## Does mathematics have a place in higher education?

A recent New York Times Opinion piece (hat tip Wei Ho), Is Algebra Necessary?, argues for the abolishment of algebra as a requirement for college. It was written by Andrew Hacker, an emeritus professor of political science at Queens College, City University of New York. His concluding argument:

I’ve observed a host of high school and college classes, from Michigan to Mississippi, and have been impressed by conscientious teaching and dutiful students. I’ll grant that with an outpouring of resources, we could reclaim many dropouts and help them get through quadratic equations. But that would misuse teaching talent and student effort. It would be far better to reduce, not expand, the mathematics we ask young people to imbibe. (That said, I do not advocate vocational tracks for students considered, almost always unfairly, as less studious.)

Yes, young people should learn to read and write and do long division, whether they want to or not. But there is no reason to force them to grasp vectorial angles and discontinuous functions. Think of math as a huge boulder we make everyone pull, without assessing what all this pain achieves. So why require it, without alternatives or exceptions? Thus far I haven’t found a compelling answer.

For an interesting contrast, there’s a recent Bloomberg View Piece, How Recession Will Change University Financing, by Gary Shilling (not to be confused with Robert Shiller). From Shilling’s piece:

Most thought that a bachelor’s degree was the ticket to a well-paid job, and that the heavy student loans were worth it and manageable. And many thought that majors such as social science, education, criminal justice or humanities would still get them jobs. They didn’t realize that the jobs that could be obtained with such credentials were the nice-to-have but nonessential positions of the boom years that would disappear when times got tough and businesses slashed costs.

Some of those recent graduates probably didn’t want to do, or were intellectually incapable of doing, the hard work required to major in science and engineering. After all, afternoon labs cut into athletic pursuits and social time. Yet that’s where the jobs are now. Many U.S.-based companies are moving their research-and-development operations offshore because of the lack of scientists and engineers in this country, either native or foreign-born.

For 34- to 49-year-olds, student debt has leaped 40 percent in the past three years, more than for any other age group. Many of those debtors were unemployed and succumbed to for-profit school ads that promised high-paying jobs for graduates. But those jobs seldom materialized, while the student debt remained.

Moreover, many college graduates are ill-prepared for almost any job. A study by the Pew Charitable Trusts examined the abilities of U.S. college graduates in three areas: analyzing news stories, understanding documents and possessing the math proficiency to handle tasks such as balancing a checkbook or tipping in a restaurant.

The first article is written by a professor, so it might not be surprising that, as he sees more and more students coming through, he feels their pain and wants their experience to not be excruciating. The easiest way to do that is to remove the stumbling block requirement of math. He also seems to think of higher education as something everyone is entitled to, which I infer based on how he dismisses vocational training.

The second article is written by a financial analyst, an economist, so we might not be surprised that he strictly sees college as a purely commoditized investment in future income, and wants it to be a good one. The easiest way to do that is to have way fewer students go through college to begin with, since having dumb or bad students get into debt but not learn anything and then not get a job afterwards doesn’t actually make sense.

And where the first author acts like math is only needed for a tiny minority of college students, the second author basically dismisses non-math oriented subjects as frivolous and leading to a life of joblessness and debt. These are vastly different viewpoints. I’m thinking of inviting them both to dinner to discuss.

By the way, I think that last line, where Hacker wonders what the pain of math-as-huge-boulder achieves, is more or less answered by Shilling. The goal of having math requirements is to have students be mathematically literate, which is to say know how to do everyday things like balancing checkbooks and reading credit card interest rate agreements. The fact that we *aren’t* achieving this goal is important, but the goal is pretty clear. In other words, I think my dinner party would be fruitful as well as entertaining.

If there’s one thing these two agree on, it’s that students are having an awful lot of trouble doing basic math. This makes me wonder a few things.

First, why is algebra such a stumbling block? Is it that the students are really that bad, or is the approach to teaching it bad? I suspect what’s really going on is that the students taking it have mostly not been adequately taught the pre-requisites. That means we need more remedial college math.

I honestly feel like this is *the perfect place* for online learning. Instead of charging students enormous fees while they get taught high-school courses they should already know, and instead of removing basic literacy requirements altogether, ask them to complete some free online math courses at home or in their public library, to get them ready for college. The great thing about computers is that they can figure out the level of the user, and they never get impatient.

Next, should algebra be replaced by a Reckoning 101 course? Where, instead of manipulating formulas, we teach students to figure out tips and analyze news stories and understand basic statistical statements? I’m sure this has been tried, and I’m sure it’s easy to do badly or to water down entirely. Please tell me what you know. Specifically, are students better at snarky polling questions if they’ve taken these classes than if they’ve taken algebra?

Finally, I’d say this (and I’m stealing this from my friend Kiri, a principal of a high school for girls in math and science): nobody ever brags about not knowing how to read, but people brag all the time about not knowing how to do math. There’s nothing to be proud of in that, and it’s happening to a large degree because of our culture, not intelligence.

So no, let’s not remove mathematical literacy as a requirement for college graduates, but let’s think about what we can do to make the path reasonable and relevant while staying rigorous. And yes, there are probably too many students going to college because it’s now a cultural assumption rather than a thought-out decision, and this lands young people in debt up to their eyeballs and jobless, which sucks (here’s something that may help: forcing for-profit institutions to be honest in advertising future jobs promises and high interest debt).

Something just occurred to me. Namely, it’s especially ironic that the most mathematically illiterate and vulnerable students are being asked to sign loan contracts that they, almost by construction, don’t understand. How do we address this? Food for thought and for another post.

Please do follow through with number 5. You could turn it into an interesting blog entry, then, perhaps, work it into a good op-ed piece.

“Finally, why did it take so long for the media to pick up on LIBOR manipulation? It tempts me to make a list of the illegal stuff that we all knew about back then and send it around just to make sure.”

Looking forward to it…..

Best,

Théo

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“First, why is algebra such a stumbling block? Is it that the students are really that bad, or is the approach to teaching it bad?”

Neither. It’s because algebra is difficult.

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No, it is not difficult, students don’t care. How many hours does it take to master a video game?

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Algebra is difficult. It’s the first soiree into the abstract world of math, away from counting on your fingers and measuring geometric shapes. It requires a developed ability in understanding and forming abstract ideas, which I would wager is where many kids struggle. They’ve tried to make it more graphic for the visual learners, but they really should be focusing on logic. When people tell me they can’t do math, what I hear them telling me is that they haven’t been taught to think logically.

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It’s not difficult. It’s basic symbol manipulation and as such is entirely rule-based. One can actually *learn* that without understanding a damn thing.

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And what could be more useless than knowing how to manipulate symbols without understanding them deeply? The real world is composed of “situation” not symbols. Everything is a word problem. Who can solve a real-world problem requiring algebra without being able to translate a situation into symbols first, THEN manipulate the symbols, THEN translate the symbols back into a situation?

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Saying “algebra is difficult” is the same as saying “students aren’t smart enough to learn algebra easily”. The former is a more polite way of putting it, but both of them boil down to the same issue of the subject being more complex than the students can handle.

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@JSF It’s also because, by that point, algebra crosses the line from general interest to focused learning that people take as they begin to select what they want to do in life. If a person is not interested in a science career (or something else that would use algebra heavily) then their commitment to learning is greatly reduced/eliminated.

Put simply: “yeah I like to eat pizza but I’m not about to become a career chef.”

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It’s also because algebra crosses the mental/emotional line from general mathematics and general learning where the only ones committed to learning it are those who are truly interested in it. If a person doesn’t wish to have a career where algebra might be important, learning it and doing well at it is that much harder.

The question is: if someone wishes to become, say, a public speaker, beyond basic mathematics, what incentive do they have to learn advanced mathematics and algebra?

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“I honestly feel like this is the perfect place for online learning…ask them to complete some free online math courses at home or in their public library, to get them ready for college. The great thing about computers is that they can figure out the level of the user, and they never get impatient.”

I disagree, and my view is essentially the same as Keith Devlin’s:

http://devlinsangle.blogspot.com/2012/07/cant-we-all-get-along.html

“The real problem is that the US (and other nations) identify mathematics learning with instruction and passing procedural tests…Sal Khan says he is trying to move into the real, conceptual learning space as well, but so far I have not seen much that would qualify, and as I noted earlier, my own interest in trying out the MOOC format notwithstanding, I have yet to be convinced that it is possible over the Web. ”

Online students learning the wrong things. What’s more, when you remove the teacher, the quality of the instructional materials becomes that much more important, and I have found most materials, online and off, to be mediocre at best.

Here’s my view on why students cannot read a credit-card statement — students cannot read. They literally cannot read technical material. (I’ve yet to hear anyone brag about it though.) I’ve sat down one-on-one several times and asked struggling students to read passages, and watched as they sort of skimmed through the symbolic elements and then started to tell me some made-up incorrect version of the context in which those symbols lies. I’ll ask them to read aloud, and they usually hesitate for quite a while before reading too quickly, skipping words as they go. They don’t have the skill to read technical material, because in 13+ years of schooling no one has ever ever ever required that they develop it.

Here’s an example: http://www.youtube.com/watch?v=BbX44YSsQ2I&feature=related

I truly don’t believe the math here is too hard for most people to understand. Instead, I believe that the contestant and most of the audience have never had to read and correctly do a problem with wording that intricate. I think most people just saw MATH, gave it a quick skim, and then guessed based on a few words they recognized in the sentence. Most students leaving twelfth grade have lost so much confidence that they will refuse to even give a task a focused try if they realize it has technical content.

I honestly believe the perfect solution to remediation is community colleges.

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@JSE, I don’t think algebra is difficult. I, for one, think it has to be the teaching approach.

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I, for another, agree.

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Probably almost none of the people who read this blog found algebra to be difficult. That doesn’t mean it isn’t difficult.

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I believe that algebra is not nearly as difficult as students suppose it to be. I guess I should have made that clear.

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I found algebra to be quite difficult, and it took me 3 or 4 years to become comfortable with it. I did well in class mostly because I had the privilege of starting algebra long before my classmates had heard of it.

In general, pronouncements of ease or difficulty should come with some context, and it should be clear that JSE is working in the context of the post, i.e., the typical student with little to no experience doing abstract manipulations.

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It will take forever for me to be “comfortable” with algebra.

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That op-ed upset me so much that I wish I could unread it. Out of the 100 things I’d like to say in response to it, I’ll stick to just a couple: First, how is it possible to have “quantitative skills, critical for informed citizenship” without understanding algebra? Does anyone know a people who fit that description? Second, most students know how to read and do arithmetic, but high school English can be just as difficult to “master” as high school math. But no one is advocating for the elimination of high school English in favor of “practical” English.

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Absolutely. Some of these comments make algebra sound like the math equivalent of a Doctorate in Shakespearean Literature.

I don’t think I was the only one in my 6th grade algebra class who didn’t have a career path all planned out yet, but surely even my classmates who were hell bent on a political science career have found some use for abstract thinking and problem solving since then.

Now, if you argued that college students in nontechnical fields should start with symbolic logic, set theory, and proofs instead of calculus, you’d be on to something.

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120 years ago, first grade teachers assumed (and rightly in many cases) that they were the last formal teacher that would teach their students.

Now, first grade teachers assume (and rightly in most cases) that their students have fifteen more years to be taught what they need to know.

I am not sure this difference makes a difference.

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what does ZEBRA’S know about Algebra?

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I spent almost 8 years as an art major, never needing or taking a math class of any sort. Just the way the curriculum dice rolled at the university of choice. As a particular irony, I then walked across the campus to what would in a year or two become the computer department but was then the math department because I realized that all of the great art patrons had died in the previous century 🙂 The transition was intense but well worth it. In the 30+ years since I’ve kept up with my art, but I’ve made my money as a programmer. Guessed early and accurately about where the countries ‘job’ needs were headed. That said, today is an entirely different place—should Algebra be dropped? Not sure, but even in the 60’s there were courses offered as an introduction to Algebra that if used today would take much of the pain out of the repair work necessitated by short-comings in their first 12 years of education. Your typical liberal arts major could stop there. Those with more math intensive needs could continue on. Notice that the intros to Algebra were not dumbed down; they were actual introductions to the basic concepts that were needed for Algebra. I imagine that it would not be particularly difficult to add ‘practical math’ somewhere in the mix for those who need to balance a check book and count change etc. Not a perfect solution, but until the first 12 years are over-hauled, it would probably do the trick.

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Today is a sad day for humanity because Algebra was considered advanced mathematics. I think history is too focused too, should that mean that one should stop teaching it?

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Algebra is not difficult. Students don’t want to learn it. They are told that some people “get” math and some people “don’t get it”. If you tell people they did good because they worked hard, they will continue to work hard and be successful. If you tell people they did good because they are smart, and they believe it, the next time it isn’t “easy” they will be much more likely to quit.

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I agree. A huge part of the problem is the approach to how things are taught in school. Trying to correct it it college is quite difficult. One thing I would do is not allow students to use calculators before college. As as teaching assistant at Syracuse university in physics I have seen students pull out calculators to do math like dividing a number by 2. Students are surprised that I remember that sin(30) = 0.5. In my opinion getting students to do calculations by using log tables and trignometric tables for at least 2 years in high school is essential to building algebra skills and demystifying math.

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Indeed, when I did maths in high school calculators were widely available; nonetheless we were taught interpolation and extrapolation with log tables.

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I wish that math was just more catered to what you are studying in college. I’m not sure I needed algebra but maybe some geometry to help with my study of perspective in my drawing classes. In the same way there is business writing I wish that math classes were more specific. And I add to this argument to take the lab science requirement out. While a knowledge of math can help in the real world I have still yet to find an application for pig heart dissection from my $4,000 biology course.

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I’m not a terribly big fan of this author. I read the book he wrote on higher education and while I agreed with many of his diagnoses of the “problems”, his “solutions” always struck me as not well thought out. (“Eliminate research professors from teaching colleges and eliminate college football,” he says, but never answers the question of how to replace the resources–monetary, academic, and cultural–these things provide.)

But focusing on this article, to me, algebra as currently conceived is like spelling and grammar lessons: they are prima facie irrelevant to our daily lives, but extraordinarily important to our understanding. Rote learning is an important component of comprehension. Just as you can’t hope to deeply understand Faulkner if you have to spend all your time with a dictionary, you can’t hope to deeply understand loan agreements if you have to spend all your time symbol pushing.

Online education has the most potential, I feel, in aiding this rote learning, whether it be in foreign language instruction or mathematics. But for most kids, it could never be “free”: somebody has to be keeping track and helping them along, somebody has to train these people (and somebody has to train them, etc.), and somebody has to be constantly updating and localizing the material. But this is the easy part. If online learning can improve rote learning outcomes (a huge, untested if!), then the hard part is remaking our curriculum so that kids understand math and use that understanding.

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An interesting article.

I have run a small law practice, taught Human Justice [criminology] classes at a Canadian University and have been a senior civil servant responsible for hiring and managing people.

My experience showed me that many young people, university educated or not, are woefully ignorant of basic knowledge and skills needed for work in all but a fast-food outlet. From my experience teaching them, many students enter university lacking basic math, reading and writing skills. When this lack is combined with laziness and disinterest, it is a recipe for disaster. Too many university graduates, especially from the so-called “Humanities” faculties, think they are ready for senior positions [and pay levels] despite the fact that the above-mentioned inadequacies are immediately noted by their employers, and their attitudes to learning often prevent them from advancing, or even retaining their jobs.

I have seen the methods used to teach them in primary and secondary education levels, and in post-secondary levels. In too many cases, these methods amount to spoon-feeding at best. Rarely are these students taught how to think, how to analyze problems or situations, and how to find solutions. I adopted methods during my teaching career designed to teach these skills and found that after the initial shock of learning to do things for themselves, my students not only enjoyed the curriculum content, but expressed amazement to me that they could actually do what I tried to teach them.

I suspect that the quality of teaching in most schools is truly abysmal and accounts for a lot of the educational disaster I see around me.

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Algebra is easy, but only if you are literate and understand basic arithmetic.

I think math education would be a much more useful (and successful) if we spent more time teaching the foundations of math. How many students are familiar with the Peano Axioms? How many know what inductive reasoning is? How many can explain what a base is?

These are the sort of things that should be taught, after rote arithmetic and before algebra, to ensure that students understand what math is. Teaching algebra seems pointless if students haven’t moved beyond rote learning and started reasoning mathematically.

TLDR version: more number theory, less rote learning

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How could algebra be easy if there are literally an unknown in every other sentence?

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Let me help you with that…

There _is_.

Every other _equation_.

Why does every other equation in algebra have to have an unknown? I didn’t realise that people were taught with alternating equations, ones with and ones without unknowns.

You may need to adjust your sarcasm detector, for the previous sentences, if it isn’t properly aligned.

On a more serious note; a solution doesn’t need to be a number, a solution to an equation can quite happily be another equation.

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You see, english is also unadequately hard and should be dropped from school programs as well as it causes too much suffering.

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Haven’t had time to read the comments yet but I don’t see a link to

http://gowers.wordpress.com/2012/06/08/how-should-mathematics-be-taught-to-non-mathematicians/

which is relevant. I spotting the need for mathematical analysis and being able to preform and understand the simpler ones is a useful skill for non-math majors than e.g. group theory.

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The anti-math sentiment in that opinion piece is not new at all. Here is a Colman McCarthy piece from 1993, and then a Richard Cohen piece from 2006. There must be one from every decade going way back.

https://groups.google.com/forum/?fromgroups#!topic/sci.math/UJlbMOtiNQk

http://www.washingtonpost.com/wp-dyn/content/blog/2006/02/15/BL2006021501989.html

There are real questions how to handle the few students who really do struggle with basic high school math. They do exist, and it makes sense to have some alternative pathway to HS graduation. Their record should make this difficulty apparent, but they have a chance to go on to a good college that can accommodate this if they are exceptional elsewhere. I’m not knowledgeable about it, but I thought in many places that’s all exactly how it plays out. Students should not be able just to dodge math because they are scared of it or have unsupportive parents….

But the vast majority of students should be able to handle the math, and get through algebra at minimum. It’s a very important life skill that really does come up, and it promotes more organized, clearer though, and better planning and objective reasoning.

I wish I had a well-founded understanding what makes math challenging for so many and how to overcome the challenges. I think key problems are some cultural factors (coolness) and the difficulty of getting good teachers who understand and love math and can convey the thinking and intuition with enthusiasm.

Just a few thoughts with no great solutions — glad to see topics like this up for discussion here.

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OK Cathy I’m wading into your blog… As you know I place a high value on equity, and also on math. I’m teaching remedial math at a Cal State – most of my students are of color and are in danger of losing their hard-won place at the university because, as most of them say, “I’ve just never been good at math.” But when I ask grade school kids of color their favorite subject (especially immigrant kids who find English frustrating!), many of them answer, “math!” If society is judging people on their ability to do math, then it should also judge junior high and high school math teachers on their amazing talent at convincing kids that they are bad at it.

And how many of us have talked to mediocre math teachers who blame “those kids” who “just aren’t able” to learn the material? Or even brag about how many they are going to fail – taking out their frustration at how badly society treats teachers, on the kids who don’t respect math because it’s mostly taught in a rote and boring way? I believe elementary math education has moved forward a whole lot, but high school math education seems to be exactly where I left it when I was a teenager.

Should colleges expect students to be able to solve percent mixture problems, and to really understand scientific notation, (ie negative and positive exponents)? These both seem crucial to participating in our society. Should colleges require students to be able to factor polynomials? These seem pretty useless. My students are atrociously bad at relatively straightforward word problems about distance/speed, but have been trained quite well at doing meaningless manipulations of algebraic symbols that I suspect they really don’t understand. Should they be required to learn how to do word problems? If you read the very simple problems that they are totally thrown off by, you can see how incredibly terrified they are by real-life math problems that you would absolutely expect anybody that you would ever want to employ to do anything to be able to solve. So yes, I would argue they need to be coaxed away from their fear to the point that they can think at least a little bit clearly about simple quantitative questions.

As to the online question – Honestly, the algebra class I’m teaching is not very hard. But students are failing it in droves, across California. I truly believe their block to learning is psychological – years of being told they are “bad at math” by peers and teachers, to the point that they repeat it back to me without fail, even with pride as Kiri points out. They really need a human being to tell them, no, you can do it, and you have to do it, and now is when you have to do it – the vast majority of 19-year-olds just aren’t mature enough to become proactive learners – so without some encouragement, by a live human capable math teacher, they will just keep on failing algebra and lose their place in college.

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Can we please stop the teacher bashing? Seriously. Teaching K-12 is a thankless, low-pay, low-prestige job and the people who do it should be treated like heros.

I have never met a single K-12 teacher who doesn’t care or who wants to do a bad job, and I have spent the past 20 years (ish) working in and around K-12 education in one form or another. They want to do a good job. They wan their kids to learn.

But teachers are beholden to ridiculous testing requirements, administrators who have never set foot in a classroom, and curricula they didn’t pick and that change every couple of years. It is nearly impossible to teach well in the current public school system in the US. I tried. Then I gave up, got my PhD, and taught at university instead.

Teachers can be our greatest asset in improving math education. Bashing them and blaming them for all of our educational ills is counter-productive, uninformed, and just cruel. And it’s a big part of why talented teachers leave in droves after 3-5 years in the classroom. It’s absolutely exhausting to work your ass off every day, and then to read about how terrible you are at your job.

(FWIW, I worked harder as a high school teacher than I have as a professor. Way, way, way harder. For about half the pay and no respect from my administrators or society. Please, thank a teacher and try to help them instead of bashing.)

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Sam,

Two things. First, a good online math program can work like a video game, with tailoring and levels etc.; this isn’t necessarily bad on people’s egos, and may be a lot better than what they experienced in high school.

Second, if you admit that there were some teachers (and peers) telling these kids they can’t do math, then what army of wonderful human beings are going to suddenly pop up and tell them they can, once they get to college? I mean, I agree that if we could find those people it would be better than online learning, probably, but it’s impractical to think they will suddenly appear, or at least enough of them will.

Love talking to you about this!

Cathy

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Cathy, a point about online teaching.

I taught introductory physics as a TA at Syracuse University and the students had to do online homework which had an extensive interactive help menu for each problem. After some free help, the more help you used the lesser points earned. I think it was a fair system that promoted learning by requiring you to use help judiciously.

We also had a physics clinic where the TA’s helped the students with concepts they had difficulties with and with offline homework problems. But a number of students would come to the clinic asking for help for the online homework as well. Refusing to help them only causes more headaches as they argue and complain and then eventually the administration brow beats the professor into changing policies. Soon we had students coming to the clinic without even trying to attempt homework and expecting solutions to be handed out.

So the work ethic part is an issue, something that should be dealt with in school by allowing teachers to demand a certain level of work and competence.

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“Practical”…

In English, what would be practical is:

– Hello!

– Do you want fries with that?

– Thank you, Good bye!

Why would an English teacher teach anything else than those three sentences? They’re quite practical, they can get a good job. And for the drop outs who can’t learn them, there’s always “Welcome to Costco, I love you.”.

Algebra and Maths are not practical and are not needed to live. Like cavemen.

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Mathematics (from the Greek word mathe, knowledge) and music and poetry and art all belonged together, and can be rejoined. Hyppasos of Metapont, 500 BC, listened to tones on his gitara, and found that two tones sound together beautifully when the ratio of the length of the strings is 2:1, and also 3:2 and also 4:3. Here the Greek people, for the first time in history, were thinking about numbers as numbers, here our western mathematics is born. Our virtual world journey project “Music and Mathematics” has as goal to show how beautifully mathe and music and poetry and arts and science all belong together.

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My carpenter dearly wishes he had paid more attention in Geometry class. Talk about practical applications!

I am pretty skilled in applied math. I have an EECS degree from MIT and I do a lot of algebra doing analog circuit design. That said, I had a hell of a time learning algebra and doing word problems when they were introduced in junior high school. My school district was hands down one of the best in town (district 66 in Omaha, NE).

The difficulty lies in the transition from concrete work with numbers to abstract work in algebra. I see no way this can be handled online; this is the sort of learning that happens far more efficiently when you can interact directly with a teacher.

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Well said, Cathy! As a mathematician by training (PhD in applied mathematics) who went on to moonlight as a writer on topics of political science and constitutional history (which fascinate me) and then went on to be a government analyst, I disagree vehemently with Andrew Hacker’s perspective. Ironically, having a university mathematics background, which included courses in logic, algebra and complex systems theory, has given me a much better perspective on issues of political science (both in what might be described as “pure” political science and in contemporary economic theory) than an equivalent observer would have without that background. So I think that even on his own turf, Hacker’s proposition fails. Furthermore I think that mastering the abstraction inherent in algebra is inherently useful in training people how to think their way through a complex problem (irrespective of the “usefulness” or otherwise of the actual exercise), like doing scales in playing a musical instrument or weights exercises for an athlete.

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“I honestly feel like this is the perfect place for online learning.”

I have to disagree here. Once a student gets to college without knowing algebra, their worldview is set: math is not something I can do.

A computer cannot be encouraging the way a good teacher can. A computer cannot help unpack and undo this worldview. A computer cannot figure out what the student likes and find ways to connect and draw him into mathematics through those likes.

Online learning is great for people who already know how to learn and who are motivated to learn something new. But for someone who looks at a page of math and completely freezes, putting that page on a screen and taking away personal interaction is only going to make it worse. And if it makes no sense when you go do your online course, and you just get everything wrong because you always do because you’ve never been any good at math… well why even bother logging on?

I know examples of remedial programs that are successful, but I don’t know a single instance of an online course that is really successful with students who get all the way to college without basic math skills. There’s too much fear & self doubt to overcome, and computers aren’t very good at that side of things.

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As someone who’s been obliged to teach an algebra class every fall semester, I can tell you what the big problem is: The kids refuse to do their homework. Heck, the kids refuse to even read the book.

Show me the kids who have both read the book and have made a decent stab at attempting the homework. If they’re having difficulties, I’ll go all out to help them. The much, much larger eighty-plus percent, the ones who find it “hard” but make absolutely no effort? And then want a a C – more likely a B?

They need to be flunked good and hard. With extreme prejudice.

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“They need to be flunked good and hard. With extreme prejudice.”

Or, you know, they need to be taught how to learn. They’re so freaked by math that they look at a page of a book and see meaningless symbols, closing the book before a panic attach sets in.

They don’t do the homework because they have seriously no idea how to get started.

So, yeah, you can flunk them good and hard. And they can try and try again until they give up. Or you could, um, *teach* them what they need to know, which includes how to read the book and how to do the homework. Just a thought.

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[sarcasm]Gee, Michelle, you know so much more about my students than I do myself. How do you manage that?[/sarcasm]

Feel free to talk about your own students that way. Don’t presume to talk about mine with any authority. These are, incidentally, the same students that have no questions to ask in class as we go over the material, the same students who never come to my office hours (except at the end of the semester when they have

very good reasonsfor not having had “the time” to do the homework), and as likely as not, the same students who are also not keeping up with reading assignments in other classes. Like, you know, basic English.Like it or not, the fact must be acknowledged that there are large numbers of students attending college who are temperamentally unable/unwilling to do the work. The problem is, there’s no other place for them to go 😦

Notice, btw, that I make no mention of intelligence or academic preparedness. This one is solely a work ethic issue.

It’s okay Michelle. You can actually talk about that one out loud.

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There’s no need to be snide. Michelle was speaking from her own experience and that may or may not apply in your case. Regardless, I’ve never heard of any cases where flunking students motivated them to work harder. Students that want to do well will try, and those that don’t care simply don’t care.

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You’re exactly right: There is absolutely no need to be snide. Which is what Michelle’s reply was, what with the ” Or, you know, they need to be taught how to learn” and the “Or you could, um, *teach* them what they need to know”. The ‘you knows’ and ‘ums’ do grate. As well as Michelle’s presumption that I don’t actually try to do those things, of course.

Despitethe fact that I specifically said so. Snide? You bet. Not that I expect her to own up to it, or expect anything other than more rude behaviour.Anyway, yes, for a certain type of student, flunking them does work: the type of student who’s been sliding and weaseling through x number of years of school without doing any work but being passed on through the system anyway. When you tell them up front that there are consequences for poor test and quiz grades, not turning in homework, etc.

and follow through on those consequences, yes they do indeed smarten up right quick. At least, that’s been my experience.I blame the parents for this one. They’ve had years to dilute the K-12 curricula; what Cathy describes here is just the latest battle. This seems to be part and parcel with a lot of schools getting away from the model of graduating well educated citizens and going more for education as a ‘product’ that is being purchased by ‘customers’. It’s also of a piece with that part of the American character where people say they value education, but in reality, view it as – at best – a necessary bit of credentialism. If it can be purchased for mere money and with no other effort involved, so much the better 😦

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“I blame the parents for this one.”

It must be awesome that everyone is to blame but you.

My students, for the most part, work very hard in my classes. They want to do well, and the believe they can do well, and (importantly, I think) they know I want them to do well.

Part of my job as a teacher is to motivate them to put in the time & effort because they see some payoff. Not that stupid grade at the end, because that’s pretty unimportant really. But payoff in the bigger sense… they feel they’ve accomplished something they weren’t sure they could do. They solve a hard problem or figure something out, or whatever.

And yes, if a student doesn’t work and fails quizzes and blows off tests and all the rest they *will* fail my class. But I don’t find joy in failing students the way you seem to, and I find the joy a bit distasteful from an educator.

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Michelle, you’re being rude and snide . . . again. So I’m not going to bother replying any further. If you want to acknowledge your poor behaviour, apologize for it, and make a sincere attempt to mend your ways I’ll reconsider.

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Cathy, I was wondering if you or Johan ever had trouble learning some types of math, especially in grad school or later, and if that helps you relate to difficulties students a having. In grad school, I struggled with algebraic geometry … I was just completely lost as to what is going on…despite taking courses on it from eminent algebraic geometers. I kept wondering what was the motivation for all these complicated abstract definitions of schemes and sheaves and why was no one (or specifically hartshorne in chapter 2 and 3) talking about solutions of polynomial equations like he did in chapter 1? I wonder if that is the same feeling some students have with algebra… the jump in abstraction throws them off, and they are missing the fact they are just doing arithmetic.

(I suppose I eventually learned how to not be scared of the most common schemes, with lots of help from advisor).

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There’s a ubiquitous Einstein poster with the quote, “Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.” The first time I saw this, I thought it was cute, and it wasn’t until later that I understood the wisdom in this: Most research mathematicians understand better than anyone else how difficult math can be and what it’s like to be completely lost, because we fight with these terrible monsters nearly every day, and on most of those days most of us lose. It is only experience and maturity that allows us to soldier on.

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We also soldier on because of the high the very few times we win give us 🙂

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Agreed. Also, we use math as a tool and realize that a task can be done with different tools with different levels of difficulty (ease). Getting this idea across to students is very challenging. The only time I have managed to do so is in one on one sessions.

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Yes,this question is absurd as mathematics is a part of our daily life then how it can’t be part of higher education.It should make the part of it.

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Regarding the question of whether algebra is “difficult,” it’s largely a semantic issue. What does “difficult” mean anyway? To offer an admittedly very flawed analogy, I think that learning algebra is difficult in the same sense that training for a marathon is difficult. It’s a lot harder for some people than others, but most people are physically capable of completing a marathon. (Personally, I find this analogy useful for imagining the mentality of a failing algebra student, because fewer things terrify me more than the idea of training for a marathon.)

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This is an excellent commentary that characterizes each author’s position well. I’ve read both and got the same messages. If you do have them for dinner, I hope you’ll follow up with a summary. I’ve also enjoyed reading many of the comments here.

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No, mathematics should not be taken from higher education. It does not make sense for anyone to think so. It would be the process of keeping people at a dumb-down level. A lack of mathematical skills shows a lack of adventure and curiosity about the world. We have always been told that mathematics is the key to the universe. I believe it is true. So, this question makes me think is this a valid question or a political question. It is true that there are problems within the education system. However, by removing a major requirement is not going to make you any smarter because the requirement no longer exists. You will appear to be smarter only to yourself and your small circle of friends. The world is much bigger than that. The world needs curious minds in order to advance itself. Having the curiosity without knowing the steps to take to solve a problem is not going to get you very far either.

The opportunity to learn mathematics should be given to all to pursue. However, all will not live up to their expectations. Nevertheless, the opportunity should be there.

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Great post! You are right that this is the perfect place to use on-line learning… and it is even a better place to use gaming strategies to engage students. John Paulos’ book Innumeracy written several years ago describes the lack of fundamental math skills in our country and bemoans the fact that most citizens don’t fully appreciate the difference between Million, Billion, and Trillion and, consequently, make ill-informed decisions when they vote. Their lack of understanding regarding the compounding of percentages also lead them to indebtedness. We need MORE mathematics not less.

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It could be we need to do basic research on how students learn (or don’t learn) math, OR take action on the research that’s already been done. I have a feeling (just a feeling) the government has probably spent millions on just such research, but the results were either useless, or unpopular with unions.

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Algebra is the essential first step for quantitative abstract thinking and therefore any modern activity which is not routine and repetitive will require some minimum algebaic abstraction.For politics or office work or planning.Any person aspiring for higher education needs some minimum skill in algebra. Best course is to make it a entry condition. They can do it online or otherwise.

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Coming in way, way late because I missed this post somehow initially.

In my and my wife’s experience teaching algebra (about 10 years between us), it’s not that students have trouble with algebra; it’s that algebra is the place where not having learned arithmetic well really clobbers them. I’ve never had a student who could reliably do a multiplication fact speed drill (all 81 facts in the multiplication table up to 9×9) in less than 5 minutes, and add/subtract/multiply fractions, who was not able to learn algebra. But if those skills have been worked around (by using calculators rather than learning them by drill), it’s almost impossible to factor or to understand why simplifying an equation works as it does.

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Why do educators, the media, and the public treat finding ways to teach practical math (balancing checkbooks, calculating an interest rate and payments, understanding polls and statistics, and the like) as harder than finding the Holy Grail? There is nothing to it, but to do it. Stop whining and start doing. It may take some trial and error, but what good discover using scientific methods doesn’t? Also, greater emphasis on educating students regarding how government works, from policies, laws, and regulations, to elections . We also need more study about the history of struggles between governments and the people who form and change them compared and contrasted to today’s political environment and realities. Show how it compares to what’s gone on before, how it affected people then, and how the choices made today will affect the people of today.

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All can verbalize by reason of the necessity to speak a language almost as usual a Natural instinct in all creatures -most humanity.

However numbers are not congruent in daily necessity -and a lye at that.

All works together but having a better understanding of maths is improvement in verbal reasoning of the mind.

Maths must be continually taught in its use and application.

However its application to currency a corruption -where we find ourself dieing.

Because we think instantly money, money and furthering life can be apart -where currency is no longer currency but merely a word of language.

The meaningful things done freely by trust and work and faith in the honesty and unity of being able to function in imaginary use of currency stroke commodity without price tag but for the needs of the counted whole in their particular unit.

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