Slashdot Mirror
Promoting Arithmetic and Algebra By Example
Capt.Albatross writes "A couple of months ago, the New York Times published political scientist Andrew Hacker's opinion that teaching algebra is harmful. Today, it has followed up with an article that is clearly intended to indicate the usefulness of basic mathematics by suggesting useful exercises in a variety of 'real-world' topics. While the starter questions in each topic involve formula evaluation rather than symbolic manipulation, the follow-up questions invite readers to delve more deeply.
The value of mathematics education has been a (recurring issue on Slashdot)."
Aside from the obvious ones in engineering, where few kids will participate...
There is the issue of "how much paint will I need to paint my house?"
Doing the math will save you money.
The reason is not algebra's application to daily life. The reason you teach algebra is because algebra teaches symbolic manipulation. Learning math teaches you not just how to add two numbers. Addition is almost unnecessary in daily life (we do have calculators). Learning algebra is critical because it teaches us to think in terms of abstractions, of models. We do not teach mathematics to teach you how to add, we teach mathematics to teach you how to solve, to teach you how to think.
Then that is your particular FIELD. In graphics, we use it all the time. It helps to understand differential geometry and Brownian motion when working with real time and ray tracing applications respectively. I mean, all higher order analysis works on concepts of stochastic processes, so really we do need people to understand algebra (At the very very least).
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But is it recursive? And more interestingly, does it converge?
"Siri, how much is 400 grams in some fucked up imperial measurement?
But teaching kids how to think is not desirable in an economy that can't provide any jobs where they need to think. Thinking leads to people not doing everything their "superiors" tell them to do, and that leads to unhappiness. You kids to grow up to be happy, obedient adults, don't you.
Andrew Hacker and nearly everyone else is missing the point.
Taking an algebra class for many students is not about the algebra, it's about learning to think. Even if you never use algebra again, the process of learning algebra is mental exercise that improves the mind. Taking a foreign language, studying biology, learning economics, studying history - it doesn't matter what the subject is, merely the more you learn the better a learner you are, and the better thinker you are.
In sports we see athletes perform all kinds of exercises that help develop skills used in their sport, but are never used directly. Ever see footage of a football player stepping through tires? Ever see one do that during the game? Ever see footage of a quarterback or pitcher throwing the ball through a hanging tire? Ever see them do that in a game? Athletics is filled with examples of training exercises done to hone one's skills for a game, yet we have difficulty accepting that mental exercises hone skills we need for life.
*jokingtroll*
I'm so sick of all these nerdy math / science posts... we need more patent litigation and mobile device war posts
*/jokingtroll*
Seriously though, I think that straying away from the mathematical fundamentals will lead to straying from linguistic fundamentals and historical fundamentals. Eventually the bulk of the education system will be 'Can you read well enough to use a computer? Congratulations you are a high school graduate".
The ability to follow through an entire equation and achieve the outcome is very useful in life. Perhaps not in the form you learn in Algebra, but in one form or another.
But lets just let our kids get less smart and more dependent on the technology that will eventually stop evolving because nobody is smart enough continue the evolution.
http://xkcd.com/1050/
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The reason you teach algebra is because algebra teaches symbolic manipulation
The tool you should be using is symbolic and philosophical logic
How do you propose to teach someone symbolic logic without teaching them what symbols are, and how to manipulate them?
I'm in IT and use math daily. I've gotten promotions because I was the only person that could do complex math easily.
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You can't teach philosophy to children. Given the philosophy classes I've taken, you shouldn't teach it to adults. It's a class in mental masturbation. The old philosophers were the Original Internet Trolls. The point of philosophy is to argue, not to learn, discover, or use. Sure, after a few millenia of arguing a truth may accidentally be stumbled upon, but that's a by product, like plastics from the space program.
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I've been a developer for about 16 years, and have had a pretty spotty math education. I've generally taught myself what I need to know as I needed to know it - 3D programming? What's a matrix? How do I rotate things with it? Developing animate graphical charts? How do I scale from business coords to pixel coords, and animate? Draw box an whiskers charts etc...
Recently, I've decided to stop doing the corporate developer gig and to go to school. As part of that, I've needed to take math a lot more seriously, so I've bought some books and been going through a more rigorous program.
One thing I've discovered through this process is that I *really enjoy it*. I'm not being pressured to learn something for a test, I'm not worried about a grade. Instead, I take my books to a coffee shop and relax and think about fascinating things, like trying to visualize the complex plane, and what the value for i really is, and what dividing by zero really means.
Instead of memorizing the quadratic equation, I spent some time learning how to derive it from basic principals. Instead of memorizing that the vertex of a parabola can be found by -b/2a, I noodled around and tried to visualize the determinant (sqrt(b^2 -4ac)), it's effect on an equation, and what happens if you zero it out.
I spend a leisurely afternoon coming up with a visual proof of the Pythagorean theorem, and was pretty excited when I finally had it, and was even more excited when I googled it and saw the same basic proof has been derived by students for a really long time - I loved the notion that I was connected back through time with a whole bunch of other people who were going through the same mental steps.
This stuff is great! And I'm only scratching the surface. I'm in baby algebra - and I'm excited to keep going.
My point is - we go about this stuff all wrong. Forcing kids to memorize equations so they can pass an exam is absolutely pointless, if not masochistic. Exploring really interesting concepts about numbers, and what they mean - this stuff should be recreation. It's great!
I see my older son struggling through his algebra course, and he hates it. He doesn't care, and hates doing the homework. But when I get excited about some math problem I'm studying, he'll come over to look over my shoulder to see what I'm doing, and we'll puzzle it out together. He forgets that we're doing math, instead we're talking about concepts and challenging each other. We'll spend an hour or two going over something that's really cool, and we both have a great time.
Ask him about math, however, and he immediately relates it to school, and he'll tell you how much he hates it.
Drinking habits can be dangerous. You can choke on the cloth and the nuns will wonder where their clothes are.
>How do you propose to teach someone symbolic logic without teaching them what symbols are, and how to manipulate them?
You are implying that algebra is the be-all and end-all of symbolic logic?
Let me introduce you to New Math. I was a "victim" of it.
It revolved around number theory, set theory, and logical operations - and, or, not. I knew venn diagrams and set notation before I knew how to find least common denominators.
New Math was widely derided by people who thought that arithmetic proficiency done through rote learning and timed tests (which was my third grade math) should be the goal of the early grades. The thing is though, when home computers showed up, I was able to teach myself programming, and a course in digital circuits wasn't as hard as it could have been, and all sorts of stuff including a summer course in Logic (with Irving M. Copi's book) that made one of my smart friends drop in a week.
My point being is that teaching philosophical logic as such is a better tool for teaching one how to think, because it's not just applicable to math. It's applicable to law and debate and other such things. It gives one an appreciation as to why, in a law, there is an "or" instead of an "and" linking clauses.
But hey, what do I know.
--
BMO
>Given the philosopy classes I've taken
And yet you skipped right over that Logic course.
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BMO
You don't need to know basic algebra to perform basic algebra. Basic math combined with the properties of real numbers (mostly intuitive themselves) makes basic algebra intuitive. Building these problems out as traditional equations involves a bunch of extra steps that serves no purpose except to satisfy a teacher who wants you to show your work. It's sort of like unit conversion. They give lessons on this and show frustrating slow processes where you put numbers over other numbers and toss in a conversion factor. But nobody would actually go through that garbage to convert a unit you added a bunch of extra crap to the problem just to cancel it out later.
A foot is 12 inches. If you have 36 inches how many feet do you have?
Anyone who understands multiplication and division can solve this problem without being taught a formal unit conversion process. If writing this down they would write 36/12 = 3.
http://www.math.com/school/subject2/lessons/S2U2L1GL.html
The amount of math used expands to fill the user's ability. The more math I learn the more uses for that math I find.
Not a sentence!
Being skilled in math makes your brain a far more capable information processing system, even if you're not aware that you're using math in your day to day life.
I suspect that algebra has value beyond its immediate application and utility. I think this is often overlooked in debates like this. When learning algebra, you are in effect modifying your brain in a particular way. You are training yourself to think a certain way. You can gain a feeling of mastery if you learn it well. And implicitly, you are taught the value of reason and logic. The very fact that you are asked to learn algebra carries the message that logic and rational thinking are valuable skills, and that people who are good at such things are particularly valuable to society. The pursuit of a topic like algebra encourages discipline and structure in the way you think about many other things.
To focus only on the immediate applications of algebra is small minded and unwise, in my opinion.
This and no other is the root from which a tyrant springs; when first he appears as a protector - Plato (423 to 327 BC)
Even people doing graphics and audio manipulation mostly just use APIs written by other people who do the hardcore math for them. If I want to perform an FFT and use the resulting data set, I don't need to know how an FFT is computed; I just call a function in an FFT library. If I want to render a 3D scene, unless I'm actually writing software for use in some specialized field, chances are I'm just going to throw a spline model at OpenGL and tell it to wrap a texture around it. I don't need to know the math of bilinear interpolation to use it. I just need to know how to set the interpolation mode for the vertex in question. And so on.
I'm guessing that probably no more than one or two percent of programmers will do any significant amount of algebra or calculus or trig during their careers, and that most of the people who do either write low-level graphics or audio code, work for NASA or the aerospace industry, or work in a handful of other highly specialized fields where exacting precision is required. The vast majority of computer programmers will never need to use math above a middle school level, even though much of the software they write will do so routinely. And this is the beauty of code reuse. It means that I can do higher-order math without having to crack open a textbook and remember how all that stuff works, because somebody else already did the heavy lifting. :-)
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Bear in mind the TFA is from a Political Scientist. That's just below an economist in my book. I'm sure we'd all welcome comments from maths teachers about Political Science (btw, why the hell is Political Science a science?)
"The greatest lesson in life is to know that even fools are right sometimes" - Winston Churchill
(At the risk of repeating one of my other comments)
Do you think the abstract problem solving you practised while learning algebra (or eg calculus later on) has subconsciously helped your programming?
I'm of the opinion that the primary benefit of learning algebra and calculus etc isn't the specific techniques you learn but the ability to think in a much more abstract way when required. Even after most of the actual techniques have faded from memory, you still have the subconscious rewiring left behind by abstract problem solving.
And it is very important for a programmer to think abstractly.