Can I choke him now?
Photo by foxphotograpy (Edited by Brian)
I have a book sitting in front of me called, Introduction to Mathematical Thinking by Friedrich Waismann; the foreword is by Karl Menger, both of whom I admire greatly.
This book has opened my eyes to something very important about math education. And it’s not because the book is so good (which it is). It’s because after about my fifth attempt at getting through the book I finally realize what it has been that’s impeding my progress.
The impediment is the same impediment that has kept me from learning math and many other things that I have considered beautiful and important, but difficult in my life.
While reading the first chapter of the book again, it finally hit me. There is some really sloppy explaining at a very basic level that, if you are the kind of person that takes things seriously and wants to really understand the deeper meaning, throws a tremendous roadblock into one’s understanding.
This problem is so pervasive in the way so many things are explained in school, at work, or in the real world, that I’m sure you’ve come up against it time and time again. But nobody really calls anyone on it. Or at least not often enough. We let this problem slide by again and again, that we hardly notice it, yet it had a detrimental effect on society, probably since the first caveman tried to explain to his neighbor how to hunt the mastodon (if that’s what they hunted).
The problem is this:
The writers in question just do not have the consideration of looking at their explanation through the eyes of someone who does not understand things the same way they understand things (or doesn’t understand them at all). They’ll use a term, and they won’t explain it; they just assume you know what it means, maybe because, “everyone knows what that means.”
(Aside: Any time you hear someone say, “Everyone knows what that means,” not only is that a lie, but it pretty much shows that the person who says it hasn’t thought about it deeply enough.)
Part of this problem also consists of giving confusing (if not downright conflicting) explanations. Or giving an explanation, and then giving an example that contradicts or twists the explanation.
Have you ever opened up an instruction booklet for something you have bought, which teaches you how to assemble the product? You know that ubiquitous phenomenon in which they say, “attach widget B to the freeble?” And then they never tell you what the goddamned freeble is?!
Or when they say, “Attach widget b to the left side of the freeble,” and then they show you an illustration of widget B being attached to the right side of the freeble?
Doesn’t that make you want to just choke the moron who wrote that? Don’t you wish every idiot who wrote instructions had to put his name and telephone number on what he wrote so you could call him at 2 a.m. on Christmas morning when you were still trying to assemble the piece of crap from China that he didn’t care enough to explain well?
I do.
So back to Friedrich Waismann; no of course he’s not a moron. His Introduction to Mathematical Thinking is a classic math text, and deservedly so. But the hacks who wrote the blurb on the back cover that, “This book, presupposes no specific training in mathematics,…” need to be taken to the courtyard and caned. That wasn’t even true in 1951, when the book was first published and when some of the general population actually knew some math.
Unfortunately in the very first sentence on the very first page of the very first chapter of this otherwise fine book, comes a typical and unforgivably vague explanation and illustration concerning natural numbers.
In that sentence he says, “The numbers presented to us at the first stage of development development are then natural or cardinal numbers 1, 2, 3, 4…” In the same paragraph he goes on to talk about how numbers can be represented on the number line. At the end of the paragraph he says, “the numbers 0, 1, 2, 3… are assigned to the points thereby generated…” This is followed by an illustration of the number line from 0 through 5. He says these points are the images of the numbers.
Which numbers? One would assume he meant the numbers he’d been talking about the entire time; the only ones he had mentioned; the natural numbers. In the very beginning of the next paragraph the first sentence says, “what properties belong to the system of natural numbers?” And proceeds to tell about them, so he is obviously still only talking about the natural numbers. But he is showing an illustration of numbers from zero onward. His definition was the numbers from one onward. So which is it, wise guy?
Can I choke him now?
Am I being overly sensitive? Would you be? If someone who is considered one of the greatest experts on the subject cannot even make something clear to a beginner on the first page of his book which presupposes, “no special training in mathematics…,” how are we supposed to understand the rest of the book?
This is not an isolated case. This case doesn’t even end here. On that same first page when he talks about the system of natural numbers and their properties, he enumerates them. In point two it he talks about the concept of “betweenness” as it applies to the natural numbers. In point three he mentions there is only one exception — “The number zero does not have a predecessor.”
Excuse me?! You originally told me that the natural numbers were from one onward. Now you really must be messing with me. Where does the freakin’ number zero enter into it? Do the natural numbers contain zero or not? There is no way a thinking person can discern that from the text. (What is a natural number will be the subject of the next post.)
No wonder so many people, myself included, grew up thinking that math class was just one big “trick question.” It was like a secret that they would test you on but they wouldn’t tell you what it was. I think this phenomenon is exactly why most of us thought that.
Can I at least choke him now?
Now I don’t want to make anybody feel that this book is not worthwhile. I’m assuming it is, although I have not gotten through all of it yet, and don’t imagine that I will in the very near future: but I am not giving up. I just resent the fact that the entire math education industry is built upon the premise that the content is so goddamned important that the people who need to learn it take second place to it.
In the last half-century or so, since the “math wars” have ravaged the educational countryside, not only has the content been more important than the people it is taught to, but now the method of teaching has become paramount. It’s so important for these pedagogical wonders to tout their méthode du jour, that not being able to see the forest for the trees is a much too gentle metaphor for their arrogant and ignorant academic-mania.
Of course not every pedagogue is an orc from the dominion of “Constructivism,” or, “No Child Left Behind.” But any teacher worth his or her salary should be very skeptical of what comes from above, and very compassionate and thoughtful towards what and whom they impart to below. That trait alone is more important than any piece of content that anyone could inculcate on the young.
Waismann is a far better expositor than the droogs that write for the “Cosmo-Demonic Scholatic Textbook Factory,” et. al. The whole industry is rife with their half-information. That’s what makes it an industry. No (sane) person could read all of them, but anyone can read enough of them to notice vaguery after vaguery. How else could an entire industry not have noticed Waismann’s error for over half a century?
There is really no hope to change the industry. It would be nice to eliminate it. There has got to be a better way to educate people than to hand that mission over to corporations and administrations.
Part of the solution is to read individual authors who write for popular consumption. I’m not talking about the kinds of new-age junk that establishes “laws” that are just some wishful-thinking pseudo-science. I’m talking about people who understand the subject well, but are brilliant at explaining.
Some of the greatest authors of mathematical explanations were neither full-time math teachers nor mathematicians. But they were full-time humans, like Martin Gardner and Isaac Asimov. Some of the great authors still writing are not just mathematicians, but philosophers and linguists.
I could be wrong, but when I read the authors that I recommended above and will recommend below, I get the feeling that they are writing because they care about their readership. They are writing to you and me. They have a mission, and the mission goes beyond “show and tell just enough to get the little tykes through the standardized hoops.”
If you want to understand the “guts” of mathematics, some of the authors you can go to are (in no particular order):
- Keith Devlin – A great place to start with him is his monthly column, “Devlin’s Angle” .l
- Jerry P. King – The Art of Mathematics
- Rosza Peter – Playing with Infinity
- Lancelot Hogben – Mathematics for the Million
- Isaac Asimov - The Realm of Numbers
(out of print but you can dig up copies if you hunt hard enough)
- Bertrand Russell – Introduction to Mathematical Philosophy
- Georges Ifrah – From One to Zero
- Kaplan – The Art of the Infinite
- Apostolos Doxiadis and Christos H. Papadimitriou – Logicomix
- G.H. Hardy – A Course of Pure Mathematics
- Karl Menger – The Basic Concepts of Mathematics (This one is hard to find, but has some great information about why we need to clear up our terminology in math.)
Not all of those books will fit any one person’s style. Please don’t just read them on this list and go order them from amazon.com until you check them out at a library or bookstore first; they may not be for you at all. But they are a good starting point, especially the Asimov book, if you can get a copy
“The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.” -G.H. Hardy
continued on the next post – More on Math Terminology Mis-Explained.
