Binary marking

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

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...

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...

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?...

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...