Jeffrey Bisanz

Pathways to Mathematics: Longitudinal Predictors of Performance.

A model of the relations among cognitive precursors, early numeracy skill, and mathematical outcomes was tested for 182 children from 4.5 to 7.5 years of age. The model integrates research from neuroimaging, clinical populations, and normal development in children and adults. It includes 3 precursor pathways: quantitative, linguistic, and spatial attention. These pathways (a) contributed independently to early numeracy skills during preschool and kindergarten and (b) related differentially to performance on a variety of mathematical outcomes 2 years later. The success of the model in accounting for performance highlights the need to understand the fundamental underlying skills that contribute to diverse forms of mathematical competence.

Cognitive arithmetic: evidence for obligatory activation of arithmetic facts.

In two experiments, obligatory activation of arithmetic facts resulted in interference on a simple number-matching task. Subjects were required to verify the presence of a probe number (e.g., 5) in a previously presented pair (e.g., 5+1). Items for which the probe was the sum of the initial pair (e.g., 5+1 and 6) were rejected more slowly than items for which the probe was not the sum (e.g., 5+1 and 3), and this effect was largest at stimulus onset asynchronies of less than 180 msec between the number pair and the probe. The results are consistent with the notion that arithmetic knowledge is represented in an associative network and accessed by means of spreading activation.

Effects of Age and Schooling on the Acquisition of Elementary Quantitative Skills.

The influence of school- and age-related variables was examined separately on 2 tasks involving elementary quantitative skills: conservation of number and mental addition. Performance on these tasks was compared by using a cutoff design with 3 groups of kindergarten and Grade 1 children who differed in age but not amount of schooling (grade), in schooling but not age, or in both age and schooling. The effects of age and schooling were distinct. On conservation of number, performance improved as a function of age but not schooling. On mental arithmetic, accuracy improved with schooling rather than age, but childrens use of various solution procedures (e.g., retrieval, counting) was not influenced by schooling. Thus, in the domain of elementary mathematical skills, the influence of schooling can be very specific, and age-related variables other than schooling play an important role in the development of elementary mathematical skills. Results illustrate the utility of the cutoff design for investigating instructional and developmental influences on cognitive development

Executive Functioning in Children with Fetal Alcohol Spectrum Disorders: Profiles and Age-Related Differences

The goal of this project was to examine the profile of executive function (EF) deficits and age-related differences among children with FASD. Twenty-nine children with FASD (8 to 16 years of age) completed 8 tests from the Delis-Kaplan Executive Function System (D-KEFS). They had difficulty on many components of EF including cognitive flexibility, inhibition, some measures of verbal fluency, abstract thinking, deductive reasoning, hypothesis testing, problem solving, and concept formation. A distinctive profile emerged with performance being poorest on the card sorting test and relatively high on category fluency, design fluency, and the tower test, indicating relative strengths on some visual-spatial EF tasks. Older children with FASD showed more difficulty (relative to the norm) on some verbal tests of EF than younger children with FASD, suggesting that difficulty on some verbal EF tasks appears to become more pronounced with increasing age.

Computational Skills, Working Memory, and Conceptual Knowledge in Older Children With Mathematics Learning Disabilities

Knowledge and skill in multiplication were investigated for late elementary-grade students with mathematics learning disabilities (MLD), typically achieving age-matched peers, low-achieving age-matched peers, and ability-matched peers by examining multiple measures of computational skill, working memory, and conceptual knowledge. Poor multiplication fact mastery and calculation fluency and general working memory discriminated children with MLD from typically achieving age-matched peers. Furthermore, children with MLD were slower in executing backup procedures than typically achieving age-matched peers. The performance of children with MLD on multiple measures of multiplication skill and knowledge was most similar to that of ability-matched younger children. MLD may be due to difficulties in computational skills and working memory. Implications for the diagnosis and remediation of MLD are discussed.

Use of the Mathematical Principle of Inversion in Young Children.

An important issue in the development of mathematical cognition is the extent to which children use and understand fundamental mathematical concepts. We examined whether young children successfully use the principle of inversion and, if so, whether they do so based on qualitative identity, length, or quantity. Twenty-four preschool children and 24 children in Grade 1 were presented with three-term inversion problems (e.g., 3+2-2) and standard problems of similar magnitude (e.g., 2+4-3). Problems were presented in three conditions to determine whether children used inversion at all and, if so, whether their decisions were based on quantitative or nonquantitative features of the problems. Both preschool and Grade 1 children showed evidence of using inversion in a fully quantitative manner, indicating that this principle is available in some form prior to extensive formal instruction in arithmetic.

The role of executive attention in the acquisition of mathematical skills for children in Grades 2 through 4.

We examined the role of executive attention, which encompasses the common aspects of executive function and executive working memory, in childrens acquisition of two aspects of mathematical skill: (a) knowledge of the number system (e.g., place value) and of arithmetic procedures (e.g., multi-digit addition) and (b) arithmetic fluency (i.e., speed of solutions to simple equations such as 3+4 and 8-5). Children in Grades 2 and 3 (N=157) completed executive attention and mathematical tasks. They repeated the mathematical tasks 1 year later. We used structural equation modeling to examine the relations between executive attention and (a) concurrent measures of mathematical knowledge and arithmetic fluency and (b) growth in performance on these measures 1 year later. Executive attention was concurrently predictive of both knowledge and fluency but predicted growth in performance only for fluency. A composite language measure predicted growth in knowledge from Grade 2 to Grade 3. The results support an important role for executive attention in childrens acquisition of novel procedures and the development of automatic access to arithmetic facts.

Information processing and cognitive development.

Publisher Summary This chapter characterizes information processing as a general framework for understanding human cognitive growth that has had enormous impact on the study of cognition and over the past decade it has been adopted by a growing number of developmental psychologists. Journals and books contain numerous articles about the development of information-processing skills in children, in sharp contrast to the recent past when such topics were only rarely mentioned. It is emphasized that information processing, as a general perspective, has considerable potential for developmental work, and some relevant characteristics and implications of information processing is described. Information processing was not influential in developmental psychology until the late 1960s and early 1970s. During this period two essentially independent events brought information processing to the forefront of developmental research. First, psychologists studying the development of attention and memory based their work, in part, on information-processing models derived from experimental psychology. Second, several psychologists from the information-processing became interested in Piagets description of childrens understanding of concepts like transitivity and class inclusion. These psychologists proposed radically different interpretations of the phenomena, and their research sometimes produced findings that were hard to reconcile with Piagets account of development.

Chapter 3 Understanding Elementary Mathematics

Summary Understanding often is defined inconsistently, ambiguously, or narrowly, and consequently the relation between understanding and cognitive processing on mathematical tasks is not very clear. Many forms of behavior are related to understanding, and a framework is needed to describe these various forms in an integrated way. A “contextual space” is proposed for classifying different types of performance related to understanding elementary mathematical concepts. The two dimensions of this space are the type of activity involved (applying, justifying, and evaluating solution procedures) and the degree of generality with which these activities are exercised. These two aspects can be used to construct a “profile” that reflects the contexts in which an individual shows various forms of understanding. Acquisition of understanding can be described in terms of the sequences of profiles that emerge, and these sequences have implications for characterizing the mechanisms underlying changes in knowledge.

The Extension of the Interference Effect to Multiplication

Using multiplication facts, this experiment demonstrated an interference effect in the number-matching task. Here, subjects verified the presence of a target number (e.g., 8) in a previously presented cue (e.g., 5 x 8) that was masked after 60 ms. The SOAs between cue and target were 100, 120, 220, and 350 ms. Subjects were slower to reject targets that were the product of the cue (e.g., 40) than to reject unrelated targets (e.g., 42), but this was true only at the 100- and 120-ms SOAs (i.e., the interference effect). This pattern is consistent with the interference effect found by LeFevre and colleagues using addition facts. Furthermore, the present result supports the interpretation that the interference effect previously found with addition facts was due to obligatory activation and not to automatic counting.