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*Subject*: Re: [xmca] a minus times a plus*From*: Andy Blunden <ablunden@mira.net>*Date*: Tue, 28 Apr 2009 13:42:57 +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=Qi+ArbY4Vn23JXPw7R0GBLo90dXZZ3qogevFxfZVBFYqjHQHKqO0HjztncLdzHUYF 7h/WbKp1CgTmPz1vfSqGg==*In-reply-to*: <30364f990904271944o2565e2e9m443d7740afc0a8ea@mail.gmail.com>*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> <30364f990904271706l114497cax5f814ffa09a51893@mail.gmail.com> <49F65663.6040607@mira.net> <30364f990904271944o2565e2e9m443d7740afc0a8ea@mail.gmail.com>*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)

Mike, according to our wonderful correspondents, the number line is the favourite "visual/practical image". I believe them. People have rightly suggested the use of this to present mutliplication tables in the form of a number line, which, to start with, stop at 0. Then you ask: "what if we extend this ...?" This looks great to me. My only point is that the teacher would use this practical/visual approach to get kids filling in the blanks correctly ... *and then* abstract the general rule "minus times minus = plus". You have to learn this as a rule at some point. Question is when and on what basis is it to be justified and understood? There is no way you can introduce that rule prior to practical/intuitive manipulations. If some appropriate intuitive image is used to present the base regularity, the kids will be blissfully unaware that stepping across the boundary into a new domain of numbers, which you have presciently placed as a continuation of the starting doamin is an unwarranted, arbitrary and unjustifable logical leap. And why not! But until kids have left behind them the intuitive justifications, they have not learnt mathematics, which is after all the object of the exercise, isn't it? Or is it? The great thing about Ed's first contribution, I think, is that he showed how foundations-of-mathematics thinking can mirror the operation used for teaching: i.e., extension of a domain. And this is a foundation on which people can reflect and learn about mathematics, as well as carry out the required opeations when asked. My only point is that kids need to first practice the operations in the narrow and then the wider domain, and then abstract the rule, and then much later learn the logical meaning of domain-extension. At bottom, the answer to "why is - x - = + ?" is "because we say so". It is no more provable than "1 + 1 = 2" and intuitive demonstrations which purport to prove it are actually fooling people. But maybe it is necessary to fool people first and explain why later, if they're interested. To return to the example that we were discussing the other day: e to the i pi = -1. This is justified in university maths classes using the domain extension rationale. It is difficult to see any other way of approaching it. And that presupposes familiarity with 4 domains of numbers. But it was fantastic at the time to already have the *Argand diagram*. The Argand Diagram looks like Cartesian x-y coordinates, and is understood as a representation of "complex numbers" in which the domain of numbers has been extended to include the square root of minus one, and represented as a "2 dimensional number". With the Argand Diagram a lot of the symmetries of this great equation become visually very compelling. Mike Cole wrote:

Andy-- Visual image PLEASE!!! mikeOn Mon, Apr 27, 2009 at 6:05 PM, Andy Blunden <ablunden@mira.net<mailto:ablunden@mira.net>> wrote:From the worst ex-maths teacher in the world ... Certainly I think Ed's explanation of "why" minus numbers behave the way they do when included in operations that make intuitive explanation impossible is right. I.e., you ask that regularities that applied in the domain so far ought to be retained when the domain is extended by adding a new group of numbers. There is no meaning for "multiplying by a negative number" that can be reliably deduced from intuitive definitions of "multiply" and "negative". So the rule is that you can grasp the idea of "multiply" intuitively through the idea of repeated addition, just as you grasped the idea of addition by repeated counting. And you can grasp the idea of negative numbers in some equally intuitive way (there are several options), but not a way which can be fitted into the idea of "repeated addition". So you take Ed's advice and rely on some general rule or visual image that worked before and require that it still work for negative numbers. In that way you move out of the bounds of intuition into mathematical thinking, guided no longer by plausible intuition, but by a mathematical rule. That still leaves open the question as to whether you can teach general rules and mathematical reasoning to someone who has had no practice in applying the rules whose jutification you claim to achieve by this "rule extension" rationale that Ed exlained. I was of a generation that learnt my times-tables by rote and had my first lesson on real mathematics in my last years as an undergrad. 15 years later, and then 6 years later was asked to teach "modern mathematics" to 13 year old kids who couldn't count and had no idea of what "1/2" meant except a 1 a stroke and a 2. I was not a happy chappy at the time. I blame Piaget and his "Genetic Epistemology" and a whole lot of absurdity that went down in the early 1970s. I say: learn to ride your bike, and then learn dynamics to make sense of it afterwards. Andy Mike Cole wrote: Great!! Thanks Ed and Eric and please, anyone else with other ways of explaining the underlying concepts. Now, we appear to have x and y coordinates here. If I am using a number line that ranges along both x and y axes from (say) -10 to +10 its pretty easy of visualize the relations involved. And there are games that kids can play that provide them with a lot of practice in getting a strong sense of how positive and negative positions along these lines work. What might there be of a similar nature that would help kids and old college professors understand why -8*8=64 while -8*-8=64? Might the problem of my grand daughter, doing geometry, saying, "Well, duh, grandpa, its just a fact!) arise from the fact (is it a fact?) that they learn multiplication "facts" before they learn about algebra and grokable explanations that involve even simple equations such as y+a=0 are unintelligible have become so fossilized that the required reorganization of understanding is blocked? mike On Mon, Apr 27, 2009 at 4:16 PM, Ed Wall <ewall@umich.edu <mailto:ewall@umich.edu>> 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 <mailto:xmca@weber.ucsd.edu> http://dss.ucsd.edu/mailman/listinfo/xmca _______________________________________________ xmca mailing list xmca@weber.ucsd.edu <mailto: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 <mailto: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

**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:*Mike Cole <lchcmike@gmail.com>

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

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

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