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Dr James Tanton is an author, a consultant, and an ambassador for the Mathematical Association of America. He has taught mathematics at both university and high-school institutions. James is a founder of The Global Math Project, an initiative set to transform the entire world’s perception of what mathematics can and should be. Over 5 million students across the planet have taken part in a common joyous piece of mathematics to see how classroom mathematics serves as a portal for human wonder, and delight.
1. What’s your earliest memory of doing mathematics?
I grew up in an old Victorian house in Adelaide, Australia, and every room of that house had a press-tinned ceiling, including my bedroom. Each ceiling had some design of some kind and mine in my bedroom was simply a five-by-five grid of squares. (Yes, lines were vines and the corners had shapes of little flowers, but the overall shape of the pattern was a geometric set of 25 squares.)
Every night I fell asleep staring at that ceiling, making up games and puzzles for myself. I counted how many squares I could see (55), how many rectangles I could see (225), and so on. But the one puzzle I made up for myself, that stumped me, was the one shown here. (You can see that I often tell this story!)
Starting at some given cell, can one walk a path with vertical and horizontal steps that visit each and every cell exactly once?
This puzzle is easy to solve if you start in the top-left corner cell or the centre cell, say. But starting from other cells, say, from the second cell on the top row, or one to the right of the centre cell, things were much trickier to solve. In fact, so tricky, that I could never solve them! After night after night of trying I was convinced it was impossible to find a valid path starting from each of these ‘tricky cells’. But part of me wondered – what if I tried just one more time? So I began to look for an iron-clad logical proof that these starting cells really were impossible starting cells. And it was that quest for a logical proof that stuck with me – for years!!
And then I had a magical moment. About six years later, as I was walking to school (I think it was year 10, so I was 15 or so), not even thinking about this problem, I was suddenly struck with an incredible visual picture entering my mind. It was a picture of my bedroom ceiling painted with a certain colouring pattern that made it absolutely clear as to why those tricky cells really were impossible starting points for paths. (Yes ... out of nowhere, a picture just popped into my brain.) This one image provided the iron-clad proof I sought. It was such an ecstatic moment. Truly, a real rush! But I was alone, and it was an experience that was just my own thing, and it had nothing to do with school. But I was on a high all that day!
And it wasn't until my university years that I realised what I had been doing with that bedroom-ceiling puzzle was mathematics, that mathematicians actually study these things (in fact, my visual proof was what mathematicians call a ‘parity argument’), and I really was a mathematician all along.
Oh, the joy and wonder of mathematics! I wish I had a photo of that bedroom ceiling. (I did write to the current owners, but the house has since been remodelled.)
When parents ask me to today what they can do to help their children with mathematics I say – and I mean it with full sincerity – “Paint a five-by-five grid of squares on your child's ceiling and SAY NOTHING!”
2. How has mathematics education changed in the time you have been involved in it?
My K-12 schooling in mathematics was, I am sad to say, just dreary. I grew up in a ‘just memorise and do’ maths teaching culture and I got the message very early on that I was not to ask questions. I had that message in primary school for sure, but I tested it in my early high-school years too.
We were ‘learning’ the Pythagorean Theorem, and each of 37 students in that class (Australian public school) were told to pull out rulers, pencils, and protractors, and draw three right triangles. Then we had to measure each side length, square those measurements, and see that "a squared plus b square equals c squared". (Yes, the sides, of course, had to be labelled a, b, and c!) I had two issues with this exercise. First, I didn't believe anyone was actually seeing this as true. (After all, no one can draw perfect triangles and no one can measure lengths exactly. But I let this first issue slide.) But for my second objection I did raise my hand. "Excuse me Mr. P., how do we know that this wasn't just a coincidence 111 times in a row?" And what was Mr. P.'s reply to my question? "Go back and draw another three triangles."
Maths education has changed drastically since then – at least in the USA where I now reside. The words ‘thinking’ and ‘understanding’ are actually in the common state standards. We want our students to think; not only to have facts but also to have tools and strategies to adapt and work their way through challenges; and to develop the resilience and the confidence to try ideas and possibilities, to fail, to keep trying, and to find success nonetheless.
3. Tell me about a time in your career when something totally flabbergasted you.
I started my career in the university world and then moved to high-school teaching. I was deeply worried about the state of K-12 mathematics education when I first started looking at it some 20 years ago – I saw then the same human-less, joyless curriculum that I had experienced back in Australia decades earlier. Nothing had changed. So rather than be one of those university professors who complains about what school teachers should be doing, I decided to become a teacher. I wanted to understand the system, the barriers to promoting deep learning and thinking and joyful doing, and then try to teach deep learning through joyful doing.
But it was that pressure to continually assess students and assign numbers to a nuanced experience. Ugh! And the thing that really flabbergasted me was the time a student who struggled with geometry finally got it in the end, truly got it, and convinced me he got it. He was an A+ student in the end. So what if it took him 10 months to get there? But, the system insisted that his end-of-year grade just had to be the average of his first semester grade (lousy) and his second semester grade (better to then spectacular). What a miserable system not to stamp that lad's passbook a golden A+. What are we doing to kiddos?
4. Do you practice mathematics differently in company?
I am a very slow thinker. Give me a maths problem and my first instinct is to go for a walk. So when I am in company doing maths I tend to remove myself from it!
5. Do you think a brilliant maths teacher is born or made?
I think the real key to being a brilliant teacher is to be your honest human self with your fabulous human students and let maths just be a conversation – a conversation of joy and play. Sure, you have a number of years of experience over your students and that is fine. Share your experience when appropriate. But mathematics is so profoundly deep and wondrous that, even if you have been teaching the same subject for 20 years, each time you think about it deeply, new wonder and intrigue emerges. So your job as a teacher is to latch on that! You don't need to be – and should not be – the ‘expert’ in the room who ekes out answers when appropriate. You need to model, instead, expert learning for your students: being curious, wondering, and demonstrating what to do when you encounter a question you genuinely do not know the answer to. And you do this by publicly sharing questions you don't know the answer to ... and to publicly fail; to try different things by pulling out the tools and facts you have at hand and adapting and tweaking them; to find progress, and to persist in moving forward; and bringing your students into this process too.
6. What’s the most fun a mathematician can have?
I think the most fun comes from just thinking deeply and intensely about one aspect of our beautiful subject and stumbling across connections and links to disparate topics. Mathematics really is a portal to absolute wonder and joy. It does not matter one whit what ‘level’ of maths you dig into deeply – the more you peer, the more wonder and insight you find. Mathematics feels profound, as something beyond our time and our physicality and our humanness. Stare at the Milky Way in the middle of the outback and you feel both insignificant and inspired at the same time. Stare into mathematics, and you feel the same.
7. Do you have a favourite maths joke?
Thank goodness this is a yes/no question! The answer is "yes", and I don't want to repeat it here...!
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