Being a math major and an occasional math tutor, I'm somewhat coerced into studying and doing math on a daily basis. I spend the better time of my day working on math. And on the weekends, I spend even more time working on math.
There are infinitely many ways to do the same math problem. For most of math education, starting from pre-school up until the second or third year of undergraduate college, math is about solving a given problem with certain conditions, of which you need to find a certain value or function.
But after these years of problem after problem of which there is only one solution and all methods are similar to each other, "real math" beings to show a somewhat different perspective; it's not so much about getting the right answer in the end but about being able to appropriately talk about how you got to that answer. It's about explaining how each step taken to get to the conclusion, not so much a single answer, is justified.
This might be confusing to understand why there's such a change in math courses. One way to look at it is to understand something that's of great interest to math tutors and educators: how do people think about math and what are their thought processes that lead them to such conclusions.
This leads to two different ideas that most people don't often think about: math isn't a universal method for everyone, but rather an expression of how you as an individual think and understand things.
When tutoring different people in math, it's evident that everyone explains math to themselves differently. They have their own idea about what numbers are, what "space" means, and what happens when you change the value or function of a certain rule or system. In this way, math is a very personal philosophy, and as such, there is truly no way to understand just how well someone does or doesn't understand math based on school, homework and grades alone.
In terms of academics, a good grade means you take the time necessary to learn how the experts in your field see things. But the way things are done changes every generation, due to the very fact that everyone sees things differently. Sure, things like the quadratic formula and Pythagorean theorem have been around for thousands of years, but that's merely because they've been time tested to work and be helpful. But in extensive trigonometry and analysis, each formula and theorem is expanded to make sense in things like curved spaces and infinite dimensions!
At the end of the day, no matter how things are worded or look, it's merely an extension of what the human mind sees around all the time. Certainly, each mind has something unique to say about how the world around us, its many patterns and wonders work.
