Mollie MacGregor

STUDENTS' UNDERSTANDING OF ALGEBRAIC NOTATION: 11–15

Research studies have found that the majority of students up to age 15 seem unable to interpret algebraic letters as generalised numbers or even as specific unknowns. Instead, they ignore the letters, replace them with numerical values, or regard them as shorthand names. The principal explanation given in the literature has been a general link to levels of cognitive development. In this paper we present evidence for specific origins of misinterpretation that have been overlooked in the literature, and which may or may not be associated with cognitive level. These origins are: intuitive assumptions and pragmatic reasoning about a new notation, analogies with familiar symbol systems, interference from new learning in mathematics, and the effects of misleading teaching materials. Recognition of these origins of misunderstanding is necessary for improving the teaching of algebra.

The Effect of Different Approaches to Algebra on Students' Perceptions of Functional Relationships

National curriculum guidelines advise teachers to begin algebra with the study of patterns leading to their description as algebra rules relating two variables. Traditional approaches are based on the use of algebraic letters to stand for specific but unknown numbers. In this paper we show that a pattern-based approach does not automatically lead to better understanding of functional relationships and algebraic rules. Approximately 1200 students in Years 7 to 10 in 10 schools were tested on recognising, using and describing rules relating two variables. Fourteen students were interviewed. Students saw a variety of patterns, many of which were not helpful for algebra. The critical steps in moving from a table to an algebraic rule are outlined.

Goals and Content of an Algebra Curriculum for the Compulsory Years of Schooling

This chapter is concerned with the significance of algebra for the broad population of students in the compulsory years of schooling and with what should constitute a basic algebra curriculum. Three broad reasons for scrutiny of the curriculum are identified: the growth of universal education, the challenges and opportunities brought about by information technology, and the concern arising from documented low student achievement. The chapter proposes that the reasons for all students to learn algebra are complex, and must go beyond simple assertions of utility. The final section gives appropriate goals for a basic algebra curriculum.

Cognitive Models Underlying Algebraic and Non- Algebraic Solutions to Unequal Partition Problems

The structure of certain word-problems can be perceived in different ways, depending on the grammatical form of presentation of the problem and the student’s expectation of how it will be solved. The results of our study involving 268 school students aged 14–16 show that, for a certain class of problems, different problem presentations promote the construction of different cognitive models of the situation described. Our data provide support for the hypothesis of Nathan et al. (1992) that in the solution of algebra word-problems there are three components of interpretation and modelling: a propositional text base, a cognitive model of the situation, and a formal model of the mathematical relationships. However we show that, for certain problems, there are two equally valid cognitive models of the situation, only one of which can be linked to an algebraic representation of relationships. For problems of this type, the lack of correspondence between a cognitive model of the situation and an algebraic representation of relationships in the problem is a powerful obstacle to the use of algebraic methods.

Learning the language of mathematics

ConclusionsThe results suggest that many students at year-nine level are not coping with generalized arithmetic or elementary algebra. The apparent reasons for failure in the test items include:1.poor word-recognition when reading;2.inadequate understanding of natural-language syntax;3.not knowing meanings of certain words, e.g., “sum”;4.poorly-developed concept of the product of two numbers;5.not knowing elementary conventions of algebraic notation, e.g., that “ab” means “a x b”;6.not knowing how to interpret an equation and the “equals” sign. Perhaps children in the lower secondary school need experience in talking, reading, and writing about numbers, both in ordinary English and mathematical notation. They need practice and guidance in constructing and expressing their own mathematical ideas. They may then develop a firm conceptual and linguistic foundation on which to build the formal structure of mathematics.

Writing in natural language helps students construct algebraic equations

An experiment was conducted to find out whether students’ success in constructing a simple algebraic equation from graphical data would he affected if they described the relation in written English first. The subjects were students in a first-year tertiary mathematics course. It was found that writing sentences helped students to write correct equations. Contrary to expectations, the most successful students were those who used common idiomatic forms of English that could not be directly translated into mathematical notation.

Difficulties of Students with Limited English Language Skills in Pre-Service Mathematics Education Courses.

Student teachers who are recent migrants to Australia from non-English-speaking countries experience difficulties in mathematics method courses and in teaching practice, but there are no published reports of research into their difficulties. This paper reports on a preliminary investigation into how students with limited English skills perceive their problems and how they think courses could be adapted to meet their needs. Information was obtained from a questionnaire, by conversation with individual students, by observation of micro-teaching sessions and teaching practice, and from a student counsellor. The factor which most concerned the repondents to the questionnaire was the difficulty which pupils might have in understanding their speech. The results of the study highlight the present lack of resources to provide intensive training for student teachers in the use of specifically mathematical English. The teacher education course was identified as the only vehicle available to them for improving their English.