The purpose of this research was to investigate students' mathematical understandings of fraction division with an ethnomathematics/STEM framework. A small group teaching study (Cobb et al., 2003) was conducted using design research methodology to examine how students interacted with the curriculum. The researcher served as interventionist and…

Researchers have reported that preservice mathematics teachers' (PMTs') conceptions for radian angle measure are dominated by conceptions of degrees, memorization, and calculational strategies, and are always expressed in terms on [pi]. In this report, I unpack the mathematical meanings of a PMT, Henry (pseudonym), as he engaged in three…

Which dimension of a set of objects is more salient to young children: number or size? The "Build-A-Train" task was developed and used to examine whether children spontaneously use a number or physical size approach on an un-cued matching task. In the Build-A-Train task, an experimenter assembles a train using one to five blocks of a…

Isomorphism and homomorphism are topics central to abstract algebra, but research on mathematicians' views of these topics, especially with respect to sameness, remains limited. This study examines open response survey data from 197 mathematicians on how sameness could be helpful or harmful when studying isomorphism and homomorphism. Using…

In the present paper I look at Edmond Halley's reconstruction of Book VIII of Apollonius's "Conic" as an example of a second-order historical text. Such texts constitute a particular class of original works whose distinction is that they present mathematicians of the past engaging with texts from their own past, as we do when…

Many disciplines make use of mathematical concepts. However, there are often discrepancies between the way mathematical concepts are understood and taught in mathematics and the way they are used in other disciplines. The literature suggests that such discrepancies might make it hard for students in mathematics service courses to make a connection…

There is growing evidence that the ability to perceive structure is essential for students' mathematical development. Looking at students' structure sense in basic numerical and patterning tasks seems promising for understanding how these tasks set the foundation for the development of later mathematical skills. Previous studies have shown how…

This study examined "backward transfer," which we define as how students' ways of reasoning about previously encountered concepts are modified when learning about new concepts. We examined the backward transfer produced when students learned about quadratic functions. We were specifically interested in how backward transfer may vary for…

Problem-based instruction (PBI) is a research-based, student-centered instructional strategy that uses rich mathematical tasks to connect students' previously acquired math knowledge to new mathematical concepts. Unlike traditional teaching and learning strategies, PBI facilitates learning by making connections between mathematical concepts and…

After waiting in a long line for your favourite cup of coffee, you finally sit down with your mug and find that the coffee is still scalding hot! How long do you need to wait before you can enjoy it? Once it cools enough, how much time do you have to enjoy it? Are there ways to speed up the process? These questions motivate the presented modelling…

Teachers' knowledge of the subject matter is considered an important component of their expertise in teaching mathematics. Yet how teachers' understanding of one content area is related to other content areas has not been investigated in depth. We explored this question by investigating teachers' knowledge of two theoretically related areas: (1)…

Rational number proficiency is predictive of later mathematics achievement, especially for rigorous mathematics courses, such as Algebra I. Students with learning disabilities (LD) in mathematics struggle to make adequate progress on rational number concepts and skills that lay the foundation for successfully completing secondary mathematics…

Across multiple age groups and academic disciplines, research has shown that incorporating multimodal signs into instruction can enhance student learning. For instance, gesture might be synchronized with speech or written signs. The present study reports on the semiotic resources used by a teacher-researcher to support kindergarten students'…

This study's purpose was to examine the effects of the concrete-representational-abstract integrated (CRA-I) sequence on the performance of students who struggled with rational number concepts. Three students in southeastern U.S. fifth grade class participated in the study. The CRA-I intervention was grounded in the principles of explicit…

In this case study we explored how a mathematician's teaching of the Cauchy-Riemann (CR) equations actualized the virtual aspects of the equations. Using videotaped classroom data, we found that in a three-day period, this mathematician used embodiment to animate and bind formal aspects of the CR equations (including conformality), metaphors,…