Wason’s Test is a good way to teach the contrapositive and to demonstrate that it is not obvious. The test is essentially a test of logic. There are 4 cards, each has a number on one side and a colour on the other. The picture below shows this set up with the playing cards I use in my lectures. (Playing cards are good as you don’t need to prepare them. These days I use a visualiser to display them but in the past I used jumbo sized playing cards. Now consider the statement “If the card is even, then the other side is red.” Which cards do you need to turn over to determine whether this is true or false? If you have never seen this before, then take a moment to find the solution. For many years I have set this as a multiple choice exercise during a lecture (the students vote using Socrative through their smartphones, tablets, etc.). The class response is typically the following: As you can see most get the correct answer (in green) but a large proportion (about quarter) make the common mistake of saying red and 8 instead of blue and 8. The blue coloured card is the tricky part. The 8 card is straightforward and to complete the test we need to realise that turning over the 3 and the red are not necessary. These non-blue cards are easily dealt with and so I don’t spend much time on them in the lecture. Dealing with the blue card is easier if we consider the contrapositive. The contrapositive statement is “If the card is not red, then the back is not even”. This, of course, is the same as “If the card is blue, then the number is...

The contrapositive is a a crucial part of logic but is often overlooked. For a statement of the form the contrapositive is the statement “not not “. For example, the contrapositive of “If I am Winston Churchill, then I am English” is “If I am not English, then I am not Winston Churchill”. Put like that it is clear that both statements are true. In fact, the contrapositive and the original statement are equivalent. This is is of course extremely useful as a method of proof. Want to prove some assumptions imply that some number is non-zero? Just assume the number is zero and proceed directly to show that the assumptions are false. It’s easier to use equalities than equalities in equations and so assuming the number is zero is much more tractable. Hence, we would like our students to understand the contrapositive and use it in their work. There are, however, at least three problems with the contrapositive. 1. Lecturers often learnt the contrapositive implicitly from repeated exposure as students. Colleagues have even told me that they didn’t know that the concept had a name — it was something they `just know’. Their contemporaries who didn’t pick it up implicitly didn’t become lecturers and so the cycle is repeated with students not explicitly taught about the contrapositive. I believe we should give the concept a name and teach it. (Once you name it you can tame it!) 2. The contrapositive is confused with the method of contradiction (both begin with contra — that doesn’t help). The example above about non-zero numbers looks very like a proof by contradiction. (I don’t believe it is as we are not assuming at the same time that the assumptions are true and the conclusion is false.) Nonetheless,...

By the time students come to me many have picked up a bad habit. They use the implication symbol, , incorrectly. This is not thought of as a symbol to use for (i.e., statement is true implies that statement is true) but more as “this is what we do next”. This is understandable as most proofs students have seen will be one deduction after another. But it does mean that I see lots of examples where is used at the start of a sentence. Steps need to be taken to overcome this. The one I use at the moment is to just tell them at the start of the introductory course that they need to be careful and shouldn’t use the symbol until we’ve studied it in the course. (This does violate my “Show, don’t tell” rule but it seems to work quite well.) There is an air of mystery about why and I think that makes them pay attention. After a few lectures we start studying it and the period of abstinence in using it gives a fresh start. I begin by observing that the symbol is rarely used in written mathematics. Almost any textbook will do to show this. For me the important thing is to get across the idea that we are connecting two statements. To do this I use everyday examples: “If I am Winston Churchill, then I am English” can be written as “I am Winston Churchill” “I am English”. With that sort of example students can see that using “ I am English” would not make sense and hence we can’t use $\latex implies $ at the start of a sentence. My usual approach to implications is: statement, converse statement and contrapositive statement. I shall discuss the latter...

Students will always need help with problems and exercises that are set.* When a student asks me for help with a particular problem, then I have a go-to question for them: “What have you done so far?”. I learned this from Bill Bruce when I was his Research Assistant many years ago. It’s a great question because it stops me launching in to an explanation only to discover that the student already knows that part of the solution — my students are generally quite polite (lucky me!) and tend not to interrupt me in full flow. It forces them to think rather than rely on me. Quite often the reply is “Nothing. I don’t even know where to start”. Even that is useful information and once oriented to what they do or don’t know I can begin follow up questions “Do you know how to do X?”, “Can you break Y it into parts?”, or “What do we know about Z?”. * If they don’t, then the questions need to be a little bit...

In yesterday’s post I touched on the need to stress the importance of definitions and gave a way to help students learn them. Today’s post is about something more subtle. It is often said that we can define a word to mean whatever we want it to mean. Of course, that does not mean we can redefine continuous function in our elementary analysis courses. Instead, it is important to let students know that a definition arises from choices. Any definition was likely a choice amongst a number of possibilities. The first place this appears is in the definition of natural number. Should we include as a natural number? Some say yes, some say no and I tell my students that they will meet different lecturers with different ideas as to whether is natural or not.* One of the first definitions met by students is that of prime number and there is an argument to be had here on what a good definition is. One has the standard discussion about why counts as a prime — it just makes statements nicer. (See One is the loneliest number for a children’s book author defending their decision to have as a prime.) Leaving aside that discussion we can argue about good taste in definitions. One can define a prime as “a natural number greater than that is divisible only by and itself” or as “A natural number with only two factors”. For the latter, is automatically excluded as a prime since it has only one factor — itself. One can then argue as to which is the better definition. They’re equivalent of course but is one more elegant than the other? There are plenty of examples to get students talking about the importance of agreeing on a...

In school mathematics the emphasis is on techniques and methods (find the roots of this quadratic, solve this simultaneous equation, use trig to find how high this clock tower is, etc). At an advanced level definitions become much more central and this school emphasis causes students to miss their significance* — they are abstract concepts to be skimmed over. But definitions do need to be learned precisely. How can you apply the definition of continuous function or abelian group in a theorem or show something is an example if you don’t know the exact details of the definition? To help students learn the precise definition, I teach what I call the cover up technique. My aim is show my students a better way to deal with definitions rather than skimming over or repeatedly reading it in the hope that it will stick. The cover up technique is simple: Read the definition while consciously attempting to observe what is written there. Next cover it up and attempt to write out the definition in full from memory. No peeking. Once done, check the written down version against the precise version. Repeating this a few times works better than just rereading until it supposedly goes in. The reason for this is that retrieving facts from memory is better for learning than mere repeated exposure. There is much evidence for the power of retrieval. For a good summary (and introduction) see James Lang‘s book Small Teaching: Everyday lessons from the science of learning by that I mentioned in an earlier post. * Please note, I’m not suggesting that there is necessarily anything wrong with this emphasis or that it should change. I’m happy for it to be a problem that teachers at advanced levels need to deal...

This is one of my favourite exercises. Actually it’s more of a competition. To emphasize the importance of converse statements I set a simple challenge for my students. In a different lecturer’s class (i.e., not in mine) all they have to do is ask the question “What about the converse?” when a statement of the form is used. Nothing more. (The statement can be in any format, e.g. “If …, then …”). The first person to do that wins a prize: one of chocolates, wine or a signed photograph of me. (Rather unbelievably, the latter has been claimed more than once.) The reasoning behind this is to send a message. Even if students, at the moment the challenge is set, do not really understand what a converse is, or that it might not be true even if the original statement is, they know that I think the concept is important and that is enough to make them pay attention to it. Usually, I tell the students they should all applaud when someone asks the question. That perplexes other lecturers when that first happens and so far I haven’t got in to trouble with my colleagues. They take the minor disruption in the spirit it is...

If a student asks me for help, say during an office hour, then the path of least resistance is to start explaining and write out a solution for them. It’s an easy process just to explain the details and ask the student if they have understood. The problem is that no one likes to say they don’t understand and so students will often write down what I write down, say they understand and leave when in fact they are hoping it all make sense when they read it back later. To avoid this, instead of writing down the explanation I hand the pen to the student and let them to write down what we’re doing. Obviously, I make sure that I don’t merely dictate the answer — the student is expected to be thinking. A benefit of this approach is that we are forced to go at their speed instead of me racing ahead. Also, I can easily tell what they are having trouble with and react accordingly. If we have being using a board rather than paper, one drawback is that the student does not have a written record of what emerged. In this case I advise them to take a photograph. (In the case of paper I let them take it away.) A word of warning. This can be a painful experience for the lecturer and student. I would advise a bit of sensitivity for a student visibly struggling or else they may not ask questions in future. This is particularly important when a student asks a question after a lecture and other students are...

In essence there are five methods of proof: Direct: A string of implications , or some straightforward calculation. Induction: true and true for all means that is true for all . Contradiction. Show a statement is true by assuming it is false and showing that this implies something that isn’t true. Contrapositive: To show implies we show false implies false. Cases: Divide the statement into separate cases. For example, , and . All proofs are constructed from these basic building blocks. In my first year course a key early aim is to make sure students know this. To achieve this I mark up some proofs during lectures noting the methods used (and where the assumptions are used, etc). They are then set the exercise of finding which methods of proof are used in a different example. The exercise has a number of benefits. One, it tests the students on the methods. Two, it exposes the students to the structure of proofs. (An added benefit is that they are more likely to remember the proof when they know the structure.) Three, having explicitly seen the breaking down of proofs into pieces they are more likely to use that in the creation of their own proofs. (For the latter I have found that, generally, a prompt is needed, for example “how could we break it down?”) As an example consider the following. To prove a statement of the form “ if and only if ” students often try . That’s a great proof and it would be wonderful if it worked all the time but, unfortunately, generally it doesn’t and that causes many students to give up. However, if they have observed* repeatedly that can been divided into the cases and and those cases can be...

This is not specific to mathematics but is so effective in my teaching that I thought I’d share it: Halfway through a lecture I take a two minute break. During that time I let the students do what they want (within reason, of course!). The idea behind this one should be obvious. All of us like a break and feel refreshed after one. Just a short period is enough to reinvigorate the students’ enthusiasm. I’m sure most of us have seen the statement that people have an attention span of about 20 minutes. So in a 50 minute lecture a break at the half way point is approximately right. I think that the other in-class activities help break up the lecture as well and so attention spans are not being over-stretched. A possible cost associated with the break and a serious point raised by colleagues is that two minutes each lecture over a term adds up to a significant amount. My reply is that the lost quantity of time is more than made up for with the greater quality of the rest of the time — students are refreshed and happier and both are good for learning. (If you are so desperate for time that you can’t spare two minutes, then there is probably too much content in the course.) Furthermore, I feel energized too and I believe that helps my teaching. In conclusion, this is one activity (non-activity!) that is easy to do and is much appreciated by students — it’s often mentioned in the end-of-teaching questionnaires as a...

For many years I have given maths talks in schools and I have regularly asked the pupils to state Pythagoras’ Theorem. The usual answer is a confident ““. Their faces turn to surprise and puzzlement when my reply indicates “not quite”. The assumptions (right-angled triangle, definitions of , and ) are usually only given with prompting from me. Years of examining also leads me to believe that focussing on the conclusion is not particular to Pythagoras’ Theorem. For example, asking for a statement of Cauchy’s Theorem I’ll get answers baldly stating There is no mention of the crucial assumptions such as closed curve, the function is holomorphic on a domain and so on. It is obvious then that students do not always realise the importance of assumptions. Furthermore, I think they don’t understand precisely what they do but I’ll leave that discussion for another day. My solution to students overlooking assumptions has been to use one of the questions from the Halmos quote that started this series: “Where does the proof use the hypothesis?” This is done in the simplest way possible: Give students a proof and ask them to find where the assumptions are used. This has been done with proofs they have just met and with proofs there have been expected to have read before class. To begin with I try to use proofs where the assumptions are used at an obvious point: “using the assumption that the group is abelian, we see that…”. Next, the occurrences are more subtle, e.g., where the abelian condition is used in the middle of a calculation without comment. The final stage is using examples where the assumption is used in reference form. For example “by Theorem 3.12 we have that…” where the assumption needed is...

Find the mistake is a simple, easy to prepare activity. I’ve used it a number of times but definitely not as much as I should despite it being very effective. During a lecture I distribute photocopies of mistakes from student exam scripts and instruct my current students to locate all the errors. Students usually look for straightforward arithmetical or algebraic mistakes such as negative sign errors and so I usually warn them to look for deeper conceptual mistakes. After a suitable period of time I ask the class what they have found. An example for complex analysis is here. Find-the-mistake has two main benefits. One, the students are active and forced to apply their knowledge. In answering a question or solving a problem students often have to construct a multi-step argument and a single small mistake can lead to messing up the whole exercise. When looking for mistakes they don’t have that problem — missing one mistake won’t prevent them finding the next one. Furthermore, a common student complaint is that they can’t get started when solving a problem. Here’s one activity where the barrier to starting successfully is lower than usual. The second benefit is that they learn what the common mistakes are. This is a perfect example of “Show, don’t tell” applied to education. Obviously, I can tell them what the common mistakes are but they won’t necessarily absorb that. This exercise shows them. Furthermore, using the same mistake from a number of scripts helps the students spot for themselves the common mistakes. Generating find-the-mistake exercises is easy. When marking exams I like to take a note of the common — or interesting — errors made and photocopy these later for this activity. Student exam scripts are usually kept for a few...

Mathematics is not a spectator sport. During my postgraduate years a paragraph helped focus my ideas about learning mathematics. It’s about reading but is applicable more generally. Don’t just read it; fight it! Ask your own question, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis? Paul Halmos, I Want to be a Mathematician (1985) How many mathematicians would disagree? Understanding mathematics involves an active struggle — it’s not a passive activity. So what does the standard mathematics lecture look like? Is Halmos’ fight evident. Generally, no. A lecturer stands in front of a board, writes something, the students copy it. In this standard scenario students are expected to be thinking and copying. But, can people do two things at once? Yes, we can drive cars and sing at the same time but we can’t walk and text. In this situation — copying and thinking — what comes first? Well, students are concentrating on the copying and not the mathematics — more likely thinking, “is that an x?” rather than “why is that an x?”. So what is the point of this method? It can’t be to give information — that would be grossly inefficient. Students could be given the notes, read a book, or watch a video. The crucial question to evaluate a teaching method is “are the students learning?”. Given that the students are copying rather thinking, the standard lecture is unlikely to foster learning. In fact, there is a very tall stack of evidence to show that the standard lecturing method (teachers writes, students copy) does not work. (I did look up some good...

D’Arcy Thompson has been of interest to me since I was a teenager. There was recently a programme on BBC World Service. Have a listen on the BBC iPlayer if it’s available in your area: Science Stories: D’Arcy...

A New Book I’ve been working on a new book about sequences and series. The original plan for my book How to Think Like a Mathematician was to include some few chapters on epsilon-delta proofs. These were written but for reasons of space I left them out and had always intended to put them into a book. That intention is now being honoured. The bare bones of my thoughts on sequences have been uploaded to my website www.sequencesandseries.com. The design is not finished and there are no exercises, specially recorded videos or extra explanations. Also, there is nothing on series yet! Only sequences so far. My plan is to make the website the best place to go for information on sequences and series. Furthermore, I want to make a public commitment to finishing the book this year. So if I don’t, then please feel free to mock me on 1st January 2019! If you are interested in this experiment of writing a book and having some of it online, then please have a look and sign up for updates. The other project I’m working on is I’m gearing up for a proper launch of my latest book Complex Analysis: An...

Here’s a question for which the answer doesn’t seem to be widely known. Certainly, no one I’ve asked so far has known. They all found the answer interesting though! Why is a matrix called a matrix? If we consult www.etymonline.com for the origins of the word matrix we find matrix (n.) late 14c., “uterus, womb,” from Old French matrice “womb, uterus,” from Latin matrix (genitive matricis) “pregnant animal,” in Late Latin “womb,” also “source, origin,” from mater (genitive matris) “mother” (see mother (n.1)). That doesn’t seem to be much help (but does explain the word matriarchy and why surgeons sometimes refer to the matrix). However, surprisingly “womb” is the origin of the word in mathematics. To see this we go back to JJ Sylvester‘s 1850 article Additions to the articles in the September number of this journal, “On a new class of theorems,” and on Pascal’s theorem in which it is used in this context for the first time. He says For this purpose we must commence, not with a square, but with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants by fixing upon a number p, and selecting at will p lines and p columns, the squares corresponding to which may be termed determinants of the pth order. So what is Sylvester talking about here? Well, his interest is in developing what we now think of as Linear Algebra. In particular, he cares about determinants and is thinking of them as arising from a square array of numbers. In the passage above he notes that from an m by n array of numbers...

Posts have been a bit scarce round here recently. Something’s coming soon…

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...

The web comic xkcd is well-known for its humour. I’ve also got a soft spot for the educational infographics, such as the following. (Not all the information should be...

This week the joint Gresham College and London Mathematical Society lecture will take place. Reidun Twarock of the University of York will give a talk Geometry: A New Weapon in the Fight Against Viruses. The details are on the Gresham College website. In case you can’t make the talk on Wednesday, then maybe there is a MathsJam near you on Tuesday. And if you can’t make either, then maybe you would like to see last year’s Gresham/LMS talk by Marcus du...

It has been a while since I posted a TED talk. There haven’t been many good ones in recent years. I enjoyed the one below (though I think he could have made clear that men respond the same way as women after asking the “What do you do?” question!). Further Reading The paper on the Honeycomb Conjecture by Thomas Hales starts off with a very good introduction before getting down to the higher level mathematics. Pappus of Alexandria Freeform Honeycomb Structures Not directly related but is an interesting paper from a conference I was at last...

The line up for this year’s London Mathematical Society Popular Lectures has finally been announced. As this year marks the 150th year of the LMS there are four lecturers instead of the usual two. We have Professor Martin Hairer, FRS – University of Warwick (and recipient of a Fields Medal last year) Professor Ben Green, FRS – University of Oxford Dr Ruth King – University of St Andrews Dr Hannah Fry – University College London The lectures will be at different times and different places, see the Popular Lectures webpage. Last year’s lectures are available...

The last post was a monster post. This one is a bit shorter as it exam time at my university so I’m a bit busy. Students these days have it easy with regards to exams. They are given far too much information about how to get good marks. When I was a lad we had to guess what an examiner wanted. Just look at the rubric of this 1980s exam (click to enlarge): I just love “Full marks may be obtained for complete answers to about FIVE questions”! And this is not an isolated case, I have a stack of exam papers like...

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....

My unscheduled summer hiatus from blogging arose from a family emergency that meant I was in stuck in Brazil for about 5 weeks. Not a bad place to be stuck but under the circumstances it was difficult to find time to finish my planned summer posts. For example, I had planned to post about the Fields Medals and as it happened lost the opportunity to report from Brazil about a Brazilian winner. The short story is that, to me, the Brazilians seemed mostly proud but surprised that a Brazilian was even in the running. The hiatus also meant that the deadline for my next textbook wooshed by. Ok, in reality I was never going to make it but I’m even further behind than expected. Anyhow, service will resume when I get back on track after all the time lost. For instance, I’ll let you know what the book is...

The London Mathematical Society runs a regular Popular Lectures Series. These are mathematics lectures by (usually) research mathematicians. In recent years a pair of lecturers has performed in London and Birmingham. This year they will be given by Julia Gog and Kevin Buzzard. (The London performance was last night and the Birmingham one is in September.) If you haven’t got tickets, then the good news is that the lectures are recorded and put online. Currently, an admittedly incomplete, collection of previous lectures is hosted in two different places: Lectures from 2008-2013 Lectures from 1986-1996 Previous speakers have included Sir Tim Gowers, Sir Roger Penrose, Reidun Twarock, Matt Parker, Mark Miodownik, Ray Hill, Vicky Neale and Dorothy...

I haven’t posted in a while for many reasons. One of which is that I’ve been trying to finish a textbook. My target date is July 31 to have the final draft finished. I’m not sure I’m going to make it… Anyhow, I noticed that the Beatles’ first film A Hard Day’s Night is being rereleased. ‘But what’s the opening chord to the title song?’ I hear you say. Funny you should ask, that gives me a chance to show again my modest attempt at solving the...

The name Martin Gardner is familiar to most mathematicians. He wrote numerous on mathematics from a culture and leisure viewpoint. (You can find his books on Amazon.) Next week Persi Diaconis will give a talk at the British Mathematical Colloquium (BMC) on the life and work of Martin Gardner. The BMC is an annual gathering of research mathematicians in the UK and beyond. Diaconis’ talk is a public lecture so anyone may attend but a (free) ticket is required. Details of the talk are here. I’ll be attending so do say hello if you see me. For all of those unable to attend but want to know a bit more about Gardner then Diaconis has co-written a biography of Gardner (as well as a great mathematical magic book). There is also a recent article in the New York...