Non-attendance to meaning by students is a prevalent phenomenon in school mathematics. Our goal is to investigate features of instruction that might account for this phenomenon. Drawing on a case study of two high school algebra teachers, we cite episodes from the classroom to illustrate particular teaching actions that de-emphasize meaning. We…

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…

I report on the findings from research and literature on (a) use of symbols in mathematics, (b) algebraic/trigonometric expressions, (c) solving equations, and (d) functions and calculus. From these, some insights and implications for teaching and learning are derived.

Teachers and researchers have long recognized that students tend to misunderstand the equal sign as an operator; that is, a signal for "doing something" rather than a relational symbol of equivalence or quantity sameness. Students' equal sign misconception has been researched for more than thirty years (Weaver, 1971, 1973) with little…

Mathematics begins in human experience thousands of years ago as empirical and intuitive experiences. It takes the deliberate naming of concepts to help crystallize and secure those observations and intuitions as abstract concepts, and to begin separating the concept of number from specific instances of objects. It takes the creation of compact…

In today's world, where the volume of knowledge everyone must deal with is increasing exponentially, many educators agree that schools must focus on developing skills for life-long learning. But what does that mean for an area such as algebra? Teachers' goal in school algebra should be to guide students to "work smarter" with algebraic symbols and…

Uses Lacanian notions to create stories on the mathematics education topics of numeracy, symbolism, and desire. (MDH)

Calculus is one of those topics in mathematics where the algorithmic manipulation of symbols is easier than understanding the underlying concepts. Around 1680 Leibniz invented a symbol system for calculus that codifies and simplifies the essential elements of reasoning. The calculus of Leibniz brings within the reach of an ordinary student…

The concept of how the definite integral is understood by the students deeply with the help of an activity and the connection between the functions [prime]integral of 1/t is described. The definition of integral of 1/t is remembered by the students by exploring the integral systematically and by application of their knowledge of rational…

In this article, the author looks at some of the aspects of the equals sign. He shares several insights from his colleagues on the meaning of the equals sign. He also discusses how children view "=" as a sign which is written before "the answer," and, in the recording of the many practical activities that form the basis for the…

Computer algebra systems (CAS) have been around for a number of years, as has dynamic geometry. Symbolic geometry software is new. It bears a superficial similarity to dynamic geometry software, but differs in that problems may be set up involving symbolic variables and constants, and measurements are given as symbolic expressions. Mathematical…

This study examined conceptions of algebra held by 30 preservice elementary teachers. In addition to exploring participants' general ''definitions'' of algebra, this study examined, in particular, their analyses of tasks designed to engage students in relational thinking or a deep understanding of the equal sign as well as student work on these…

A method of helping students attack the components of mathematics as a language is discussed; examples from arithmetic and from calculus are illustrated. (DT)