This research report describes the findings of a study on teachers' ways of interpreting student responses to tasks involving equivalent expressions. The teachers in this study were engaged in model-eliciting activities designed to promote the development of their knowledge and reveal their models (or interpretations) of their students' algebraic…

A common approach used for introducing algebra to young adolescents is an exploration of visual patterns and expressing these patterns as functions and algebraic expressions. Past research has indicated that many adolescents experience difficulties with this approach. This paper explores teaching actions and thinking that begins to bridge many of…

A teaching intervention study was conducted with sixth grade students to explore the interconnections between students' growing understanding of arithmetic expressions and beginning algebra. Three groups of students were chosen, with two groups receiving instruction in arithmetic and algebra, and one group in algebra without arithmetic. Students…

This paper presents an initial attempt to define structure sense for high school algebra and to test part of this definition. A questionnaire was distributed to 92 eleventh grade students in order to identify those who use structure sense. Presence and absence of brackets was examined to see how they affect use of structure sense. The overall use…

Computer algebra system (CAS) calculators are becoming increasingly common in schools and universities. While they offer quite sophisticated mathematical capability to teachers and students, it is not clear at present how they may best be employed. In particular their integration into students' learning and problem-solving remains an issue. In…

Students who learn mathematics with CAS calculators are likely to develop new practices for doing and recording mathematics. Students discussed in this paper were able to use CAS calculators in examinations, making their own decisions about what to document as written records for solutions to problems. A comparison of some features of written…

The paper examines how community was established in a professional development institute that focused on algebra content knowledge for middle school mathematics teachers. This qualitative study was framed within a situative perspective. We analyzed multiple data sources to identify the ways in which community was established. Results indicate…

This article shows that the recognition of the dualities in equality (operator-equivalent) of the minus sign (unary-binary) and the zero (nullity-totality) during the transitional process from arithmetic to algebra by 12-13 year-old students constitutes a possible way to achieve the extension of the natural number domain to the integers. (Contains…

The study of linear functions is important as it provides students with their first experience of identifying and interpreting the relationship between two dependent variables. This paper, which builds on previous research, reports a study undertaken with 64 year 9 students from two Australian schools. Linear functions were introduced to these…

In this study, we explore six students' conceptions of Z[subscript n] in an effort to understand students' conceptions of quotient groups in general. We discovered that there were three different ways our students thought about Z[subscript n], namely as infinite sets, element-set combinations, and representative elements. We explore how…

As part of a larger investigation of preservice teachers' use of, and movement amongst, various modes of representing exponential relationships, this report focuses on one case study, that of Mike, whose facility in moving amongst representational registers was not matched by conceptual understanding of the underlying mathematical ideas as he…

This paper presents an analysis of high school and starting college students' work with algebraic expressions and problems involving parameters. We suggest that parameters should be considered as general numbers that are used to make second order generalizations. The Three Uses of Variable model (3UV model) is then used as theoretical framework to…

Our teaching conception acknowledges the teacher's central role as a decision maker, influenced by knowledge, beliefs, and emotions. We believe that teachers' education must be focused on teachers' awareness of the complexity of the teaching process, of the incidence of these factors in it, and of the importance of looking at theory as a strong…

We present two vignettes of classroom episodes that exemplify new activity structures for introducing core algebra ideas such as linear functions, slope as rate and parametric variation within a new educational technology environment that combines two kinds of classroom technology affordances, one based in dynamic representation and the other…

This paper presents the conceptualisation of infinity as a multi-faceted concept, discussing two examples. The first is from history and illustrates the work of Euler, when using infinity in an algebraic context. The second sketches an activity in a school context, namely students who approach the definite integral with symbolic-graphic…