Concrete vs. Abstract

I spent the whole day reading other people’s blogs psyching myself up to do this, so now I only have a few minutes to do it, so this one’ll be short.

In April 2008 the NY Times published a little article announcing a study by researchers at Ohio State University claiming to find that “it might be better to let the apples, oranges and locomotives stay in the real world and, in the classroom, to focus on abstract equations.” The study was published in no less a forum than Science Magazine. Knowing that something must be terribly wrong, I found myself inexorably drawn to read the Science article for myself.

“The Advantage of Abstract Examples in Learning Math”
by Jennifer Kaminski, Vladimir Sloutsky, and Andrew Heckler
Science 25, April 2008, Vol. 320. no. 5875, pp. 454-455

The article summarizes the author’s findings that when they taught a certain mathematical concept through concrete examples, students did not do a good job applying the concept in a novel situation, but when they taught it through an abstract representation, the students successfully transferred the concept to a new situation. They concluded that “If a goal of teaching mathematics is to produce knowledge that students can apply to multiple situations, then presenting mathematical concepts through generic instantiations, such as traditional symbolic notation, may be more effective than a series of ‘good examples.'” (p.455)

The bad news is that the actual details of the study design are not found in the article. The good news is that they are found in a supplemental thingydoo on the Science website. The bad news the article costs $15 to download. The good news is that the supplemental materials don’t cost anything. Even better, since I paid for and printed out everything and it’s next to me right now, you can benefit from what I found out without even leaving this site.

If you’re a math teacher you already know there must be something rank in here somewhere. The authors are actually claiming to have scientifically shown that it’s better to introduce a math idea in a way totally disconnected from reality than to base the idea on anything the students already understand about the universe. Well, a look thru the online support material makes the rankness available to all:

First of all, when the authors say they gave students abstract training in one condition and concrete training in a different condition, what they mean is that in both conditions, the students stared at words on a computer screen. In the abstract condition, the words went like this:

“On an archaeological expedition, tablets were found with inscriptions of statements in a symbolic language. The statements involve these three symbols: {circle} {diamond} {flag} and follow specific rules.

“Rules for combining symbols.

“Rule 1. The order of the two symbols on the left does not change the result.

For example {diamond}, {flag} -> {diamond}

is the same thing as {flag}, {diamond} -> {diamond}


In one of the three concrete conditions (the other two were similar), the words went like this:

“A pizzeria takes orders for one, two or three slices represented on individual cards as {1/3 of a pizza}, {2/3 of a pizza}, and {3/3 of a pizza}.  Multiple orders are placed at a time; and the cook systematically burns a portion of each group order.  Antonio needs help to determine how much pizza is burned.  There is never more than 1 whole pizza burned.  So the burned amount will always be {1/3}, {2/3} or {3/3}.

“Rules for finding how much pizza is burned stated by Antonio:

“Rule 1.  What I order first or second doesn’t matter.  The same amount gets burned.  For example, if I order this {3/3} first and then this {1/3}, then this much {1/3} is given to us burned.  The same thing happens if I order this {1/3} first and then this {3/3}.  We get this much {1/3} burned.”

In other words, the instruction in the two conditions was substantively identical, except that in the “concrete” situation there was some additional distracting information.

But the whole point of using concrete examples in teaching math is to connect the mathematical structure to a comprehensible reality in the students’ world, and this was not done at all, in either condition.  So the question that the study authors, and the NYT article, addressed themselves to (“Is it better to use concrete or abstract teaching methods?”) wasn’t answered.  The question that got answered was “Do people learn better when you don’t throw in distracting info?”  (Oh, you mean they do?  You’re kidding, right?)

I’ve already made myself late but I want to tell you more detail just to have it on record, so I’ll post again soon.


7 thoughts on “Concrete vs. Abstract

  1. When I was a student, I found mathematics much easier to understand when it was separated from “what I already understood about the universe”. I still find theoretical or abstract mathematics much easier to comprehend than something that smacks of finance or physics. I tutor mathematics and find that often students understand it better when it’s introduced in an abstract fashion.

    I have been reading a lot of blogs lately from math teachers, and find their methods of introducing math in concrete ways to be interesting, and I take a lot away from those blogs. But my own personal experience also suggests there may be some truth to this study. You clearly started out with a specific bias against the concept of the study, rather than keeping an open mind.

  2. The methodology of the study is ridiculous. The study authors have no idea how to design a lesson around a concrete example – their “concrete” lesson doesn’t make any sense. (I wrote a more detailed dissection here.) So their findings don’t apply to any thoughtful implementation of a concrete example. Whatever you think of the conclusion on its own merits, the study doesn’t back it up at all.

    As an aside, I’d like to press on the way you’ve expressed the distinction between abstract and concrete. “What you already understand about the universe” is a very broad category, and for a person (as I imagine you to be) with a substantial level of mathematical development, it includes a lot of “pure math.” To me concrete doesn’t especially mean physics or finance. Just anything you already understand; anything you can imagine or visualize clearly; anything you can hold and manipulate comfortably in your mind. For a lot of students, that means their daily physical and social world, which is one reason why I object so strongly to the study. But the more mathematics a student has learned (and I mean really learned, internalized, owned), the less necessary it is to look outside mathematics for the concrete reality that give problems a sense of life. I’ve discussed this at some length here.

    (For me personally, finite groups, rings and fields of small order are wonderfully concrete; topological spaces less so unless they can be realized as subsets of R^n; and categories are airy, insubstantial things that I don’t understand. If you want to teach me about categories, I am excited to learn – but please use groups and rings as concrete examples.)

  3. Hi there, I just wrote about this study and Sue VanHattum alerted me to the fact that you had too. I completely disagree with you and invite you to discuss the issue over at my place.

  4. the political dimension is missing
    from this discussion altogether…
    but for me, it’s *what’s going on*.

    in my parts of the universe… college math classrooms
    and math-ed blogs… it’s widely known that students
    that can, for instance, easily solve a system of two
    linear equations in two variables will resist understanding
    how to *set up* such problems from exercises given
    in paragraph form. “word problems” are the bane
    of one heck of a lot of algebra 101 students…
    and this is very well known.

    in particular, students *do not want to*
    define variables based on “concrete” data.
    on my model, this is because “finding out
    how confused one has been so far” is
    humiliating. take a “coin” problem.
    some pedant keeps insisting that
    one must write out “n = number of nickels”
    (or “d = # dimes” or what have you)
    and *won’t settle for* “n= nickels”.

    the instructor wants to make the point
    that, say, the value-of-all-the-nickels-in-cents
    can only “really be understood” as 5n
    if we clearly know that “n” itself
    represents a number of coins;
    there’ll be an equation about how-many-coins
    and an equation about how-much-money;
    units must match across equations;
    etcetera… (of course everyone *here*
    gets it).

    a lifetime of being urged to perform
    meaningless tasks for incomprehensible
    reasons makes it a moral certainty to
    many students that whatever point
    the teacher thinks they’re making
    is irrelevant to whatever allows those
    slightly-more-advanced students
    that breeze through the set-up process
    *to* breeze through the set-up process
    (after all, many of *those* students
    *don’t define their variables at all*).

    “n” and “d” don’t need to mean
    *anything at all* (in the “concrete”
    sense… the so-called “real world”)
    for “n+d = 12 and 5n+10d = 105”
    to have a solution. so it’s just
    intrinsically *harder* to work
    the same problem if we have
    to attach meanings… particularly
    if we are, not to put to fine a
    point on it, barely literate.

    the problems are *already assumed*
    real-world-useless… it’s *math class*,
    for hecksake. so there’s a very widespread
    belief that throwing word problems in
    is *nothing but* more difficulty-for-its-own-sake;
    one is being set up to take another fall in another
    “weeder” course because they never tell you
    what they *really* want… they keep moving
    the finish-line.

    very few problems in math ed can be as well-known
    in my neck of the woods as this, actually… the
    “remedial algebra” courses are paying a lot of
    the bills at the community college that recently
    grew tired of me and most of the faculty will
    have to face this issue with at least a few classes
    every year. what then are we to do.

    well, we could *cheat*: here are some “coin problems”;
    set ’em up *this* way. (here are some “boats in a
    stream” problems; this way. etcetera.) yes it’s
    pandering. yes it defeats the stated purpose
    of having a section on “real-world” problems.
    nevertheless. students must be passed along
    and books must be sold (for a great deal more
    than their worth given the glut on *old* books
    with the same god-damn misguided pedagogy
    [and slightly more clarity]).

    *should* students learn to set up equations
    based on short paragraphs? i say yes,
    of course. but not in *this way*.
    which, and here’s the “political dimension”
    i mentioned, is in some sense
    the *industry standard*.

    if i propose to deviate from it even in my own classroom
    i’ll risk reprisals. if i propose to take steps in the direction
    of *changing the industry standard*, i’ll need a better place
    to stand than working-classroom-teacher.

    for instance, one could seek to carve out a place in
    the “textbook wars”… which appears to me to be
    at least a major site (if not the *only* site) for
    questions like “more or fewer ‘concrete’ examples?”.

    now, the “more real world examples” party seems
    to have done very well over my teacherly lifetime
    and i’ll bet a lot of my colleagues will agree
    (in the hallway but shut up when the committee
    meetings get going) so naturally there’s a backlash.
    we’re being made to use designedly-assbackwards
    methods after all… and in the *name* of “concreteness”.

    meanwhile, there’s this “data-driven” fad
    (tell more expensive and more confusing lies;
    look how well it works in the financial industry).
    *naturally* there will emerge studies telling
    people that their prejudices (on *either* side)
    are justified by “studies”.

    it just isn’t at all useful or interesting
    to most of the poor SOBs that have to meet
    the classes and it never will be or can be.
    nothing to do with math-ed rightly
    so-called and everything to do
    with “follow the money”.

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