I do not see how anyone can evaluate a logical argument without basic algebra. Or perhaps I mean that anyone with a basic understanding of logic does know algebra?
Basic math skills, like algebra and geometry are as fundamental as syntax and vocabulary.
What? We have to take things for granted instead of thinking about whether they really hold up and possibly re-evaluating our positions? Why?
Quite possibly he’s right. However as a left-brained kind of person, I would say that the bias of society is very much towards pointless right-brained things. Algebra, or taken more to the abstract set theory, is pure logic. It is rationality. And society could do with a lot more rationality about a whole deal of topics.
In other words, fuck you all, rightites!
As I said, you could call different things “algebra”. And I would feel differently about those different things.
“If 5x + 7 = 17, what is x?”. Fine. Everyone should have some comfort with this. Some basic arithmetic operations are defined to invert others, and are ubiquitously useful enough to force people to grow familiar with the process of such inversion. (Of course, children are already being exposed to this sort of thing even in first grade; it’s just a matter of firming that up.)
The intricate manipulation and analysis of rational functions over the reals… hey, I wish this was everyone’s cup of tea, but if reality departs from my wishes here, what’s the harm in letting some people live their lives not having to pretend to care? The vast majority of people get by in their lives without caring a whit about it once they finish their last math exam. Now, perhaps they would get by even better if they were better at it, but what’s the evidence for that assertion, and does it counterbalance the opportunity cost of forcing them to spend so much time on such classes rather than other material or activities of potentially greater interest to them?
Well, you can understand some level of each of these topics with some level of formal algebraic familiarity. Of course, to really be intricately comfortable with them, you need to really be intricately comfortable with the manipulation and analysis of rational functions. But not everyone wants to understand statistics, calculus, or geometry at that level.
One of the better defenses of algebra I’ve seen is this one by Keith Devlin, in which he talks about what algebraic thinking is, how it’s different from arithmetical thinking, and why it’s important.
I only have anecdotal evidence myself, but in my experience, the non-scientist-or-otherwise-highly-technical people I know, almost without exception (including tutoring some kids – like flinging myself against a brick wall), did not really understand algebra at all. So far from building mental muscles, it seems actually to be building mental blocks.
I really think that I’d much rather people come out of high school really understanding a bit about elementary statistics (when is it appropriate to use the mean vs. the median? what are the tricks people use to make graphs say the things they want to say?), and a bit about logic (A implies B does not mean that B implies A; correlation is not causation), and maybe – I know i’m pushing it here – a bit about probability, than having to take algebra and geometry.
Of course, who would teach these classes?
Also, weirdly, I kind of think people should read Shakespeare and The Great Gatsby even if they end up hating them. So I guess I’m not entirely consistent with what I think the educational system ought to do… I think it’s that I’ve met a whole lot more people who say things to me like, “Wow, math. I don’t get math at all, I never understood those algebra classes” than who say things like “Man, I never figured out English class at all.”
That’s really strange. I’m a progammer and I’ve had to use algebra and trig in my work all the time. My work is advanced but hardly bleeding edge. I work on navigation systems.
That’s a nice cliche, but really I’m pretty unlikely to start another Civil War even if I had somehow failed to learn the causes of the last one. That’s the sort of history kids learn in high school, after all.
To be clear, I’m not saying we shouldn’t learn history. I just don’t think it’s more useful than mathematics. Maybe for making conversation (e.g., most people are more likely to reference some historical event than some mathematical theorem), but not really beyond that.
But I’m glad to have some education in both, and over a decade after the fact I wish my education in history had been better. My high school classes emphasized dates and names more than why things happened the way they did, which left me with no interest in pursuing the subject further in college. Now I know there’s a lot more too it than that… but still not so much that I would have sooner studied history than algebra (which I have actually used a great deal in my career.)
Algebra, generally speaking, is how numbers relate and interact with each other. It is the foundation of every application of math in the real world. Teaching arithmetic without any algebra is almost a waste of time, because just quantifying things is not terribly useful. Even simple grocery-store applications of math like “Is item X a better value than item Y?” or even just “How do I make change for this bill?” are, at their heart, algebra problems.
I did all right in first-year algebra and geometry. I pretty much washed out of second-year algebra midway through. I still find polynomials hard to do properly. Obviously (?) I never really bothered with calculus. (And I never cared about “classic literature,” preferring Asimov and New Wave science fiction. I was a bad student.)
When I was in school, we often let students promote through without making sure they had the material covered; I know that happened to me at times. Of course, then, most–not some, most–students will find the higher levels too hard to do because they never got through all the intermediate prerequisite steps.
We do need to set our sights moderately low and teach what we can teach fully. And even then, we won’t get everyone. But yes, there should be a requirement for algebra as such, not just as one handout of information in another class, even if many students never really master anything more complex than the quadratic equation.
Oh, and I’m not a programmer, but I really appreciated what familiarity with algebra I had when I was doing a lot of hypertext template coding several years ago.
If you can come up with a metric for what it even means to be an educated, well-rounded, functioning member of society, let alone a proof that (say) teaching history leads to such a result, I’d like to hear it.
Yes, we can measure things like whether teaching technique X on subject Y leads to good retention after Z years. But good luck finding evidence that reading Shakespeare leads to better communication skills. Or that learning history will help us avoid war. Or that learning algebra helps people understand their tax forms.
Let me pose a pair of questions:
A) What is the value of mandating that everyone, regardless of interests*, be comfortable manipulating and solving quadratic equations by hand?**
B) Why do we not similarly require everyone be able to solve cubic equations by hand?
*: With acknowledgement that interests are fluctuating and malleable over life.
**: And does this mandate actually achieve this goal?
The specifics don’t interest me much. The point is to be comfortable with the notion of a variable, manipulating those variables, and with the basics of functions (roots, etc.). Memorizing the quadratic or cubic formulas don’t contribute to that, but working through problems involving those formulas will.
You can always look up formulas later. But evaluating those formulas or even knowing which formulas to pick requires some level of basic education.
I really, REALLY sucked at math in high school. But I can understand how a society that puts such high value on producing technology would want everyone to have a robust math/science background. I’d love to go back and work on some math in my spare time, but I wouldn’t even know where to begin at this point.
It seems to me the solution would be to implement a strategy like the Germans have where you decide what kind of secondary education you want based on your future career goals. There are some people who, at the age of 13, know that they would rather have their teeth broken out than do quadratic equations. Don’t make them do more than the most banal, simple math.
I’m a music major turned law student; I can tell you harmonic ratios for intervals, and solve for ‘x’ on simple algebraeic problems. Beyond that, even most of the terms being bandied around in this thread are over my head. And that makes me sad. 
Alright. That all seems agreeable enough to me. Like I said, this conversation is difficult, because there’s different levels of what “algebra” might mean; at the most basic level, I have no qualms requiring everyone to go through it, and at other levels, I think it’s incredibly pointless.
Well, right. I don’t think Shakespeare, war history, or algebra, beyond the utter basics, should be imposed on students against their will. I also think anyone who becomes interested in them, at any time, should be able to learn about them.
(And, of course, I’m all for making sure people understand what these topics are prerequisite for, and ensuring a sufficient level of exposure to these topics that people have an opportunity to discover, even to their surprise, whether they might find some interest in these topics… so long as there is a genuine ability for students to nonetheless opt-out of any such courses rather than slog onerously through should they ever decide “Yeah, thanks, but I’m not interested”.)
Well, if you’re up for it, we could help you out, perhaps. What sort of thing are you interested in learning about and what sort of background do you have?
The answer I can come up with right away—and I’m not sure it’s the best answer, or even a good one—is that it’s important to understand and appreciate that not everything is linear. So they should get some experience working with nonlinear equations, of which quadratics are perhaps the simplest.
I have no idea why such a large emphasis on calculus is made in CS education. Perhaps in analogy to the education that mechanical and electronic engineers receive, to try to make it appear that CS or SE is a legitimate form of engineering, or an institutional bias amongst CS academics towards subfields like computer graphics and machine learning which make use of calculus, but at least from my viewpoint the overwhelming majority of research that I read is based on discrete mathematics and there’s whole subfields of the subject which are essentially being shafted by the insane focus on derivatives and integration in undergraduate years.
Aren’t you basically programming math if you’re programming navigation? I think that’s probably different than what a lot of programming is about.
I should note that my sister’s BS and MS degrees are both from UW Madison, so she did in fact go through all these math hoops. Perhaps she’s understating the math cause she’s found all the advanced stuff to be so useless to her.
Also, I read a link upthread about algebra and Excel, and I certainly agree that algebra is necessary for that, and by extension a necessary general office skill. And while I don’t work as a programmer, I have taken several courses and played around as an amateur. Certainly algebraic logic helps with much of it.
This is my understanding. Historically, it seems that many CS programs developed within an already existing mathematics program, and so the requirements for the degree were very similar to those for a math degree. And once on this track it’s very hard to get off without offending the “old guard” – and since CS is so relatively new, there’s probably some of the old guard left to offend.