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,…

This study investigated how three fifth-grade students' understanding of fraction and decimal magnitude evolved over the course of a five-week teaching experiment. Students participated in teaching and learning sessions focused on developing concepts of fraction and decimal magnitude. The following questions guided this study: (1) How do fifth…

Teachers across the United States struggle with meeting students' needs in their classrooms amidst the growing diversity of children in their classrooms and the amplification of standardized testing. Time and again, teaching with the heart and action of inclusivity is placed on the back burner as teachers feel the pressure for students to perform…

This paper presents findings from a study that examined the strategies that children, who had only been taught the part-whole fraction sub-construct at school, used for finding the fraction associated with solving varied partitive quotient problems. A qualitative, microgenetic research design was used involving nine year 5 (aged 9-10) children…

Mathematics instruction can work to open or constrain children's opportunities to learn, and the busy, interactive nature of instruction makes it challenging for researchers to identify patterns in high-quality opportunities to learn. While researchers have documented how whole-group discussions and small-group work can contribute to such…

The use of different registers to represent mathematical concepts enhances understanding. For example, rational numbers can assume pictorial, symbolic and natural language representations and this kind of change improves learning. Based on these assumptions, a teaching experiment for the learning of rational numbers by 2nd grade students was…

This study investigates the use of manipulatives by elementary students working on a fraction task. Extending previous work on the role played by the manipulatives in students' activity, we aim at describing how the choices made for the task design disrupt students' activity, creating opportunities to learn. The theoretical underpinnings allow…

In the present study, we tested the hypotheses that: a) there are individual differences in secondary students' conceptual and procedural fraction knowledge, and b) these differences are predicted by students' approach (deep vs. surface) to mathematics learning. We used two instruments developed and evaluated for the purposes of the study which…

Through their work on the Fractions Project, Steffe and Olive (2010) identified a progression of fraction schemes that describes students' development toward more and more sophisticated ways of operating with fractions. Although several quantitative studies have affirmed this progression, the question has remained open as to whether it is specific…

Fractions are perceived as one of the most difficult areas in school mathematics to learn and teach. The most frequently mentioned factors contributing to the complexity are fractions having five interrelated constructs (part-whole, ratio, operator, quotient and measure) and teachers' pedagogical approaches to address these constructs. The present…

In this chapter, I propose a stance on learning fractions as multiplicative relations through reorganizing knowledge of whole numbers as a viable alternative to the Natural Number Bias (NNB) stance. Such an alternative, rooted in the constructivist theory of knowing and learning, provides a way forward in thinking about and carrying out…

Learning trajectories are built upon progressions of mathematical understandings that are typical of the general population of students. As such, they are useful frameworks for exploring how understandings of diverse learners may be similar or different from their peers, which has implications for tailoring instruction. The purpose of this…

Purpose: This paper presented the learning process of a group of primary mathematics teachers who participated in two iterations of lesson design, enactment and reflection in a Chinese Lesson Study. Design/methodology/approach: An expansive learning theory was employed to examine the teachers' learning process in lesson study (LS) on representing…

Self-efficacy in mathematics is related to engagement, persistence, and academic performance. Prior research focused mostly on examining changes to students' self-efficacy across large time intervals (months or years), and paid less attention to changes at the level of lesson sequences. Knowledge of how self-efficacy changes during a sequence of…

There is a broad consensus that beginning teachers of mathematics need a strong foundation in mathematical knowledge for teaching (MKT), defined as the mathematical knowledge required to recognize, understand, and respond to the mathematical work of teaching one must engage in. One recurrent challenge in teacher education is how to provide support…