Subsections

Epilogue

There is a lot of jargon introduced in this book: symbol motion, chunking, fusion, morphing, subimages, and so on. As far as I know, all of them are new. I wrote this book by carefully examining what I do inside when I solve equations. I did my best to put in writing how it happens, using language you are already familiar with. Sometimes I came across concepts that just didn't seem to already have labels neatly defining them. I think each of them have probably been defined somewhere, by someone. But I haven't found them in the context they are in this book. Even if I had, these concepts are not (yet!) well known. In these cases, I invented terms to describe them, and gave them (hopefully) useful definitions.

Perhaps as a result of how they were ``invented'', the divisions among these terms are not sharply defined. Symbol motion and fusion do seem to be pretty different concepts, and once you are familiar with them you will never confuse the two. But it is not a great leap to see each of those as a kind of morph. Morphing as described in chapter is something you apply to an expression or a small group of symbols, but there is also a more general sense of the word. The general sense implies a systematic change in the structure of an image. In that larger sense, a symbol motion is just a particular kind of morph of the whole equation. Fusion is a morph of an image of an expression into another expression that takes less ink to write. You can't get too loose with this. The mathematical logic must always be preserved, no matter what, or we forsake the whole point. But assuming we are smart enough to stay grounded there, we could summarize everything in this book with the sentence ``You can solve equations by visualizing them and morphing that image''. We introduce symbol motion, chunking, and so on because they are sub-patterns of morphing that are commonly useful and relatively easy to learn. It's good to occasionally remember that these are all simplifications of what is going on.

You can invent your own patterns. If you practice the techniques in this book, and pay attention to what you are doing and how, you will start to notice subtle things about how your mind works. You may notice that you can do something, but aren't sure how you do it. Maybe you can factor certain polynomials easily, for example, and you usually think nothing of it because it seems normal to you. Then one day you may think about it, and wonder how you can do that, and surprisingly you won't immediately be able to say. If you keep looking at it, you may eventually learn how. Maybe it's something you can articulate and maybe it's not. Sometimes things that happen inside of you are like that. It will be known to you, though, and it may open your awareness to how you do some things mathematically. These will be things you already knew, you just forgot you knew them.

If you work with equations in a way that you find yourself executing a particular procedure over and over, it will start to become more automatic. It could be almost anything that is a repeated pattern. Maybe you do a particular sort of symbol motion, followed by a fusion of the chunk and the symbol beside it, followed by a morph. Or it could be something much simpler. Whatever, you may find that you are using this pattern in many different equations and different situations. At some point, instead of seeing it as a series of operations, you will sense it as one ``thing''. It will be one mental operation that you apply to the situation in one fell swoop. That is exactly how I came up with symbol motion. I just observed what I was doing, and noticed that I was doing something similar in many different circumstances. I formalized it a bit, slapped a label called ``symbol motion'' on the concept, and wrote a chapter in this book detailing it. You can do the same thing. Maybe it will be something that others can use, that you can describe and communicate to people in a way that they can adopt it. Or maybe it is something individual to you, that you find valuable but others won't really find useful because they are not you. It is kind of interesting how it works out.

Everything in this book is just a suggestion. In the community of people who write software using the Perl programming language, there is an expression, TMTOWTDI, pronounced ``tim toady''. It stands for ``There's More Than One Way To Do It.'' They say that because the language is designed so that there are often several different, equally valid ways to code an algorithm. The idea is that you do it the way that works for you in that situation. The same applies here, only to a greater degree, because what goes on between your ears is much more expressive than something like a language can be. Who decides what the ``correct'' way is? You do. If you are following some of the instructions in this book, and one day you notice that there is another way to do something, who decides if it is all right to do that? You do. This may seem a weird way to learn math. Maybe it is, but you have the ability to do it this way.

Thank you for reading this book. I hope you enjoy using it as much as I enjoyed writing it.

Exercise

Find something you can do mathematically and which most people cannot do. Teach someone how to do it.