Susan Pirie

The Equation, the Whole Equation and Nothing But the Equation! One Approach to the Teaching of Linear Equations

There exists an extensive range of research looking at the teaching and learning of linear equations, resulting in many papers highlighting a range of teaching approaches and illustrating a variety of significant cognitive problems and stumbling blocks to the learning of linear equations with understanding. Building on this literature, this paper presents some of the results of a case study which looked at the mathematics classroom of one particular teacher, Alwyn, trying to teach mathematics with meaning to less able pupils at secondary school level. Our interest here is those lessons which dealt specifically with the learning of linear equations, in which firstly a different approach was utilised and secondly many of the problems referred to in the literature were not present. We contrast this method with the teaching of linear equations to a variety of ability levels in several other classrooms that we have studied and we attempt through use of the Pirie-Kieren model, to analyse and account for the successful growth of understanding of the lower ability, year eight pupils in one particular classroom.

Recursion and the Mathematical Experience

Mathematics for children, as they ‘do’ it and build onto their own knowledge, is a complex phenomenon. As with personal knowledge of mathematics for any person, it is multifaceted, a view supported by mathematicians (e.g. Davis and Hersh, 1986) and psychologists interested in mathematics education (e.g. Vergnaud, 1983). There are many ways to observe and hence say something about children’s mathematics. One could consider mathematical thought mechanisms used by children. One could analyze specific aspects of the complexity. However, in this essay we try to observe and talk about children’s mathematical experience as a complex whole.

The role of collecting in the growth of mathematical understanding

Folding back is one of the key components of the Pirie-Kieren Dynamical Theory for the Growth of Mathematical Understanding. This paper looks at one aspect of folding back, that of collecting. Collecting occurs when students know what is needed to solve a problem, and yet their understanding is not sufficient for the automatic recall of useable knowledge. They need to recollect some inner layer understanding and consolidate it through use at an outer layer in the light of their now more sophisticated understanding of the concept in question. The collecting phenomenon is described and distinguished through exemplars of classroom discourse, and implications for teachers and learners are discussed.

Mathematical Discussion: Incoherent Exchanges or Shared Understandings?.

Abstract The data examined in this paper are from one group of those collected for a project addressing the question of whether pupil‐pupil discussion aids mathematical understanding. They represent episodes which were initially coded as ‘incoherent’ by the observers, but which on closer inspection appeared, for the participants, to be relevant to the mathematical discussion taking place. A variety of features, which lead to this external appearance of incoherence, were revealed through deeper analysis and the key phenomenon seemed to be that of unarticulated, but shared, meanings. One of the major problems for the listener occurs when the shared meanings are in fact in some way erroneous. This can manifest itself in misuse of mathematical language or invention of personal vocabularies, in reliance on memories of earlier, possibly misconstrued, events and in reference to shared, previously created, visual images for which they do not need explicit language. The features which this paper exposes give rise ...

Making Images And Noticing Properties: The Role Of Graphing Software In Mathematical Generalisation

This paper discusses the growth of mathematical understanding of two students, Graham and Don, as they use a computer graphing program to explore the properties of quadratic equations. Through analysing extracts of video data using the Pirie-Kieren theory, we discuss the ways in which the mathematical understanding of the students grows and how their interactions occasion, facilitate, and restrict this. We consider four ‘clips’ of their mathematical working, highlighting different aspects of their developing understanding, and use of the graphing software. Although we are talking about a computer based graphing package, our conclusions are equally relevant to the use of graphing calculators.

The Use of Talk in Mathematics

The statement ‘talk is but one element of language use’ becomes highly significant when considering mathematics classrooms. Whereas in other areas of the curriculum the words and syntax used in oral discourse are closely related to those employed in written communication, this is untrue for mathematics. Here the written forms are symbolic rather than verbal, and one cannot therefore assume that notions concerning relationships between oral and written language derived from other disciplines will necessarily transfer to the learning and teaching of mathematics. The learning process becomes more complex, because, for understanding to take place, children have to construct mathematical meanings from experiences that, at least initially, are embedded in everyday language and “everyday language” can have slippery, changing meanings sometimes encompassing more, sometimes less, than the sought after mathematical meaning. Only after this can mathematical symbols be introduced (Pirie, 1997). There is no one-one correspondence between the written and the oral. Indeed, this is not the only linguistic transition that the mathematics learner is expected to master. Mathematics also has its own verbal register, which is composed of a combination of mother-tongue words, many with distorted or specialised meanings, and additional new vocabulary, quite specific to mathematics (Pimm, 1987). Consider the differences between: “Four threes are twelve”, “the product of three and four is twelve”, and 3 4=12. From the perspective of the teacher, how can one know what and how much understanding is encapsulated in students’ own use of the mathematics register, unless the students can also express themselves in everyday language and function with the symbolic representation? The three facets of mathematical language are inextricably woven together.