Mathematical tasks,study approaches,and course grades in undergraduate mathematics: a year-by-year analysis

Students approach learning in different ways, depending on the experienced learning situation. A deep approach is geared toward long-term retention and conceptual change while a surface approach focuses on quickly acquiring knowledge for immediate use. These approaches ultimately affect the students’ academic outcomes. This study takes a cross-sectional look at the approaches to learning used by students from courses across all four years of undergraduate mathematics and analyses how these relate to the students’ grades. We find that deep learning correlates with grade in the first year and not in the upper years. Surficial learning has no correlation with grades in the first year and a strong negative correlation with grades in the upper years. Using Bloom's taxonomy, we argue that the nature of the tasks given to students is fundamentally different in lower and upper year courses. We find that first-year courses emphasize tasks that require only low-level cognitive processes. Upper year courses require higher level processes but, surprisingly, have a simultaneous greater emphasis on recall and understanding. These observations explain the differences in correlations between approaches to learning and course grades. We conclude with some concerns about the disconnect between first year and upper year mathematics courses and the effect this may have on students.     

The Mathematics Attitudes and Perceptions Survey: an instrument to assess expert-like views and dispositions among undergraduate mathematics students

One goal of an undergraduate education in mathematics is to help students develop a productive disposition towards mathematics. A way of conceiving of this is as helping mathematical novices transition to more expert-like perceptions of mathematics. This conceptualization creates a need for a way to characterize students' perceptions of mathematics in authentic educational settings. This article presents a survey, the Mathematics Attitudes and Perceptions Survey (MAPS), designed to address this need. We present the development of the MAPS instrument and its validation on a large (N = 3411) set of student data. Results from various MAPS implementations corroborate results from analogous instruments in other STEM disciplines. We present these results and highlight some in particular: MAPS scores correlate with course grades; students tend to move away from expert-like orientations over a semester or year of taking a mathematics course; and interactive-engagement type lectures have less of a negative impact, but no positive impact, on students' overall orientations than traditional lecturing. We include the MAPS instrument in this article and suggest ways in which it may deepen our understanding of undergraduate mathematics education.     

Implementation and outcomes of inquiry-based learning in mathematics content courses for pre-service teachers

This mixed-methods study describes classroom characteristics and student outcomes from university mathematics courses that are based in mathematics departments, targeted to future pre-tertiary teachers, and taught with inquiry-based learning (IBL) approaches. The study focused on three two-term sequences taught at two research universities, separately targeting elementary and secondary pre-service teachers. Classroom observation established that the courses were taught with student-centred methods that were comparable to those used in IBL courses for students in mathematics-intensive fields at the same institutions. To measure pre-service teachers' gains in mathematical knowledge for teaching, we administered the Learning Mathematics for Teaching (LMT) instrument developed by Hill, Ball and Schilling for in-service teacher professional development. Results from the LMT show that pre-service teachers made significant score gains from beginning to end of their course, while data from interviews and from surveys of learning gains show that pre-service teachers viewed their gains as relevant to their future teaching work. Measured changes on pre-/post-surveys of attitudes and beliefs were generally supportive of learning mathematics but modest in magnitude. The study is distinctive in applying the LMT to document pre-service teachers' growth in mathematical knowledge for teaching. The study also suggests IBL is an approach well suited to mathematics departments seeking to strengthen their pre-service teacher preparation offerings in ways consistent with research-based recommendations.     

Opportunities for learning: the use of variation to analyse examples of a paradigm shift in teaching primary mathematics in England

We demonstrate the power of Variation Theory as an analytical tool used to understand the underlying conceptual structure of mathematics lessons taught by English primary school teachers. We study excerpts of three lessons that are posted on a professional website. We show how lesson analysis using variation allows us to focus on what is made available to be learnt in the lesson excerpts. We identify some differences in their use of dimensions of variation and the associated ranges of change and discuss how suitable patterns of variation and invariance might differ according to the nature of the learning focus. We reflect on the value of our analytical approach.