Gestures--hand movements that accompany speech and express ideas--can help children learn how to solve problems, flexibly generalize learning to novel problem-solving contexts, and retain what they have learned. But does it matter who is doing the gesturing? We know that producing gesture leads to better comprehension of a message than watching…

In this study, hypothetical samples of students' work on a task involving pattern generalizations were used to examine the characteristics of the ways in which 154 elementary prospective teachers (PSTs) paid attention to students' work in mathematics. The analysis included what the PSTs attended to, their interpretations, and their suggestions for…

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…

Conjectures are a key component of mathematical inquiry, a process in which the students raise conjectures, refute or dismiss some of them, and formulate additional ones. Taking a design-based research approach, we formulated a design principle for personal feedback in supporting the iterative process of conjecturing. We empirically explored the…

Mental models are representations of students' minds concepts to explain a situation or an on-going process. The purpose of this study is to describe students' mental model in solving mathematical patterns of generalization problem. Subjects in this study were the VII grade students of junior high school in Situbondo, East Java, Indonesia. This…

This article describes sixth-grade students' engagement in two model-eliciting activities offering students the opportunity to construct mathematical models. The findings show that students utilized their knowledge of fractions including conceptual and procedural knowledge in constructing mathematical models for the given situations. Some students…

This paper explores how a group of pre-service elementary school teachers training to become mathematics teachers for elementary schools arrived at generalizations based on patterns. Two representative problems were investigated with these preservice teachers. The focus of this study was how these preservice teachers analyze and symbolize…

Results indicate that training in semantic awareness increased problem-solving ability for both analytic and synthetic problems. (Author)

The operation of means-ends analysis (MEA) involves attempts at reducing differences between problem states and the goal state. It was paradoxically found that the more problem solvers knew of the goal state, the less they learned of the problem structure during the solution process. (PN)

This paper describes the rule-assessment approach to cognitive development. The basic question that motivated the rule-assessment approach is how people's existing knowledge influences their ability to learn. Research using the rule-assessment approach is summarized in terms of eight conclusions, each illustrated with empirical examples.…

Imagine a wooden cube painted and cut into "unit cubes." A typical activity consists of predicting the number of unit cubes with exactly three faces painted, two faces painted, etc. This article presents extensions of this activity designed to help students develop analyzing abilities and powers to generalize. (JP)

Students often fail to apply knowledge and skills learned in one context to other situations. Although the implicit assumption in educational practice has been that transfer takes care of itself, a knowledge of the mechanisms of transfer can enable educators to help their students integrate general and local knowledge. (TE)

Numerous mathematical examples are presented which illustrate and raise questions about students' tendencies to overgeneralize. (BB)

Research on Vygotsky's zone of proximal development theory was used as a foundation to investigate differences in the learning and transfer ability of 30 students with learning disabilities (LD) and 30 students with matched low achievement (LA), from grades 7 and 8. The two groups were assessed on their problem-solving ability on a balance scale…

This issue, part of an annual series prepared for Maryland educators, presents the problem solving aspect of learning behavior by reviewing some of the variables which affect how individuals confront intellectual problems. Selected studies and a bibliography which are helpful in developing instructional methods which take into account individual…