Based on Piaget's theory of logico-mathematical knowledge, 126 students in grades 2-5 were asked 6 questions about elapsed time. The main reason found for difficulty with elapsed time is children's inability to coordinate hierarchical units (hours and minutes). The educational implications drawn are that students must be encouraged to think about…

This article is the story of a very non-standard, absolutely student-centered multivariable calculus course. The course advocates the so-called problem method in which the problems used are a bridge between what the learners know and what they are about to know. The main feature of the course is a unique conceptual story that runs through the…

Discussion of science topics can be generated using sport and fitness as a context, either in relation to the Olympic Games this summer or more generally. In this article a new set of Concept Cartoons focused on sport and fitness, produced as a partnership publication between the Association for Science Education (ASE) and Millgate House…

This study examined Advanced Placement Calculus students' mathematical understanding of rate of change, after studying four years of college preparatory (integrated or single-subject) mathematics. Students completed the Precalculus Concept Assessment (PCA) and two open-ended tasks with questions about rates of change. After adjusting for prior…

Since slope is a fundamental topic that is embedded throughout the U.S. secondary school curriculum, this study examined standards documents for all 50 states to determine how they address the concept of slope. The study used eleven conceptualizations of slope as categories to classify the material in the documents. The findings indicate that all…

Science teachers know that the mathematics concepts taught in the Common Core are critical for students' understanding of science. But what can a teacher do when his/her students lack the necessary mathematics skills to master science content? There may be other reasons besides students not paying attention in their math courses. Maybe the…

Probabilistic misconceptions are common among students and teachers alike. This study involved 18 preservice secondary mathematics teachers and documents some of the common probabilistic misconceptions held by these preservice teachers. Data was collected through a survey involving eight probability problems and two semi-structured interviews.…

This article discusses results from interviews investigating students' understanding of probabilistic independence and mutual exclusivity. Three students compared several sets of events in various sample spaces. Data collected from these interviews gives evidence of a temporal conception wherein students think of independence as reliant on a…

Three clinical interviews were conducted with each of 15 sixth grade students to test conjectures about the relationship between their level of multiplicative reasoning and their solution of combinatorics problems that could involve single and multi-digit multiplication. The problems that involved multi-digit multiplication were designed with the…

In this paper, we review the rules and game board for "Chutes and Ladders", define a Markov chain to model the game regardless of the spinner range, and describe how properties of Markov chains are used to determine that an optimal spinner range of 15 minimizes the expected number of turns for a player to complete the game. Because the Markov…

The Nobel prize winning physicist Richard Feynman (2007) famously enthused about "the pleasure of finding things out". In day-to-day classroom life, however, it is easy to lose and undervalue this pleasure in the process, as opposed to products, of mathematics. Finding things out involves a journey and is often where the learning takes place.…

We provide several constructions, both algebraic and geometric, for determining the ratio of the radii of two circles in an Apollonius-like packing problem. This problem was inspired by the art deco design in the transom window above the Shadek Fackenthal Library door on the Franklin & Marshall College campus.

In this article, I explore tensions of care in the context of school mathematics by examining two accounts of a classroom moment involving labelling an angle. In particular, I draw attention to how caring for students and caring for mathematical ideas interplay in complex ways by inquiring into the two accounts through ideas of care and…

We propose an alternative to the standard introduction to the derivative. Instead of using limits of difference quotients, students develop Taylor expansions of polynomials. This alternative allows students to develop many of the central ideas about the derivative at an intuitive level, using only skills and concepts from precalculus, and…

Much reference has been made to Paul Ernest's "philosophy of mathematics education" to legitimise a strong fallibilist trend in mathematics education. This article presents the argument that: (1) This philosophy makes unwarranted assumptions that have been taken as "given". For example, that "absolutist" or "Platonist" views of mathematics…