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...

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...

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 ,...

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...

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...

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...