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RE: [xmca] a minus times a plus

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


-----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?)


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
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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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