Showing papers in "Mathematics Education Research Journal in 1998"

An investigation of U.S. and Chinese students’ mathematical problem posing and problem solving

Abstract: This study explored the mathematical problem posing and problem solving of 181 U.S. and 223 Chinese sixth-grade students. It is part of a continuing effort to examine U.S. and Chinese students’ performance by conducting a cognitive analysis of student responses to mathematical problem-posing and problem-solving tasks. The findings of this study provide further evidence that, while Chinese students outperform U.S. students on computational tasks, there are many similarities and differences between U.S. and Chinese students in performing relatively novel tasks. Moreover, the findings of this study suggest that a direct link between mathematical problem posing and problem solving found in earlier studies for U.S. students is true for Chinese students as well.

Cognition in the Formal Modes: Research Mathematics and the SOLO Taxonomy.

Abstract: Mathematics researchers put considerable cognitive effort into trying to expand the body of mathematical knowledge. In so doing, is their cognitive behaviour different from those who work on more standard mathematical problems? This paper attempts to examine some aspects of mathematical cognition at the highest level of formal functioning. It illustrates how the structure of a mathematician’s output—and, to a certain extent, its cognitive complexity—can be characterised by the SOLO taxonomy. A number of cognitive and philosophical issues concerning mathematical functioning at the research level will also be discussed.

Associations between teacher-student interpersonal behaviour and student attitude to mathematics

Abstract: This article reports on research using a convenient questionnaire designed to allow mathematics teachers to assess teacher-student interpersonal behaviour in their classrooms. The various forms of the Questionnaire on Teacher Interaction (QTI) are discussed, and its use in past research is summarised. The article provides validation data for the first use of the QTI with a large sample of mathematics classrooms and examines the relation of teacher-student interpersonal behaviour to student attitude. It also describes how mathematics teachers can and have used the questionnaire to assess perceptions of their own teacher-student interpersonal behaviour, and how they have used such assessments as a basis for reflecting on their own teaching. The QTI may thus provide a basis for systematic attempts to improve one’s own teaching practice.

Development of angle concepts: A framework for research

Abstract: 36 Grade 4 children were interviewed to find how they interpreted the angles implicit in six realistic models. Angle recognition varied considerably across the six situations, as did children’s tendency to recognise angular similarities between them. Based on these and other results, a framework for further research on the development of children’s angle concepts is proposed. It is suggested that children first classify their everyday angle experiences intophysical angle situations; they then group situations to formphysical angle contexts; and then they gradually group contexts intoabstract angle domains. Each classification step leads to the formation of a corresponding angle concept.

A developmental multimodal model for multiplication and division

Abstract: This paper presents an analysis of young students’ development of multiplication and division concepts based on a multimodal SOLO model The analysis is drawn from two sources of data: a two-year longitudinal study of 70 Grade 2 to 3 students’ solutions to 24 multiplicative word problems, and examples from a problem-centred teaching project with Grade 3 students An increasingly complex range of counting, additive, and multiplicative strategies based on an equal-grouping structure demonstrated conceptual growth through ikonic and concrete symbolic modes The solutions employed by students to solve any particular problem reflected the mathematical structure they imposed on it A SOLO developmental model for multiplication and division is described in terms of developing structure and associated counting and calculation strategies

Sequential development of algebra knowledge : a cognitive analysis

Abstract: Learning to operate algebraically is a complex process that is dependent upon extending arithmetic knowledge to the more complex concepts of algebra. Current research has shown a gap between arithmetic and algebraic knowledge and suggests a pre-algebraic level as a step between the two knowledge types. This paper examines arithmetic and algebraic knowledge from a cognitive perspective in an effort to determine what constitutes a pre-algebraic level of understanding. Results of a longitudinal study designed to investigate students’ readiness for algebra are presented. Thirty-three students in Grades 7, 8, and 9 participated. A model for the transition from arithmetic to pre-algebra to algebra is proposed and students’ understanding of relevant knowledge is discussed.

Students' Attitudes towards Mathematics in Single-Sex and Coeducational Schools.

Abstract: This paper examines students’ attitudes towards mathematics at the secondary school level. Using five of the Fennema-Sherman scales, the attitudes of boys and girls in Grades 8 to 12 in four schools were compared: a single-sex boys’ and a single-sex girls’ private school, and a state and a private coeducational school. Multivariate analysis of variance was used to guide an exploration of how students’ attitudes varied according to grade, sex and educational setting. There were no differences between students in the two coeducational schools. In general, students’ attitudes were found to be less positive in more senior grades; and overall, boys had more positive attitudes than girls. There were clear differences between boys and girls on the Mathematics as a Male Domain scale, with girls being less stereotyped in their perceptions than boys. Except for this scale, effects related to the sex of the student were small, and effects relating to grade level and school type on all variables were also small. Implications are drawn for future research in this area.

Preservice teachers’ problem-solving processes

Abstract: The purpose of the study reported in this paper is to explore some of the common difficulties with mathematical word problems experienced by preservice primary teachers. It examines weaknesses in students’ content and procedural knowledge, with a particular focus on how they apply these aspects of knowledge to solving closed word problems. The SOLO Taxonomy (Biggs & Collis, 1982, 1991) is used to classify the processes used by students who attempted to solve a group of word problems of varying difficulty. Other characteristics of the students’ processes that are analysed include the way they used the cues provided in the problem, the way they brought in additional concepts or processes, and the types of errors they made.

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

Abstract: 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.

Teaching Problem Solving to Year 6 Students: A New Approach

Abstract: This study was designed to investigate the effects of an instructional intervention derived from the Garofalo and Lester (1985) cognitive-metacognitive framework on the problem-solving performance of Year 6 students with different ability levels. A quasi-experimental design was employed using one experimental and two control classes. Four different techniques were applied to identify above average, average, and below average students. There was a significant improvement in problem-solving performance for the experimental class compared with both control classes. Furthermore, higher ability students appeared to gain more from the experimental instruction than lower ability students. Implications for instruction and research are explored.

Covering Shapes- with Tiles: Primary Students' Visualisation and Drawing

Abstract: Students’ early area concepts were investigated by an analysis of responses to a worksheet of items that involved visualising the tiling of given figures. Students in Years 2 and 4 in four schools attempted the items on three occasions and some of the students completed ten classroom spatial activities. Half the students had difficulty visualising the tiling of shapes, but students who participated in spatial activities were generally more successful in determining the number of tiles that would cover a shape. Students’ drawings indicated a varying awareness of structural features such as alignment and tile size. Students who drew the tilings were more likely to be successful on the items involving trapezia. The tiling items were part of a test of spatial thinking, Thinking About 2D Shapes, and scores on the overall test were very highly correlated with results for the tiling items.

The accessing of geometry schemas by high school students

Abstract: In this study I examine the question, what is the nature of prior mathematical knowledge that facilitates the construction of useful problem representations in the domain of geometry? The quality of prior knowledge is analysed in terms ofschemas that provide a measure of the degree of organisation of prior knowledge. Problem-solving performance and schema activation of a group of high- and low-achieving students were compared. As expected, the high achievers produced more correct answers than the low achievers. More significantly, schema comparison indicated that the high achievers accessed more problem-relevant schemas than the low achievers. In a related task which focused on the problem diagram, both groups accessed almost equal numbers of geometry schemas. The results are interpreted as suggesting that high achievers build schemas that are qualitatively more sophisticated than low achievers which in turn helps them construct representations that are conducive to understanding the structure of geometry problems.

Probing the Links between Language and Mathematical Conceptualisation

Abstract: Although it is well known that the number-naming grammars of some Asian languages affect the number conceptualisation and operational capacity of the young children who use them, language structure in other fields of the mathematical register appears to have been underemphasised in explanations of the differential mathematical attainment of East Asian and western students. This study probed the conceptual structures which samples of Year 9 students in Korea and Australia had constructed for various geometrical terms. The two samples of students exhibited major differences in definition ranking preferences on angle, and in semantic profiles for parallelogram, triangle, square, circle, cylinder, trapezium and line. It is hypothesised that these differences may be due in part to the noun phrase grammar differences between Korean and English, and to the syntax and semantic transparency of some Korean geometrical names. Results indicated reinforcement for the Vygotskian view that culturally imposed language structures directly influence the cognitive development of the learner.

Learning and teaching mathematics: A cognitive perspective

Abstract: We are delighted to have had the privilege of being editors for this special issue which has a focus on cognitive perspectives on teaching and learning mathematics. It has taken a long time to come to fruition. It was about 4 years ago that I (Gillian) suggested to the then editor, Nerida Ellerton, that I would be happy to edit an issue with such a focus. In late 1997, I was asked if I would be prepared to edit the special edition for 1998. I agreed enthusiastically, because I believe that it is very important to focus on student cognition and the learning aspects of any teaching situation and that ongoing research and subsequent publications are needed in mathematics education journals in this area. There are many researchers in MERGA who undertake research in cognitive aspects of mathematics education, and the special issue would give some of them an opportunity to present significant aspects of their work. I asked Lyn if she would be co-editor with me because of her undoubted expertise in the area. Lyn has written summary chapters of work in cognition for MERGA publications which have highlighted the diverse interests of Australian researchers in this field. This special issue is an opportunity to explore some of these interests in greater depth. More specifically, the present issue indicates the continued appeal of the SOLO (Structure of Observed Learning Outcomes) model for analysing cognitive development in mathematics. The SOLO model was originally advanced by Biggs and Collis (1982) and has since been refined in their later works (e.g., Collis & Biggs, 1991). The use of the SOLO model in research in mathematics is strong in Australia, where it was developed, and this is reflected in the articles in this issue. In this issue, Watson and Mulligan, Watson and Moritz, and Chick apply the model to mathematical learning among young children, older students, and adults, respectively. Watson and Mulligan present an analysis of young students' development of multiplication and division concepts based on the SOLO model. Their research has identified an increasingly complex range of counting, additive, and multiplicative strategies based on an equal-grouping structure, demonstrating conceptual growth through the ikonic and concrete symbolic modes. The authors present a SOLO developmental model for multiplication and division, described in terms of developing structure and associated counting and calculation strategies. Watson and Moritz apply the SOLO model to analysing the understanding of …

Alive and well: A review of reviews of Mathematics Education Research in Australasia

Abstract: Research in Mathematics Education in Australasia 1992-1995 is, in the first instance, a testament to the extensive and wide-ranging research in the field emanating from Australasia, and to the success of the Mathematics Education Research Group of Australasia (MERGA) as the major organisation giving voice and impetus to such activity. The book is the product of reviews of research work in mathematics education in Australasia published between 1992 and 1995, with a lengthy reference list at the end of each chapter (notwithstanding references to the same research article in a number of different chapters). As a text with the major purpose of indicating the \"scope and focus of mathematics education research conducted in this part of the world and to identify sources for researchers who may wish to examine particular aspects of the research in more detail\" (p. 1) it meets its goal admirably. It is indeed a \"comprehensive guide to research\" (p. 5) and thus an invaluable resource, particularly for newcomer researchers who need to access what has been done, by whom and where. The fifteen chapters are arranged into four main parts, starting with the broad context in which mathematics education is located and hence through which it needs to be interpreted; then becoming more and more specific as the chapters move from research into curriculum and instruction, to teacher and adult education and, finall~ to topic areas in mathematics such as early arithmetic, algebra and geometry. The book begins with brief introductions to MERGA and its goals. This is