We report on an innovative design of algorithmic analysis that supports automatic online assessment of students' exploration of geometry propositions in a dynamic geometry environment. We hypothesized that difficulties with and misuse of terms or logic in conjectures are rooted in the early exploration stages of inquiry. We developed a generic…

Demonstrates that an elementary understanding of computer programing and the development and illustration of the structure of computer program analysis will be valuable assets in the understanding and teaching of mathematics and may be used as modeling devices to simulate the stages of learning. (Author)

The hypothesis investigated is that understanding of the long division algorithm requires a higher cognitive level than understanding of fundamental division concepts. Sixth-grade children were tested on performance and understanding of a given algorithm and concepts of division. (MP)

A process for helping students in the elementary grades develop their own algorithms for subtraction with carrying is described. Pupils choose their own times and ways to move from manipulative materials to written notation. (MP)

The time at which component skills are taught is studied to see whether it is a significant instructional variable. This research involved the instruction of a multiplication algorithm to 15 below-average first graders. (MP)

Describes types of learning and organizational strategies and suggests that pattern noting--a technique to organize information in a relational manner showing linkages between topics--is a powerful organizational strategy. An algorithmic procedure for semantically analyzing pattern notes and various pattern note applications are described.…

The cognitive method, an algorithmic method for transforming verbal word problems into equations, equation systems, inequalities, etc., is presented and discussed. The psychological and mathematical aspects of the cognitive method are also reviewed. (MP)

Proposes a criterion for selecting a sequence of objectives from a learning hierarchy and describes a sequence generating algorithm which is designed to generate a minimal memory load sequence from a learning tree. Its effectiveness in generating low memory load sequences from hierarchies that are not trees is also discussed. (RP)

A study to investigate one of the mechanisms teachers may use to convince themselves incorrectly that students have learned science concepts requiring formal operational ability is presented. The investigation indicates instructors may actually teach and test for memorization of algorithms rather than understanding. (MP)

Addresses the distinction between conceptual and algorithmic learning and the clarification of what is meant by a second-tier student. Explores why novice learners in chemistry and physics are able to apply algorithms without significant conceptual understanding. (DDR)

Considers possibility of developing a computer tutor around an explicit concept-learning theory derived from machine learning techniques. Some problems with using the focusing (and similar) algorithms in this role are discussed and possible solutions are developed. Design for a guided discovery learning system for tutoring concepts is proposed.…

A survey of general mathematics students whose teachers were taking an inservice workshop revealed that they had not yet mastered division. More direct introduction of the standard division algorithm is favored in elementary grades, with instruction of transitional processes curtailed. Weaknesses in transitional algorithms appear to outweigh…

Children's difficulties remembering correct algorithms for operations with fractions are traced to difficulties in the instructional sequence. Many teachers "teach mistakes" by failing to provide sufficient referents for their pupils. (MP)

The question is raised: What comes first: rules of calculation or the meaning of concepts? The pressures on the teacher to teach and simplify knowledge to algorithms are discussed. The relation between conceptual and procedural knowledge in school mathematics and consequences for the teacher's professional knowledge are considered. (DC)

Provides the transcript of an interview with Dr. Lev Landa that addressed issues related to his algorithmico-heuristic theories of learning and instruction, called Landamatics. Highlights include teaching thinking versus knowledge; algorithms; instructional design; improving training and performance in industry, business, and government;…