What is ironic about this blog/story is that my daughter suffered from deep anxiety/depression 2 years ago — brought on by many factors, including the pressure of school, and to perform to expectations/grades, etc.
Mathematics was something that added to her pain.
My daughter now sees mathematics as something that is quite the opposite — almost beautiful. It is beginning to feel for her what the hopeful and human essence of my last and final mathematics book was about.
When my mentor Peter Harrison retired from teaching 20 years ago, he left me with some hard to swallow wisdom — that mathematics was more or less dying in education. It was a provocative statement back then. It is perhaps even more so today.
And yes, he was correct. Except it’s not dying. It’s more or less dead.
However, I should clarify what about mathematics he thought was in a terminal stage. Surely there is plenty of mathematics being discussed and shared today. In fact, because of social media there is far more mathematics discussed today. Unfortunately, quite a bit of it is a political tug-of-war in terms of fighting for how to teach it.
How to teach it.
There it is. That is the difference between the time of Peter Harrison’s entire career(and most of mine) and today. Pedagogy, which as a word, didn’t exist in his time, is now the bread and butter in math education. The actual content has almost become immaterial, as there is this almost incurable fever to center everything around how to teach, deliver, unpack, consolidate, and test mathematics.
The problem is that few are actually discussing the quality of the mathematics.
Great pedagogy has only been around maybe a decade. Great math content over 5000 years. Seems a tad fucked up to give the leading actor role to an understudy that barely knows their lines, and relegate mathematics to the margins. The danger of allowing pedagogy center stage is that it has allowed a group like Science of Math to wade into the discussions — because the waters of mathematics have become shallow. Prior to, they wouldn’t have had the temerity to crown pedagogy with “Science” — especially since Gauss already crowned mathematics as the queen of the sciences. He also called number theory the queen of mathematics.
Number theory. Also known as mathematics not taught in K to 12.
Algebra, which for the longest time, was a jewel in learning mathematics, has been pawned off so many times in the last decade in its importance, that its value inside the echo chamber of math education is not much more than a velvet painting of dogs playing poker being sold at a Flea market.
That’s because everything in math education has been reduced to a commodity, only having value if it can give students a competitive edge in a competitive world for a competitive life. Stop me if the narrative I am painting is false.
That’s a really myopic, depressing, and capitalistic purpose for learning mathematics. Drilling down further, how we teach mathematics really means to prepare our students for success on tests.
Pedagogy is only a bad thing if it gets partially weaponized to steer mathematics discussions only to the performance culture of testing.
Love? Please. Romance in learning mathematics got hit by the bus of competition and grades many years ago.
When my kids were born, I was committed to a few important things. You’re going to know about Pink Floyd and you’re going to know about calculus. I don’t care what you do as a career — street poet, social worker, sous chef, or statistician — I am going to show you the beauty, elegance, and algebra of calculus.
Sure enough, just a few days ago, I showed my daughter Raya, in grade 9, this:
I didn’t start there. In fact, I didn’t even finish there.
I started by drawing a straight line on graph paper that had a slope of 4 that could easily be calculated by counting the “rise” and “run” squares. I then drew a whole more bunch of lines — sloppily — that looked close to a slope of 4. I wanted her to have a strong image of what slopes that were approximately 4 looked like.
After that, I asked her to get some output values for this function:
The kid can rattle off squares until 729, so knocking off squaring -3, -2, -1, 0, 1, 2, 3, was easy.(The reason I stopped at 729 is that it has the property of being able to have its square and cube root taken. Which means it must be able to written with “6” as an exponent.). We sketched out the graph by hand on graph paper. I did this on purpose, as I wanted to increase the error in finding our tangent slope. Raya learned some new terminology which seemed intuitive in her mind, and simultaneously problematic as there appeared several hand-drawn tangents that seemed plausible. All this doubt was important to underscore the sharpness that calculus was going to deliver in a few minutes.
I told her we want to find the slope of the tangent at the point (2, 4). She didn’t hesitate to say it looked really close to “4”. Without even doing a stitch of calculus/algebra, we got the answer out of the way. We could now focus on the poetry of mathematical logic that lay just up ahead.
I referenced the point (2, 4) as a marble. I told Raya I am going to roll that marble up and down the curve to new points, so that I know have three on this curve. I connected (2, 4) to these points to generate two secant lines.
I asked Raya to discuss the slopes of these lines just qualitatively — being more or less than are mysterious tangent line. Again, without too much hesitation, she said one line was steeper than the tangent one and one was less steep.
About 15 minutes later, we did all this. I wasn’t interested in technicalities at this point, just the story of the “unmasking” of the mysterious slope.
I only had to look at Raya’s face to know she was pretty blown away by the calculus she saw. Most of the mathematics that we needed, she already knew — slope formula, expanding brackets, collecting like terms, and basic factoring.
I then paused. I said there is even a faster way…
I told her calculus can create a slope machine to find the slope at any point along any curve. Before I went further, I asked her what the slope would at x = 0. The horizontal ruler quickly alerted her. She first said “nothing”, but then tightened up her answer to “zero”. I said exactly. What about at x = -3.5? All I want to know if it is going to be positive or negative, and maybe talk about the steepness however you want. She said “it will a steep negative”.
I then went to show her the slope machine for our quadratic function. I said we need to drop down the exponent 2 and multiply it by the coefficient that sits in front of the x² term. Raya looked baffled to the “simplicity”.
Our slope machine was equal to 2x.
I told her that this will tell us exactly the tangent slope at any point along the curve — how things are changing precisely at that moment.
We plugged in our x-coordinate of 2. I didn’t even need to calculate. Raya was already bug eyed.
2 x 2 = 4.
As the wheels of bewilderment churned in her head, I asked her would you like to know how/why this works — the “dropping down of the exponent and multiplying it by the coefficient”?
She said “Yes” quite enthusiastically.
I said we will wait a couple of years. You will be in a better place to understand the mathematics.
No. I didn’t show her the image above. Mathematics is also part mystery, in addition to its inherent awe, wonder, and magic.
This is why I think mathematics is dead. The romance is long gone. The rush to blow through topics has been an epidemic for years. The discussion of actual mathematics, replete with history and a storied importance of algebra, started losing its currency measurably when measurement and treating mathematical success as discrete outcomes became the fashion.
I have no problem teaching “dead mathematics” to my kids. It has already brought much life to them. Don’t ask me. Ask them.
We created this monster of what mathematics looks like now. We have no choice but to feed it — teach to the test, extricate all things superfluous, and just get down to skeletal remains of the subject.
Just don’t say you want students to experience the joy of mathematics through the exhausting journey of bean-counting, rubric-matching, and score keeping of its driest parts.
Nobody should be finding joy in this journey. At least, let’s stop lying to ourselves and to our students.