In 2004, on the cusp of the social media “Big Bang” — Facebook, Twitter and YouTube would all come to birth over the next 2 years — a book was released that foretold of the unhealthy communication hyperactivity that would follow. The book was called In Praise of Slow: How a Worldwide Movement is Challenging the Cult of Speed. It’s author, Carl Honore, is now a highly sought after speaker on the Slow Movement. His TED Talk on this subject has been viewed over 2 million times. There is no longer a cult. The world has been overtaken by speed and skimming for the purpose of volume and efficiency.
The documentation of the detrimental effects of unmanageable velocity in our lives has been vast. Unfortunately, the news has not reached the towers of math education — which is ironic in the most painful sense. The exploration and discovery of mathematics is the original slow movement! The best example of this is the historic search for the proof of Fermat’s Last Theorem.
Spanning five centuries of arduous investigation by the top mathematical minds, a solution was finally pieced together — over a grueling 7 years — by Andrew Wiles. But his marathon became a relative sprint because of related breakthroughs in elliptic curves in the mid-to-late 20th century.
Speaking of slow, there are currently six Millennium problems in mathematics that remained unsolved. Even the enticing carrot of one million dollars offered by the Clay Mathematics Institute for a solution cannot accelerate answers to mathematics’ hardest problems. And, this is the natural and organic, default narrative of mathematics — it takes bloody time! Not just time to offer a solution…but just time to think, ponder and reflect.
A recent article in the Huffington Post finally gave closure on where the focus of math education should be in the 21st century — number sense. Jo Boaler from Stanford University has done wonderful work in this area and James Tanton of the Mathematical Association of America has been a vocal advocate for joyful playing with numbers.
But, the other half of the learning puzzle — adequate time — remains regrettably absent in most classrooms. That is because there is curriculum to get through — which really means to defer to speed. They say speed kills. While this reference usually means being behind a motorized vehicle, it could be borrowed here to mean speed kills interest and deep understanding in mathematics. That’s because a false narrative of mathematics is being followed. When students are not told of the humanness of mathematical discovery — that it is rooted in failure and large consumption of time — and they themselves are on the clock to finish tasks and and answer questions constantly, what do you think will be left in the wake of all of this? I think Paul Lockhart answered that question around the time of Carl Honore’s book:
“Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity — to pose their own problems, to make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs — you deny them mathematics itself.”
The above thinking is natural — and human —i n mathematics and in problem-solving in general. Soon as you pull out the stopwatch, the denial of an authentic mathematical experience begins.
In 2002, I had the pleasure of teaching a challenging grade 12 course called Geometry and Discrete Mathematics with one of my key teaching mentors, Peter Harrison(who happened to be one of the authors of the textbook we were using). Towards the back of the textbook were the toughest problems in the book. However, that wasn’t the interesting part. The interesting part was the editorial commentary about the questions in this section — that they might take a week or longer to solve. When have you ever come across a textbook that came close to honoring the natural time lines of intense mathematical problem solving?
Ironically, both Peter and I were presented with an opportunity to implement an almost unheralded window of time to think about questions on a final exam — 30 days.
It was the middle of May, almost all the students in our classes had gotten early acceptances in the university of their choice. The fatigue of the last year of high school was setting in. Combine that with the beautiful weather and summer holidays now on the horizon, it seemed senseless to give bright math students a cliched 2 hour exam on the “greatest hits” of the course. They didn’t care…and luckily for them…neither did their teachers!
Peter and I selected 12 questions from the Challenge Problems in the book. Gave them to the students in the third week of May and said “5 of these 12 questions will be on the final exam. You have a month to work on the problems….go!” The class average for the final exam was over 90%. That was the irrelevant part. The cool part was some of the stunning and creative solutions we got back — which would have been impossible had they not had a chance to marinade with time and positive collaboration.
But, its more than just having time. It’s using it in a leisurely way that honestly to attempts to stop and smell the roses that are replete in math’s garden. It’s not important to see every single “flower”. Hardly. Sometimes just being fixated with a petal or two has tremendous benefits. For some inexplicable reason, there is this “race to the moon” feel when children start learning about numbers. Quite unnecessary. If you ask most children what “55” means to them, they will give you host of answers — or maybe, none at all…just a shrug indicating “I got nothing”.
I think if we spent more time “smelling roses” — which in this case is joyful play with numbers — more children would remark that fifty-five is the sum of the first ten numbers. With even more time and play, 1 2 3 4 5 6 7 8 9 10 could be refashioned to look like: 1 2 3 4 5 and 10 9 8 7 6. Slide the second set under the first and you have 5 sets of 11. Ahhhh…the bouquet of math:)
If the above sounds too innocent and simple…that’s because it is. The meaningful and happy adventure in mathematics is rooted in sandlot play of numbers. It also means sitting there for hours without a care in the world.
I wish, for mathematics sake and for its stakeholders, that we lived in that world. Well, they say “the world is what you make it”.
Let’s make it with time being reunited with mathematics — in our classrooms.