Lara Alcock

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.

On Mathematicians' Different Standards When Evaluating Elementary Proofs

In this article, we report a study in which 109 research-active mathematicians were asked to judge the validity of a purported proof in undergraduate calculus. Significant results from our study were as follows: (a) there was substantial disagreement among mathematicians regarding whether the argument was a valid proof, (b) applied mathematicians were more likely than pure mathematicians to judge the argument valid, (c) participants who judged the argument invalid were more confident in their judgments than those who judged it valid, and (d) participants who judged the argument valid usually did not change their judgment when presented with a reason raised by other mathematicians for why the proof should be judged invalid. These findings suggest that, contrary to some claims in the literature, there is not a single standard of validity among contemporary mathematicians.

Challenges in Mathematical Cognition: A Collaboratively-Derived Research Agenda

This paper reports on a collaborative exercise designed to generate a coherent agenda for research on mathematical cognition. Following an established method, the exercise brought together 16 mathematical cognition researchers from across the fields of mathematics education, psychology and neuroscience. These participants engaged in a process in which they generated an initial list of research questions with the potential to significantly advance understanding of mathematical cognition, winnowed this list to a smaller set of priority questions, and refined the eventual questions to meet criteria related to clarity, specificity and practicability. The resulting list comprises 26 questions divided into six broad topic areas: elucidating the nature of mathematical thinking, mapping predictors and processes of competence development, charting developmental trajectories and their interactions, fostering conceptual understanding and procedural skill, designing effective interventions, and developing valid and reliable measures. In presenting these questions in this paper, we intend to support greater coherence in both investigation and reporting, to build a stronger base of information for consideration by policymakers, and to encourage researchers to take a consilient approach to addressing important challenges in mathematical cognition.

Mathematics lecturing in the digital age

In this article, we consider the transformation of tertiary mathematics lecture practice. We undertake a focused examination of the related research with two goals in mind. First, we document this research, reviewing the findings of key studies and noting that reflective pieces on individual practice as well as surveys are more prevalent than empirical studies. Second, we investigate issues related to the transformation of lecture practice by the emergence of e-lectures. We discuss the latter in terms of claims about the efficiencies offered by new technologies and contrast these with possible disadvantages in terms of student engagement in a learning community. Overall findings indicate that while survey results appear to trumpet the value of e-lecture provision, empirical study results appear to call that value into question. Two explanatory theoretical frameworks are presented. Issues concerning the instructional context (e.g. the nature of mathematical thinking), inherent complexities and recommendations for implementation are discussed.

Undergraduates’ Example Use in Proof Construction: Purposes and Effectiveness

Abstract In this paper, we present data from an exploratory study that aimed to investigate the ways in which, and the extent to which, undergraduates enrolled in a transition-to-proof course considered examples in their attempted proof constructions. We illustrate how some undergraduates can and do use examples for specific purposes while successfully constructing proofs, and that these purposes are consistent with those described by mathematicians. We then examine other cases in which students used examples ineffectively. We note that in these cases, the purposes for which the students attempted to use examples are again appropriate, but the implementation of their strategies is inadequate in one of two specific ways. On this basis we identify points that should be borne in mind by a university teacher who wishes to teach students to use examples effectively in proof-based mathematics courses.

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’.

Experimental methods in mathematics education research

This is a guest editorial. It was published in the journal, Research in Mathematics Education [© British Society for Research into Learning Mathematics] and the definitive version is available at: http://dx.doi.org/10.1080/14794802.2013.797731This special issue of Research in Mathematics Education (RME) is devoted to research designs that involve experimental methods. Few recent studies in mathematics education have taken advantage of such methods. In fact, during 2012 only 3% of papers published in leading mathematics education journals reported experimental studies, a remarkably low figure. In this special issue, we aim to address this imbalance, demonstrating that careful and inventive experimental designs can illuminate a variety of questions of interest to the mathematics education community. Specifically, experimental research designs can be of great value in our field because many of the questions that mathematics education researchers seek to address are, by their nature, causal. Interestingly, the semiotician Charles Sanders Pierce, who has had a heavy influence in mathematics education, was one of the first to pioneer experimental methods in social science research. Peirce and Jastrow’s (1885) great insight was that, by using randomisation, researchers could avoid the danger of incorrectly inferring a causal relationship between two unconnected factors. Experimentation is still preferred by many empirical researchers from a range of social science disciplines for exactly this reason: it permits causal inference in a way that other research designs do not. The structure of a basic between-subjects experiment is simple. The researcher wishes to know whether there is a causal relationship between an independent variable and a dependent variable. He or she sets up two conditions, in each of which the independent variable is set to a different level. Participants are randomly assigned to one condition or the other, and the dependent variable is measured in both conditions. Random assignment means that the two groups are probabilistically identical on all variables except one: the level of the independent variable. So, provided that the experiment is well controlled, and provided that the probability of the researcher’s results occurring by chance is sufficiently low, we can be sure that it is the between-conditions difference in the independent variable that caused any observed difference in the dependent variable. This straightforward design can reveal effects of relevance for policy-making, and part of its value in this context is in its capacity to reveal undesirable effects. Such was the case in a recent experiment we conducted in our own research, designed to test an educational intervention. Based on insights from the mathematics education literature, we designed multimedia presentations to combat undergraduates’ wellknown difficulties with understanding mathematical proofs (for details, see e.g., Alcock and Wilkinson 2011; Alcock and Inglis 2010). These presentations, which became known as e-Proofs, appeared highly successful: student feedback was remarkably positive, both in large-scale surveys and focus group settings, and lecturing staff were keen to incorporate the technology in their own classes. To Research in Mathematics Education, 2013 Vol. 15, No. 2, 97 99, http://dx.doi.org/10.1080/14794802.2013.797731

Interactions Between Teaching and Research: Developing Pedagogical Content Knowledge for Real Analysis

In this chapter, I describe five practices that I use in teaching undergraduate mathematics: regular testing on definitions, tasks that involve extending example spaces, tasks that involve constructing and understanding diagrams, use of resources for improving proof comprehension and tasks that involve mapping the structure of a whole course. I describe the rationale for each of these practices from my point of view as a teacher and relate each to results from mathematics education research. The discussion regularly returns to two overarching themes: the need for students to develop skills on multiple levels and the question of how best to use available lecture time. In a final section, I discuss these themes explicitly, focusing on their relevance to the work of teaching at all levels.

Support with caveats: advocates’ views of the Theory of Formal Discipline as a reason for the study of advanced mathematics

ABSTRACT The Theory of Formal Discipline (TFD) suggests that studying mathematics improves general thinking skills. Empirical evidence for the TFD is sparse, yet it is cited in policy reports as a justification for the importance of mathematics in school curricula. The study reported in this article investigated the extent to which influential UK advocates for mathematics agree with the TFD and their views on the arguments and evidence that surround it. Quantitative and qualitative analysis of data from structured interviews revealed four themes: broad endorsement of the TFD; reference to supportive employment data; the possibilities that mathematics education might not always effectively develop reasoning and that study of other subjects might have similar effects; and concerns about causality and the extent of the evidence base. We conclude that advocates broadly support the TFD despite being aware of its limitations.

Have you seen this?… Students and Proof: Bounds and Functions. A DVD for Mathematicians

The DVD contains material from two research interviews , in each of which a student attempts to answer proof tasks about upper bounds of sets of real numbers and about functions and maxima. The video content is specially annotated so that half of the screen shows the student as they work on the problem and the other half is white and has subtitles appearing in blue at the bottom and a printed version of what they are writing on the page at the top (see Fig 1 for a screen shot). This allows the viewer to follow the student’s thinking without being distracted by having to switch between document types. The content is also broken up into short (approximately 2-minute) excerpts, each of which is followed by a screen with suggestive questions to facilitate reflection on the material and/or discussion between colleagues.