How to Think Like a Mathematician

It’s the blog’s birthday! It’s also the week before the start of the university year here and so it’s a good time to shamelessly advertise my best selling book. The success of How to Think Like a Mathematician has taken me by surprise. It has sold nearly 10,000 copies since it was published. Ok, I’m not going to rival JK Rowling but given that the statistic often quoted in the publishing world is that 95% of books sell less than 5000 copies, I am really happy that the book has found an audience and I get emails from all round the world saying how much they like the book. It has been translated into French, German, Turkish and a Korean version will appear soon. (See links below.) It has been adopted as a textbook throughout the world – I’ve lost count of the places that use it. Enough of the puff, what’s it about? Well, the book was written for anyone studying a mathematics degree or a mathematically based subject. My aim when I wrote it (it took me about 17 years but that’s another story) was to let students in on some of the secrets behind actually thinking like a mathematician so that learning mathematics becomes easier. The book has six parts: Study Skills for Mathematicians. This shows you how to read and write mathematics. The latter is an easy to learn skill and will set your work apart from most other students. How to Think logically. Mathematics is well known to be dependent on logic. Here you learn the basics from a very practical perspective. Definitions, theorems and proofs A key difference between university and pre-university level mathematics is that it is now not just about calculating. Definitions are made precise, mathematical truths (called theorems) are stated and they have to be shown to be unequivocally true by giving a proof. In this part you learn how to pull apart definitions, theorems and proofs so that you understand them. Techniques of proofs. Now that you have to prove things you need some techniques to bring out as the need arises. Mathematics that all good mathematicians need. The title is self-explanatory. Many other books introduce university level mathematics but then forget to include the maths you will be studying in your course. Closing remarks. We pull together everything learned so that rather than rely on superficial rote learning you develop deep understanding. Some of the tips in the book are simple but effective. The following is a taster: Many students asked to show that an equation holds will merely rearrange it an haphazard way to produce some other equation they know is true such as 1=1. (The reason why this is just plain wrong is given in Chapter 21, Some Common Mistakes. Hint: it is to do with logic.) Here’s a simple tip that works so much better: Take the most complicated side and do something to reduce it to the other side. This works. It stops you tying yourself in knots when randomly rearranging terms from one side to the other in the hope that something will happen. Instead it forces you to think about what you can do with the terms. It’s a simple trick that changes behaviour. If you are interested in owning a copy, then just follow the links below: How to Think Like a Mathematician: A Companion to Undergraduate Mathematics (UK) How to Think Like a Mathematician: A Companion to Undergraduate Mathematics (US) Wie man mathematisch denkt: Eine Einführung in die mathematische Arbeitstechnik für Studienanfänger (Deutschland) Comment penser comme un mathématicien (France) Matematikçi Gibi Düşünmek (Türkiye) How to Think Like a Mathematician: A Companion to Undergraduate Mathematics (Italia) How to Think Like a Mathematician Paperback: A Companion to Undergraduate Mathematics (España) (Some of the above are affiliate...

Maths problems tweets...

I’m not a big user of twitter as I prefer to read material that has more than 140 characters. Nonetheless, often something interesting appears. Dan Meyer set off an intriguing set of responses to his call for maths problems that can be posed within Twitter constraints. The results can be seen here: Tweet-sized...

The number of spots in a deck is 365?...

Today’s TED video is an augmented reality card trick by Marco Tempest. Now, I’m a fan of card tricks but this was a bit too cutesy for me. However, I draw your attention to his computer’s claim (at 3:20) that if you add up all the spots on the cards in a deck then you get 365 which is the number of days in a year. I make the same claim in one of my school talks but I follow it by pointing out that obviously it can’t be right. (Obvious in the sense that you don’t need to get a deck and count but just use some simple maths.) Usually at least one of the pupils gets it. Anyhow, here’s the...

Maths projects

Last week I updated my personal and work web sites for the first time in far too long and I changed the look of this blog. It doesn’t work perfectly yet as some of the colours and spacing are wrong. And I should get rid of that banner at the top – it’s looking a bit old. On a different topic, I’ve been criticized over on the TES forum for my mechanic analogy. I thought it was good analogy, but PaulDG disagrees: As with all analogies, and having only read those few paragraphs of his work, I’m afraid it looks to me as if he’s the sort of “expert” who’s got us into the problems we’re currently in. Ouch. He then gives a straw man a thorough thrashing and finishes with Houston appears from his analogy to be in favour of understanding without skills. This seems rather illogical: Houston wants students to do more of A and less of B therefore he doesn’t want students to do any B. To be honest I can’t see how anyone could have such a reading of what I said. I was tempted to jump into action with a shout of “Someone is wrong on the internet, I must intervene” but after some reflection I decided to just read the rest of the thread. So, the thread itself is about a very important topic — maths projects in schools. As someone who is trying to prepare a projects module for second year students I am aware how difficult it is to prepare good projects. The problem at all levels, school or university, is that it is very hard to set “medium strength” projects – they are either too easy or too hard. I think the key difficulty is...

Pi is 4 (again)

I’ve been busy this week. On Monday I gave my new Auto-Tune talk to the School of Maths (I spent the previous Saturday reading the patent for Auto-Tune. Very interesting.) And yesterday I was in London to give a talk on Teaching Analysis. The meeting was also interesting but poorly attended as just about every project is running conferences and workshops. During my talk I showed my Pi is 4 video as an example of something I am using in my lectures. You can see it here. Anyhow, Vi Hart has a video on the subject which is worth...

Teaching Introductory Mathematical Analysis...

One for lecturers: I’ve been asked to speak at a free workshop entitled Teaching Introductory Mathematical Analysis. This HEA MSOR-funded event will take place at De Morgan House in London (home of the London Mathematical Society) on Wednesday May 2nd. Leaving aside this humble scribe it has a cracking line up: Lara Alcock (University of Loughborough), download her book with Adrian Simpson Camilla Jordan (The Open University), Joe Kyle (University of Birmingham), Chris Sangwin (University of Birmingham). Further details can be found at...