Is Algebra Necessary?

Great Debates
1

Andrew Hacker thinks that it is not (nytimes). We’ve had this conversation before, but I don’t recall any cogent external links, so why not have it again? In essence, “Making mathematics mandatory prevents us from discovering and developing young talent. In the interest of maintaining rigor, we’re actually depleting our pool of brainpower.” Youth are excluded from further education (they can’t get into a good college, or perhaps can’t graduate) due to their poor performance in a topic that has little relevance to their talents that, if developed, could better society. Quantitative reasoning is certainly important, but polynomials? Less so.

Or at least I think that’s what he’s getting at.

I have knee-jerk discomfort with this argument. I’ve always viewed algebra as a rather elementary topic that really ought to be learned before high school, and that it is not learned due to poor math instruction, imploding home life, etc. This opinion is colored by being at a school where even the lackluster students had a year of algebra by the time they finished 8th grade, and from playing catch-up with many a struggling student whose problem was not algebra, but the basic math functions leading up to it. While it’s satisfying to have a student that the public school says is just shy of retarded, and take him up to and past grade level (he’s in college now and doing just fine), it’s frustrating to think of how many students like that couldn’t get the extra help.

That said, how much math do we really need? My scoffing at what I see as math for 13-year-olds doesn’t mean that spending time on it is worthwhile. But if it is, why not set the cutoff higher? Why not expect linear algebra, calculus (single and several variable), and differential equation for all? I didn’t take some of those, and I’m not hurting for it (how do I really know that though?), but are they any less valuable than the two years of algebra we ask for from children? Why not set the cutoff lower? The author writes, “Of course, people should learn basic numerical skills: decimals, ratios and estimating, sharpened by a good grounding in arithmetic.” But how much of that is enough? I really don’t know. Should we set the cutoff for what is useful to the typical adult? That would be low, I think. Or should we set it higher so students (who most likely have no clue what they’ll do when they grow up) have more options later in life? If we just want to teach analytical skills, does it have to be in a math class, or could it be through another method?

I’ll leave things at that, as my head is sufficiently tangled by this point. I must apologize in advance; I’m leaving for a work trip this afternoon and will only be checking in here once a day at most.

It’s too much for this thread, but we could certainly have interesting spin-off conversations for any topic that is or could be taught. Science, history, linguistics, languages, etc.

2

I’ve seen the argument made that the real problem is that we tend to teach students math that is unlikely to be useful to them, and avoid math that would be useful. Statistics for example would be of more general use than lgebra or geometry; people try to fool you with bad statistics far more than they do with algebra or geometry.

3

While basic statistics instruction is scattered throughout other classes, I certainly found the year of high school stats that I took to be valuable. I often recommend it to young folk who have a class period to fill.

4

If we’re going to pick what to teach based on necessity, then most of a standard high school curriculum is going to get thrown out. No one needs to analyze a poem or balance a chemical reaction, so why teach any of that?

The goal of a secondary education is (or at least should be) to develop a certain level of cultural literacy so that they can function in society after graduation, and to prepare students to be able to study whatever they want at a higher level. It’s easy to understand why you need to know some basic math to keep your options open post-high school, but it’s not that much harder to understand why you need basic math skills in your daily life. Yes, being able to solve a quadratic equation is far beyond what most people will need, but understanding how you get to the solution requires you to have really solid quantitative literacy, and that’s where the value is.

And let’s not kid ourselves into thinking that the notion that you can study statistics at any reasonable level without being able to do algebra is anything other than pure fantasy.

5

Yes, exactly what I was going to post. Algebra is such a foundational math discipline, ISTM that abandoning it is the same as saying, “Forget about math in all its forms more complex than multiplication tables.” There are lots of science and math paths that are immediately closed off if algebra is off the table. There’s a world of stuff that disappears as an option if algebra is not in one’s bag of tricks.

Grit your teeth and focus, children–you’ll get through it! It’s not as bad as all that.

6

That is a poor argument. Learning to solve an equation is not necessarily understanding and you can’t do either in statistics without algebra.

7

Let’s question some of those assumptions. I went to a really bad high school but also went to some really good colleges. Algebra I & II were required in high school but I basically refused to participate because I knew the teacher wouldn’t fail me. That was mostly true although I came close my senior year in Algebra II when he threatened to fail me but passed me with a ‘D’ instead. I had already gotten into college though so I didn’t care. I know extremely little pure Algebra and hate the way it is presented.

I was told that not knowing Algebra would only screw me over in math based college courses and I was terrified. I was forced to take statistics first during my Sophomore year. Not only did I not see any disadvantage to not knowing Algebra when it comes to learning statistics, I got one the the highest A’s in the class because I was willing to work hard and the concepts were new to all the students. I went on to take several other statistics classes at the undergraduate and graduate and even taught some of it in grad school each time doing quite well. I never heard any other students say they were thankful for algebra knowledge outside of algebra.

I found the same thing true with computer programming where I make my living now. You always hear people say you need abstract classes like Algebra before you can move on to the more useful applications. I strongly dispute that. I think the best way to teach logic and math skills would be to replace Algebra and other abstract math classes with few later uses with a combination of Statistics and basic level computer programming. The skills in the latter are more relevant to today’s world and also provide very practical skills that should last a lifetime and possibly even lead to a career. Algebra can still be available later for those that want to go into fields that require it.

8

OK, so that’s one self-reported data point. Odds are that you did pick up a fair amount of algebra along the way without realizing it, or that you struggled through some parts that wouldn’t have been so bad if you had a stronger algebra background.

9

My attitude is that people who don’t understand mathematics are at the mercy of the people how do. Innumeracy goes a long way to explaining a lot of actions of Congress and various state legislatures. When it comes to quantitative decision making, a lot of people are just faking their way through life.

We just had a long discussion on this topic when Rick Roach, a local school board member, came up.

http://hereandnow.wbur.org/2012/01/30/test-rick-roach

10

I was considering going after a computer science degree at UW Madison, which requires 3 semesters of calculus and a semester of math at an even higher level. I didn’t know there was a higher level. :wink: Anyway, I asked my sister, who has a Masters in CS and worked in the field, “What level of math do you really need for computer programming?”

Her answer – “Addition”. That is all, unless you are working at what they call the “bleeding edge” of the field, doing stuff like designing ever-faster computers and new computer languages.

Still, I disagree with the story in the OP. Algebra is an essential tool IMO for developing problems solving and logic skills, even if you don’t directly ever use algebra again.

11

True but algebra is far from the most of the useful math disciplines to the average person especially when it comes to mathematical literacy in everyday life. In a perfect world, everyone could study everything but the amount of knowledge needed to function in the world continues to grow. I would argue that the typical person, even college bound, does not need to know much about algebra, trigonometry, or anything but basic geometry (geometry proofs are very odd way to teach logic for example yet almost all high schools insist on it).

I believe people who aren’t going to become engineers or physicists would be much better served if they were taught statistics, computer programming, and financial math. Those teach math and logic concepts too and they also have the benefit of being immediately useful in life. High school math teachers are notoriously bad at answering the question “Why do we need to learn this?” They usually mumble something learning logic and teaching how to think rather than having a practical purpose. I say that there are math and logic disciplines that teach the same things and are useful to everyone.

12

Basic algebra and geometry teach students to think critically.

13

That is what I am talking about. So do statistics, computer programming, and related subjects plus they are useful to most everyone. Why do so many people insist on teaching people to think critically with marginally useful subjects as the vehicle when there are other alternatives?

14

Is history?

There are more career paths that I can think of that require a knowledge of algebra (e.g., any kind of engineering or mathematical science) than that require a knowledge of, say, U.S. history (e.g., um… historian? history teacher?). But most people won’t use either one in their eventual job.

It seems to me there are some things that we as a society have decided you just need to know to be considered educated. One could certainly argue that this approach is flawed, but I don’t see any reason to single out math as particularly useless.

15

Of course a vast oversimplification.

But I will grant that linear algebra is probably more important to a CS major than calculus. It’s a debate I’ve seen among math professors. I’ve come around to that viewpoint, actually. More than calculus, I think all CS majors (and pretty much all engineering students) could do well by taking a basic linear algebra course. And, of course, a lot of linear algebra will only make sense if students have been introduced to some basic algebra.

Even there, even if you go to software engineering rather than computer science, a basic understanding of derivatives and integration goes a long way. It’s debatable (obviously) how much the formality of an actual course helps vs picking up bits and pieces here and there, but getting exposed to a formal mode or train of thought often expands students minds, even if they never see that particular material ever again.

16

Why not just teach courses in Logic? That is the real question. Logic, statistics and some type of home math (budgets, etc) are what’s needed.

17

Even more than that, they teach students to think logically and in an organized fashion. Yes, algebra is necessary.

18

Algebra is a slightly more abstract version of arithmetic, and geometry is logic applied to a limited domain where it’s possible to draw pictures. Going straight into a formal logic course would be an even bigger leap than what we’re expecting now.

19

“Algebra” can encompass many things

For example, being able to see that 5x + 7 = 17 means the same thing as x = 2. This is the sort of thing often called “algebra”. But you might also say understanding this is the same as understanding subtraction and division; that it is just arithmetic by a different name. Such reasoning, at any rate, seems a basic level of numeracy which is ubiquitously useful to an extent worth requiring everyone to train in it.

Then there’s something like factoring (x^3 - 2x^2 - x + 2)/(x^3 - 3x^2 + 3x - 1) into (x - 2)(x + 1)/(x - 1)^2, recognizing that this causes the rational function to be positive on three intervals and negative on one intervals, with two zeros and one essential discontinuity as it approaches arbitrarily negative values from both sides, while asymptotically approaching 1 at both extremes, and so on, and so on. It would be a sad human who was not capable of eventually grasping this, but it is also, I will readily admit, a rare human who will actually find this to enrich their life in any meaningful way, and a common one who has no particular interest in this sort of thing.

I’m not sure what good it does to have a great many people walking around with “(-b ± sqrt(b^2 - 4ac))/2a” in ritualistically memorized form in their heads, and no actual interest in or use for its derivation or applications. This kind of thing, I don’t see why it should be mandatory for everyone to suffer through, though of course everyone should have the opportunity to learn it if they like, at any point in their life that they like, and be well-informed as to how it can serve as prerequisite for other areas of mathematics, science, and engineering.

20

That is spoken like an academically oriented person who can master such skills regardless of the way that they are presented. That is good for you but not everyone is like that. I especially question its worth at the high school level. It is like insisting that all high school students learn Latin and German before diving into essay writing because that is the only way to truly understand the roots of the English language. It is true in a way but also great overkill and unnecessary.

My position is that you learn useful skills by doing useful things and working your way up through more and more difficult problems that all have real-world applications. Algebra as it is commonly taught fails that test. I know a lot about Set Theory for example only because I became a relational database expert which depends on that understanding. I didn’t even know I was doing learning advanced math. I just knew I needed to solve certain problems that I was working on.

You can teach the skills you are targeting in a way that is both immediately relevant and instills long-term retention much better than the way that Algebra is currently taught.