I always have trouble creating problem sheets so I’m trying to classify the types of exercises and problems I can use: Routine exercises. A routine exercise is usually a calculation or follows some given worked example. Good way to start a sheet. Problems. Solutions to these require more than just regurgitating some worked example or simply applying a theorem. Extend the theory. Prove a theorem that there has not been time for. Usually done via a set-by-step process. True/False or Always, Sometimes, Never. See previous post. Generalise a statement. Give a more general statement than in the notes say. Is it true or false? Create examples. Give an example of some mathematical object. Counter examples. Create counter examples to statements. Challenges. These are hardest to create but should be on the problem sheet for the strong students. What other types of questions are there? Please leave comments below. If I get quite a few, I’ll describe them in another...

## Creating exercises: Don’t give all the information...

posted by Kevin Houston

One problem with setting routine exercises is that students will often develop what might be called a `plug and play’ approach to solve them. That is, they look for the common features with a worked example they already know and look for some data in the question. You know the sort of question: “A radioactive material has a half life of . Currently there are 25g. How many grammes are left after 4 days?”, “A tap fills a barrel at a rate of …” or “A bacterial colony grows at a rate…”. The student knows that they have to grab the specific parameters from within the question and stick them into a formula. The exercise has become more about decoding exercises rather than solving a problem. One way to stop this behaviour is to not give all the information. For example, I once set a geometry question that required students to use the lengths of the semi-axes of the elliptical orbit followed by the Earth going round the sun. I deliberately left this information out of the question and chaos ensued. Many students couldn’t answer the question despite the fact that it takes seconds to find the relevant numbers online. One of the tutors on my module even told the students that the question was unanswerable! To me this showed that the standard way to set mathematical problems is broken. Any question where there is ambiguity — even if it can be resolved by a quick information search — causes problems. How can we create more of these questions? Well, we could rewrite the radioactive material question as “We have 25g of Fermium-252. How many grammes are left after 4 days?”. The required information is easily accessible via the internet and so the exercise...

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

posted by Kevin Houston

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...

posted by Kevin Houston

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...

posted by Kevin Houston

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...