The concepts of cognitive development and objectivity are explored preparatory to a discussion of a suitable theory of development to be used as the foundation for developmentally-based standardized tests. It is hypothesized that Piagetian theory is the most appropriate basis for such tests and two possible approaches for constructing…

A model of qualities of mathematical activity was developed within a framework of learning goals. A clinical application of the model is described, in which ninth-grade students were given assignment projects. A hierarchy of effective guidance was established, permitting diagnostic evaluation and progressive development of mathematical activity.…

This article recommends a holistic approach to student problems in mathematics. Teachers need to be aware of all paradigms (e.g., developmental, behaviorist, psychoanalytic) explaining causes of problems in mathematics. Examples of misunderstandings resulting from insufficient knowledge, insufficient mathematical vocabulary, and overgeneralization…

Two experiments examined possible negative transfer in nonexperts from the use of pictorial examples in a laboratory design problem-solving situation. In Experiment 1, 89 participants were instructed to "think aloud" and were assigned to 1 of 3 conditions: (a) control (standard instructions), (b) fixation (inclusion of a problematic…

This study investigated how preadolescents and adolescents solve problems involving three temporal dimensions. Specifically examined was the question of whether speed and space information would influence the time judgments of 90 subjects 9, 12, and 15 years of age who solved 16 word problems describing the displacements of two cars. The problems…

The purpose of this study is to contribute to an understanding of how errors could be employed in mathematics instruction so that the students use them constructively in support of their learning of mathematics. A teaching experiment was designed to create an ideal context in which the pedagogical approach to errors as springboards could be…

Implementation of an adaptive achievement test for teaching signed-numbers operations to junior high students is described. A computer program capable of finding 240 basic errors in signed-number computations was written on the PLATO system and used for analyzing a 64-item conventional test, as well as an adaptive test of addition problems. The…

Forty-three first-grade children, who had received no formal instruction in addition and subtraction, were individually administered 20 problems that could be solved using addition or subtraction. The problems were selected to represent the following semantic types: joining, separating, part-part-whole, comparison, and equalizing. Responses were…

The need to provide understandable problems and ways to help children understand problems are explored. An interview with a sixth grader depicts his incorrect strategies and leads to suggestions for teaching problem solving using a range of mathematical models for each operation. (MNS)

We report a study of 10-14 year old children's use of additive strategies while solving ratio and proportion tasks. Rasch methodology was used to develop a diagnostic instrument that reveals children's misconceptions. Two versions of this instrument, one with "models" thought to facilitate proportional reasoning and one without were…

The problem-solving strategies used by 10 radiologists and 10 radiology residents during the interpretation of difficult mammograms were examined under two experimental conditions. In the "authentic" condition, standard unmarked mammograms were used. On a second set of the same cases for the "augmented" condition, mammographic…

Discusses concurrent, or parallel, programming languages; explains the design of the MultiLogo programming language; and describes a study of fourth and fifth graders that used MultiLogo to control robotic devices built out of Lego bricks. Three categories of MultiLogo programming bugs are described, and implications for changes in design and…

Reports a case study exploring how 2 16-year olds were enabled to capitalize on the potential of errors to stimulate and support mathematical inquiry. Discusses variations within the strategy of using errors to enhance mathematics instruction. Appendix includes list of instructional episodes and error activities. (Contains 56 references.) (MKR)

We present a methodology for designing better learning environments. In Phase 1, 6th-grade students' (n = 223) prior knowledge was assessed using a difficulty factors assessment (DFA). The assessment revealed that scaffolds designed to elicit contextual, conceptual, or procedural knowledge each improved students' ability to add and subtract…

This paper describes the various procedures associated with the individual interviews that are part of the data gathering processes of the Coordinated Study being carried out by the Mathematics Work Group of the Wisconsin Research and Development Center for Individualized Schooling. The first major section describes the six basic verbal addition…