge in students' understanding of rational numbers. The present study tests the effect of the "whole" or "natural number bias" (i.e., the…

This paper discusses findings from an ongoing study investigating mental mechanisms involved in the conceptualisation of linear transformations from the perspective of Action (A), Process (P), Object (O), and Schema (S) (APOS) theory. Data reported in this paper came from 44 first-year linear algebra students' responses on a task regarding the…

Students entering sixth grade operate with three different multiplicative concepts that influence their reasoning in many domains important for middle school. For example, students who are operating with the second multiplicative concept (MC2 students) can begin to construct fractions as lengths but do not construct improper fractions as numbers.…

This paper builds on our previous research and investigates how students' fractional competence and reasoning can provide clear evidence of non-symbolic algebraic thinking and its progressive transition towards fully generalised algebraic thinking. In a large-scale study, 470 primary students completed a written paper and pencil test. This…

In the present study, we illuminate students' multiplicative reasoning in the context of their units-coordinating activity. Of particular interest is to investigate students' use of three levels of units as given material for problem-solving activity, which we regard as supporting a more advanced level of multiplicative reasoning. Among 13 middle…

Mathematical coherence is a goal within the Common Core State Standards for Mathematics. One aspect of this coherence is how student mathematical thinking is developed across concepts. Unfortunately, mathematics is often taught as isolated ideas across grades. The multiplicative field is an area of study that needs to be examined as a space to…

Students in developmental math classes have been documented to struggle with arithmetic. This dissertation research was a response to the calls in the literature to qualitatively understand what students are learning in a classroom. Number sequences have been well researched to document the connection of student understanding of math in elementary…

The learning of basic facts, or the sums and products of numbers 0-10 and their related differences and quotients, has always been a high priority for elementary school teachers. While memorization of basic facts has been a hallmark of elementary school, current recommendations focus on a more nuanced development of fluency with these facts.…

The algorithm for fraction multiplication is relatively easy to memorize and implement with accuracy. The simplicity of the algorithm masks the conceptual complexity involved in making sense of what fraction multiplication means in quantitative situations. One important interpretation of fraction multiplication involves fraction composition,…

Visuospatial representations of numbers and their relationships are widely used in mathematics education. These include drawn images, models constructed with concrete manipulatives, enactive/embodied forms, computer graphics, and more. This paper addresses the analytical limitations and ethical implications of methodologies that use broad…

This study examines the multiplicative problem-solving strategies used by a 14-year-old student with autism spectrum disorder and intellectual disabilities during an instructional process based on the Conceptual Model-based Problem Solving (COMPS) approach. The instruction aimed to enhance conceptual comprehension of problem-solving by the use of…