Learning to Work Like a Mathematician

Doug Williams

Keynote Address
Annual Conference
Mathematical Association of Victoria
December 5th 2002

Green Line

Introduction

Thank you for choosing to come to this keynote address. I recognise that there is an additional degree of responsibility associated with being invited to deliver a keynote address and for some time now I have been pondering the credentials that have brought this responsibility on me. I have come to the conclusion that it must be
... age.
My age was brought home to me not long ago at the local shopping centre. I was dragged along to push the trolley for my Princess. Not such a bad thing, but it was supposed to be a shortish excursion away from the computer, so I certainly didn't dress for the occasion. I was comfortable; not the best jeans and my very favourite, very bulky, zipper up the front cardigan.

Our supermarket has one exit that leads to a back car park via a side corridor. Shopping done we headed in that direction, but hadn't gone ten steps when:

Oh I forgot...

I volunteered to wait in the side corridor and meandered into it slothfully pushing a relatively co-operative cart. I backed into a wall, slouched on the trolley and settled down to let my mind wander back to what I had left on the computer.

Suddenly around the corner from the car park direction came two young men in Armaguard uniform. One was pushing a cart of some sort and was very focussed on where they were going. The other was hovering around him, hand on the Smith & Weston holstered on his hip, eyes darting in all directions. He spied me ... and I suspect sized my up in a millisecond as non-threatening. As they flashed past he greeted me with:

G'day digger!
G'day digger! I was flabbergasted. Couldn't he see it was my father who fought in the war - and even that was the second one!

However, it wasn't the event itself that made me realise I had earned my age credentials. It was actually my reaction to it. Before my Princess turned up with the missing item, I realised that in my mind my first response had actually been:

Why you young whipper-snapper!

Story Telling

So, credentials established, one thing old people are allowed to do is tell stories, and that, as described in the blurb for this address is all I intend to do for the next fifty minutes or so. Some of the stories will be from classrooms and some I will take from this book, which is
... the second greatest story ever told.
Let me take a moment now to establish the credentials of my fellow story tellers.
Fermat's Enigma, Walker and Company, New York, 1997

Author: Simon Singh, Ph. D. in particle physics, mathematician, member of BBC science department for many years.

Foreword: John Lynch, Editor BBC Horizon Series

Companion Films:

  • Fermat's Last Theorem, BBC, UK (Co-produced and directed by Simon Singh.)
  • The Proof, PBS, Nova, USA
Why would I choose this story - the story of the search for the solution of the world's hardest ever mathematics problem - to share the platform with my stories from everyday classrooms? Simon and John can answer that:
(The story of Fermat's Last Theorem) provides a unique insight into what drives mathematics, and perhaps more important, what inspires mathematicians.
Simon Singh, Preface, p. xv

The story of Fermat's Last Theorem is unique. By the time I first met Andrew Wiles, I had come to realize that it is truly one of the greatest stories in the sphere of scientific or academic endeavour.
John Lynch, Foreword, p. viii

For some time in my research I looked for a reason why the Last Theorem mattered to anyone but a mathematician, and why it would be important to make a program about it. Mathematics has a multitude of practical applications, but in the case of number theory the most exciting uses that I was offered were in cryptography, in the design of acoustic baffling, and in communication from distant space craft. None of these seemed likely to draw in an audience. What was far more compelling were the mathematicians themselves, and the sense of passion that they all expressed when talking of Fermat.
John Lynch, Foreword, p. ix

My hope is that these storytellers can help us learn how to work like a mathematician. Perhaps if we grasp that a professional mathematician does NOT go into the office in the morning, turn on the computer and do all the exercises down the left-hand side of the screen, we will be in a better position create an alternative environment for learning mathematics in our schools from K through 12.

But who is this Andrew Wiles?

In 1963, when he was ten years old, Andrew Wiles was already fascinated by mathematics. "I loved doing the problems in school. I'd take them home and make up new ones of my own. But the best problem I ever found I discovered in my local library." ... Andrew was drawn to a book with only one problem, and no solution. The book was The Last Problem by Eric Temple Bell.
p. 5

"It looked so simple, and yet all the great mathematicians in history couldn't solve it. Here was a problem that I, a ten-year-old, could understand and I knew from that moment that I would never let it go. I had to solve it."
p. 6

So, a ten year old could understand the statement of Fermat's Last Theorem. Let's see if we can. To do so, we have to go a little further back in history than 1963...

Stating Fermat's Theorem

Around 530 BC, Pythagoras, learning from the Egyptians and Babylonians who preceded him and working within a mathematical community that we call his school, proved a geometric theorem. Today, in Western cultures, we call it Pythagoras' Theorem. I will demonstrate an example of it with Task 97 from the Task Centre Project, called Pythagoras Rods.

First make a triangle with the light green, pink and yellow Cuisenaire Rods. It turns out to be a right-angled triangle. Pythagoras 1
Then use more rods to complete the square on each side. Pythagoras 2
Rearrange the rods from the smaller squares to exactly cover the larger one. Pythagoras 3

Of course this demonstration is not a proof. It is only a signpost to something that may be worthy of proof. A hint that in a right angle triangle, the square built on the hypotenuse may be the sum of the squares built on the other two sides. The Egyptians and Babylonians provided the signposts, but Pythagoras provided the proof that this was always true. And once proven, the theorem became true forever. There is a straight forward reconstruction of Pythagoras' general proof in an appendix of Singh's book.

My interest is in what mathematicians might do with this knowledge. What can they learn from it?

  • Can the construction work another way?

  • How many ways can we find to make it work?

  • To generate a square you only need to know one side (called the root of the square, or square root). For example if the square root is 3 you build 3 row of 3 , if 4, build 4 rows of 4 and if 5, build 5 rows of 5. So, in general, if the right angle triangle has sides x and y and hypotenuse z, we can write a numerical equivalent of the geometry as:
    x rows of x + y rows of y = z rows of z ... or ... x2 + y2 = z2
  • Squares on the hypotenuse. What about pentagons, hexagons or even semi-circles?

  • How about cubes on the hypotenuse? To generate a cube we also only need one side, the cube root. For example, 3 rows of 3 stacked 3 high. Could it be true that:
    x3 + y3 = z3 ???
  • True or not, could there be x, y, z that solve:
    x4 + y4 = z4 ???
    x5 + y5 = z5???
    ...
    xn + yn = zn ???
Mathematicians have played with all these extensions of Pythagoras' Theorem, but this last one is where Fermat gets into the act. Through mathematical history from 530BC to 1637AD, a proposition arose that:
For n > 2, there are no x, y, z such that: xn + yn = zn
Mathematicians thought this was true, because in the passage of around 2000 years they hadn't found any values that had worked. This was not a proof of course, because there could have been a set of suitable values waiting around the next mathematical corner.

Fermat was an amateur mathematician. His profession was actually law, but he pursued mathematics as a hobby. One of his pastimes was to work through an ancient mathematical manuscript written by Diophantus around 250AD - remember there was no television in 1637.

Fermat came across many things in the manuscript which excited his mathematical imagination and he frequently annotated his copy of the manuscript with what later turned out to be new mathematics. However he is most famous for writing in the margin next to this proposition:

For n > 2, there are no x, y, z such that: xn + yn = zn

I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain.

Unfortunately Fermat died before he did find a margin that was big enough .. and if it were not for the fact that every other observation made by this amateur turned out to be true - even though it took decades before some were proven - then this one would have caused no more interest.

However, after his death, mathematicians began to respect Fermat's mathematical inspiration and having proven all his other conjectures believed that this last one to submit - Fermat's Last Theorem - would soon also be proven ... or disproven. They were wrong.

History of Solution

To DISPROVE Fermat's Last Theorem, mathematicians only had to find one set of x, y, z and n that WOULD work, because Fermat claimed there were no such sets.

To PROVE Fermat's Last Theorem mathematicians only had to show that for an infinity of possible sets of x, y, z and n the equation WOULD NOT work.

2000 years of history had not yet turned up a case that would work. Option 1 was looking shaky. however, the second option of proving every possibility looked a bit daunting, so mathematicians applied the strategy of trying a simpler case. I will use excerpts from Singh's book to outline some of this history.

Remember, my proposition is that by becoming aware of the way professional mathematicians work, we may be empowered to create a mathematics learning environment that is more relevant, challenging, satisfying and fruitful for our students.

A century after Fermat's death there existed proofs for only two specific cases of the Last Theorem. Fermat had given mathematicians a head start by providing them with the proof that there were no solutions to the equation:
x4 + y4 = z4.

Euler had adapted the proof to show that there were no solutions to:
x3 + y3 = z3.

After Euler's breakthrough it was necessary to prove that there were no whole number solutions to an infinity of other equations:
...
Although mathematicians were making embarrassingly slow progress, the situation was not quite as bad as it might seem at first sight.
...
The proof for case n = 3 is particularly significant because the number 3 is an example of a prime number.
...
This seems to lead to a remarkable breakthrough. To prove Fermat's last theorem for all values of n, one merely has to prove it for the prime values of n. All other cases are simply multiples of the prime cases and would be proved implicitly.

p.88 - 90 (excerpts)

In terms of my proposition, we can already we see that mathematicians:
  • learn from each other
  • build on each other's work
  • engage in high order thinking
  • are content with partial solutions
  • expect to take time to solve a problem

But the story continues, this time through the work of a great female mathematician who was the first to succeed with a general proof as opposed to proving particular cases:

By the beginning of the nineteenth century, Fermat's Last Theorem had already established itself as the most notorious problem in number theory. Since Euler's breakthrough there had been no further progress, but a dramatic announcement by [Sophie Germain] was to invigorate the pursuit of Fermat's lost proof. p. 97 In her letter to Gauss she outlined a calculation that focused on a particular type of prime number p such that (2p + 1) is also prime. ... For values of n equal to these Germain primes, she used an elegant argument to show that there were probably no solutions to the equation
xn + yn = zn.

In 1825 her method claimed its first complete success thanks to the work of [Dirichlet and Legendre] ... Both of them independently were able to prove that the case n = 5 has no solutions, but they based their proofs on, and owed their success to, Sophie Germain.

Fourteen years later the French made another breakthrough. Gabriel Lame made some further ingenious additions to Germain's method and proved the case for the prime n = 7. p.106 (excerpts)

So now the problem had been trimmed even further. Germain's method disposed of a class of prime numbers and only the irregular primes were left.
[In 1847] Kummer pointed out that there was no known mathematics that could tackle all these irregular primes in one fell swoop. However he did believe that ... they could be dealt with one by one ... [But] disposing of them individually would occupy the world's community of mathematicians until the end of time.

Kummer had demonstrated that a complete proof of Fermat's Last Theorem was beyond the current mathematical approaches. pp. 116 & 117

Kummer's work was also to add a different layer to what was becoming the romance of Fermat's Last Theorem. Yet another amateur mathematician, Paul Wolfskehl, comes into the story in late 1800s. Regrettably he was rejected by a beautiful woman and since he could no longer see the point of living, he appointed a time to execute himself. In true Prussian style he put his affairs in order and wrote his suicide note; but he was too efficient. All was in readiness some hours before the appointed time to shoot himself in the head.

What was a man to do? Wolfskehl was a well-to-do industrialist with an extensive library, so he drifted to that room to bide his time. He spied Kummer's paper on the shelf and while skimming through it thought he had discovered a fault in the logic. He decided a worthy way to spend his last few hours was to attempt to correct the error - remember there was no TV in the latter part of the nineteenth century either!

By dawn his work was complete. The bad news, as far as mathematics was concerned, was that Kummer's proof had been remedied and the Last Theorem remained in the realm of the unattainable. The good news was that the appointed time of the suicide had passed, and Wolfskehl was so proud that he had corrected a gap in the work of the great Ernst Kummer that his despair and sorrow evaporated. Mathematics had renewed his desire for life.
p. 123
Could it be that learning to work like a mathematician can renew our students' desire for mathematics?

Following his failed suicide Wolfskehl rewrote his will and on his death in 1908 his family discovered that he had bequeathed 100,000 Marks to the person who could prove Fermat's Last Theorem. At the time that prize would have been worth around $1 million in today's terms.

From around Kummer's time until the late 1940s, mathematicians were basically stumped by Fermat. However, there was no implication that they couldn't do mathematics. They just put the problem aside for now and worked on different problems. Being unable to solve one problem was not an indication of being less than satisfactory as a mathematician.

With the arrival of the computer awkward cases of Fermat's Last Theorem (those of the irregular primes) could be dispatched with speed, and after the Second World War teams of computer scientists and computer scientists and mathematicians proved the Last Theorem for values of n up to 500, then 1000, and then 10,000. In the 1980s Samuel S. Wagstaff of the University of Illinois raised the limit to 25,000, and more recently mathematicians could claim that Fermat's Last theorem was true for all values of n up to 4 million. p. 158
That is, there were no values of n up to 4 million for which an x, y and z could be found to solve:
xn + yn = zn.

However mathematicians knew that this success was merely cosmetic. Demonstrating something up to the umpteenth million case does not prove anything about umpteen million and one. As recently as 1988 this was point was driven home again.

Euler's Conjecture
When working on Fermat, Euler developed the view that there are no solutions to:
x4 + y4 + z4 = w4

For 200 years no-one could prove or disprove this conjecture.

Noam Elkies' Disproof
Then at Harvard University 1988, Professor Elkies showed that:
2,682,4404 + 15,365,6394 + 18,796,7604 = 20,615,6734

In fact Elkies proved that there were infinitely many solutions to the equation. The moral is that you cannot use evidence from the first million numbers to prove a conjecture about all numbers.
p. 160

A Little Child Shall Lead Them

Now let's return to the story of our 10 year old boy.
For over two centuries every attempt to rediscover the proof of Fermat's Last Theorem had ended in failure. Throughout his teenage years Andrew Wiles had studied the work of Euler, Germain, Cauchy, Lame and finally Kummer. He hoped he could learn by their mistakes, but by the time he was an undergraduate at the University of Oxford he confronted the same brick wall that faced Kummer.
p. 118

Wiles was not prepared to give up: Finding a proof of the Last Theorem had turned from a childhood fascination into a full-fledged obsession. Having learned all there was to learn about the mathematics of the nineteenth century, Wiles decided to arm himself with techniques of the twentieth century. p. 119

As with all great stories the plot becomes more complex the deeper you get into the story. I don't intend to detail every fascinating step from here. Suffice it to say that:
  • Unknown to Andrew at the time, the tool he needed was the Taniyama-Shimura conjecture that was proposed by two Japanese university students following the second world war.
  • It turned out that to prove Fermat he only had to prove the Taniyama - Shimura conjecture.
  • For seven years he worked alone in his attic during the working day. He went to sleep with Fermat and woke up with Fermat ... and there was television in the 1980s.
  • He relearned every piece of mathematics that had ever been applied to Fermat and when he couldn't make any of it work any better than anyone else, he went hunting for new techniques.
  • All the time he was doing mathematics because he was trying to solve a problem. That fact that he did not get the answer was no judgement on whether he could do mathematics.
  • His perseverance paid off and on June 23rd, 1993 he presented his proof in a series of lectures to a world assembly of mathematicians at Cambridge, England.

Andrew appeared on the cover of Time magazine and was lauded in newspapers and magazines all over the world.

But he had made a mistake.

Three months after the lecture, a colleague pointed out the error in his logic. The mathematics community allowed him time and opportunity to correct it. Not so easy!

For fourteen more months he hid himself away, this time with a colleague and worked through every step to correct the problem. He was frustrated time and again because neither of his main tools, called the Kolyvagin-Flach method and the Iwasawa theory, were achieving what his intuition thought they should. Then:

"I was sitting at my desk one Monday morning, September 19, examining the Kolyvagin-Flach method. It wasn't that I believed I could make it work, but I thought that at least I could explain why it didn't work. I thought I was clutching at straws, but I wanted to reassure myself. Suddenly, totally unexpectedly, I had this incredible revelation. I realized that, although the Kolyvagin-Flach method wasn't working completely, it was all I needed to make my original Iwasawa theory work. I realized that I had enough from the Kolyvagin-Flach method to make my original approach to the problem from three years earlier work."
p. 274-75

"It was so indescribably beautiful; it was so simple and so elegant. I couldn't understand how I'd missed it and I just stared at it in disbelief for twenty minutes. Then during the day I walked around the department, and I'd keep coming back to my desk looking to see if it was still there. It was still there. I couldn't contain myself, I was so excited. It was the most important moment of my working life. Nothing I ever do again will mean as much."
p. 275

Is it possible that we can create happy, healthy, cheerful, productive, inspiring classrooms where all students can experience that same joy of discovery?

Working Mathematically Curriculum

The answer to that question is an unequivocal YES.

Mathematics Centre contains many anecdotal stories and a range of research data to support this statement. I will draw information from the largest and longest running of the research projects to provide a little evidence for now. You are invited to follow up on the site for yourself.

Improving Numeracy for Indigenous Secondary School Students (INISSS), a project of the Tasmanian Aboriginal Education Unit, invested in its teachers over a four year period by providing extensive professional development time and linked resources. From the beginning, an evaluation plan was established to monitor the program. Also from the beginning, a professional film maker was employed to visit classrooms and video the changes in teaching practice. These tapes were an important component in the discussion and visualisation of what it meant to develop an inclusive Working Mathematically curriculum.

(NB: At this point in the address segments were played from the 100s of hours that have been recorded. They showed the same teacher using two quite different teaching styles, both of which were noticeably different from the traditional mathematics classroom environment.)

INISSS teachers taught within a scaffold of learning to work like a mathematician. They were originally resourced to do this with the Task Centre Kit for Aboriginal Students, then later with Maths300 membership and a range of other hands-on learning tasks. Teachers learnt to 'walk the walk and talk the talk' of the Working Mathematically process as originally described by the Task Centre Project.

Given that the 'problems down the left side of the screen' image I used earlier is clearly not applicable, the project asked mathematicians to explain how they worked. This is the composite response. I expect you can see signs of this process in the retelling of Fermat's story.

Working Mathematically Process

When mathematicians become interested in a problem they:
  • Play with the problem to collect & organise data about it.
  • Discuss & record notes and diagrams.
  • Seek & see patterns or connections in the organised data.
  • Make & test hypotheses based on the patterns or connections.
  • Look in their strategy toolbox for problem solving strategies which could help.
  • Look in their skill toolbox for mathematical skills which could help.
  • Check their answer and think about what else they can learn from it.
  • Publish their results.
Questions which help mathematicians learn more are:
  • Can I check this another way?
  • What happens if ...?
  • How many solutions are there?
  • How will I know when I have found them all?
When mathematicians have a problem they:
  • Read & understand the problem.
  • Plan a strategy to start the problem.
  • Carry out their plan.
  • Check the result.
A mathematician's strategy toolbox includes:
  • Do I know a similar problem?
  • Guess, check and improve
  • Try a simpler problem
  • Write an equation
  • Make a list or table
  • Work backwards
  • Break the problem into smaller parts
  • Act it out
  • Draw a picture or graph
  • Make a model
  • Look for a pattern
  • Try all possibilities
  • Seek an exception
  • ...

    If trying one way doesn't work they just start again another way.

(PDF version of Working Mathematically Process)
The INISSS teachers learnt to use tasks as an invitation to students to work like a mathematician, and whole class investigations, such as those in Maths300, to model how a mathematician works. They also learnt to embed their skill practice work in what came to be called toolbox lessons. Skills being developed for the purpose of solving problems, and in many cases developed in parallel with solving a problem.

You might ask whether the teacher in the video was born, or somehow gifted, to teach this way. His answer would be absolutely not. As a result of the INISSS project he, and his colleagues, changed their teaching practice and the students won. That is why we confidently say that all our projects are 100% professional development.

INISSS Results

INISS Report Cover The results of the INISSS professional development project are unique. In spirit, I am now joined in my story telling by Rosemary Callingham and Patrick Griffin. Their summary report is available at the Mathematics Task Centre.

The graphs below illustrate the trends in the data. They are not graphs of actual data, but they do show the statistically significant trends. The graphs were examined by Rosemary Callingham before publication. Comments with the graphs are taken from an email on July 19th 2002 giving approval to use them in this form.

Pre-1998
Mathematics
Tasmania
Years 7 to 10

...the picture ... is a generally accepted one. Certainly in 1997 only about 50% of Indigenous kids were achieving Year 8 outcomes (and this was in Year 9) compared with about 80%+ of non-Indigenous kids.
Pre-1998


1998 - 2001
Mathematics
INISSS
Years 7 to 10

The overall picture of INISSS results is okay if a little exaggerated. But it does convey a picture of the improvement.
1998-2001


1998 - 2001
Mathematics
INISSS Girls
Years 7 to 10

The girls graph is good. By the end of the 4th assessment in Year 10 the Indigenous girls were doing better than the non-Indigenous boys and as well as non-Indigenous girls.
1998-2001 Girls


Rosemary added:

You might like to mention that we have replicated the results with INISSS B - a completely separate group (of teachers) - which gives me confidence in the overall picture.
There were actually 3 groups involved in INISSS.
  • 2000 INISSS A students (about 5% of whom were Indigenous). INISSS A teachers began their professional development program in 1998.
  • 1000 INISSS B students. Their teachers began their PD in 2000.
  • 750 control group students whose teachers were not involved in INISSS professional development.

Evaluation Program

INISSS A INISSS B
Year 7, 1998 - Statewide Testing  
A1, Year 8, March 1999 - Performance Tasks B1, Year 8, March 2000 - Performance Tasks
A2, Year 8, October 1999 - Performance Tasks B2, Year 8, October 2000 - Performance Tasks
A3, Year 9, October 2000 - Performance Tasks B3, Year 9, October 2001 - Performance Tasks
A4, Year 10, October 2001 - Performance Tasks &
Multiple Choice Tests
maths & literacy
linked to 1998
control group
 

Official Results

  • All tests, including the performance assessment tasks were highly reliable.

  • No tests, including the performance assessment tasks, showed bias against any sub-groups.

  • The performance assessments were measuring the same construct in the same way across all schools and all teachers who marked them.

  • On the performance tasks Indigenous students, and girls in particular, made gains that "closed the gap".

  • On the conventional multiple-choice tests, comparisons with the control group showed overall statistically significant differences on both numeracy and literacy.

  • On conventional multiple-choice tests, Indigenous students made the same gains as the control group in numeracy but much greater gains in literacy.

  • Structural equation modelling suggests a "method" effect - how the assessment was carried out makes a difference.

Coming Together

Finally, how do I bring these stories together ... and in the best tradition, leave the door open for a sequel?

Firstly, if you would like to dig more deeply into the activities and approach that has brought student success in Tasmania and elsewhere, then investigate Maths on the Move or other Professional Development Partnerships from the Mathematics Task Centre.

Secondly, could I suggest that you read Simon Singh's book - it is one of the best human interest stories you will ever read. As you do, note the statements that reflect how a mathematician works. For example:

Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost.
p. 71, W. S. Anglin

During the era of Fermat, mathematicians were considered amateur number-jugglers, but by the eighteenth century they were treated as professional problem-solvers.
p. 73

Pure mathematicians just love a challenge. They love unsolved problems. When doing math there's this great feeling. You start with a problem that just mystifies you. You can't understand it, it's so complicated, you just can't make head nor tail of it. But then when you finally resolve it, you have this incredible feeling of how beautiful it is, how it all fits together so elegantly.
p. 146, Andrew Wiles

Mathematics has its applications in science and technology, but that is not what drives mathematicians. They are inspired by the joy of discovery.
p. 146

Do our classrooms offer this joyful opportunity to all students, or are they more likely to reflect Ancient Egyptian thinking:
Pythagoras observed that the Egyptians and Babylonians conducted each calculation in the form of a recipe that could be followed blindly. The recipes, which would have been passed down through generations, always gave the correct answer and so nobody bothered to question them or explore the logic underlying the equations. What was important for these civilizations was that a calculation worked - why it worked was irrelevant.
p. 7
Thirdly, I promised in the blurb for the session that I would tell stories in word, picture and (perhaps) song. Well, guess which one you haven't had ... yet!

Please feel free to join in at any time to save me from myself.

Working Mathematically: The Anthem

(Tune: Advance Australia Fair)
Right Click on the tune and choose Save Link As... to download a Midi file.

Maths teachers all let us rejoice
Our subject is not trite!
It's far more than the daily toil
Of "Is this wrong or right?"
The theme we weave each time we teach
Must challenge students to
Engage with problems in the way
That math'maticians do!
In joyful classrooms let us work
Like math'maticians do!

Alternative arrangements of the tune:

More Maths Education Web Papers