Is abstract thinking necessary?

Noted social scientist, Andrew Hacker, wrote a provocative opinion piece in the New York Times Sunday arguing that we relax mathematics requirements for higher education. Here are some excerpts from his piece:

New York Times: A TYPICAL American school day finds some six million high school students and two million college freshmen struggling with algebra. In both high school and college, all too many students are expected to fail. Why do we subject American students to this ordeal? I’ve found myself moving toward the strong view that we shouldn’t.

…There are many defenses of algebra and the virtue of learning it. Most of them sound reasonable on first hearing; many of them I once accepted. But the more I examine them, the clearer it seems that they are largely or wholly wrong — unsupported by research or evidence, or based on wishful logic. (I’m not talking about quantitative skills, critical for informed citizenship and personal finance, but a very different ballgame.)

…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.

The expected reaction from some of my colleagues was understandably negative. After all, we live in a world that is becoming more complex requiring more mathematical skills not less. Mathematics is as essential to one’s education as reading. In the past, I too would have whole heartedly agreed. However, over the past few years I have started think otherwise. Just to clarify, Hacker does not (nor I) believe that critical thinking is unimportant. He argues forcefully that all citizens should have a fundamental grounding in the concepts of arithmetic, statistics and quantitative reasoning. I have even posted before (see here)  that I thought mathematics should be part of the accepted canon of what an educated citizen should know and I’m not backing away from that belief. Hacker thinks we should be taught a “citizen’s statistics” course. My suggested course was:  “Science and mathematics survival tools for the modern world.”  The question is whether or not we should expect all students to master the abstract reasoning skills necessary for algebra.

I’ll probably catch a lot of flack for saying this but from my professional and personal experience, I believe that there is a significant fraction of the population that is either unable or unwilling to think abstractly.  I also don’t think we can separate lack of desire from lack of ability. The willingness to learn something may be just as “innate” as the ability to do something. I think everyone can agree that on the abstract thinking scale almost everyone can learn to add and subtract but only a select few can understand cohomology theory.  In our current system, we put high school algebra as the minimum threshold, but is this a reasonable place to draw the line? What we need to know  is the distribution of people’s maximum capacity for abstract thinking. The current model requires that  the distribution be  almost zero left of algebra with a fat tail on the right. But what if the actual distribution is broad with a peak somewhere near calculus?  In this case, there would be a large fraction of the population to the left of algebra. This is pure speculation but there could even be a neurophysiological basis to abstract thinking in terms of the fraction of neural connections within higher cortical areas versus connections between cortical and sensory areas. There could be a trade-off between abstract thinking and sensory processing. This need not even be purely genetic. As I posted before, not all the neural connections can be set by the genome so most are either random or arise through plasticity.

To me, the most important issue that Hacker brings up is not whether or not we should make everyone learn algebra but what should we do about the people who don’t and as a result are denied the opportunity to attend college and secure a financially stable life. Should we devote our resources to try to teach it to them better or should we develop alternative ways for these people to be productive in our society? I really think we should re-evaluate the goal that everyone goes to college. In fact, given the exorbitant cost and the rise of online education, the trend away from traditional college may have already begun. We should put more emphasis on apprenticeship programs and community colleges. Given the rapid rate of change in the job market, education and training should be thought of as a continual process instead of the current model of four years and out. I do believe that a functional democracy requires an educated citizenry. However, college attendance has been steadily increasing the past few decades but one would be hard pressed to argue that democracy has concomitantly improved. A new model may be in order.

6 thoughts on “Is abstract thinking necessary?

  1. […] However, there are numerous confounding factors in trying to relate the teaching of abstract reasoning in schools to any of these purported benefits. I don’t see any way to causally link the study of algebra to any of these effects in a controlled way. For instance, being able to master algebra may be related to other factors that also affect its supposed benefits. These factors could be something we call intelligence, persistence, or parental guidance. Even more likely, they are a combination of all of these and many others. I cannot speak with any certainty about how the capacity to reason abstractly affects people outside academia. Few in this debate have pointed out how poorly we understand the issues involved (but see Carson Chow’s interesting take on the issue). […]


  2. I look at this issue perhaps from a different perspective. For example, you mention Simon (of chern-simons theory—which i’ll have to google since i can’t recall it, though it may be affiliated with (the famous) s. chern of china who wrote a brief review in am math monthly years ago i liked). And, that he runs a hedge fund.
    This may be biased, but too often when i think of hedge funds i think of people profiting (using advanced math methods) from essentially selling alot of junk (eg junk food for emerging markets, lottos, etc.) . Obviously its not all that—some investment may be made in, say, MIT opencourses, wikipedia….

    But alot of math applied in the economy may require the existence of a ‘dumbed down’ mathematically illiterate population to support those who sell its applications. So when one says ‘what is the benefit of math or any other (eg critical thinking) knowledge’, this partly depends on who ‘we’ refers to. And, what ‘society’ ‘needs’—again society is subjective (like legal citizen’ or ‘human rights’) as are ‘needs’ (eg following ronald reagen, i refer to the Koch family as the ‘truly needy’ because apparently they need billions of $—-maybe i can prove this to be a fact following Rawls and Amartya Sen).

    Alot of the current push for education seems often to derive from a few people to need some people to have an intermediate range of scholastic skills—yeah, you need algebra, maybe some programming or software skills, but this is required essentially to do ‘lower level’ tasks required for ‘higher level’ tasks performed by say ‘managers’. (eg a Recent post by Ben Goldhager on Bad Science blog suggets that over 50% of neuroscience papers published make an elementary statistical error (this was published in PLOS). No dought these people have many competencies, but not apparently in statistical inference—hence one can even question why they are doing research and being supported by a staff who are not encouraged to think they could be involved in ‘higher order’ tasks such as research design. (This might as well be theology—-the blind leading the blind, and getting paid to lead as well.)

    I see research all the time (eg from Georgetown u ctr on labor supply) about how ‘ we’ or ‘our society’ projects we need mostly home health workers, service industry workers, etc, and maybe not so many scientists. Its possible the republican congress and fox news don’t need anyone who understands statistics; same would be true of many religious evangelists. (the blog ‘math under the microscope’ also says basically a vanishingly small % of people need to know any advanced math—its only required for research; other than that its pop entertainment—maybe the kinduh stuff jonah lehrer did (i read his new yorker article on E O Wilson, which was interesting, if somewaht incomplete from a population genetics point of view—tho it wasnt in J Theor Biol or TPB.).

    i guess maybe i’m a biased keynsian—people should be organizing society and needs based on less busy and destructive work, and more towards developing advanced capabilities (eg sen) whether valuing nature, or arts, or sciences…).

    It may be possible that rather than ‘society needs stupid people’ some people are say ‘genetically’ stupid and so they will always prefer Rush Limbaugh and junk food, or this may be an emergent phenomena neccesarily correlated with ‘laws of large numbers’ (eg perhaps like the ‘junk dna’ or the failed histories seen in the fossil record). But ‘another end of history’ may be possible, even if at present ‘there are no academic jobs’ because most future work will be maybe be in hotel work at conferences on global warming, or airports transporting gore to a film screening.

    another kind of related question is, for example, if one glances at the variety of theories on arxiv (in math, cs, logic, physics…) how many of these are needed apart from those who either produce them for work or produce them as a labor of love (or maybe hate—eg the various dissenting views, eg Dyson fo example showed schwinger and feynman were redundant, and someone did that for heisenberg and schrodinger. Different perspectvies have value in various contextx (as likely do forms of language or music or art or recipes, economic skills) but when is redundancy innefficient (eg too many school grads of various types) and possibly a ‘socially unnecesary’ ‘scam’ (eg new form of junk food).
    also, one wonders what academics as a subset of the public should know—how many can read or understand what the person in the next office or next building or who wrote the book next to their in the library wrote, and what should be the criteria for deciding what they should comprehend.
    again one ends up with the problem regarding what people can actually know, whether geneticlly or socially constructed. (if that equation of consiousness discussed on the blog–tonis?—is true, then maybe a solution of that might be the answer.)


  3. If you want to think of better reasons to not teach math, you should take a look at the concept of digital wisdom, rather than Mr. Hacker’s view. You might also consider that many Chinese music students seem to have perfect pitch, much more than statistically expected. Why is that? Perhaps some things can be taught, it just takes effort. I do empathize with people that have difficulty with math. I personally get nauseous when I open a quantum chemistry textbook … the only course I dropped out of in college because I could not handle it.


  4. Chinese is a tonal language which may explain why Chinese students have a higher proportion of perfect pitch but that would be a training effect.


  5. I have dyscalcula which is math blindness. Interestingly, the only math I got was algebra, got A;s all the way. Do I remember any of that now? Ha.

    I am also verbally gifted. I was fortunate to be in school during the 1940’s and 50’s and not subject to the demands of today’s world. My teachers honored what I could do and passed me with D’s in all my math courses. Colleges accepted me on my combined IQ score. Today, all that would be denied.

    Next point: I read many years ago that from 30 to 50% of the world’s population cannot think abstractly. Think Jerome Kagan made that point which is politically incorrect and no one dares mention it today. I do because I think expecting people to do what they cannot do is abusive. But then what do I know I am just a cranky old lady.


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