Wednesday, January 14, 2015

Why Learning Math is Important (General)

A common question I get from tutees is, why learn math at all? Many others on the internet have provided satisfactory answers to this question, so I'll try my best to come up with a couple new points.

(A side note to those who are highly educated in humanities, social sciences, etc.: I am not claiming that every one of these benefits is unique to mathematical learning. Those disciplines are useful for critical thinking as well!)

Problem-solving practice

If you don't go into a technical career, the odds that you'll have to use algebra or calculus every day are slim. However, no matter what you do in life, you will have to know how to effectively solve problems. It's unfortunately difficult to practice and develop good problem-solving skills on their own---that often comes with experience---but doing relatively simple math problems is a good substitute.

How would that work? When you start a new videogame, the game doesn't immediately drop you into the final boss fight; you start by doing the tutorial instead. This is what schools are hoping to accomplish by giving you simple problems to solve: they may be in a weird format, and you may not be sure how they connect with day-to-day life, but you're being given them because these number problems are some of the simplest problems that are possible to solve. Furthermore, it's not a bad thing that algebra problems are disconnected from your real life---if they were, then there would be much greater punishments for failure to solve them. I'm certain you'll agree a broken friendship or broken arm is much worse than losing a couple of test points!

To summarize: think of grade school math problems as the tutorial level to real-life problems, and their disconnect from regular life as a protective safety net.

Increased ability to communicate abstract ideas

Math can be seen as not only a tool, or a scientific discipline, but also a language. Understanding and communicating mathematical ideas requires a set of symbols and vocabulary that people wouldn't learn just by going through life. Ideas related to math do pop up from time to time, and it feels great when you know what the answer is, and how to explain it!

Here's an example: let's say you and some friends are trying to get to a frozen yogurt place a block away. We call your current location A and the location of the frozen yogurt place B. Also, let's call the frozen yogurt place Froyomorphism for the sake of puns. Your friends, whose favorite colors are purple, green and blue, suggest the following three paths on the map:

Now you are asked to choose the best path. Because you are good at judging distances, you know that the purple and green paths are the same length, but the blue path is much longer. Could you communicate this concept to your friends without using math? Without using the word 'sum' or 'length' or evoking a visual proof? Probably, but it would be much harder. The point is: even if you're right, you may end up taking the blue path if no one can explain to Blue why the other two paths are shorter.

If two people have a shared vocabulary that can be used to talk about abstract objects, they can exchange information about what essentially amounts to different lines of thought. This is how people get smarter and better at problem solving.

Protection against being exploited

Most people think they are smart. However, as I'm sure you've figured out by now, not everyone is. Several people/institutions/etc. have realized this and use people's lack of mathematical awareness to make a living. Gambling is a classic example: also see the Monty Hall Problem or Bertrand's Box Paradox for situations where common sense can be deceiving.

However, not only can ill-meaning people use your unwillingness to think about mathematics (and academic prospects in general) to separate you from your money, they can twist information to separate you from your ideals and beliefs as well. Most people like the idea of experiments being able to prove, disprove, support, or refute ideas, but don't want to dig through heavily written academic papers to find the point. This is where exploitative people come in. If they can bank on the audience being too busy or unable to read the source material, they can make their audience believe whatever they want---even if it comes at the expense of the audience! See Flaws and Fallacies in Statistical Thinking for tons of real-world examples; Stephen Campbell explains this better than a blog post ever could.

The only way to protect yourself against this is to be able to read and analyze scientific papers without needing someone to tell you what you mean. In many cases, this requires some knowledge of statistics (math), experimentalism versus mathematical modeling and the implications, or what conclusions can be drawn from the data presented (logic, which is part of math).

Not looking like a tool on the internet

If you've spent any amount of time on the internet, at all, you may have come across someone who is angry at their opponents for not understanding "logic" and "reason." You may have seen someone make a statement along the lines of "that doesn't make any logical sense" without noting what the error is (or invoking a fallacy incorrectly). You may have seen someone who is incapable of understanding that a smart person may disagree with them, and who concludes that if someone disagrees with them, that person is stupid.

Judging by the relatively low proportion of people with bachelor's degrees in mathematics or philosophy, it stands to reason that very few of these people have had real training in formal logic. Someone with completely illogical arguments would have no way of knowing so (i.e., a special case of the Dunning-Kruger Effect.) On the other hand, someone who has studied higher-level mathematics can recognize what is and is not logically consistent, which affects how they act in everyday life as well. This makes everyday life a lot---a lot!---easier. (I could go on for days about this; but that's a story for another blog post... or twelve.)

Yet embarrassing oneself is often caused by a lack of empathy---what about that? Math has no relation to that, unfortunately. [Sad face.]

Concluding remarks

Oh, and talking about math is awesome, and we have the best jokes.

(Check back in the future for specific examples of how some topics you may have seen are actually used by mathematicians and scientists!)

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