Student difficulty in constructing proofs: The need for strategic knowledge

The ability to construct proofs is an important skill for all mathematicians. Despite its importance, students have great difficulty with this task. In this paper, I first demonstrate that undergraduates often are aware of and able to apply the facts required to prove a statement but still fail to prove it. They thus fail to construct a proof because they could not use the syntactic knowledge that they had. By comparing doctoral students and undergraduates constructing proofs in abstract algebra, I have hypothesized four types of `strategic knowledge' – knowledge of how to choose which facts and theorems to apply – which the doctoral students appeared to possess and undergraduates did not. The doctoral students appeared to know the powerful proof techniques in abstract algebra, which theorems are most important, when particular facts and theorems are likely to be useful, and when one should or should not try and prove theorems using symbol manipulation. This revised version was published online in July 2006 with corrections to the Cover Date.     

Why and how mathematicians read proofs: further evidence from a survey study

In a recent paper (Weber & Mejia-Ramos, Educational Studies in Mathematics, 76, 329–344, 2011), we reported findings from two small-scale interview studies on the reasons why and the ways in which mathematicians read proofs. Based on these findings, we designed an Internet-based survey that we distributed to practicing mathematicians working in top mathematics departments in the USA. Surveyed mathematicians (N?=?118) agreed to a great extent with the interviewed mathematicians in the exploratory studies. First, the mathematicians reported that they commonly read published proofs to gain different types of insight, not to check the correctness of the proofs. Second, they stated that when reading these proofs, they commonly: (a) appeal to the reputation of the author and the journal, (b) study how certain steps in the proof apply to specific examples, and (c) focus on the overarching ideas and methods in the proofs. In this paper, we also report findings from another section of the survey that focused on how participants reviewed proofs submitted for publication. The comparison of participant responses to questions in these two sections of the survey suggests that reading a published proof of a colleague and refereeing a proof for publication are substantially different activities for mathematicians.     

The Contribution of Educational Psychology to Intervention Research and Practice

Advances in psychological theory have made significant contributions to the considerable progress that has been achieved in intervention research and practice, especially in the past two decades. This article presents a discussion of the major conceptual and methodological issues that have arisen in intervention research, with a focus on interventions for children with learning problems, particularly reading problems. Examples from our own research are presented to illustrate how conceptual and methodological factors in intervention research can be taken into consideration in the design of intervention studies.     

A case study of the development of WebPA: An online peer-moderated marking tool

Peer moderation of group work in higher education is rapidly advancing through the use of technological developments and is increasingly being informed by pedagogical research. The highly successful WebPA online assessment system has gone through a number of development phases over its 15-year history and has now evolved into a relatively mature and flexible tool for facilitating group work. This paper presents a case study of the approaches that have been used in the system's development from the technical perspective, and describes how the scope of the project has massively expanded and that the development has been continually backed by a sound and wide-ranging pedagogy. The benefits of using the online system are shown to be underpinned throughout by examples of good practice in the supervision of academic group work. A critical evaluation of the tool and surrounding pedagogical practices highlight future areas for technical expansion.     

What university teachers teach and how they teach it

In this article we make three related arguments. The first isthat different teachers have different intentions concerning whatstudents will learn and consequently in their teaching they constitutethe topic or subject to be taught quite differently. The second is thata teacher's intentions concerning what it is that students should learnis closely aligned with a teacher's expectation of how students learnand how they can be helped to learn through teaching. The third is thatwhen teachers focus specifically on the teaching of a particular topic,within a specific context, there is a close relationship between theirintentions and their teaching practice. In this article we explore thesearguments through an empirical study which considers the different waysin which 26 university teachers intended to constitute a subject ortopic for their students to study, how they then taught the subject andsubsequently how consistent were their intentions and their practice.The analysis shows that when the context of teaching and learning istightly defined there is a clear relationship between a teacher'sintention and their practice. In particular, university teachers whoadopt more conceptual change and student-focussed approaches to teachingconstitute objects of study which are more relational and focus on thestudent's knowledge. Approaches which are more information transmissionand teacher-focussed constitute objects of study which are moremulti-structural and have a focus on knowledge which is as constitutedas being external to the student.     

Initial hydration status, fluid balance, and psychological affect during recreational exercise in adults

There is little information on the impact of hydration status on the psychological response to exercise despite potential implications for adherence to an exercise programme and for overall health and fitness. We investigated initial hydration status, fluid balance, and psychological responses associated with a typical recreational exercise session in healthy adults. Fifty-two participants performed a freely chosen gymnasium-based exercise session at a fitness centre, with ad libitum access to fluids. Urine samples were collected on arrival for analysis of osmolality. Sweat loss was estimated from the change in body mass after correction for fluid intake and urinary losses. Subjective psychological ratings were recorded before and after exercise. Pre-exercise urine osmolality was above 900 mOsmol · kg(-1) (used as a threshold for hypohydration) in 37% of participants. Fluid intake during exercise was 390 ± 298 mL, while estimated sweat loss was 794 ± 391 mL. The percentage change from pre-exercise body mass was -0.62 ± 0.20%. Physically active adults who arrived to take part in exercise hypohydrated reported more negative changes in psychological affect in response to their subsequent freely chosen recreational exercise session than those classified as euhydrated prior to exercise (-0.2 ± 0.7 vs. 0.8 ± 0.7; P < 0.005).     

Religious beliefs in science classrooms

The question of the relationship between science and religion assumes importance for many secondary school students of science, especially but not exclusively for those in Christian schools. Science as presented in many school classrooms is not as objective and value free as it might seem on first examination, nor does it represent adequately the range of beliefs about science held by students and teachers. This paper reports part of a larger research study into beliefs about science and religion held by students, teachers and clergy in a Lutheran secondary school. Results indicate that participants in the study was the relationship between science and religious belief in ways unforeseen and unappreciated by traditional school science programs. The stories of selected participants are told and they frame a discussion of implications of the study for science teaching.     

Cognitive Neuroscience and Educational Psychology: What Season Is It?

Byrnes and Fox get much right in their review of the relevance of cognitive neuroscience for educational psychology. They are correct that theoretical reduction is to be welcomed. They are also right that neurophysiological findings can constrain psychological theory and that this too is to be welcomed. Their review is on the mark in recommending that educational psychologists must become bilingual in their understanding of neurological terms and their analogous psychological constructs. The one difference in our positions that I do highlight is one of calibration rather than strong disagreement—my level of enthusiasm for scientific reduction in domains related to education is somewhat more muted than that of Byrnes and Fox because of wariness bom of the premature reductive attempts that litter our field.