Derivations in lectures: Using interactive questions to find focus

Photo credit: Unsplash, Edwin Andrade, CC0

In this post, Dr Job Thijssen, Lecturer, and Dr Ross Galloway, Senior Lecturer, both in the School of Physics and Astronomy, argue that using interactive tools in large classes might democratise student participation…

Feedback from our own students on traditional physics courses suggests that they want the lecturer to do more derivations on the blackboard. In fact, it seems to be a common theme across cohorts, physics disciplines and institutions [1]. To some degree, this feedback is perfectly understandable: students are asked to do derivations at exams and observing how a professional physicist tackles a derivation is not dissimilar to an apprentice observing a master when learning a craft. However, the educational benefit of doing more derivations on the blackboard is still under debate [2-5]. One criticism often raised is that it tends to be a passive form of learning, in which student and teacher take their traditional roles of receivers and providers of knowledge. Moreover, the lecturer runs the risk of covering material the students already know.

Mid-semester feedback for one of the authors’ 2018 Honours-level courses suggested again that students wanted more derivations on the blackboard and more tutorial-style content. Given limited resources, the suggested solution was a non-traditional approach to in-class derivations: interactive Top Hat quizzes. In Top Hat, the students are given a starting scenario, for example a random walk consisting of N steps. The students are then given ample time, usually five minutes or so, to derive a useful property (the mean end-to-end distance of the random walk in the example below). The multiple-choice answers for this Top Hat question are equations representing stages of the derivation. Hence, the students’ answers give the lecturer a good sense of which part(s) of the derivation they understand and which part(s) they struggle with. The lecturer can then skip the former and focus their attention on the latter, thereby devoting more time to explaining those steps in the derivations that the students struggle with.

A screenshot of using interactive Top Hat quizzes

During Peer Observation of Teaching (POoT), it was observed that this approach to derivations seemed to have really worked. First, students were engaged during the five minutes they tried the derivation themselves. Second, the majority of students got stuck at the same stage of the derivation, allowing the lecturer to skip the first part of the derivation and focussing the remainder of the lecture on the part that the students initially did not understand. This allowed the lecturer to go through the crucial mathematical steps one by one, provide additional qualitative explanations, and discuss some transferable derivation skills, including checking the units of the answer and confirming its correct behaviour in some more easily predictable situations.

In a way, this approach is essentially a creative way to extend the use of clicker questions beyond the sort of question that can be answered with just a few moments’ thought. It provides a medium by which the whole class has the potential to engage with the development of a derivation, rather than just the few who might be willing to shout out suggestions. As such, it seems to be a successful innovation in helping to make extended ‘board work’ genuinely interactive. End-of-semester student feedback also suggested that this approach had worked well. Having said that, this ‘Top Hat derivations’ approach has only been trialled in a relatively small class, so the next step is to try it in one of the large first or second-year physics courses, to see whether it translates well to a larger class.

Furthermore, while active learning approaches are now fairly well established in physics [6], nevertheless they remain more prevalent in early-years, introductory courses. While some institutions are developing approaches for more advanced courses (e.g. the University of Colorado Boulder’s work on upper division teaching [7]), there remain many open questions about how students can effectively engage with very advanced, complex material during lecture-style classes without becoming overwhelmed. The interactive quiz-based approach to derivations that we describe here provides one avenue for introducing meaningful active student engagement into an Honours-level course while still allowing the lecturer to ‘show the path ahead’ with their experience and expertise when required.

In short, rather than considering active learning and derivations on blackboards as opposing features in a lecture, this Top Hat approach to derivations blends the two. This allows the lecturer to focus on the steps of the derivation that the students struggle with. In turn, this frees up time to teach additional skills, for example checking whether a derived mathematical expression makes physical sense. The appropriate use of digital tools can further enhance this approach. For example, a common criticism of derivations in lectures is that they are better done in a textbook, so the student can go over it again, but lecture recording that also captures the blackboard allows the student to do just that with an in-class derivation as well. Creatively combining Top Hat derivation questions with lecture recording may also help to bring traditional ‘board work’ to interactive distance learning.


[1] Ball, P. (2017). The power of the blackboard, Physics World June.

[2] Freedman, R.A. (1996). Challenges in Teaching and Learning Introductory Physics. In: B Cabrera, H. Gutfreund and V. Kresin (eds) From High-Temperature Superconductivity to Microminiature Refrigeration. Springer, Boston, MA.

[3] Frank, A. (2013). Sometimes, The Old Ways Are The Best Ways, (date accessed 4 October 2019)

[4] Mulvihill, M. (2014). Save the blackboard: an endangered species that is essential to scientific success, The Irish Times, 13 Nov 2014.

[5] Galloway, R. (2019). A clean slate, (date accessed 4 October 2019)

[6] Meltzer, D. E. (2012). Resource Letter ALIP–1: Active-Learning Instruction in Physics, American Journal of Physics, 80, 478.

[7] Physics Education Research Group, University of Colorado Boulder, (date accessed 16 January 2020)

Job Thijssen

Job Thijssen is a Lecturer in the School of Physics and Astronomy. He teaches on undergraduate courses in Soft Condensed Matter Physics and Experimental Physics. In addition, he supervises Senior Honours, MPhys and PhD research projects. His research focuses on the physics and application of soft materials, especially those in which interfaces are crucial and those that have potential for applications in coatings and energy materials. He is also a co-investigator on the University’s Inclusion Matters eBase project.

Ross Galloway

Ross Galloway is a Senior Lecturer in the School of Physics and Astronomy. He teaches on the undergraduate programmes in physics and also conducts pedagogic research as a member of the Edinburgh Physics Education Research group (EdPER). His research interests include the development of student problem solving skills, diagnostic testing, and flipped classroom pedagogies.

One comment

  1. From a psychological point of view, students like black-board derivations because they like the teacher’s performance. There is no doubt that today students can learn everything themselves, thanks to online materials. Last year I taught blackboard lectures and students liked them a lot. From my experience, books do not provide proper derivations or explanations, a teacher can always explain difficult parts using a language that students can understand.

    Besides, it is always possible to involve students in discussions and derivations. There is also a question, what if 10 % of students could not derive an equation correctly. Do you go to the next question? This would probably the case for larger classes. In this case, 5 min spent on the quiz and then 5 min spent to explain the same for 10 % of students. In this case, it becomes not effective. A usual black-board explanation would help all audience including those 10%.

    Maybe the question is about engaging students. It is proved that students lose attention after 15 min of doing something the same (listening to a lecture). So, I think a quiz could be used here to just let students a short break. I am also interested if students consider derivations consistent. I mean, whether breaking blackboard derivations into quiz peaces has any negative effect on understanding a whole picture of a topic of derivation.

    Interesting paper to discuss.
    Thank you

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