Katherine M. Robinson

Children's Understanding of Addition and Subtraction Concepts.

After the onset of formal schooling, little is known about the development of childrens understanding of the arithmetic concepts of inversion and associativity. On problems of the form a+b-b (e.g., 3+26-26), if children understand the inversion concept (i.e., that addition and subtraction are inverse operations), then no calculations are needed to solve the problem. On problems of the form a+b-c (e.g., 3+27-23), if children understand the associativity concept (i.e., that the addition and subtraction can be solved in any order), then the second part of the problem can be solved first. Children in Grades 2, 3, and 4 solved both types of problems and then were given a demonstration of how to apply both concepts. Approval of each concept and preference of a conceptual approach versus an algorithmic approach were measured. Few grade differences were found on either task. Conceptual understanding was greater for inversion than for associativity on both tasks. Clusters of participants in all grades showed that some had strong understanding of both concepts, some had strong understanding of the inversion concept only, and others had weak understanding of both concepts. The findings highlight the lack of developmental increases and the large individual differences in conceptual understanding on two arithmetic concepts during the early school years.

A Microgenetic Study of the Multiplication and Division Inversion Concept

This microgenetic study investigated the discovery and development of the multiplication and division concept of inversion. Little is known about multiplicative concepts relative to additive concepts, including the inversion concept. Grade 6 participants (mean age = 11 years 6 months) solved multiplication and division inversion problems (e.g., d x e/e) for several weeks. In the final week they solved inversion, modified inversion (e.g., e x d/e), and lure problems (e.g., d/e x d) to investigate transfer of knowledge. Despite years of formal arithmetic instruction and repeated exposure to inversion problems, over a third of the participants failed to discover the inversion-based shortcut whereas another third used the shortcut almost exclusively. Almost all participants had difficulty appropriately generalising the inversion concept. Current theories of mathematical understanding may need to be modified to include the developmental complexities of multiplicative concepts.

Children's Use of Arithmetic Shortcuts: The Role of Attitudes in Strategy Choice

Current models of strategy choice do not account for childrens attitudes towards different problem solving strategies. Grade 2, 3, and 4 students solved three sets of three-term addition problems. On inversion problems (e.g., 4

A microgenetic study of simple division.

How simple division strategies develop over a short period of time was examined with a microgenetic study. Grade 5 students (mean age = 10 years, 3 months) solved simple division problems in 8 weekly sessions. Performance improved with faster and more accurate responses across the study. Consistent with R. S. Sieglers (1996) overlapping waves model, strategies varied in their use. Direct retrieval increased, retrieval of multiplication facts remained steady, and addition facts, derived facts, and special tricks marginally decreased. Consistent with previous research, multiplication fact retrieval was the most common strategy, although it was slower and more error prone than direct retrieval. Strategy variability within and across individuals was striking across all of the sessions and underscores Sieglers (1996) assertion that development is in a constant transitional state.

Children’s understanding of additive concepts

Most research on childrens arithmetic concepts is based on one concept at a time, limiting the conclusions that can be made about how childrens conceptual knowledge of arithmetic develops. This study examined six arithmetic concepts (identity, negation, commutativity, equivalence, inversion, and addition and subtraction associativity) in Grades 3, 4, and 5. Identity (a-0=a) and negation (a-a=0) were well understood, followed by moderate understanding of commutativity (a+b=b+a) and inversion (a+b-b=a), with weak understanding of equivalence (a+b+c=a+[b+c]) and associativity (a+b-c=[b-c]+a). Understanding increased across grade only for commutativity and equivalence. Four clusters were found: The Weak Concept cluster understood only identity and negation; the Two-Term Concept cluster also understood commutativity; the Inversion Concept cluster understood identity, negation, and inversion; and the Strong Concept cluster had the strongest understanding of all of the concepts. Grade 3 students tended to be in the Weak and Inversion Concept clusters, Grade 4 students were equally likely to be in any of the clusters, and Grade 5 students were most likely to be in the Two-Term and Strong Concept clusters. The findings of this study highlight that conclusions about the development of arithmetic concepts are highly dependent on which concepts are being assessed and underscore the need for multiple concepts to be investigated at the same time.

The Understanding of Additive and Multiplicative Arithmetic Concepts

Children’s understanding of arithmetic has been gaining increased attention from both researchers and educators as it plays an integral role in how children perform in both early and later mathematical tasks. However, most of the research on arithmetic concepts has focused on additive concepts. This chapter discusses not only the development of children’s additive but also multiplicative concepts by specifically examining the concepts of inversion and associativity. Not only are there developmental changes in conceptual understanding, but children’s and adults’ understanding of arithmetic concepts varies across arithmetic operations. Further, marked individual differences in conceptual knowledge exist and are related to factors, such as working memory and inhibition, as well as the roles of attitudes and educational experiences. Finally, future directions for researchers and educators are discussed.

Understanding arithmetic concepts: Does operation matter?

Most research on childrens arithmetic concepts is based on (a) additive concepts and (b) a single concept leading to possible limitations in current understanding about how childrens knowledge of arithmetic concepts develops. In this study, both additive and multiplicative versions of six arithmetic concepts (identity, negation, commutativity, equivalence, inversion, and associativity) were investigated in Grades 5, 6, and 7. The multiplicative versions of the concepts were more weakly understood. No grade-related differences were found in conceptual knowledge, but older children were more accurate problem solvers. Individual differences were examined through cluster analyses. All children had a solid understanding of identity and negation. Some children had a strong understanding of all the concepts, both additive and multiplicative; some had a good understanding of equivalence or commutativity; and others had a weak understanding of commutativity, equivalence, inversion, and associativity. Associativity was the most difficult concept for all clusters. Grade did not predict cluster membership. Overall, these results demonstrate the breadth of individual variability in conceptual knowledge of arithmetic as well as the complexity in how childrens understanding of arithmetic concepts develops.

Frameworks for Effective Screen-Centred Interfaces

At the union of the humanities and technology, computer interfaces are often studied technically and from a psychological point of view, but more rarely do such studies include a broader perspective connecting cultural theories and cognitive processes to the transformation of user interfaces as the screen real estate changes. This paper introduces a research framework for such research that the authors have developed for repeatable, broadly scoped experiments aimed at the identification of the relationship between screen-centered cultures and user interface semantics. A first experiment based on this framework is then illustrated. Although the experiment described has not come to an end yet, the aim of this paper is to propose the framework as a collaborative tool for researchers in the humanities, social sciences, sciences and engineering that allows an integrated approach identifying interdisciplinary contributions and discipline transitions, with the clearly positive gains that such an approach affords.