Two-hour long developmental teaching interviews were conducted with each of 14 sixth grade students, ages 11-12. The purposes of the interviews were to investigate how students solved arrangement problems (APs), and how their solutions of these problems differed from their solution of Cartesian product problems (CPPs). The 14 students represented…

Unitizing, a cognitive support for number sense, describes how students conceptualize quantities in terms of units. Given the importance of this skill, a formative assessment tool was developed for array multiplication problems, including arrays that model the distributive property found in typical classroom curricula. Video of 77 fourth grade…

Quantitative reasoning plays a crucial role in students' and teachers' successful modeling activities. In a semester-long teaching experiment with an undergraduate student, we explore how her conception of a graph plays a role in her ability to quantify and maintain quantitative structures. We characterize here Lydia's conception of a graph as one…

In this paper, we explore how students' algebraic noticing's and explanations changed across a two-year period with the introduction of designed instructional material. The data in this report is drawn from n=53 Year 7-8 students' responses to a free-response assessment task across two different years. Analysis focused on how students noticed and…

Whole number computations are a critical skill that serves as a foundation upon which higher-order concepts in mathematics are taught to children. To facilitate their instruction, educators often use multiple representations to support a child's cognition. Representations with physical manipulatives are widely studied through a graduated…

Conceptual model-based problem solving (COMPS) was tested for its efficacy in teaching a student diagnosed with autism spectrum disorder to solve word problems involving multiplication and division. A single-case, multiple-baseline across behaviors design was conducted. The ability to solve each of three types of multiplication problems examined…

In the current study, it is aimed to determine the activities that need to be done to eliminate the mistakes made by primary school fourth grade students in multiplication and division operations and to present solution suggestions for eliminating these mistakes. The study employed action research, one of the qualitative research methods. The…

In three experiments, we used a masked prime in a verification task to investigate the processing stages occurring during multiplication fact retrieval. We aimed to investigate the retrieval process by overlapping its execution with the processing of a masked prime consisting of a number. Participants evaluated the correctness of multiplication…

This study investigates college students' understanding of arithmetic fraction operations as defined by the Common Core State Standards in grades three through six Mathematics. This study is meant to extend upon an extensive body of research regarding elementary and middle grades students' understanding of fraction operations and see if recognized…

A grandmother attempts to teach her four-year-old granddaughter the multiplication tables using simple repetition, but they repeatedly start over at 'three fives'; the child keeps coming up with 'thirty-five'. We consider three possible explanations: self-perpetuating frequency of behavior, saliency of memory and Vygotsky's next or proximal zones…

This article examines the mathematics learning experiences in proportional reasoning I created for pre-service teachers in my teacher education course in Canada. The approach aimed to enable pre-service teachers to work on and with their prior mathematics understandings, make connections that were particular to what emerged during group work, and…

This paper aims to explore how three secondary school teachers make sense of multiplication based on their conceptions of multiplication. The framework of supportive and problematic conceptions in making sense of mathematics proposed by Chin (2013) is exemplified in this study. Initially, data were collected through a survey assessment that…

Whole-number arithmetic is a core content area of primary mathematics, which lays the foundation for children's later conceptual development. This paper focuses on teaching whole-number multiplication (WNM) to build a stepping stone for children's proportional reasoning. Our intention in writing this paper is to obtain a practice-based perspective…

Literature on learning trajectories is reviewed, and challenges in developing designs to support responsive instruction are highlighted. A distinctive form of design called a "multi-dimensional instructional framework" is introduced. These frameworks span key dimensions of mathematisation, such as notating and distancing the setting,…

Previous research demonstrated that some children inappropriately solve multiplicative missing-value word problems additively, while others inappropriately solve additive missing-value word problems multiplicatively. Besides lacking skills, children's preference for additive or multiplicative relations has been shown to contribute to those errors.…