To what depth can you learn math?

[okunyg]

What do people mean when they say learn math in depth, and not just memorizing formulae? How is "mathematics in depth" defined?

I want to make sure I learn "in depth", so I try to play around with the rules of math, for example multiplication with exponents: How come a^m*a^n = a^m+n?

Obviously, this example was very simple. But when it comes to more advanced methods: How are you sure that you're actually _learning_ and not memorizing?

 

[BSMSMSTMSPHD]

It sounds like you're confronting the age-old discussion of procedural knowledge vs. conceptual knowledge (see https://www.amazon.com/dp/0898595568/?tag=pfamazon01-20 for the archetypal description.)

For me (a graduate student) the difference between the two comes with understanding the "bigger picture". When I learn a new concept, definition, or theorem, it's not enough to just know it. I must understand its position within mathematics, and how it relates to other concepts, definitions and theorems.

Once these inter-relations are well known, it's not necessary to memorize anything, since the structure is what is really important.

As for the example you provided, the answer depends on how you define a, m and n. If they are positive integers, it's obvious that n copies of a multiplied by m copies of a will produce m + n copies of a. But what if the numbers are fractions? Irrational numbers? Complex numbers? What if a is a matrix, or a function? This "rule" may not always be true, depending on the situation.

 

[Office_Shredder]

When it comes to more advanced methods, you're pretty sure you're learning when someone can give you a problem you've never seen before, and you can do it.