The development of proficiency in the fraction domain : affordances and constraints in the curriculum

G. Bruin - Muurling

    Research output: ThesisPhd Thesis 1 (Research TU/e / Graduation TU/e)

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    Proficiency in basic mathematical skills is a topic of heated discussions in many countries. International comparative studies on mathematical skills such as PISA and TIMMS have lead to concerns about the mathematics curriculum especially in countries with a relatively low rating in the summary reports (the league-tables effect). In the Netherlands –as in many other countries– these discussions concentrate on the questions whether and to what extend there is reduced mastery, what causes this problem and how curriculum reform might remedy this. Given the lack of data to answer these last three questions, there is a need for further research. This doctoral dissertation therefore explores the nature and causes of problems with basic mathematical skills in the Dutch curriculum, taking the transition from primary to secondary education as its main focus. More concretely, this dissertation investigates the development of the students’ proficiency in the fraction domain from grade 4 through 9 and analyzes how textbooks support students in the transition from primary to secondary education and in reaching the formal understanding that is needed for the transition from arithmetic to algebra. In doing so, possible footholds for improvement of instruction and contributions to theory building are investigated. This dissertation follows two lines of research. In the first part, the proficiency of students is studied using the test results of 1485 students from grades 4 through 9. The second part concentrates on textbooks and how the curriculum can account for the result we found in the first line of research. In the final chapter, these two perspectives on the curriculum are reflected upon. We started our study with the development of a test to provide us with detailed empirical data on the proficiency of the students (Chapter 2). This test has been the basis of our analysis of student proficiency. The test is designed as a paper and pencil tests to allow for efficient assessment of a large number of students in grades 4 through 9. The theoretical framework of this test is based on literature on the learning of fractions and the nature of the fraction domain itself. We identify five so-called ‘big ideas’ that describe the domain at the level of underlying concepts: relative comparison, equivalence, reification, from natural to rational numbers and relation division-multiplication. A list of so-called ‘complexity factors’ describes the external characteristics of tasks that influence their difficulty. The systematic construction of the test, using our framework of big ideas and complexity factors, allows for an analysis of test results at two different levels. The first level of analysis involves an item per item analysis of the types of tasks the students can or cannot answer correctly. The second, more complex level of analysis involves the combination of tasks, aiming at insight in the students’ understanding of the concepts underlying fraction proficiency. In Chapter 3 we exemplify how our test meets three pre-set criteria: (1) Is it possible to construct a single linear scale for fraction proficiency on which all our items can be ordered according to their difficulty?, (2) Can the development of proficiency from grade 4 to 9 be described with the results of this test?, and (3) Does this provide diagnostic data which can be used for improving instruction? A single linear scale for fraction proficiency is created employing a Rasch model on the data from the fraction test. This analysis and evaluation of the results regarding all 1485 students and 169 test items results in positive conclusions about the validity and the reliability of the test. The data fits the Rasch model and we conclude that our test measures one latent trait. The Rasch scale based on our test serves as scale for proficiency in fractions in the remainder of our studies. The examples of possible analyses focus on the proficiency of students in grade 6 regarding the addition of fractions and three of the big ideas, namely unit, equivalence and the development of rational number. These examples show how the development of proficiency can be described at the levels of tasks and items and that this provides footholds for improving instruction. The analyses show that the students were able to add and subtract fractions with a common denominator, but had not yet mastered the addition and subtraction with unlike denominators. Regarding the big idea of unit, most students were able to answer items on naming parts and representing fractions with part-whole models. Most of the students did not master conceptual mapping in a context or in relation to the numberline. Regarding the idea of equivalence, the students were able to reduce fractions to its lowest term, but had difficulty in recognizing fractions as equivalent when there was no whole number factor between numerators. Finally, regarding the development of rational number, the students mastered improper fractions on the numberline, but were not able to use improper fractions/mixed numbers in multiplication and division tasks. For the students, improper fractions appeared to have not become rational numbers that have the character of object like entities that can be used as a number. Overall, the analyses show that the students were capable to solve tasks that ask for reproduction and procedural use of symbols and operations. However, tasks that differ from standard and ask for more conceptual understanding surpassed the ability of most of the 6th grade students. In Chapter 4 we analyse the results of secondary education. Our expectation was to find that students would have deepened their understanding thanks to their experience in using fractions in a variety of tasks. However, it shows that there was no significant progress in fraction proficiency in lower secondary education. This result mirrors the lack of explicit attention to fractions in textbooks. We find problems in two different lines. On the item level we find that the grade 9 students had not developed generalized strategies for the basic operations. Instead these students developed number specific strategies, which they could not generalize over various situations. This finding corresponds with our findings on textbooks (Chapter 5 and Chapter 6). The analysis at concept level indicates that the students lacked the notion of a fraction as a division and the interpretation of a fraction as both two numbers and one, an interpretation that relates to the so-called ratio-rate duality. We conclude that the proficiency level of most students is insufficient for the transition to algebra, since the students in grade 9 are still in the process of mastering general ways of solving fraction tasks that involve addition, subtraction, multiplication and division. That is, for algebra, plain knowledge of these general rules for arithmetic as well as conceptual understanding of these operations and their relations with other concepts underlying the domain of fraction, are a prerequisite. Although our results are based on the students of one school and one textbook series, we argue that the results are representative. Our analysis of textbooks shows that the two major textbook series (with a current estimated total market share of 90% do not differ much in respect to the explicit attention to fraction arithmetic and understanding (Chapter 5). Furthermore, the results are corroborated by data that we collected in tertiary education, where we tested 97 first year science students using a selection of our items. In this dissertation not only students’ proficiency but also the formal curriculum is analyzed. In the analyses in part 2, we consider textbooks as representative for the curriculum. We analyze the texts and inscriptions on fractions in mathematics textbooks for grade 6 and 7 and study how these textbooks relate to prototypical instructional sequences and tasks produced in design research, and to the principles of realistic mathematics education (RME).We assume that aspects of an educational system are reflected in the textbooks. We further assume that at the same time textbooks determine instructional practices to a large extent. In the Netherlands this connection between educational practice and the content of the textbooks may even be tighter than in other countries. Analysis of Dutch textbook use in mathematics education shows a strong connection between the textbook and the practice of teaching, both in primary and secondary education. The link between textbook and educational practice is dialectic, in that teachers not only rely heavily on the textbook regarding content and pedagogy, textbooks can also be regarded as the product of the culture among teachers. Given this strong relationship between instructional practices and textbooks, we assume that actual incoherencies between practices in primary and secondary mathematics education can be observed in the textbooks. In Chapter 5, we focus on the transition from primary to secondary education starting from a CHAT perspective. We hypothesize that part of the problems with fraction proficiency may be explained by the fact that the fraction curriculum stretches over a large number of years, in the Dutch curriculum even over primary and secondary education. Among others, the confusion of tongue in the public debate gave reason to believe that some of the problems stem from differences in the educational practices in the two educational systems. We hypothesize that the cultural and historically determined traditions of both primary and secondary education, will express themselves in the educational practices and the use of artifacts, leading to confusing differences in the epistemological messages expressed. We have found that differences between these educational systems result in a gap between primary and secondary education textbooks with regard to fraction multiplication. We argue that this gap is difficult for students to overcome. Given the different traditions in both educational systems, we illustrate how similar artifacts get a different meaning in the context of each of these traditions. In secondary education textbooks, students are expected to use different strategies for solving tasks which require an understanding of fractions, which is more formal than can be expected on the basis of the primary school textbooks. Our analysis also points to differences between primary and secondary education implying that students develop an "operational conception" of fractions in primary school, whereas they are expected to reason with a "structural conception" of fractions in secondary school. In neither systems students are supported in making the transition to a structural conception. Moreover, primary school textbooks aim at training number-specific solution procedures which are likely to become a barrier for the generalization and formalization that is required for coming to understand fractions at a more formal level. However, differences stay hidden because textbooks in primary and secondary school use similar artifacts, with different meanings. Furthermore, such differences are usually unnoticeable for participants in each of the educational systems, because these participants bring with them different frameworks of reference. In Chapter 6 we address the content of the textbooks in primary school and its relation to prototypical work in design research on fractions and the basic principles of RME. It shows that the way this prototypical work has been incorporated in the textbooks leads to a number of very distinct procedures for multiplying fractions. That is, multiplication of fractions became a set of number specific procedures. We distinguish four compartmentalized interpretations of the multiplication sign. In the textbooks, tasks on a "whole number times proper fractions" are related to repeated addition. Tasks on "proper fraction times whole number" are to be interpreted as part of that whole number. Tasks on "proper fraction times proper fraction" are connected with multiplication as an area and finally "multiplication with mixed numbers" is related to splitting. The test results show that this ompartmentalization is still observable in the performance of students in grade 6 and 7. From the outcomes of the analysis of the empirical data it follows that the success rate and type of errors are related to the order of operands and the abstraction of the problem formulation. We conclude that this compartmentalization may hinder the process towards more formal understanding of multiplication of fractions. Thus, the versatile approach of mathematical concepts is not yet elaborated in such a manner that it allows students to come to grips with the conventional mathematical method for multiplying fractions as intended. In reflection, we show that the use of contexts has two sides. On the one hand it both affords and supports students’ in building upon informal knowledge. On the other hand however, contexts can hinder mathematization when their inherent characteristics become constraints in the generalization process. In conclusion, we find that problems in the fraction domain start already early in education and continue in later grades. Although the proficiency of the students developed considerably from grade 4, analysis of the students’ proficiency level at the end of grade 6 reveals a lack of deeper understanding of fractions. Corresponding to the textbooks, the teaching of fractions and fraction arithmetic in primary education is directed towards procedural problem solving of standard tasks. The students did not develop deeper understanding of underlying concepts such as unit, rational number and equivalence. Moreover, strategies that students did learn appeared to be number-specific. In the transition from primary to secondary education, problems arise because of an apparent misunderstanding of the intended level of proficiency at the end of primary school. From the results of both our test (Chapter 4) and the analysis of primary and secondary school textbooks (Chapter 5) it follows that secondary school teachers, and textbook authors do not realize that the fraction curriculum is not completed in primary school. In addition, insufficient attention is given to generalizing and formalizing in secondary education. This dissertation provides new insights in the Dutch curriculum for fractions especially regarding the transition from primary education, compartmentalization in primary education and preparation for basic algebraic skills. In Chapter 7 we reflect on these findings and discuss the broader significance of the results. We argue that our results represent more general problems in the mathematics curriculum. We discuss the influence of the educational setting by identifying problems in proceduralization, local analysis, contexts, and the role of teachers. Further, we discuss where mathematics education deviated from the nature of academic mathematics. We argue for more emphasis on structure, coherence and ambiguity. That is, we think that there is a need for more attention to the ambiguity of mathematical symbols and concepts, for more attention to mathematical structures and to the relation between concepts instead of an exclusive attention for algorithms to solve bare arithmetic tasks. Finally, we recommend changes for textbooks, the transition from primary to secondary education, and the transition to higher education.
    Original languageEnglish
    QualificationDoctor of Philosophy
    Awarding Institution
    • Eindhoven School of Education
    • Gravemeijer, Koeno, Promotor
    • Jochems, Wim, Promotor
    • van Eijck, Michiel, Copromotor
    Award date21 Dec 2010
    Place of PublicationEindhoven
    Print ISBNs978-90-386-2400-6
    Publication statusPublished - 2010


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