Strategies for Teaching Concepts by Using Definitions

Abstract

Reflection indicates that there is a kinship between terms which designate, and concepts. A teacher would do much the same thing if he were teaching the concept of a rectangle as if he were teaching the meaning of the term rectangle. Accepting the risk of oversimplification, we suggest that a term is a name of the concept denoted by the term. It follows that defining a term is one way of teaching the concept denoted by the term.

A Comparison of U.S. and Chinese Mathematics Textbooks and Teaching: Concept Definition, Conceptual Information, and Classroom Instructions

“…Another limitation of the current study was that often the sample size was small. This effect was caused by the fact that only three textbooks were coded in the current studies, and that it was not likely to expand the selection of content because previous research indicated that the definition patterns might vary for different topics (Ginther & Henderson, 1966). For example, in U.S. textbooks, a larger proportion of algebraic definitions tended to be defined denotatively than geometric definitions, while a higher percentage of geometric definitions tended to be defined connotatively than ones in algebra.…”

“…Specifically, the generalization of mathematical ideas, rules and principles (hereinafter referred to as Mathematical Generalizations, or MGs) and links between lessons are the two main focus besides definitions. The coding scheme will be developed based on previous work in mathematical taxonomy (Ginther & Henderson, 1966) and TIMSS video study ( Jacobs et al, 2003). Second, another area of research interest is whether or not and how often teachers emphasize conceptual understanding in class and whether such patterns correspond to the characteristics identified in textbooks.…”

“…The primary content of all selected textbooks was pre-algebra, including rational numbers, monomials and polynomials, and first-order equations with one unknown. The content was controlled because previous study suggested that the distribution of definition types and strategies vary on different topics (Ginther & Henderson, 1966). Thus, Sections 1.1 to 4.8 and Sections 12.1 to 12.6 in Bennett et al (2008); Sections 1.1 to 6.2, Section 8.1 and Sections 12.1 to 12.5 in Larson et al (2008); and Chapter 1 to 3 in Kecheng Jiaocai Yanjiusuo ( 2007) were selected for the coding purpose in this study, so that the coded content was the same across different textbooks.…”

“…The coding scheme that was used in this study was modified based on the existing mathematical taxonomy research that was discussed earlier (Ginther & Henderson, 1966). In this study, a definition was defined as the sentence containing a bold or highlighted concept that includes (1) examples, (2) the necessary and sufficient conditions for an object to be classified as the concept being defined, and/or (3) synonyms for the concept.…”

“…An explanation of a concept definition must serve for at least one of the following purposes: ( 1) providing examples of the concept, (2) formally or informally defining a term that was a part of the concept definition body, or (3) specifying the scope to which the concept could apply. Based on the locations of the explanation of a definition, four strategies are identified in this coding scheme (Ginther & Henderson, 1966). An explanation-definition strategy has only an explanation proceeding a definition, a definition-explanation strategy has only an explanation following a definition, an explanation-definition-explanation strategy has explanations both before and after a definition, and a definition strategy does not have any explanation before or after a definition.…”