This paper presents data from the first of three iterations of teaching experiments conducted with secondary teachers. The purpose of the experiments was to investigate how teachers' combinatorial reasoning could support their development of algebraic structure, specifically structural relationships between the roots and coefficients of…

Generalisation is a skill that enables learners to acquire knowledge in general, and mathematical knowledge in particular. It is a core aspect of algebraic thinking and, in particular, of functional thinking, as a type of algebraic thinking. Introducing primary school children to functional thinking fosters their ability to generalise, explain and…

Ambridge argues that the existence of exemplar models for individual phenomena (words, inflection rules, etc.) suggests the feasibility of a unified, exemplars-everywhere model that eschews abstraction. The argument would be strengthened by a description of such a model. However, none is provided. I show that any attempt to do so would immediately…

Learning trajectories are built upon progressions of mathematical understandings that are typical of the general population of students. As such, they are useful frameworks for exploring how understandings of diverse learners may be similar or different from their peers, which has implications for tailoring instruction. The purpose of this…

Ambridge argues that there is widespread agreement among child language researchers that learners store linguistic abstractions. In this commentary the authors first argue that this assumption is incorrect; anti-representationalist/exemplar views are pervasive in theories of child language. Next, the authors outline what has been learned from this…

This paper presents a conceptual analysis for students' images of graphs and their extension to graphs of two-variable functions. We use the conceptual analysis, based on quantitative and covariational reasoning, to construct a hypothetical learning trajectory (HLT) for how students might generalize their understanding of graphs of…

Empirical arguments rely on examples without necessarily addressing all cases. Students should be skeptical of empirical evidence and should seek more secure arguments for generalizations, such as those that explain why a generalization is true for all cases. Generalizing on the basis of patterns in data is an important mathematical practice;…

Among the 12 documents selected for this annotated bibliography of documents and journal articles in the ERIC database is an expert's argument that the brain's multipath and multimodal capacities are ignored by educators. Another writer fears that the "back-to-basics" movement may have eclipsed the prominence earlier accorded to thinking…

The concern of the experiment is to find out the roles of abstraction and generalization in the learning of mathematical structures. The basic question is whether to generalize before abstracting or vice-versa in order to maximize transfer. The experiment involves four mathematical tasks and a transfer of activity. Experimental procedures are…

This paper describes the methods and techniques called Conceptual Analysis (CA), a rigorous procedure to generate (without involuntary omissions and repetitions) knowledge bases for the development of knowledge-based systems. An introduction is given of CA and how it can be used to produce knowledge bases. A discussion is presented on what is…

The ADAPT program for freshmen emphasizes building an experiential base in introductory courses before moving to abstract formulations. The goal is to develop thinking processes that are integrated with the student's experiences. Students in the ADAPT program outperformed two comparison groups on a variety of thinking tasks. (Author)

The generalization of "trained" creativity in easel painting to untrained creativity in blockbuilding was examined in two preschool boys. Verbal reinforcement of every different form painted in a picture increased both the number of different forms per picture and new forms (those appearing for the first time in the total output of…

Thirteen learning disabled and 15 remedial high school students were taught reasoning skills using computer-assisted instruction and were given basic or elaborated corrections. Criterion-referenced test scores were significantly higher for the elaborated-corrections treatment on the post- and maintenance tests and on a transfer test assessing…

Three experiments using university students explored what constitutes an event and what determines its boundaries. Results supported the hypothesis that sequences of stimuli repeating in different contexts are cut out to become cognitive entities ("things" with a beginning and an end) in their own right. Results suggest that the…

This study investigated the effects of K. J. Klauer's (1989) inductive reasoning training program of teaching children. Effects of training and the range of transfer of the training were assessed. The subjects were 34 third-grade Dutch children of average ability, matched on age, sex, and IQ. Children from the training condition (N=17) received…