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 better to write material when I had gained more experience of online teaching — I didn’t want to write some material and then re-write it after term had started. That seemed wasteful.)

2. Don’t alter too much from previous years.
Before Covid-19 we had decided to make fairly big changes to some of the modules I was going to teach. This was a bad idea. We should have just kept it the way we did before for one more year. The changes led to too many instances of having to create new materials such as a new set of questions. Or it led to the sudden realisation that some concept was needed before it had been taught. For example in one of the parts moved from Semester 2 to Semester 1 I needed the concept of bijection before it had been taught in one of the parallel courses.

3. Weave the theory into the worked example videos more.
My students liked having videos (even if they preferred the notes) but they favoured worked examples videos over theory videos. Perhaps this is not surprising. Students often — mistakenly, I believe — latch onto worked examples as the most important part of learning. Hence, if I were to do the videos again, I would weave the theory into the examples much more.

4. Do the videos in one take.
Not editing videos was a great time saver. Ok, it meant I had to be on top of the material before I recorded but recording, exporting and uploading took so long that missing out the editing stage was a huge time saver.

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 stamps on uploads can be used but these can be easy to forge so care needs to be taken.
• Mathematics has a higher proportion of neurodiverse students and this may need special consideration.

Setting take away open book exams

• The simplest solution to the problem of setting a take away examination is to use a standard examination with reduced marks for bookwork (e.g., state the definition/theorem, give the proof of famous theorem). This clearly is less than desirable.
• The next simplest solution is to remove the bookwork but this may mean not all the learning objectives of the course are assessed.
• Do a web search for answers to questions you set. Assessors should do the same.
• Plagiarism can sometimes by avoided by setting questions with different coefficients for different students.
• Mixing the order of questions between students helps combat collusion but makes marking and collating the results harder. (One can’t just tell a marker to mark Q1 as there is no consistent Q1.)
• Now for types of questions one can ask.
• Ask “Why does the proof of a theorem go wrong for this particular example?”.
• Students create an example that meets certain constraints and show it really is an example. This is hard to set if you have not set this type of question before. E.g., Give an example of a function with singular point x=3. The student answer f(x)=0 is valid and can show that the student really understands the definition. However, some students realise that using the zero function and other trivial examples often gives a suitable answer.
• Similar questions can be written asking for non-examples, i.e., examples which don’t satisfy a certain set of constraints.
• Ask for a book review. This has to be an online text to which everyone has free access to. This can be done for a textbook. Students demonstrate how much they have learned by identifying weak parts, good parts, or describing the contents of a chapter in their own words, etc.
• Some questions can be for discussion. For example, “A metal bar is heated in the middle. Discuss.” This type of question needs good mock examples and very strict instructions on length and scope. Very easy to set but also very easy for students to misunderstand.
• Ask students to explain the most important theorem (or example) in the course explaining why they think it is the most important.
• Use incorrect answers. Either create your own or (quicker) use exam scripts from previous years and (suitably anonymised) present a student’s incorrect answer that the current students can critique. Number each line and not just equations.

Other links

• Numbas Good for random questions in assessment and is client-side based so doesn’t suffer so much from student internet problems. Update: A page on how Numbas can help during Covid-19 Crisis
• Modifying closed-book exams for use as open-book exams Advice from Toby Bailey, George Kinnear, Steven O’Hagan and Chris Sangwin at the School of Mathematics, University of Edinburgh.

I did learn about the e-assessment packages Stack and Numbas. I will definitely be trying these in the future.

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 general.

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 not sufficient as detailed feedback is crucial to improving the students. This is easily done by referencing the relevant section in my book. For example, “Don’t start sentences with a symbol, p37”.

Writing well is a good skill to have beyond writing for mathematics and so training in this area is very useful. Furthermore, well written work is so much easier to mark that there is a strong incentive for the lecturer to ensure that students receive that training.

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 important.

Take a proof and space out separate sentences (or two or three sentences). For each student or small group of students, print and cut so that each sentence is on a separate strip. Shuffle the strips. Then, let the students assemble the pieces in order to make the proof.

This is an active way of learning as it forces the students to read and understand each strip and then make a decision as to where it should go. This forces them to think about what is there and makes them consider the structure. For example, it should hopefully be clear that the “Let” statements should come at the start.

No doubt this could be done electronically but I’ve only ever done it with paper. If you know of a good piece of software then comment down below!

This shouldn’t be a surprise but it is just their lack of familiarity. No one tells them what all the symbols are but as mathematics students they are expected to know them. If they don’t pick up the less common ones after hearing them just a few times, then embarrassment stops them asking — after all, they’re mathematicians, they’re expected to know.

Since the failing is that they are not taught the symbols, the easy solution is to explicitly teach them. Give them a pronunciation guide such as the one in my book, How to Think Like a Mathematician. After a few weeks, get them to test each other in pairs or threes. Doing it in small groups saves them from too much embarrassment. Saying the words is very important, students need to be comfortable saying them and the saying also helps them remember.

As a side note, the guide in my book is for British pronunciation. I have had some queries from the USA that were sceptical as to whether the pronunciations were correct (or should that be skeptical?). This is a problem that I may rectify in the next edition. However, I’m no expert in American pronunciation. We say bee-tah, whereas in USA they say something like bay-tah, and even bay-dah. I may need help…

Asking students to turn words in symbols and symbols into words is a useful exercise for this. One of the best places to use it is in analysis particularly where quantifiers are grouped together in subtly different combinations. For example,

The activity can be used in other areas to show how mathematical concepts are tied together in a single package. For example, “Any group with order equal to the square of a prime is Abelian” can be written as ““. The former is much more compact and readable.

The aim of this example is to improve mathematical writing skills by showing contrasting statements. This can then be developed into asking for the perfect blend of words and symbols.(I don’t really believe that there is a perfect answer.) A good compromise for the above might be “For any prime , a group of order is Abelian”. Students can learn a lot about their preferences and those of others from discussing this.

This activity should be used sparingly. It is only effective in the service of some other goal. It is particularly good for making students observe what is written, for developing mathematical writing skills, and identifying weak areas of understanding.