Sven Trenholm

Individual differences in students' use of optional learning resources

We investigated ways in which undergraduates use optional learning resources in a typical blended learning environment. Specifically, we recorded how often students attended live face-to-face lectures, accessed online recorded lectures, and visited a mathematics learning support centre during a multivariate calculus course. Four distinct study strategies emerged, but surprisingly none involved making heavy use of more than one resource. In contrast with some earlier research, the general strategy a student adopted was related to their academic achievement, both in the multivariate calculus course, and in their degree programme more widely. Those students who often accessed online lectures had lower attainment than those who often attended live lectures or the support centre. We discuss the implications of these findings and suggest that ‘blended teaching environments’ may be a more accurate description for what have previously been called ‘blended learning environments’.

A Review of Cheating in Fully Asynchronous Online Courses: A Math or Fact-Based Course Perspective

Un-proctored fully asynchronous1 online courses now appear to be a reality in all discipline areas. With full degree programs being offered online, it has become a badge of honor, to some colleges, to graduate students that have never stepped foot on the physical college campus nor have never before met any college representative, faculty, or staff. To many involved in managing, teaching, and implementing these online courses, the idea of academic integrity related to courses taught in a fully online un-proctored modality is therefore of great interest and concern. This review represents a deliberate effort to distinguish the issue of academic integrity in fully asynchronous online courses into two categories: “Writing-Based” (WB) vs. “Math or Fact-Based” (MFB) courses. WB courses, which tend to be subjective in nature, include discipline areas such as English, history, psychology… . MFB courses, which emphasize calculation and/or factual recall, have a tendency to be highly objective in nature, and include discipline areas such as math, science, business, computers, medical technology… . While there are certainly significant areas of overlap in how different assessment instruments are administered in these two categories, most math and many science courses, for instance, tend to focus on calculation and fact-based assessment instruments (such as mid-term and final exams) and most English and history courses, for example, tend to focus on writing-based assessment instruments (such as written assignments and term papers). Based on this distinction, the notion that pedagogical mechanisms (apart from proctored assessments) are sufficient to ensure academic integrity in online MFB courses is questioned and challenged. Recommendations are given to advance the use of proctored assessments in online MFB courses through the creation of universal proctoring standards and the expansion of current and new testing center networks.

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.

Tertiary Mathematics Education

Mathematical and statistical education research relevant to students in tertiary settings is reviewed. This is an expanding field and is evolving as researchers shift their attention from the reporting of innovations in lecturing practice and course design to include a deeper consideration of the experiences of educators and learners in this space. The purposeful inclusion of group work and discussion, focus on concepts, authentic problem solving, interactions in lectures with student response systems and online learning are all changing the way mathematics and statistics are taught at this level. The authors note that traditional measures of achievement in the form of exam marks are still relied upon, and call for theory-based and theory-building research including investigations of depth of understanding, and of transfer of knowledge and skills to new situations. An emphasis on the learner’s experience and the employment of cross-disciplinary teams of researchers are further suggestions.

Prospective primary teachers’ perceptions about the use of mathematics homework

ABSTRACT The benefit of homework (HW) has been the subject of ongoing debate among various stakeholders. Within this debate, and in relation to teacher development, prospective teachers’ views of HW have received limited attention. In this study, we survey primary pre-service teachers’ (‘PSTs’) views of HW use (n = 45 teaching grades 2–5; n = 39 teaching grades 6 and 7) in a single discipline (mathematics) at one large Australian university. The literature suggests students in upper primary and secondary grades will benefit from the use of mathematics HW. Results of the study, in contrast, suggest our PSTs teaching early primary grades planned to give proportionally (but not significantly) more mathematics HW than those teaching later grades. Reasons for this disparity are discussed. We argue there is a need in teacher development to equip PSTs with a better understanding of how classroom and home-based learning work together in the development of mathematical understanding.

A Review of Fully Online Undergraduate Mathematics Instruction through the Lens of Large Scale Research (2000-2015)

Abstract The use of fully online instruction has been proliferating in higher education. In undergraduate mathematics, arguments about the effectiveness of fully online instruction have been met with contrasting claims about its potential. Some assert that fully online instruction can promote a deeper and richer conceptual understanding of mathematical ideas while others claim it is difficult to teach mathematics effectively in a fully online context. The aim of this study is to examine these claims by documenting relevant empirical evidence, the findings of which we hope will inform current efforts, by both faculty and administrators, to improve fully online mathematics instruction.

Symbiosis Between Subject Matter and Pedagogical Knowledge in Geometry

Teacher knowledge that supports effective mathematics teaching has come under scrutiny alongside associated theoretical developments in the education field. Amongst these developments, the Mathematics Knowledge for Teaching (MKT) framework by Ball et al. (J Teacher Educ 59(5):389–407, 2008) has been one of the most influential. While MKT has been useful in helping us identify the knowledge strands teachers need for effective practice, the interplay among MKT’s knowledge strands during the course of teaching has received less attention. In this study, we address this issue by exploring interaction between Subject Matter Knowledge (SMK) and Pedagogical Content Knowledge (PCK) in the domain of secondary geometry. We provide results of a preliminary study of SMK and PCK in the context of a teacher teaching students how to construct and bisect an acute angle with the aid of compass and ruler only. Our analysis suggests future research needs to consider (a) the particular characteristics of the discipline of geometry and (b) the developmental knowledge trajectories of teachers of geometry in order to better understand how teachers’ SMK influences and influenced by PCK.

Investigating undergraduate mathematics learners’ cognitive engagement with recorded lecture videos

ABSTRACT The use of recorded lecture videos (RLVs) in mathematics instruction continues to advance. Prior research at the post-secondary level has indicated a tendency for RLV use in mathematics to be negatively correlated with academic performance, although it is unclear whether this is because regular users are generally weaker mathematics students or because RLV use is somehow depressing student learning. Through the lens of cognitive engagement, a quasi-experimental pre- and post-test design study was conducted to investigate the latter possibility. Cognitive engagement was operationalized using the Revised Two-Factor Study Process Questionnaire (R-SPQ-2F), which measures learning approaches on two major scales: surface and deep. In two mathematics courses at two universities, in Australia and the UK, participants were administered the questionnaire near the course start and finish. Overall findings were similar in both contexts: a reduction in live lecture attendance coupled with a dependence on RLVs was associated with an increase in surface approaches to learning. This study has important implications for future pedagogical development and adds to the sense of urgency regarding research into best practices using RLVs in mathematics.