This article describes a variety of activities that ask students to identify, describe, compare, and classify symbol strings (algebraic expressions and equations). The activities use a collection of twelve symbol strings on cards. (Contains 2 figures.)

In this paper, we distinguish between two ways that an individual can construct a formal proof. We define a syntactic proof production to occur when the prover draws inferences by manipulating symbolic formulae in a logically permissible way. We define a semantic proof production to occur when the prover uses instantiations of mathematical…

The modular exponentiation, y[equivalent to]x[superscript k](mod n) with x,y,k,n integers and n [greater than] 1; is the most fundamental operation in RSA and ElGamal public-key cryptographic systems. Thus the efficiency of RSA and ElGamal depends entirely on the efficiency of the modular exponentiation. The same situation arises also in elliptic…

Students "doing" mathematics ultimately results in students reading mathematics. This reading of mathematics is manifested in students reading words, symbols (including numerals), and visuals such as diagrams and graphs. Furthermore, students' successful response to word problems engages them in reading mathematics in the context of real-life or…

We report a study of repairs in communication between workers and visiting outsiders (students, researchers or teachers). We show how cultural models such as metaphors and mathematical models facilitated explanations and repair work in inquiry and pedagogical dialogues. We extend previous theorisations of metaphor by Black; Lakoff and Johnson;…

For online mathematics instructors, synchronous communication can be a challenge due to the need for specialized symbols, graphics, or notations to explain key concepts. While software tools exist to support such communication, they can often be cumbersome and time-consuming for users to adopt in online exchanges. As one way of addressing this…

Mathematics can be perceived as being a difficult subject to learn due to the conceptual leaps required to understand particular mathematical topics. In some areas of mathematics, part of the difficulty may be associated with applying sufficient imagination to visualize a particular mathematical concept, and applying sufficient visio-spatial…

Some students can perform operations on fractions when using physical materials, but are unable to transfer this ability to operations with symbolic expressions. They seem to view the manipulative and paper-and-pencil problems as different and are not surprised to get different answers. (SD)

This study was designed to determine whether (1) rules are more easily memorized when stated in mathematical symbolism or when stated verbally, and whether (2) the ability to use constituent symbols correctly, assuming mastery of the underlying grammar, is a necessary and/or sufficient condition for applying a learned rule statement. Twenty-four…

This paper deals with the problems of rigorous research within the dynamic organizational context. Definitions and axioms establish the natures of a social problem, a social intervention program, continuous program monitoring, a method of describing alternative treatments, treatment effect testing, and programmatic change. These definitions and…

Two distinct strategies for teaching the solution to verbal problems were compared. Programs of instruction were prepared that reflected the Polya Method (read and understand the problem, plan for a solution, carry out the plan, and check the result) and the Dahmus Method (translate the verbal statements into mathematical symbols prior to solving…

The new, and prevalent, raised-dash notation (for subtraction) appearing in school mathematics texts is examined, especially for its effects on students' computational skills. Reasons for a return to the standard notation of the centered dash are presented. (Author/MN)

The author argues for a clarification of standard notation rather than the use of flow diagrams to make the meaning of algebraic formulas more obvious to the beginning algebra student. (MN)

Based on the premise that meaningful learning in mathematics occurs when children link their understandings with the symbols and procedures they are taught, three places are identified where children can link symbols and their referents, procedural rules and conceptual knowledge, and answers to problems with other knowledge about what is…

Demonstrates that basic algebraic symbols can be used to illustrate fundamental concepts in technical writing, such as hierarchical organization, sentence structure, modification, and punctuation. Suggests that students in calculus, chemistry, and physics may benefit from this application of familiar concepts to those of technical writing. (HTH)