While the general planning advice offered to mathematics teachers seems to be to start with simple examples and build complexity progressively, the research reported in this article is a contribution to the body of literature that argues the reverse. That is, posing of appropriately complex tasks may actually prompt the use of more sophisticated…

Part of the research community that has followed the Early Algebra paradigm is currently delimiting the differences between arithmetic thinking and algebraic thinking. This trend could prevent new research approaches to the problem of learning algebra, hiding the importance of considering an arithmetico-algebraic thinking, a new approach which…

In this paper, we report a study in which nine research mathematicians were interviewed with regard to the goals guiding their reading of published proofs and the type of reasoning they use to reach these goals. Using the data from this study as well as data from a separate study (Weber, "Journal for Research in Mathematics Education" 39:431-459,…

This study examines patterns in middle-grade boys' and girls' written problem solving strategies for a mathematical task involving proportional reasoning. The students participating in this study attend a coeducational charter middle school with single-sex classrooms. One hundred nineteen sixth-grade students' responses are analyzed by gender…

As part of individual interviews incorporating whole number and rational number tasks, 323 grade 6 children in Victoria, Australia were asked to nominate the larger of two fractions for eight pairs, giving reasons for their choice. All tasks were expected to be undertaken mentally. The relative difficulty of the pairs was found to be close to that…

This conceptual research framework addresses the problem of rote learning by characterising key aspects of the dominating imitative reasoning and the lack of creative mathematical reasoning found in empirical data. By relating reasoning to thinking processes, student competencies, and the learning milieu it explains origins and consequences of…

The purpose of this study is to investigate the ways in which Swedish seventh grade students (12 and 13 years old) handle chance encounters. Four groups of students working in pairs participated in the study. In the group discussions, which were tape-recorded and fully transcribed, the students were encouraged to explore strategies for winning a…

Describes an earlier study on the main characteristics and background of undergraduate students' difficulties when trying to solve mathematical tasks. Focuses on and extends part of an earlier study that concerned task solving strategies. Indicates that focusing on what is familiar and remembered at a superficial level is dominant over reasoning…

Addresses the initial understanding that advanced undergraduate mathematics students exhibit when presented with a new concept in an environment requiring self-generation and self-validation of examples of the concept. Concludes that students who used an example generation learning strategy were more effective in attaining an initial understanding…

Analyzes video recordings of three undergraduate students' textbook-based homework. Focuses on the ways in which their exercise reasoning is mathematically well-founded or superficial. Indicates that most strategy choices and implementations are carried out without considering the intrinsic mathematical properties of the components involved in…

Presents a theoretical framework based on interviews with undergraduate students and written questionnaires to study how undergraduate students cope with abstract algebra concepts. Indicates that students' responses can be interpreted as a result of reducing the level of abstraction. Examines the theme of reducing abstraction based on three…

Defines visualization as the product and the process of creation, interpretation, and reflection upon pictures and images. Analyzes, exemplifies, and reflects upon the many different and rich roles that it can and should play in the learning and doing of mathematics. Discusses limitations and possible sources of difficulties visualization may pose…

Explores the relationship between learners' actions, visualizations, and the means by which these are articulated. Describes a microworld called Mathsticks which is designed to help students construct mathematical meanings by forging links between the visual and symbolic representations they develop and their actions. Presents a case study of two…

Focuses on the role of tools, provided by a computer microworld (C.AR.ME), on the strategies developed by 14-year-old students regarding the concept of conservation of area of a non-convex polygon. Indicates that the nature of the tools used affected the nature of solution strategies that students constructed. (Author/KHR)

Presents a teaching approach related to the concept of function that lead students working within a graphic milieu to producing, discussing, and testing the validity of mathematical statements and theorems. Indicates that students experimenting with the approach became able to think of functions as objects and engage in questions of the validity…