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?...

posted by Kevin Houston

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?...

posted by Kevin Houston

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...

## Ask the same questions repeatedly...

posted by Kevin Houston

One the simplest techniques I use is to ask certain questions repeatedly through a course. Very few people hear an idea once and internalise it. It can take repeated exposure to absorb some facts or methods. Hence, the repetition ultimately helps understanding. For example, we often need to show that an equality holds. A lot of students take the equality and manipulate it into something they know is true. (I show them that this is a dubious method via The latter we know is true and hence . In other situations the implications can be reserved and we are fine.) A better strategy is to take one side and proceed to show, by a string of equalities (or similar), that it equals the other side. Hence when presenting an equality I don’t just say “Let’s pick this side and change it into the other side”. Instead I ask “How do we show an equality?” After repeated exposure they know the answer is “Pick the complicated side and do something with it”. Note that the question requires retrieval which is better for learning. This initial approach is in fact too simplistic. For example, we need to ask which is the complicated side — sometimes it is not obvious and is often open to debate. The other important consideration is that sometimes the best strategy to show is to show . (Showing and is also an alternative.) Hence a follow up question is “Ok, so what should the complicated side be?” This is better than “Which is the complicated side?”. Another question I use repeatedly is “What about the converse?” I have already described a competition I run regarding this. Quite often the converse is not true and so this becomes a great exercise in finding...

## Ask specific questions...

posted by Kevin Houston

The easiest question to ask a class is “Any questions so far?”. This rarely produces an answer. Some people tell me that they have a better response with framing this to indicate that questions are expected: “What are your questions?” or “What questions do you have?” for example. Similarly, I’ve given up asking at the end of the course “What do you want me to go over in the revision session?” since some wag usually says “everything”, we all chuckle and that ends the matter. To encourage a response I have found that it is better to ask specific questions, “Are you happy with affine transformations?”, “Do you understand equivalence relations?” The prompt forces them to consider something concrete and it reminds them of what they have been taught. This method can work at any time of the course. “Which exercises are you having trouble with” can be replaced “Do you have trouble with question 6?” The latter forces students to focus. They either do or don’t have a problem with it. I usually pick a question which they will benefit from discussing so even if students are just saying yes to get on with the tutorial at least it will be productive. In summary: Vague general questions are the enemy. Make questions...