## Creating exercises: Get students to do it...

One interesting technique I’ve tried is to ask students to create an exercise and its solution. The advantages of this are many. One, creating exercises is hard and I can use all the help I can get. The best exercises I can use in the following year or for revision purposes. Two, the students get practice in calculating when they are creating the solutions. Three, students have to think deeper. Sometimes a lot deeper. This gives them much more insight into the technique(s) being practised. For example, when teaching vector calculus to engineers, I asked the students to produce an exercise on the calculation of a surface integral. Constraints were important as a shamelessly flippant student could set the exercise of integrating the constant function over a square. Hence, the surface should not be flat and the function should not be constant or consisting of only linear terms. This worked quite well with most of the students being able to get a good exercise from it. Admittedly, many did not stray far from the type of question they had already seen and essentially just changed some coefficients. Nonetheless, they got a sense of achievement from this. The task was frustrating for some students though. This was particularly true if during working out the answer to their question they discovered that they have created something too hard and had to simplify. Their assessment was that the previous work is somehow wasted. Interestingly, when questions were created to give to others, some students tried to set the hardest possible question that they themselves could answer in an attempt to torture their peers. I also tried this to generate problems involving eigenvalues and eigenvectors for and determinants. I’m sure that there are many other possibilities but I...

## Useful explanations in solutions...

I always provide students with solutions to the homework exercises. One simple but effective tactic to improve student understanding is to include helpful asides in square brackets. These are bits of information that I would not expect in a handed in solution but give students insight into what I am thinking while answering the question. For example, “[We can use Method A here but Method B is actually quicker. If you like, use Method A and compare.]” or “[We can’t use the Ratio Test directly as some elements of the series are zero.]”. If could also be information that expands on the answer to show something extra: “[If we take the imaginary parts rather than the real parts, then we get another equation we can use.]”. This is a very simple technique but appreciated by students as it allows them to see what is actually required for the final answer. Some subjects require more explanation than others. For example, when teaching analysis I dispensed with the square brackets and had two sections for each question in my model solutions. One was labelled Thoughts and the other Written Solution. In the Thoughts section I discussed some ideas for answering the problem including explanations of why some ideas won’t work but usually the section contained working that would be discarded in writing up. For example, in analysis, many proofs involve finding an (or similar!) given an . The usual procedure is to find the (dependent on ) by some calculation but when writing up we give the required in the first line and show that this satisfies our required condition. This is problematic for students as without the initial working they often can’t see where the comes from. This method shows where it comes from and...

## Beyond True/False: Always, Sometimes, Never...

A classic method of assessment is to set true or false questions. These are easy to mark and students can even mark them without difficulty. There is another method called Always, Sometimes, Never that goes beyond straightforward true/false.* For example, take the statement The sum of any two integers is odd. We could clearly frame this as a true or false statement in an assessment. But now look at it from the perspective of whether it is true always, sometimes true or never true. In this case it is sometimes true and then we can ask what assumptions are necessary for the statement to be true. This particular statement is true when exactly one of the numbers is odd. Another classic statement in this area: The square root of a number is less than or equal to that number. Again, this is sometimes true and we can ask for a condition to make it true and also, in this case, when it is false. This format helps with stressing the importance of assumptions. For example consider For any integer , we have . Students should recognise this as part of Fermat’s Little Theorem. However, the crucial assumption that is prime is missing. Hence, the correct answer is not Always but Sometimes. In this example we have a problem if we ask students to determine precisely when the statement is true. Certainly, the statement is true for prime. But it is not the case that it is false for not prime. When is not prime the precise conditions for the statement are difficult to state. Pseudo-primes exist and we need to start talking about and being coprime and so on. Hence, when setting a question with a Sometimes answer we need to be careful about...

## Peer Marking I

Assessment is probably the most important task we do for our students. It encourages students to learn by setting a deadline and the feedback stimulates improvements. Having said that, a good idea is to let students mark each other’s work. We shall refer to this as peer marking. This would at first seem to be a bad idea. Students are not experts so will they have the necessary knowledge to mark correctly? How can they give good feedback without expert knowledge? (Anyway, isn’t this precisely the sort of work we are paid to do?) Well, there are a number of advantages. The time saved by the lecturer could be used more profitably in other assessments. Students are able to compare their work with that of others — maybe it will spur them on to do better or help them realise that their work is not as bad as they thought it is. They get an insight into how hard marking is. (I admit that sounds like a self-serving reason!) For me one of the best advantages concerns writing mathematical arguments. When entering university almost no student can write mathematics well. (And that’s ok. My belief is that it is something I have to teach — I am not blaming the teachers at pre-university level!) When students peer mark they can see how difficult marking is when the work is not written in sentences or is placed all over the page instead of in a nice flow. This is another example where showing is far better than telling. Peer marking shows them the type of mistakes that they are probably making too. So how can peer marking be done? One possibility is to leave one exercise from a weekly exercise sheet unmarked. At the next...

## What does this symbol mean?...

Another useful question is to ask students is “What does this symbol mean?”. Mathematics uses a lot of symbols and even a small symbol may be representing a very complicated concept. And, as was observed yesterday, the same symbol may be representing different concepts! In particular, the zeros in are different. One is a vector, one a scalar. More generally, consider the symbols in the equation below taken from a course on fluid dynamics: Asking the students what the symbols mean can be enlightening as to what they are having trouble with. The behaves differently when combined with (the dot). Do students know that? Some symbols represent vectors, some scalars. It’s hard to know whether students grasp this until I ask what the symbols mean to them. Even asking a student to read out such an expression can be interesting. I have met students who are unsure of how to pronounce the Greek alphabet and will refer to — admittedly with some embarrassment — any symbol they don’t know as “that funny symbol”. In conclusion, it is important to ask the students to describe what the symbols...

## Can you give me an example?...

Probably the most powerful question during tutorials and lectures to ask of students is “Can you give me an example?”. Students often don’t see the assumptions in a theorem — their focus is on the conclusion — and so they miss what the theorem actually applies to. Thus after stating the theorem it is a good idea not to launch immediately into the proof. Far better to ask for examples of the objects in the assumptions by asking “Can you give me an example?” This will help bring the statement to life. Does it apply to lots of examples we know? Is it telling us something about examples we know or ones we don’t know? Just asking the question forces students to think about the statement and observe what is there. Only things that are consciously observed are remembered. Also, forces them to retrieve examples and retrieval is, of course, good for learning. It is a good idea to push students for plenty of examples or else they will believe that the theorem only applies to the discussed examples. So, the assumption that is a differentiable function should produce constants, polynomials and even smooth non-analytic functions from students. The question works for definitions as well. Consider linear independence — a definition that traditionally students find difficult. The condition within the definition, has a lot going on. (In particular, those zeros are different types of zeros — one is a vector, the other an element of a field!) To help students we can ask “Can you give me an example of ?” Hopefully students will have no problem coming up with examples such as . But how many would think of as being a function? By repeatedly asking them for examples such as functions they...