“I will remark, in some horror and exasperation with the modern educational system, that I do not recall any math-book of my youth ever once explaining that the reason why you are always allowed to add 1 to both sides of an equation is that it is a kind of step which always produces true equations from true equations.”
I can now say that my K-12 education was, at least in this one way, better than yours. I must have been 14 at the time, and the realization that you can do that hit me like a ton of bricks, followed closely by another ton of bricks—choosing what to add is not governed by the laws of math—you really can add anything, but not everything is equally useful.
E.g., “solve for x, x+1=5”
You can choose to add −1 to the equation, getting “x+1+-1=5+-1”, simplify both sides and get “x=4“ and yell “yay”, but you can also choose to add, say, 37, and get (after simplification) “x+38=42” which is still true, just not useful. My immediate question after that was “how do you know what to choose” and, long story short, 15 years later I published a math paper… :)
I absolutely remember being taught that the reason we could add something to both sides, subtract something from both sides, divide or multiply… was because it preserved the equality. I think 7th grade. And when I helped my kids on the relevant math (they did not inherit my “intuitive obviousness” gene) I would repeat this over and over.
Of course we all want to blame the teacher or the course when we haven’t learned something in the past, but I have seen too many people not learn things that I was repeating and emphasizing and explaining in as many ways as I could imagine or read, and still they didn’t get it. I think we have all head the experience of having someone we were explaining something to finally get it and exclaim something like “why didn’t you just tell me that in the first place.”
I’d probably be interested in reading that paper, since I’ve never myself figured it out and still have to (grudgingly) rely on the school-drilled method of “do more exercises and solve more problems until it’s so painfully obvious that it hurts”. AKA the we-have-no-idea-but-do-it-anyway method.
My mat class provided the simple “how to choose” heuristic that you want X to be alone. So if you have “x+1″ on one side, you’ll need to subtract 1 to get it by itself. X+1-1=5-1. X=4.
I can see how this wouldn’t get explicit attention, since I’d suspect it becomes intuitive after a point, and you just don’t think to ask that question. I can’t see how one could get through even basic algebra without developing this intuition, though o.o
Yes, clearly, a bit after I asked, I learned how to use intuition, and at some point, it became rote. But the bigger point is that this is a special case—in logic, and in math, there are a lot of truth-preserving transformations, and choosing a sequence of transformations is what doing math is. That interesting interface between logic-as-rigid and math-as-something-exploratory is a big part of the fun in math, and what led me to do enough math that led to a published paper. Of course, after that, I went into software engineering, but I never forgot that initial sensation of “oh my god that is awesome” the first time Moshe_1992 learned that there is no such thing as “moving the 1 from one side of the equation to the other” except as a high-level abstraction.
I can’t see how one could get through even basic algebra without developing this intuition, though o.o
And yet so many do. Numbers are horribly scarce in this area, though. Sometimes I get desperate.
Anecdotally, I do remember saying something very similar to high school peers, in a manner that assumed they already understood it, and seeing their face contort in exactly the same manner that it would have had I suddenly metamorphosed into a polar bear and started writing all the equations for fourier and laplace transforms using trails of volcanic ash in mid-air.
This was years after basic algebra had already been taught according to the curriculum, and we were now beginning pre-calculus (i.e. last year of secondary / high school here in québec, calculus itself is never touched in high school level courses).
If you’re trying to find the value of x and there’s only one x in the equation, it’s simply a matter of inverting every function in the equation from the outside in. It’s harder if you have multiple x’s because you have to try to combine them in some reasonable way—and sometimes you actually want to separate them, e.g. “x^2 − 9 = 0” into “(x − 3)(x + 3) = 0″.
Solving this problem appears equivalent to writing a computer program to solve algebraic equations.
For polynomial expressions of the third order or lower in only one variable, the process is trivial. I believe that in every general case for which a solution is guaranteed, there is also a trivial method which will always generate such a solution.
Are any decisions required to be made in their implementation? If so, they are nontrivial. If not, they are trivial.
I was saying that for every general case in which a zero of a polynomial is guaranteed, there is a deterministic path to determine what the guaranteed zeros are.
Hrm. Is there a short proof of this claim? It’s not completely obvious to me. How do you know there isn’t some set of n-th degree polynomials, for which there’s a non-constructive proof that they have roots, but where there isn’t any known algorithm for finding them?
I suppose you could always do numeric evaluation with bisection or somesuch, since polynomials are smooth. But I think you’re saying something more than “iterative numerical methods can find zeros of polynomials over one variable.” I also imagine that you don’t want to include “explore all possible decision trees” as a valid solution method.
Part of my hesitation is that there are classes of equation (not polynomial) for which it’s believed that there aren’t efficient algorithms to find solutions. For instance, suppose we have a fixed g, k, n. There’s no known efficient technique to find x such that g^x = k, mod n. (This is the discrete logarithm problem.) It seems thinkable to me that there are similar cases over the reals, but where you can’t do exhaustive search.
Yes. I’m aware of them. But that’s not quite the question here. The question was whether there were equations that have solutions, that can be found, but where there’s no algorithm to do so. So by definition the numbers of interest are definable in some way more succinct than their infinite representation.
As a nitpick: My understanding is that algebraic just means a number is the root of some polynomial equation with rational coefficients. But that’s not the same as “cannot be described except with an infinite decimal string.” A number might be described succinctly while still being transcendental. The natural logarithm base is transcendental—not algebraic—but can be described succinctly. It has a one-letter name (“e”) and a well known series expression that converges quickly. I think you want to refer to something like the non-computable reals, not the non-algebraic reals.
I only got K-8[\6] (I’m not a highschool dropout, thank you, I just never went in the first place) so I’ve got no idea what people learn when they’re 14.
One thing that I’ve always wanted to ask you, but for some reason never have, is what kind of form your autodidactism took. Did you just read books (if so, what kind? Textbooks?), did you get a tutor, did your parents help?
If you consider the ideas of John Taylor Gatto, the 1991 New York State teacher of the year, regarding schooling, you might hesitate before using the word “schooling” to mean something credible.
Yeah, I’ve read some of what he says. He’s right and he’s wrong; American public education is indeed designed to produce workers, but we still have far more entrepreneurs per capita than, say, China.
Hmmm. Not to ignore the fact that this essentially sidesteps my point that school should not necessarily be associated with credibility, but you’ve succeeded in making me curious: what do you think the difference is between the US and China that would explain that?
Actually, I think I’m wrong on the facts here; The GEM 2011 Global Report has some tables about this, but they don’t copy/paste very well. The U.S. is ranked first in “early-stage entrepreneurial activity” (percentage of the population that owns or is employed by a business less than three and a half years old) among nations characterized as “innovation-driven economies” (a category that includes most of Europe, Japan, and South Korea) at 12.3%, but “efficiency-driven economies” (which includes China, most of Latin America, and much of Eastern Europe) tend to have higher values; China is listed at 24.0%,
I think it’s a good thing you didn’t receive more schooling. I’m an autodidact, too. I wish I had embarked on a self-education journey sooner. Reasons.
“I will remark, in some horror and exasperation with the modern educational system, that I do not recall any math-book of my youth ever once explaining that the reason why you are always allowed to add 1 to both sides of an equation is that it is a kind of step which always produces true equations from true equations.”
I can now say that my K-12 education was, at least in this one way, better than yours. I must have been 14 at the time, and the realization that you can do that hit me like a ton of bricks, followed closely by another ton of bricks—choosing what to add is not governed by the laws of math—you really can add anything, but not everything is equally useful.
E.g., “solve for x, x+1=5”
You can choose to add −1 to the equation, getting “x+1+-1=5+-1”, simplify both sides and get “x=4“ and yell “yay”, but you can also choose to add, say, 37, and get (after simplification) “x+38=42” which is still true, just not useful. My immediate question after that was “how do you know what to choose” and, long story short, 15 years later I published a math paper… :)
I absolutely remember being taught that the reason we could add something to both sides, subtract something from both sides, divide or multiply… was because it preserved the equality. I think 7th grade. And when I helped my kids on the relevant math (they did not inherit my “intuitive obviousness” gene) I would repeat this over and over.
Of course we all want to blame the teacher or the course when we haven’t learned something in the past, but I have seen too many people not learn things that I was repeating and emphasizing and explaining in as many ways as I could imagine or read, and still they didn’t get it. I think we have all head the experience of having someone we were explaining something to finally get it and exclaim something like “why didn’t you just tell me that in the first place.”
I’d probably be interested in reading that paper, since I’ve never myself figured it out and still have to (grudgingly) rely on the school-drilled method of “do more exercises and solve more problems until it’s so painfully obvious that it hurts”. AKA the we-have-no-idea-but-do-it-anyway method.
My mat class provided the simple “how to choose” heuristic that you want X to be alone. So if you have “x+1″ on one side, you’ll need to subtract 1 to get it by itself. X+1-1=5-1. X=4.
I can see how this wouldn’t get explicit attention, since I’d suspect it becomes intuitive after a point, and you just don’t think to ask that question. I can’t see how one could get through even basic algebra without developing this intuition, though o.o
Yes, clearly, a bit after I asked, I learned how to use intuition, and at some point, it became rote. But the bigger point is that this is a special case—in logic, and in math, there are a lot of truth-preserving transformations, and choosing a sequence of transformations is what doing math is. That interesting interface between logic-as-rigid and math-as-something-exploratory is a big part of the fun in math, and what led me to do enough math that led to a published paper. Of course, after that, I went into software engineering, but I never forgot that initial sensation of “oh my god that is awesome” the first time Moshe_1992 learned that there is no such thing as “moving the 1 from one side of the equation to the other” except as a high-level abstraction.
And yet so many do. Numbers are horribly scarce in this area, though. Sometimes I get desperate.
Anecdotally, I do remember saying something very similar to high school peers, in a manner that assumed they already understood it, and seeing their face contort in exactly the same manner that it would have had I suddenly metamorphosed into a polar bear and started writing all the equations for fourier and laplace transforms using trails of volcanic ash in mid-air.
This was years after basic algebra had already been taught according to the curriculum, and we were now beginning pre-calculus (i.e. last year of secondary / high school here in québec, calculus itself is never touched in high school level courses).
If you’re trying to find the value of x and there’s only one x in the equation, it’s simply a matter of inverting every function in the equation from the outside in. It’s harder if you have multiple x’s because you have to try to combine them in some reasonable way—and sometimes you actually want to separate them, e.g. “x^2 − 9 = 0” into “(x − 3)(x + 3) = 0″.
Solving this problem appears equivalent to writing a computer program to solve algebraic equations.
For polynomial expressions of the third order or lower in only one variable, the process is trivial. I believe that in every general case for which a solution is guaranteed, there is also a trivial method which will always generate such a solution.
There are closed form solutions to all cubic or quartic polynomial equations, but they’re quite complicated. Do those solutions count as “trivial”?
Are any decisions required to be made in their implementation? If so, they are nontrivial. If not, they are trivial.
I was saying that for every general case in which a zero of a polynomial is guaranteed, there is a deterministic path to determine what the guaranteed zeros are.
Hrm. Is there a short proof of this claim? It’s not completely obvious to me. How do you know there isn’t some set of n-th degree polynomials, for which there’s a non-constructive proof that they have roots, but where there isn’t any known algorithm for finding them?
I suppose you could always do numeric evaluation with bisection or somesuch, since polynomials are smooth. But I think you’re saying something more than “iterative numerical methods can find zeros of polynomials over one variable.” I also imagine that you don’t want to include “explore all possible decision trees” as a valid solution method.
Part of my hesitation is that there are classes of equation (not polynomial) for which it’s believed that there aren’t efficient algorithms to find solutions. For instance, suppose we have a fixed g, k, n. There’s no known efficient technique to find x such that g^x = k, mod n. (This is the discrete logarithm problem.) It seems thinkable to me that there are similar cases over the reals, but where you can’t do exhaustive search.
There exist what is called non-algebraic reals, which cannot be described with anything other than the infinite string of decimals they are.
Yes. I’m aware of them. But that’s not quite the question here. The question was whether there were equations that have solutions, that can be found, but where there’s no algorithm to do so. So by definition the numbers of interest are definable in some way more succinct than their infinite representation.
As a nitpick: My understanding is that algebraic just means a number is the root of some polynomial equation with rational coefficients. But that’s not the same as “cannot be described except with an infinite decimal string.” A number might be described succinctly while still being transcendental. The natural logarithm base is transcendental—not algebraic—but can be described succinctly. It has a one-letter name (“e”) and a well known series expression that converges quickly. I think you want to refer to something like the non-computable reals, not the non-algebraic reals.
Yes, the distinction is this:
http://en.wikipedia.org/wiki/Computable_number
http://en.wikipedia.org/wiki/Algebraic_number
I only got K-8[\6] (I’m not a highschool dropout, thank you, I just never went in the first place) so I’ve got no idea what people learn when they’re 14.
Homeschooled! The status-preserving term is “homeschooled”!
For the people who got Thiel Fellowships, the status-preserving term is “stopped-out”.
Heck, at that level even “drop-out” works as a form of counter-signaling.
Homeschooling is what Christians do. I’m an autodidact.
One thing that I’ve always wanted to ask you, but for some reason never have, is what kind of form your autodidactism took. Did you just read books (if so, what kind? Textbooks?), did you get a tutor, did your parents help?
Homeschooled? That is status-preserving? Really? I think I’d prefer “dropout”.
“Homeschooled” is what you say to college admissions people.
“Dropout” implies that you’re not interested in continuing your education.
If you consider the ideas of John Taylor Gatto, the 1991 New York State teacher of the year, regarding schooling, you might hesitate before using the word “schooling” to mean something credible.
His perspective is that schooling is not education and can help or hinder learning (explained under the heading “Learning, Schooling, & Education”). The criticisms he provides in “Dumbing us Down” are pretty nasty. I gave the gist of it and linked to an online book preview here. I highly recommend reading at least the first chapter of that book.
Yeah, I’ve read some of what he says. He’s right and he’s wrong; American public education is indeed designed to produce workers, but we still have far more entrepreneurs per capita than, say, China.
Hmmm. Not to ignore the fact that this essentially sidesteps my point that school should not necessarily be associated with credibility, but you’ve succeeded in making me curious: what do you think the difference is between the US and China that would explain that?
Actually, I think I’m wrong on the facts here; The GEM 2011 Global Report has some tables about this, but they don’t copy/paste very well. The U.S. is ranked first in “early-stage entrepreneurial activity” (percentage of the population that owns or is employed by a business less than three and a half years old) among nations characterized as “innovation-driven economies” (a category that includes most of Europe, Japan, and South Korea) at 12.3%, but “efficiency-driven economies” (which includes China, most of Latin America, and much of Eastern Europe) tend to have higher values; China is listed at 24.0%,
I think it’s a good thing you didn’t receive more schooling. I’m an autodidact, too. I wish I had embarked on a self-education journey sooner. Reasons.