Human Calculators
It's a widely known fact that school math sucks.
It's lesser known fact that in order to take spacecraft to the moon, many women were used as the "calculators" to meticulously calculate trajectories (or something like that -- sorry haven't watched/read Hidden Figures).
It's even lesser known that human calculators were used to calculate artillery fire launching angles in around the 1800s (i think).
It's even lesser known that the first square root tables were modeled by polynomials that were determined by a "high-ranking" mathematician, and the lower level workers would bash out the numbers in order to accurately approximate the square root of a number.
And it's even lesser known that schools are turning students into calculators.
Schooling is great except when it's not and students walk into class and end up caring more about how there's a weird spot on their desk more than why a tangent to a circle is at a right angle with the center.
I argue this is because students are being turned into calculators.
I distinctly how in elementary school, we were rewarded based on how fast we could add or multiply numbers together. Not like 2nd place gets a massive prize or anything, but rather to earn a good grade you had to do a specific amount of problems in a specific amount of time.
Now, arguably, addition and multiplication have to be memorized in some sort of way -- there isn't any real way to invent multiplication (to 7 year olds) other than just by doing repeated addition and storing the resulting value in their brain for later use.
But the problems start emerging in later grades -- when we start solving for variables or try to solve x + 2 = 3, there's no notion of exploration being presented. Rather, it's more like a history class where students are told information to absorb, steps and processes to remember, and math class becomes nothing more than the daily chore of memorizing some information.
Math is also very precise. There's only one real answer to the equation x + 2 = 3 (given no other information), and so math is precise. This leads students to look at the information they've memorized and instead of trying to draw dots or "make conclusions," and in the process, fail, they instead just try to get to the answer with the steps given, putting their brain on autopilot instead of actually thinking (which is very hard).
In the book Grit, Duckworth gives the example of a student struggling to identify what sort of rule or identity the equation below is.
$ 3 + 0 = 3 $
Essentially, the student keeps on trying (guessing) answers from the bank of answers that they know (it's gotta be either identity, associative, commutative, zero product rule) -- selecting one at random and just trying it out to see if they get positive reinforcement. The teacher takes this as "learning" (which it technically is considering AIs built on RL or reinforcement learning essentially do this x1000000), but fails to recognize that instead of actually learning the concept, the student is merely learning to associate the additive identity rule with a series of symbols with a 0 and a plus sign in it (which isn't bad per se, but in this case, the essence of the subject is lost - the student merely regurgitates (as many people call it) the fact and does not realize any significance).
Following on to this point, instead of trying to actually experiment with kids and have them fail until they learn something, our society instead fills their brains with information, some useful, some rubbish.
For example, the classic, the quadratic formula. Or the y = mx + b. These two things are merely means to an end, that is, a way to solve specific equations that appear in very specific situations. And I condone teaching these two concepts.
However, instead of actually teaching the ideas that are behind these two formulae/equation (quadratic equation is completing the square with variables, y = mx + b refers to the fact that a line is determined by two points), and the thinking required to create these ideas, we instead merely give our students these facts as something to memorize. Instead of giving them ideas and roadmaps, we drive them to the destination while they are free to watch TikTok on the whole journey there. I would also like to point out that all the identities and stuff I mentioned above don't actually matter until you get to the mystical land of group theory and algebraic structures, rendering them essentially useless for 2nd graders. No one cares. And if they do care, they can find their own way to learn this information.
To end off, here's an equation for you:
Students who are not willing to think critically} + Teachers only being able to teach state mandated (bullshit) curriculum) + An emphasis on the facts themselves rather than the methodologies and ideas that lead up to it + Useless information --> Students that function equivalently to calculators.
And don't even talk about word problems. Word problems are just a test to convert English (or any other language) text into math symbol equation stuffs.