Caroline Yoon

Student Perceptions of Effective Use of Tablet PC Recorded Lectures in Undergraduate Mathematics Courses.

Tablet PCs have been increasingly used in undergraduate courses to create recorded lectures that are close copies of the live lectures. Research has shown that students are largely positive about the availability of tablet PC recorded lectures. However, there is some concern that the availability of such faithful recordings may encourage students to miss live lectures, which may in turn lead to lower achievement in the course. In this study, we surveyed students on their use of recorded lectures in two large undergraduate mathematics courses. We investigated patterns in their use of recorded lectures and live lecture attendance, how and why they used recorded lectures and how this use was associated with their final grade. The results suggest that the practice of missing live lectures intentionally because the recordings were available was not associated with final grade. However, those respondents who intended to watch more recorded lectures than they actually did achieved significantly lower grades.

How high is the tramping track? Mathematising and applying in a calculus model-eliciting activity.

Two complementary processes involved in mathematical modelling are mathematising a realistic situation and applying a mathematical technique to a given realistic situation. We present and analyse work from two undergraduate students and two secondary school teachers who engaged in both processes during a mathematical modelling task that required them to find a graphical representation of an anti-derivative of a function. When determining the value of the anti-derivative as a measure of height, they mathematised the situation to develop a mathematical model, and attempted to apply their knowledge of integration that they had previously learned in class. However, the participants favoured their more primitive mathematised knowledge over the formal knowledge they tried to apply. We use these results to argue for calculus instruction to include more modelling activities that promote mathematising rather than the application of knowledge.

It's Not the Done Thing: Social Norms Governing Students' Passive Behaviour in Undergraduate Mathematics Lectures.

Students often play a passive role in large-scale lectures in undergraduate mathematics courses: they observe the lecturer demonstrate mathematical procedures, but they rarely engage in authentic mathematical activity themselves. This study uses semi-structured interviews of undergraduate students to investigate the implicit and explicit social norms and expectations that influence students to maintain their passive roles during lectures. Students were aware that their passivity was influenced by social norms, but perceived these norms as necessary for allowing the lecturer to get through the content in the allotted lecture time, while enabling students to avoid being publicly embarrassed in the lecture. However, the students appreciated opportunities to work on examples in small groups during lectures. We argue that the success of small group interactions during large-scale lectures depends on students and lecturers establishing supportive social norms, and adjusting their lecture goals from ‘covering the content’ to ‘developing mathematical understanding’.

Gestures and insight in advanced mathematical thinking

What role do gestures play in advanced mathematical thinking? We argue that the role of gestures goes beyond merely communicating thought and supporting understanding – in some cases, gestures can help generate new mathematical insights. Gestures feature prominently in a case study of two participants working on a sequence of calculus activities. One participant uses gestures to clarify the relationships between a function, its derivative and its antiderivative. We show how these gestures help create a virtual mathematical construct, which in turn leads to a new problem-solving strategy. These results suggest that gestures are a productive, but potentially undertapped resource for generating new insights in advanced levels of mathematics.

Undergraduate mathematics students’ reasons for attending live lectures when recordings are available

With the proliferation of new affordable recording technologies, many universities have begun offering students recordings of live lectures as a part of the course resources. We conducted a survey to investigate why some students choose to attend lectures in person rather than simply watching the recordings online, and how students view the two types of lectures. Students attending live lectures in five large undergraduate mathematics lecture streams were invited to respond to the survey. A significant number of respondents viewed recorded lecture as superfluous to their needs which were met upon attending live lecture. Surprisingly, however, an equally large number of students described compelling reasons for watching both live and recorded lectures. A number of factors were identified as determining students’ perceptions of live and recorded lectures as competing or complementary: personal learning styles, study habits, esteem for the lecturer, and the possibility of interaction in the lecture.

Mapping Mathematical Leaps of Insight

Mathematical leaps of insight—those Aha! moments that seem so unpredictable, magical even—are often the result of a change in perception. A stubborn problem can yield a surprisingly simple solution when one changes the way one looks at it. In mathematics, these changes in perception are usually structural: new insights develop as one notices new mathematical objects, attributes, relationships and operations that are relevant to the problem at hand. This paper describes a novel analytical approach for studying these insights visually using “mathematical SPOT diagrams” (SPOT: Structures Perceived Over Time), which display evidence of the mathematical structures students perceive as they work on problems. SPOT diagrams are used to compare the conceptual development of two pairs of participants, who investigate whether a gradient (derivative) graph yields information about the relative heights of points on its antiderivative; one participant pair experiences a leap of insight, whereas the other does not. Each pair’s SPOT diagrams reveal key differences in the structural features they attend to, which can account for the disparate outcomes in their conceptual development.

The FOCUS Framework: Characterising Productive Noticing During Lesson Planning, Delivery and Review

Enacting the work of diagnostic teaching is challenging and demands that teachers pay attention to mathematical details when designing tasks, orchestrating discussions and reflecting on their lessons. This chapter presents the FOCUS Framework on teacher noticing, which can be used to characterise teachers’ efforts to notice productively during all three phases of diagnostic teaching: lesson planning, delivery and review. Using the two key components of the framework, the focus and its focusing, we provide snapshots of a teacher’s mathematical noticing in each of the phases. The findings from this research suggest that productive noticing in all the three phases is highly consequential, and illustrates how the FOCUS Framework can be used to analyse a teacher’s mathematical noticing.

Visualisation for Different Mathematical Purposes

Visualisation is often suggested as a useful heuristic for generating new ideas when one is stuck on a problem. Yet generating ideas is just one aspect of mathematical activity. Visualisation can also help students generalise mathematical discoveries and communicate mathematical ideas.

Visualising Cubic Reasoning with Semiotic Resources and Modeling Cycles

Diagrams and physical manipulatives are often recommended as useful semiotic resources for visualising area and volume problems in which nonlinear reasoning is appropriate. However, the mere presence of diagrams and physical manipulatives does not guarantee students will recognise the appropriateness of nonlinear reasoning.