Before I get to worrying about algebra, Andrew Hacker’s essay in the Sunday NYT made me worry about writing and research. As in, “This is poorly written” and “I don’t think he did much research”.

If I were marking the column up like a first-year student’s paper, I’d immediately be all over the meandering, confused structure of the essay and its tone-deaf alternation of tremulousness and tendentiousness (a combination that a lot of Hacker’s writing in recent years has demonstrated). And then I’d mark it up for the weakness of the research behind it. All of the questions he’s asking have been asked before and debated at length in the history of American education in general and mathematics education in specific, but much of the affect of Hacker’s own essay is of the discovery of some long-ignored or never-asked question. Most crucially, he never really asks (or looks into) the basic question, “So why do most mathematics educators believe so strongly that algebra is an important objective in K-12 education?”

The way that Hacker frames the issue is consistent with the corrosive form of populism he’s been peddling lately, that he is uncovering a sort of “educators’ conspiracy” which has no real explanation other than the self-interest of the educators. If he were to frame it as, “This is an interesting on-going debate where the various sides have coherent or well-developed arguments that have both technical and philosophical underpinnings, and here’s the side that I’m on”, he’d be doing a public service. As it is, he’s just yanking some chains, either calculatedly or out of feeble cluelessness.

If you were going to reassemble the column so that it built up to a genuine argument, I think it might look something like this:

1. Algebra is a common part of the mathematical education of most Americans as well as in other school systems around the globe. By way of introduction, here’s what algebra is. (Baseline definition.)

2. Algebra is a common stumbling block for American students, far more than other subject they study in K-12 education. (Evidence thereof, which Hacker cites fairly well.)

3. Why do we believe algebra is an important educational objective? What do mathematicians, educators and others say about this? How did it get into the common K-12 sequence?

3a. Because algebra is believed to be an important conceptual precursor to every other form of advanced mathematical thought and inquiry.

3b. Because algebra is believed to be an important practical precursor to mathematical skills used in many professions. E.g., because “you will need it later in life”.

3c. Because algebra is believed to be “good to think”, a way to get high school students to regard mathematics as a form of critical and imaginative thought rather than an area of rote calculation.

Here’s where Hacker really falls down: these questions aren’t evaluated or explored a remotely systematic or coherent way.

4. Are these assertions true?

4a. Could you learn other fields of advanced or practical mathematics without any knowledge of algebra? Or is there a simple knowledge of algebra that is sufficient for certain kinds of progression?

4b. Do many careers really use algebra, or have knowledge of algebra assumed in their use of quantitative skills and data? Are there everyday uses of algebra that are important to an educated citizenry?

4c. Is algebra really useful for quantitative forms of critical or imaginative thought?

5a. If 4a. and 4b are in fact true, is there a different or better way to teach algebra that would allow more students to progress successfully through it, or at least to mitigate or excuse their inability to do so? If 4a and 4b are not true, why do we believe them to be true?

5b. 4c presupposes that the goal of high school is progression towards critical and imaginative thought. Are we sure that should be the case? If it’s not the case for math, shouldn’t that be true for everything? Maybe this is an argument against high school *in toto*, at least as it commonly exists?

5a is where Hacker’s essay seemed to me to just crash and burn. He fumbles around in the dark when he concedes that yes, it’s important for people to be quantitatively literate both as citizens and for their employment prospects but that no, you don’t need algebra for either. I’m not particularly quantitatively literate myself, but I think trying to read and work with statistical data with no knowledge whatsoever of algebra would be very difficult. I don’t know how you’d do anything with algorithms without having some conceptual grasp of algebra. (Just to mention two of the things that Hacker agrees citizens and employable people ought to be able to do.)

It might be that there is a different way to approach algebra that helps high school students glean some of its conceptual value, that there is a problem with how it is commonly taught or imagined. I’m sympathetic to that general question about most high school education. For example, while I think the study of literature or history and the craft of analytic writing should have a *progression* throughout high school, there’s plenty of room to question what *kinds* of literature students should read, or what ways they ought to study and know history. But this sort of terrain is way too sophisticated and subtle for Hacker, who is really doing a lot to degrade the brand value of expertise lately.

I can’t disagree with a single thing you say about the value of learning/knowing algebra. But I can’t get around the fact that for me it was simply impossible to learn; even more difficult than the rest of math, which was excrutiatingly difficult. I have no idea why. The requirement to learn (or rather, to pass) algebra was entirely counterproductive in my case, and I doubt I’m alone. Imagine being required to step up to the roof and soar into the sky–repeatedly. It’s Kafka-esque: a regime of mindless punishment.

People who had little trouble with algebra have neither patience nor sympathy for this kind of argument, I know. But it does zero good to force people do do something they cannot do. I made a blog post about it.

I completely feel your pain on this, as I struggled horribly with it myself, and barely managed to get through Algebra II. (For some reason, I had no problem at all with Geometry.) I think it was a combination of how my mind works on a fundamental level and how the subject was taught (in multiple ways) when I was in high school. It wasn’t the right time for me to learn it and I might not have been or never will be the right person to learn it. But I understand now far better what it’s for and what you can do with it, and I think it’s a terrible mistake to argue against it as Hacker does–his argument, such as I can extract it out of that piece, would be an argument against ANYTHING taught as exploratory, conceptual, or imaginative–it is an argument that all subjects should be taught in grimly vocational and utilitarian ways. Which, judging from his other recent attacks on higher education, might be exactly what he believes.

Isn’t it the “baseline definition”, step 1 in your proposed outline, where he’s really stumbling?

If he agrees that quantitative reasoning and statistics and algorithms are things everyone needs, both as part of their job and just so they can participate in public discussions as citizens, but he thinks that “algebra” is something arcane that’s less necessary, then that almost certainly means he’s using the word “algebra” to mean something other than what you or I might mean by it.

Maybe he would say that symbolic manipulation of quantitative things is important but that most people will never need to factor a quadratic equation. If that’s what he means, then he might well be right. Quite possibly we should restructure math education, and quite possibly this means that some of the things we typically teach in 9th grad algebra ought to be taught earlier and others later, or not at all.

Yeah, that’s why I stuck that point in there, because I have the same suspicion you do–that he is taking some particular part of algebra and having it stand in for the whole. There might well be questions of sequencing, pedagogy, and so on that are totally legitimate.

I also think that Hacker gives no consideration at all to education as the development of potentiality–or that what consideration he’s giving is creepy and disturbing. E.g., if you take him seriously, he’s saying, “If you find algebra difficult at the first go, then you should track immediately into the ‘I don’t need algebra’ classes. If unfortunately for you at age 19 you start to find computer science kind of interesting abstractly, you are then totally hosed. We expose everyone because we don’t want to take on the task of presorting people into social classes at age 13. Hacker is apparently quite willing to do so, which is another reason to call his appeals to populism into question.

It was my understanding that one of the biggest problems with math education in the US is that we try to teach quantitative and abstract concepts before children are ready for those concepts. His solution? Lower math standards, teaching only what everybody needs, and start teaching it earlier. If we took his argument to other subjects, we’d only teach photoshop in art class, because that’s all that’s practical. I’m not sure we’d teach history at all.

It boggles the mind.

Timothy,

I completely agree with you. It seems weird that he would posit the problem is with teaching algebra, and not the timing, sequencing, or pedagogy behind math education. Bjorn, I might pose the opposite question: Why do we teach algebra so late? Why don’t we teach abstract concepts at a point in which the mind can most readily accept it, like at 4 or at 6? My niece is 7 years old, and entering a pilot program in which basic math and physics concepts are introduced in third grade. Up until this point she has learned up to two digit multiplication. While I am only speaking anecdotally, is there room for a movement towards introducing math concepts earlier (as it has been shown to be beneficial to begin language education as soon as 2 or 3 years old?)? Maybe our problem with math is that we begin to reach difficult and seemingly abstract subjects right as our brains are hard wired to think about learning and to adapt to learning in a very strict way. I’m just throwing that idea out there. I don’t have studies right off the bat to support that.

In most schools, math concepts are taught early. I’d say by 3rd grade, many students are learning algebra concepts even if they aren’t called that. Word problems, for example, often use concepts from algebra. In the younger grades, students use different strategies than they do in older grades.

I’d actually advocate–selfishly–for teaching computing sooner. Then you have a very practical field to use algebra with. Making a simple game like pong requires serous algebra, and then you can see why it’s useful. It’s also fun to get the computer to do all your calulations for you, but you still have to tell it what to do.

I do feel like students should be able to forgo math after algebra, which in most schools, they can’t. Maybe they’ll come back to it later in life when it makes sense. I also think, in addition to computing, more schools should offer courses in statistics or data analysis, more good applications of math.

Funny that anyone would ask that question at all. Because some people might not be good at a subject then we should dumb it down or not teach it at all?? I teach many students at the high school level and many of them are in calculus and above (which I would have never thought possible when I was in high school). Many students do well at math and if encouraged and fostered in the right way, I think many students would do better. Parents and teachers as a whole have failed in supporting students by encouraging them to perform at a higher level than what is expected at a public school. They fail by accepting failure.

I don’t think anyone making the claim that algebra shouldn’t be taught just because students might be bad at it, is making an intelligent argument. Although, I think it is safe to say that many of us don’t use algebra or higher level math on a regular basis, there does come a certain satisfaction from the fact that if confronted with it we would not be at a complete loss. Just the same as reading classic American literature – I hated much of it, but was required to learn about it anyway. Does that mean I shouldn’t have been exposed to it? If his argument had been that the way math was taught needed to be changed and had some suggestions as to what that might look like, then possibly he might have a good point. However, doing away with a core subject because it may set people up for disappointment is much like saying everyone should get a participation award. Real life is full of disappointments and that is what teaches us to strive to do better. Encourage children to do better and support their ability to think and they will do just that.

There’s another aspect of 5a where the essay really seems to fall, short: the question of whether our teaching is at fault in so many people’s inability to get algebra even as it is currently formulated. At this point I think a look at the data from a bunch of different nations/educational systems is in order. I seem to recall that the US is especially bad in math compared to some other things relative to other places. Obviously this is complicated, one would need to sort out the affects of tracking on the test population vs those of teaching on the scores, etc. But it probably too easy (and self-congratulatory) to assume US scores are lower a higher percentage of people take the tests – other places now have more economic mobility, is this correlated with more democratic educational systems.

Then of course we could get interesting: if tracking is more acceptable in Germany than the US is it because the financial stakes in the choice between professional-managerial class and other kinds of work are less stark? How is our school system shaped by its ideological function of demonstrating that income inequality is based on merit (or in what amounts to something very similar, by the hopes we place on it for solving the problem of poverty)? As for Hacker’s animus towards educators, I think it appeals (if it isn’t driven by) the way teachers function as gate-keepers for mobility.

I did well in and enjoyed English and history and struggled with and disliked math for most of elementary and high school. Subsequently I ended up going to graduate school in physics and am currently co-investigator on a grant from the mathematics division of the NSF. So I think I have some perspective on the “math is hard” issue from both sides.

I think quadratic equations keep getting mentioned because they play such an outsized role in high school algebra. I can recall endless homework sets that consisted of solving various quadratic equations over and over again with no context whatever. I can see how that would discourage people (it certainly discouraged me).

Hacker’s piece is worse than useless because it distracts from a real issue (can we do a better job at teaching algebra in high school?) and replaces it with “who needs that stupid old algebra anyway?”

BTW, did the New York Times editorial page always hate education as such as much as they have in the last 5 years? It seems like anyone who a) has never taught a class, and b) is willing to say that we should abolish tenure/get rid of algebra/crush teachers unions/replace everything with YouTube videos, is welcome to some space on the editorial page these days.

I like Laura’s suggestion very much, partly because it coincides with an argument I’d make about literature and history in junior high and high school–that when those subjects alienate students, it’s often because they’re taught using materials that a teenager can’t generally connect to or taught with no sense of their relevance or importance to the world. I’d almost argue that taking a break from math for three months in the fifth grade and trying to program Lego Mindstorms robots would be a better lead in to algebra than anything else they’re asking kids to do.

Stephanie (in particular), I’m old enough to have experienced “new math” in the 60s. The intent was blameless, to introduce concepts like set theory and use them as the underpinning for arithmetic. (“Why do we carry the one?”) Unfortunately the whole project wound up crashing and burning: as I dimly remember it, there were issues ranging from teacher training to overall skepticism about the whole idea. And guess what, not everybody is good at everything, and some kids turned out to be unable to grok set theory, and it all went down in flames. I would not be surprised if there’s lingering skepticism in the mathematical educational community.

I’m a software engineer so I don’t really have a dog in this fight: my profession requires facility with algebra and “numeracy” in general. But every time you face a question like, “my car gets 32 MPG, and I’ve gone 300 miles, and I have an 11 gallon tank, so am I in trouble yet?” you’re facing algebra, aren’t you? Even if you call it something different?

Is it possible that this is Andrew Hacker’s second (on record) unnoticed satire? http://news.google.com/newspapers?id=8hpQAAAAIBAJ&sjid=7VYDAAAAIBAJ&pg=2514%2C2554084

His other articles seem to offer a much different perspective on the ability of all students to participate in college-level education. http://www.thedailybeast.com/articles/2011/08/28/college-rankings-2011-everyone-should-go-to-college.html

This is mostly not new, but I’d like to pull out a strand of criticism in some of the above comments and flag it as being a particularly salient flaw in the article, one where I thought it went completely off the rails to begin with in its initial framing. What is this unified monolith called “algebra” that we are either to keep in toto or eliminate completely?

Also, it’s silly to discuss this in isolation, without exploring its impact on other parts of the curriculum. You’re also making a decision about (e.g.) what physics you will teach.

Standard disclaimer: didn’t have an American education and don’t (for instance) have a clue what “Algebra II” means.

Well, neither did I, and I actually took the class!

Anyway, what I recall is that in the early 1980s it was a ton of polynomials, quadratic equations, graphing complex numbers, logarithms.

At the private school I teach it, we definitely shifted from “everybody has to take calculus” to everybody has to take statistics in the last two year. We adopted “Everyday Math” about 6 years ago and our first kids who started it should be getting to algebra soon. We’ll see if it makes a difference. But it’s not like there hasn’t been a ton of research on Everyday Math (and other competing programs) that teach math concepts and real world applications. I’m pretty pleased with the way it’s played out for my kids so far (1st and 3rd grade). However, we have had a ton of teacher training in everyday math for our lower school teachers and have a super math coordinator (who also teaches in the Upper School when needed). We have enough confidence in our Math teachers in MS that they were told to cut the time teaching for everyday math in MS from 1 hour to 40 minutes a day without sacrificing content and pulled it off no problem.