Finally, I question in MO I feel qualified to answer!

I am a PhD student in Ireland doing an amount of lecturing. As a first remark, I am lucky in the sense that undergraduate maths was never especially easy for me and therefore I empathise with the average student. My second remark is that I hope for a career lecturing in the Irish Institute of Technology sector where the role in primarily teaching as opposed to the university sector where research is the primary role. Hence I am acutely interested in the skills as a mathematics teacher.

The second half of the answers here are closer to my philosophy than the first. A particular distinction must be put on the classroom environment and facilities. Regardless, my first instinct is that slides alone is sub-optimal.

The alternative to this is to produce everything on the blackboard. I did this last year in a differential calculus module (the students were maths studies --- by and large headed towards a career as "high school" mathematics teachers). The emphasis in this course is to convey to the students that although differential calculus is a relatively intuitive subject with the motivation coming from geometric concerns, as mathematicians we must also be rigorous, logical and precise in our thinking. Hence, we are not merely making a series of calculations and passing exams --- we must understand the content. When I wrote blackboard after blackboard of notes, the students did not have any chance of understanding the material. While I am a fervent believer that exercises and reflection are the best way for a student to achieve this aim, I am reminded of my undergraduate experience where certain obstacles lay in the path of me putting in this work and luckily my presence at lecture-time was sufficient for me to grasp the general theory and progress (eventually with first class grades) despite less than exemplary exam results in previous years. Put simply, ordinary students do not have the faculties to take down written notes and consider the important comments of the lecturer in real time.

However, slides do not work because mathematics is not a spectator sport (not a cliche when the average student is first interested in passing exams --- its is the goal of the educator to transcend this). It takes a superlative lecturer and a cohort of motivated and enthusiastic students to assimilate a lecture purely by ear. At least once I had a lecturer of this standard but I would vouch that were engineering, scientific or humanities students subjected to his fantastic delivery and questioning, they would simply fall asleep. It is a curse but a fact (among my students at least --- none of which are Math majors), that the average student does not have that aptitude to bask in such splendour.

My compromise, therefore is the very similar to what has been suggested above. I produce a set of notes (available soft-bound in a local printing house), with gaps which we fill in during the class (I print the notes onto an acetate sheet which I project onto a screen and can write on with a marker). All the theorems are writ-large, and everything else is teased out per a blackboard with suitable prior fillings in to both give the students a sneak preview and for the practical reasons of properly spacing out my scribblings. Does the need arise, I can put more complicated graphics in this set of notes. Today we introduced implicit differentiation and I projected this Wikipedia page list of curves onto the screen and this was but a two minute interlude.

The issue of students looking ahead was served by a motivation at the start of term (we are studying continuous and differentiable (smooth) functions. We draw a picture. We translate these geometric pictures into an algebebraic ones and never lose sight of this fact).

I have covered more content this year than last using this method, the first continuous assessment results showed a marked improvement and I am ahead of schedule despite being able to allocate a lot more time to comments and explanation of subtleties.

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