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*Subject*: Re: [xmca] a minus times a plus*From*: Andy Blunden <ablunden@mira.net>*Date*: Tue, 28 Apr 2009 17:36:57 +1000*Cc*: "eXtended Mind, Culture,Activity" <xmca@weber.ucsd.edu>*Delivered-to*: xmca@weber.ucsd.edu*Domainkey-signature*: a=rsa-sha1; s=2007001; d=ucsd.edu; c=simple; q=dns; b=JmeiKaSMbvf7KUvfGThS8TbHy+mn7cUCp64NzXDkyBNcaLyL8TjpNJJLVSGJomMK8 +9MTYihLZJo8yq5OUq4nA==*In-reply-to*: <3B19033D3E2EC34C97DF364119A79A61C90CE1@EXVS1.its.uidaho.edu>*List-archive*: <http://dss.ucsd.edu/mailman/private/xmca>*List-help*: <mailto:xmca-request@weber.ucsd.edu?subject=help>*List-id*: "eXtended Mind, Culture, Activity" <xmca.weber.ucsd.edu>*List-post*: <mailto:xmca@weber.ucsd.edu>*List-subscribe*: <http://dss.ucsd.edu/mailman/listinfo/xmca>, <mailto:xmca-request@weber.ucsd.edu?subject=subscribe>*List-unsubscribe*: <http://dss.ucsd.edu/mailman/listinfo/xmca>, <mailto:xmca-request@weber.ucsd.edu?subject=unsubscribe>*References*: <30364f990904271547o5b4df21eifca69bf8318483f2@mail.gmail.com> <2C46D7A7-AD94-441C-AABD-269045835E3D@umich.edu> <0193A85F-03A7-4811-B912-217722A5770B@uvic.ca> <C23F99F4-D4AC-463B-AF11-30585E74F7CF@me.com> <30364f990904271949r3cfef116m89d39ea419d112b2@mail.gmail.com><p06240805c61c2a15a97a@[192.168.1.65]><49F68AEE.40502@mira.net><3B19033D3E2EC34C97DF364119A79A61C90CDD@EXVS1.its.uidaho.edu> <49F69438.4050900@mira.net> <3B19033D3E2EC34C97DF364119A79A61C90CE1@EXVS1.its.uidaho.edu>*Reply-to*: ablunden@mira.net, "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>*Sender*: xmca-bounces@weber.ucsd.edu*User-agent*: Thunderbird 2.0.0.14 (Windows/20080421)

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 plusEmily, I quite candidly introduced my earlier message as"the world's worst maths teacher". I developed this identitypartly by being given the task of teaching "New Maths" toalmost-innumerate kids in Brixton in the 1970s.I was an Engineering PhD who could solve integral equationsbut couldn't sing, and had no teacher training.I was asked to teach for example, the algebra oftransformations of a figure in 3 dimensions (eg rotating by90deg 4 times = null). This was not my choice. That was thesyllabus! But because of my own background, I couldn'tunderstand what they found so difficult. :)Later I had a seminal chat with the English teacher who toldof how he only learnt to understand the workings of thedifferential (those things on the back axle of motor carswhich allow the 2 wheels to go at different speeds), byhaving someone tell him in words, and going over and overthose words. The diagrams meant nothing to him. My firstglimmer 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 plusPersonally, Jim and Linda, I would find ideas like negativeas reflection or negative as contradiction, easy tounderstand if *you are already familiar* with -x-=+ but Iwould have thought that the leap from visual or rhetoricalrelationships to numbers is asking too much of children whohave not yet learnt to count backwards from zero.Why is a number line easier to understand than thesymmetries of topological or rhetorical transformations? Idon't know, but intuitively I think it is. But differentpeople think differently. Maybe the Linda's rhetoricalexplanation is easier for a verbal thinker than a spatialthinker? (is this mumbo-jumbo?)Andy Jim Levin wrote:Perhaps it would be helpful to think about multiplying by -1 asreflection. 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 thenumber line (imagine putting a mirror at the 0 point - it then lookslike the number is on the other side of the zero point). So if you havean odd number of these reflections, you end up on the other side; if youhave an even number, you end up on the same side. Perhaps you'll want todraw 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 goestoward themathematical foundations, the other goes toward everyday transactions. I still want a picture to help me out the way + cartesian coordinates APPEARS to help us understandaddition and subtraction involving mixture of positive and negativenumbers.every puzzled in san diego. mikeOn 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 veryheart of how to think in terms of how a typical CNC milling machinemoves.These are large milling machines that, without making this morecomplicated, essentially move in three axes, x, y and z. Theprograms workby "zeroing out" each axis at a given point in relationship to the partbeing cut, and then commanding the cutter to go to specific pointsalong thex, y and z axes to create cutter paths that machine the part. CNC milloperators (in 1980's-90's generation technology) usually have tomanuallymove the machine to set cutters, otherwise have to move the machinearoundfor a variety of reasons, including sometimes making rather intricatetooling moves, and adjust cutter paths during cuts when the part iscomingout too thick or thin, or wide or narrow.Getting plus and minus right really, really matters. Go the wrongway -get minus and plus mixed up - even just a small fraction of an inch,and youcan scrap the part, break the cutter, damage the tooling surfaces,wrecktools, knock the spindles out of alignment, put the machine down forhoursor 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 hadsomething of a knack for it. One of the reasons Boeing needed someof thistraining was to transition a whole bunch of "conventional" millingmachineoperators 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 ofcourse came up all the time. I am a little off-topic here in thatit israrely, if ever, necessary to **multiply** anything by a negativenumberwhen running a spar mill. But just **adding** and **subtracting**negativenumbers - especially in complex successions - and moving the machineexactlywhere you wanted to go could get surprisingly complicated to wrap one'sbrain around. Mistakes were common, and very experienced operatorscouldmake them, too.In classes for the conventional operators who had never been exposedto CNCtechnology, 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 alwaysbe aspecific distance from where you started. That is basically how therelative register on the machine works, and most grasped the idea assecondnature. There is also an absolute register on the machine, whichhas apermanent zero position for each axis, so it is important to knowwhich oneis which when you are looking at the screen for axis position.Where andhow that permanent position gets set is still another question.The biggest trick in running a spar mill or any such CNC machine isto beable to think in terms not of just moving in one, but *three* axesat once.In the case of our usual spar mill, we ran with two spindles atonce, eachwith their own axis system - which made everything more complicated,nowhaving to keep in mind things like whether your spindles are beingprogrammed in "mirror mode" so the two parts come out as mirrorimages ofeach other.The operators have to routinely keep precise track of whichdirection isminus and plus, and precisely which axis or axes they are concernedwith ina given circumstance. Running a mill is something of an ongoingmulti-dimensional puzzle with lots of little tricks andmisdirections tofool the operator into thinking one thing when something else istrue, orlooking in one direction and forgetting to look in another. Believeme,under pressure, in these conditions of complexity, and where can be solittle or no room for error, keeping minus and plus straight all dayis notalways so easy!Working with many operators over the years, I could see that somewere verysharp about how the CNC axis system worked, could read programsfluently,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 himtrying to figure out a way to get negative numbers to make sense tohim as a"theory" or "concept" so he could do some routine things moreconfidently,and not run into trouble, as he not infrequently did - often byplaying itsafe and just not doing some things, which sometimes causedproblems, too.I tried everything I could, using paper and pencil, moving the machinearound and watching the screen together, using rulers and numberscales andmoving things back and forth over them, and anything else I couldthink of.He would get lost as to "where he was" as the objects or symbolsmoved backand forth along an axis, especially if he started at a negativenumberedposition. Something about the concept just wouldn't stick.I finally made a breakthrough when I related moving the machine backandforth along an axis - to money in his pocket! "Suppose you startwith $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 knewEXACTLY how much money he had! The idea of negative numbers startedto fallinto 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'sthinking. We can think of the following general function (operator in physics) that produces an image y of x operated upon by A. y = Axif x is from the domain of positive integers, then A = -1 wouldproduce animage 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 projectedopposite of the origin on a number line, and, if the number isunequal 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, thenegative integers (and, if you wish, zero) were added to thenatural numbersin a way that preserves (in a sense) their (the natural numbers) usualarithmetical regularities. It would be unfortunate if somethingthat wastrue in the natural numbers was no longer true in the integers,which is aextension that includes them. Perhaps the easiest way to thenegative xpositive business is as follows (and, of course, this can be madeopaquelyprecise - 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 happensnext? This thinking works for, of course, for negative timesnegative. Theopaque 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 numberequationsof the form y = b ('b' a natural number)Multiplying these two equations in the usual fashion within thenaturalnumbers 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(whichI'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, couldsomeoneplease 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_______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca _______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca_______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca_______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca_______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca

-- ------------------------------------------------------------------------ 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/>. _______________________________________________ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca

**Follow-Ups**:**RE: [xmca] a minus times a plus***From:*"A.Bakker" <A.Bakker@fi.uu.nl>

**References**:**[xmca] a minus times a plus***From:*Mike Cole <lchcmike@gmail.com>

**Re: [xmca] a minus times a plus***From:*Ed Wall <ewall@umich.edu>

**Re: [xmca] a minus times a plus***From:*Wolff-Michael Roth <mroth@uvic.ca>

**Re: [xmca] a minus times a plus***From:*Steve Gabosch <stevegabosch@me.com>

**Re: [xmca] a minus times a plus***From:*Mike Cole <lchcmike@gmail.com>

**Re: [xmca] a minus times a plus***From:*Jim Levin <jalevin@ucsd.edu>

**Re: [xmca] a minus times a plus***From:*Andy Blunden <ablunden@mira.net>

**RE: [xmca] a minus times a plus***From:*"Duvall, Emily" <emily@uidaho.edu>

**Re: [xmca] a minus times a plus***From:*Andy Blunden <ablunden@mira.net>

**RE: [xmca] a minus times a plus***From:*"Duvall, Emily" <emily@uidaho.edu>

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