The aim of this study is to describe what kind of knowledge base is needed when pre-school teachers work goal-oriented with children's mathematical learning. The question of this study aims to answer is: What kind of knowledge base do pre-school teachers need when they work with the object of learning in the pre-school context? Both as a…

The purpose of this study was to validate inferences about scores of one task designed to measure subject matter knowledge and three tasks designed to measure aspects of pedagogical content knowledge. Evidence for the validity of inferences was based on two expectations. First, if tasks were sensitive to expertise, we would find group differences.…

The ability to represent numerical quantities in symbolic form is a necessary foundation for mathematical competence. Variables are particularly important symbolic representations for learning algebra and succeeding in higher mathematics, but the mechanisms of how students link a variable to what it represents are not well understood. Research…

The language of mathematics is unique and complex. One feature of the mathematical register is the use of symbols and abbreviations. Whilst it may be possible for a student to think mathematically in the absence of symbols, the written communication of mathematical ideas cannot be achieved concisely without the use of mathematical symbols.…

This study reports on the implementation of an intermediate algebra curriculum centered on a framework of student-centered questions designed to investigate algebraic procedures. Instructional activities were designed to build discourse in the small-group discussion meetings of the course. Students were assigned writing prompts to emphasize the…

This article introduces an optimization task with a ready-made motivating question that may be paraphrased as follows: "Are you smarter than a Welsh corgi?" The authors present the task along with descriptions of the ways in which two groups of students approached it. These group vignettes reveal as much about the nature of calculus students'…

Advisory committees of mathematics consider abstract algebra as an essential component of the mathematical preparation of secondary teachers, yet preservice teachers find it challenging to connect the topics addressed in this advanced course with the high school algebra they must someday teach. This study analyzed the mathematical content…

Middle-school students are expected to understand key components of graphs, such as slope and y-intercept. However, constructing graphs is a skill that has received relatively little research attention. This study examined students' construction of graphs of linear functions, focusing specifically on the relative difficulties of graphing slope and…

This paper examines ten K-8 preservice teachers' visual representations of fraction operations using the four main pattern blocks. Data consist of figures made using the pattern blocks, drawn colored representations, and detailed written comments and algebraic formalism. The theoretical framework is drawn from representational theories and…

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.

Basic brain function is not a mystery. Given that neuroscientists understand its basic functioning processes, one wonders what their research suggests to teachers of developmental algebra. What if we knew how to teach so as to improve understanding of the algebra taught to developmental algebra students? What if we knew how the brain processes…

Augustus De Morgan (1806-1871) was a significant Victorian Mathematician who made contributions to mathematics history, mathematical recreations, mathematical logic, calculus, and probability and statistics. He was an inspiring mathematics professor who influenced many of his students to join the profession. One of De Morgan's significant books…

The purpose of this study was to identify the degree to which college algebra students' value mathematical skills in their prospective careers. A survey was administered to N = 144 students in 6 college algebra classes at a mid-sized doctoral granting university. Students in half the classes completed a data analysis project, and half of the…

This study examined ways in which students make use of a graphing calculator and how use relates to comfort and understanding with mathematical symbols. Analysis involved examining students' words and actions in problem solving to identify evidence of algebraic insight. Findings suggest that some symbols and symbolic structures had strong…

Using a novel design approach, a randomized experiment (RE) was embedded within a regression discontinuity (RD) design (R-RE-D) to evaluate the impact of developmental mathematics at a large midwestern college ("n" = 2,122). Within a region of uncertainty near the cut-score, estimates of benefit from a prospective RE were closely…