Teachers and mathematicians hold different perspectives on the teaching and learning of whole number arithmetic. Though these perspectives may be complementary, sharing them across communities is challenging. An unusual professional development course for primary school teachers, initiated and taught by research mathematicians, provided a setting…

In the transition from secondary to tertiary mathematics, students try to participate in tertiary mathematics by replicating familiar school mathematical discourses. The objective of this case study is to investigate the conditions and affordances under which students proceed from familiar school mathematical discourses to new, tertiary discourses…

A design experiment with 18 students in a regular seventh grade math class was conducted to investigate how to differentiate instruction for students' diverse ways of thinking during a 26-day unit on proportional reasoning. The class included students operating with three different multiplicative concepts that have been found to influence rational…

Composing and decomposing numbers with base-ten blocks depends on children being able to see ten both as ten units and as one group of ten units (a long), cognizant that its value is the same in either case. Being able to see, or deciding when to see, an object or collection of objects as a unit is a key skill that children must develop to solve…

As the number of independent tosses of a fair coin grows, the rates of heads and tails tend to equality. This is misinterpreted by many students as being true also for the absolute numbers of the two outcomes, which, conversely, depart unboundedly from each other in the process. Eradicating that misconception, as by coin-tossing experiments,…

In this article, we examine various strategies used to express cardinal numbers in Yucatec Maya Sign Languages (YMSLs) from three historically unrelated communities in Yucatán, Mexico: Chicán, Nohkop, and Cepeda Peraza. Our findings describe some numeral strategies, which remained unattested in previous accounts, and demonstrate that YMSL numerals…

Numerical and symbolic representations are used extensively in physics problems. However, relatively little is understood about how students respond to these two representational formats when they are solving problems. This study examined the effect of the representational format (numeric vs symbolic) on the problem-solving performance,…

Developmental theory has long emphasized the importance of linking perception, cognition, and action. Techniques designed to record the spatial and temporal characteristics of hand movements (i.e., "manual dynamics") present new opportunities to study the nature of these links across development by providing a window into how perceptual,…

Practicing teachers as well as researchers, mathematicians, and teacher educators have offered opinions and theoretical critiques of the multiple models used to teach integer arithmetic. Few studies, however, have investigated what students learn with models or empirically compared affordances and constraints of integer models. This led me to…

We show that children in the Tsimane', a farming-foraging group in the Bolivian rain-forest, learn number words along a similar developmental trajectory to children from industrialized countries. Tsimane' children successively acquire the first three or four number words before fully learning how counting works. However, their learning is…

In this article, I present the notion of a set-oriented perspective for solving counting problems that emerged during task-based interviews with postsecondary students. Framing the findings within Harel's "ways of thinking", I argue that students may benefit from this perspective, in which they view attending to sets of outcomes as…

In the first year of life, infants possess two cognitive systems encoding numerical information: one for processing the numerosity of sets of 4 or more items, and the second for tracking up to 3 objects in parallel. While a previous study showed the former system to be already present a few hours after birth, it is unknown whether the latter…

In human history, the origin of the numbers came from definite practical needs. Indeed, there is strong evidence that numbers were created before writing. The number "1", dating back at least 20,000 years, was found as a counting symbol on a bone. The famous statement by the German mathematician Leopold Kronecker (1823-1891), "God…

We provide evidence of validity for a newly developed diagnostic competence model of operation sense, by both (a) describing the theoretically substantiated development of the competence model in close association with its use within a large-scale formative assessment and (b) providing empirical evidence for the theoretically described cognitive…

To increase opportunities for students to take more advanced math courses in high school, many school districts enroll grade 8 students in Algebra I, a gateway course for advanced math. But students who take Algebra I in grade 8 and skip other math courses, such as grade 8 general math, might miss opportunities to develop the foundational…