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...