# Re: [xmca] a minus times a plus

```regarding sand and multiplication:-
another crazy analogy i was toying with.  i imagine Mr Sand-man
piling up sand (+) and taking away sand (-).  i imagine his
counter-part Mr Hole-man living upside down in the ground, who
treats holes as "real stuff" and sand as absence of holes.  When
Mr Sand-man digs a hole say 6 feet below ground (-6), he's piling
up something "real" for Mr Hole-man.  If Mr Hole-man living
upside-down below ground uses his magic spade to dig away one
feet of hole "-(-3)", he's in fact filling up one feet of sand
(+3).  If Mr Hole-man does this twice, then it is (-2 × -3), same
as filling up the hole (2 × 3).
it seems we got to tweak our ontologies to get this to work.

regarding discs:-
i was part of a Singapore team that did the AlgeTools software
that included AlgeDisc, an electronic version of "number discs".
You can do this just as well with coins (heads vs tails) or
coloured discs.  There are 2 types of discs e.g. blue (positive)
and red (negative).  a blue and a red disc forms a zero-pair, so
a blue and a red disc annihilate each other.  you can also
put in pairs of blue and red discs at will with impunity.
so put in say 6 such zero-pairs.  -2 × -3 means two times
_removing_ sets of 3 red discs.  that leaves the 6 blue discs
i.e. the operation has the same effect as adding two sets of
blue discs (2 × 3).

regarding structure and agency, arbitrariness:-
i think now it's time for me to pop this question that has been
bugging me for some time.  i am convinced that mathematics is
socially constructured, but i am not so convinced that mathematics
is _merely_ socially constructured.  if we vary across cultures
and different human activities, we might find different ways
in which patterns and structure can be expressed and yet we might
find commonalities / analogies.  the question i am asking is:
is maths just a ball game determined by some group of nerds who
happen to be in power and dominate the discourse, or is there some
invariant, something deeper in maths that can transcend and unite
language, culture, activity .... ?

Foo Keong,
NIE, Singapore

2009/4/29 Duvall, Emily <emily@uidaho.edu>:
> Thanks for reminding us of the sand, David!
> In return, here is how happiness can be found in an equation:
>
>
> I'm actually going to take a math class this summer to see what the
> state of Idaho plans on teaching its 4th-8th graders. Hopefully we'll
> tackle the negative/positive issue. For me, the issue is really not so
> much negatives and positives, but how to explain multiplying two
> negatives = positive... all my hands on, minds on goes out the window.
>
> ~em
>
>
>
>
>
> -----Original Message-----
> From: xmca-bounces@weber.ucsd.edu [mailto:xmca-bounces@weber.ucsd.edu]
> On Behalf Of David Atencio
> Sent: Tuesday, April 28, 2009 7:00 AM
> To: eXtended Mind, Culture, Activity
> Subject: Re: [xmca] a minus times a plus
>
> Emily and Andy (and others),
> Your recent exchange in this thread and in particular Karprov and Gindis
>
> notion of cross domain internalization reminds me of a good example of
> Rogoff's notion of "bridging the new to the old" as a component of
> guided participation.
>
> In the movie "Stand and Deliver",  the character of Jaime  Escalante is
> at the board trying to explain the concept of positive and negative
> integers. He asks the students to remember be being at the beach and to
> imagine scooping up some sand, he explains that the amount scooped out
> that is piled atop the surface is the positive value, the hole that is
> left is the negative value.
>
> So getting back to Mike's original question, multiplication in a sense
> is about how many times something is repeated.  No matter how many times
>
> (i.e., a X n, n +1, n +2, ...), the "hole" or vacant space  is repeated
> or what call in the math world multiplied, it will never become pile of
> sand, it will only become more vacant spaces.
>
> David
>
>
>
> 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
>>>>>
>>>>>  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
>>>>>  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
>>>>>  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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>>>>>
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