What can I say? 2020, Covid, unprecedented, etc. You already know this. Well, since I’m co-organising a Teaching and Learning Mathematics Online (TALMO) workshop today on lessons learned following the sudden switch to online teaching I decided to write down some of my lessons learned. The TALMO meeting can be found at Tackling Term 2: Top Tips from Term 1. Last term, apart from a single hour right at the start, my teaching was online. As expected it didn’t go perfectly. Actually, some of it went quite badly and I’ll talk about that on another day. For the moment I’ll concentrate on the lessons learned. 1. A decent set of notes still counts for a lot. I have a theory that, at least in normal years, students will forgive most things if they have been given a decent — i.e., well-organised and clearly explained — set of notes. Certainly, a set of incoherent notes does not magically “challenge students” and “lead them to study harder” as I’ve sometimes heard. This theory seems to hold true for online teaching as well. In-course questionnaires indicated that students were using their notes for the bulk of learning and were not using the videos as much as expected. My conclusion was that it was important to tailor the videos to the notes and not tailor the notes to the videos. I had initially wondered if students would use the videos more than the notes but that did not appear to be the case. (Aside: I wished that I had worked on my notes more before teaching started. I was teaching on modules that were (mostly) new to me and so I was always going to have to produce new notes this. My mistake was to think it was...

## Mathematics Take Away Open Book Assessment

posted by Kevin Houston

With the current Covid-19 crisis many Higher Education Institutions (HEIs) are moving to take away exams which by their nature are open book exams. Many mathematicians do not have experience of take home open book assessments and this post is intended to help deal with this. This has been put together quickly and is intended to be practical rather than theoretical. I hope you find it of use. I will update when I can so if you have any suitable links or ideas, then please leave a comment. Principles Throughout the next few months, students and staff will be sick, caring for the sick, living in isolation, perhaps mourning loved ones, and generally suffering from the stress and strain. Keep this in mind. Mathematics lecturers and students are generally not familiar with take home or open book assessment and will need more instruction/guidance than usual. Furthermore, if they had known they were to be assessed in this way, then they would have approached their learning differently. Assessment is likely to be online. Students sending scripts through the post is generally infeasible as HEI campuses are likely to be closed. Plagiarism, collusion, and impersonation are difficult problems that will require thought (and likely some compromise). Keep it simple. Practicalities Online assessment is not straightforward. Some HEIs have suggested assignments should be submitted in, say, Word. However, it is not unreasonable for students to submit handwritten work. See How to have students submit handwritten work in an online setting by Robert Talbert. Gradescope are (at time of writing) offering free access to new courses. Gradescope is worth investigating if you have not yet seen it. It was invented by some tutors tired of marking mathematics. It can help with assessment of handwritten mathematics! Involve external examiners. They will be dealing with the same problems and often have useful advice and experience. Make it clear to students what assessment will be used and what a good answer looks like. In an ideal world, students should be given a model paper and a mock paper to try. The model paper shows them what a good answer looks like. The mock paper should be administered like the real one. This gives students a chance to experience the log in, upload and so on so that they are familiar with the system. Length of time for take away: Students may be in different time zones, not have access to equipment, work in noisy, shared spaces (siblings may not be at school), have special needs, and so on. Hence, a short timescale for uploading answers is problematic. A time of at least 24 hrs can mitigate many such problems. Shorter exams can help reduce the opportunity for plagiarism. Longer exams means asking for answers in Latex more reasonable. Such answers can be checked by plagiarism detectors such as TurnitIn. Traditionally, student work which appeared not to be the work of the student could be checked by giving the student a viva. In the current situation this will be difficult and limited by resources. It may not be possible to interview all students on all modules. Also, conducting a viva by video conference is significantly harder than in person as mathematics may need to be written down and held up to the camera. Furthermore, the resulting picture may be low resolution due to bandwidth issues. Consider whether you are assessing for pass/fail or giving a mark. Obviously a mark unseen by students can be useful to determine pass/fail. It is important to keep the marks in case of appeal. Check with accreditation bodies what their policies are during this crisis. In particular, their advice may conflict with long time frames for assessments. Many students have special needs to consider and may need more time to complete assignments. Hence, a system releasing an exam and expecting all students to complete at the same is likely best avoided. Times...

## E-assessment

posted by Kevin Houston

I’ve been attending a conference for the last few days and so didn’t have much time to write a post. I did learn about the e-assessment packages Stack and Numbas. I will definitely be trying these in the...

## Proof: From the particular to the general...

posted by Kevin Houston

A common technique in teaching mathematics is to go from the particular to the general. We can look at a few examples and then state and prove the general result. We can do this for proofs as well. Instead of giving students a proof of a statement, give them the proof for a specific example of the statement and ask them to prove the general from that. A classic example of this is to prove the irrationality of and then that of . Once the common features have been determined the proof for irrationality of for prime is (hopefully) straightforward. For another example, consider a statement involving vectors in a space of dimension . Prove it for particular vectors in . Then, set the exercise of proving it for the general case. It is, of course, important to stress to students that this particular case proof is not a proof of the general case. The benefit of this technique is that the students are usually able to understand the particular case (so feel good about themselves) and can then use it as a scaffold to prove the...

## Marks for writing

posted by Kevin Houston

Writing mathematics clearly is a crucial part of a student’s education. Too often students begin higher education with the misconception that writing mathematics merely involves putting down a sequence of symbols without paying too much attention as to whether they make sense to the reader. As long as the answer is there in some form and underlined, that’s ok, the thinking goes. Since the marker can see that the answer is there, this may be almost acceptable for problems such as finding the derivative of some product. But it does not work for writing proofs where a logical and coherent argument is required. I have been interested in this aspect of mathematics for many years and so Chapters 3 and 4 of my book are dedicated to writing mathematics. (The chapters are freely available on my website for people to use.) An important benefit of requiring students to produce good mathematical writing is that it forces them to think. If they don’t understand something, then it is likely that they cannot explain it. The converse is not true but can give a useful hint as to what students are struggling with. Using the principle that students are strongly motivated by marks, an effective marking scheme out of 10 is to give 7 marks for the content of a coursework assignment and 3 marks for how well written the work is. My experience is that many students score 0 in the first week but most are scoring 3 by the end of an 11 week course. This is achieved by being quite harsh to begin with. This can be justified by the fact that my students are new to university and automatically assume that that is how things are done at university. The mark is...

## Binary marking

posted by Kevin Houston

An idea recently introduced to me is binary marking. Students receive either a 0 or a 1 for their weekly coursework mark. This makes marking easier — no need to decide if this question is worth 2 or 3 out of 4 — and no need for adding up at the end. Only a simple “Have they made a reasonable attempt or not?” is required. This could be determined by how much of the coursework they have attempted or by how well, overall, it has been done. Students can receive feedback on their work in the usual manner. The theory behind this is that the precise mark is not important — it is the feedback that is...