If your goal is to become a sculptor, it’s worth your while to try chiseling a wheel out of stone. Just for practice, you know. Reinventing the wheel isn’t always a bad thing. It gives you a lot of insight and skill in an area you’re starting out in. I believe that for an Autodidact, retracing past work can be invaluable. Studying any field in a freshman textbook, it’s easy to get the impression the material just materialized out of a magic hat. Like Isaac Newton sat down one day and penned calculus in one sitting. The textbooks provide a valuable perspective: a highly-polished, highly efficient presentation, often aimed at people who are only studying it because of a course requirement. But to gain a fundamental, deeper understanding, it’s absolutely crucial to have a grasp of how the material was developed.
In the past year, I finally rented a web-server and started teaching myself web programming. My first endeavor, I didn’t even know what a database was– I spent hours carefully programming a C program to generate romaji dictionary files, only to discover that servers aren’t usually happy about people uploading hundreds of thousands of text files 
The point is, I didn’t know squat about programming for the web. I could’ve grabbed a textbook, but that would go against the whole point: I write and develop things for fun, and last time I checked, elementary coding manuals were anything but! So I did some wheel re-inventorating. I coded my own forums from scratch, and even the wiki-like Connections Project. It would’ve been ten thousand times easier to download phpBB and wikimedia. But I wouldn’t have learned nearly as much.
By doing those projects from scratch– reinventing the wheel– I learned so much about how the internet functions! And the knowledge which I gained is emotional, not just intellectual. If I read some textbooks, even took a class in PHP or SQL, I’d gain the knowledge intellectually, but it wouldn’t have been etched into my very intuition like it has since become.
One thing about reinventing wheels is, your wheel is gonna really suck. It probably won’t roll straight, if it even rolls at all. Until today, my forums were an eyesore. I’m great at coding now, but I’m not great at web design, the art of rearranging and resizing things so they’re all purrty-looking. Maybe I’ll master that next The forums are still a long way from being as advanced as the common forum engines out there… they still suck, just not quite as much as yesterday! It might take years ’til I catch up. But when I finally do, I’ll have enough momentum, I’ll whoosh right past all the open-source forums 
That’s ok, though. Once you’ve developed the skills, you can easily refine your wheel and make it better. Today, for example, I completely renovated the forums, so they use the same layout as the blog. (Now that I look back on it, that seems so obvious it’s painful. Isn’t that always the way with good ideas?) Tomorrow I’ll do the same to my Connections Project. In the process of marrying forums and blog, I made dozens and dozens of modifications to WordPress to make it more efficient and better-performing. Most of these, I never would’ve spotted if I didn’t have the on-your-feet intuitive programming skill I got from building some projects from scratch.
One of the surprising things about mathematics curriculum at the university level, is that “History of Math” is usually a senior-level course, or even early graduate level. If you think about it, this class could be renamed, “Reinventing the Math”. It isn’t so “advanced” because it necessarily requires mathematical maturity. If they wanted, they could adjust the course to be doable by freshman or high school students. The thing is, most people who aren’t majoring in mathematics, just want to pass their math requirements and that’s it. In order to study and really benefit from the history of math, you have to be really interested in mastering math. Otherwise, it just looks like algebra except the notation is even worse! Thus the requirement for “History of Math” shouldn’t be so many years as a math major– it should be so much desire to learn the subject. The benefit of learning it is, you gain a much deeper understanding and appreciation for those theorems that come from the magic hat.
Chiseling your first project from stone isn’t always the best strategy. If you need to do something under a deadline, go with whatever you can get your hands on. If you have a long-term interest in solidly understanding something, your roots established deep in a profound understanding, then start from the beginning: start chipping!
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