Does it make sense to learn mathematical concepts as you encounter them rather than in a fixed progression?

Question

I understand the fundamentals of algrebra, but have very limited knowledge of geometry and trigonometry. I wish to learn calculus at this point.

Is it reasonable to begin learning calculus, and learn these other concepts as I encounter them rather than learning the prerequisites in the standard fixed progression (i.e. Algebra I -> Algebra 2 -> Geometry -> Trigonometry, etc)? How difficult will it be to learn these concepts without the prescribed linear progression?

Answer 1

Here's a description of how Peter Scholze (a Fields medalist) learns math:

At 16, Scholze learned that a decade earlier Andrew Wiles had proved the famous 17th-century problem known as Fermat’s Last Theorem, which says that the equation $x^n + y^n = z^n$ has no nonzero whole-number solutions if $n$ is greater than two. Scholze was eager to study the proof, but quickly discovered that despite the problem’s simplicity, its solution uses some of the most cutting-edge mathematics around. “I understood nothing, but it was really fascinating,” he said.

So Scholze worked backward, figuring out what he needed to learn to make sense of the proof. “To this day, that’s to a large extent how I learn,” he said. “I never really learned the basic things like linear algebra, actually — I only assimilated it through learning some other stuff.”

So, I'd say that this backtracking approach to learning math is fine if you enjoy it and it comes naturally to you. But be sure to aim for true, deep understanding of the topics you're interested in rather than rote memorization.

Answer 2

Calculus mostly deals with functions: that is what their derivatives are amd what their integrals are.

Whereas trigonometry deals with triangles and angles between two lines, and geometry deals with circles lines, ellipses, again triangles. As these are something one has encountered in life as opposed to functions geometry and trigonometry are easy to see why they could be useful what their applications are.

So it is advisable to study trigonometry and geometry. All illustrations in calculus are mainly using functions of trigonometry and geometry (tangents of curves). In this order understanding would be easier.

Answer 3

How difficult will it be to learn these concepts without the prescribed linear progression?

It's not really a linear progression, not at all. All of these subjects actually deal with the same thing. Mathematics is all connected.

Algebra (as it's defined by educators), geometry and trigonometry are taught separately because someone down the line thought it is the best way to teach. It's not. Those subjects just offer different methods/instruments to solve the same problems.

However, you don't really need to know everything that's taught in geometry/trigonometry by heart. Mostly it's all just some particular theorems which are hardly ever used in practice.

What you really need is to have a good grasp on the connections between all of these subjects. For example, all of these complicated trigonometric formulas are derived with the help of algebra together with a small set of rules, connecting $\sin$ and $\cos$.

Any geometric problem is eventually reduced to an algebraic one, and any algebraic problem (for example, solving an equation) can be represented in geometric terms. Not to mention, that $\sin$ and $\cos$ are usually first defined through their geometric meaning.

An example: the general solution for a cubic equation can be defined in radicals (with lots of square and cubic roots) or it can be defined in trigonometric form, which looks much neater.

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Now, in practice, if you want to study a new subject and are not sure about the prerequisites, just start learning (through textbooks or some other way) and see if you encounter some unclear concepts or methods you don't know. Then use the internet or the help of a tutor to see what you need to learn/remember to understand them.

That's what I do when I need to solve an unfamiliar problem now that I am not longer a student: I just try and search for new methods if needed. It's a fine way to learn, but then again, I do have years of formal education behind me, which helps to recognize the connections I have mentioned.

Answer 4

I'd like to add that learning is not linear no matter what. Even if you learn things in a sequence at the end of the sequence there might be some gain (sometimes huge ones) in revisiting older stuff later on with new conceptual tools and a broader understanding of a field.

There's no obvious path to optimally learn, but I'd say there is at least one mistake that is crucial to avoid: don't hold yourself back thinking that if you don't go deep enough or devote enough time to something than it's not worth looking at. Allow yourself to peek and spend some time in multiple direction. You don't need to learn all there is to know about trigonometry or be highly proficient in using prostaferesis formula to benefit from it. Sit down one afternoon and do a bit of studying and practice some simple trig equation and definition. When you get bored or concerned with time pick up your calculus and go ahead with it. At some point you'll find an integral of a cosine and you will benefit a great deal from just having seen a cos(x) function once. Also, remember that stopping learning a specific tool today doesn't stop you from picking it up again later on.

The more you advance, the more you'll be able to discriminate between when it's time to go deep and what you want to just skim over and use.

Answer 5

Having a personal goal in mind very motivating, and backtracking to learn what you need for specific goals as they come up is fantastic way to learn... as long as you have the time for it.

If you're in a hurry to get something done, though, it can be incredibly frustrating to interrupt your progress for long periods of time.

If you're trying to keep up with your peers in calculus, it can be incredibly frustrating if you have stop, go back, and learn some basics while your peers are pushing forward. Then you have to rush to catch up.

Nonetheless, I think goal-oriented learning is generally much more efficient. The benefit of having a broad base of basic knowledge in a wide variety of subjects is that you never have to backtrack too far.

To answer your specific case: yes, you can learn the fundamentals of calculus without knowing too much about geometry or algebra. You will be at a disadvantage, mostly due to lack of practice on algebra and geometry problems, but it will be easier to work though that than to practice algebra and geometry first with no specific goals in mind.