
Re: Please remind me why 3^2 = 9
Posted:
Oct 18, 2012 11:21 PM


On Oct 18, 2012, at 8:58 PM, Philip Mahler wrote:
> It does occur to me that stressing the ?correct? interpretation of ? > 3^2 with students develops no understanding of operations that I can > see, and it does consume an amount of time and create more > consternation than it is worth.
I agree because the difficulty is elsewhere.
(1) There is no such thing as the "correct" interpretation of, say,  and the understanding which ought to be conveyed to the students is that here, as in other places, different "encodings" of the real world are possible and that what is most important is to agree on what encoding is to be used. For instance, in Polish notation, we would write 4+5 as +3,5 and in reverse Polish notation we would write 4+5 as 3,5+. Another example is that where analysts write f(x), algebraists write (x)f or, better yet, xf. Thus, not only is there no "correct" coding, there is not even a best coding.
This is completely similar to what happens with natural languages: what we call "table" in English and French is called "taulukko" in Finnish, "Tisch" in German, "tavola" in Italian, etc.
(2) Another, equally important, point to be made is that the meaning of symbols cannot be conttextfree because that would require a lot more symbols than is possible. Thus the students should be made to learn always to explicit the context.
For instance,  is used to denote subtraction among "plain", i.e. "unsigned", numbers, but also in the names of integers to code one of the ways they can go, say up or down or left or right, etc. But then it is also used for subtraction for "signed" numbersas well as for other things. It is useful to use, temporarily, different symbols to make things contextfree.
Thus, my developmental students write
3 ominus 5 = 3 oplus opposite 5 (by definition of ominus) = 3 oplus +5 (by definition of opposite) = +2 (by definition of oplus)
where oplus is written as + within an o and ominus as  within an o.
The students do not of course write the stuff in parenthesis but should be able to invoke it should they be challenged with "Why?"
By the way, note that the definition of oplus, and therefore its understanding, involves  both as sign of negative numbers and  as subtraction of "plain" numbers.
(3) What complicates matters is that in mathematical languages there is a lot of stuff that "goes without saying", i.e. that there is a lot of stuff that is being handled by "default rules" which, unfortunately, are rarely stated. For instance, while 5 is emphatically not the same as +5 since 5 does not encode the same reality as +5, when 5 is used in the context of "signed" numbers, it is as a shorthand for +5 and there is a default rules that says that, in the context of "signed" numbers, where there no sign in front of the plain number that encodes its "size", + is to be be understood to go without saying.
So, when we write, say, 5  3, it can in the context of "plain" numbers or it can be in the context of "signed" numbers where it is then a shorthand for
+5 opluse 3 or for +5 ominus +3.
This becomes obvious when we look at 4  7: In the context of "plain" numbers 4  7 stands for a subtraction that cannot be done but in the context of "signed" numbers 4  7 is a shorthand for
+4 oplus 7 or for +4 ominus +4.
Here, it is most important to let the students realize that the default rules are made in such a way that, regardless of the context, the right symbols will come out of the computations.
Regards schremmer
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