Teachers should explain this point to students. When I was in secondary school, the teachers called the practice tasks "problems". But they obviously already knew the answers, so the problematic nature of the tasks was entirely artificial. The name felt like they were gloating about being permitted to force you to do pointless work. Calling the tasks "exercises" and explaining the analogy to physical exercise would have been much better.
My teachers never called practice tasks "problems", and I still didn't like doing repetitive tasks like "do 20 long-form divisions". I think students do understand that doing something over and over will make you better at that task, and it doesn't motivate them much.
That's not the mental model I had, and I doubt I'm the only one. I believed knowledge to be something you either had or didn't have. I understood that it was possible to learn things incorrectly, and to forget things (hence the necessity of revision), but I had no appreciation of "fluency" as a concept until I was an adult.
As far as I understood it, the main limiting factor in education was teachers deliberately holding back knowledge. The main purpose of "problems", and especially of homework, was a show of social submission, designed to persuade the teachers to reveal the next secret. School one essentially one giant hazing ritual.
There's one memory that stands out as the greatest moment in my education. My school had Acorn Archimedes computers, which came with BBC BASIC, and we were allowed to program them during lunch breaks. One of the older students programmed a Space War clone. I very much wanted to write one myself, but I didn't know how to calculate rotations. The older student refused to tell me the secret, which made perfect sense according to my dominance hierarchy model of education.
I begged my math teacher for the secret, and to my surprise he actually explained how to rotate points using trigonometry. This felt like something incredibly subversive, where the teacher was giving me knowledge I hadn't earned. For this reason I still think of that teacher more favorably than any other, despite other teachers objectively doing more for me. I went on to write a superior Space War clone.
> The main purpose of "problems", and especially of homework, was a show of social submission, designed to persuade the teachers to reveal the next secret.
Of all the weird misunderstandings kids have about the world, this might be the weirdest I've ever heard.
I'm kinda fascinated, do you recall at what age you believed this? What was the context you grew up in (country, culture, school). Do you recall where the seed of this idea got planted?
Hard to say exactly, but I'd guess ages 11 to 16. UK, boys' grammar school (selective state school). When you're not aware of the Ebbinghaus forgetting curve it's the most obvious explanation for how education works. I made the common assumption that quality of education is measured by exam scores, and the most effective method of passing exams is obviously cramming. And if cramming is the best method of learning, it must be possible to learn an entire syllabus in a single day, so the teachers must be intentionally limiting you.
I feel this is the most traditional and natural model of education. It's like medieval guilds, where you had literal secrets that would be revealed only after sufficient demonstration of loyalty.
