The dread of teaching pointless math ruins generations

Posted on Thu 15 February 2024 in teaching, mathematics

I recently started helping as a Mattecoach in a local high school in Stockholm, and by the third time I was struck by the pointlessness of the mathematics these guys are forced to study. Not so much about the *math* in itself: math is beautiful in all its shapes really; but the attitude these kids show towards it.

Obviously, students that come to afternoon hours are not the strong students. I don't stand behind the usual denigration foreigners like to throw at the swedish education system: it *is* a system where it's easier not to *fail*, but I know there are skilled and ambitious students in it as well, I've met them when I was teaching at university and I would get last-year high school students for a 2-week summer project in cryptography at the math department: 25% of them were students with a great potential, that enjoyed being pushed and challenged and who did give back. The other 75% was an anonymous average, not particularly competent at anything, with knowledge gaps on many topics, with a *wish* to get interested into the topics I covered but an acknowledgment that they, somehow, *couldn't*. I really don't understand why anybody would expect otherwise.

Those were however students that had chosen to be there, to spend 2 weeks of their summer with me in a math classroom. I always gave them the *longest* lunch break and encouraged them to head to the nearby beach, but still. My afternoon sessions at school have instead brought all the struggling students to me. It's 15 year olds who are unsure whether two points are enough to determine a line; lost at the question of taking the percentage of a number; incapable of multiplying/dividing by powers of ten without a calculator; blank at the request of calculating the distance between two points (and wrong in placing those points on the cartesian place), and then hesitant at the mention of Pythagoras's theorem ("is that the *c squared b squared a squared thing*? How was it again?").

I step into the room into a landscape of workstations all screaming equal *desperation*. They all have a book of cryptic hieroglyphs pushed as much away as possible, a laptop open on who knows what, a scientific calculator, a high-end smartphone, and a blank paper to stare at while fiddling with a pen. And they *scroll*. I watch a girl take a stab at deciphering hieroglyphs for 30 seconds, acknowledge that she's still unable to crack the language that page is speaking with, unlock her phone and satisfy her addiction. It's one of the modern apps I don't know: she clicks on her contacts one by one to reveal their latest selfies with a raised eyebrow, with their tongue out, with their hair combed, with whatever it takes to look like they're fitting in. Then she takes a selfie herself. They all look *the same*: humans manufactured in series. Then she goes back to stare at her white sheet. I'm tempted to exercise physical violence against the next overage person that claims that smartphones are *just another form of entertainment*. They are not *destroying* new generations -- they have *destroyed* them already.

I sit beside a student that dares ask for help after an hour of idle scrolling and book browsing and he's almost unable to speak. He points at an exercise and only says "I don't know". I read it quickly out loud, then ask him what he thinks of it, how he has approached it so far. He struggles to articulate a thought and is probably put off by what he may regard as a sadistic display of his ignorance. The exercise says something like "The city has run a survey over 10000 people. The average age is 50, and the standard deviation 10. How many participants are between 40 and 60 years of age?", and I start by asking what pops in his mind when he reads this, what's the average value and the standard deviation, how he thinks of using them. I could have as well asked him the etymology of russian words. I encourage him to flip a few pages back and consult the manual of hieroglyphs to at least find out what those words mean. We discover that just one page away from where the exercise is, there is a worked out example that is exactly the same -- it takes him a while to map them together, but he acknowledges that yes, they are similar. I leave him with the task of making sense of the worked out example.

I come back to him 15 minutes later to check his progress. He has copied a drawing of a gaussian in a way that evidently shows he has no idea what it represents. I go through the symbols, the μ and the σ, what values map to them, what's a standard deviation *away* from the mean. And I know it is all pointless. It's like somebody with infrared glasses helping you get out of a swamp at night: you find the shore with their help, but you're as clueless about the landscape as you were while in it. As soon as you'll fall in again, you're again in deep waters. These students only *stack* exercises on each other, but it's a new swamp every time: there's no learning *path*, they don't *build* anything, they *grope around* in the darkness all the time.

But it's not their fault, it's ours. The back of the book contains the solution to each exercise, implying that it doesn't matter *how* you get there, only the final digit. The journey doesn't matter. There's a general acknowledgement that modern math education should be based on practical problems, so all their exercises talk about surveys, engineers building bridges, the evolution of movies tickets throughout the decades. Everything is grounded in reality, with the assumption that they may be able to apply their common sense to the problems. That's not the case: they've learnt that they are all artificial contexts to make them do some math operation, and project these problems into an abstract world, one that they find inaccessible.

Furthermore, there's always some trick that betrays the fact that there's no reality here. To carry through his exercise to the end, that student needed to know that the percentage of observations one standard deviation away from the mean is *32.1%*. I want to jump at the throat of whoever has had complacency of thinking that any 15-year-old would find that number relevant to their life, or even to think that has *any* mathematical value. They imagine these kids being Feynman-like creatures who compute logarithms in their heads and that are going to be able to solve *so many problems* with that piece of info.

I encourage this student to visualize these 10000 people as *actual* people, and give an estimate of what number 68.2% of those might be, but I could have as well asked what color is god wearing in the sky right now. It's an unfathomable information: he doesn't think of *people*, and he doesn't try to relate that question to anything he might know. He says he's not sure about percentages. I ask him what 68% of 100 is, and he hesitantly answers 68, fearing a trick. But no, I encourage him that he *can* percentages. We continue with 68% of 10 and then of 1000, and he then easily infers 68% of 10000. He's so relieved to have solved a point, but hits again a wall when I ask him to find a general formula to calculate percentages. I suggest he should combine 68 and 10000 with another number, to get the 6800 he inferred. He suggests to subtract 68 from 10000, and again I am struck by the *absolute lack of number sense*, at how one could think that 10000-68 can yield something in the ballpark of 6k. It turns out he can't multiply and divide by powers of ten in his head, so he can't know how to fabricate 6800 from 68. I show him, and there I leave him. I know I've brought him to shore and given him the tiniest torch, but I also know it's gonna blow out in an hour, and he's gonna be again in deep waters tomorrow.

One minute later he unlocks his phone and scrolls -- he needs a *dose*, of course. He hasn't understood gaussians, he has only given the book the number it wanted. He thinks math is about knowing about numbers like *68.2%*, and how to blame him -- that's the takeaway from the chapters on gaussians in his textbook. I'm enraged but also full of pity, and think about what a wasted opportunity all the hours he spends in this building are.