I have a lot of sympathy for this approach, and of course you learn more by having fun and continuing, then being meticulous and stopping.
However if at each fork, you choose the fun road, you might get a long way until you run out of options (which may be fine).
But the terrible difficulty is necessary knowledge that is not explicitly stated. These gaps ("mathematical maturity") are difficult to even identify, let alone fill. Story time:
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In high school, I missed a week or two, and later got stuck on a problem in calculus. Discussed it with the teacher, and he eventually seemed to see my difficulty was, but wouldn't tell me, instead saying "you can work it out". I couldn't.
Years later, I started asking online, and people responded helpfully, but didn't actually help. I read wikipedia and math websites about it. Watched videos about it, that were interesting and gave a new perspective, but didn't help my particular issue. I looked it up in famous textbooks (Spivak, Hardy), still nothing.
Finally, I did all the problems in the derivatives section of Khan Academy, but when I came to this problem, it still didn't address my specific issue.
Thinking I had gaps in my knowledge, some known to me, and suspecting others unknown, I went back to even earlier material.
Some time later, I reviewed the problem again, and realized that the issue was a completely trivial part of limits - an earlier section. Which was what I missed in those 2 weeks of high school, all those years ago.
However if at each fork, you choose the fun road, you might get a long way until you run out of options (which may be fine). But the terrible difficulty is necessary knowledge that is not explicitly stated. These gaps ("mathematical maturity") are difficult to even identify, let alone fill. Story time:
---
In high school, I missed a week or two, and later got stuck on a problem in calculus. Discussed it with the teacher, and he eventually seemed to see my difficulty was, but wouldn't tell me, instead saying "you can work it out". I couldn't.
Years later, I started asking online, and people responded helpfully, but didn't actually help. I read wikipedia and math websites about it. Watched videos about it, that were interesting and gave a new perspective, but didn't help my particular issue. I looked it up in famous textbooks (Spivak, Hardy), still nothing.
Finally, I did all the problems in the derivatives section of Khan Academy, but when I came to this problem, it still didn't address my specific issue.
Thinking I had gaps in my knowledge, some known to me, and suspecting others unknown, I went back to even earlier material.
Some time later, I reviewed the problem again, and realized that the issue was a completely trivial part of limits - an earlier section. Which was what I missed in those 2 weeks of high school, all those years ago.
A nice maths teacher can save you years.