The Effects of Precision Teaching and Self-regulated Learning on Early Multiplication Fluency

Fluent recall of basic facts is essential to the development of more complex math skills. Therefore, failure to develop fluency with basic facts may impede the development of these skills. The present study used a between-group experimental design to investigate whether a basic facts fluency program, implemented within a self-regulated learner (SRL) framework, could lead to increased fluency with multiplication facts for Year 5 and Year 6 New Zealand students (9–10 years old). This study also investigated the extent to which the SRL program altered students’ basic facts practice behavior outside of school hours. The study found that the SRL program resulted in rapid fluency development that was maintained over time. Nomothetic and idiographic analysis confirmed that the program was suitable for use within Tier 1 of the response to intervention framework. In addition, the study also found that students who received the program altered their practice behavior outside school hours. The results from this study show how elements of self-regulated learning and precision teaching can be successfully combined to enhance students’ mathematics achievement.

     

Studies of Diagnosis and Remediation with High School Algebra Students

Despite extensive discussion in the literature about the diagnosis and subsequent remediation of students' errors, few studies have compared the effects of different styles of error-based remediation. Swan (1983) found that a conflict approach (pointing out errors made by students and demonstrating their consequences) was more effective than simple reteaching, but Bunderson & Olsen (1983) found no difference between error-specific remediation and reteaching. More studies are needed in order to understand the factors which lead to successful remediation. The three studies discussed in this article compared error-specific or model-based remediation (MBR) with reteaching in algebra. MBR bases its remediation on the model inferred for an individual student before reteaching the correct procedure. Reteaching simply shows students the correct procedure without addressing specific errors. The results show that MBR and reteaching are both more effective than no tutoring; however, MBR is not clearly more effective than reteaching. The results are discussed in terms of stability of errors, their relevance to educational practice and to intelligent tutoring systems (ITS). Although the studies were carried out using human tutors, the results suggest that for the purpose of remediation in the algebra domain, when taught procedurally, “classical” computer-assisted instruction (CAI) would be as effective as an ITS.     

An attempt to understand students' understanding of basic algebra

This paper reports the results obtained with a group of 24 14-year-old students when presented with a set of algebra tasks by the Leeds Modelling System, LMS. These same students were given a comparable paper-and-pencil test and detailed interviews some four months later. The latter studies uncovered several kinds of student misunderstandings that LMS had not detected. Some students had profound misunderstandings of algebraic notation: Others used strategies such as substituting numbers for variables until the equation balanced. Additionally, it appears that the student errors fall into several distinct classes: namely, manipulative, parsing, clerical, and “random.” LMS and its rule database have been enhanced as the result of this experiment, and LMS is now able to diagnose the majority of the errors encountered in this experiment. Finally, the paper gives a process-oriented explanation for student errors, and re-examines related work in cognitive modelling in the light of the types of student errors reported in this experiment. Misgeneralization is a mechanism suggested to explain some of the mal-rules noted in this study.     

Accommodating Surprise in Taxonomic Tasks: The Role of Expertise

This paper reports a psychological study of human categorization that looked at the procedures used by expert scientists when dealing with puzzling items. Five professional botanists were asked to specify a category from a set of positive and negative instances. The target category in the study was defined by a feature that was unusual, hence situations of uncertainty and puzzlement were generated. Subjects were asked to think aloud while solving the tasks, and their verbal reports were analyzed. A number of problem solving strategies were identified, and subsequently integrated in a model of knowledge‐guided inductive categorization. Our model proposes that expert knowledge influences the subjects' reasoning in more complex ways than suggested by earlier investigations of scientific reasoning. As in previous studies, domain knowledge influenced our subjects' hypothesis generation and testing; but, additionally, it played a central role when subjects revised their hypotheses.