My new book, Complex Analysis: An Introduction, is nearly finished. To help my students with revision I created a list of common mistakes and this forms a chapter in the book. As a lecturer with many years of experience of teaching the subject I have seen these mistakes appear again and again in examinations. I’m sure that, due to pressure, we’ve all written nonsense in an exam which under normal conditions we wouldn’t have. Nonetheless, many of these errors occur every year and I suspect something deeper is going on. What follows is not intended to be a criticism of my students, who, luckily for me, are generally hard-working and intelligent. Nor is it an attempt to mock or ridicule them. Instead the aim is to identify common mistakes so that they are not made in the future. And if this post seems negative in tone, the a later one is more positive as it delves into techniques that improve understanding. Imaginary numbers cannot be compared The first mistake is the probably the most common: the comparison of imaginary numbers. For example, students write for a complex number. This cannot be right. If were what does it mean for to be less than ? What is usually intended is the modulus of , i.e., . The point is, unlike real numbers, we cannot order the complex numbers. For example, which is bigger or ? This is difficult to decide! Since complex numbers can be identified with the plane ordering them is equivalent to ordering the points of the plane and clearly this can’t be done — at least not in any useful or meaningful way. One last point needs to be made. Although is incorrect, note that expressions like can be true if is...

## How to get a good degree 3: How to become a straight-A student...

posted by Kevin Houston

Usually I am against books offering general study advice, I favour those that focus on a particular subject. (Which is why I wrote How to Think Like a Mathematician. Mathematics students may also be interested in Lara Alcock’s How to Study for a Mathematics Degree.) I’ll make an exception for Cal Newport’s How to Become a Straight-A Student. Let’s get the bad stuff out of the way first. The subtitle is The Unconventional Strategies Real College Students Use to Score High While Studying Less which, along with the book’s other marketing, makes the scam-like sounding promise that you can do less work and get better grades. This can of course happen but does lend the book a feeling of too-goo-too-be-true. Furthermore, the book could do with some trimming of excess material though, at 216 pages, it is quite short for this type of book. The book could do without the regular mentions of partying and beer swigging but I suspect I’m not the target market for those bits. But leave aside those problems. Why would I recommend the book? Essentially, most of the advice is good. There is the standard stuff that all students know that they should do: get plenty of rest, eat properly, do your work in the morning between lectures to gain a sense of accomplishment. The non-standard stuff is good too. He doesn’t advocate lots of highlighter pens or even a highly detailed to do list. Instead he mostly focusses on the methods for efficient and effective learning. For example, on page 105, he talks of the Quiz-and-Recall method, [emphasis mine] Whether it’s philosophy or calculus, the most effective way to imprint a concept is to first review it and then try to explain it, unaided, in your own words....

## How to Think Like a Mathematician

posted by Kevin Houston

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...

## Estimating large numbers the Fermi way...

posted by Kevin Houston

Here’s a really good introduction to estimating numbers and includes Enrico Fermi‘s famous “How many piano tuners are there in Chicago?” You can see more like this at...

## Scott Young on Lectures...

posted by Kevin Houston

I follow a number of education writers and those who write about learning. One of these, Scott Young, is a high achieving recent graduate who seems to be making a living by giving learning advice. His latest claim to fame is his MIT Challenge where he studied a complete 4-year MIT course in one year as a way of learning new skills but also, presumably to prove the effectiveness of his methods. My opinion is that some of the methods are debatable as they don’t foster really deep understanding of a topic, a number are just ways of quickly mastering the necessary ideas to pass an exam. As an example, in one of the key documents he uses to advertise his Learning on Steroids programme he shows how someone used his methods to learn the mathematical concept of limits. There is a scan on page 7 of the notes. The student writes in a section headed “What is a limit, REALLY?” that “A limit is like a stalker, forever getting close to the target. Forever trying to close the distance between it and the target, but rarely ever succeeds.” This analogy is deeply flawed. Worse than that, leaving aside the word stalker, it is precisely what I have to stop students thinking a limit is. When I set an exam question on the definition of a limit, many students will give rather incoherent answers about “numbers getting close to other numbers but never quite reaching”. However, what I want students to realize is that the limit of 0,0,0,0,0,0,… is 0, i.e., every element of the sequence is equal to the limit — the sequence doesn’t get closer and closer to the limit. Another good example is interweaving this sequence with 1/n. I.e., 0, 1,...

## Four strategies for better study...

posted by Kevin Houston

In Cal Newport’s Study Hacks blog he recently had a post about a piano student called Jeremy. Jeremy’s Strategies for Becoming Excellent… Strategy #1: Avoid Flow. Do What Does Not Come Easy. “The mistake most weak pianists make is playing, not practicing. If you walk into a music hall at a local university, you’ll hear people ‘playing’ by running through their pieces. This is a huge mistake. Strong pianists drill the most difficult parts of their music, rarely, if ever playing through their pieces in entirety.” Strategy #2: To Master a Skill, Master Something Harder. “Strong pianists find clever ways to ‘complicate’ the difficult parts of their music. If we have problem playing something with clarity, we complicate by playing the passage with alternating accent patterns. If we have problems with speed, we confound the rhythms.” Strategy #3: Systematically Eliminate Weakness. “Strong pianists know our weaknesses and use them to create strength. I have sharp ears, but I am not as in touch with the physical component of piano playing. So, I practice on a mute keyboard.” Strategy #4: Create Beauty, Don’t Avoid Ugliness. “Weak pianists make music a reactive task, not a creative task. They start, and react to their performance, fixing problems as they go along. Strong pianists, on the other hand, have an image of what a perfect performance should be like that includes all of the relevant senses. Before we sit down, we know what the piece needs to feel, sound, and even look like in excruciating detail. In performance, weak pianists try to reactively move away from mistakes, while strong pianists move towards a perfect mental image.” Of course as we are talking about studying for a public performance of music these strategies don’t translate immediately or perfectly to...