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Research shows that students, and sometimes teachers, have trouble with fractions, especially conceiving of fractions as numbers that extend the whole number system. This paper explores how fractions are addressed in undergraduate mathematics courses for prospective elementary teachers (PSTs). In particular, we explore how, and whether, the instructors of these courses address fractions as an extension of the whole number system and fractions as numbers in their classrooms. Using a framework consisting of four approaches to the development of fractions found in history, we analyze fraction lessons videotaped in six mathematics classes for PSTs. Historically, the first two approaches-part-whole and measurement-focus on fractions as parts of wholes rather than numbers, and the last two approaches-division and set theory-formalize fractions as numbers. Our results show that the instructors only implicitly addressed fraction-as-number and the extension of fractions from whole numbers, although most of them mentioned or emphasized these aspects of fractions during interviews. [ABSTRACT FROM AUTHOR]

In this paper, we focus on Finnish pre-service elementary teachers’ ( N = 269) and upper secondary students’ ( N = 1,434) understanding of division. In the questionnaire, we used the following non-standard division problem: “We know that 498:6 = 83. How could you conclude from this relationship (without using long-division algorithm) what 491:6 = ? is?” This problem especially measures conceptual understanding, adaptive reasoning, and procedural fluency. Based on the results, we can conclude that division seems not to be fully understood: 45% of the pre-service teachers and 37% of upper secondary students were able to produce complete or mainly correct solutions. The reasoning strategies used by these two groups did not differ very much. We identified four main reasons for problems in understanding this task: (1) staying on the integer level, (2) an inability to handle the remainder, (3) difficulties in understanding the relationships between different operations, and (4) insufficient reasoning strategies. It seems that learners’ reasoning strategies in particular play a central role when teachers try to improve learners’ proficiency. [ABSTRACT FROM AUTHOR]

When we consider the gap between mathematics at elementary and secondary levels, and given the logical nature of mathematics at the latter level, it can be seen as important that the aspects of children's logical development in the upper grades in elementary school be clarified. In this study we focus on the teaching and learning of “division with decimals” in a 5th grade classroom, because it is well known to be difficult for children to understand the meaning of division with decimals, caused by certain conceptions which children have implicitly or explicitly. In this paper we discuss how children develop their logical reasoning beyond such difficulties/misconceptions in the process of making sense of division with decimals in the classroom setting. We then suggest that children's explanations based on two kinds of reversibility (inversion and reciprocity) are effective in overcoming the difficulties/misconceptions related to division with decimals, and that they enable children to conceive multiplication and division as a system of operations. [ABSTRACT FROM AUTHOR]

This paper explores written calculation methods for division used by pupils in England (n = 276) and the Netherlands (n = 259) at two points in the same school year. Informal strategies are analysed and progression identified towards more structured procedures that result from different teaching approaches. Comparison of the methods used by year 5 (Group 6) pupils in the two countries shows greater success in the Dutch approach, which is based on careful progression from informal strategies to more structured and efficient procedures. This success is particularly notable for the girls in the sample. For the English pupils, whose written solutions largely involved the traditional algorithm, the discontinuity between the formal computation procedure and informal solution strategies presents difficulties. [ABSTRACT FROM AUTHOR]

This article deals with the roles of variation problems ('one problem multiple solution' and 'one problem multiple changes') as used in Chinese textbooks. It is argued that variation problems as an 'indigenous' Chinese practice aim to discern and to compare the invariant feature of the relationship among concepts and solutions. This practice also aims to provide opportunities for making connections, since comparison is considered the pre-condition to perceive the structures, dependencies, and relationships that may lead to mathematical abstraction. In the first part of the article, the 'indigenous' practice is discussed against its philosophical Daoism and Confucianism backgrounds. To grasp its distinctiveness, a comparison between Chinese and American textbooks is carried out. In the second part of the article, the focus is on the manner in which fraction division is articulated in an important Chinese textbook. A framework to understand variation practice is introduced and some educational implications are suggested. [ABSTRACT FROM AUTHOR]

Inversion is a fundamental relational building block both within mathematics as the study of structures and within people's physical and social experience, linked to many other key elements such as equilibrium, invariance, reversal, compensation, symmetry, and balance. Within purely formal arithmetic, the inverse relationships between addition and subtraction, and multiplication and division, have important implications in relation to flexible and efficient computation, and for the assessment of students' conceptual understanding. It is suggested that the extensive research on arithmetic should be extended to take account of numerical domains beyond the natural numbers and of the difficulties students have in extending the meanings of operations to those of more general domains. When the range of situations modelled by the arithmetical operations is considered, the complexity of inverse relationships between operations, and the variability in the forms that these relationships take, become much greater. Finally, some comments are offered on the divergent goals and preoccupations of cognitive psychologists and mathematics educators as illuminated by research in this area. [ABSTRACT FROM AUTHOR]

Researchers have speculated that children find it more difficult to acquire conceptual understanding of the inverse relation between multiplication and division than that between addition and subtraction. We reviewed research on children and adults' use of shortcut procedures that make use of the inverse relation on two kinds of problems: inversion problems (e.g., $$ {9} \times {24} \div {24} $$) and associativity problems (e.g., $$ {9} \times {24} \div {8} $$). Both can be solved more easily if the division of the second and third numbers is performed before the multiplication of the first and second numbers. The findings we reviewed suggest that understanding and use of the inverse relation between multiplication and division develops relatively slowly and is difficult for both children and adults to implement in shortcut procedures if they are not flexible problem solvers. We use the findings to expand an existing model, highlight some similarities and differences in solvers' use of conceptual knowledge across operations, and discuss educational implications of the findings. [ABSTRACT FROM AUTHOR]