Conceptions of function promoted by seventh- and eighth-grade textbooks from eighteen countries

Abstract

This study investigated the conceptions of function enacted by problems and exercises in 35 mathematics textbooks for seventh- and eighth-grade students from 18 countries participating in the Third International Mathematics and Science Study (TIMSS). The notion of conception used was that of Balacheff: a quadruplet consisting of problems, operations, representation systems, and control structures. A coding system was developed that had 10 codes for problems (given by the use of function in the problem), 35 for operations, 9 for representation systems, and 9 for control structures. Von Eye's Configural Frequency Analysis was used to determine types and antitypes of configurations.

Five conceptions of function were identified as promoted in the textbooks: symbolic rule, ordered pair, social data, physical phenomena, and controlling image. The different characteristics of the conceptions suggested that different school practices were associated with each conception. Groups of countries were identified whose textbooks shared similar characteristics. Across countries, the textbooks fell into four clusters according to the predominant conceptions and uses of function: rule oriented, abstract oriented, abstract oriented with applications, and applications oriented. The results suggested that (a) there is no canonical curriculum for teaching function and (b) there are no traditions of organizing mathematics textbook content on function.

Ten items from the TIMSS achievement test were coded and compared with the tasks in the textbook clusters. Performance on the item by students in countries using textbooks promoting the same conception was also examined. The results suggested that (a) the test did not reflect any country's distribution of conceptions and (b) using a textbook belonging to a particular cluster or promoting a certain conception did not provide an advantage to students. Thus, at the micro level of textbook content, there is no evidence that one organization is better than other in terms of student achievement. Arguments linking student achievement on the TIMSS test to the use of specific textbooks were challenged.

The study illustrated the application of the four-dimensional definition of conception to three questions about textbook content in mathematics. It suggests possible applications to other mathematical notions and other areas of research in mathematics education.