Posted in Educational Reform, Math

Math Is A Creative Endeavor

Many of us come away from our compulsory math education with the impression that math and creativity have nothing to do with each other, that math is the epitome of convergent thinking: there is just one right answer, and just one way to find it. At best, this is only half true.

Mathematical statements are precise, and have binary true/false values (if they are well-defined), but those of us who enjoy math see it as a creative exploration of logical relationships. The answers in math may be convergent, but the ways of arriving at them are infinitely divergent.

For example, consider the question “what is 5 + 5?” Ostensibly, it has just one answer, 10. But how many ways are there to arrive at this answer? Some people might do it by counting. Others might do it by multiplying 5 times 2. Others might do it by looking at their hands. Many probably know it by rote memorization. In terms of the inner cognitive process of computation, there is literally no limit to the variations of thought involved even with such a simple calculation.

Now consider something a bit more sophisticated (yet still relatively simple), like a proof of the Pythagorean theorem. It has been known at least since the Babylonians, definitively proved at least since Euclid, yet over the centuries hundreds of proofs of the Pythagorean theorem have been recorded, including an original one by president James Garfield. The question “how many ways are there to prove the Pythagorean theorem” is a classic example of divergent thinking in action.

If the idea of math as a creative endeavor seems surprising to you, don’t worry, it’s not your fault. It’s simply a result of outdated teaching methods. When I teach math, I do it in a way that engages both sides of your brain, so that it is actually engaging, interesting, satisfying, and yes, creative.

Posted in Books, Physics

Feynman On Memorization

Excerpted from Feynman’s Tips on Physics, by Richard P. Feynman:

It will not do to memorize the formulas, and to say to yourself, “I know all the formulas; all I gotta do is figure out how to put ’em in the problem!”

Now, you may succeed with this for a while, and the more you work on memorizing the formulas, the longer you’ll go on with this method – but it doesn’t work in the end.

You might say, “I’m not gonna believe him, because I’ve always been successful: that’s the way I’ve always done it; I’m always gonna do it that way.”

You are not always going to do it that way: you’re going to flunk – not this year, not next year, but eventually, when you get your job, or something – you’re going to lose along the line somewhere, because physics is an enormously extended thing: there are millions of formulas! It’s impossible to remember all the formulas – it’s impossible!

And the great thing that you’re ignoring, the powerful machine that you’re not using, is this: suppose Figure 1 – 19 is a map of all the physics formulas, all the relations in physics. (It should have more than two dimensions, but let’s suppose it’s like that.)


Now, suppose that something happened to your mind, that somehow all the material in some region was erased, and there was a little spot of missing goo in there. The relations of nature are so nice that it is possible, by logic, to “triangulate” from what is known to what’s in the hole. (See Fig. 1-20.)

And you can re-create the things that you’ve forgotten perpetually – if you don’t forget too much, and if you know enough. In other words, there comes a time – which you haven’t quite got to, yet – where you’ll know so many things that as you forget them, you can reconstruct them from the pieces that you can still remember. It is therefore of first-rate importance that you know how to “triangulate” – that is, to know how to figure something out from what you already know. It is absolutely necessary. You might say, “Ah, I don’t care; I’m a goodmemorizer! In fact, I took a course in memory!”

That still doesn’t work! Because the real utility of physicists – both to discover new laws of nature, and to develop new things in industry, and so on – is not to talk about what’s already known, but to do something new – and so they triangulate out from the known things: they make a “triangulation” that no one has ever made before. (See Fig. 1-21.)

In order to learn how to do that, you’ve got to forget the memorizing of formulas, and to try to learn to understand the interrelationships of nature. That’s very much more difficult at the beginning, but it’s the only successful way.

Posted in Books, Educational Reform, Math

Schneider On The Personalities Of Numbers

Excerpted from A Beginner’s Guide to Constructing the Universe, by Michael S. Schneider:

It’s a shame that children are exposed to numbers merely as quantities instead of qualities and characters with distinct personalities relating to each other in various patterns.  If only they could see numbers and shapes as the ancients did, as symbols of principles available to teach us about the natural structure and processes of the universe and to give us perspective on human nature.  Instead, “math education” for children demands rote memorization of procedures to get one “right answer” and pass innumerable “skill tests” to prove superficial mastery before moving on to the next isolated topic.  Teachers call this the “drill and kill” method.  Even its terminology informs us that this approach to math is full of problems.  It’s no wonder countless people are innumerate.  We’ve lost sight of the spiritual qualities of number and shape by emphasizing brute quantity.

Posted in Math, Tips for Students

Why Memorization Is Worthless In Math

Well, not quite, but almost. To understand why, a distinction between “memorizing” and “remembering” is in order.

When you use something in math, like a formula, enough times, you will remember it. Remembering comes from familiarity.

Remembering is part of learning. You can’t claim to have learned something unless you can remember it. However, learning also implies understanding the meaning of something, as well as how it is related to other things. As you use something repeatedly, you will come to understand what it means and how it is related to other things.

“Memorizing” means repetitively exposing yourself to something *for the express purpose of committing it to memory*. Memorization involves time spent dedicated to remembering something, and does not imply understanding meaning or relatedness.

Now, in math certain facts and formulas are used more frequently than others. Generally speaking, the more often you use something, the more important it is. Therefore, the more important something is, the more likely it is that you will remember it, as well as come to understand its meaning and relatedness, because you will be using it more frequently.

Because all of the knowledge available to humankind is instantly accessible to all of us via the Internet, anything worth memorizing is worth looking up whenever you need to. If you look something up enough times, you will *automatically* remember it.

Take, as an example, the formula for the area of a circle:

 circle

and the formula for the volume of a sphere:

 sphere

The formula for the area of a circle is fairly important, and will come up repeatedly in many different applications. Therefore, if you study much math, you will be using it over and over again, and don’t have to bother memorizing it. You can just look it up whenever you need to, and eventually, if you look it up enough times, you will remember it and not need to look it up any more.

The formula for the volume of a sphere, on the other hand, is not as important (depending on how far you progress with your math studies, at least). You probably won’t use it as often, and therefore any time you spend memorizing it will be wasted. You can just look it up whenever you need it, and if you don’t end up looking it up enough times to remember it, then you didn’t need to anyway.

What it comes down to is this: given any fact or formula you encounter, you will either need to use it again in the future, or you won’t. If you need to use it again, you can look it up again, until you don’t need to any more. If you don’t ever use it again, then it wasn’t worth memorizing.

The one exception I can think of is the multiplication table. Taking the time to memorize the multiplication table up to 9 x 9 = 81 will definitely pay off for anybody in terms of the ability to quickly and easily do mental arithmetic.

Posted in Math, Teaching & Learning

Learning By Instruction Vs Learning By Discovery

Learning by instruction is like following along while somebody connects the dots for you. Learning by discovery is like connecting the dots yourself.

Learning by discovery requires context, and builds context for further discovery at the same time. It grows, and grows out of, a solid foundation of understanding.

Instruction can lead you to new and unfamiliar places, but there is no guarantee that you will be able to find your way back to them.

Instruction typically requires 50 to 100 exposures in order to reliably stick. On the other hand, you only have to discover something 3 to 5 times to remember it for the rest of your life. Take, as an example, the beloved quadratic formula that so many of us memorize in school:

quadratic equation

This formula is typically taught by instruction, so that you can get to work putting it to use. Various mnemonic devices and memorization schemes are employed. You might even remember it for a long time. But can you derive it? If you forgot it, could you deduce it again from basic principles? If somebody asked you what it is for or why it works, could you explain it? Could you teach them to do the same?

When you learn by discovery, you not only remember what you have learned, you also gain the ability to re-teach it.

Another difference between discovery and instruction is that instruction needs a teacher, but discovery doesn’t. Discovery is something you can do for yourself and by yourself.

The two are not mutually exclusive, however. In fact, the best kind of instruction is assisted discovery.