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I've posted a number of diaries about math, statistics, and math education.  Today, I'd like to distinguish two subjects that get taught as if they were the same thing, when they are not.  It's as if we thought literature and spelling were the same subject.

We call both these things math.  But one of them is what mathematicians do, and hte other is not.  And, while many people who are good at one are good at the other, it's not always the case.

If you'd like to read more, join me below the fold.

One subject is mathematics.  This is almost never taught before high school, and often not until college.  That's a shame.  It could be taught much earlier.  Mathematicians do lots of things, but perhaps the one central idea is PROOF.  Paul Erdos, one of the best and most prolific mathematicians of the 20th century, said

 A mathematician is a device for turning coffee into theorems

Bertrand Russell, another great mathematician, said

 Mathematics is the only subject in which we never know what we are talking about, nor whether what we are saying is true

Doesn't sound much like math, does it?

That's because what is taught in school isn't really math.  It's quantitative reasoning.  It's about numbers.  In high school, you might get a little abstract with algebra and geometry and maybe even trigonometry, but you probably won't be introduced to formal proofs.  

Here are two simple bits of real math, that don't require more than 2nd or 3rd grade 'math'....

  1.  You have a stadium full of people, and a bunch of sacks of bags of candy.  How do you tell if you have more candy or people?

The quantitative way to do this is to count each.  A royal pain.  You'll probably count wrong.  Then you'll have to check.  Oy.
The math way uses one-to-one correspondence: Give each person a bag.  If you run out of bags, you have more people than bags.  If you have more bags left at the end, you have more bags than people.  If they match, the numbers are equal.

Note that no counting is involved.  No calculation of any kind.

preliminary info for problem 2:  A prime number is one that is divisible only by itself and 1; all other numbers are composite.  e.g 5 is prime.  6 is composite, because 2*3 = 6

  1. Are there an infinite number of prime numbers, or do they stop?

A.  Assume there are a finitie number of primes.  If so, there must be a biggest one.  Call it p.
B.  Find all the primes smaller than P.  (They knew how to do this even when Euclid came up with this proof).  
C.  Multiply all the numbers in B together.
D.  Add 1; call this number Q.  Clearly Q is bigger than P
now, one of two things must be true:
   a) Q is prime, in which case P is not the largest prime
   b) Q is composite.  But, if it is, then it must have a factor larger than P, because, if you divide by P or any of the primes that are less than P, you will have a remainder of 1.  
 

Note, again, that no calculations need be done.  

Quanttative reasoning is different.  That's a 'number sense'.  A simple example is this:

Suppose you are helping your son with his homework.  He has the following problem:

   54 + 32  + 20 + 98

and he has given an answer of 752.   If you have good number sense, you know that this is wrong before you even add anything.  All the numbers are less than 100, there are 4 of them.  The total must be less than 400.  

Of course, both math and QR get more complex than this; and, of course, there are SOME connections.  But by teaching these two subjects as if they were one, we do a disservice to everyone.  Math, as I've written before, should be taught because it is beautiful.  QR should be taught because it is useful.

Originally posted to plf515 on Tue Dec 19, 2006 at 05:48 AM PST.

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Comment Preferences

  •  Maybe I went to an unusual high school (3+ / 0-)
    Recommended by:
    Ahianne, illinifan17, plf515

    but we had formal proofs coming out of our ears; almost entirely Euclidean geometry and Boolean logic. The logic proofs were somewhat interesting, but I can't say the same about the geometry ones.

    Now, if the curriculum had taken the Euclidean axioms, looked at the fifth postulate and used that to segue into some simple non-Euclidean geometries, that might have caught my interest. Space-warps and all that...

    -dms

    Having trouble finding stuff on Daily Kos? This page has some handy hints and tricks.

    by dmsilev on Tue Dec 19, 2006 at 06:05:16 AM PST

    •  That sounds pretty unusual all right (4+ / 0-)
      Recommended by:
      dmsilev, vcmvo2, plf515, Cronesense

      We did do geometric proofs, but almost nobody understood them because nobody ever explained the concept of a proof to us before then (if they even did in that class--I don't remember).

      Fortunately I got it immediately and had a great time.  That class ruled.

      What is wanted is not the will to believe, but the will to find out, which is the exact opposite. -- Bertrand Russell

      by RequestedUsername on Tue Dec 19, 2006 at 06:16:18 AM PST

      [ Parent ]

    •  Sounds nothing like most high school (1+ / 0-)
      Recommended by:
      Cronesense

      curricula that I've seen.

      What are you reading? on Friday mornings
      stats_geeks_of_daily_kos

      by plf515 on Tue Dec 19, 2006 at 06:48:01 AM PST

      [ Parent ]

      •  It is/was good public high school (2+ / 0-)
        Recommended by:
        plf515, Cronesense

        in the Boston area, in a suburb that has a disproportionately large population of academics. That probably skews things somewhat, in terms of the expectations that the parents had of the school.

        I was also in the AP track for math, but the lower tiers also covered similar material, though in less depth.

        -dms

        Having trouble finding stuff on Daily Kos? This page has some handy hints and tricks.

        by dmsilev on Tue Dec 19, 2006 at 06:57:58 AM PST

        [ Parent ]

    •  God, I hope not (2+ / 0-)
      Recommended by:
      Ahianne, plf515

      Forty years ago, everyone learned the basics of formal proof in the "plane geometry" class, usually taught to sophomores or juniors.  Don't you all remember?  Stuff link "angle-side-angle" for congruent triangles?
      If a mandatory year of plane geometry has been dropped from most curricula, then we've got a problem.

      Don't be a DON'T-DO... Be a DO-DO!

      by godwhataklutz on Tue Dec 19, 2006 at 07:41:41 AM PST

      [ Parent ]

  •  Anecdote (4+ / 0-)
    Recommended by:
    illinifan17, plf515, Cronesense, karmsy

    I was extremely disturbed when I recently used a fast food drivethrough because their computers were down.  They were taking orders face to face but as I started to place mine, they stopped me.  

    "Our computers are down so we can't do it if you have a lot cause we are doing the math by hand and we aren't sure we will get it right."

    We are talking about addition and multiplication of decimals (simple multiplication at that).

    It disturbs me that so many of the students I see can NOT do this on paper (much less in their head) and are completely dependent upon a calculator.  It disturbs me even more that no one else in my building seems frustrated by this.   What's even worse is with the new math curriculum we teach, you know the ones were the concept and process are more important than the calculation, they don't even know how to set up the problem with or without a calculator

    •  What? (1+ / 0-)
      Recommended by:
      plf515

      What's even worse is with the new math curriculum we teach, you know the ones were the concept and process are more important than the calculation, they don't even know how to set up the problem with or without a calculator.

      I think you need to expand on this, because I don't understand how concentrating on concept and process would make one less able to set up a problem.

      What is wanted is not the will to believe, but the will to find out, which is the exact opposite. -- Bertrand Russell

      by RequestedUsername on Tue Dec 19, 2006 at 06:17:51 AM PST

      [ Parent ]

      •  I agree (0+ / 0-)

        an expansion of that point would be very interesting

        But pumpkinlove's example is not unfamiliar to me.  IMHO, it's not just the curriculum, but the fact that it's mis-tuaght, often by teachers who don't really understand it.  I am NOT ragging on teachers - the vast majority are hardworking, dedicated, and working for way too little money and respect.

        But expecting a person with the training that most teachers get to learn the sort of math curriculum they are supposed to use would be like expected someone who learned a little English from a couple classes they took while in HS in Tokyo to be able to teach Shakespeare given a curriculum written by AL Rowse.

        What are you reading? on Friday mornings
        stats_geeks_of_daily_kos

        by plf515 on Tue Dec 19, 2006 at 06:52:30 AM PST

        [ Parent ]

        •  It goes beyond that I think (1+ / 0-)
          Recommended by:
          plf515

          Every last one of the students I've had who are successful in these curriculums either get outside tutoring OR have highly educated parents or both.

          Even the NCTM has backed off this type of curriculum but it will be another decade before these textbooks are replaced throughout the country.

      •  it doesn't work (2+ / 0-)
        Recommended by:
        plf515, Cronesense

        This particular curriculum focuses on students finding their own solution to problems.   There is no repetition and no direct instruction.  

        The problem is that most of the students can not do it.  Most of the kids who are successful in this curriculum either recieve outside traditional tutoring or have parents who are good at math themselves. There is a reason there are basic algorithms and that they have been taught for years.

        We are no longer teaching them.  Instead we hand them a calculator and ask them to make their own meaning.

        It doesn't work.

        •  Let's don't throw out the baby with the bathwater (1+ / 0-)
          Recommended by:
          plf515

          I agree that handing a first grader a sheet of story problems (or "math stories" as they call them now) and a calculator is ineffective, but I don't see why that means we have to stop teaching concepts.  It just means we have to also drill on fundamentals.

          Maybe your state has different standards, but my second grader has been getting "number fact" flashcard-type things to do every day for two school years now.  

          What is wanted is not the will to believe, but the will to find out, which is the exact opposite. -- Bertrand Russell

          by RequestedUsername on Tue Dec 19, 2006 at 07:12:04 AM PST

          [ Parent ]

          •  Drill is overrated (0+ / 0-)

            I think.

            The problem with what pumpkinlove describes is that it doesn't teach the facts OR the concepts, in fact, it does not teach ANYTHING.

            In Hebrew, they have cases of verbs (e.g. causative, reflexive)....."To teach" = lelamed, IS to cause to learn - lilmod.

            A teacher (melamed) IS someone who causes learning.

            What are you reading? on Friday mornings
            stats_geeks_of_daily_kos

            by plf515 on Tue Dec 19, 2006 at 07:16:05 AM PST

            [ Parent ]

            •  No, drill is definitely necessary (2+ / 0-)
              Recommended by:
              hestal, plf515

              It develops the "muscle memory" that higher level things are built on.  It's years of doing algebra with pencil and paper that now allow me to estimate if a given ugly problem is going to end up with anything nice or if I've got a monster on my hands.

              What is wanted is not the will to believe, but the will to find out, which is the exact opposite. -- Bertrand Russell

              by RequestedUsername on Tue Dec 19, 2006 at 07:24:49 AM PST

              [ Parent ]

              •  Perhaps we are using 'drill' to mean (0+ / 0-)

                different things.

                What I meant was the endless worksheets of 2-digit by 2-digit multiplication problems my 4th grade teacher thought vital to master before teaching division

                I agree that solving many problems is the only way to learn algebra (or, for that matter, calculus, trig, statistics or whatever)

                What are you reading? on Friday mornings
                stats_geeks_of_daily_kos

                by plf515 on Tue Dec 19, 2006 at 07:46:58 AM PST

                [ Parent ]

                •  Well, I don't know about 'endless' (1+ / 0-)
                  Recommended by:
                  plf515

                  I'd say a large number of 2 digit multiplication problems would be a good idea when learning how to multiple N digit numbers.  I don't know what that large number should be, though.

                  Shouldn't art students be required to pick a pencil/brush/chisel once in a while to get the feel of the tool?  Shouldn't sports teams run exercises and play scrimmages?

                  What is wanted is not the will to believe, but the will to find out, which is the exact opposite. -- Bertrand Russell

                  by RequestedUsername on Tue Dec 19, 2006 at 07:50:06 AM PST

                  [ Parent ]

                  •  I disagree (0+ / 0-)

                    at least somewhat.

                    The drill should happen until it is clear that the student knows what he/she is doing, not until he/she gets only a few errors.

                    It is more important to know WHEN to multiply (as opposed to add, subtract, or whatever) and to know how to take an estimated guess at the answer (to know when you've pushed the wrong button on the calculator) than to get the multipication tables down cold.

                    Addition and subtraction are different, because they are SO needed in everyday life.  (e.g. making change) that using a calculator would be a serious hindrance.

                    What are you reading? on Friday mornings
                    stats_geeks_of_daily_kos

                    by plf515 on Tue Dec 19, 2006 at 08:22:41 AM PST

                    [ Parent ]

                    •  That answer fails to please either direction (2+ / 0-)
                      Recommended by:
                      Ahianne, plf515

                      I think I'd be happier if you just said we didn't need to drill on anything, rather than that there are a couple operations that are important and the others, pfff who cares.

                      If you only pick two to drill on, I agree that addition and subtraction should be the two.  But I don't agree that multiplication and division are so esoteric that the average person rarely needs them.  

                      Every time I make change, it's because I've made a purchase.  How did I know which item to purchase?  I made a unit cost evaluation, i.e. I divided.  How did I know if I could afford as many as I needed?  I multiplied the price times the count.

                      Furthermore, the higher up the "chain" one drills, the more everyday questions one can answer without having to do much thinking.  With algebra, geometry, trigonometry and (to a lesser extent) calculus being almost second nature, I can answer many questions that come up in my everyday life.  

                      What is wanted is not the will to believe, but the will to find out, which is the exact opposite. -- Bertrand Russell

                      by RequestedUsername on Tue Dec 19, 2006 at 08:52:49 AM PST

                      [ Parent ]

                    •  disagree (1+ / 0-)
                      Recommended by:
                      plf515

                      fractions, algebra.. all depend upon mult./div.

                      You can't progress without the foundations.

        •  In that case (0+ / 0-)

          yeah, it's stupid.

          Let's see....it took the best math minds a couple thousand years to get to the point where negative numbers were really understood and accepted.  

          Now we either expect kids to figure this out, or, if they ask, we go to some version of

          "Minus times minus equals plus.
          The reasons for this, we needn't discuss"

          A teacher who cannot explain why minus times minus equals plus is not a math teacher, he or she is a place holder.  

          What are you reading? on Friday mornings
          stats_geeks_of_daily_kos

          by plf515 on Tue Dec 19, 2006 at 07:13:13 AM PST

          [ Parent ]

  •  Great examples (1+ / 0-)
    Recommended by:
    plf515

    I like your distinction between math and qr.  I am not sure I agree with your aesthetic or practical conclusions.  I think we don't teach enough qr either.  We teach calculation processes.  And, yes, math is beautiful, but some of us enjoy the beauty of numbers too (and we spend way too much time playing fantasy baseball).

    So I see only tatters of clearness through a pervading obscurity - Annie Dillard -6.88, -5.33

    by illinifan17 on Tue Dec 19, 2006 at 06:08:28 AM PST

    •  I agree with this (0+ / 0-)

      perhaps my diary isn't really clear on that, or overgeneralizes.

      Some of us (e.g. me, and apparently you) have fun playing with numbers.  It can be really neat stuff.  

      I also agree that we don't really teach math or QR, but we try to teach QR, only we call it math.

      What are you reading? on Friday mornings
      stats_geeks_of_daily_kos

      by plf515 on Tue Dec 19, 2006 at 06:54:29 AM PST

      [ Parent ]

  •  math is the language of the universe (2+ / 0-)
    Recommended by:
    illinifan17, plf515

    My school did approach the subject all through theoroms and set theory for second-year level algebra (title of book had theroms in it) and for geometry. Scared the s%^t of me. The I sucked at calculus (well I would have if I done any homework; did not do much of anything freshman year) in college. Then I discovered (well was introduced to) probability and stats. the rest is his-tory.

  •  John Allen Paulos makes this distinction (2+ / 0-)
    Recommended by:
    illinifan17, plf515

    in his excellent Innumeracy, among other books.  (BTW, what ever happened to him--no books in like 15 years.)  I believe he even gives the example of a mathematician who can't balance his own checkbook and needs a calculator when he adds "tricky 7s".

    Another central idea to mathematics (rather than computation) is abstract structures.  The Greeks seem to kind of implicitly realized this by doing all their mathematics via geometry.  

    What is wanted is not the will to believe, but the will to find out, which is the exact opposite. -- Bertrand Russell

    by RequestedUsername on Tue Dec 19, 2006 at 06:13:37 AM PST

  •  Here is a challenge (1+ / 0-)
    Recommended by:
    plf515

    A mathematics colleague of mine used it in a teaching seminar recently.  Is the following statement true.

    Two basketball teams played a regulation game.  Team A had a higher field goal percentage in the first half and in the second half, therfore, thy must have had a higher fg percentage for the whole game.

    I know the calculation process to prove it wrong.  I just play with numbers until I get a set that is a counter-example:

    Team A  5/20=25%  6/10=60%  11/30=36.7%

    Team B  4/20=20%  11/20=55% 15/40=37.5%

    What is the math way to do it, with no numbers or calculations?

    So I see only tatters of clearness through a pervading obscurity - Annie Dillard -6.88, -5.33

    by illinifan17 on Tue Dec 19, 2006 at 06:17:46 AM PST

    •  The average of the sums is not (0+ / 0-)

      the sum of the averages.

      (A1 + A2 + ... + AN)/N =
      A1/N + A2/N + ... + AN/N !=
      A1/1 + A2/1 + ... + AN/1

      What is wanted is not the will to believe, but the will to find out, which is the exact opposite. -- Bertrand Russell

      by RequestedUsername on Tue Dec 19, 2006 at 06:21:07 AM PST

      [ Parent ]

  •  I could pick on you for giving the definition... (1+ / 0-)
    Recommended by:
    plf515

    ...of irreducaible as the definition of prime, but since they are the same in the integers, I could also let you slide. :-)

    Teacher's Lounge opens each Saturday, sometime between 10am and 12 noon EST

    by rserven on Tue Dec 19, 2006 at 06:33:18 AM PST

    •  Wow (2+ / 0-)
      Recommended by:
      danielbiss, rserven

      I didn't even know there were definitions of prime for non-integers....trying to see how that would work....what's a prime in the rationals? or whatever?

      What are you reading? on Friday mornings
      stats_geeks_of_daily_kos

      by plf515 on Tue Dec 19, 2006 at 06:57:18 AM PST

      [ Parent ]

      •  By definition, a prime is a non-zero, non-unit... (2+ / 0-)
        Recommended by:
        danielbiss, plf515

        ...with the property that if it divides a product of two numbers, then it must divide one or the other ("divides" means the quotient is also an element of the ring).  Every non-zero element of the rationals is a unit (divides 1) and hense there are no prime rationals.  However, the integers may be extended by attaching roots of polynomials, to for instance Z[√-3], where elements look like a + b√-3, a,b ∈ Z, to form interesting rings.  In that ring, 4 = 2 * 2 = (1 + √-3)(1 - √-3) is not uniquely factorable as a product of nonunits. 2 cannot be factored into a product of nonunits in the ring and hence is irreducible, but divides the product of two elements without dividing either of the two elements, and hence is not prime.

        Teacher's Lounge opens each Saturday, sometime between 10am and 12 noon EST

        by rserven on Tue Dec 19, 2006 at 09:05:58 AM PST

        [ Parent ]

  •  We have a very serious problem (2+ / 0-)
    Recommended by:
    hypersphere01, plf515

    in this country with the way mathematics is taught. That's why we're so pitifully behind all the other rich countries, and even the developing countries, in math and science literacy.

    One problem we have, is that it's assumed that only an elite few of our students are "smart enough" to learn anything beyond arithemtic. For the rest (most students, actually), we have bonehead courses--you know, a two-year program to cover what should be taught in a one-year course in beginning algebra.

    It makes me furious.

    I personally was told at the end of eighth grade, "Oh, honey, don't bother with algebra. Business math is the place for your pretty little head."

    Right, the clown who made this recommendation called himself a math teacher? He drew a salary? Or maybe the problem wasn't him as much as the context in which he functioned.

  •  I'll throw this in for what it's worth (1+ / 0-)
    Recommended by:
    plf515

    My math education was severely screwed up because I moved during my grade school years to 4 different schools in two states, in my junior high years to 2 different schools in 2 countries, and in high school to 3 different schools in two countries.

    Wherever I went taught something different than where I had been. For example, one place would be starting trig while the next would be finished with trig. After a few years, I had no clue and I don't to this day.

    It's something to pay attention to because Americans move on the average about every 7 years.

    utahgirl

  •  In the mid 1960s (3+ / 0-)
    Recommended by:
    hypersphere01, plf515, ticket punch

    there was a movement to teach grade school math from a theoretical background. The School Math Study Group, SMSG, developed an entirely different set of books to support that approach. It was heavy on set theory.

    Tom Lehrer addressed it in "New Math:" "It's so simple, so very simple, that only a child can do it!"

    There was a backlash against the approach and a return to "basics."

  •  I am a mathematician. I taught (1+ / 0-)
    Recommended by:
    plf515

    high school math for a time and went into computers when they were just getting started.  Over three decades I hired and worked with people who were usually math or engineering majors.  It was overwhelmingly true that the engineers knew more math.  And the reason is that they quickly get involved in solving real world problems using math as a tool.  

    In our public schools we should be teaching math by giving our students real world problems to solve using mathematics.  It is much more effective to go from the specific, in which the student is interested, to the general than vice versa.  

    After decades in the computer business I taught high school math again for a brief time at one of the best school districts in Texas.  The Algebra texts and classes were the same as when my mother was a student in the 1930's, the same as when I was a student, and the same as when I taught.  A period of 60+ years had elapsed and nothing had changed.  Students were still told to memorize the Quadratic Formula because they would be tested on it.  

    When I tried to insert practical problem solving into the classes I taught I was rebuffed by the department and the principal.  Interest and application are the keys to teaching to mathematics.

    If you don't have an earth-shaking idea, get one, you'll love building a better world.

    by hestal on Tue Dec 19, 2006 at 07:44:01 AM PST

  •  plf515........ (1+ / 0-)
    Recommended by:
    plf515

    I tried here and here.  Does this mean I failed to earn my stat groupie badge?

    You make a living by what you get and a life by what you give. W. Churchill

    by Cronesense on Tue Dec 19, 2006 at 11:23:29 AM PST

    •  You've got the badge (0+ / 0-)

      I'm not sure of the design, or the color, or anything....I think you're the only member of the stat groupies.

      But thanks for your support.

      No rescue, no jotter list......oh well, cfk will, at th end of December, have another chance for us to nominate our own diaries.

      What are you reading? on Friday mornings
      stats_geeks_of_daily_kos

      by plf515 on Tue Dec 19, 2006 at 12:17:28 PM PST

      [ Parent ]

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