# Mathematics – A way of knowing.

But a way of knowing what?

This is a set of resources that are predominantly linked to the IB Diploma Theory of Knowledge course, and about Mathematics

There is a presentation with some links to it at the end. The other resources can be used before or after doing the presentation or at any time really. They are just interesting and in some cases fun!

Numberwang Episode 1

Any discussion of Maths should begin with an episode of Numberwang. Feel free to watch several more episodes to get you thinking. I recommend starting each lesson with one. Work up to The History of Numberwang with its special appearance by Bertrand Russell and of course The Numberwang code.

Mitchell and Webb at their crazy best.

These two could be homework

The More or Less podcast on the art of counting

This BBC series is a wonderful look at current statistics and their fallibility, too frequent inaccuracy and ability to mislead. This short clip interviews the writer of a book on the art of counting, indicating that what you count and how you count it matters more than the numbers you end up with. Any statistic requires you to ask who is doing the counting, what are they counting and why.

Fermat’s Last Theorem

If you only watch the first minute and a half of this wonderful BBC documentary on Andrew Wiles, you will see the passion of discovery and never again believe people who say that Maths is unemotional. I still well up in tears before the opening credits! And for good measure, you even see the great John Conway if you go on a bit as well!

The invention of complex numbers

An excellent video on the history of the search for the solution to the cubic and Bombelli’s decision. The people who made this are called Veritasium. They also have an excellent one on Godel’s incompleteness theorem

The next two are activities I suggest you do in class before or after the introductory session

The Objects of Mathematics

This investigates, in a very simple way, what we mean by the things we use in mathematics. Are they actually real?

A Set of Dots

This is a great activity which I recommend for everyone. Can you count? What is counting? Why Mathematical truth has absolutely nothing to do with evidence. Possibly the only are of knowledge not requiring evidence.

It ain’t what you prove,,,

… It’s the way that you prove it

This may be the most important link in the list. We are often faced with assertions that the character of Maths is that it is irrefutably correct. As the play shows in trivial examples, that is not the case. It is an evolving growing changing area of knowledge. I think this play also shows it is more of an activity than a body of knowledge. The misconception often crops up when non-mathematicians ask if it is invented or discovered.

This is the full play, in both acts. I freely credit Imre Lakatos with the inspiration, but this version is designed for High School students. It should be fun, understandable and thought-provoking. I always used to read it through in class with the students taking the main roles. I have also staged it in various schools with the Maths teachers taking the roles of students. Fun!

The next few are referred to in the session on Maths as a way of knowing. If you have inquisitive students who want to know what a 2 ½ sided shape is or why there are more irrationals than rationals, then these are relatively simple activities and explanations.

This shows you that the Pentagram is actually a 2 ½ sided shape and invites you to explore other shapes with sides that are rational rather than whole numbers

Trouble in Mathemagic Land

This is an exercise to prove that the Golden Ratio is not a number in Pythagorean language, or is an irrational number in modern language. There is a case to be made that this was the great leap into formal abstraction, and indeed abstract thought itself. In that sense, I would be prepared to argue that it was the dawn of civilisation.

Infinity and Beyond

This is a very short intuitive proof that there are infinitely many more irrational numbers than there are rational numbers. If you look at the set of all numbers there are almost no whole numbers and fractions

Two Infinities and beyond

This is a video I made during lockdown about another case of mathematicians finding the current numbers that we have are insufficient and so creating new ones. In this case the development of Transfinite numbers by Georg Cantor. It was highly controversial at the time and he was called a madman and worse. But it is now accepted as one of the most beautiful areas of mathematics. It is introduced through the story of The Infinite Hotel.

Those last few pieces are all on my website www.mathsthoughts.com along with lots more different and interesting mathematical ideas.

Rain Man counting toothpicks

This is the clip from the film.

Related to this, and clearly the inspiration for the film is the case of the Twins described by Oliver Sachs in his book: “The Man Who Mistook His Wife For A Hat”.

This is the chapter

And this is a brief synopsis

Numberwang Ep 2

Numberwang Ep 3

Nummerwang in German

The Numberwang Board Game

A History of Numberwang

The Numberwang Code

Presentations

This is the link to the Prezi file for Windows and for Mac.

The part where I do the Maths using other programmes is on a video here.

The presentation for the Maths and Ethics part is here.

# Testing Covid Statistics

I was listening to a story on the radio about how a message from a government expert to the Prime Minister of Great Britain had been leaked. This was not the story, as it happens regularly, but I was alerted by what he was supposed to have said. He was informing the PM that the results of the covid testing should be regarded carefully. He claimed that even with quite accurate tests of over 90% reliance, the results are misleading and that, of the people given a positive test, probably only 25% actually have the disease. Furthermore, he said that this figure could be even lower in some regions, and could be as low as 3%.

This seems bizarre. It means if I take a test which is considered highly reliable, and I get a positive result, then if I live in certain areas, I still have 97% chance of not having the disease. 97% of the results could be false positives. Really?

And, why does the region of the country matter? It makes no sense! Or doesn’t it?

The adventurous should go and do the Maths. The key variables are, of course, the accuracy of the test, and most crucially the level of incidence of the disease in the population. This last one varies from region to region.

If you want some structure to your investigation, there is a prompt sheet here.

The key is the incidence of the disease. If a large enough proportion of the population do not have the disease, then the small percentage of the errors in this large group can be much greater than the large percentage of correct results in the the small population who have the disease. This means that as the disease wanes in a population, an accurate test will show up more and more false positive results.

# Infinity and beyond

We mathematicians know that there is more than one type of infinity, and that some infinities are infinitely larger than other infinities. Non mathematicians struggle to understand how this can be possible. This is a simple thought experiment that demonstrates this. I thought of it many years ago and have used it with children and adults ever since. People seem to understand it.

I call this a proof, but when I say proof, it probably doesn’t have the rigour of Cantor’s diagonalisation proof.

### ​The proof

Imagine that you have a ten sided dice with the numbers 0 to 9 on the faces.
On a piece of paper write down 0. (That’s zero point).
Roll the dice and write down, after the decimal point, the digits you get. You are gradually writing the decimal for a number between 0 and 1. You are going to keep doing this forever.

What is the chance that you will roll a fraction?

Well, to get a fraction you need to either have a decimal that terminates, in which case you need to roll zeros for a very long time, or you need to have a recurring decimal, so that’s repeating the same pattern over and over for ever.

You can think about and discuss this prospect. Let’s say you have something that looks like a fraction after 10,000 rolls. This is promising, but to get the next digit correct you have a 1/10 chance. To get the next two you have 1/100 etc. The probability diminishes very quickly to become almost zero.

Technically the chance of getting a fraction tends to zero as the digits mount up. In other words, in all the numbers our dice can roll there are almost no fractions. The infinite number of real numbers is so much greater than the infinite number of rationals that the chance of picking a rational is almost zero.

Can we consider an experiment that is technically impossible? You cannot roll a dice an infinite number of times. I would suggest that the same argument can be applied to Cantor’s proof. He envisages writing down infinitely long decimals, doing this an infinite number of times and then creating an infinitely long diagonal. At least I have only one infinite process.

What do you mean by “almost” zero? Well technically this means an event has probability zero but is not impossible. It is just so unlikely that almost certainly cannot happen. In schools they like to equate probability zero with impossible, but it is not quite the same thing.

Can we have a ten sided dice? There is no regular platonic decahedron! The answer is a ten faced prism that you roll.

This all reminds us of the infinite number of monkeys who will type the complete works of Shakespeare.

“To be or not to be. That is the gezornenplaz” Bob Newhart

“The quality of mercy is not strnen” The Mekons first album

## A Conversation with Dr Stephen Wolfram

Stephen Wolfram has some serious credibility in the Mathematics world. He wrote Mathematica. He created Wolfram Language. He created Wolfram Alpha and many other things. If you don’t know what any of these are then look them up. You can also look up his TED talk. It is fair to say that his language and his maths software have revolutionised the actual use of real maths in the real world. No engineer, scientist, statistician or indeed anyone needs to crunch the number, the algebra or the calculus anymore. There are tools to do it.

He came to our school, Markham College in Lima – Peru, while visiting his Wolfram research team for South America that has its home in Lima. He chatted with a group of us before giving an inspirational talk to over 300 students. In the conversation he described how computational thinking was taking over from mathematical thinking. He despaired that Maths education in schools and universities was still mired in the formal processes and that not only was it increasingly irrelevant but was designed to put kids off Maths. These are my words for his ideas, but I am not too far off I think. In any case it set me thinking about what he was saying. Is it really so radical? Does it question what we do in schools? Is it about time we changed? The answers to these questions are No, Yes and Yes. I will explain.

Wolfram began as a physicist in the white heat of the 70s and 80s when those looking at particles discovered or invented (take your pick depending on your philosophy) Quarks and their friends. He told us that some of his work still used at CERN in testing the bits that fly off when they collide particles at near light speed. He maintains that particle Physics has not really moved on from there in the last 30 years, which is an opinion I am not qualified to judge on.

What has changed has been computational thinking, partly due to the technological advances that have allowed Mathematica, for example, to become so sophisticated and powerful. It is also due to a change in the way people have thought about how to solve problems. It has taken a long time. I remember reading an article by David Tall in about 1982 (I guess) saying that the way we teach calculus was redundant already. He foresaw correctly that no practical mathematician would solve a differential equation analytically. They would use numerical methods, and the technology would give them a much more useful answer. The arguments were largely threefold. Numerical methods are easier to use. They do not require a set of clever tricks and they are universal. This means that computational methods can integrate Sin(Sin(x)) just as easily as they can integrate Sin(x). Note that there is no analytical solution for the former. No function that we can write in algebraic syntax differentiates to Sin(Sin(x)). The classical mathematician has to give up. The computational mathematician doesn’t see the problem.

Wolfram also argues that the mathematical way of thinking puts a lot of children off the subject. In his book on coding the concept of a variable does not arise until some 30 chapters in! This is shocking to a mathematician who would hinge understanding of the processes to an understanding of variables.

What can we draw from this? I certainly do not believe Maths is dead, but have long thought that the formal, syntactical way we teach it puts people off. As many mathematicians will have experienced there is an often repeated response when “normal” people hear what my subject is. “! never understood Maths” or even “I hated Maths” often followed by “I know it’s important, I just can’t do it”. The sad times are when this comes from students who steadfastly refuse to see the beauty that we see in the subject. I agree with Wolfram that the way we teach it contributes greatly to this.

Wolfram’s favourite rule 30 drawn with a spreadsheet!

I was talking to a friend and colleague on the weekend about this. He is a literature teacher and struggled to understand what I was saying about Maths teaching until I gave the following analogy. Imagine, I suggested, that in teaching literature you spent years on vocabulary, grammar, syntax, the techniques of narrative, of analogy, of plot and character construction but did not allow the students to read any book or poem until these were mastered. This is how we, all too often, teach Maths. We constrain our students to master numerical, algebraic and calculus techniques long before we let them loose on any voyage of mathematical discovery. Of course my literature minded friend found the idea that years would be spent on techniques before discovering the books was ridiculous. So why do we do this?

Perhaps the answer is that we Maths teachers are part of the small group of people who have succeeded through our ability to understand, appreciate and dare I say love this formal manipulation and expression. It is, then, no surprise that we cling to this as being the heart of Mathematics; its soul and its very nature. Wolfram might then respond that it was time it was mercifully killed off.

But, if so, is that the death of Maths? Well I guess that depends on how you define Maths. I sympathise with Wolfram’s story that he never knew his times tables until middle age, as I had a very similar experience, so perhaps my view of Maths is not the normal one. I got very excited in the early 80’s when I discovered the fascination and beauty of exploring iterative processes, as Wolfram did with cellular automata. Mandelbrot and Feigenbaum were my heroes. I created pictures and made music, some of which can still be heard on these pages. Mathematics was about the exploration of the consequences of the rules we set ourselves. It was a journey into the unknown that drew me along for hours on end. It still does!

One key question for educators is whether we need the great technological tools to be able to discover Mathematics. I would say the answer is a resounding No, although there are some wonderful things that the technology helps us to do. Investigative, exploratory Maths can come from a pencil and paper. The skills acquired in these explorations are the same as when looking at iterative processes on a computer, just the tools are different.

So, Maths is not dead. It just needs to wake up and smell the changed planet in which it resides.

# A Music Pi

This piece of music has been written to celebrate Pi Day, the 14th March 2017.

The violin drone follows circular functions Sin modulates the pitch and Cos the volumes of each instrument. The cycles should feel a bit like breathing! The piano uses the digits of the decimal expansion of Pi to generate the notes of the chords and their timing.

The picture evokes Archimedes approximations of Pi with polygons approaching a circle.

I am not entirely happy with it, but it needs to be out for tomorrow. One day I will improve it.

# Music and Maths

## Is there a magical connection? (There is some mathematical music at the end of this to listen to!)

I remember having this discussion at home as a child on many occasions. My father was a successful musician with an interest in Maths. Lancelot Hogben’s “Mathematics for the Million” was one of his favourite books. I was a budding mathematician with an interest in Music. It is remarkable how many Mathematicians turn out to be excellent musicians, and vice versa.

My father saw a mathematical form in the music he wrote. One piano piece was called “Upside Downside”. It was written as a leaving present to the minor prep school down the road in Surrey that I left at the age of 9, called Downside School. The clever bit was that you could turn the score upside down, rotating the page through 180˚ and play it from that viewpoint, and it was exactly the same. For him the challenge of making this work was a mathematical puzzle. He did a similar thing when he and some other composers were asked to write short pieces as birthday presents for Prince Charles (his 21st? Not sure.). Charles played the cello and the clarinet. My Dad wrote a cello part that became the clarinet part when you turned it upside down. He loved the mathematical aspects of Bach’s fugues as much as Schoenberg’s serial music.

There seem to be various aspects to the link between Music and Maths that make sense to me. One is this mathematical pattern forming. In tonal music the harmonies are numerical patterns and the repetitions, inversions and key changes are mathematical transformations. Serial music is even closer to an algorithm with its insistence on strict rules of which notes can be used. Even probability gets called on. Mozart created a parlour game where you created pieces from his fragments with the roll of dice, and this has been echoed and extended into the 20th century by Cage and Stockhausen.

There is a more philosophical point as well. Bertrand Russell described Maths as “The subject where we do not know what we are talking about nor whether what we are saying is true”. He meant that it is an abstract creation and that the objects of mathematics do not actually exist in the real world. There are no perfect circles or straight lines except in our imagination. Where will you find a number Five? You can see symbols that indicate it, you see sets or lengths or areas that have it as a property, but where is the Five itself? What colour is it? What does it feel like to touch? What does it look like? Not the numerical symbol for five, of which there are many, but an actual Five. These mathematical objects do not betray any of the properties that real objects hold.

Music has a similar sort of abstraction. If I look at the score of a piece of music, it is precisely that. It is of the music but it is not the music itself which hides uneasily behind that score. In the same way that we might have a performance of the music, or a recording of the music. Each gives a property of a representation of something that we cannot grasp in our hands. This is different from the visual arts where a piece of art is exactly what it is. A painting, sculpture or film is the artwork. It is not a proxy for the hidden abstract form. I think that this connection of a shared abstraction is a powerful link between Music and Mathematics. It is why musicians like my father have no difficulty understanding concepts such as “The Beauty of Mathematics” that GH Hardy alludes to when he says “I am interested in mathematics only as a creative art”

So to some music. My pieces based on the Logistic Mapping and the Lorenz Butterfly map are already on these pages. I thought I would say a bit about the process behind them. I am doing this because I have been invited to do so in a Maths course we offer by my friend and colleague Russell Scott. He teaches Mathematical Genie as part of our Elements Programme at Island School. In Genie they cover all the really interesting bits of Maths that most syllabi leave off. He offers starters, but the students choose their research area and explore it, whether it be Game Theory, Puzzles, Large Scale Euclidean Modelling, Fractal Geometry or wherever the muse takes them. Elements courses are mostly like this. There are 56 of them at the moment, all with a high degree of student ownership of content and process. The classes are all mixed ages and the courses are not constrained by anyone except the teacher and the students.

The Music Process

Music, at least digitally produced music, is just a set of numbers. Note values, note lengths times etc are all numbers. So I use a mathematical tool to generate the numbers and then assign the variables to them. The most recent pieces have been created using the Fibonacci process. The programme creates sets of notes in chords or individually by repeatedly asking for the next number in the Fibonacci sequence. Doing that actually can sound quite dreadful, so decisions need to be made. What instruments? What range of values can I use for the notes? What range for the times between notes?

Inspired by Brian Eno’s “Music for Airports” I wanted to make something that had that ambient feel. This is it. All note values, the sizes of the chords, the times notes appear or change are generated by the next Fibonacci number. These numbers are recycled modulo a very large number, as otherwise they will overflow my computer! The modular base has to be large enough for the series not to repeat in the time of the music. Or does it? I need to think about that? Anyway, you can create any number of different pieces by choosing different starting numbers. 1 and 1 give the normal series, but 1 and 3 give a different series hence a different piece.

This next piece is basically the same but less ambient as it has a time function that is quadratic. It speeds up as the time variable is the ordinate of a parabola, giving a value that rises to a peak in the middle of the music and falls again to the end.

Is there an aesthetic element to writing these pieces of music? I would say definitely yes. There is a constructual similarity to the process of writing 12 tome music, although this is very different from that. The algorithm provides only a set of numbers and several decisions need to be made in order to turn it into a piece of music that I am satisfied with.

My theory is that I can mix my mathematical rules and my aesthetic values to adjust and change the piece until I like it, but with the mathematical rule at the core. The Logistic and Butterfly pieces, found on this page, are done in similar fashion, as is the Reich inspired minimalist piece Ford Circles.

# A New Curriculum

I was talking to Jennie Wathall, author of Concept Based Mathematics the other day, and she was asking my opinion about things I though should be included on the next IB Higher level Syllabus. As I thought about this it occurred to me that we were doing things the wrong way round. I have always been rather suspicious of the way in which a syllabus, and particularly an examined syllabus, limits genuine creativity and exploration. Perhaps we could have a curriculum that didn’t depend on a list of content in a syllabus and an exam at the end.

As I thought more about this, I thought about the role of the examiner, or Quiz Master. You may have discovered and learned millions of wonderful things but, if I am the Quiz Master, I decide what I ask you questions on. Let’s get rid of the Quiz Master, and get rid of the list of topics that limits what you learn.

So, here is my new Mathematics Curriculum in outline. It is only a draft, but even in this incomplete form it has two major problems which will mean it will never be adopted by organisations like the IB. The first problem is that it is mathematical, with all the clunky and non linear learning that entails. The second is that the lack of traditional assessment means no external body can take control. The students are in control, which could never be accepted. Oh, but they would have so much fun, and learn so much.

# Concept Based Mathematics – a book review

I don’t normally review books, but this one is worth a plug.

### Concept Based Mathematics – Teaching for Deep Understanding in Secondary Classrooms – by Jennifer Wathall

I need to declare my interest here. Jennie is the Head of Maths at Island School in Hong Kong where I am principal.

So, having got that off my chest, what is this book about and what is it like? Well, it is an interesting and balance between theory and practice. Many educational books are very general and discuss goals, aims and philosophies in fairly abstract terms. Often Maths books, on the other hand, have lots of examples but very little of the theory behind them. Jennie’s book has roughly equal parts of both.

The theoretical background of all of this comes from Lynn Ericksen’s work on concept based learning. Lynn writes the forward and the whole project has been supported by Lynn and Lois Lanning. This is where the book starts with definitions of the terms addressing the question of what we mean by concept based learning in Mathematics. There are the usual diagrams of Erickson and Lanning but annotated by Maths examples. Then we get the first key idea which is that inquiry led learning supports a deeper understanding of the concepts. Once we have that then it all flows from there. Jennie maintains there are different levels of inquiry and gives mathematical examples of activities at different levels,

Chapters often have intriguing questions as their titles. What are generalisations in Mathematics? What does a concept based classroom look like? How do I captivate students? The eight strategies for engaging the hearts and mids of students are offered as an answer to the last question. Every time a suggestion is put forward, it is accompanied by real examples of activities to set students, and ways to deliver them.

It is a book that is mathematical in its structure and style as well as its content. Jennie writes in a logical way, the prose is straightforward and well put together. Altogether it is an excellent synopsis of some really good practice in teaching Maths, underpinned by solid theory and demonstrated by useful examples. A great book for any Maths teacher.

# Quiz Questions

I haven’t put anything on this site for a while, so here are some quiz questions I put together for the end of term quiz.

Some Maths Quiz Questions

Question 5 is fun, but you must only put it up on a screen. It is far too easy on paper. It is great for a discussion about whether Maths is evidence based, as the clear evidence of doing this is that th number of things in a set depends on the person counting!

Some of these are referred to in The Resources section of this site.

Another fun question is this year’s GCHQ puzzle. Here.