Scientific Understanding and Mathematical Abstraction

This paper argues for two related theses. The first is that mathematical abstraction can play an important role in shaping the way we think about and hence understand certain phenomena, an enterprise that extends well beyond simply representing those phenomena for the purpose of calculating/predicting their behaviour. The second is that much of our contemporary understanding and interpretation of natural selection has resulted from the way it has been described in the context of statistics and mathematics. I argue for these claims by tracing attempts to understand the basis of natural selection from its early formulation as a statistical theory to its later development by R.A. Fisher, one of the founders of modern population genetics. Not only did these developments put natural selection of a firm theoretical foundation but its mathematization changed the way it was understood as a biological process. Instead of simply clarifying its status, mathematical techniques were responsible for redefining or reconceptualising selection. As a corollary I show how a highly idealised mathematical law that seemingly fails to describe any concrete system can nevertheless contain a great deal of accurate information that can enhance our understanding far beyond simply predictive capabilities.

The effects of cumulative practice on mathematics problem solving

This study compared three different methods of teaching five basic algebra rules to college students. All methods used the same procedures to teach the rules and included four 50-question review sessions interspersed among the training of the individual rules. The differences among methods involved the kinds of practice provided during the four review sessions. Participants who received cumulative practice answered 50 questions covering a mix of the rules learned prior to each review session. Participants who received a simple review answered 50 questions on one previously trained rule. Participants who received extra practice answered 50 extra questions on the rule they had just learned. Tests administered after each review included new questions for applying each rule (application items) and problems that required novel combinations of the rules (problem-solving items). On the final test, the cumulative group outscored the other groups on application and problem-solving items. In addition, the cumulative group solved the problem-solving items significantly faster than the other groups. These results suggest that cumulative practice of component skills is an effective method of training problem solving.     

The influence of activity choice on problem behaviors maintained by escape versus attention

This study assessed whether the function of an individual's problem behavior was related to the effectiveness of an intervention involving choice among tasks. Analogue functional analyses were conducted with 7 students with various diagnoses to determine whether problem behaviors were maintained by escape or attention. Following identification of the function of each student's problem behavior, reversal designs were used to assess the effectiveness of an intervention that allowed the students to choose their own instructional tasks. Results showed that students who displayed escape-maintained problem behavior showed substantial reductions in such behavior when they were provided with opportunities to choose among tasks. On the other hand, students who displayed attention-maintained problem behavior did not show any effects as a result of the choice intervention. These findings are discussed in terms of the effective use of behavior management programs involving choice and the reduction of problem behavior.     

Age, memory load, and individual differences in working memory as determinants of class-inclusion reasoning.

We studied the effects of individual differences in speak-span scores and variations in memory demands on the class-inclusion performance of 10-, 13-, and 15-year-old children. The speak-span task was an age-appropriate modification of Daneman and Carpenter's (1980) reading-span task and was considered to be a measure of global resources. The age variable was assumed to be a global index of skill development, and some of the specific skills hypothesized to be important in class-inclusion reasoning were estimated using a mathematical model. The results from both regression analyses and the mathematical model indicated that differences in age, speak span, and memory load all affected performance. Surprisingly, the effects of speak span and memory load were independent. However, the effects of each of these variables depended on the age level of the participants. Based on these findings, we argued that (a) resources vary continuously with age, (b) both skill level and global resources should be varied in developmental studies of problem solving, and (c) resource theories (e.g., Norman & Shallice, 1986) should be modified to account for developmental change.     

Sibling collusion and problem behavior in early adolescence: toward a process model for family mutuality

Sibling collusion is a process by which siblings form coalitions that promote deviance and undermine parenting. Collusive sibling processes were identified and measured using macro ratings of videotaped family interactions. Hypotheses were tested on a multiethnic sample of urban youth, with a target child identified as either high risk (n = 26) or normative (n = 26), and their families. Siblings in families with a high-risk target child showed reliably higher rates of collusion than those in families with a normative target child. Sibling collusion also accounted for variance in problem behavior after controlling for involvement with deviant peers. Findings suggest that deviant conduct forms a common ground among siblings, potentially amplifying risk of mutuality in problem behavior during early adolescence. These data also indicate that attention to sibling relationship processes is relevant to family interventions designed to mitigate the development of behavior problems.     

Spatial breakdown in spatial construction: evidence from eye fixations in children with Williams syndrome

We investigated the role of executive and spatial representational processes in impaired performance of block construction tasks by children with Williams syndrome (WS), a rare genetic defect that results in severely impaired spatial cognition. In Experiment 1, we examined performance in two kinds of block construction tasks, Simple Puzzles, in which block faces contained a single color, and Complex, in which some block faces contained an arrangement of two colors. WS and control children were comparable in their ability to solve simple puzzles, and showed similar eye-fixation patterns, suggesting that basic executive processes were intact. However, WS children were severely impaired in their ability to solve complex puzzles. In these puzzles, WS children fixated the complex puzzle models and checked their partial solutions less often than normal children, but they were comparable in their ability to detect errors in their copies and almost exclusively made repairs to copies that were, in fact, incorrect. We conjecture that the abnormal fixation patterns were a consequence of impoverished spatial representations, rather than a cause of it. This conjecture was tested in Experiment 2, where we examined children's capacity to match and place individual blocks without engaging the complex executive processes required to carry out a complete puzzle solution. We found serious deficiency among WS children in both aspects of spatial representation. Moreover, estimates of the errors in representing the identity and location of model blocks derived from Experiment 2 provided a good account of the observed errors in the block construction task of Experiment 1.     

Functional analysis of problem behavior: a review

Functional analysis methodology focuses on the identification of variables that influence the occurrence of problem behavior and has become a hallmark of contemporary approaches to behavioral assessment. In light of the widespread use of pretreatment functional analyses in articles published in this and other journals, we reviewed the literature in an attempt to identify best practices and directions for future research. Studies included in the present review were those in which (a) a pretreatment assessment based on (b) direct observation and measurement of (c) problem behavior was conducted under (d) at least two conditions involving manipulation of an environmental variable in an attempt (e) to demonstrate a relation between the environmental event and behavior. Studies that met the criteria for inclusion were quantified and critically evaluated along a number of dimensions related to subject and setting characteristics, parametric and qualitative characteristics of the methodology, types of assessment conditions, experimental designs, topographies of problem behaviors, and the manner in which data were displayed and analyzed.     

Global optimality of the successive Maxbet algorithm

The Maxbet method is an alternative to the method of generalized canonical correlation analysis and of Procrustes analysis. Contrary to these methods, it does not maximize the inner products (covariances) between linear composites, but also takes their sums of squares (variances) into account. It is well-known that the Maxbet algorithm, which has been proven to converge monotonically, may converge to local maxima. The present paper discusses an eigenvalue criterion which is sufficient, but not necessary for global optimality. However, in two special cases, the eigenvalue criterion is shown to be necessary and sufficient for global optimality. The first case is when there are only two data sets involved; the second case is when the inner products between all variables involved are positive, regardless of the number of data sets.The authors are obliged to Henk Kiers for critical comments on a previous draft.     

小学四年级数学诊断性测验的编制与研究

诊断性测验是诊断性评价的工具。本研究以教育测量与评价的基本原理为指导,按照诊断性测验编制的方法和步骤,在详细制定了小学四年级数学教学目标体系的基础上编制出《小学四年级数学诊断性测验》并获取了该测验的信度和效度证据。本研究对中小学各科诊断性测验的编制具有普遍的指导意义,所编测验对于诊断学生学习困难、弥补教学不足有实际应用价值。