The part-whole multiplicative relationship, as a topic that gives rise to the concept of fraction, is fundamental in education at the primary school level, and must therefore be included in training courses for prospective primary school teachers (PSTs). In this paper, we introduce a first study of a larger project, which aims to understand the…

Multiplicative reasoning is essential for students' engagement with various mathematical concepts. Although the field's understanding of children's multiplicative concepts has grown over the past 30 years, relatively few studies have examined the development of multiplicative concepts with whole numbers, and even fewer have studied this phenomenon…

Knowledge representations that result from practicing problem solving can be expected to differ from knowledge representations that emerge from explicit verbalizing of principles and rules. We examined the degree to which the two types of learning improve problem-solving knowledge and verbal explanation knowledge in classroom instruction. We…

Math fact fluency is foundational for later mathematics education. Unfortunately, many students across the nation continue to struggle with these core skills. Computer-assisted instruction may be a potentially valuable tool for improving math fact fluency due to its ability to differentiate instruction at the student level, provide added practice…

It is important for pre-service teachers to know the conceptual difficulties they have experienced regarding the concepts of multiplication and division in fractions and problem posing is a way to learn these conceptual difficulties. Problem posing is a synthetic activity that fundamentally has multiple answers. The purpose of this study is to…

This study compared the effectiveness and efficiency of an error-correction procedure, response repetition, to a prompting procedure, simultaneous prompting, on the acquisition and maintenance of multiplication facts for three typically developing 3rd grade students. This study employed an adapted alternating treatments design nested in a multiple…

Strategy flexibility, adaptivity, and the use of clever shortcut strategies are of major importance in current primary school mathematics education worldwide. However, empirical results show that primary school students use such shortcut strategies rather infrequently. The aims of the present study were to analyze the extent to which Dutch sixth…

The "Reasoning Algebraically about Operations Casebook" was developed as the key resource for participants' Developing Mathematical Ideas seminar experience. The thirty-four cases, written by teachers describing real situations and actual student thinking in their classrooms, provide the basis of each session's investigation into the…

In this article, we use a two-dimensional assessment to examine the experimental impacts of a mathematics learning trajectory-oriented formative assessment program on student strategies for problems involving multiplication and division. Working from the theory that the development of students' multiplicative reasoning involves improvements in…

Mathematics instruction for students with intellectual disabilities and autism is important. However, it is imperative for researchers and practitioners to focus on the maintenance of mathematical concepts and not just acquisition for these students. Through a single-case multiple probe across participants study, researchers explored an…

This study explores components of ambitious teaching practices for teaching multiplicative properties that can give teachers opportunities to learn through a quick images activity in a cycle of enactment and investigation. Throughout the cycle, it is found that the participants have opportunities to learn the following components of ambitious…

This study explored how teachers interpreted and responded to their own student work during the process of formative assessment. The study involved a purposefully selected sample of 32 teachers in grades K-5 who had been trained by the Ongoing Assessment Project (OGAP) to use learning progressions to analyze and respond to evidence in student…

This article examines the development of professional knowledge in pre-service mathematics teachers. From the discussion of a task associated with the multiplication of consecutive integer numbers, generalization is recognized as a process that allows to explore, to explain, and to validate mathematical results, and as an essential ability to…

Fraction arithmetic is among the most important and difficult topics children encounter in elementary and middle school mathematics. Braithwaite, Pyke, and Siegler (2017) hypothesized that difficulties learning fraction arithmetic often reflect reliance on associative knowledge--rather than understanding of mathematical concepts and procedures--to…

Productive struggle leads to productive classrooms where students work on complex problems, are encouraged to take risks, can struggle and fail yet still feel good about working on hard problems (Boaler, 2016). Teachers can foster a classroom culture that values and promotes productive struggle by providing students with challenging tasks. These…