In this paper two studies are reported in which two contrasting claims concerning the intuitiveness of the law of large numbers are investigated. While Sedlmeier and Gigerenzer ("J Behav Decis Mak" 10:33-51, 1997) claim that people have an intuition that conforms to the law of large numbers, but that they can only employ this intuition…

Students with mathematics difficulty demonstrate lower mathematics performance than typical-performing peers. One contributing factor to lower mathematics performance may be misunderstanding of mathematics symbols. In several studies related to the equal sign (=), students who received explicit instruction on the relational definition (i.e.,…

A lack of fractional understanding is a well-documented obstacle to student achievement in upper elementary and middle school math (National Center for Educational Statistics [NCES] 1999; Lamon 1999; National Research Council [NRC] 2001). Lamon (1999) notes that one major conceptual hurdle that students must overcome is the idea that fractions are…

This article reports on an exploratory study designed to investigate the reasoning behind algebraists' selection of examples. Variation theory provided a lens to analyze their collections of examples. Our findings include the classes of examples of groups and rings that algebraists believe to be most pedagogically useful. Chief among their…

In this article, Scott Smith presents an innocent problem (Problem 12 of the May 2001 Calendar from "Mathematics Teacher" ("MT" May 2001, vol. 94, no. 5, p. 384) that was transformed by several timely "what if?" questions into a rewarding investigation of some interesting mathematics. These investigations led to two…

Since the 1950s, the understanding of how the base 10 system works has been encouraged through alternative base systems (Price 1995; Woodward 2004). If high school students are given opportunities to learn other base systems and analyze what they denote, we believe that they will better understand the structure of base 10 and its operations…

This paper discusses findings from a research study designed to investigate calculus instructors' perceptions of approximation as a central concept and possible unifying thread of the first-year calculus. The study also examines the role approximation plays in participants' self-reported instructional practices. A survey was administered to 279…

Manipulatives are ubiquitous in early childhood classrooms; yet, findings regarding their efficacy for learning mathematics concepts are inconsistent. In this article, we present four general principles that have emerged from cognitive science about ways to ensure that manipulatives promote learning when used with young children. We also describe…

In this paper, we analyze individual semi-clinical interviews conducted with one kindergarten and one first-grade student. We build on prior research to offer evidence, via excerpts from these interviews, that children as young as kindergarten have a powerful, intuitive sense of generality and indeed naturally draw upon it to reason through…

Teaching students how to use and interpret various representations of functions remains an enduring challenge for educators. Providing students access to such representations is a key feature of providing them access to mathematical participation and communication. This study is based on a teaching experiment, which aims to provide students access…

Fraction and decimal arithmetic are crucial for later mathematics achievement and for ability to succeed in many professions. Unfortunately, these capabilities pose large difficulties for many children and adults, and students' proficiency in them has shown little sign of improvement over the past three decades. To summarize what is known about…

Having students address mathematical inquiry problems that are ill-structured and ambiguous offers potential for them to develop a focus on mathematical evidence and reasoning. However, students may not necessarily focus on these aspects when responding to such problems. Argument-Based Inquiry is one way to guide students in this direction. This…

The guiding philosophy of this theoretical work lays in the argument that mathematics teachers' professional knowledge is the integration of various knowledge facets derived from different sources including teaching experience and research. This paper goes beyond past trends identifying what the teachers' knowledge is about (content) by providing…

Background: Students' ability to construct and coordinate units has been found to have far-reaching implications for their ability to develop sophisticated understandings of key middle-grade mathematical topics such as fractions, ratios, proportions, and algebra, topics that form the base of understanding for most STEM-related fields. Most of the…

This paper reports the performance of one entire cohort of Singapore prospective secondary school mathematics teachers in a mathematics proficiency test. The prospective teachers were admitted to the teacher education program specializing in teaching secondary school mathematics. The strengths of the prospective teachers' content knowledge, their…