Solving the Spider Problem

A few days ago, while searching for tweets containing the word 'math', I came across this problem:

Who wouldn't attempt to solve it after that commentary? Poor spider, though. That must have been tiring.

I'm certain any readers would want to attempt this for themselves as well, so my solution (and the accompanying story!) can be found after the jump break.

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!)

Math Test-taking Strategy

Math testing strategy varies a bit from the strategies that would normally be useful for classes requiring a lot of rote memorization. This is mostly because, in addition requiring a student to interpret and regurgitate information beforehand, math tests also involve a performance element that tests short-term problem solving ability. You may recognize this as an important skill to have! Before we list some specific ways to prepare, keep in mind that:

• Mathematical ability can be improved. Some people have a tendency to give up on math if they aren't good at it immediately. However, if you spend more time on math than your classmates; whether it is going to math camp, working with a private tutor, thinking about outside problems, or figuring out how things work; you will get better faster. Nobody comes out of the womb being able to solve every type of math problem. It's like playing a musical instrument or learning to dance: the more practice, the better.
• Getting an A in most classes requires a level of understanding that is not taught in the class. In American schools, an A grade is meant to signify that a student is going above and beyond what is required, even if it looks like all of the testing material is being taught in the class. There is often far more to the material! For example, a lot of students understand the general material being taught, but get slammed on small mistakes such as minus sign errors. Catching and being aware of these errors is something that the student must develop on their own, and it is hard to explicitly teach. Small things like this are often the difference between an A and a C.

Got that? Good! If you're studying for a test and aren't quite sure what to start with, there are several 'levels' of understanding the material, which, for your purposes, we'll express in three separate categories:

• Basic understanding. You're at this level if you can read and understand everything in the textbook chapter, and know how to do the problems that aren't word problems.
• Familiarity. You're at this level if you understand what the answers should 'look' like, why the answers 'look' that way, and how to fix something if you've made a mistake. To get to this level, you have to be observant and look for patterns in the work you're doing. Knowing where mistakes can arise in a certain type of problem is very powerful!
• Creative application. If basic understanding is like knowing how to get home, and familiarity is knowing how to get home even if you've taken a wrong turn, creative application is like getting home by parkour. You're at this level if you understand the technique so well that even when it's not mentioned, you know when it has to be used. This type of understanding is the most important for word problems.

Let's use (scalar) multiplication as an example. Someone would have basic understanding if they could multiply 45x25 on paper (or in their head! Can you?). They're familiar with how multiplication works if they can explain why the answer cannot be 725, or 329670, or 2621. Lastly, they will have mastered multiplication as a concept if they can use it to solve problems such as 'how much do 45 vending-machine gumballs cost'? or use multiplicative identities to prove the exponent rules.

So, in a nutshell, if you understand the book and can do all the problems, you're still not guaranteed to do well on tests. Nooo! How can this be?

The primary problem is that very few books go into detail on how things work and how to recognize mistakes. Yet recognizing where mistakes happen and how to fix them is vital---not to mention that knowing how methods work is the only way to understand word problems! Here are a few helpful things you can do to improve your math testing ability:

• Know exactly where your abilities are for each type of problem. Can you do mental multiplication? Can you do basic algebra problems? Can you prove that L^2 is complete? Being honest about where your weaknesses are makes it much easier to conquer them. Needing more practice on a specific type of problem isn't a bad thing, and you'll save time if you focus on only the hardest problems.
• Develop 'sanity checks'. 'Sanity checks' are what I call pieces of information you remember to check whether you've made a mistake in the problem. Using the multiplication example above, a good example of a sanity check would be 'an odd number times an odd number cannot be an even number.' This helps build your familiarity with the material and could save you from losing tons of points.
• Build the test. Even if you don't know which specific problems will be on the test, you can generally work out how many of each type of problem will be on the test. This will help you direct your attention towards whatever will win you the most points. For example, you might be okay with everything on the test except one very, very hard type of problem: would it be better to focus on figuring out the hard problems, or making sure you don't mess up on the moderate problems? This depends on how many of the hard problems will be on the test.
• Talk with friends. We don't mean about videogames. There aren't a lot of ways to develop creative thinking for mathematics other than 'think about math a lot and try to come up with and solve math problems outside of school', but this is one of the more fun ones. Your friends may have some insight about math that hasn't occurred to you yet, or you may be able to solve a hard problem by working together. In any case, talking directly to people who know more than you will teach you a lot.
• Look for patterns. If you don't want to talk to your friends about math, or have some pride about developing things on your own, remember that math is all about finding and exploiting patterns. Once you've found a pattern, try to figure out where it comes from. This line of thinking often leads to developing newer, faster ways of solving problems. See our mental math post for some simple examples of pattern exploitation.
• Hire a private tutor. We said earlier that talking to people who know more than you will improve your skills very quickly: well, private tutors know a lot of math and they can teach you a lot about math. There's no shame in needing one! If someone is very good at the guitar and wants to become even better, they hire a private guitar teacher. The same is true for math. This is how you should see a private math tutor: someone who can help you improve very, very quickly and understand mathematics far beyond what you are taught in class.

These are all things you would do before the test happens (we've left out the obvious "read the textbook and do the problems" advice.) While the test is happening, try to:

• Read the test beforehand and do the easiest problems first.
• Temporarily improve performance by stretching, chewing gum, drinking caffeine, or listening to energetic music.

Now you should be focused and in the moment!

Lastly: even if progress seems slow sometimes, or you're having trouble catching up, don't feel hopeless. If you spend time thinking about how the methods work, you will improve.

Good luck, and happy testing!