Linguistic Pointers to Students' Understanding in Introductory Algebra: A Cognitive Approach

Teachers use "extremely subtle pragmatic interpretive judgements [...] regularly in the course of mathematics teaching and learning..." (Pimm, 1987, p.167). The form of their discourse – the coherence, the structure and modality, characteristics of natural language in use – indicates the commitment of students to the truth-value of their statements. Hence, the listener might infer the extent of students' confidence in their understanding. In this study, linguistic features were identified that could be aligned with the conceptual growth of students in the context of introductory algebra. The aim was to devise a model that provided explicit, objective evidence to support the subtle, interpretive judgements made by teachers. Secondary students in Years 8 and 9 (13-15 year olds) from three schools in a NSW regional centre (N=222) participated in the study. The study consisted of two phases of data collection. The first was the collection of quantitative data from students' responses to a survey (test) of 40 algebra items drawn from the algebra syllabus for the first four years of secondary schooling in NSW. Survey data provided information about algebra concepts, and conceptual development demonstrated by the students, through Rasch modelling of the responses and an analysis of errors. The Rasch model indicated items and students clustered around significantly different estimates of, respectively, difficulty and ability. Clustering indicated groups of items requiring similar levels of conceptual development to be addressed successfully, and the corresponding groups of students who demonstrated this development. End-points of clusters indicated where conceptual change was necessary for further success on items, and the students who could achieve this

An Undergraduate Student's Understanding of Differential Equations through Concept Maps and Vee Diagrams

The paper presents the case of a student (Nat) who participated in a semester long study which investigated the impactof using concept maps and vee diagrams on students' understanding of advanced mathematics topics. Through the constructionof concept maps and vee diagrams, Nat realized that there was a need for him to deeply reflect on what he really knows,determine how to use what he knows, identify when to use which knowledge, and be able to justify why using valid mathematicalarguments. He found that simply knowing formal definitions and mathematical principles verbatim did not necessarily guaranteean in-depth understanding of the complexity of inter-connections and inter-linkages between mathematical concepts and procedures. During seminar presentations and one-on-one consultations, Nat found that using the constructed concept maps and vee diagrams greatly facilitated discussions, critiques, dialogues and communication. The paper discusses the results from this case study and some implications for teaching mathematics

Concept Maps & Vee Diagrams as Tools For Learning New Mathematics Topics

This paper presents the data for 6 students who participated in a study that investigated the use of the metacogntivetools of concept maps and vee diagrams in learning and solving problems for selected mathematics topics. The six students usedthe tools to learn about new mathematics topics. Initially, students struggled to understand their new topics and the tools.However, with independent research and progressively mapping their findings on concept maps and vee diagrams, and withcritiques and feedback from others, students eventually developed enhanced and deeper understandings of their chosen topics

Rates of Change and an Iterative Conception of Quadratics

Investigating students' conceptions of covariation patterns between quantities situated within contextual settings engenders enriched, deep understandings of functional relationships. This paper presents data from a case study of a student (Mary) who solved quadratic contextual problems. Mary's schemes, constructed from quadratically related quantities and patterns of additive rates, fostered the development of an iterative, summative conceptualisation of quadratics in contrast to the product view. Findings support the use of contextual problems to motivate students to think reflectively and mathematically

Developing a more conceptual understanding of matrices and systems of linear equations through concept mapping and vee diagrams

The paper discusses one of the case studies of a multiple-case study teaching experiment conducted to investigate the usefulness of the metacognitive tools of concept maps and vee diagrams (maps/diagrams) in illustrating, communicating and monitoring students' developing conceptual understanding of matrices and systems of linear equations in an undergraduate mathematics course. The study also explored the tools' role in scaffolding and facilitating students' critical and conceptual analyses of problems in order to identify potential methods of solutions. Data collected included students' progressive maps/diagrams, journals of reflections and justifications of revisions, and final reports and researchers' annotated comments on students' maps/diagrams and anecdotal notes from presentations. Findings showed that students developed more enriched, integrated and connected understandings of matrices and systems of linear equations as a result of continually organizing coherent groups of concepts into meaningful networks of propositional links, critically reflecting on the results against feedbacks from critiques and negotiations for shared meanings, and crystallizing these conceptual changes and nuances where appropriate as revised or additional propositional links. Verifying and justifying solutions were greatly facilitated through the combined usage of concept maps and vee diagrams. Findings suggest that students' classroom experiences in working, thinking and communicating mathematically can be enhanced by incorporating these metacognitive tools into students' repertoire of effective learning strategies

Students' conceptions of models of fractions and equivalence

A solid understanding of equivalent fractions is considered a steppingstone towards a better understanding of operations with fractions. In this article, 55 rural Australian students' conceptions of equivalent fractions are presented. Data collected included students' responses to a short written test and follow-up interviews with three students from each year. This exploratory study found most participating Years 4, 6 and 8 students were familiar with geometric area models, particularly circles, and able to explain equivalent fractions when presented geometrically as area models but had difficulties when equivalents were presented numerically as a/b

'Not just another face in the crowd': Report from SiMERR New South Wales

More than 6.7 million people, or one third of all Australians, live in New South Wales. Of these, about 75% live in the 2.5% of the state referred to as the Great Metropolitan Region (GMR), comprising Sydney, the Illawarra and Lower Hunter regions (Department of the Environment and Conservation, 2003). The rest of the population live on the coastal strips outside the GMR (10%), and in the regional and rural areas of inland NSW (15%).The SiMERR NSW team visited four schools located in these regional and rural areas, interviewing teachers, parents and students about their perceptions of science, ICT and mathematics education. This chapter presents and discusses the results of these interviews, providing detailed and contextualised insights into what these groups identified as key issues affecting educational outcomes in the three subject areas.The chapter begins with a description of the schools, their communities and contexts. The three subsequent sections present and discuss the responses of teachers, students and parents to the interview question. The final section presents a summary of the findings, along with the SiMERR NSW team's reflections on those findings