Learning progressions (LPs) describe the development of domain-specific knowledge, skills, and understanding. Each level of an LP characterizes a phase of student thinking en route to a target performance. The rationale behind LP development is to provide road maps that can be used to guide student thinking from one level to the next. The validity…

The concept of pattern recognition lies at the heart of numerous deliberations concerned with new mathematics curricula, because it is strongly linked to improved generalised thinking. However none of these discussions has made the deceptive nature of patterns an object of exploration and understanding. Yet there is evidence showing that pattern…

The study was designed to probe students' thinking about which numerical values can be assigned to algebraic letters. The data from students in Year 7 (n = 533), Year 8 (n = 377) and Year 9 (n = 172) was analysed using response patterns. The data confirmed that each year contained students with two misconceptions; "Different Letter means…

Initial exposure to algebraic thinking involves the critical leap from working with numbers to thinking with variables. The transition to thinking mathematically using variables has many layers, and for all students an abstraction that is clear in one setting may be opaque in another. Geometric counting and the resulting algebraic patterns provide…

This article draws out many mathematical ideas connected with a familiar game often used by middle school teachers. (Contains 5 figures and 1 table.)

Learning in educational settings emphasizes declarative and procedural knowledge. Studies of expertise, however, point to other crucial components of learning, especially improvements produced by experience in the extraction of information: perceptual learning (PL). We suggest that such improvements characterize both simple sensory and complex…

A geoboard-related extremal problem dealing with both concave and convex polygons is discussed and a proof is given of the solution. (MP)

A method of summing finite sequences by use of formal power series techniques is described. (SD)

The author shows how a rapid computational "trick" can lead to an investigation of Fibonacci-type sequences. (SD)

A coordinate system is imposed on a geoboard to form a table that leads to the discovery of the formula n[(n+1)(n+1)] (n+2)/12. Proof is by induction. (MP)

An algebraic development of the Fibonnaci sequence, appropriate for use in beginning algebra classes, is presented. (SD)

With the advent of computers and electronic calculators, the role of logarithms in the curriculum is changing. An intuitive approach to logarithms, stressing the notion of isomorphism, is discussed. (SD)

A familiar Christmas song described as a second-year algebra problem is presented in "The Twelve Days of Christmas". Five strategies for solving inequalities are discussed in "Solving Algebraic Inequalities". (JT)

A method for solving certain types of problems is illustrated by problems related to Fibonacci's triangle. The method involves pattern recognition, generalizing, algebraic manipulation, and mathematical induction. (MP)

Learning in educational settings most often emphasizes declarative and procedural knowledge. Studies of expertise, however, point to other, equally important components of learning, especially improvements produced by experience in the extraction of information: "Perceptual learning." Here we describe research that combines principles of…