I sat with my 10 year-old daughter tonight for 15 minutes doing math — like I do most nights. I also sit with my 12 year-old son for the same amount of time. Thankfully, they don’t get a truckload of homework delivered by some cargo van from the 20th century. So, these moments are just random, happy excursions in mathematics that we always look forward to…:)
We did square numbers tonight, going all the way up to 256. I first asked her to draw a square with one tiny box. I then drew the next size square, by adding an “L-shaped” 3, to get a 2 x 2 square. I then asked her to make the next bigger square. She drew a 3 x 3 correctly. Although, and rightfully so, she completely missed the idea of the “L-shape” of the 3. I drew the next one, once again, adding an “L”. This time it was a 7…
While she enjoyed drawing the squares, she was quite content to start adding consecutive odd numbers. It was a nice displacement for drawing.
So the square sizes we had so far were 1, 4, 9, 16, 25, and 36.
I asked her if she knew what 7 x 7, 8 x 8, etc. were. She got to 12 x 12 is 144.
I then asked her what did we add to 1 to get to 4. She quickly responded “3”. I then asked the question again for the jumps from 4 to 9, 9 to 16, etc. Funny thing, she got excited because she identified the “jump numbers” of 3, 5, and 7 as prime before citing them as odd(she has a fascination with primes!) While she felt a little deflated seeing a “9” next, she did like the pattern of consecutive odd numbers being added to produce square numbers.
And, when I showed her that these odd numbers were fashioned into the 12th letter of the alphabet, the “L”, she gave that affirmative…”that’s cool!”.
Every square number after 12 x 12, we built from scratch. She just didn’t memorize 13 x 13. She knew she had to add by this time a “25” to 144. No calculator. She did it in her head. We kept doing this all the way to 256.
I could have given her a list of square numbers, recited them with her like those who champion the memorization of facts, but I didn’t. I took time to show her the patterning and help her figure out a strategy for adding 29 to 196 — add 30 and then subtract one.
I spent considerable time on just one tiny shard of mathematics. Why?
Being a math educator has taught me many things. But, the single most important idea has been the idea of patience and exploring every nook, cranny, and crevice of a problem. Besides making math more interesting, it also allows me to look forward to moments of irresolution. Yes. To look forward to them!
Dan Meyer made those word famous in his 2010 TED Talk, “Math Class Needs A Makeover”. That one of the large stumbling blocks for students — and, I suppose teachers sometimes — is an impatience for irresolution. That somehow not getting an answer in the time it takes to boil an egg means some kind of failure or inability to do/understand mathematics. This myth is not only false, it has done almost irreparable damage to how/why mathematics gets taught in schools.
Everything is a bloody race. If math education were a pizza, it would have 5000 toppings on it. The curriculum is almost morbidly obese. Every test seems to be crammed to the nines with as many questions you can get on 5 sheets of double-sided paper.
And then we are flummoxed as why students don’t understand math, don’t like math, and in the end, fear math — which becomes some gargoyle creature that moves fast in unexplained directions, trampling over students.
The gargoyle image isn’t an exaggeration. We have distorted the natural narrative of mathematics understanding/development — that it takes lots and lots of time — and turned it into something whose value is measured by speed, efficiency, and cold calculations. If the goal was to have kids hate math, mission accomplished. Paul Lockhart said it far more powerfully over a decade ago.
Back in the early 2000’s, I had the honor of teaching one of the best math courses created in high school — and, with one of the authors, Peter Harrison. Just reading the introduction in the Cumulative Problems section still gives me a shiver as to how the authors Peter Taylor and Peter Harrison correctly saw that time, patience, and resilience are required to solve math problems.
Crafting the environment for patience is one thing; having it be organically transmitted to our students is quite another. As such, we as teachers have to be patient with our own understanding of mathematics. And, that anything that is unfamiliar to us shouldn’t be a cause of negative reactions. Confusion and frustration are the early badges of learning mathematics — for everyone.
Thankfully, things are changing in this democratized era of information. The powerful idea of math equity will almost ensure that traditional math education of outdated ideas — which served as barriers for many students — will eventually die off. This will hopefully include the soul-crushing notion that the stopwatch and standardized tests have any place in math education.
Mathematics will wait for us. It has waited for everyone. There is no idea, in my opinion, that needs to get rushed over. It’s more than just shooting for understanding. I want people to like mathematics. Getting good at it is something defined by math education — that somehow mastery of a thousand details entails success. Maybe it does.
But, it doesn’t mean that there was any jubilation or enthrallment with the subject. Some might say those things are not important. Well those who do might believe that not everyone is capable of doing or appreciating mathematics.
I know that I certainly do, as well as thousand of other teachers.
I have learned to give much patience for doing mathematics. I tend to have little for doing things which distort and insult the most important criteria in doing mathematics — for everyone — that it takes time.
The Slow Movement, which was documented in the book In Praise of Slow by Carl Honore almost 20 years ago, seems to have reached every valuable part of our society — except math education.
If mathematics is ever going to be understood at any meaningful depth, then we have to afford more patience with students and teachers in that process. And, if we desire that mathematics be a pleasurable experience, then anything short of giving back leisurely amount of time will continue the status quo experiences of most students— stressful and uneventful.
Students tally about 500 hours in their school career with mathematics. Surely, we can do a much better job than aiming for volume and efficiency — like a Drive-Thru at a McDonald’s.
If we can’t, then we have not only betrayed our teaching community, but we have betrayed mathematics…nobody will be capable of having a good time.
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Sunil Singh is the author of Pi of Life: The Hidden Happiness of Mathematics(2017) and co-author of Math Recess: Playful Learning For The Age of Disruption(2019)