Loading ...
Sorry, an error occurred while loading the content.
 

Re: Absolute Values inside an Absolute Value

Expand Messages
  • adh_math
    ... Brian, So far in this group you ve called factoring and the quadratic formula worthless and/or uninteresting, and declined (aside from one sentence about
    Message 1 of 10 , Nov 1, 2004
      --- In mathforfun@yahoogroups.com, "brianejensen" <brianejensen@p...>
      wrote:
      > Most of the math I see including these absolute values has no
      > purpose. But understanding graphs is very important.
      >
      > [snip]
      >
      > So much math is worthless, yet when students get out of school,
      > they don't know how to calculate the important things.
      >
      Brian,

      So far in this group you've called factoring and the quadratic formula
      worthless and/or uninteresting, and declined (aside from one sentence
      about graphs above) to say what mathematics you yourself use regularly
      or find important. Judging from your posts, much of mathematics *is*
      useless or uninteresting *to you*; that's your business. However, when
      you make these comments as if they were factual, it seems an attempt
      (perhaps unconscious) to instill students with mistrust or contempt of
      theoretical considerations. (Perhaps you're aiming at plodding and/or
      ignorant school teachers, but if so you're shooting wide of the mark.)

      You've mentioned software on multiple occasions, but seem to ignore
      that fact that someone had to write the software, and therefore had to
      understand how mathematics works theoretically and be able to verify
      that the software does what it claims to do. I'd be sorry to see
      students oblivious to the fact that computers are tools (not thinking
      machines), and taught to treat all computer answers as definitive.
      Training students to use a particular piece of software instead of
      teaching them to think with logic, clarity, and organization would be
      an educational disaster, in my opinion.

      vcamt1510's absolute value questions may not have direct mathematical
      relevance, but they serve at least two obvious pedagogical purposes:

      * Applying a definition recursively.
      * Organizing cases.

      In my experience, students are often surprised that W-shaped graphs
      can be described by a single formula, so a third result might be
      "breaking down conceptual misperceptions".

      Regards,

      adh
    • jason1990
      Well said, adh. As for myself, I was wondering why someone who thinks so much math is worthless would even be a member of this group. As I think I ve
      Message 2 of 10 , Nov 1, 2004
        Well said, adh. As for myself, I was wondering why someone who
        thinks "so much math is worthless" would even be a member of this
        group.

        As I think I've mentioned before, I'll be teaching a class called
        Introduction to Brownian Motion and Stochastic Calculus next
        semester. It will be the most advanced and theoretical class I have
        so far taught in my short career. And yet, I've been told by fellow
        faculty that most of my students will not be math majors. They will
        be finance majors, biologists, and other applied scientists.

        For whatever reason, these "real world" folks find this kind of math
        useful. And you can rest assured that they all will know how to
        factor, apply the quadratic equation, and deal with absolute values.
        The idea that these tools are worthless is just so incredibly naive.

        I want to write more, but I feel I'm probably wasting my "breath".
        Also, comments like this probably deserve less attention than
        they've gotten. So enough said. But hopefully younger members of the
        group will come to their own, independent conclusions about the
        value and interrelationship of all parts mathematics.

        --- In mathforfun@yahoogroups.com, adh_math <no_reply@y...> wrote:
        >
        > --- In mathforfun@yahoogroups.com, "brianejensen"
        <brianejensen@p...>
        > wrote:
        > > Most of the math I see including these absolute values has no
        > > purpose. But understanding graphs is very important.
        > >
        > > [snip]
        > >
        > > So much math is worthless, yet when students get out of school,
        > > they don't know how to calculate the important things.
        > >
        > Brian,
        >
        > So far in this group you've called factoring and the quadratic
        formula
        > worthless and/or uninteresting, and declined (aside from one
        sentence
        > about graphs above) to say what mathematics you yourself use
        regularly
        > or find important. Judging from your posts, much of mathematics
        *is*
        > useless or uninteresting *to you*; that's your business. However,
        when
        > you make these comments as if they were factual, it seems an
        attempt
        > (perhaps unconscious) to instill students with mistrust or
        contempt of
        > theoretical considerations. (Perhaps you're aiming at plodding
        and/or
        > ignorant school teachers, but if so you're shooting wide of the
        mark.)
        >
        > You've mentioned software on multiple occasions, but seem to ignore
        > that fact that someone had to write the software, and therefore
        had to
        > understand how mathematics works theoretically and be able to
        verify
        > that the software does what it claims to do. I'd be sorry to see
        > students oblivious to the fact that computers are tools (not
        thinking
        > machines), and taught to treat all computer answers as definitive.
        > Training students to use a particular piece of software instead of
        > teaching them to think with logic, clarity, and organization would
        be
        > an educational disaster, in my opinion.
        >
        > vcamt1510's absolute value questions may not have direct
        mathematical
        > relevance, but they serve at least two obvious pedagogical
        purposes:
        >
        > * Applying a definition recursively.
        > * Organizing cases.
        >
        > In my experience, students are often surprised that W-shaped graphs
        > can be described by a single formula, so a third result might be
        > "breaking down conceptual misperceptions".
        >
        > Regards,
        >
        > adh
      • dragonregret
        ... purpose. Sorry you feel that way Brian. They do have a purpose though: Look outside of your window. It s a non-linear world out there. And Mathematics is
        Message 3 of 10 , Nov 1, 2004
          --- In mathforfun@yahoogroups.com, "brianejensen"
          <brianejensen@p...> wrote:
          > Most of the math I see including these absolute values has no
          purpose.

          Sorry you feel that way Brian. They do have a purpose though:

          Look outside of your window. It's a non-linear world out there.
          And Mathematics is a refelction of that non-linearity. Know about
          the "butterfly effect" huh? The smallest of inconsequential,
          seemingly unimportant concepts in math can lead to profound effects,
          a bifurcation if fact. Now they want to control hurricanes by . . .
          but I digress. "The Mandelbrot set broods in silence at the center
          of the complex plane". Thus began a revolution.
          I recently found another use for the quadratic formula . . . and RSA
          (I'm doin' that now). You know, if you know phi(n) then you can
          find p and q by the . . .

          Dragon
        • Adam
          For many of these problems, the case-by-case method works well. See also other post re: the importance of organizing cases. However, one should also recall
          Message 4 of 10 , Nov 1, 2004
            For many of these problems, the case-by-case method works well. See
            also other post re: the importance of organizing cases. However,
            one should also recall that there is an algebraic (non-piecewise)
            definition for the absolute value. abs(x)=sqrt(x^2). So we could
            write sqrt[(1-abs(5x)^2]=sqrt[x^2] then square both sides to get (1-
            abs(5x))^2=x^2, and et cetera. Like I said, for this particular
            problem, I think the case by case method is going to be easier.

            Note also that some of these problems are identities, e.g., #3, abs
            (x)>=0 always, so abs(x)+2>=2 and so abs(abs(x)+2)>=2 and is always
            >1. #2 is similar except abs(2-abs(x)) might equal 0, when abs(x)
            =2, when x=+/-2.

            Good luck!

            Adam
            --- In mathforfun@yahoogroups.com, "vcamt1510" <destinygurl15@h...>
            wrote:
            >
            > how would I start off these problems?
            >
            > 1) |1 - |5x|| = |x|
            >
            > 2) | 2 - |x| | > equal 0
            >
            > 3) ||x| + 2 | > and equal to 1
            >
            > 4) |5 - |x| | > and equal to 1
          Your message has been successfully submitted and would be delivered to recipients shortly.