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...