A Comparative Analysis of the Addition and Subtraction of Fractions in Textbooks from Three Countries

Abstract

In this paper, we report on a comparison of the treatment of addition and subtraction of fractions in primary mathematics textbooks used in Cyprus, Ireland, and Taiwan. To this end, we use a framework specifically developed to investigate the learning opportunities afforded by the textbooks, particularly with respect to the presentation of the content and the textbook expectations as manifested in the associated tasks. We found several similarities and differences among the textbooks regarding the topics included and their sequencing, the constructs of fractions, the worked examples, the cognitive demands of the tasks, and the types of responses required of students. The findings emphasized the need to examine textbooks in order to understand differences in instruction and achievement across countries. Indeed, we postulate that within a given country there may exist a recognizable “textbook signature.” We also draw on the results and the challenges inherent in our analysis to provide suggestions and directions for future textbook analysis studies.

This paper is an application of a synthesis of research conducted by the first three authors while taking a class on curriculum in mathematics education taught by the last author. We opt for an alphabetical listing of names, but contributions were at comparable levels. Preliminary reports on this work were presented at the annual meeting of the American Educational Research Association, in Chicago, 2007 and the 31st meeting of the International Group of the Psychology of Mathematics Education, in Seoul, 2007. We thank the editor and three anonymous reviewers for their comments on an earlier version of this paper and the publishers of the books, who granted us permission to reproduce the illustrative excerpts in this article.

Notes

¹Or the publisher's interpretation of those goals and objectives.

²For the purposes of this paper, opportunity to learn refers to a description of the complexity of the mathematics that is present in the textbooks that students would use in their classrooms, thus limiting its definition to the mathematical content as opposed to the more general notion that includes school factors that most directly affect student learning. For an historical description of the evolution of this notion, see McDonnell (1995) and Tate and Rousseau (2006).

³Interestingly, however, the topic has been studied less extensively than multiplication or division of fractions (Verschaffel, Greer, & Torbeyns, 2006, p. 65).

⁴In our literature search, we found this type of textbook analysis frequently, not only in mathematics but in other disciplines as well. For example, Beck, McKeown, and Gromoll (1989) analyzed the presentation of specific topics in four series of social study textbooks for fourth and fifth grade.

⁵Our discussion of the textbooks used in the three countries is grounded in a set of criteria from the “Background Information” category of the Horizontal Analysis dimension of the framework presented in Figure 1 (i.e., title of the textbooks, number of books, number of pages per book, publisher and year of publication, and accompanying materials).

⁶Defined by the highest level of numbering, the tasks in both Taiwanese textbooks tended to include only one or two exercises/problems (more than 90% of the textbook tasks). In contrast, more than 40% of the tasks in the CT and more than 50% of the Irish textbooks included tasks with five or more exercises/problems.

⁷A more detailed description of these constructs as used in this study appears in Charalambous and Pitta-Pantazi (2007).

⁸We are aware that the cognitive demands of the tasks depend on students' prior experiences. However, we decided not to include such considerations in our analysis because exploring the textbooks of previous grades was not considered sufficient for determining students' prior knowledge; the presence alone of a topic in the textbook of previous grades does not ensure students' prior knowledge of that topic. Because we decided not to take into consideration students' prior experiences and because we do not know how these tasks might actually be enacted during a lesson, we talk about potential cognitive demands.

⁹We used Cohen's κ because it allows assessing inter-rater reliability when there are two coders and the variables have several categories. Because this coefficient does not count agreement that is simply due to chance, it is more stringent than simply calculating the rate of agreements to the total of agreements and disagreements. According to Landis and Koch (1977), κ = 0.40 to 0.59 reflects moderate inter-rater reliability, 0.60 to 0.79 substantial reliability, and 0.80 or greater outstanding reliability.

¹⁰For instance, some tasks in TTA (vol. 2, p. 72) presented students with strips and asked them to figure out the combined length of two strips joined end-to-end or the difference in length between two strips. Although these tasks can be solved by a simple addition or subtraction, we decided to code them as procedures with connections because they implied some connection to meaning.

¹¹In Chinese, these twin questions are both phrased using the terms “how many?” (i.e., “how many bags of tomatoes?” and “how many tomatoes?”). The original version of this problem as it appears in Chinese is “一袋蕃茄有12個，用了2/6袋煮湯後，再用多少袋蕃茄，就會把一袋蕃茄都用完呢？是多少個？”

¹²The concept of unit fraction, as used by ancient Egyptians, integrates three fundamental ideas—equipartitioning, unit, and quantity—all of which underlie all fraction constructs (cf. Carpenter, Fennema, & Romberg, 1993, p. 3). Brousseau (1997, pp. 104–105) also comments on the importance of unit fractions for the addition of fractions.