Adrian Simpson

What is the object of the encapsulation of a process

Abstract Several theories have been proposed to describe the transition from process to object in mathematical thinking. Yet, what is the nature of this “object” produced by the “encapsulation” of a process? Here, we outline the development of some of the theories (including Piaget, Dienes, Davis, Greeno, Dubinsky, Sfard, Gray, and Tall) and consider the nature of the mental objects (apparently) produced through encapsulation and their role in the wider development of mathematical thinking. Does the same developmental route occur in geometry as in arithmetic and algebra? Is the same development used in axiomatic mathematics? What is the role played by imagery?

A Search for Understanding

Abstract We discuss an important breakthrough for us in our search for a technical meaning of “understanding” in mathematics education. In this article, we describe the background to this discovery, the catalyst for the breakthrough, and a concise explanation of our new definition. In discussing the consequences of this definition, both in terms of the theoretical implications for the internal characteristics and external manifestations of understanding with some initial practical consequences for a teachers attempts to model a learners understanding, we begin to describe our ongoing search for a more comprehensive theory.

The Warwick Analysis Project: Practice and Theory

The number of new concepts in analysis, coupled with the new standards of rigour in university mathematics, makes the learning ofanalysis difficult. No course aiming to cover the standard amount of work in the standard amount of time could hope to change this. However, the intuitive ideas of the mathematicians who developed and taught on this course have led to a new pedagogy: small classes, collaborative learning, questions that encourage students to develop the mathematical content and arguments for themselves. The new course provides for negotiation of a new didactic contract, fast feedback from fellow students and from experienced and sensitive staff and the answering of questions which emphasise the manipulation of definitions. Through this it encourages students to amend their evolutionarily developed general cognitive strategy, which is such a powerful way of thinking outside formal mathematics, with a new awareness vital to understanding university mathematics: therigourprefix.

Visible Learning: A Synthesis of over 800 Meta-Analyses Relating to Achievement. By John A.C. Hattie

Eysenck described ill conducted meta-analyses as ‘an exercise in mega-silliness’ (Eysenck, 1978). However, at their best, meta-analyses save lives. When they focus on very similar interventions (e....

The misdirection of public policy: comparing and combining standardised effect sizes

Abstract Increased attention on ‘what works’ in education has led to an emphasis on developing policy from evidence based on comparing and combining a particular statistical summary of intervention studies: the standardised effect size. It is assumed that this statistical summary provides an estimate of the educational impact of interventions and combining these through meta-analyses and meta-meta-analyses results in more precise estimates of this impact which can then be ranked. From these, it is claimed, educational policy decisions can be driven. This paper will demonstrate that these assumptions are false: standardised effect size is open to researcher manipulations which violate the assumptions required for legitimately comparing and combining studies in all but the most restricted circumstances. League tables of types of intervention, which governments point to as an evidence base for effective practice may, instead, be hierarchies of openness to research design manipulations. The paper concludes that public policy and resources are in danger of being misdirected.

When does a way of working become a methodology

We discuss the question of how to determine what are valid methodologies in mathematics education and the problems associated with developing new ways of working into acceptable research methods. We develop the idea of a two-stage validation process and look at the consequences for communicating between what we term methodological neighbours, an alternative diffusion model that explores the relationship between teachers and researchers, and the role of the PhD within the validation process. We then describe our own way of working and suggest how it is beginning to evolve into a methodology.

Classification and Concept Consistency

This article investigates the extent to which undergraduates consistently use a single mechanism as a basis for classifying mathematical objects. We argue that the concept image/concept definition distinction focuses on whether students use an accepted definition but does not necessarily capture the more basic notion that there should be a fixed basis for classification. We examine students’ classifications of real sequences before and after exposure to definitions of increasing and decreasing; we develop an abductive plausible explanations method to estimate the consistency within the participants’ responses and suggest that this provides evidence that many students may lack what we call concept consistency.RésuméCet article analyse jusqu’à quel point les étudiants se servent systématiquement d’un seul mécanisme comme base pour la classification des objets mathématiques. Selon nous, la distinction entre image du concept et définition du concept permet de déterminer si les étudiants utilisent une définition reconnue, mais elle ne reflète pas nécessairement l’idée plus fondamentale que la classification doit se faire sur la base de certains critères fixes. Nous analysons la classification des séquences réelles de la part des étudiants avant et après leur formation sur la définition des concepts de ‘croissant’ et ‘décroissant’. En outre, nous élaborons une méthode inductive ‘d’explications plausibles’ pour apprécier la cohérence logique des réponses des participants, et les résultats indiquent que de nombreux étudiants manquent de ce qu’on pourrait appeler ‘cohérence du concept’.

The surprising persistence of Biglan's classification scheme

Within higher education systems, different institutions deliver different patterns of disciplines. A simple analysis of the structure of that pattern of disciplines across institutions in one higher education system uncovers a surprising relationship. That is, the key dimensions which describe that structure align nearly perfectly with dimensions discovered in a very different context: the Biglan classification scheme. This paper explains correspondence analysis as a mechanism for uncovering structure in simple contingency tables, shows the accuracy of the fit with Biglans scheme and demonstrates that the analysis also has a measure of predictive validity in its ability to classify previously unclassified disciplines. The study not only acts as a more accurate validation of Biglans scheme than those previously undertaken, but indicates that a scheme developed in the USA in the 1970s has current validity in a very different higher education system and suggests disciplines as a core genotype of institutions.

University students’ perceptions of summative assessment: The role of context

Abstract We report on a mixed-method study that compared students’ perceptions of summative assessment across two distinct disciplines – education and mathematics, at two research-intensive institutions in the UK. The disciplines chosen represent opposing positions in Biglan’s classification of academic disciplines, as well as having very different assessment practices. Results suggest that these education students prefer to be assessed by methods they perceive to discriminate on the basis of academic abilities. Moreover, they perceive the traditional closed-book examination as inadequate to assess the capabilities which are key to being successful in their subject, which fits some but not all of the general findings in the literature. However, comparing these results with those of an identical study with mathematics students, we find that the perceptions of summative assessment are very different. We account for that difference by suggesting that students’ epistemic beliefs play a role in shaping these perceptions and conclude that, in designing summative assessment in higher education, generalised and centralised forces for change need to be tempered by contextual and disciplinary factors.

Students’ views of oral performance assessment in mathematics: straddling the ‘assessment of’ and ‘assessment for’ learning divide

This paper explores the views of a group of students who took an oral performance assessment in a first-year mathematics module. Such assessments are unusual for most subjects in the UK, but particularly within the generally homogenous assessment diet of undergraduate mathematics. The evidence presented here resonates with some, but not all, of the existing literature on oral assessment and suggests that, despite concerns about anxiety and fairness, students see oral assessments as encouraging a focus on understanding, being relatively authentic and reactive to their needs. We argue that, suitably implemented, oral assessment may be a viable assessment method for straddling the ‘assessment for’ and ‘assessment of’ learning divide in higher education.