Elementary pseudo-algebra

Discussion in 'General Math' started by Dom, Mar 7, 2006.

  1. Dom

    Dom Guest

    The Jan. 2002 issue of The College Mathematics Journal contains the
    following item in Ed Barbeau's column, "Fallacies, Flaws, and
    Flimflam," page 39.
    ==============================

    FFF#183. Dimensions of a yard.

    Problem. The distance around a rectangular yard is 228 feet. If the
    length of the yard is 6 feet more than the width, find the dimensions
    of the yard.

    Solution (by a student). Divide 228 by 2 to get 114. Knowing that the
    length is 6 feet more than the width, I subtracted 6 from 114 which
    gives me 108 and add the 6 I subtracted onto the length which makes
    120. I then divided the 108 by 2 to get the result which is 54 feet
    long. By dividing the 120 by 2, I would get 60 which is the same as the
    length.
    ================================

    This student solution is a variation of the "strategy/algorithm" that
    is becoming more and more widespread under the guise of "solving
    algebra problems" and "algebra for all." In my opinion, far from
    teaching any meaningful concepts, these mechanical recipies are doing
    little more than enhancing the pseudo-education of students.

    In the U.S., this type of pseudo-education is being promoted--at
    conferences, workshops, minicourses, and training sessions--by people
    who promise to boost scores on assorted "mastery tests" and other
    standardized tests. These promotions are being adopted mindlessly by
    administrators and teachers, whose bonuses and other financial rewards
    are based on the results of these tests.

    As long as these types of recipies continue to be promoted, the
    pseudo-education of American students will continue unabated.

    Dom Rosa
     
    Dom, Mar 7, 2006
    #1
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  2. Dom

    porky_pig_jr Guest

    I vaguely remember that sort of solutions when I was in elementary
    school, *before* we were ever introduced to algebra. It was rather
    frustrating since every solution was sort of ad hoc. Once we got into
    algebra, that crap disappeared like a nightmare.

    I think you should never ever give any problem that can't be solved in
    some systematic way but only with some ad hoc approach, before you've
    learned that systematic way.
     
    porky_pig_jr, Mar 7, 2006
    #2
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  3. Dom

    fishfry Guest

    That's a great phrase. Maybe the government should have a Department of
    Pseudo-Education. Or maybe we already do.
     
    fishfry, Mar 8, 2006
    #3
  4. Dom

    Dom Guest

    Yes, we already do! In October 1999, U.S. Secretary of Education
    Richard Riley announced his department's endorsement of 10 K-12
    mathematics programs selected
    by an "Expert Panel" of educators. Subsequently, 200 mathematicians and
    scientists signed a highly critical letter addressed to Riley. This
    letter, published as a full-page advertisement in The Washington Post
    on Nov. 18, 1999, called the recommendations "premature" and asked
    Riley to withdraw them. "These programs are among the worst in
    existence," said David Klein, a professor of mathematics at California
    State University Northridge, who helped write the letter and organize
    the signing
    campaign. "It would be a joke except for the damaging effect it has on
    children."

    What is even more disturbing is the extensive funding that the National
    Science Fondation has been awarding to the promoters of these "reform
    math" programs.

    DR
     
    Dom, Mar 8, 2006
    #4
  5. Dom

    Dave Rusin Guest

    It's very popular to criticize "mechanical recipes", but isn't it
    exactly the _lack_ of a "recipe" that has you hot under the collar here?
    (I mean, what we expect the student to do is to follow Standard Operating
    Procedure: let the width be w so the length is w+6 and the
    perimeter is 2w + 2(w+6) ; set equal to 228, solve for w .)

    I absolutely agree that IF the student is unable to follow SOP here,
    he or she will be unable to solve more complex problems later.
    But that's a big "if". What if the student saw the answer immediately
    in his/her way? I have to say quite frankly that my reaction to
    the problem was almost the same as the student's: it's clearly a
    nearly-square yard, but for the 12 extra feet, so the short ends
    are (228-12)/4 feet long and the others of course are 6 feet longer.

    Kind of a sneaky twist to use feet and yards in the same problem,
    I have to say! I'll remember that.

    dave
     
    Dave Rusin, Mar 8, 2006
    #5
  6. Dom

    Herman Rubin Guest

    The key part of algebra, the use of variables not just for
    numbers, can be taught with beginning reading, and should
    be so taught. Once this is done, and one uses the one rule
    of equality, that the same operation performed on equal
    entities gets equal results, the reduction to straight
    arithmetic is not only easy but natural.

    One does not have to memorize lots of special procedures and
    rules to understand something, and generally that memorization
    interferes with understanding.
     
    Herman Rubin, Mar 9, 2006
    #6
  7. Dom

    Herman Rubin Guest

     
    Herman Rubin, Mar 9, 2006
    #7
  8. Dom

    Herman Rubin Guest

     
    Herman Rubin, Mar 9, 2006
    #8
  9. Dom

    porky_pig_jr Guest

    The key part of algebra, the use of variables not just for
    I agree 100%. The algebraic approach to solving the problems should be
    taught as early as possible. I see no reason why one should spend a
    couple of years doing things in a hard way only to learn much simpler
    way and never going back. It's a very practical thing to learn.
     
    porky_pig_jr, Mar 9, 2006
    #9
  10. Dom

    Bill Dubuque Guest

    The solution seems fine. It simply appeals to the obvious method
    to recover two numbers from their sum S and difference D, namely

    S,D = L +-W -> L,W = (S +-D)/2


    S = L + W L = (S + D)/2
    i.e. ->
    D = L - W W = (S - D)/2

    Here it is clear that the sum S is equal to half of the perimeter.

    Why does Barbeau think the solution is Fallacious, Flawed, or Flimflam?
    Why does Dom Rosa think the solution is based upon pseudo-mathematics?
    Does anyone understand their objections?

    --Bill Dubuque
     
    Bill Dubuque, Mar 10, 2006
    #10
  11. Dom

    Dom Guest

    Teaching students recipes like the above completely defeats the reason
    for doing these "silly" problems--which is to teach elementary algebra
    concepts: identifying the given information in terms of standard
    variables, determining the applicable formula, using the substitution
    principle, performing elementary algebraic operations.

    Dom Rosa
     
    Dom, Mar 10, 2006
    #11
  12. Dom

    Bill Dubuque Guest

    The logic of equality is governed by more than just one rule.
    Namely, it employs the rules of Substitution and Replacement
    in addition to the Equivalence Relation rules for equality.
    See my prior post [1] for further discussion.

    While I'm being pedantic, I should add that it is sad
    that budding algebraic minds are being corrupted with
    overloaded language such as "variable". The damage
    caused by this is so tremendous that many students
    find it impossible to later appreciate the powerful
    concept of purely *formal* objects such as polynomials,
    rational functions, power series (generating functions).
    Apparently the active viewpoint of X as a "variable"
    becomes entrenched so deeply in their mathematical mind
    that they cannot help but envision F(X) as a function.
    The consequent loss of generality leads to embarrassing
    algebraic impotence (or, even worse, dense remedies
    achieved by popping little blue topological pills)
    Emmy Noether would be rolling in her grave (or ROTFL).
    Evidence that this variable plague is pandemic can be
    seen almost every day on any mathematical newsgroup,
    e.g. see the current thread [2] on partial fractions.
    My posts there contain further discussion and references.

    --Bill Dubuque

    [1] http://google.com/groups?selm=
    [2] http://google.com/groups?selm=
     
    Bill Dubuque, Mar 10, 2006
    #12
  13. Dom

    toto Guest

    I think though the point is that the student was NOT taught this as an
    algorithm, but figured out the solution using his own strategy and
    method.

    Why is it better to teach a recipe like let w be the width, then the
    length is w +6 so the perimeter is 2(w) + 2(w+6) and then solve
    for w to get the answer. If the student doesn't understand the
    procedure they are no better off with this algorithm then with the
    strategy the student used.


    --
    Dorothy

    There is no sound, no cry in all the world
    that can be heard unless someone listens ..

    The Outer Limits
     
    toto, Mar 10, 2006
    #13
  14. It's the problem with NCLB. Now all kids are being taught to do is,
    well, pass a test.
     
    man_in_black529, Mar 10, 2006
    #14
  15. Dom

    Dom Guest

    Based on my experiences, students ARE "taught this as an algorithm."
    Too many American students are being cheated out of an education in
    elementary algebra. In my opinion, this is a national disgrace, and it
    is important that people speak up. DR
     
    Dom, Mar 11, 2006
    #15
  16. I regret to say that, looking at Barbeau's recent FFF, I have the same
    question about one of the items. It's FFF#250. Minding the technology.
    CMJ 37:2 (March 2006) pp. 122-123, sent in by Paul H. Schuette. He used
    three computer algebra systems (CAS) to compute the indefinite integral of
    1/(x sqrt(1 - x^2)) wrt x. Someone casually reading FFF#250 would think
    that Maple, Derive and Mathematica all give results which are incorrect.
    But in fact, all three give results which are perfectly correct!

    Before examining that integral however, let's consider a simpler and far
    more common one which poses exactly the same "problem": the indefinite
    integral of 1/x wrt x. All CAS known to me give log(x) as the answer, while
    all elementary calculus texts known to me give instead log|x|, plus an
    arbitrary constant of integration. Why the discrepancy? CAS are normally
    designed to give results which are applicable in the complex, not just the
    real, realm. There, log(x) is a correct answer, while log|x| is not. But of
    course, for the purpose of elementary calculus, log|x| is perfectly fine
    and arguably nicer than just log(x). Nonetheless, log(x) is not wrong in
    that context. The two expressions, log(x) and log|x|, differ when x < 0,
    but that difference is just a _constant_ (albeit an imaginary one, pi i)
    and so we may think of it as being harmlessly "absorbed" in the arbitrary
    constant of integration.

    Now let's go back to FFF#250.
    For the indefinite integral of 1/(x sqrt(1 - x^2)) wrt x, Schuette reports
    that Maple (versions 7 and 9.5) gives -arctanh(1/sqrt(1 - x^2)), while
    Derive (version 6) gives ln((sqrt(1 - x^2) - 1)/x) and Mathematica (version
    unspecified) gives ln(x) - ln(1 + sqrt(1 - x^2)). These are _all_ correct
    antiderivatives. In particular, for 0 < |x| < 1, they have the same real
    part. Their only differences, for -1 < x < 0 or for 0 < x < 1, are
    merely _constants_, albeit imaginary ones.

    There are at least two lessons to be learned here.
    1. Warn students in elementary calculus, if they use CAS, that those
    systems are designed to work with complex numbers. As such, an
    antiderivative from a CAS may well differ from an antiderivative in the
    text (or as obtained by the student or instructor) by a _complex_ constant.
    2. Be careful about what you claim to be FFF. (Otherwise, it may be your
    claim itself which is FFF.)

    Regards,
    David W. Cantrell
     
    David W. Cantrell, Apr 17, 2006
    #16
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