A study of the cognitive and affective characteristics of high and low achievers in Year 10 algebra

This study focused on successful performance in school algebra. It sought to determine what high achievers in Year 10 algebra knew and felt about algebra that enabled them to succeed while their peers were less successful. The study aimed to identify essential cognitive and affective variables which associate significantly with successful performance in algebra and to develop a model to explain how these variables interact to facilitate that success. Implications for instruction were then drawn and recommendations made relating to curriculum design, teacher education and further associated research.

The literature study indicated that three basic concepts, which are representation, generalisation, and functionality, appear to underlie algebraic thinking. A conceptual understanding of a subject together with a range of problem solving abilities distinguish expert from novice intellectual behaviour generally. In addition a range of affective variables including belief in the value of the task, self-concept and self-efficacy beliefs, together with attitudinal variables, impact significantly on successful performance generally.

A two-stage research design was chosen for the study, involving a gender-balanced sample of 54 Year 10 extension (top stream) mathematics students - approximately half from a state school, and the remainder from a private school - from a semi-rural area of Australia. In Stage One all subjects were given a written test of algebra attainment and a series of written tests of the identified basic concepts, that is representation, generalisation and functionality. A questionnaire survey was conducted to determine their beliefs about and attitudes towards, themselves, algebra and mathematics in general. The questionnaire also asked about classroom experiences, perceived scholastic abilities, parental, and perceived teacher and peer group influences on the subjects' learning of algebra. The data were analysed quantitatively and qualitatively and comparisons made between the responses of a group of ten high and a group of ten low achievers in algebra. In Stage Two gender and school-balanced sub-samples from the high and low achievers groups of Stage One were selected for interview. These groups participated one-on-one in a one hour audio/video recorded think-aloud mode algebra problem solving interview conducted by the researcher. The subjects articulated theirthinking procedures and feelings as they solved a range of routine to novel algebra problems. The data gathered from this interview were analysed using both quantitative and qualitative techniques and triangulated with that of Stage One.

The Stage One findings confirmed that high achievers had a significant command of the concepts identified as basic to algebraic thinking. The questionnaire data showed high achievers held positive beliefs and attitudes about algebra and about their own capabilities. These beliefs and attitudes were significantly different from those of their less successful peers. High achievers were generally positive and confident about algebra. They valued the subject for aesthetic and practical reasons. They reported enjoying algebra and believed it enhanced their general thinking and problem solving capabilities. High achievers were self-aware and held strong self-efficacy beliefs. They knew they worked hard and were respected by their teachers and peers and were supported by their parents. Subjects who were less successful in algebra also reported parental support and generally good teacher relations, but reported not being inspired by algebra and not being confident that they could succeed in the subject. No gender or school effects (private or state) were found.

The data gathered from algebra problem solving interviews conducted in Stage Two confirmed the high and low achiever ratings on the identified basic algebra concepts. This was particularly evident with generalisation and with functionality. Noticeably, however, even high achievers at this Year 10 level had difficulties with the function concept. The qualitative data analysis of the interview transcripts was aided by a grounded theory methodology which identified three stages in the algebra problem solving process, these are, information gathering, information processing and information reporting. The analysis identified eight cognitive and eight affective variables which associated significantly with successful performance in Year 10 algebra.

Based on these findings an algebra learning success model was developed. The model postulates that success in algebra results from the reciprocal interaction of positive beliefs and attitudes towards algebra, a sound knowledge of the identified basic concepts of representation, generalisation, and functionality from and the application of a range of problem solving abilities.

These abilities are developed, it is postulated, through the confidence building and optimistic experiences gained from early and sustained comprehension of and success with algebra. Implications for curriculum design and teacher education have been identified, presented and discussed, together with suggestions for further research relating to the algebra learning success model.