Once in sixth grade mathclass, I misplaced a graded quiz on fractions. Mrs. Locatelli called me to her desk while we were doing work on our own, and started discussing what we were gonna do, because by sheer misfortune, she had managed to delete the grade as well. The situation looked difficult, until I reminded her of my grade on the quiz: 2 out of 50. "Oh, then I don't need to see that," she said, "I'll take your word for it." Nowadays, I'm a 3rd year math Ph.D. student at the Ohio State University, and to pay my way, I teach the stuff.
The problem back in sixth grade wasn't that I was naturally bad at maths. As a matter of fact, noone is. It's just that I thought of myself as terrible at it. So what made the change? That's an interesting story, actually.
One day I saw a very interesting-looking paperback in my best friend's parents' room. "Develop Your Psychic Abilities", by Litany Burns. To a young preteen, that book may as well have been titled "The Coolest Thing Ever Written In The Whole History Of The Universe". My friend's dad was a professional psychic hotline operator, lending an air of credibility to the literature he read. I was too embarrassed to ask about it, so, being a little shit, I snatched the book while noone was watching.
I could write a whole separate article about the adventures which followed, but to make a long story short, it opened my eyes to the idea that I can teach myself things, and that it's awesome. The autodidact within me was born. (I had already taught myself some things before, like the BASIC programming language when I was no taller than your waist, but it had never been about reading books!)
At the same time, I was also interested in Dungeons and Dragons. Psionicists show up in this game, with the stereotype that they're brainy and that their psychic powers are directly correlated with their intellectual powers. Consequently, a D&D psionicist will always be studying things like abstract math and logic. Think "Vulcans" from Star Trek. This further stirred up the sudden love of knowledge in me which, a mere year ago, had lain dormant.
I started studying anything that seemed particularly abstract or arcane. That quickly lead me to Euclid's Elements. The Elements is the second most widely read and widely sold piece of literature in the world, second only to The Bible. It is the original geometry textbook, written thousands of years ago and still read today. Unlike modern public school mathbooks, written with the assumption that the reader is stupid, Elements is written with the assumption that the reader is intelligent. Unlike modern texts which fail utterly in their pathetic attempts to be "relevant" or "hip", the Elements is timeless. I took one glance through the almost mystical-looking illustrations, and I was hooked.
While it's tempting to reap the praise of being a child genius or something, the fact is, I really didn't understand the Elements at all. The first proposition-- instructions for how to draw an equilateral triangle-- made sense (after reading it a million times), and that's about it. But that one proposition blew my mind, the way the instructions were laid out so logically and beautifully, and then how the guy proved that the construction worked. It was my first taste of the Olympian-Nectar which is formal proof, the real heart and soul of mathematics.
I didn't understand Euclid's works, but I sure loved flipping through the pages and looking at the pictures. It was like a game, trying to find just how amazing an illustration could be. I took a ruler and a compass and some paper and I started making geometric figures of my own. I couldn't understand what Euclid was doing, so I was doing it myself! Of course, I failed to discover anything, but what happened there was magical: my self-image metamorphosized from that of someone who sucked at math, into that of someone who was good at it.
I wasn't born a mathematical genius. It's just that when we get a self-image in our heads, we tend to make it real by any means necessary. This has proven true in my life so many times, I'm convinced if I could just solidify a real, genuine "multibillionaire" self-image, and really believe it, it would materialize. Once I was drawing those geometric figures and emulating Euclid, people started seeing me as a kid who was good at mathematics. It was an illusion, but over the next few years, it was set to become real.
My parents bought me some math books for my birthday that year. I got a precalculus textbook and a couple calculus ones. I devoured them, along with an ancient algebra textbook we had lying around in the house. In the precalc book, I didn't understand everything. I didn't understand trigonometry hardly at all. If I had to take a midterm on the stuff, I'd've failed it, but what was important was that I was "learning the language". Most people who take precalculus in college, they don't read the textbook at all, besides doing assigned exercises. Reading it made me feel really smart and intelligent. And even if I didn't understand something, I was picking up the general "style" of mathematical texts. Symbols which once looked like alien writing, were transformed, if not into symbols I understood fully, at least into symbols I'd seen before. Symbols I could at least pronounce.
Conventional wisdom in mathematics says, you have to always build on what comes before. I found this to only partly true, and more than half the time, false. It's an example of ideology, which pollutes knowledge. But since I was self-teaching, I escaped the misleading ideology. There is certain truth to the ideology, but, in mathspeak, it's a "local truth", which means that it generally holds true within a section or a chapter. But often you can skim one chapter and jump right to the next and you're fine. (You can go back and pick up the hard stuff later, when your mathemuscles are bigger)
When my parents took me to interview for a radical fundamentalist private Christian highschool, we sat down with the lady who would become my ninth grade algebra teacher. She asked me if I had any questions about algebra. At the time, I was really struggling with the how the quadratic equation works. I was good at solving linear equations, it's just a matter of "keeping the scales balanced", but there was a certain "trick" to quadratics (completing the square), and I didn't have the mathematical maturity to grapple with it. I asked her about it. She showed it to me, but having a person explain it in person no longer added any new light that the books did not. I was sufficiently liberated from the crutch of teachers, that I would never again need them to translate texts for me. (At least not until I started getting into real journal articles in grad school!)
Literally days after the interview, I suddenly "got" it. The process of completing the square suddenly made sense. And once I "got" it, it was so simple I was amazed that I had been stumped for so long. Suddenly, I didn't even need the quadratic formula to solve quadratic equations, I could do it manually. But it wasn't because of some natural born talent or anything. Completing the square was a big ol' dumbbell that I'd been straining at forever with no progress. But meanwhile, I was also training with smaller dumbbells-- reading ahead, I mean. Then one day, with muscles enhanced by the smaller weights, I returned to my old nemesis and suddenly found I could lift it with ease.
I can't emphasize enough the importance of getting a textbook which treats you like an intelligent person. Anything in the undergrad section of a college bookstore will be trash, because it's written for people who are only studying it because they have to. Also, this might sound a little shallow, but I think it's important the text should have a professional look and feel to it. That rules out most textbooks with their cheesy covers, always making some futile attempt at "relevance" or "hipness". Look, it doesn't matter what's on the cover of a calc book, people are gonna think it's an arcane tome of mystic spells-- so you may as well make it look like an arcane tome of mystic spells! At least then it's impressive and cool, and I even think you learn more from it, just because you somehow take it more seriously. Mom and dad did a great job with my birthday gift, because one of the calculus books they got me was Swokowski's Calculus, the Classic edition.
I didn't know it at the time, but Swokowski is actually one of the most well-respected calculus texts around. Compared to some of the trash I've seen undergrads forced to buy here at OSU, it's like heavenly script.
That interview with the future algebra teacher took place a while before the school season would actually start. The next time I saw her, I was well beyond the quadratic equation. Once my mathematical maturity was good enough to swallow completing the square, it was good enough for me to rocket through most of basic calculus almost nonstop. Not much in calculus is actually fundamentally harder than solving quadratic equations manually, it's just more intimidating. By being a self-teacher, I was able to pierce right through the illusionary intimidation factor.
It didn't take very long before I knew more mathematics than anyone in that cheap crappy private highschool. Rather than praise me for teaching myself the rudiments of analysis, it actually became a sore spot in my already very poor relations with the teachers. In my article "You might be an autodidact if...", I jokingly wrote that you might be one if you tuned out your highschool algebra teacher to study calculus. But actually, that line was based on my real life experience! I'm one of a very select few who can boast that their ninth grade principal confiscated a textbook from them.
The Summer after ninth grade, my math autodidact days came to an end as I started taking calc classes at Mira Mesa Community College (in California, the state will pay your tuition if you take college classes as a high schooler). It was a temporary end, as I would start teaching myself far more advanced mathematics years later while I was in the Air Force. But that's a story for another day ;)
FURTHER READING
I wrote an article with some hints about being better at mathematics. Five Ways To Be Better At Math.
Read about self-teaching in general at my article, Autodidact: Be A Self-Teacher. There are many lifelong advantages to becoming a self-teacher. Everything suddenly becomes a whole lot easier.
Should rote memorization be used in mathematics? The knee-jerk response these days among math educators is a resounding "NO", and I used to agree, but lately I've been really questioning that. Sometimes, some rote memorization is good. Read more in Rote Memorization In Mathematics.
The problem back in sixth grade wasn't that I was naturally bad at maths. As a matter of fact, noone is. It's just that I thought of myself as terrible at it. So what made the change? That's an interesting story, actually.
One day I saw a very interesting-looking paperback in my best friend's parents' room. "Develop Your Psychic Abilities", by Litany Burns. To a young preteen, that book may as well have been titled "The Coolest Thing Ever Written In The Whole History Of The Universe". My friend's dad was a professional psychic hotline operator, lending an air of credibility to the literature he read. I was too embarrassed to ask about it, so, being a little shit, I snatched the book while noone was watching.
I could write a whole separate article about the adventures which followed, but to make a long story short, it opened my eyes to the idea that I can teach myself things, and that it's awesome. The autodidact within me was born. (I had already taught myself some things before, like the BASIC programming language when I was no taller than your waist, but it had never been about reading books!)
At the same time, I was also interested in Dungeons and Dragons. Psionicists show up in this game, with the stereotype that they're brainy and that their psychic powers are directly correlated with their intellectual powers. Consequently, a D&D psionicist will always be studying things like abstract math and logic. Think "Vulcans" from Star Trek. This further stirred up the sudden love of knowledge in me which, a mere year ago, had lain dormant.
I started studying anything that seemed particularly abstract or arcane. That quickly lead me to Euclid's Elements. The Elements is the second most widely read and widely sold piece of literature in the world, second only to The Bible. It is the original geometry textbook, written thousands of years ago and still read today. Unlike modern public school mathbooks, written with the assumption that the reader is stupid, Elements is written with the assumption that the reader is intelligent. Unlike modern texts which fail utterly in their pathetic attempts to be "relevant" or "hip", the Elements is timeless. I took one glance through the almost mystical-looking illustrations, and I was hooked.
. The Elements is the second most widely read and widely sold piece of literature in the world, second only to The Bible. It is the original geometry textbook, written thousands of years ago and still read today. Unlike modern public school mathbooks, written with the assumption that the reader is stupid, Elements is written with the assumption that the reader is intelligent. Unlike modern texts which fail utterly in their pathetic attempts to be "relevant" or "hip", the Elements is timeless. I took one glance through the almost mystical-looking illustrations, and I was hooked.While it's tempting to reap the praise of being a child genius or something, the fact is, I really didn't understand the Elements at all. The first proposition-- instructions for how to draw an equilateral triangle-- made sense (after reading it a million times), and that's about it. But that one proposition blew my mind, the way the instructions were laid out so logically and beautifully, and then how the guy proved that the construction worked. It was my first taste of the Olympian-Nectar which is formal proof, the real heart and soul of mathematics.
I didn't understand Euclid's works, but I sure loved flipping through the pages and looking at the pictures. It was like a game, trying to find just how amazing an illustration could be. I took a ruler and a compass and some paper and I started making geometric figures of my own. I couldn't understand what Euclid was doing, so I was doing it myself! Of course, I failed to discover anything, but what happened there was magical: my self-image metamorphosized from that of someone who sucked at math, into that of someone who was good at it.
I wasn't born a mathematical genius. It's just that when we get a self-image in our heads, we tend to make it real by any means necessary. This has proven true in my life so many times, I'm convinced if I could just solidify a real, genuine "multibillionaire" self-image, and really believe it, it would materialize. Once I was drawing those geometric figures and emulating Euclid, people started seeing me as a kid who was good at mathematics. It was an illusion, but over the next few years, it was set to become real.
My parents bought me some math books for my birthday that year. I got a precalculus textbook and a couple calculus ones. I devoured them, along with an ancient algebra textbook we had lying around in the house. In the precalc book, I didn't understand everything. I didn't understand trigonometry hardly at all. If I had to take a midterm on the stuff, I'd've failed it, but what was important was that I was "learning the language". Most people who take precalculus in college, they don't read the textbook at all, besides doing assigned exercises. Reading it made me feel really smart and intelligent. And even if I didn't understand something, I was picking up the general "style" of mathematical texts. Symbols which once looked like alien writing, were transformed, if not into symbols I understood fully, at least into symbols I'd seen before. Symbols I could at least pronounce.
Conventional wisdom in mathematics says, you have to always build on what comes before. I found this to only partly true, and more than half the time, false. It's an example of ideology, which pollutes knowledge. But since I was self-teaching, I escaped the misleading ideology. There is certain truth to the ideology, but, in mathspeak, it's a "local truth", which means that it generally holds true within a section or a chapter. But often you can skim one chapter and jump right to the next and you're fine. (You can go back and pick up the hard stuff later, when your mathemuscles are bigger)
When my parents took me to interview for a radical fundamentalist private Christian highschool, we sat down with the lady who would become my ninth grade algebra teacher. She asked me if I had any questions about algebra. At the time, I was really struggling with the how the quadratic equation works. I was good at solving linear equations, it's just a matter of "keeping the scales balanced", but there was a certain "trick" to quadratics (completing the square), and I didn't have the mathematical maturity to grapple with it. I asked her about it. She showed it to me, but having a person explain it in person no longer added any new light that the books did not. I was sufficiently liberated from the crutch of teachers, that I would never again need them to translate texts for me. (At least not until I started getting into real journal articles in grad school!)
Literally days after the interview, I suddenly "got" it. The process of completing the square suddenly made sense. And once I "got" it, it was so simple I was amazed that I had been stumped for so long. Suddenly, I didn't even need the quadratic formula to solve quadratic equations, I could do it manually. But it wasn't because of some natural born talent or anything. Completing the square was a big ol' dumbbell that I'd been straining at forever with no progress. But meanwhile, I was also training with smaller dumbbells-- reading ahead, I mean. Then one day, with muscles enhanced by the smaller weights, I returned to my old nemesis and suddenly found I could lift it with ease.
I can't emphasize enough the importance of getting a textbook which treats you like an intelligent person. Anything in the undergrad section of a college bookstore will be trash, because it's written for people who are only studying it because they have to. Also, this might sound a little shallow, but I think it's important the text should have a professional look and feel to it. That rules out most textbooks with their cheesy covers, always making some futile attempt at "relevance" or "hipness". Look, it doesn't matter what's on the cover of a calc book, people are gonna think it's an arcane tome of mystic spells-- so you may as well make it look like an arcane tome of mystic spells! At least then it's impressive and cool, and I even think you learn more from it, just because you somehow take it more seriously. Mom and dad did a great job with my birthday gift, because one of the calculus books they got me was Swokowski's Calculus, the Classic edition.
, the Classic edition.I didn't know it at the time, but Swokowski is actually one of the most well-respected calculus texts around. Compared to some of the trash I've seen undergrads forced to buy here at OSU, it's like heavenly script.
That interview with the future algebra teacher took place a while before the school season would actually start. The next time I saw her, I was well beyond the quadratic equation. Once my mathematical maturity was good enough to swallow completing the square, it was good enough for me to rocket through most of basic calculus almost nonstop. Not much in calculus is actually fundamentally harder than solving quadratic equations manually, it's just more intimidating. By being a self-teacher, I was able to pierce right through the illusionary intimidation factor.
It didn't take very long before I knew more mathematics than anyone in that cheap crappy private highschool. Rather than praise me for teaching myself the rudiments of analysis, it actually became a sore spot in my already very poor relations with the teachers. In my article "You might be an autodidact if...", I jokingly wrote that you might be one if you tuned out your highschool algebra teacher to study calculus. But actually, that line was based on my real life experience! I'm one of a very select few who can boast that their ninth grade principal confiscated a textbook from them.
The Summer after ninth grade, my math autodidact days came to an end as I started taking calc classes at Mira Mesa Community College (in California, the state will pay your tuition if you take college classes as a high schooler). It was a temporary end, as I would start teaching myself far more advanced mathematics years later while I was in the Air Force. But that's a story for another day ;)
FURTHER READING
I wrote an article with some hints about being better at mathematics. Five Ways To Be Better At Math.
Read about self-teaching in general at my article, Autodidact: Be A Self-Teacher. There are many lifelong advantages to becoming a self-teacher. Everything suddenly becomes a whole lot easier.
Should rote memorization be used in mathematics? The knee-jerk response these days among math educators is a resounding "NO", and I used to agree, but lately I've been really questioning that. Sometimes, some rote memorization is good. Read more in Rote Memorization In Mathematics.
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1 comments:
I like this article. I'm 52 and am teaching myself calculus. I hated math when I was young, saw the need for it when I was in college, but didn't learn well under the time pressure. I have had an old calculus text that I bought in cambridge, ma. 25 years ago. It was written by a prof at MIT, and even something about the color of the book drew me in in some mystical fashion.
I had always wanted to go through this book but assumed I needed to have all kinds of background in math to do it. So I didn't do it, out of fear and the feeling that I couldn't take the time to get the background. But as you note in your article, one can dive in without having to know all the steps. So I'm progressing through the book and it's a real joy--what learning should be. When I reach sections in which I need background, I just "go get it" from other old algebra and pre-calc texts I have laying around and had never read. I'm not sure why I have the ability to go find what I need now, and didn't when I was younger.
I would still like to take university mathematics, but am afraid the pace and pressure would once again kill the inspiration and leave me with anxiety.
I have an almost mystical love for what I'm doing now.
As an aside, your site has some similarities to Steve Pavlina's. You have similar backgrounds and I suspect one of you is influencing the other. The nice thing about both your blogs is that the articles are longer than the fluff norm, and one feels like one is getting ones moneys worth for the time put in. You both take occasional forays into new-agey stuff, which makes me wonder if you've gotten whacked out. Will you be talking about poly-amorism next? If so, I'm gone, but thanks for the articles that were useful. ha ha.
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