107 research outputs found
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
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
Process- and object-based thinking in arithmetic
Many influential theorists have proposed that learners construct mathematical
objects via the encapsulation (or reification) of processes into objects. These processto-
object theories posit that object-based thinking comes later in the developmental
path than process-based thinking. In this paper we directly test this hypothesis in the
field of early arithmetic. An experiment is reported which studied 8 and 9 year-old
children’s use of the inverse relationship between addition and subtraction. We
demonstrate that a subset of children were unable to solve arithmetic problems using
process-based thinking, but that, nevertheless, they were able to use the inverse
relationship between addition and subtraction to solve problems where appropriate.
The implications of these findings for process-to-object theories are discussed
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
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
Watching mathematicians read mathematics
This report contributes to the debate about whether expert mathematicians skim-read
mathematical proofs before engaging in detailed line-by-line reading. It reviews the conflicting introspective and behavioural evidence, then reports a new study of expert mathematicians' eye movements as they read both entire research-level mathematics papers and individual proofs within those papers. Our analysis reveals no evidence of skimming, and we discuss the implications of this for research and pedagogy
Representation systems and undergraduate proof production: a comment on Weber
Weber (2009) suggested that counterexamples can be generated by a syntactic proof production, apparently contradicting our earlier assertion (Alcock & Inglis, 2008). Here we point out that this ostensible difference is the result of Weber working with theoretical definitions that differ slightly from ours. We defend our approach by arguing that Weber’s relies upon an as yet unspecific metric for gauging the amount of work conducted in each representation system, and that it does not recognize an important asymmetry between the status of representation systems in the context of undergraduate mathematics
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
Intelligence and negation biases on the Conditional Inference Task: a dual-processes analysis
We examined a large set of conditional inference data compiled from several previous studies and asked three questions: How is normative performance related to intelligence? Does negative conclusion bias stem from Type 1 or Type 2 processing? Does implicit negation bias stem from Type 1 or Type 2 processing? Our analysis demonstrated that rejecting denial of the antecedent and affirmation of the consequent inferences was positively correlated with intelligence, while endorsing modus tollens inferences was not; that the occurrence of negative conclusion bias was related to the extent of Type 2 processing; and that the occurrence of implicit negation bias was not related to the extent of Type 2 processing. We conclude that negative conclusion bias is, at least in part, a product of Type 2 processing, while implicit negation bias is not
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