Studies have found that some teacher professional development programs that are based on Cognitively Guided Instruction (CGI) can increase student mathematics achievement. The mechanism through which those effects are realized has been theorized, but more empirical study is needed. In service of this need, we designed a novel measure of…

How do engineering students approach mathematical modelling problems and how can they learn to deal with such problems? In the context of a course in mathematical modelling and problem solving, and using a qualitative case study approach, we found that the students had little prior experience of mathematical modelling. They were also inexperienced…

A theoretical model of metacognition in complex modeling activities has been developed based on existing frameworks, by synthesizing the re-conceptualization of metacognition at multiple levels by looking at the three sources that trigger metacognition. Using the theoretical model as a framework, this study was designed to explore how students'…

This paper argues that design fixation, in part, entails fixation at the level of meta-representation, the representation of the relation between a representation and its reference. In this paper, we present a mathematical model that mimics the idea of how fixation can occur at the meta-representation level. In this model, new abstract concepts…

A theory that there is a correspondence between Piagetian conservation operations and groups of symmetry transformations, and that these symmetry transformations may be used in explaining human problem solving behaviors, is developed in this paper. Current research in artificial intelligence is briefly reviewed, then details of the symmetry…

Discusses the nature of tasks used to elicit the development of such systems by middle school students. Analyzes mathematical reasoning development of students across tasks and the diversity of thinking patterns identified on problem tasks. Discusses student reasoning about the relationships between and among quantities and their application in…

This study examined effects of individual differences in speak-span scores and variations in memory demands on class-inclusion performance of 10-, 13-, and 15-year-olds. Results from regression analyses and the mathematical model indicated that differences in age, speak span, and memory load affected performance. Effects of speak span and memory…

Proposes a new conceptual framework, embedded in a dynamic model, that accounts for children's failure to reason transitively. Examines five different models of transitive reasoning. Develops a model of how children initially represent and then use the ordered information available in the transitive inference model and how these processes change…

Reviews the history of the concept of function, looks at its relationship with other sciences, and discusses its use in the study of real world situations. Discusses the process of constructing mathematical models of function and emphasizes the importance of the roles of the numerical, graphical, and algebraic representational forms. (MDH)

This study investigated a theoretical structure which attempts to explain the problem-solving strategies used by first-grade students who are successful in solving simple arithmetic story problems. The structure used describes the problem-solving procedure as consisting of a three-stage model building process: (1) comprehending the story, (2)…

This investigation used a clinical interview technique to identify the difficulties of kindergarten children who are unable to develop models or representations of simple arithmetic story problems. It is hypothesized that effective human problem solvers first generate some type of "physical model" and use this model to create a…

The intuitive simplicity of probability can be deceiving. Described is a dialogue that presents arguments for conflicting solutions to a seemingly simple problem determining the probability of having two boys in a two-child family knowing that one child is a boy. Solutions contain multiple arguments and representations. (MDH)

Applies the Lesh Translation Model to develop conceptual understanding by showing relationships between five modes of representation proposed by Lesh to learn multiplication of fractions. Presents five teaching activities based on the translation model. (MDH)

Concrete experience should be a first step in the development of new abstract concepts and their symbolization. Presents concrete activities based on Hyde and Nelson's work with egg cartons and Steiner's work with money to develop students' understanding of partitive division when using fractions. (MDH)

Described is a learning activity that requires students to observe, read, and interpret graphs and organize and describe data. Included are the grade level, materials, objectives, prerequisites, directions, answers to questions, and copies of handouts. (KR)