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Is Algebra Hurting America?

Andrew Hacker argues that, while quantitative skills are “critical for informed citizenship and personal finance,” making kids master algebra to graduate high school has disastrous consequences.

Making mathematics mandatory prevents us from discovering and developing young talent. In the interest of maintaining rigor, we’re actually depleting our pool of brainpower. I say this as a writer and social scientist whose work relies heavily on the use of numbers. My aim is not to spare students from a difficult subject, but to call attention to the real problems we are causing by misdirecting precious resources.

The toll mathematics takes begins early. To our nation’s shame, one in four ninth graders fail to finish high school. In South Carolina, 34 percent fell away in 2008-9, according to national data released last year; for Nevada, it was 45 percent. Most of the educators I’ve talked with cite algebra as the major academic reason.

Shirley Bagwell, a longtime Tennessee teacher, warns that “to expect all students to master algebra will cause more students to drop out.” For those who stay in school, there are often “exit exams,” almost all of which contain an algebra component. In Oklahoma, 33 percent failed to pass last year, as did 35 percent in West Virginia.

Algebra is an onerous stumbling block for all kinds of students: disadvantaged and affluent, black and white. In New Mexico, 43 percent of white students fell below “proficient,” along with 39 percent in Tennessee. Even well-endowed schools have otherwise talented students who are impeded by algebra, to say nothing of calculus and trigonometry.

California’s two university systems, for instance, consider applications only from students who have taken three years of mathematics and in that way exclude many applicants who might excel in fields like art or history. Community college students face an equally prohibitive mathematics wall. A study of two-year schools found that fewer than a quarter of their entrants passed the algebra classes they were required to take.

[…]

Another dropout statistic should cause equal chagrin. Of all who embark on higher education, only 58 percent end up with bachelor’s degrees. The main impediment to graduation: freshman math. The City University of New York, where I have taught since 1971, found that 57 percent of its students didn’t pass its mandated algebra course. The depressing conclusion of a faculty report: “failing math at all levels affects retention more than any other academic factor.” A national sample of transcripts found mathematics had twice as many F’s and D’s compared as other subjects.

Nor will just passing grades suffice. Many colleges seek to raise their status by setting a high mathematics bar. Hence, they look for 700 on the math section of the SAT, a height attained in 2009 by only 9 percent of men and 4 percent of women. And it’s not just Ivy League colleges that do this: at schools like Vanderbilt, Rice and Washington University in St. Louis, applicants had best be legacies or athletes if they have scored less than 700 on their math SATs.

This comports with my own experience. I excelled at math through my early years of school and was even on the math team as a high school sophomore. While my skills have eroded somewhat owing to technology, I’m still pretty good at the sort of math that’s useful in everyday life, including estimation and statistical interpretation.

In my own case, it wasn’t algebra that got me but calculus. I graduated high school in 1984, just before the Advanced Placement craze swept the country. (Indeed, it had probably already swept the country, as it hit my rural Alabama high school the next year.) No advanced mathematics classes were necessary to graduate but I nonetheless took two years of algebra, geometry, and something called “advanced math” (essentially, pre-calculus) as electives.  While I’m not sure I learned much useful—in the sense that I actually used any of it outside school, much less still remember it–I was able to get through that sequence with good grades, although I did stumble a semester or two and wound up with a B+ rather than an A-.  But it was mostly a matter of remembering formulas by rote and applying them in cookbook fashion; I never truly understood what I was doing in the way that I did arithmetic.  That last year was at a level of abstraction that truly baffled me—such that I don’t even really remember what it was that I was exposed to.

When I got into West Point that summer, despite solid SAT and ACT scores in math, I did poorly enough on their post-acceptance placement exam that I was placed in a what was derisively called “rock math,” a remedial course in an institution that was still at heart an engineering school, rather than calculus.  I never really recovered. I got through that and Calculus I, although the latter only because they were grading on a curve and I was above it. I failed Calculus II and barely got through Physics and wound up washing out.  It was just as well, as there were still several more math and engineering courses ahead of me on the curriculum, including Differential Equations, a course which ended the careers of many a cadet.

Now, I suspect that I’d have been able to pass Calculus II and do better at Physics at the University of Alabama than at West Point, simply because I wouldn’t have also been drowning in extracurriculars and the stresses of a pressure cooker that aimed at driving out a third of each class.  Then again, I likely wouldn’t have been in those classes to begin with as a poli-sci major. I would go on to take several graduate courses in statistics as part of my doctoral study. While they were not as intuitive as my poli-sci classes, I nonetheless understood the concepts and got A’s in the classes.

Despite these frustrations and the fact that the lack of proficiency in abstract math was never an issue outside the classroom, I’ve nonetheless defended the notion that forcing kids to grapple with the subject was useful. Six years ago, in addressing a Richard Cohen column, I wrote a post titled “Is Algebra Worthless?” and argued in the negative.

If a degree is merely a credential for employment, there’s not much argument for requiring algebra for those not aspiring to scientific and technical fields. Ditto literature and the arts for those who are not headed in that direction. Or foreign languages for those not intending to travel. If it is about broadening the mind-in a sense, an end into itself rather than merely a means-then all those things must be part of the curriculum.

English (or whatever written and spoken language predominates in a given society) and mathematics are the two essential languages of education. One is simply not educated without a solid foundation in both.

But Hacker isn’t arguing that we should take math out of the curriculum.

Nor is it clear that the math we learn in the classroom has any relation to the quantitative reasoning we need on the job. John P. Smith III, an educational psychologist at Michigan State University who has studied math education, has found that “mathematical reasoning in workplaces differs markedly from the algorithms taught in school.” Even in jobs that rely on so-called STEM credentials — science, technology, engineering, math — considerable training occurs after hiring, including the kinds of computations that will be required. Toyota, for example, recently chose to locate a plant in a remote Mississippi county, even though its schools are far from stellar. It works with a nearby community college, which has tailored classes in “machine tool mathematics.”

That sort of collaboration has long undergirded German apprenticeship programs. I fully concur that high-tech knowledge is needed to sustain an advanced industrial economy. But we’re deluding ourselves if we believe the solution is largely academic.

[…]

Algebraic algorithms underpin animated movies, investment strategies and airline ticket prices. And we need people to understand how those things work and to advance our frontiers.

Quantitative literacy clearly is useful in weighing all manner of public policies, from the Affordable Care Act, to the costs and benefits of environmental regulation, to the impact ofclimate change. Being able to detect and identify ideology at work behind the numbers is of obvious use. Ours is fast becoming a statistical age, which raises the bar for informed citizenship. What is needed is not textbook formulas but greater understanding of where various numbers come from, and what they actually convey.

What of the claim that mathematics sharpens our minds and makes us more intellectually adept as individuals and a citizen body? It’s true that mathematics requires mental exertion. But there’s no evidence that being able to prove (x² + y²)² = (x² – y²)² + (2xy)² leads to more credible political opinions or social analysis.

Many of those who struggled through a traditional math regimen feel that doing so annealed their character. This may or may not speak to the fact that institutions and occupations often install prerequisites just to look rigorous — hardly a rational justification for maintaining so many mathematics mandates. Certification programs for veterinary technicians require algebra, although none of the graduates I’ve met have ever used it in diagnosing or treating their patients. Medical schools like Harvard and Johns Hopkins demand calculus of all their applicants, even if it doesn’t figure in the clinical curriculum, let alone in subsequent practice. Mathematics is used as a hoop, a badge, a totem to impress outsiders and elevate a profession’s status.

It’s not hard to understand why Caltech and M.I.T. want everyone to be proficient in mathematics. But it’s not easy to see why potential poets and philosophers face a lofty mathematics bar. Demanding algebra across the board actually skews a student body, not necessarily for the better.

I WANT to end on a positive note. Mathematics, both pure and applied, is integral to our civilization, whether the realm is aesthetic or electronic. But for most adults, it is more feared or revered than understood. It’s clear that requiring algebra for everyone has not increased our appreciation of a calling someone once called “the poetry of the universe.” (How many college graduates remember what Fermat’s dilemma was all about?)

Instead of investing so much of our academic energy in a subject that blocks further attainment for much of our population, I propose that we start thinking about alternatives. Thus mathematics teachers at every level could create exciting courses in what I call “citizen statistics.” This would not be a backdoor version of algebra, as in the Advanced Placement syllabus. Nor would it focus on equations used by scholars when they write for one another. Instead, it would familiarize students with the kinds of numbers that describe and delineate our personal and public lives.

It could, for example, teach students how the Consumer Price Index is computed, what is included and how each item in the index is weighted — and include discussion about which items should be included and what weights they should be given.

This need not involve dumbing down. Researching the reliability of numbers can be as demanding as geometry. More and more colleges are requiring courses in “quantitative reasoning.” In fact, we should be starting that in kindergarten.

This strikes me as a sensible approach. My experiences teaching college and engaging in blog comment thread discussions over the years have highlighted how bad most people are at this sort of “walking around math.” And people who are attending college and commenting on blogs are an elite slice of our society; I shudder to think how innumerate the bottom half must be.

At the same time, while most of us will never need to solve a quadratic equation, much less do whatever it is that Calculus is used for, at least one in twenty will. How will our future mathematicians, physicists, chemists, and engineers discover their interest and aptitude for those endeavors if they’re not exposed to abstract math in high school? Maybe there’s some middle ground solution that allows people to graduate high school and college taking courses in practical math and science but offering non-punitive opportunities for students to see whether they have the aptitude for the more abstract varieties? Part of the problem is that our entire system is geared around semester-long sequences that result in the earning of credit hours. So, there’s no way to take algebra or calculus—or, for that matter, introductory philosophy or Latin—for a few weeks, struggle to stretch one’s mental facilities–and then move on to something else without penalty. That means that either we make mandatory courses that everyone doesn’t strictly “need” or else we make optional courses that everyone should at least be exposed to.  Instead, we should allow for broad exposure and familiarization with the opportunity to move on without penalty into subjects where one’s natural talents and aptitudes lie.

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About James Joyner
James Joyner is the publisher of Outside the Beltway, an associate professor of security studies at the Marine Corps Command and Staff College, and a nonresident senior fellow at the Atlantic Council. He's a former Army officer and Desert Storm vet. He earned a PhD in political science from The University of Alabama. Views expressed here are his own. Follow James on Twitter.

Comments

  1. Jim Henley says:

    I’m a budgeting and dashboarding nerd in the finance department of an internet marketer, and now and then I find myself tittering at just how much of my job is: addition, subtraction, multiplication and division. About twice a year I do find myself having to solve a “two equations and two variables” problem. That’s mostly it for explicit algebra.

    I think that spreadsheet programs have fundamentally transformed a lot of real-world math by making algebra implicit rather than explicit. The bulk of what most of us do with Excel is, of all things, solving word problems! The bane of most school-age math students. But we do it in a fundamentally different way – solving for every intermediate value, basically – than transforming the equations themselves and only calculating a single numerical answer at the end.

    Now it’s possible that the reason I’m good at my job is that I understand basic algebra well and there’s a lot of implicit algebra in using Excel or a dashboarding tool to solve a real-world problem. But damn if it doesn’t seem like I’m just multiplying, dividing, adding and subtracting all day.

    Like or Dislike: Thumb up 8 Thumb down 0

  2. Tony W says:

    These abstraction skills are indeed needed daily in the engineering fields, but I have often said that the problem with schools is that they are run by people who went to college.

    Poor performance in maths is part of why we have two unemployment problems in the United States – unskilled people with no prospects, and tens of thousands of open positions at the most senior technical levels remaining unfilled and dependent upon foreigners who can get through our visa system.

    When we try to train *everyone* for those lucrative engineering positions we needlessly frustrate those without natural engineering skills. Europe and elsewhere has long had a two-track education system in which people are selected at some point (often middle school age) for the “college” track or the “trade school” track. While the discussion of importing anything European to the United States results in fallacious attacks from the right, I maintain this might be something with which we might begin experimenting.

    Like or Dislike: Thumb up 8 Thumb down 1

  3. Murray says:

    What nonsense. Algebra is central to natural sciences and engineering. It’s the branch of mathematics that deals with describing things.

    As for the “I don’t use it in real life” argument: what bull. How much are the humanities useful?

    Discarding algebra to focus only on the quantitative is like discarding grammar and focusing only on spelling. You’ll end up with people who can spell anything they hear, but unable to build a sentence.

    Hot debate. What do you think? Thumb up 24 Thumb down 19

  4. superdestroyer says:

    There are many who have argued that schools, when faced with poor performance, will lower educational standards instead of actually trying to improve the teaching. Many of the students fail algebra because those students have never developed the underlying mathematical skills in earlier grades. Does anyone really believe that students who never memorized their multiplication tables are ever going to master algebra.

    I keep thinking of the scene in the movie version of “The Wizard of Oz” where the wizard gave the scarecrow a diploma in order to make up for the scarecrow not having a brain. Too many educators believe that the point of high school is to give people pieces of paper that say that students are educated instead of actually educating students.

    Like or Dislike: Thumb up 11 Thumb down 3

  5. Tsar Nicholas says:

    The funny thing about an article like Hacker’s — in a black comedy sense, of course — is that it could have been ghost written by the Chinese and Indian governments. Seriously. Literally. Our loss is their gain. That’s how competition works. Real world competition. Outside the academic bubble.

    One can’t help but extrapolate to where this line of thinking inevitably will take us. Having slain the algebra boogeyman the academe no doubt would find others. Physics? Chemistry? The MCAT? The LSAT? The GMAT?

    Given that having the public directly and indirectly subsidize the likes of “potential poets and philosophers” is the dumbest misallocation of public resources in history I think it’s quite fortunate that, despite our nation’s collective shame, the hard science of algebra will continue to serve as a basic and useful screening mechanism. We’re FUBAR in any event, but at least we’ll remain above the level of national catatonia. For now.

    Like or Dislike: Thumb up 5 Thumb down 2

  6. James Joyner says:

    @Jim Henley: Agreed. Additonally, there’s algebra and then there’s Algebra. That is, there’s the basic “solve for X” that I learned and mastered in 8th grade and use pretty much daily and the quadratic equation and more advanced form of the subject that I studied under the Algebra rubric in 9th and 11th grades (geometry was 10th, for some reason) and have largely forgotten even though I did reasonably well in it.

    @Murray: I’m not sure what it is you think you’re arguing.

    @superdestroyer: Actually, I don’t think this is right. It’s possible to be really good at abstract math and really lousy at memorization—especially in the age of the calculator. And the converse is true is well. I memorized my multiplication tables in, what, 3rd grade and still can apply the Commutative, Associative, and Distributive principles I learned in 5th grade. I can do the math required for algebra, trigonometry, and calculus—I just don’t understand what it is that I’m doing; it’s rote.

    @Tsar Nicholas: Most Chinese and Indian students can’t do algebra, either. They just have a larger population. Is it your belief that, if we just demand it, most Americans will be able to master complex mathematics and become engineers?

    Like or Dislike: Thumb up 7 Thumb down 1

  7. Moosebreath says:

    I tend to agree with Murray. How often do people need to know how to diagram sentences? How often do they need to be able to find Albania on a map? Should grammar and geography no longer be needed for graduation?

    Like or Dislike: Thumb up 4 Thumb down 2

  8. Jim Henley says:

    @Tsar Nicholas & @Murray:

    I don’t read Hacker or James as saying we should give up on instilling mathematical literacy. I’m certainly not saying that. I read them as saying we need to be instilling a different kind of mathematical literacy. I think there’s a strong case for that.

    Also, I use my humanities education – from proper grammar to critical reading skills to basic psychology – every day. It’s at least as useful to me as my math background, which goes up to multi-variable calculus. And outside of work it is more enriching. Knowing something about music, art, poetry and fiction makes my life better.

    @James Joyner: And that basic algebra starts in grade school! I loved showing my 12-year-old daughter that by taking an equation like 6 + X = 10 and just replacing “X” with a triangle or blank line, like she’d been seeing since 1st or 2nd grade.

    Like or Dislike: Thumb up 2 Thumb down 0

  9. gVOR08 says:

    “…while most of us will never need to solve a quadratic equation…much less do whatever it is that Calculus is used for…”. Amongst other things, calculus, or at least an understanding of the concepts calculus represents, is useful for looking at graphs of unemployment and realizing that whatever happened, it’s clearly not Obama’s fault. If one has a concept of change over time, one is less prone to the cum hoc, ergo propter hoc error.

    I’d really want to know a lot more about the history of math education and about math education in other countries, before I decideed whether dropping algebra makes sense, or is just another capitulation to the dumbing down of the country.

    Like or Dislike: Thumb up 1 Thumb down 1

  10. James Joyner says:

    @Moosebreath: The argument isn’t simply “you don’t need that in real life” but “you don’t need that in real life but a whole lot of kids are being denied a high school diploma because they can’t learn it.” I don’t know that the ability to diagram a sentence or locate Albania on a map has kept anyone from getting a diploma. Whether those skills, particularly the former, ought be part of the curriculum is, of course, debatable.

    Like or Dislike: Thumb up 1 Thumb down 0

  11. James Joyner says:

    @gVOR08: /a>:

    Amongst other things, calculus, or at least an understanding of the concepts calculus represents, is useful for looking at graphs of unemployment and realizing that whatever happened, it’s clearly not Obama’s fault. If one has a concept of change over time, one is less prone to the cum hoc, ergo propter hoc error.

    I got a C in Calculus I but remember literally nothing from the course. I failed Calculus II. Yet, somehow, I can intuitively interpret a graph of unemployment numbers; ipso facto, a knowledge of calculus is not required to interpret said graph. And I learned about the post hoc ergo propter hoc fallacy in introductory philosophy, not any of my math classes.

    Like or Dislike: Thumb up 5 Thumb down 2

  12. Modulo Myself says:

    If you’re making the distinction between basic algebra and the quadratic formula, you probably aren’t getting how much people can panic over maths. Something very basic, like backing 15% out of a gross cost, which algebra can explain directly, can be tough for people with college degrees.

    Like or Dislike: Thumb up 1 Thumb down 0

  13. James Joyner says:

    @Modulo Myself: I see “backing 15% out of a gross cost” as basic math. Yes, it’s a rudimentary form of algebra but it doesn’t necessarily need to be taught using algebraic equations. Indeed, I essentially taught myself basic algebra because the way the teacher explained it didn’t make any sense to me, so I reverse engineered the problems.

    Like or Dislike: Thumb up 1 Thumb down 0

  14. Ben Wolf says:

    Seems to me that the real lesson is mathematics is taught in such an ad hoc fashion students are effectively being penalized for not having these concepts down beforehand. Each mathematics teacher I had in school taught in such radically different ways with such varying degrees of communicative ability, it’s a wonder we learned anything. It seems a field screaming for an holistic approach.

    Like or Dislike: Thumb up 3 Thumb down 0

  15. Modulo Myself says:

    @Ben Wolf:

    In my experience, bad math teachers often had special tricks that they forced a class to learn, tricks that to me made simpler problems far more complicated. I still remember learning fractions in 5th grade and not being able to understand the teacher’s ‘trick’ method for adding together anything that went over the value of ‘1’. It got to be a problem because I just added 3/4 and 2/3 together rather than the show whatever the work would have been.

    Thinking right now, I wonder if the anxieties teachers have over teaching math are taught to their students in addition to the actual material.

    Like or Dislike: Thumb up 2 Thumb down 0

  16. But my sense is that the essay was written on the wrong level with the wrong assumptions. I would think more about brain structure and function. Doing things, whether that is practicing tennis or doing math exercises, changes the brain.

    Millions of students have climbed an educational ramp, k-12, and the impact I see with those algebra classes is that suddenly they are doing abstract symbol manipulation. To dumb it down, after adding and subtracting apples, suddenly it is x, y, and z placeholders.

    So while on the one hand I can see something like a split HS diploma, one college prep and one not, or something, I wouldn’t be ready to say that the whole algebra process was bad for non college bound brains.

    (I did pretty well in math, until I hit the 381 course at university. It had something “nonlinear” and lots of matrix math (by hand in those days), and an annoying little professor. So I’ve done lots of math that I don’t use much and life and have since forgotten.)

    Like or Dislike: Thumb up 4 Thumb down 0

  17. Argon says:

    If schools were serious about math education, students should complete algebra by the 5-6th grade and be handling calculus well before 9th. Social understanding and skills develop slowly but math capability is already there pretty early in development. And the early it’s taught, the better one can learn the sciences later.

    The problem isn’t that some kids just can’t pick it up or that it’s ‘hard’. That’s baloney. It’s that far more often than not, math is poorly taught in most schools and taught by teachers that barely understand the basis of mathematics themselves.

    Superdestroyer, I disagree about the excessive need for rote memorization in math. I’ve a friend who runs a very successful math program — It’s not memorization that’s the key to success (e.g. contra the Indian and Chinese education program) but developing the curriculum necessary to develop critical thinking and learning how to reason in mathematics, and to understand the basis of math. This friend has also guest taught at Indian schools and found their rote method of learning detrimental — It doesn’t produce problem solving capabilities, just people who mindlessly perform the same formulas and gack at anything novel. He says that great Indian mathematicians arise *despite* their system. Same for most of our math education.

    Like or Dislike: Thumb up 7 Thumb down 1

  18. @James Joyner:

    I got a C in Calculus I but remember literally nothing from the course. I failed Calculus II. Yet, somehow, I can intuitively interpret a graph of unemployment numbers; ipso facto, a knowledge of calculus is not required to interpret said graph.

    Better (for the brain) to have failed Calculus II than to have never Calculus II’d at all?

    Like or Dislike: Thumb up 3 Thumb down 1

  19. @john personna:

    Hey, that comment leads with “but” because I decided I buried my lede and swapped the first paragraph to the last.

    Like or Dislike: Thumb up 0 Thumb down 0

  20. From “Development of the adolescent brain:
    implications for executive function and social
    cognition”

    “Neural plasticity of the developing brain may
    underpin different propensities for learning new
    skills, such as problem solving, at different stages of
    the life cycle. For example, sensitive periods for
    learning phonemes of one’s mother tongue occur in
    the first six months of life (Kuhl et al., 1992) and the
    ability to learn a second language declines with age
    (Hakuta, Bialystok, & Wiley, 2003). Logical reason-
    ing required to solve mathematical problems activ-
    ates both parietal and frontal cortex in both
    adolescents and adults. An fMRI study that required
    subjects to solve algebraic equations before and after
    a practice period demonstrated differential activa-
    tion patterns after four days of learning in adoles-
    cents and adults (Luna, 2004b; Qin et al., 2004).
    Both adolescents and adults showed an increase in
    prefrontal, parietal and motor activation while solv-
    ing the equations. Both groups also demonstrated a
    reduction in prefrontal areas after practice. However,
    adolescents, as distinct from adults, additionally
    demonstrated a reduction in parietal regions after
    the practice period.
    The authors proposed that the
    parietal cortex represents an ‘imaginal’ component
    necessary for this sort of abstract reasoning task.
    They proposed that after the learning period, the
    adolescents are less reliant on this area than the
    adults. However, the directions of cause and effect
    remain ambiguous. It is unclear whether the ado-
    lescents’ decrease in parietal activation with practice
    was a result of an immature parietal cortex and
    hence a higher relative dependence on the prefrontal
    cortex”

    Like or Dislike: Thumb up 2 Thumb down 0

  21. PD Shaw says:

    @john personna:

    Millions of students have climbed an educational ramp, k-12, and the impact I see with those algebra classes is that suddenly they are doing abstract symbol manipulation. To dumb it down, after adding and subtracting apples, suddenly it is x, y, and z placeholders.

    I’m about James’ age and that’s how I was taught. A curtain was pulled back at 8th grade and mysterious knew symbols were introduced.

    My grade school kids in a regular public school, however, are taught to use symbols from at least by second grade. They can be asked to identify a pattern : 3, 7, 11, ___, 19, asked to solve the missing number, learn to describe the pattern as X +4.

    That’s a different trajectory, that appears to assume kids will need algebra to go to college, as opposed to kids must learn lengthy multiplication/addition calculations so they can at least handle simple tasks at the cash register.

    Like or Dislike: Thumb up 0 Thumb down 0

  22. Franklin says:

    I basically agree with Jim Henley that what people tend to solve in real-life is word problems (we called them “story problems” when I was a kid), and I agree with Joyner that many of these are rudimentary algebra problems. I do think people absolutely need to be able to do this at a basic level to function in society.

    Anything beyond this level boils down to: should we require teaching kids more than they really need to know? To me, it just seems like an awfully slippery slope to reduce standards when the standards are already so low.

    Like or Dislike: Thumb up 4 Thumb down 0

  23. JKB says:

    So are we educating humans or teaching monkeys? That seems to be the question. Either we are developing cognitive abilities or we are teaching skills. In the beginning, they both look a lot alike and as has been pointed out by inference, many in education don’t know the difference. Yes, math has a lot of rote work at the beginning, so does baseball. You drill, drill with the explanation lost, either because you aren’t ready to see the big picture or the coach/teacher don’t know it themselves.

    Mathematics is a formalize way of thinking. It is far better than other fields for the mere fact that there is a right answer and you cannot live in your own mental fallacy as is so often the case in liberal arts. Education is less about the fact picked up along the way although a common set is useful and more about training the mind to order thoughts and articulate them. Yes, Algebra is hard, it’s a new way of thinking. Yes, Calculus is as well. They are a foreign language and the training is cumulative. And while you may not use the formulas in the future, you will always have the formalized reasoning skills.

    If we wish to dispose of something, let us get rid of critical analysis of literature. All I got from that was a deep, abiding aversion to Shakesphere, etc. Until the student has read many works and many analyses, they are unqualified to discuss most of what they are forced to essentially fake in such paper anyway. Isn’t it ironic that making great literature a farce by requiring incompetent criticism reduces the likelihood of future readings. Did I use that right?

    Like or Dislike: Thumb up 3 Thumb down 0

  24. Dave Schuler says:

    The prescription is pedagogically unsound for the reasons that john personna suggests. Further, since some significant minority of students will, in fact, require the advanced skill for which algebra is a foundational skill, not requiring the foundational skill of all students assumes some level of tracking, beginning at quite an early age.

    I took algebra when I was 13. I don’t know what most people’s experience is. I think that assigning students to different tracks starting at 13 or before is, if not an error, pretty inconsistent with our aspirations.

    Let me put it another way. Should reading be taught? The reading that most people do in their jobs does not closely resemble their first grade readers. If employers need their employees to be able to read, they could collaborate with the local community college (why would there be a local community college?) to create a course, say, “Reading for machine tool operators”.

    Like or Dislike: Thumb up 3 Thumb down 1

  25. @PD Shaw:

    That’s a different trajectory, that appears to assume kids will need algebra to go to college, as opposed to kids must learn lengthy multiplication/addition calculations so they can at least handle simple tasks at the cash register.

    Starting earlier & etc., may be great, but my point with all that above is that education has long since moved toward being “child development” oriented rather than skills oriented.

    Hacker’s essay, and most of the responses, respond to the skills aspect from personal experience. That’s fine, but that probably isn’t where educators are thinking these days.

    … or at least you need to pair the two. You don’t want to short-change brain development, and yeah you want to teach some useful skills.

    Like or Dislike: Thumb up 1 Thumb down 0

  26. Franklin says:

    Like PD Shaw, my boy could do simple fill-in-the-blank or solve-for-X by 1st or 2nd grade in a regular public school. Admittedly, he’s good at math like his old man, but given there were 10 more years after that to high school graduation, I would hope that all of his classmates could surpass that level.

    Like or Dislike: Thumb up 1 Thumb down 1

  27. JKB says:

    @Modulo Myself:

    I read a similar complaint on a site that taught self defense skills. The author pointed out that in martial arts some instructors will develop a “power through” trick that was hard for others, not that fast/strong to replicate. Also, they would drop small elements of forms, so that although the form looked lovely and got points in competitions, they left you open to attack in a real fight. Side by side pictures were quite dramatic in illustrating how the “modified” form left you unprotected if your opponent chose not to follow your lead.

    Like or Dislike: Thumb up 0 Thumb down 0

  28. @James Joyner:

    “you don’t need that in real life but a whole lot of kids are being denied a high school diploma because they can’t learn it.”

    This.

    And life in our society sans a high school diploma means being in the underclass for the most part.

    I can speak as someone with a family member who not only struggles with math, but who has an identified and diagnosed processing problem. This has meant a sincere concern that this person might not make it out of high school because, mainly of the requirement to take Algebra II (not because he lacks the overall intelligence or ability to make it through school or to function in life). Now, I am obviously pro-education, but my knowledge of this situation has led me to realize that a) we get a bit too cookie-cutter in our approach to education, and b) God help a child with problems of this nature who does not have actively involved parents.

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  29. Modulo Myself says:

    @JKB:

    It’s terrifyingly unfortunate that literature is taught simply so it can be explained or analyzed away. Shakespeare, Eliot or Melville can be as abstract as maths–which is why so many people are bored and find no pleasure in Lear, The Wasteland or Moby-Dick. Great literature is, on its surface, as arbitrary and capricious as a page’s worth of differentials that need to be solved.

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  30. Sandra says:

    Actually, I have found that algebra has come in handy more than once a week and geometry (when buying a house or redecorating) is an essential skill. Statistics, less so, except around the times of Presidential elections (nothing like knowing the margins of error that you can drive a Mack truck through). And calculus… wish I paid more attention, I would have gone the electrical engineering route instead of the “educator” track after all.

    Word problems, the torture from junior high, are what we face daily, and for many with a degree of math illiteracy, or dyscalculia (what dyslexia is to letters, dyscalculia is to numbers) something they learn to work at avoiding. About the same or slightly more people have this, as have dyslexia. However, having one does not mean you have both.

    The thing is, that many teachers in the primary grades have some form of it themselves (and are the reason why they went into the primary grades where strong math skills ARE NOT REQUIRED). This then leads to “uncomfortable” feelings and lack of skills being passed on to generations of students. In turn, this sets up a “math-phobia” that continues throughout a person’s life, unless they learn to address it.

    Math is not that hard, but HOW it is taught can make it HARDER. In addition, social progression before mastering concepts sets up a failure feedback.

    I tutor mostly middle school students in math concepts and HOW TO SOLVE real problems. Fractions are barely being taught, but competency in the skills and knowledge of operations involving fractions IS ESSENTIAL to algebra, geometry, trigonometry, calculus and statistics.

    Many of the kids I volunteer with are “at risk” of dropping out of school and falling into delinquency and criminal behavior. The math skills of what adults that are around (usually grandmothers that did not complete school) are not that well developed and they cannot assist with homework. Frustration from NOT understanding, leads to the development of “attitude” that further sets up for failure. It starts with having difficulty with completing homework assignment. This not understanding = fail quiz. Allowed to continue, fail tests, fail examinations, fail class, failing in school = dropout.

    It is not that the kids are not able to learn, it is the fear of ridicule, and the tacit acceptance of “well baby, maybe it is too hard after all;” from teachers, school administrators and the public.

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  31. JKB says:

    Of course, the problem is with the get it or get left behind way of educating. I saw a talk by Salman Khan where he made a simple but profound observation. Everyone gets stuck at some point in their schooling. Things just don’t click. But we have an education system that leaves those needing more time behind. If the stumble happens early enough or at one of those moments when they are sorting gifted from others, you can literally be sidetracked into to courses that don’t even offer the chance to catch up once you’ve gotten over the rough spot. Once tracked for the advanced programs, you get help to get over the tough parts because the system has a stake in not looking like they made a mistake.

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  32. @Steven L. Taylor:

    Don’t you have a harsh choice then? Either you dumb down the HS degree, or you create an option … the “underclass diploma”

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  33. Sandra says:

    @Sandra: I forgot to add that I agree with mathematics should be incorporated in ALL curricula. not segregated to it’s own separate “track.”

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  34. @john personna: There are some hard choices, yes.

    There are also issues of curricular choices the needn’t be that stark.

    Part of what I am getting at is that even someone on a college track doesn’t necessarily need as much math (since that is what we are talking about) as standard curricula tend to require.

    Indeed, in a broader discussion that goes beyond math: high school is more constraining than college. For example: one has to take more comp and lit in HS (usually 4 years) than you do in most undergraduate degree programs (where you might have to take 6 or 9 hours–the equivalent of 1/1.5 year(s) in HS). If one is good at math but struggles in lit, HS is harder than college in that regard. Granted, the likelihood that one is good at math but would fail lit is lower than the reverse scenario.

    Still, one has leeway in college in terms of avoiding things one isn’t good at than is the case in HS. I took more math in HS than I did in college, for example.

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  35. PD Shaw says:

    A family friend graduated from HS shortly after the recession, wasn’t sure he was college material, so he applied to a new large destination retail operation to work in the stock room. He was required to take an employment exam and was surprised by the amount of algebra on it. He handed it to the test-taker, saying he was surprised he would ever use this stuff, and the testaker said, you should see the old guys when they see that on the test, some of them leave on the verge of crying.

    What is the relationship between algebra and stocking goods? Is it merely a means of weeding down a large applicant pool? Is it representative of the ability to think analytically about space? Is it actually used on a day-to-day basis?

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  36. @Steven L. Taylor:

    Three things.

    First, if we acknowledge “child development” goals and that course plans are not for skills alone, then we must acknowledge that any “light” path to a college degree is going to have less development.

    Second, if we offer an easy option some number of students who were capable of more will “slack” into it.

    Third, I know that California high schools have taken a “pass them out” path, and as a result the Cal State system has had to increase remedial freshman classes.

    That doesn’t seem like a good result to me. A certain level of HS dropout may be preferable. Better than that though, we really should go the two path approach, with the “workers” HS diploma carrying less status than the “college bound” version.

    We need that status division because people, including adolescents, understand status and work for it.

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  37. @john personna: My only point would be that curricular choices are not set in stone and there is nothing wrong with reassessing them.

    Further, some requirements, including in math, often are built around the idea that not only are all students going to college, but that they are going to major in something that requires advanced mathematics skills. Speaking for myself by way of example, I did not need the pre-calc stuff I took in high school–and yet I managed to get a lot of education.

    Regardless, as per the conversation above, it is not that I am stating that we shouldn’t teach math, but that perhaps we might reevaluate how and what we teach.

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  38. sam says:

    @JKB:

    while you may not use the formulas in the future, you will always have the formalized reasoning skills.

    If the goal is to teach formalized reasoning skills, then why not make Algebra I and II electives, alongside of the propositional calculus and the first-order predicate calculus? These last two have the virtue of being more obviously connected to language than algebra.

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  39. @Steven L. Taylor:

    That strikes me as a “skills based” comment, referring only obliquely to “development based” discussion elsewhere.

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  40. Drew says:

    Another interesting and thought provoking essay from James.

    In a prior life an engineer, and even with a BS and MS in engineering, I’m rather ambivalent about the issue. If I understand JPs essential thrust, I think I’m with him. It’s more an exercise of the mind, and an ability to develop a way of thinking.

    In my original field, basically chemistry, heavy duty math was not required like in, say, big time physics or electrical engineering. You had basic rate equations (kinetics: how fast does it happen ?). And equilibrium equations (once we get there, what’s the state of play?) You of course had things like fluid flow, with mathematical descriptions of how things worked in corners, at the end of a pipe, at the boundary, or in the middle of the pipe. But the math wasnt important, it was the concepts. And you can use calculus in the kinetics and fluid flow or residence time in a reactor stuff but only for elegance. And in the engineering field they bastardized all these equations up so much with “factors” and “constants” etc that what’s the point?

    From where I stand, I’d mandate from the education system a rock solid understanding of arithmetic, a reasonable dose of algebra and maybe calculus for conceptual reasons……………and a big focus, and a much heavier focus than we have now, on statistics. I don’t care what field you are in, from housewife/husband to businessman to technical specialist…….statistics is a concept you run into every single day. Every damned day. And from where I sit, as a nation we are statistically illiterate.

    As for sentence diagramming……are you shxttn me? Maybe that’s why in don’t speak so no good….

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  41. Nat says:

    @Ben Wolf: Ben, we do have holistic curricula, but they are tremendously controversial. Google “math wars” or “mathematically correct”. We are still fighting those. There is a set of the population who fervently believes that the only way to increase math test scores is to increase rote learning. This is partially because of poorly implemented reform curricula and partially because some successful mathematicians believe that the way they learned is the only way.

    Modulo: “Thinking right now, I wonder if the anxieties teachers have over teaching math are taught to their students in addition to the actual material.” Not so much, starting in middle school. Teachers are frustrated, but tend to know their stuff. There are definitely those who should not still be teaching, but that is a different discussion. There is a huge problem with elementary teachers who need to be teaching number sense and basics, but who have no idea what they are doing.

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  42. James Joyner says:

    @Ben Wolf: @Ben Wolf: I agree that part of the problem is simply bad math teachers. It’s not as prevalent as bad social science teachers, but it’s a real problem. But I’m not at all confident that we can actually produce enough really good math and science teachers as we’d need, even if the threshold is to simply get the James Joyners of the world over the calculus hump. That is, I was at something like the 87th percentile on the SAT in math and yet found myself unable to learn it (granting a somewhat unique environment).

    s@JKB: @Dave Schuler: This is sort of where I was rambling toward with the close. I think we need to figure out a way to expose as many people to higher mathematics as possible without making failure to master it a career stopper.

    @Sandra: As noted earlier in the thread, we’re talking about different things. I absolutely agree that we use basic algebra and geometry on a regular basis. You’re talking about the sort of mathematics I mastered in grade school. I’m talking here about the sort of Algebra and Geometry I had in high school, which was much more abstract and complex equation based.

    @john personna: @Steven L. Taylor: As noted in the post and above thread, I think we need to expose everyone to higher math. The question is whether it should be a show stopper. To use my own example: I managed to do reasonably well in school, getting straight A’s through PhD level in political science, despite failure to master Calculus II. Should I have been forced to either master Calculus II or drop out of college? Given that I’ve managed to somehow get by for the past quarter century without such mastery, I’m hard pressed to see why.

    Again, I think the answer is some sort of exposure without risk path for several of the harder subjects, whether it be higher math, hard science, or philosophy.

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  43. @James Joyner:

    Again, I think the answer is some sort of exposure without risk path for several of the harder subjects, whether it be higher math, hard science, or philosophy.

    Exactly.

    As noted earlier in the thread, we’re talking about different things. I absolutely agree that we use basic algebra and geometry on a regular basis. You’re talking about the sort of mathematics I mastered in grade school. I’m talking here about the sort of Algebra and Geometry I had in high school, which was much more abstract and complex equation based.

    Also agreed.

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  44. @James Joyner:

    We are familiar with the Bryan Caplan style signaling versus learning discussion for education. While we may not weigh American education as mostly about signalling, we should acknowledge it as a component. Some places, with some degrees, and value networks to maintain, will want you to pass X to receive certification Y even if it is not “needed” only because it maintains the signalling value.

    At any level of education removing any arbitrary requirement for complete reduces signalling value.

    The contradiction I see in “Education, now with less Math!” is that proponents are trying to get acceptance and prestige for the student .. with a lower value degree, with lower signalling power.

    (Again, I think the answer is multiple tracks for education, doing what we can to let people switch to a higher track when they are late bloomers.)

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  45. James Joyner says:

    @Sandra: @Steven L. Taylor: To put it in a different way: It’s hard to see how someone is an educated citizen if they can’t figure out how many square feet of carpet they need for a square room. On the other hand, one can go through life just fine not being able to formulate a 12-step proof of why that math works.

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  46. (That is not to say that I am backing away from “forcing algebra into reluctant minds.” As I’ve said, I’d want to see arguments rooted in child development for that decision. Making everyone try real hard, with a passing grade and a diploma as a goal, may be very important: developmentally.)

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  47. Drew says:

    PS

    And to be clear. The single most important concept that the every day person needs to get from calculus is the concept of rate vs acceleration. Derivatives.

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  48. @James Joyner:

    On the other hand, one can go through life just fine not being able to formulate a 12-step proof of why that math works.

    If you keep saying things like this, is it because you reject the developmental aspect?

    On the other hand, one can go through life just fine not being able to formulate a 12-step proof of why that math works.

    When Steven says this does he not understand that he can never know what that other person, “Steve, with less Math!” was like?

    The key that you seem to be missing is that doing lots of repetitive algebra in adolescent years may change your brain in positive ways, but ways you can’t tie back through a process of memory now.

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  49. @john personna:

    Making everyone try real hard, with a passing grade and a diploma as a goal, may be very important: developmentally

    I agree. That does not tell us. however, what it is that they should be trying really hard to do, yes?

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  50. @this:

    Oops, for Steven’s quote I wanted:

    Speaking for myself by way of example, I did not need the pre-calc stuff I took in high school–and yet I managed to get a lot of education.

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  51. @john personna:

    The key that you seem to be missing is that doing lots of repetitive algebra in adolescent years may change your brain in positive ways, but ways you can’t tie back through a process of memory now.

    True. But then again that could be true about having students engage in more hours studying poetry, in shop class, or doing any number of other things.

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  52. @Steven L. Taylor:

    Right, and as has been my theme above, I think you get that from child development studies.

    You don’t get it from retrospectives of “math I did not use.”

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  53. @john personna:

    The key that you seem to be missing is that doing lots of repetitive algebra in adolescent years may change your brain in positive ways, but ways you can’t tie back through a process of memory now.

    Fair enough.

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  54. @Steven L. Taylor:

    Again, of course. If child development studies tell us that we get smarter and happier people out of poetry classes, go for it.

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  55. Nat says:

    @Steven L. Taylor: Some repetition is fine: the fluency that comes with practice is wonderful. That’s one of the reasons I play music. What repetition in math does not do is impart a useful understanding of the math itself or of its uses, and therefore removes most of the motivation for learning it. At least when I get enough time in on the guitar I can play a cool song for my friends.

    Being able to solve complex algebraic equations or to find the derivative quickly does not mean that a person has an understanding of what they are doing, why they are doing it, or how it might be useful, which is one of the reasons you forgot your calculus. As soon as you put the math in context it begins to make a lot of sense and become useful. The derivative is just a rate of change. You use it intuitively every time you visit fivethirtyeight and look at a graph.

    One big problem here is that we are trying to teach kids a zillion processes which are very useful and highly interlinked, but without context. If you have a couple of minutes, please read Lockhart’s Lament.

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  56. James in LA says:

    I use boolean algebra every day. All software developers do. But I think young people in school would be better served by courses in formal logic than math-based algebra. The value of abstract thinking cannot be underestimated.

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  57. Drew says:

    As an aside, and this may surprise some, music is considered to be steeped in intuitive mathematical concepts. All those “quants” Wall Street was hiring to do derivatives etc? Lots of music majors. More than math or physics majors according to a Goldman friend of mine.

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  58. wr says:

    @JKB: “If we wish to dispose of something, let us get rid of critical analysis of literature. I got from that was a deep, abiding aversion to Shakesphere, etc. ”

    And here is the entire Republican party summed up in two sentences: “I was too stupid to understand the subject, so clearly it’s worthless and should be banned.”

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  59. Modulo Myself says:

    @Drew:

    That’s no surprise. Music and maths are fields lit up by prodigies.

    Music is a lot like math in that you don’t have to know what the nature of something is in order to use it. With music, most people can hear music that is diatonic, and know, almost intuitively, why a musical phrase sounds ‘complete’ without knowing anything about tonics, dominants or dissonances.

    Math is similar. People can count and abstractly understand that ‘4’ is applied to apples, oranges, dogs, computers. To use Wittgenstein’s example, we seem to ‘know’ that somebody who has learned to count can conceivably count forever, without there being proof that they have grasped any specific mathematical rule.

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  60. stonetools says:

    Well, according to Mr. Heinlein:

    Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes, bathe and not make messes in the house.
    –Lazarus Long, “Time Enough for Love”, (Robert A. Heinlein)

    At the risk of disagreeing with him,I would say that for most folks, learning math beyond solving for basic quadratic equations is unnecessary.

    @john personna

    Haven’t such studies been done? This is not the first time that people are asking questions like , ” Is algebra necessary?”, ” Is learning Latin necessary”, “Is learning Shakespeare necessary”, etc.

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  61. Drew says:

    @Modulo Myself:

    This reminds me of a scene from “Amadeus”. The then ruler claims that Mozarts musical piece had “too many notes.”. Mozart inquires”which ones?”.

    The ruler has no clue, and makes up an answer. Mozart, of course, knows its just right.

    Perfect.

    Switching genres. (quite a bit). I’m also reminded of a video I have of critics speaking about Led Zeppelins first album. So the critic is going on about the phrasing and timing of a song………….and then just smiles and looks into the camera and says “perfect.” Nothing quantitative. You just know it.

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  62. george says:

    @wr:

    And here is the entire Republican party summed up in two sentences: “I was too stupid to understand the subject, so clearly it’s worthless and should be banned.”

    Is it the Republican party that’s trying to remove algebra from high school? Not a snark, just wondering. I suspect that idea is as popular among Democrats as it is among Republicans.

    Ultimately, most people use almost nothing of any subject they learn in school except reading, right, and arithmetic. Presumably we should only be in school until grade four?

    The thing is, most of our learning is increasing our mental capacity (like John Personna points out above); the point of doing pushups and running laps isn’t that you’re going to have to do pushups and run in day to day life, but that they increase your physical capacity and health. Same for studying algebra and Shakespeare in terms of mental capacity – as studies of neural development show.

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  63. al-Ameda says:

    Mathematics instruction can have great value to any student.

    I was an okay mathematics student in high school – not great, but good. I was tempted to stop taking math courses once it was not required – but I kept on because I was too lazy to disagree with my counselor who kept signing me up for the next course.

    In college I was a liberal arts major, however I took a 5 quarter series of courses including calculus, matrices, and power series, as well as 3 quarters of upper division statistical analysis.

    While I have had some occasion to use statistical methods in my professional career, much of the math I have not used. But that does not mean that it had no value. In fact I credit that instruction in mathematics and statistics as providing me with a more disciplined approach to solving problems of many types, how to organize myself to systematically solve a problem.

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  64. george says:

    @stonetools:

    At the risk of disagreeing with him,I would say that for most folks, learning math beyond solving for basic quadratic equations is unnecessary.

    For most folks, everything they learn in every subject past grade four is unnecessary. So is being able to do a pushup, or run a block … after all, that’s what cars and elevators are for. I think there was a Simpson’s episode which examined this (ended up with Homer wondering why he had to go through the effort of breathing, when an iron lung could do it for him).

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  65. Jonathan Stiegler says:

    I think some of the discussion above is more illuminating of how overpaid a lot of Americans are for what they do on a daily basis than anything wrong with having algebra as part of a standard curriculum. I wouldn’t want to be your kids growing up now if you think middle management jobs where you don’t do much more than arithmetic and push some emails will be as plentiful for your kids as they were for us.

    Regardless of whether algebra or calculus are still in the standard curriculum in the next 20 years, I know that my kids will damn well graduate high school with a rock solid foundation in both, I will make sure to that myself.

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  66. Nat says:

    @george: There are necessary skills critical (or at least helpful) to survival, though: being able to write a persuasive essay, being able to reason through a monthly budget, etc. I put quadratics in the same place as diagramming sentences or remembering the elements that compose DNA: they offer a glimpse into the building blocks of a topic, and offer next steps for the intrepid. It is up to educators to put those in context and make them interesting. I put push-ups and running a block solidly in the survival category: PE is the basis for skills and habits that allow us to maintain our bodies as we grow up and sit in front of computers all day.

    The iron lung example is excellent: imagine school being interesting enough and useful enough that all kids actually want to be there and learn, and are gaining success there! It would look radically different than what we have now, which is how this thread started, I think.

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  67. stonetools says:

    @Nat:

    Another Heinlein quote:

    “A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly.”

    There’s a lot you can do to develop a mind-and maybe learning to build a wall or balance an accountnt may be as good or better for development than advanced algebra

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  68. Jonathan Stiegler says:

    It’s pretty hard to be an accountant without algebra.

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  69. Drew says:

    @Jonathan Stiegler:

    Really? How so?

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  70. stonetools says:

    @Jonathan Stiegler:

    But you can certainly balance an account without algebra. The point is, there are plenty of things that can be thought to develop the mind-including many practical skills. Its not either teach higher level algebra or watch our kids’ minds turn to mush.

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  71. george says:

    @stonetools:

    There’s a lot you can do to develop a mind-and maybe learning to build a wall or balance an accountnt may be as good or better for development than advanced algebra

    Or studying literature, or history, or social studies, or geography, or anything past basic reading and writing. If the idea of school is just to pass on the bare minimum skills that will be needed by everyone, we’re wasting kids time after grade four.

    The reason we teach all those things, including algebra, is because the learning process both exercises the mind, and exposes students to the range of what can be learned.

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  72. DRE says:

    @Jim Henley: …there’s a lot of implicit algebra in using Excel or a dashboarding tool to solve a real-world problem. But damn if it doesn’t seem like I’m just multiplying, dividing, adding and subtracting all day.

    This seems backward to me. I love spreadsheets because they mean that I never have to multiply, divide, add or subtract at all. Excel does that for me. But it would be mostly useless if I didn’t understand much more complex mathematical concepts. If tools like spreadsheets were used extensively in math education, in order to overcome the drudgery of calculation and record keeping, I think it would be a lot easier to teach more advanced concepts. Much of what I see in my kids homework seems so buried in the drudgery that the concepts are hard to recognize, much less understand.

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  73. If a STEM professor released an op-ed along the lines of “while reading skills are critical for informed citizenship and communication, making kids master understand literature to graduate high school has disastrous consequences”, it’s unlikely people would take them seriously. Likewise, for a prominent scientist to admit they could not read would make them a laughingstock.

    It never thus ceases to amaze me the degree to which supposed educated people from non-STEM fields have absolutely no shame about publicly flaunting a shocking degree of innumeracy.

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  74. DRE says:

    @James in LA: “But I think young people in school would be better served by courses in formal logic than math-based algebra.”

    I think this is the most critical thing. The basics of logic should be taught along with arithmatic, and should dominate math classes in middle school. This would make it much easier to teach algebra in high school. Combine it with some programming, and teach kids to write programs to do their homework. This would provide extremely useful tools to those who don’t continue with advanced math, and allow a more direct teaching of advanced concepts to those who do.

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  75. James Joyner says:

    @Stormy Dragon:

    Likewise, for a prominent scientist to admit they could not read would make them a laughingstock.

    It never thus ceases to amaze me the degree to which supposed educated people from non-STEM fields have absolutely no shame about publicly flaunting a shocking degree of innumeracy.

    You’re comparing apples and pomegranates. Hacker and I repeatedly call for rigorous mathematical literacy (numeracy), including strong grounding in statistics. No one is arguing that mathematics isn’t important. No. One.

    What we are arguing is that making mastery of advanced algebra or pre-calculus a condition of graduating high school—or even a non-STEM college degree—makes little sense. I would say the same thing about, say, diagramming a complex sentence or identifying the symbolism of a haiku. There’s a vast difference between basic walking around knowledge and complex mastery of specialized subjects.

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  76. @James Joyner:

    Hacker and I repeatedly call for rigorous mathematical literacy (numeracy), including strong grounding in statistics.

    If you don’t know pre-calc or algebra, you’re not grounded in statistics. You may have memorized some canned algorithms for doing things that you trust to do what the person who presented them to you says they do, but you have no real understanding of the subject.

    or identifying the symbolism of a haiku

    Yes, but we still expect children to learn how to do this. And any educated person who said something like “oh, I just don’t get all the themes in Harry Potter, I never had a head for reading” would be a joke. Someone who says “oh, I never figured out basic algebra, I never had a head for math” should be considered just as much of a joke.

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  77. Likewise, for a prominent scientist to admit they could not read would make them a laughingstock.

    And I should note here I was thinking of a NPR piece I heard last year where a reporter with a journalism degree was trying to figure out how steep the hill in a local bike race was and actually brought on a mather professor from UPENN to calculate the slope of the hill for her. This was, in my mind, the equivalent of having to get someone to read a sign for her, and I was rather disgusted by the way she acted like it was some amusing joke.

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  78. Nat says:

    @Stormy Dragon: I think that if we taught reading the way we taught math, we would have that problem. If we taught English like we taught math, students would spend an entire year analyzing the mechanics of the traditional forms of poetry, but would perhaps write 3 or 4 poems (advanced students only!), and never actually read any literature.

    I also think our expectations are skewed. Reading HP is comparable to 5th or 6th grade arithmetic. Finding and understanding the themes in HP is comparable to a solid understanding of the subject matter in an algebra 1 or geometry course. I think people would raise a stink if students had to do any in-depth literary analysis to get out of high school, but that is essentially what we are asking of math.

    We do have a cultural stupidity in math, but that is because we set up our math courses to fail all “unworthy” students.

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  79. @Nat:

    I agree that the way we teach mathematics is horrible. But there’s a difference between “we should be ashamed at the broad inability of people in our society to learn basic math and we need to look into how our educational system can fail so broadly” and “we should stop trying to teach that because no one really needs to know it anyways”.

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  80. @Nat:

    I think people would raise a stink if students had to do any in-depth literary analysis to get out of high school, but that is essentially what we are asking of math.

    This leaves me wondering what you did in High School English since discussion (either in class or in the form of essays) of the themes in whatever we just read was the majority of what I did in grades 9-12.

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  81. JKB says:

    @James Joyner:

    The problem is, what does the diploma represent. If passing Algebra isn’t required for the high school diploma then employers will have to test some other way. Not to mention, the graduate who suddenly discovers that skipping Algebra competency in 10th grade has left them unemployable or nearly so.

    However, we do see to have a building solution with Khan Academy. Rather than track students in lock step, they could all be required to take math every year, learning online then having more like a math lab where the teacher provides individualized help on the lessons. There is no real reason to track the whole grade at some arbitrary pace or even pigeonhole students based on some rough patch they encounter. There could also be video discussions by the most capable mathematicians to provide the theoretical framework the formula fit into. This would provide time for kids to work through the rough spots and permit them to race ahead when it starts to flow.

    Perhaps some annotation on math competency on the diploma would be better than the vagaries of curriculum difficulty we have now.

    BTW, math starts making a whole lot more practical sense when you get into physics but you need the Calculus which requires Algebra, etc. Up until physics with calculus, there is to much waving of the hands.

    As for Trig and Geometry, well, that could have been reinforced with practicality if some idiot hadn’t decided to get rid of woodshop and other manual arts. A year of pen and paper drafting would also help this as well as train the student how to visualize in 3-D. The magic boxes, such as CAD and Excel, are great for productivity but they don’t train the mind or improve the mental capabilities of the student.

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  82. Alex says:

    At the same time, while most of us will never need to solve a quadratic equation, much less do whatever it is that Calculus is used for, at least one in twenty will. How will our future mathematicians, physicists, chemists, and engineers discover their interest and aptitude for those endeavors if they’re not exposed to abstract math in high school?

    Hell, American kids who are going into a lot of STEM fields aren’t exposed to enough abstract math in American high schools. I was the most accelerated math student in my high school class – my teacher thought I seemed really bored in Algebra II, so she let me take the final exam in October. I passed and thereby leapfrogged ahead by a year. (To me, even then, the ideal of memorizing the quadratic formula seemed ridiculous – you can easily derive it from scratch in a minute or two by the method of completing the squares.) I ended up graduating with both Calculus I and Calculus II (as an independent study) under my belt. In undergrad, as a physics major, I took eight or math courses and got as far as tensor calculus and a bit of measurement theory.

    Despite all that, I found myself woefully unprepared in grad school (physics again) when compared to European or Asian students. They came into grad school already knowing Lie group theory and a bit of algebraic topology – things that are considered advanced graduate student courses in the US.

    These days I’m working in the software industry and although I’ve forgotten more math than most people have ever learned in the first place, I’m completely boggled by computer programmers and even electrical engineers who aren’t able to do what I consider really elementary stuff like calculating the square root of i (itself the square root of -1). Electrical engineers are supposed to be comfortable with complex numbers.

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  83. michael reynolds says:

    I know nothing about algebra. My math ends at long division.

    I also don’t know how to diagram a sentence. No, really: no idea.

    I’m comfortably within the 1%, have authored around 150 books, have two movie deals in the works, more work than I can keep up with, and most people think I’m fairly bright.

    Put away the cookie cutters, math folk, there is no formula here. It’s past time to rid ourselves of the notion that all kids develop in lockstep and should all hit certain stops on a prescribed path at a given date. Humans are not machines, they cannot be built on an assembly line.

    We have the tools to judge kids as individuals, and to teach them as individuals, so why don’t we begin to evolve past grades and ages and mass-produced curricula?

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  84. superdestroyer says:

    @Argon:

    Do you really believe that kids who never learned that 3 x 4 = 12 will ever be able to do fractions or factoring? Do you really believe that kids who never learn to add and multiply fractions will ever be able to be analytical?

    Do you really think that kids that cannot divide 100 by 5 without a calculator will ever learn algebra, statistics, or calculus?

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  85. JKB says:

    @michael reynolds:

    The problem with your plan is it means the high school diploma means nothing, nor the college degree. Either you meet a standard or the credential is worthless.

    Now, this doesn’t have to be a terrible thing. We could dispose of the diploma and create competency records that records satisfaction of competency testing. Schools could become more like labs with experts to assist the independent study for those kids needing working hours warehousing. The revolt against the failed public education with homescholers and such might push this along.

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  86. george says:

    @michael reynolds:

    What should we be teaching kids past the ‘read,’rite, and ‘rithmetic stage then?

    I’m also comfortably in the one percent, though as an engineer. In my case I do use algebra daily, but I’ll concede that’s hardly typical. But other than that, I don’t think there’s really anything I use other than the three R’s on a daily basis.

    Literature? I read a lot of fiction and non-fiction, but that’s just one of the 3 R’s. I don’t do literary analysis of it.

    I write reports and so on regularly, but again, 3 R’s.

    History? Not often I need to know what led to say the Civil War, or say the “glorius revolution”. Hobby at best.

    Geography? Other than for travelling or hobby, never.

    Social studies? Same as history, never comes up except as personal interest (hobby).

    My point is that if the purpose of education is to give students tools they’re going to directly use, we might as well stop at grade four for most students – they’re just not going to use anything past that in day to day life unless they specialize.

    But if you take post grade four education as serving to increase mental capacity, in the same way that phys.ed, while not teaching direct skills (how many folks need to be able to play basketball in day to day life for their livelihood), builds up healthier minds and bodies, and to expose students to the range of intellectual possibilities, then all of the above serve a very real purpose.

    And algebra is the basis of every math beyond arithmetic. If students are having a hard time passing it, I’d argue a case could be made for easing the testing for those who aren’t planning on doing a STEM degree. But I would no more remove it from the curriculum than I’d remove English or history or geography.

    In fact, the proposal to remove algebra strikes me as the continuation of the argument that the humanities are a waste of time and should be dropped from the curriculum because they don’t lead to direct business or day to day life skills, and is wrong for the same reason.

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  87. george says:

    Though I’ll note the whole idea of elminating teaching subjects not used daily would remove a lot of political conflicts. Not one person in a ten (and probably far fewer) uses evolution in day to day life. Nor for that matter chemistry, physics, or any science). Removing them from the elementary and high school curriculum would probably go over well in some parts.

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  88. FWIW, Tyler Cowen points to this wide-ranging reply:

    Yes, algebra is necessary.

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  89. @michael reynolds:

    This advice is equivalent to an NBA player suggesting kids drop out of school and focus on playing basketball because that’s all he ever needed. It may have been true in his case, but it’s such an unlikely avenue to future success that you have to be nuts to suggest it as widescale model.

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  90. @michael reynolds:

    I also don’t know how to diagram a sentence. No, really: no idea.

    To some extent, you must. Otherwise how do you know how to write one properly? The purpose of diagramming sentences isn’t because diagramming sentences per se is a useful skill, but that it’s the best way of getting people to internalize the structural mechanics of written english. Until they do this, they’re not going to be able to write anything beyond the most simple sort of sentence.

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  91. James Joyner says:

    @Stormy Dragon:

    The purpose of diagramming sentences isn’t because diagramming sentences per se is a useful skill, but that it’s the best way of getting people to internalize the structural mechanics of written english.

    While I was actually pretty good at diagramming sentences (a skill that’s long since perished from atrophy) my ability to write came almost entirely from reading widely and mimicking. I’m actually rusty or never learned a lot of rules of grammar but have managed to go quite a ways on the basis of “what looks right.” Whether this is some unique talent or widely transferable, I couldn’t say.

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  92. @James Joyner:

    What I’m saying is that your intuitive ability to figure out “what looks right” is based in part on the time you spent as a child diagramming sentences. You no longer do it conciously, but the process of decoding what your read and in constructing what you right make use of it internally.

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  93. @Stormy Dragon: Not to get too pedantic in this discussion, but I would point out that depending on when or where one went to school, one may never have learned to diagram sentences–it is not the only way to learn grammar.

    My wife went to early highs school in CA and I went in TX. I learned to diagram sentences, she didn’t. We both can write and speak in English.

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  94. @Steven L. Taylor:

    The question wasn’t whether there are alternatives to diagramming sentences, it was whether time spent doing it was wasted time. Both you and your wife did some exercise as a child that caused you to internalize the mechanics of english grammar. Whatever that exercise was, it’s unlikely you still do it as an adult, but you still benefitted from it because it built the ability you do use today.

    In essence, diagramming sentences is kind of like academic scaffolding. There’s a lot of ways to put up scaffolding up, and the fact the scaffolding is no longer there once the building is completed doesn’t mean the effort spent in putting it up was a waste.

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  95. @Stormy Dragon: Yes, but isn’t what you are describing what this whole thread is about: what the appropriate scaffolding ought to be? As James noted above: no on is suggesting we stop teaching math.

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  96. Sandra says:

    @James Joyner: “I’m talking here about the sort of Algebra and Geometry I had in high school, which was much more abstract and complex equation based.”

    From small seeds, grow big trees…

    I did most of my college math in a large chunk, going from one class to another to another with only the term breaks. I went from a “generalized” review of everything to trigonometry in one class and worked my way through differential forms, integral forms, vector analysis, and multivariate calculus. All accomplished in my 40s. I had severe math phobia, in high school, but out of necessity picked up “things” to do my military duties. It wasn’t until I retired from active duty that I could focus on “schooling” for myself, and getting our teen kids through high school.

    What I had found in my personal observations over the years, was that those that had the greatest difficulty in higher math (and sometimes in other areas as well), had NOT mastered some pretty basic skills you develop while learning “rote” stuff like memorize your times tables, or learning enough Greek and Latin to understand the vocabulary, or the order of operations, or the additive and distributive rules. Things that to those that are math literate, are “look, see, finger snap!” Instinctive by the time you are an adult.

    You want to know where we lose future scientists, engineers, and competent workers in all fields; we lose them in the middle school years where “tracking” begins.

    @superdestroyer:

    Many of the students fail algebra because those students have never developed the underlying mathematical skills in earlier grades. Does anyone really believe that students who never memorized their multiplication tables are ever going to master algebra.

    Children in the primary grades are already exposed to Algebra, and begin learning the various orders of operations. Learning basic skills like “reading a clock.” (Not a digital timepiece, but one of those old circular, round clocks with two hands and 12 numerals) sets up the basis for learning both fractions (quarter, half, thirds) but also angles. Until I started to study how to teach mathematical skills and concepts, I had no idea about how much more difficult it has become.

    Having classes that reinforce these concepts is important too! Most of the kids I tutor have failed one or more chapter tests on operations with fractions. A few afternoons spent cooking or baking with me, learning to change the ratio of ingredients for example… suddenly the “light” comes on and they have made the jump to understanding fractions. Middle school students no longer have home economics or “shop” classes anymore. However, those classes re-enforced other areas of learning. Many of the kids have no one that cooks from scratch in their homes… so where does this practical application of theory happen.

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  97. michael reynolds says:

    @george:

    I think the answer is that we don’t quite know yet what to teach or how. I think this is an evolving situation thanks to leaps in technology – internet, apps, online interactivity, so on.

    But I think it’s time to start looking at the future of education rather than focus on tweaks to the past system. What does education mean in a world where all data is available all the time to everyone? That’s a different educational system entirely. I think we need to focus on teaching context, research skills, critical judgment. Not to say we don’t still need to teach facts, but the focus should reflect the fact that we do in fact have this thing called the internet.

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  98. michael reynolds says:

    @Stormy Dragon:
    As others have pointed out above: there is no necessary connection between diagramming sentences and writing them. If anything, it may hurt in my line of work. I don’t use language according to rules, I use language to convey meaning, emotion, etc… I probably violate some grammatical law in every page I write.

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  99. @Steven L. Taylor:

    Yes, but isn’t what you are describing what this whole thread is about: what the appropriate scaffolding ought to be? As James noted above: no on is suggesting we stop teaching math.

    Not Really. Diagraming sentences is a method of teaching a subject (mechanics). Hacker was making an argument along the lines of “stop teaching mechanics”, rather than “we need to teach mechanics differently”.

    @michael reynolds:

    Part of mastery is learning when to break the rules. But the master should still know what the rules they’re breaking are.

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  100. michael reynolds says:

    @Stormy Dragon:

    Part of mastery is learning when to break the rules. But the master should still know what the rules they’re breaking are.

    Seems logical, but I don’t think it’s true. A great number of writers have little if any formal education. I think what happens more often is that people become captured by and intimidated by the “rules.” When I talk to aspiring writers I give them one rule: tell your story.

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  101. grumpy realist says:

    @Alex: I had to pound group theory into my head the first year of grad school….A lot of the commentary in L&L Quantum Mechanics helped out a great deal. Recommend highly.

    Trying to pound differential geometry into my head and tensor dragging, which I had to do several years later…..ouch! I think the worst was trying to visualize a non-trivial SO(3) mapping in the fifth dimension.

    And right now I’m going back and dealing with quantum statistics for qubits. Expect another shredded brain for a while.

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  102. JKB says:

    As a humorous aside on why political science professors shouldn’t expound on math, turns out no one remembers “Fermat’s Dilemma”

    UPDATE: Thread-winner from Orin Kerr: “Fermat’s Dilemma is whether to admit that you don’t know the proof for a theorem or just to pretend you know the proof but you don’t have space in the margin to explain it.”

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  103. EP says:

    I am a German, teaching at an American university. Allow me still a few comments:
    1) High school algebra is very basic stuff for European standards. Although standards are also declining in Europe, in most European countries this content is still taught to kids when they are 12. I find that US students are mathematically the least prepared when they enter college, among all nations we have.
    2) The example given about the consumer prize index reminds me about an old joke about New Math. “Exercise: Five red, three green, and two blue balls are in a bag, from which one is chosen randomly. Underline the word “green” and discuss with your neighbor the unfairness of having only two blue balls in the bag.”

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  104. grumpy realist says:

    @Drew: I think your friend was having you on. The level of mathematics that is used by quants isn’t the sort of thing you learn in a music course. All the quants I know (including myself) come from mathematics or theoretical physics disciplines and at the Ph.D. level. It was in fact quite amusing how often I felt like “oh, yeah, that equation again!” when studying quantitative finance.

    Someone with a Ph.D. in physics/mathematics who was also passionate about music and a double-major in math and music at the undergrad level? That I will believe. Plopping someone from Julliard down in the middle of Moody’s to figure out CDOs? No. That I do not.

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  105. Katie says:

    Is the issue really that students are failing Algebra or that educators are failing to teach it properly? I was a straight A student for most of my early academic career and managed to excel in courses like Trig and Calc. But back in middle school my parents got me a tutor because our math teacher had no clue what he was doing and the entire class was failing Algebra. With that extra help I was soon breezing through the tests and helping the rest of my classmates gain a better understanding of the course materials. So before we decide that mathematics aren’t worth the bother maybe we should look at how these courses are being approached by schools and educators? If the teachers understood the materials and were instilling basic skills from the ground up we wouldn’t have these issues. I’m not trying to outrage the entire teaching community – obviously there are many wonderful teachers out there. But sadly, they are in the minority.

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