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…

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…

To test the hypothesis that a higher tendency to "s"pontaneously "f"ocus "o"n "m"ultiplicative "r"elations (SFOR) leads to improvements in rational number knowledge via more exact estimation of fractional quantities, we presented sixth graders (n = 112) with fraction number line estimations and a…

In this study, we decomposed the broad practice of deciphering multiple solution strategies. We conducted interviews with 11 preservice elementary teachers, in which we asked teachers to decipher students' standard and nonstandard strategies to multiplication and division problems. We examined what teachers said and did--what we refer to as…

In this study, we decomposed the broad practice of deciphering multiple solution strategies. We conducted interviews with 11 preservice elementary teachers, in which we asked teachers to decipher students' standard and nonstandard strategies to multiplication and division problems. We examined what teachers said and did--what we refer to as…

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,…

A growing body of literature supports the effectiveness of Modified Schema-Based Instruction (MSBI) to improve mathematical problem-solving for students with autism spectrum disorder (ASD) and intellectual disability (ID). MSBI is an intervention package that teaches students to identify the problem structure and use a problem-solving heuristic to…

Introducing the distributivity property of multiplication over addition is a well-known challenge in mathematics education, especially in primary school. As a contribution, this paper presents the results of a cycle of design research that focuses on the design, implementation and evaluation of a modelling activity in which 2nd-grade students are…

The present study is motivated by a significant body of research documenting teachers' perennial difficulties with a critical swath of topics related to multiplication. In response, we track how Nina, a future middle grades mathematics teacher, made progress constructing explanations across topics by reasoning with measurement-based definitions of…

Three iterative, 18-episode design experiments were conducted after school with groups of 6-9 middle school students to understand how to differentiate mathematics instruction. Prior research on differentiating instruction (DI) and hypothetical learning trajectories guided the instruction. As the experiments proceeded, this definition of DI…

There is a growing body of research that suggests that video-based interventions, such as video modeling and video prompting, are effective tools for teaching academic skills to struggling learners. This study used a single subject, multiple-baseline-across-subjects design to evaluate whether a video-prompting intervention could effectively assist…

The development of conceptual multiplication knowledge will assist students in making progress within current mathematics standards. Previous research has shown the concrete-representational-abstract (CRA) sequence to be successful in teaching multiplication with regrouping with an emphasis on conceptual understanding while developing fluency and…

This study investigates the teaching and learning of single-digit whole number multiplication in China. Analysis of data from documents, classroom teaching, and semi-structured interviews revealed three salient characteristics of emphasizing "oral calculation," "calculation speed," and "understanding" across…