Adam K. Dubé

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.

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.

Intuitive or idiomatic: An interdisciplinary study of child-tablet computer interaction

Using Luhmanns communication framework, we examine the interaction implications for kindergarten to Grade 2 students using mathematics applications on four types of tablet computers. Research questions included what content is communicated between the child and the tablet computer and how engaged are children in the interaction. We found that mathematics applications developers have focused on creating applications for the practice of a priori knowledge, rather than on creating instructional applications. Results show preliminary evidence that child‐tablet communication is generally successful, but this success comes at the cost of richer, multimodal interactions. Tablet computer application developers are being cautious in offering a variety of options for children to interact with the devices, and we suggest that there is scope for a broadening of communicative interaction modes.

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.

Are Games a Viable Home Numeracy Practice

The promise that mathematics can be taught from a young age through fun, engaging games is an enticing proposition, so enticing that considerable effort has been expended to determine whether gaming is an effective teaching tool. This effort includes not just individual investigations but multiple meta-analyses with a combined evaluation of over 500 studies testing the effectiveness of educational games. Despite all of this interest and the resulting concerted research effort, there is no clear answer as to whether games are a viable tool for improving young children’s numeracy ability. This ambiguity is due to math game research not addressing two critical questions: what is a game and how does learning occur within the context of a game. Answering these questions is the first step in determining whether math games can be both fun and formative.

Intuitive or Idiomatic? An information‐cognitive psychology study of child‐tablet computer interaction

Using Luhmanns communication framework we examine the interaction implications that arise when kindergarten-grade 2 students use mathematics applications on four types of tablet computers. We asked a) what content is communicated between the child and the tablet computer, b) how is content communicated, c) how engaged are children in the tablet-child interaction, and d) what factors influence this engagement. We found that mathematics applications developers across the four platforms have focused on creating applications for the practice of a priori knowledge rather than on creating instructional applications. Also, the overall the emphasis in the applications studied was not to ‘gamify’ mathematics by providing entertainment but to offer a more traditional pedagogy, and that currently there is a low degree of diversity from developers in making use of the range of affordances of tablet computers. This communication studies-psychology interdisciplinary study offers a new conceptual approach to the study of child-tablet interaction.