Introduction

There has been little overt discussion of the experimental philosophy of logic or mathematics. So it may be tempting to assume that application of the methods of experimental philosophy to these areas is impractical or unavailing. This assumption is undercut by three trends in recent research: a renewed interest in historical antecedents of experimental philosophy in philosophical logic; a “practice turn” in the philosophies of mathematics and logic; and philosophical interest in a substantial body of work in adjacent disciplines, such as the psychology of reasoning and mathematics education. This introduction offers a snapshot of each trend and addresses how they intersect with some of the standard criticisms of experimental philosophy. It also briefly summarizes the specific contribution of the other chapters of this book

Proof in mathematics education: research, learning and teaching

This is a book review of Proof in mathematics education: research, learning and teaching, by David Reid with Christine Knipping, Rotterdam, Sense Publishers, 2010, 266pp., £35 (paperback), ISBN 978-94-6091-244-3

How do you describe mathematics tasks?

Colin Foster and Matthew Inglis ask what it means to describe a task as ‘rich’

Dual processes in mathematics: reasoning about conditionals

This thesis studies the reasoning behaviour of successful mathematicians. It is based on the philosophy that, if the goal of an advanced education in mathematics is to develop talented mathematicians, it is important to have a thorough understanding of their reasoning behaviour. In particular, one needs to know the processes which mathematicians use to accomplish mathematical tasks. However, Rav (1999) has noted that there is currently no adequate theory of the role that logic plays in informal mathematical reasoning. The goal of this thesis is to begin to answer this specific criticism of the literature by developing a model of how conditional “if…then” statements are evaluated by successful mathematics students. Two stages of empirical work are reported. In the first the various theories of reasoning are empirically evaluated to see how they account for mathematicians’ responses to the Wason Selection Task, an apparently straightforward logic problem (Wason, 1968). Mathematics undergraduates are shown to have a different range of responses to the task than the general well-educated population. This finding is followed up by an eve-tracker inspection time experiment which measured which parts of the task participants attended to. It is argued that Evans’s (1984, 1989, 1996, 2006) heuristic-analytic theory provides the best account of these data. In the second stage of empirical work an in-depth qualitative interview study is reported. Mathematics research students were asked to evaluate and prove (or disprove) a series of conjectures in a realistic mathematical context. It is argued that preconscious heuristics play an important role in determining where participants allocate their attention whilst working with mathematical conditionals. Participants’ arguments are modelled using Toulmin’s (1958) argumentation scheme, and it is suggested that to accurately account for their reasoning it is necessary to use Toulmin’s full scheme, contrary to the practice of earlier researchers. The importance of recognising that arguments may sometimes only reduce uncertainty in the conditional statement’s truth/falsity, rather than remove uncertainty, is emphasised. In the final section of the thesis, these two stages are brought together. A model is developed which attempts to account for how mathematicians evaluate conditional statements. The model proposes that when encountering a mathematical conditional “if P then Q”, the mathematician hypothetically adds P to their stock of knowledge and looks for a warrant with which to conclude Q. The level of belief that the reasoner has in the conditional statement is given by a modal qualifier which they are prepared to pair with their warrant. It is argued that this level of belief is fixed by conducting a modified version of the so-called Ramsey Test (Evans & Over, 2004). Finally the differences between the proposed model and both formal logic and everyday reasoning are discussed

La fuerza de la aserción y el poder persuasivo en la argumentación en matemáticas

El análisis de la construcción y evaluación de argumentos en matemáticas se ha convertido en una parte importante de la investigación en Educación Matemática a nivel universitario. En este artículo notamos que hasta el momento este análisis se ha hecho desde una perspectiva restringida que se concentra en argumentos construidos para eliminar toda duda en torno a una conjetura. Discutimos una perspectiva más amplia de la argumentación en matemáticas, que toma en consideración las distintas maneras por medio de las cuales tanto estudiantes como matemáticos califican sus conclusiones y se sienten persuadidos por argumentos en matemáticas. Esta perspectiva esta basada en el esquema de argumentación propuesto por Toulmin (1958) y permite analizar argumentos construidos par areducir el nivel de incertidumbre asociado a una conjetura, al mismo tiempo que permite el análisis de distintos tipos de persuasión en la evaluación de argumentos en matemáticas

Sampling from the mental number line: how are approximate number system representations formed?

Nonsymbolic comparison tasks are commonly used to index the acuity of an individual’s Approximate Number System (ANS), a cognitive mechanism believed to be involved in the development of number skills. Here we asked whether the time that an individual spends observing numerical stimuli influences the precision of the resultant ANS representations. Contrary to standard computational models of the ANS, we found that the longer the stimulus was displayed, the more precise was the resultant representation. We propose an adaptation of the standard model, and suggest that this finding has significant methodological implications for numerical cognition research

The problem of assessing problem solving: can comparative judgement help?

School mathematics examination papers are typically dominated by short, structured items that fail to assess sustained reasoning or problem solving. A contributory factor to this situation is the need for student work to be marked reliably by a large number of markers of varied experience and competence. We report a study that tested an alternative approach to assessment, called comparative judgement, which may represent a superior method for assessing open-ended questions that encourage a range of unpredictable responses. An innovative problem solving examination paper was specially designed by examiners, evaluated by mathematics teachers, and administered to 750 secondary school students of varied mathematical achievement. The students’ work was then assessed by mathematics education experts using comparative judgement as well as a specially designed, resourceintensive marking procedure. We report two main findings from the research. First, the examination paper writers, when freed from the traditional constraint of producing a mark scheme, designed questions that were less structured and more problem-based than is typical in current school mathematics examination papers. Second, the comparative judgement approach to assessing the student work proved successful by our measures of inter-rater reliability and validity. These findings open new avenues for how school mathematics, and indeed other areas of the curriculum, might be assessed in the future

Indexing the approximate number system

Much recent research attention has focused on understanding individual differences in the Approximate Number System, a cognitive system believed to underlie human mathematical competence. To date researchers have used four main indices of ANS acuity, and have typically assumed that they measure similar properties. Here we report a study which questions this assumption. We demonstrate that the Numerical Ratio Effect has poor testretest reliability and that it does not relate to either Weber fractions or accuracy on nonsymbolic comparison tasks. Furthermore, we show that Weber fractions follow a strongly skewed distribution and that they have lower test-retest reliability than a simple accuracy measure. We conclude by arguing that in future researchers interested in indexing individual differences in ANS acuity should use accuracy figures, not Weber fractions or Numerical Ratio Effects

Conditional inference and advanced mathematical study

Many mathematicians and curriculum bodies have argued in favour of the theory of formal discipline: that studying advanced mathematics develops one’s ability to reason logically. In this paper we explore this view by directly comparing the inferences drawn from abstract conditional statements by advanced mathematics students and well-educated arts students. The mathematics students in the study were found to endorse fewer invalid conditional inferences than the arts students, but they did not endorse significantly more valid inferences. We establish that both groups tended to endorse more inferences which led to negated conclusions than inferences which led to affirmative conclusions (a phenomenon known as the negative conclusion effect). In contrast, however, we demonstrate that, unlike the arts students, the mathematics students did not exhibit the affirmative premise effect: the tendency to endorse more inferences with affirmative premises than with negated premises.We speculate that this latter result may be due to an increased ability for successful mathematics students to be able to ‘see through’ opaque representations. Overall, our data are consistent with a version of the formal discipline view. However, there are important caveats; in particular, we demonstrate that there is no simplistic relationship between the study of advanced mathematics and conditional inference behaviour