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...

## Jigsaw proofs

posted by Kevin Houston

Writing a proof is very difficult — even for experts. Here’s an idea that allows students to think about and engage with a proof without starting from nothing. 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...

## Teach the Greek alphabet...

posted by Kevin Houston

Many students don’t know what to call certain Greek symbols. They are happy with common ones like and but show them a or and they get a little shy. 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...

## Words into symbols, symbols into words...

posted by Kevin Houston

Mathematicians use a lot of symbols. Everyone knows that. A symbol may represent a simple constant ( is the constant of integration) or represent a collection of difficult concepts ( is a group). Students need to be able to unpack the meaning from symbols and to do the reverse — bundle up concepts into a few symbols. 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, is profoundly different to Translating these into words serves at least a couple of functions. First, students get a different perspective — it is almost as if a different part of the brain is being used. Second, and this is the most important, they observe what is there. It is all too easy for an eye to glide over a symbol and not have it consciously perceived. This activity prevents that. Third, they are forced to describe what a symbol means and that requires them to know the material. Being unable to translate helps locate areas of weakness. 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 ,...

## What goes in the gap?...

posted by Kevin Houston

Lara Alcock wrote about Tilting the Classroom in the London Mathematical Society Newsletter. One of the activities she does that I don’t do but should is what I’ll call What goes in the gap? from her section on deciding. The example she gives is which symbol of , , goes in the gap in Clearly one can do this with many other statements not just symbolic ones For , is orthogonal is an isometry. Alcock recommends that students vote on the answers to encourage engagement and notes that sometimes multiple rounds of voting are...

## Minute notes

posted by Kevin Houston

The main reason for writing these posts is because I am giving a talk about teaching at the British Mathematical Colloquium next week. This has been an opportunity to prepare for the talk by reflecting on my teaching, particularly over the last few years. During this time while consulting books and websites I have come across many great ideas about task and activities that I do not do and perhaps I should do. So until the end of this month I’ll be writing a post every day about these. It should be borne in mind that I have not tested many of these and so they come with no guarantees! (The reason for stopping at the end of the month is that in May I will be finishing my next book. The final draft has been sitting in a metaphorical drawer since January waiting for me to come to it with fresh eyes. All the horrible bits and typos should jump out and so I’ll move onto Final Draft Version 2…) So here’s the first activity: MINUTE NOTES One book I highly recommend to new mathematics lecturers is How to Teach Mathematics by Steven G. Krantz. The Second Edition is the one to find as it contains some great appendices written by critics of the First Edition. For some reason these have been removed from the Third Edition. The book itself contains huge amounts of helpful advice about teaching higher level mathematics. I don’t agree with all he says but he asks many of the important questions and draws together many practical solutions to teaching problems. The activity Minute Notes (p.120, 2nd Ed) gives students a minute every week or so to write on a piece of paper about what they are having trouble...

## Teaching Problem Solving...

posted by Kevin Houston

Learning mathematics involves solving problems. So, is it a good idea to teach students how to solve problems in the sense of an abstract process called “problem solving”. That is, should we provide a course on solving problems where we go through problems – not necessarily of a mathematical nature (for example, the classic you have a lit candle, a box of tacks. How do you attach the candle to the wall?) and then analyse and reflect on the solution? Polya’s classic book How To Solve It is probably the most famous example of attempting to teach methods for problem solving. His four point plan is as follows: Understand the problem. Make a plan. Carry out the plan. Look back. The problem with this is that the second and third parts are basically “solve the problem”. Also, the four point plan perhaps generates unreasonable expectations of how problems get solved. Problem solving is messy with many false starts and dead ends. Part (i) and (iv) are certainly useful tasks and should be emphasised to students. In his Foreword to Polya, Ian Stewart admits to wanting to cheerfully strangle students who fail to understand a problem by failing to know what a word a means. Reflection — the looking back — is something I find very powerful. Ruminating on a problem afterwards can produce some new insight that produces a better solution or how to make the solution more general. Leaving aside the slightly dubious plan, there are many ideas within Polya’s book that are worth instilling in students. For example, I often tell students to “Draw a diagram”. This rule forces them to pay attention to the details and to understand the problem. However, I would never advocate having a course dedicated to solving...

## Creating exercises: Types of questions...

posted by Kevin Houston

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...

## Peer Marking I

posted by Kevin Houston

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...

## Plan ahead for your questions...

posted by Kevin Houston

Yesterday’s post on asking a question about which method to use is also a failure on my part to prime the students. If I had spent more time stressing which techniques are the important ones, they would probably have known which method I was looking for. For example, the ratio test is a simple and reliable test that is a good method to try first when testing for convergence of series. I find it useful to repeatedly note when we have used it: “We’ve used the ratio test again. It’s a good one to go for first when testing for convergence.” This primes the students so that they will use it in their own work (perhaps with a bit of prompting). Another example: To prove convergence of a sequence we show that the sequence is bounded above and increasing. It is well worth drawing attention to the use of this result. Then when faced with students who can’t show convergence ask them which method we used repeatedly. This does not have be for particular results. It could be an overarching principle. For example, in projective plane geometry using determinants solves many problems. One can prime students by pointing this out. In this case the prime is that “Note how we recast the problem into one involving determinants”. Thus when planning a lecture course, I ask myself, “Which results should I prime the students to...

## Read my mind!

posted by Kevin Houston

Asking student questions is an art. In any reasonably sized class student ability will be very variable and so it is difficult to pitch a question which is hard enough to not be insultingly trivial and easy enough to be answered by most. The sweet spot is where the level of difficulty is just hard enough to stretch them. One method is to start with a difficult question and then keep simplifying it (bit like a Dutch auction — start high and progress downwards). This can be a good approach but if there is a specific answer in my head and I am merely indulging in a guessing game, then that is a mistake. I call this the “Read my mind” error. For example, let’s say that in a tutorial I want them to tell me that we should use the simplex method to solve a particular question. I may start with “How can we answer this?” and when greeted by silence or blank stares I try something simpler. “What methods do we know to answer this type of question”. Blank stares. This then continues “What method did we do last week? … The what method’? … The simp… method? … No one? The simplex method perhaps?” There are two mistakes occurring with the read-my-mind error. First, there is a right answer in my head and I am going to keep questioning until students find it. (Students often believe that there is always a right answer anyway and, to be honest, that is regularly the case in mathematics.) The other mistake is that it conditions students to expect the question to be simplified and besides “he’s going to give us the answer anyway” so there is no point in answering. If I find myself...

## Tutorials: Students doing the work 3...

posted by Kevin Houston

Another alternative method to encourage students to present in tutorials is to invite a student to write on the board with the twist that the other students have to provide the solution. The student at the board is merely an amanuensis. This takes the pressure off the student at the board. Half the battle of making students present is to make it feel safe. In this method the stakes are low and it allows them to get comfortable with being the centre of attention but without the perceived danger of making a fool of themselves by getting the answer...

## Tutorials: Students doing the work, Part 2...

posted by Kevin Houston

An alternative to having students prepare a presentation of a solution is to randomly call on the students during a tutorial to present. For me, again this has worked well for groups of good students. It has worked particularly well for groups that experienced the weekly advanced warning version. That is, in Semester 1 members of the tutorial group were chosen to present in a week. Once comfortable with that they could be chosen at random during the tutorial to...

## Tutorials: Students doing the work...

posted by Kevin Houston

For simplicity I shall define a tutorial as a timetabled meeting for discussing exercises and problems. One activity I have tried with varying degrees of success is to get students to present during a tutorial. Each week I would choose a student and one of the easier exercises for them to present at the board the following week. During the presentation the other students were allowed to ask questions or make comments. Once every student had presented we would cycle back to the first one and keep going like this until the end of term. The outcomes varied immensely. For a group of strong students the presentation would stimulate good discussions. Even the students not presenting would take it seriously and participate. For weaker groups I had serious problems with absenteeism. Being selected to present had a high correlation with not attending the tutorial the following week. In fact, one year I had a group of six students for which, in twenty tutorials, we did not do a complete cycle. Hence, I only use this with stronger groups. This is unfortunate as the students really benefit from the practice they get...

## Practice. Practice. Practice....

posted by Kevin Houston

Practice is generally considered good for learning mathematics. That’s probably not controversial. The disagreements arise when we talk about the type of practice. There is much talk about deliberate practice and the famous 10,000 hours to achieve mastery. (Both fantastic areas for a discussion but I’ll come back to them on a different day.) Following on from yesterday it is probably clear that it is easy to provide practice in exercises but less easy for problems. (Also, one should distinguish between practice and exam practice.) A problem that arises most years for me is a student who feels that they did a lot of work during the course but did poorly in their exam score and want to know why. In conversation with them, I often find that they have done a lot of work — lots of practice. Digging deeper it emerges that it is the wrong sort of practice. They have focussed on the bits of the course that they like and can do and have done exercise after exercise (and sometimes problem after problem) on those particular areas. This I think arises from two aspects of human nature. First, we like to do what gives us pleasure. Solving a problem successfully gives a little bit of satisfaction that encourages us to repeat the action. And so we get into a cycle. The second relevant aspect of human nature is a tendency to find reasons to procrastinate: “I’ll need a big block of time to tackle that difficult area I don’t understand. To fill up the short time I have, I’ll do some more exercises on that thing I like.” Doing the mental work on something new, something hard, is less appealing and so that doesn’t receive the attention it needs. Hence,...

## Exercises versus Problems...

posted by Kevin Houston

Learning mathematics is closely associated with doing exercises. Most of us will have experienced mathematical learning as: receive some information on a new (to us) area of mathematics and then do some exercises to increase and test that learning. In this post I want to make a distinction between between exercises and problems. For me exercises are routine while problems involve deeper thinking. Thus, an exercise may be a straightforward calculation, or involve checking that something satisfies the conditions of a definition, or require the application of a theorem and so on. It may be something for which a worked example is in the notes. A problem, on the other hand, requires combining a number of ideas or seeing some concept in a different way to produce the solution. It would not be possible to give a worked example for a problem. The distinction between the two can be in the eye of the beholder though. For stronger students a problem may be an exercise and for weaker students an exercise may be a very difficult problem. However, I wish to distinguish the two for later while accepting the dividing line between them is not strict. The practical advantage of the distinction is that it becomes easier to set exercise/problem sheets for students: Start with exercises and increase the difficulty level to...

## Examples vs Definitions...

posted by Kevin Houston

One tradition in mathematics is to make a definition and then give examples. I’m going to argue that this is the wrong way round. Consider how new mathematics is developed. A definition is introduced to help put a name to a concept that has been useful — for example, a group. We don’t often hear of an important definition that was made on a whim, explored and found to be useful. Are there any? (Suggestions in the comments, please!) Thus examples precede definitions and are the result of seeing a unifying feature. In lectures we often give the definition, in other words the unifying feature, and then show the examples and the consequences following from it. In doing so we miss providing the students with a motivation for the definition. Certainly, it can be argued that there isn’t time to motivate everything (and there isn’t — there is never enough time to tell the students everything we think that they should know!). It can also be argued that mathematics research is a messy business with lots of dead ends and false starts. By putting definitions first we are tidying up the mess. That’s true but next time, if you don’t already do so, maybe try as an experiment to put the definition after the examples. This was successful for me when teaching topology. By giving a diagram with lots of examples at the start of the very first lecture I was able to clearly motivate definitions such as homotopy and homeomorphism. Let’s return to the definition of group. Considering examples first means we can have non-commutative examples such as matrix multiplication. When write try to write a definition that brings together all the examples and extracts their important features it is then clearly a...

## Teach Wason’s Test...

posted by Kevin Houston

Wason’s Test is a good way to teach the contrapositive and to demonstrate that it is not obvious. The test is essentially a test of logic. There are 4 cards, each has a number on one side and a colour on the other. The picture below shows this set up with the playing cards I use in my lectures. (Playing cards are good as you don’t need to prepare them. These days I use a visualiser to display them but in the past I used jumbo sized playing cards. Now consider the statement “If the card is even, then the other side is red.” Which cards do you need to turn over to determine whether this is true or false? If you have never seen this before, then take a moment to find the solution. For many years I have set this as a multiple choice exercise during a lecture (the students vote using Socrative through their smartphones, tablets, etc.). The class response is typically the following: As you can see most get the correct answer (in green) but a large proportion (about quarter) make the common mistake of saying red and 8 instead of blue and 8. The blue coloured card is the tricky part. The 8 card is straightforward and to complete the test we need to realise that turning over the 3 and the red are not necessary. These non-blue cards are easily dealt with and so I don’t spend much time on them in the lecture. Dealing with the blue card is easier if we consider the contrapositive. The contrapositive statement is “If the card is not red, then the back is not even”. This, of course, is the same as “If the card is blue, then the number is...

## Explicitly teach the contrapositive...

posted by Kevin Houston

The contrapositive is a a crucial part of logic but is often overlooked. For a statement of the form the contrapositive is the statement “not not “. For example, the contrapositive of “If I am Winston Churchill, then I am English” is “If I am not English, then I am not Winston Churchill”. Put like that it is clear that both statements are true. In fact, the contrapositive and the original statement are equivalent. This is is of course extremely useful as a method of proof. Want to prove some assumptions imply that some number is non-zero? Just assume the number is zero and proceed directly to show that the assumptions are false. It’s easier to use equalities than equalities in equations and so assuming the number is zero is much more tractable. Hence, we would like our students to understand the contrapositive and use it in their work. There are, however, at least three problems with the contrapositive. 1. Lecturers often learnt the contrapositive implicitly from repeated exposure as students. Colleagues have even told me that they didn’t know that the concept had a name — it was something they `just know’. Their contemporaries who didn’t pick it up implicitly didn’t become lecturers and so the cycle is repeated with students not explicitly taught about the contrapositive. I believe we should give the concept a name and teach it. (Once you name it you can tame it!) 2. The contrapositive is confused with the method of contradiction (both begin with contra — that doesn’t help). The example above about non-zero numbers looks very like a proof by contradiction. (I don’t believe it is as we are not assuming at the same time that the assumptions are true and the conclusion is false.) Nonetheless,...