RE: [xmca] a minus times a plus

["A.Bakker" <A.Bakker@fi.uu.nl>]

Interesting question, Mike!

Hans Freudenthal used the James Hasty method to explain the negative times
negative:
4 * -5 = -20
3 * -5 = -15
2 * -5 = -10
1 * -5 = -5
0 * -5 = 0
-1 * -5 = 5
-2 * -5 = 10
-3 * -5 = 15

On -(-5)=5:
In the Dutch math textbooks there is a 'story' about a witch that puts small
ice cubes into and out of a bucket of water. If she puts one cube into the
water, the temperature goes down by one degree: 8+ (-1) = 7. So if she takes
one ice cube out of the water, the temperature rises: 7 -(-1)=8.
Children aged about 11-13 generally get the idea, and many teachers use it,
but as you can imagine, real mathematicians and physicists dislike it:
physically it is such non-sense.

Arthur


> -----Original Message-----
> From: xmca-bounces@weber.ucsd.edu [mailto:xmca-bounces@weber.ucsd.edu] On
> Behalf Of Andy Blunden
> Sent: dinsdag 28 april 2009 9:37
> Cc: eXtended Mind, Culture,Activity
> Subject: Re: [xmca] a minus times a plus
> 
> Ha, ha, ha, ha, ha! LOL! Mathematics was surely invented
> only for the purpose of making jokes, Emily!!
> 
> I don't doubt Emily that children learning maths, must begin
> with some process of reflecting on practical experience. But ...
> 
> * When you get past the elementary level you can't rely on
> generalization directly from everyday experience. Students
> must be inducted into an existing culture which does not
> arise directly from 'practical' experience. But having
> learnt to use and 'understand' a certain level of
> mathematical abstraction, those operations can provide a
> basis in experience for induction into higher levels of maths.
> 
> * What is the point of learning maths? If it is just to make
> sure that you get the right change at the supermarket, why
> bother with negative numbers? Isn't there a value in
> learning to do maths at a level far beyond what is required
> at the supermarket? of learning to work with this kind of
> abstraction?
> 
> Andy
> Duvall, Emily wrote:
> > Hi Andy,
> > Well, most of your response is over my head... :-)
> >
> > Seriously though, going back to Leonti'ev (1997), spontaneous concepts
> take us from the concrete to the abstract... so perhaps some folks are
> still working towards an understanding of the mathematical concepts
> involved from everyday concepts, while others who are working from the
> abstract to the concrete are working a path of consciousness with regard
> to scientific concept formation.
> >
> > I am definitely in the realm of the everyday... whereas I would suspect
> that you are in the realm of the scientific.
> >
> > As to what sign system I am most comfortable in...
> >
> > scroll to "A diabolical new testing technique: math essay questions"
> > although you may prefer: "But this is the simplified version for the
> general public"
> >
> > http://www.geocities.com/CapeCanaveral/Hangar/7773/comics.html
> >
> >
> >
> >
> > ~em
> >
> > -----Original Message-----
> > From: xmca-bounces@weber.ucsd.edu [mailto:xmca-bounces@weber.ucsd.edu]
> On Behalf Of Andy Blunden
> > Sent: Monday, April 27, 2009 10:29 PM
> > Cc: eXtended Mind, Culture,Activity
> > Subject: Re: [xmca] a minus times a plus
> >
> > Emily, I quite candidly introduced my earlier message as
> > "the world's worst maths teacher". I developed this identity
> > partly by being given the task of teaching "New Maths" to
> > almost-innumerate kids in Brixton in the 1970s.
> >
> > I was an Engineering PhD who could solve integral equations
> > but couldn't sing, and had no teacher training.
> >
> > I was asked to teach for example, the algebra of
> > transformations of a figure in 3 dimensions (eg rotating by
> > 90deg 4 times = null). This was not my choice. That was the
> > syllabus! But because of my own background, I couldn't
> > understand what they found so difficult. :)
> >
> > Later I had a seminal chat with the English teacher who told
> > of how he only learnt to understand the workings of the
> > differential (those things on the back axle of motor cars
> > which allow the 2 wheels to go at different speeds), by
> > having someone tell him in words, and going over and over
> > those words. The diagrams meant nothing to him. My first
> > glimmer of thinking about thinking.
> >
> > What sort thinking designed that maths syllabus?
> >
> > Andy
> >
> > Duvall, Emily wrote:
> >> I think you bring up an important point, Andy. In what ways do we
> understand and convey concepts?
> >> I go back to Karpov & Gindis (2000) and the levels of problem solving,
> an hierarchical arrangement that suggests to me that it is not so much
> that we think differently but that perhaps we have come to accept
> different levels of understanding... yet our level of understanding could
> be developed:
> >>
> >> Symbolic or abstract
> >> Visual or visual-imagery
> >> Concrete or visual-motor
> >>
> >>
> >> Karpov, Y. & Gindis, B. (2000). Dynamic assessment of the level of
> internalization of elementary school children's problem-solving activity.
> In: C. Lidz & J. Elliott (Eds.), Dynamic assessment: Prevailing models and
> applications.(pp.133-154). Oxford, UK: Elsevier Science
> >>
> >> ~em
> >>
> >> -----Original Message-----
> >> From: xmca-bounces@weber.ucsd.edu [mailto:xmca-bounces@weber.ucsd.edu]
> On Behalf Of Andy Blunden
> >> Sent: Monday, April 27, 2009 9:50 PM
> >> To: eXtended Mind, Culture, Activity
> >> Subject: Re: [xmca] a minus times a plus
> >>
> >> Personally, Jim and Linda, I would find ideas like negative
> >> as reflection or negative as contradiction, easy to
> >> understand if *you are already familiar* with -x-=+ but I
> >> would have thought that the leap from visual or rhetorical
> >> relationships to numbers is asking too much of children who
> >> have not yet learnt to count backwards from zero.
> >>
> >> Why is a number line easier to understand than the
> >> symmetries of topological or rhetorical transformations? I
> >> don't know, but intuitively I think it is. But different
> >> people think differently. Maybe the Linda's rhetorical
> >> explanation is easier for a verbal thinker than a spatial
> >> thinker? (is this mumbo-jumbo?)
> >>
> >> Andy
> >>
> >> Jim Levin wrote:
> >>> Perhaps it would be helpful to think about multiplying by -1 as
> >>> reflection.  So, if you separate out the -1 part of the
multiplication,
> >>> you get:
> >>> -a * -b = -1 * a * -1 * b = -1 * -1 * a * b
> >>> (an even number of reflections)
> >>>
> >>> -a * b = -1 * a * b
> >>> (an odd number of reflections)
> >>>
> >>> a * b = a * b
> >>> (no reflections)
> >>>
> >>> Each -1 multiplier "reflects" the result across the 0 point of the
> >>> number line (imagine putting a mirror at the 0 point - it then looks
> >>> like the number is on the other side of the zero point).  So if you
> have
> >>> an odd number of these reflections, you end up on the other side; if
> you
> >>> have an even number, you end up on the same side. Perhaps you'll want
> to
> >>> draw a number line for your granddaughter, Mike, and find a small
> mirror?
> >>>
> >>> At 7:49 PM -0700 4/27/09, Mike Cole wrote:
> >>>> This is great, Steve. Back to Eric's strategy of personal experience
> and
> >>>> money exchanges.
> >>>> We are dividing in what I think is a productive way. One way goes
> >>>> toward the
> >>>> mathematical foundations,
> >>>> the other goes toward everyday transactions.
> >>>>
> >>>> I still want a picture to help me out the way   + cartesian
> coordinates
> >>>> APPEARS to help us understand
> >>>> addition and subtraction involving mixture of positive and negative
> >>>> numbers.
> >>>> every puzzled in san diego.
> >>>> mike
> >>>>
> >>>>
> >>>> On Mon, Apr 27, 2009 at 7:19 PM, Steve Gabosch <stevegabosch@me.com>
> >>>> wrote:
> >>>>
> >>>>>  Thinking in terms of minus and plus along number axes goes to the
> very
> >>>>>  heart of how to think in terms of how a typical CNC milling machine
> >>>>> moves.
> >>>>>
> >>>>>  These are large milling machines that, without making this more
> >>>>>  complicated, essentially move in three axes, x, y and z.  The
> >>>>> programs work
> >>>>>  by "zeroing out" each axis at a given point in relationship to the
> part
> >>>>>  being cut, and then commanding the cutter to go to specific points
> >>>>> along the
> >>>>>  x, y and z axes to create cutter paths that machine the part.  CNC
> mill
> >>>>>  operators (in 1980's-90's generation technology) usually have to
> >>>>> manually
> >>>>>  move the machine to set cutters, otherwise have to move the machine
> >>>>> around
> >>>>>  for a variety of reasons, including sometimes making rather
> intricate
> >>>>>  tooling moves, and adjust cutter paths during cuts when the part is
> >>>>> coming
> >>>>>  out too thick or thin, or wide or narrow.
> >>>>>
> >>>>>  Getting plus and minus right really, really matters.  Go the wrong
> >>>>> way -
> >>>>>  get minus and plus mixed up - even just a small fraction of an
inch,
> >>>>> and you
> >>>>>  can scrap the part, break the cutter, damage the tooling surfaces,
> >>>>> wreck
> >>>>>  tools, knock the spindles out of alignment, put the machine down
> for
> >>>>> hours
> >>>>>  or days, and otherwise cause thousands or more dollars in damage -
> and
> >>>>>  otherwise ruin your day.
> >>>>>
> >>>>>  I did a lot of training of operators on spar mills at Boeing over
> the
> >>>>>  years.  I was just an operator myself, but enjoyed doing this, and
> had
> >>>>>  something of a knack for it.  One of the reasons Boeing needed some
> >>>>> of this
> >>>>>  training was to transition a whole bunch of "conventional" milling
> >>>>> machine
> >>>>>  operators to CNC (computer numerical controlled) machines.  This
> was
> >>>>>  interesting because the idea of running a machine along numerically
> >>>>>  designated axes was often new to them.
> >>>>>
> >>>>>  So I got to look inside the heads of a lot of people who were
> grappling
> >>>>>  with numbers and axes in a new way, and this minus and plus
> business of
> >>>>>  course came up all the time.  I am a little off-topic here in that
> >>>>> it is
> >>>>>  rarely, if ever, necessary to **multiply** anything by a negative
> >>>>> number
> >>>>>  when running a spar mill.  But just **adding** and **subtracting**
> >>>>> negative
> >>>>>  numbers - especially in complex successions - and moving the
> machine
> >>>>> exactly
> >>>>>  where you wanted to go could get surprisingly complicated to wrap
> one's
> >>>>>  brain around.  Mistakes were common, and very experienced operators
> >>>>> could
> >>>>>  make them, too.
> >>>>>
> >>>>>  In classes for the conventional operators who had never been
> exposed
> >>>>> to CNC
> >>>>>  technology, I used the idea of a trip odometer on a car that only
> moved
> >>>>>  forward and backward 100 feet.  I explained that you can zero it
> out
> >>>>  > anywhere, and move back and forth all you want, and you will
> always
> >>>> be a
> >>>>>  specific distance from where you started.  That is basically how
> the
> >>>>>  relative register on the machine works, and most grasped the idea
> as
> >>>>> second
> >>>>>  nature.  There is also an absolute register on the machine, which
> >>>>> has a
> >>>>>  permanent zero position for each axis, so it is important to know
> >>>>> which one
> >>>>>  is which when you are looking at the screen for axis position.
> >>>>> Where and
> >>>>>  how that permanent position gets set is still another question.
> >>>>>
> >>>>>  The biggest trick in running a spar mill or any such CNC machine is
> >>>>> to be
> >>>>>  able to think in terms not of just moving in one, but *three* axes
> >>>>> at once.
> >>>>>   In the case of our usual spar mill, we ran with two spindles at
> >>>>> once, each
> >>>>>  with their own axis system - which made everything more
complicated,
> >>>>> now
> >>>>>  having to keep in mind things like whether your spindles are being
> >>>>>  programmed in "mirror mode" so the two parts come out as mirror
> >>>>> images of
> >>>>>  each other.
> >>>>>
> >>>>>  The operators have to routinely keep precise track of which
> >>>>> direction is
> >>>>>  minus and plus, and precisely which axis or axes they are concerned
> >>>>> with in
> >>>>>  a given circumstance.  Running a mill is something of an ongoing
> >>>>>  multi-dimensional puzzle with lots of little tricks and
> >>>>> misdirections to
> >>>>>  fool the operator into thinking one thing when something else is
> >>>>> true, or
> >>>>>  looking in one direction and forgetting to look in another.
> Believe
> >>>>> me,
> >>>>>  under pressure, in these conditions of complexity, and where can be
> so
> >>>>>  little or no room for error, keeping minus and plus straight all
> day
> >>>>> is not
> >>>>>  always so easy!
> >>>>>
> >>>>>  Working with many operators over the years, I could see that some
> >>>>> were very
> >>>>>  sharp about how the CNC axis system worked, could read programs
> >>>>> fluently,
> >>>>>  write their own, etc.  And then there were a few who had trouble
> even
> >>>>>  grasping the basic idea of negative numbers.
> >>>>>
> >>>>>  There was one fellow, then in his 40's, who was having a hell of a
> time
> >>>>>  with the concept of negative numbers.  I spent a few sessions with
> him
> >>>>>  trying to figure out a way to get negative numbers to make sense to
> >>>>> him as a
> >>>>>  "theory" or "concept" so he could do some routine things more
> >>>>> confidently,
> >>>>>  and not run into trouble, as he not infrequently did - often by
> >>>>> playing it
> >>>>>  safe and just not doing some things, which sometimes caused
> >>>>> problems, too.
> >>>>>
> >>>>>  I tried everything I could, using paper and pencil, moving the
> machine
> >>>>>  around and watching the screen together, using rulers and number
> >>>>> scales and
> >>>>>  moving things back and forth over them, and anything else I could
> >>>>> think of.
> >>>>>   He would get lost as to "where he was" as the objects or symbols
> >>>>> moved back
> >>>>>  and forth along an axis, especially if he started at a negative
> >>>>> numbered
> >>>>>  position.  Something about the concept just wouldn't stick.
> >>>>>
> >>>>>  I finally made a breakthrough when I related moving the machine
> back
> >>>>> and
> >>>>>  forth along an axis - to money in his pocket!  "Suppose you start
> >>>>> with $100,
> >>>>>  and then you spend $30 ... and then I give you $40 ... following
> me?
> >>>>> (yes)
> >>>>>  ... and then you spend another $20 ..."  Without fail, he always
> knew
> >>>>>  EXACTLY how much money he had!  The idea of negative numbers
> started
> >>>>> to fall
> >>>>>  into place after that.
> >>>>>
> >>>>>  Luckily, I didn't have to try to explain to him why -8 * -8 = 64.
> >>>>>
> >>>>>  - Steve
> >>>>>
> >>>>>
> >>>>>
> >>>>>
> >>>>>
> >>>>>
> >>>>>
> >>>>>  On Apr 27, 2009, at 5:34 PM, Wolff-Michael Roth wrote:
> >>>>>
> >>>>>    Can't you think like this---perhaps it is too much of a
> physicist's
> >>>>>>  thinking. We can think of the following general function (operator
> in
> >>>>>>  physics) that produces an image y of x operated upon by A.
> >>>>>>  y = Ax
> >>>>>>
> >>>>>>  if x is from the domain of positive integers, then A = -1 would
> >>>>>> produce an
> >>>>>>  image that is opposite to the one when A = +1, the identity
> operation.
> >>>>>>
> >>>>>>  Conceptually you would then not think in terms of a positive times
> a
> >>>>>>  negative number, but in terms of a positive number that is
> projected
> >>>>>>  opposite of the origin on a number line, and, if the number is
> >>>>>> unequal to 1,
> >>>>>>  like -2, then it is also stretched.
> >>>>>>
> >>>>>>  The - would then not be interpreted in the same way as the +
> >>>>>>
> >>>>>>  Cheers,
> >>>>>>  Michael
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>  On 27-Apr-09, at 4:16 PM, Ed Wall wrote:
> >>>>  >>
> >>>>>>  Mike
> >>>>>>
> >>>>>>     It is simply (of course, it isn't simple by the way) because,
> the
> >>>>>>  negative integers (and, if you wish, zero) were added to the
> >>>>>> natural numbers
> >>>>>>  in a way that preserves (in a sense) their (the natural numbers)
> usual
> >>>>>>  arithmetical regularities. It would be unfortunate if something
> >>>>>> that was
> >>>>>>  true in the natural numbers was no longer true in the integers,
> >>>>>> which is a
> >>>>>>  extension that includes them.  Perhaps the easiest way to the
> >>>>>> negative x
> >>>>>>  positive business is as follows (and, of course, this can be made
> >>>>>> opaquely
> >>>>>>  precise - smile):
> >>>>>>
> >>>>>>  3 x 1 = 3
> >>>>>>  2 x 1 = 2
> >>>>>>  1 x 1 = 1
> >>>>>>  0 x 1 = 0
> >>>>>>
> >>>>>>  so what, given regularity in the naturals + zero) do you think
> happens
> >>>>>>  next? This thinking works for, of course, for negative times
> >>>>>> negative. The
> >>>>>>  opaque proof is more or less as follows.
> >>>>>>
> >>>>>>  Negative numbers are solutions to natural number equations of the
> form
> >>>>>>  (I'm simplifying all this a little)
> >>>>>>
> >>>>>>                      x + a = 0    ('a' a natural number)
> >>>>>>
> >>>>>>  and likewise positive numbers  are solutions to natural number
> >>>>>> equations
> >>>>>>  of the form
> >>>>>>
> >>>>>>                     y = b          ('b' a natural number)
> >>>>>>
> >>>>>>
> >>>>>>  Multiplying these two equations in the usual fashion within the
> >>>>>> natural
> >>>>>>  numbers gives
> >>>>>>
> >>>>>>
> >>>>>>             xy + ay = 0
> >>>>>>
> >>>>>>  or substituting for y
> >>>>>>
> >>>>>>
> >>>>>>       xy + ab = 0
> >>>>>>
> >>>>>>  so, by definition, xy is a negative number.
> >>>>>>
> >>>>>>  Notice how all this hinges on the structure of the natural numbers
> >>>>>> (which
> >>>>>>  I've somewhat assumed in all this).
> >>>>>>
> >>>>>>  Ed
> >>>>>>
> >>>>>>
> >>>>>>
> >>>>>>  On Apr 27, 2009, at 6:47 PM, Mike Cole wrote:
> >>>>>>
> >>>>>>  Since we have some mathematically literate folks on xmca, could
> >>>>>> someone
> >>>>>>>  please post an explanation of why
> >>>>>>>
> >>>>>>>  multiplying a negative number by a positive numbers yields a
> negative
> >>>>>>>  number? What I would really love is an explanation
> >>>>>>>  that is representable in a manner understandable to old college
> >>>>>>>  professors
> >>>>>>>  and young high school students alike.
> >>>>>>>
> >>>>>>>  mike
> >>>>>>>  _______________________________________________
> >>>>>>>  xmca mailing list
> >>>>>>>  xmca@weber.ucsd.edu
> >>>>>>>  http://dss.ucsd.edu/mailman/listinfo/xmca
> >>>>>>>
> >>>>>>>
> >>>>>>>
> >>>>>>  _______________________________________________
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> >>>>>>
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> 
> --
> ------------------------------------------------------------------------
> Andy Blunden http://home.mira.net/~andy/
> Hegel's Logic with a Foreword by Andy Blunden:
>  From Erythrós Press and Media <http://www.erythrospress.com/>.
> 
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