But there's another flaw that gets overlooked most of the time, which is that we're raising kids to believe that "why are you teaching me something that you're not 100% sure I will need in my day-to-day life" is a sensible question, when it really isn't.
Outside of my 2-year stint in the game development industry, I never really needed most of what I learned about trigonometry in my day-to-day life. But that doesn't mean it wasn't useful.
Yes, we should make the subject matter more approachable to kids, but we should also try to shift the paradigm so that kids are more open to learning new things.
Agreed.
> I think adults should be able to justify why we’re using what boils down to the threat of force (if we’re honest) to make them sit in classrooms and listen to us.
Disagree. The justification for why they should learn $FOO may never be understood by a mind that we are teaching $FOO to.
There's good justification for learning to read, but not one that would be understood by a 6 year old.
There's similarly good justification for teaching Maths, but you'd be hard pressed to convince a 16 year old of the value in practicing abstract reasoning, using Maths as the vehicle.
Sometimes, the only good answer to give a kid is "you'll see the value when you're older".
Maybe awesome, but I doubt it.
I also disagree that there needs to be justification. I don't think students' minds work like that. What's needed is something different and probably many kinds of something different since there's many kinds of learners.
So far, a huge percentage of students are getting left behind when teachers and material fail to have a good answer.
With an insufficiently developed brain, there is no answer that leads to engagement or interest.
Sometimes you'll find yourself telling kids "How do you know you won't like it unless you try it?"
If you, personally, claim to have never told a kid that specific sentence (regardless of context), I have serious doubts that you actually have kids.
Sometimes engagement and interest only come after the kid has been forced through a little bit of it.
They are children; you can't always reason with them because they have not yet developed sufficient reasoning skills. Making the claim that reasoning is all you need to get children to do the right thing is plain nonsense.
> I also disagree that there needs to be justification.
Sounds like we're in agreement, after all? I also don't think there needs to be a justification for "You need to learn Maths". This is why I said an answer along the lines of "you'll understand why later" is all you can do when asked for a justification.
Well, I use other words bit that's my message anyway :-)
I think the truth is a lot simpler. Most kids won’t use trig in real life.
Advanced math came in handy just once in my life. My keys fell in the toilet, and I realized the best tool was a wire bent like an integral.
Later at university I complained about the lack of applications in the textbook, and my classmates became very upset. One of them responded, "we are mathematicians, we do not concern ourselves with applications."
Just look at software. It's undeniably useful with many applications. Still, some people treat it playfully and in an enjoyable way, they learn about algorithms they won't ever use, just for the elegance, read Knuth, even if reading some React handbook may be more useful for their day-to-day. There are more considerations than "but how will this make my employer richer?"
Especially when "useful" includes "to get a future job" or study something else much more useful and interesting later."
At some point I had made a small space ship and was able to make it turn around with the wonderful angle command [1]. However, I could not figure out how to make it move "forward" regardless of the angle.
I was also attending an after hours computer graphics club, mostly about Deluxe Paint, taught by a 20-something student (who much later went on to found a GPU company and got acquihired by ATI/AMD). He would help me occasionally, and in this case he took a tiny slip of paper and wrote down a couple of lines about sin and cos. No questions, no explanations, no gatekeeping.
Just like that I internalized this foundational piece of trig - later when it arrived in school maths it was easy and obvious for me. I had a practical application, but even more I think was because it started as a need I had, and when given to me, felt like a gift and an enabler.
Still much later I studied Seymour Papert's pedagogy and understood I had lived it. I consider myself fortunate.
But IMO that pushes you to making the curriculum more flexible and not forcing every student into the same sequence where math is the core IQ test for STEM. If a kid doesn’t find it interesting I struggle to justify forcing them.
Personally I also think trig and calculus are far inferior to statistics for most people. If you have an intuition for probability distributions, precision/recall, and a few other basic concepts, you’ll be guaranteed to apply them everywhere in your life.
Of course if you are interested in STEM then calculus must be available too, but it’s pretty specialized in practice.
Your suggestion is interesting but I am not convinced that a student would be helped by aligning the examples with their interests. I could see a student asking how trig relates to computer games and the example the LLM generates becoming much more involved.
I see no problem with the examples being boring. The people that developed these techniques had such fundamental problems to solve and the wonder to me is the human mind that came up with these methods.
All this to say, maybe we lack appreciation for the fundamental sciences that underpin every aspect of our modern lives.
The trouble is a lot of those practical examples fall into the, "why would I care category". I had a high school physics teacher who described his university antics, one of which included a funny story of a bunch of his friends climbing on top of each other to measure the height of a flag pole. I guess the profs got tired of dealing with students scaling flag poles because I was measuring the height of mountains on the moon at the same university a couple of years later. The thing is nobody really cares about the height of a flag pole, while only a few would care about the height of the mountains on the moon.
The reality is the interesting applications are much more involved. They either require a depth of thought of process or a depth of knowledge that isn't appropriate for a textbook question. Take that trigonometry in games example. The math to do it was in my middle school curriculum, but it becomes obvious that computer graphics is more than trigonometry the moment you try to frame it as an example. I had linear algebra in high school. That will take you pretty far with the mathematics, but it will also be clear that a knowledge of computer programming is involved. Even knowing how to program isn't going to take you all of the way because few are interested in rendering verticies and edges ...
And that is just the obvious progression of knowledge in a simple application. Physics itself involves buckets full of trigonometry in extremely non-obvious ways, non-geometric ways.
Rambling off-topic, but I remember being very impressed at how a uniform hanging rope makes a catenary [0] shape which is related to making strong structural arches.
So maybe if the students were somewhere where the class could design and make an igloo... :p
So your off-topic rambling isn’t off-topic at all :)
I'm in agreement with this point; the examples are boring, but that's not really relevant. After all, we're mostly talking about Maths ITT, not history or social sciences.
1. Some foundational study is needed before you get to the really interesting problems at a higher grade/level/school/university.
2. Who cares if they are boring? A spectacular facility to learn Maths which is demonstrated by high marks indicates better abstract reasoning skills, making it easier for specific trades to decide who is more suitable.
3. How will the kids know whether they like Maths or not if they skip trig in high school?
(Sidenote: Am I the only one who finds trig easy and everything else in Maths hard?)
There are much deeper flaws in this question:
1. That you only learn what you are guaranteed to use in everyday life. Education should leave you with choices. You learn trigonometry so you can later choose to do things that require it.
2. That everything you learn in school has to be something that is likely to be useful. When did you last use history or literature or art?
How many paving stones do I need to buy for the walkway I'm building in my backyard?
How far from the top of the roof should I start attaching this gutter so that I still have roof to nail it to 30 feet later?
How big of a ladder do I need to get to that branch I want to cut down in that tall tree?
Will I be able to get this couch up the stairs, around the corner, and through the door?
I do think there is pedagogical value in showing where these concepts can be used practically and the advantage of LLMs is that you can transform the examples to what you're actually interested in. For example the Red Blob Games series on A* pathfinding are really good at showing how Dijkstra and graph traversal algorithms work, for a use-case (video games) that is appealing to a lot of nerdy people.