Dietmar Kuchemann

Using patterns generically to see structure

A common approach to generalization activities in school algebra is to set them in the context of figurative patterns (e.g., a row of touching squares formed of matchsticks, with a relationship between the numbers of squares and matchsticks; or a row of touching tables surrounded by chairs, with a relationship between the numbers of tables and chairs). This approach allows students to see and explain numerical relationships in terms of the figurative patterns structure. However, patterns are all too often presented, in textbooks and by teachers and researchers, in the form of a systematic sequence of elements, with the result that students generate a systematic set of ordered pairs for which they try to induce an empirical relationship, divorced from the structure of the pattern that produced them. In this paper, I argue that one reason that students appear to have difficulty in seeing structure is that they are not sufficiently initiated into this way of thinking. Further, I argue that the search for structure can often be done by focusing on an individual element of a pattern and treating it as a prototype, that is, generically. I begin by looking at some of the influences that might have fostered an empirical approach and point to manifestations of this approach in a recent special issue of ZDM – The International Journal on Mathematics Education. I then discuss how well various generalization tasks are suited to a generic approach. Last, I report on a short intervention where students were encouraged to use a generic approach, with positive results.

Children's understandings of algebra 30 years on

In this paper, we outline the design and method of the research project Increasing Student Competence and Confidence in Algebra and Multiplicative Structures (ICCAMS). Phase 1 consists of a large-scale survey of attainment in algebra and multiplicative reasoning, using test items developed during the 1970s for the Concepts in Secondary Mathematics and Science (CSMS) study (Hart, 1981). This will enable a comparison of the current attainment of students aged 11-14 with that of 30 years ago. Phase 2 consists of a collaborative research study with 8 teachers extending the investigation to classroom / group settings and examining how formative assessment can be used to improve attainment. Although the focus of this paper is on reporting the research design, some early analysis of data from the initial survey data from 2008 (n = 2400) is reported.

Lower secondary school students' knowledge of fractions.

Increasing Student Competence and Confidence in Algebra and Multiplicative Structures (ICCAMS) is a 4-year research project funded by the Economic and Social Research Council in the UK (Hodgen et al. 2008). In this paper, we report and discuss early findings of the study regarding students’ understanding of fractions. In particular, we compare Key Stage 3 (ages 11-14) students’ performance in 2008 and 1977 on items probing their understanding of fractions and decimals. We then discuss some interview responses to a fractions item, with particular reference to their use of diagrams.

Lower secondary school students' attitudes to mathematics: evidence from a large-scale survey in England

Increasing Student Competence and Confidence in Algebra and Multiplicative Structures (ICCAMS) is a four-year research project funded as part of an initiative aimed at identifying ways to increase ...