Abandoning Algebra Is Not the Answer

http://blogs.scientificamerican.com/observations/2012/07/30/abandoning-algebra-is-not-the-answer/

by tokenadult • 14 years ago

91 66 14 years ago

blogs.scientificamerican.com

385668• 14 years ago

Hacker sounds like kind of an idiot. I really liked the point made that being able to analyze "The Old Man and the Sea" doesn't come up very often in day to day life, but is still a good thing to be exposed to. Additionally, I appreciate the valid opinion that most teachers, particularly elementary, and to a lesser extent, middle school teachers have little to no math education. My mother actually changed majors and became a teacher because she did not have the prior education to pass calculus, and did not think she would ever be able to pass it. She's a very good teacher, and very smart, but it's for the best that she teaches HS English, and isn't required to teach math.

natep• 14 years ago

As he says, Hacker's essay had so many flaws it's hard to know where to start. On the other end of the spectrum, this (and the linked) essays raised so many good points that it's still hard to know where to start.

rossjudson• 14 years ago

I'm not going to defend removing some of the math. What I will say is that improving understanding of statistics, generally, is something that most democratic countries desperately need.

zarify• 14 years ago

From the article, I particularly liked the bit about math being useful to doctors so that they can learn to succeed in the face of a challenging subject... as if medicine isn't enough of a challenge in itself.

I would certainly like to see more statistics in high school math, since people are bombarded with bad stats every day through the media and just swallow it up without thinking. It's hard to agree that other aspects should be dropped though (although I am in favour of repurposing some math courses to better meet the needs of the students).

comicjk• 14 years ago

This essay is still buying into the idea that algebra is "higher level," or that only people who identify as mathematicians would be affected by this. There are 1.5 million engineers in the US and 3.1 million programmers, and without algebra they would be as helpless as lawyers who couldn't read.

This essay says "Hacker is probably right that very few people use high-level math directly in their work." If algebra is counted as high-level, then no, Hacker is dead wrong. His position does not deserve even the modicum of respect which this essay gives it.

ak217• 14 years ago

Stupefying.

With friends like Andrew Hacker, does this country need enemies?

On the other hand, perhaps we can take this as an opportunity to discuss the accessibility of math education, the level of engagement and hands-on examples that teaching materials give, and how this can be improved with modern teaching aids (tablets). I know I certainly struggled with engineering-grade math in college (math was one of my majors, but not a favorite one), and I blame the combination of my laziness, poor professors (rewarded for publications, not teaching), and poor teaching materials (far removed from practicality).

I'm really excited by how much better kids' aptitude and quantitative cognitive skills will improve over the coming years for the ones who have access to tablets. New apps are popping up that make learning elementary math a joy... apps that can keep a kid's attention without adult help - starting at age 4. This will be a serious boost to whichever nations/groups emphasize it.

dinkumthinkum5• 14 years ago

I agree completely with the first part. The tablets, not so much. Perhaps they will just be using more apps like imgfave. I don't think the issue is tools or needing things like Khan Academy or anything like that. I think it is a combination of the cultural view of education, the educational, social, and economic status of those teaching, and, perhaps most importantly, parental engagement.

There are a lot of cultural issues at play here. Children aren't born afraid of math. It's not a coincidence that hundreds of years ago educated people tended to be polymaths (if not experts in many fields).

Are we likely to fix all these issues anytime soon? I fear probably not. But we don't have to go backwards.

rimantas4• 14 years ago

  > I'm really excited by how much better kids' aptitude and quantitative cognitive skills
  > will improve over the coming years for the ones who have access to tablets.

I say we will see zero improvement caused by tablets (or any other technology of this kind). This is not a technological problem it cannot and wont be solved by technology. Technology might aid a tiny bit with it, but just that.

darkestkhan• 14 years ago

This is much more complicated. Technology may add a lot, not just a tiny bit, but it has to be used in such way. While certainly apps are being created [and soon they may reach much harder topics, like indefinite integrals] the core issue is in using them - there is no point in technology alone if it is not used. It depends heavily on parents so that they can enable and encourage children to use the app but it also depends on children to use [or rather play] the app.

Technology is not solution to the problem per se - Technology only enables us to solve the problem.

jeffdavis6• 14 years ago

"Math certainly is incomprehensible to many students, but from where I sit, poor teaching is often the reason."

I'll paraphrase what I said in the last thread: algebra isn't harder than other subjects, it's more objective. Therefore, it tends to make educational fraud more visible. People rarely fail algebra and succeed in other subjects; they fail in all subjects and algebra is the only one where it can't be ignored any longer.

adharmad• 14 years ago

Can't upvote you high enough! Same can be said about any subject, mastering of which requires applying it to problems with objective answers. (Physics, Geometry...)

dinkumthinkum5• 14 years ago

Exactly, many of these students are doing very poorly in their other subjects (not all obviously, but we don't destroy the educational system over outliers) but are just being passed along. High school algebra is pretty basic, OK. I think we all know that if we are honest. In some ways, it's just too simplistic, many other nations would agree wholeheartedly. Students that are failing these classes, I think, for the most part, have other things going on, perhaps other educational deficiencies, problems at home (even if it is just a bad study atmosphere), unable to prioritize, caving into other pressures of just giving up, etc.

Parents have to be engaged in education. It's much harder without that. I suspect without parental engagement, those that succeed are really motivated self-starters and autodidacts, outliers.

saraid2162• 14 years ago

> I suspect without parental engagement, those that succeed are really motivated self-starters and autodidacts, outliers.

There's a fourth category: natural talent, which carries you so far without real effort but no further. Still an outlier, though.

tkahn62• 14 years ago

It may be more objective but it's not wholly different than how other subjects are taught. You're still memorizing a bunch of different facts (how to graph, how to change base of a logarithm, etc.). It's not like high school algebra actually gets into the underlying mathematics of the operations and why they happen to work specifically as they do on real numbers.

mrxd4• 14 years ago

In the humanities, you are taught how to understand what people are trying to do with language, so I can't help reading your comment from that perspective.

You say people rarely fail algebra and succeed at other subjects. To put that another way, let's assume that there are four sets. Set A consists of a group of people who are successful at one or more subjects, and Set B consists of people who are successful at no subjects. Set C consists of people who are good at math, and Set D consists of everyone else. Your statement implies that, for the most part, A = C and B = D.

Thus you seem to be arguing that excellence at math is the sine qua non of an educated person -- it is a defining characteristic.

What might you be trying to do with this line of reasoning? It seems fair to assume that you yourself are skilled at math, so really you are making a claim about yourself and other people who are similar to you. Your model of an educated person is based on yourself. In society, educated people are people whose judgments should be trusted and listened to, so we can conclude that the purpose of your argument is to try to get those benefits for yourself and people like you. Conversely, we can also conclude that you are trying to have those benefits removed from people who are different from you.

einhverfr2• 14 years ago

Exactly. I have a liberal arts degree and studied math basically only through first year calculus. The strange thing is, compared to most people around me, I am relatively skilled at math. I can do algebra, some calculus perhaps at least pulling some estimates of integrals and derivatives, but my real love is in humanities.

The reason why algebra is important has nothing to do with the GP's idea that it is the most objective of studies. Algebra is useful. That's it. People who know basics of algebra can use it constantly. I could see replacing geometry with a deductive logic class since that's all one studies with HS geometry anyway. I could see teaching less plane trig and more spherical trig too (I tried to teach myself spherical trig in order to better follow some writings of Ptolomy and others, and failed). But these aren't going to happen. But if you don't know algebra these doors are all closed.

dinkumthinkum5• 14 years ago

No, I think you take this too far. No claim is being made proficiency at mathematics is the sine qua non of an educated person. The issue at principii is teaching basic algebra. An educated person, in general, should not have trouble with basic algebra. We're not discussing PDEs, Cantor's diagonal argument, topology, numerical analysis, or any of that.

From a philosophical perspective, I think this is a strawman.

jeffdavis6• 14 years ago

"Thus you seem to be arguing that excellence at math is the sine qua non of an educated person -- it is a defining characteristic."

No[1]. I am saying that a person who is on a path to being educated won't let something as simple as algebra stand in their way.

The context is that some people are failing in pretty major ways in life, and the question is whether that's due to algebra requirements or not. I say that's ridiculous -- nobody that's on a path toward education will give up because of something as simple as algebra... there are other problems going on there that we are not seeing. And we should be grateful that algebra does show us that the student has fallen off the path to education, and we should be trying to identify that situation even earlier.

I am willing to be proven wrong empirically, if you can make the case that some significant fraction of people are well on their way to an education, but then it gets derailed because of an algebra requirement.

[1] I am inclined to believe that algebra (or something very similar) is fundamental to an education, particularly those parts of an education that involve reasoning, but I don't state that as a fact or even my opinion.

"Conversely, we can also conclude that you are trying to have those benefits removed from people who are different from you."

No, you can't. There are a long string of false assumptions and logical flaws in your post. You should reflect on your own reading and writing.

InclinedPlane• 14 years ago

I think you're reading the argument exactly backwards. The perception is that algebra (and advanced mathematics) is much harder than other subjects and the perception is that people can be good at a lot of things and still be bad at math. His argument is that in reality everything is taught poorly but it's easier to fudge the results when grading is necessarily subjective.

dinkumthinkum5• 14 years ago

So, the answer to people failing at basic mathematics and being unable to pass high school or college is to simply lower the standards? Why stop at mathematics? Why not stop short of anything beyond being able to write Tweets or just figure out where the signature box on a loan application is (but not understand all those funny things that come before it)?

Many want to say college degrees aren't necessary. I say fine, great that's awesome. For this NY Times article, I say, well of what utility is a high school education? I think it benefits society for people to be able to read at least enough for Google to sell people text-based advertising that most citizens can modestly understand so we should have training to a fourth grade level or so. But really, once you know "1+1" then just leave it that and it's time to go to the Nissan plant.

This just seems like someone saw the movie Idiocracy and decided "that's a good plan."

We have a bad education system in some ways in the U.S? Much of it is probably related to poverty and, perhaps, to some extent, parental irresponsibility. This is just pathetic. The idea that we should just throw away any standards is hard to take serious. For many people in this audience, it would probably benefit us greatly, financially, but if you care about society as whole, it doesn't sound very appealing.

rimantas4• 14 years ago

I am not that knowledgeable in US education and policies, but there is/was some "no child left behind" thing going on and I may be completely off, but my impression was it has something to do with all this. My (again, possibly completely wrong) understanding of it, that the end effect was "let's pretend we don't have dumb students" instead of "let's help the bright students".

If someone can shed more light on it I'd appreciate that a lot.

stinky613• 14 years ago

I may not be much better suited to answer your question than you, but my understanding goes more like: the intent was "let's use standardized testing to identify and then help weak students and to support schools that yield good overall students (according to the test results)" and the result was schools trying to game the tests to maintain/gain funding.

ten_fingers• 14 years ago

Let's see: The author of the essay is in political science. Here notice 'science'!

Now, I have to wonder: (1) In political polling, could he give a solid definition of a simple random sample? Could he suggest how to select one? Could he explain just why we often want a simple random sample? (2) Could he explain why in political polling we average to estimate a population proportion? That is, the average is a statistical estimator. Just why is it the estimator we want? (3) If he is to estimate variance, does he know which of the two common algebraic expressions is biased and which is unbiased? Can he define a biased estimator? (4) A common topic in political science is principal components analysis, and key there is orthogonality in vector spaces. Can he suggestion how to give a solid description of that topic without algebra? (5) Political science likes to construct quantitative measures, and there two important issues are 'reliability' and 'validity'. Can he give solid discussions of these two issues without algebra? (6) Political polling reports results with an 'margin of error'. Could he define that? The usual calculation is based on the central limit theorem; could he describe that? (7) One of the more important research tools in political science is statistical hypothesis testing, especially nonparametrics. There the core concepts are Type I error, Type II error, power of a test, and a most powerful test. How can he describe this material without algebra?

Uh, a first course in algorithms commonly described heap sort and points out that its execution time, on sorting n items, is O( n log(n) ) in both worst case and average case and, thus, is best possible in that it meets the Gleason bound. Without algebra, how is a student to understand the significance of that material?

The world is moving on to more in automation. The main approach here is computing where we describe what we want and how to get it in terms that are essentially mathematical -- e.g., stiffness of arbitrary space frames, finite element analysis in solid mechanics, Reynolds number in fluid flow, sines, cosines, and exponentials in A/C circuit analysis with passive components, force, torque, energy, power, and momentum in mechanical engineering, etc. All these subjects commonly assume first and second year algebra, plane and solid geometry, trigonometry, analytic geometry, calculus, and ordinary differential equations.

For political science, we should include linear algebra and multivariate statistics.

That author wants less algebra but more statistics? That's like saying he wants to lose weight but eat more!

Yes, to too great an extent, public K-12 is more about babysitting, i.e., keeping the kids off the streets, than about education. Moreover, really, math is too difficult for essentially all of the US K-12 educational community.

scotty79• 14 years ago

USA already abandoned knowledge of evolution as visible here: http://www.youtube.com/watch?v=UkBmhM0R2A0

Now it's time for math as imagined here: http://www.youtube.com/watch?v=9QBv2CFTSWU

wikkiwa• 14 years ago

Let's just have the chinese do our math and then pay them with the money we borrow from them.

tkahn62• 14 years ago

Any discussion about mathematics that fails to distinguish between computation-based and proof-based mathematics is not going to be a very informed discussion.

There's a reason that so many would-be math majors switch out when they reach Intro to Proofs. Up until then it's just memorizing formulae and recipes; you're just a glorified TI-89.

biggfoot• 14 years ago

When you start defining education & learning 'objectively' you can kiss progress goodbye. That we don't directly apply most of what we learn is a bad way to look at why we learn stuff in the first place.

nyrulez• 14 years ago

Instead of getting all riled up, it's fair to realize that there is such a thing as free market, demand and the corresponding motivation to supply. Folks write all sort of controversial opinions (there was this essay a few months back from a UCSD professor in Bloomberg about how CS education is overrated and the folks in CS don't make enough money).

My response to these kind of articles is: everything is fair game. If someone or some community thinks CS or math is not for them, it's their right. It's just that the demand will be filled up by folks who do believe in it (like folks from outside US which constitute a significant portion of most CS/tech firms inside US). Ultimately, if some folks in US realize they are losing too much potential out there by refusing to raise their bar, they might re-orient towards the demand. If they don't realize this/stuck in their stubbornness/incompetence, then they didn't probably deserve it anyway (or don't care).

Agreed that there is short term pain for folks who get effected by these kind of policies outside their choice.

retrogradeorbit2• 14 years ago

Free market. Seriously? There are no free markets anywhere in the world today.

tsotha• 14 years ago

>Hacker’s first main point is that math is difficult, and the poor grades that result prevent too many people from graduating high school or college.

Yeah, great plan. Dumb down high school and college even further so even a monkey can end up with a BA. Then we'll all have great jobs.

thisrod• 14 years ago

Forcing rooms of teenagers to factor cubic polynomials for hours on end is pointless, of course; these essayists fail to consider that pointlessness might be the point of school, as it is of boot camp. Read John Gatto instead.

retrogradeorbit2• 14 years ago

I found the late George Carlin had an even better explanation of the New York Times piece.

https://www.youtube.com/watch?v=9dY4WlxO6i0

WalterBright• 14 years ago

My accounting prof used to be a used car salesman. I was talking about that with him one day, and he said that a car dealership doesn't make money selling cars, but makes it selling financing.

The people who really pay through the nose for financing are people who do not grasp mathematics. He said it wasn't just a matter of them being taken advantage of - even when he explained the lower cost option, they'd insist on the higher cost one, based on faulty notions about math.

If you don't learn math, it's going to cost you plenty your whole life, and you won't even realize it. Math is necessary, even if you don't go on to get a STEM degree.

saraid2162• 14 years ago

Well, based on your story, the important thing is not really to know math, but to not be incorrect about knowing math.

theorique2• 14 years ago

a car dealership doesn't make money selling cars, but makes it selling financing.

This also scales to the production level. Prior to the recent troubles, GMAC, the financing arm, was larger and more profitable than GM, the auto maker.

tpr1m• 14 years ago

For years and years I've wasted time in the classroom with Algebra, a skill I will _never_ need in my lifetime. As a computer science major, I find it perfectly equatable to make my math teacher sit through the same amount of time in some of my major-specific classes, since he will get just as much value out of them. Not to mention I have to pay for it.... difficult or not, most math classes simply are not useful. This differs from other "general" classes, such as English.

rimantas4• 14 years ago

As a programmer I use the most basic skill algebra teaches— manipulating abstractions—every day. I don't get how programmers claim that they have no use in thinking.

jinfiesto3• 14 years ago

I find this to be short-sighted. I also program, and have found Algebra to be very useful. Solving a problem trivially with Algebra can sometimes save you lots of programming work. Granted I don't use Algebra in my day to day, I'm glad I know how to do it and have found it useful often enough to say I'd be less of a programmer without it.

theorique2• 14 years ago

A computer science major who never uses algebra? I call shenanigans.

jinfiesto3• 14 years ago

This was an interesting read. I think the author of the original paper though has some valid points and that this response labeling the original paper "anti-intellectual" is dishonest in that regard.

In my opinion, the problems we have with Mathematics education in this country stem more from how the curriculum is structured than from particularly bad teaching or lack of natural ability. Looking back, it seems bizarre that all of elementary school gets devoted to performing arithmetic and then Algebra is suddenly introduced sometime in middle or early high school. In my opinion, arithmetic can be taught much faster than it is (Trachtenberg system et alia) and basic Algebra should really be introduced concurrently.

The problem with this approach is that you can no longer sell neatly compartmentalized textbooks labelled "Addition," "Subtraction," ... , "Algebra" etc...

The other thing that bugs me is the bizarre choice of topics taught in Algebra classes. I remember, I was forced to memorize the binomial theorem. At the time, it really made no sense. I'm now of the opinion that you shouldn't be taught math you don't have the machinery to develop on your own. Knowing arithmetic, you can build Algebra (not very rigorously, but the idea seems pretty obvious.) It's a much bigger leap to develop the binomial theorem, which requires more advanced machinery, which is generally not taught.

Of course, once I could derive the binomial theorem on my own, it made much more sense.

I did very well in math throughout my academic career, but didn't ever really feel like I had very much mathematical skill. Only now that I've developed an interest in doing math in a recreational capacity that I really feel like I've developed any sort of substantial mathematical ability. I'm really now convinced that the problem solving approach is the way to go and that this is how early math should be taught.

mangodrunk2• 14 years ago

Now that you mention it, it does seem that too much emphasis and time is spent on counting and calculations. A symptom may be how some people equate doing fast calculations with being good at math.

jinfiesto3• 14 years ago

It's funny how that escapes detection. I only realized looking back how baroque that was. I think people do equate doing fast calculations with being good at math. While I think everyone should certainly know how to add, subtract, multiply and divide without error, there's really no reason to be able to do so obscenely quickly, given the wide availability of calculators. If you did want to be able to do very fast mental math, there are much better systems than what are typically taught in schools.

Really though, my point was that the school system spends far too little time on solving substantial problems in Math classes. I think an integrated problem solving approach that touches on many of the important branches of Math (graph theory, number theory etc...) would serve most people much better than the current system, where we spend 8 years building up to Algebra and then engage in a smattering of Geometry and Trigonometry.

In the current system, if you're smart, you do AP Calculus, which is obscenely dumbed down. If you want to proceed to study Math in college, you'll almost certainly have to re-teach yourself calc to be successful.

johnx123-up• 14 years ago

Site crashes my Chrome and IE. Probably as Xdm.swf is pinging for every second.

Safe URL: http://viewtext.org/article?url=http://blogs.scientificameri...

scrame• 14 years ago

Innumeracy should be treated like Illiteracy.

cafard• 14 years ago

I buy the "no bad students, only bad teachers" argument--though I must say that might teachers from 8th through 11th grade were good to very good. I buy the "more objective" arguments I see here and elsewhere. The "push-ups for your brain" maybe not so much.

ctdonath2• 14 years ago

The biggest un-addressed problem is sheer lack of will. A significant percentage of students will not do the work. They'll show up, they'll go thru motions, but will not do the actual work required. Telling them they'll fail won't help: they either don't care or don't comprehend. Punishments won't work because punishments are forbidden. Offering more assistance and resources won't help because those are tools, useless if unused. They do nothing, life goes on in an acceptable manner. They have no reason to do the work, so they will not do the work, and thus will fail.

ctdonath2• 14 years ago

As a counter to any complaint about how hard algebra is: recall the recent HN-featured app DragonBox. It strips away the boring/confusing irrelevancies of the subject, replacing the appearance with cute tiles and animations - creating a game where a kid learns the core concepts. If my 4-year-old (and even my 2yo) will sit there for prolonged periods playing and picking up rules & techniques on her own, then surely high school students can grasp the subject - if they choose to.

http://www.hnsearch.com/search#request/submissions&q=dra...

alter8• 14 years ago

> Eliminating abstract math education in the early school years, or allowing young students to opt out of rigorous math classes, will only serve to increase the disparity between those who “get it” and those who don’t. Those who have a grasp of mathematics will have many career paths open to them that will be closed to those who have avoided it.

This is currently true on programming, which is (was?) left out of the basic curriculum.

imgabe• 14 years ago

The real problem I think, or at least part of it, is that it's considered acceptable for kids to be declared "just not good at math" and left at that. It's presented as a skill that you're either born with or not, instead of one that can be improved with practice. We wouldn't find it acceptable to declare a kid "just not good at reading" and give up on teaching him to read, so we shouldn't do the same for math.

mangodrunk2• 14 years ago

It's a little different though. I think a better analogy would be writing, and I have seen people claim to not be good at writing as others do with math in general. Reading would be like counting in math, and most people can do that. I agree though, that with practice many would probably benefit from it for either math or writing. But there will be some people, who will just be not good at math short of some upgrades to their brains.

alberthartman• 14 years ago

Math, especially algebra, is essential to all technology. Counting and measuring and projecting forward are foundational skills. The thing is that there are two basic aspects to algebra, one is conceptual and the other is mechanical. The conceptual part is interesting and worthwhile to most people. The second thing is what drives people off - the long detailed mechanical aspect of accurately doing long equations, perfect transcriptions, looking for simplifications and cancellations. But it is precisely this laborious mechanical aspect that has been drastically improved with modern symbolic math software like Mathematica and Mathcad, et al. Factor a twenty term formula, do any integral, or differentiate a messy equation in about, oh, half a second? The failure is that we don't teach these software skills right at the start.

How boring would writing be if we didn't let students use word processors with their perfect erasers, spelling checkers, scissors, tape, carbon paper, and endless new blank sheets? Or if we forced all architects and engineers to use rulers, pencils, Leroy lettering guides, and E-size paper on drafting boards every time they wanted to design something for others to build, not allowing them to use Solidworks or Autodesk, or equivalent modern tool to help? Ditto for movie making, or pretty much everything else.

Algebra is great. Teach it along with teaching the modern symbolic math tools that go along with it.

numeromancer• 14 years ago

The original article is another iteration from the ancient and venerated Academics with Inferiority Complexes Society. Here is an article discussing an earlier effort from this time-honored society:

http://www.sourcetext.com/grammarian/newslettersv06/6.1.htm

This article is by the late malcontent newsletter The Underground Grammarian, a diabolical work which has cleverly trapped me into spending entire nights in the wicked attractions of its addictive prose.

An à propos quote from another part of the site:

However, while the retreat from the measurable provides comfort for the educationist[and, apparently, the occasional “political scientist”], it makes it hard for him to claim, as he would so dearly love to, that "education"[...] actually is a body of knowledge and that his Faculty Club card should not be stamped: "Valid only when accompanied by an adult." What a dilemma.

dougabug• 14 years ago

Sad that the march to abandon reason has reached this far, where even the most rudimentary mathematical knowledge is scorned. Maybe degrees should be given out based on attendence and self esteem. Without algebra, people could not evaluate even at a broad level the tenability of a scientific or technical claim, they would be denied any form of meaningful understanding of probability, algorithms, complexity, physics, chemistry, biology, astronomy, even linguistics has deep roots in algebraic notions. To abandon the teaching of algebra would be to condemn the public to a life of ignorance and obedience to crowd think.

Abandoning Algebra Is Not the Answer

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