
Re: Mathematical literacy and the form a/b
Posted:
Sep 30, 2010 3:13 PM


On Wed, Sep 29, 2010 at 2:41 PM, Paul A. Tanner III <uprho@yahoo.com> wrote:
>> I guess I'm just not seeing it yet Paul, how division >> should or could >> be treated as an inverse function of multiplication, so >> consider me >> one of those confused elementary school teachers. > > First: If you recall what I wrote earlier, you know that this is not for elementary school teachers. What would be for them, as I said, would be a watered down version, reduced to the real number context. >
You started with a kind of lament, that certain things elementary math teachers are meant to communicate are not getting through. It looks like you're trying to address this lack upstream someplace, in the vicinity of group theory, which has been expanded into many more layers over time, to where now we have groupoids or magmas as they were apparently called by Bourbaki.
On the topic of French mathematics and linking to food standards in the cafeteria (a parallel thread at the time of this writing), I was informed by a peer teacher last night that "what's for lunch" on a typical French school menu is way better than what you'd find almost anywhere in New York. Perhaps Haim would care to confirm? It's just a part of the culture, to care about food more.
> The existence of invertible elements implies the existence of cancelable elements, which implies the existence of an injective function, which implies the existence of an inverse function. Note: Since a function has an inverse on it domain and codomain if and only if the function is bijective, notice that the domain and range of an injective function form a bijective function when the range is redefined to be also a codomain. Yes the domain and range do not change, but we just redefine what the codomain is. >
I'm not sure this is where I'd be coming from if pressed to talk about subtraction as the inverse of addition. The model there would be and is the number line, with hops. There's insufficient attention to "point of view" (i.e. camera position), such that the positives are depressingly always off to the right (from the reader's point of view). Concerns about left and right handedness are too much suppressed, e.g. if you swing around to view a clock from the back, the "clockwise" becomes "counterclockwise" (unless you stand on your head...).
So with plus and minus, I'd be looking at twisting a knob +9, then 9, back to where we started. Left and right, clockwise / counterclockwise, that kind of thing. Motor skills matter. Especially in K3 and like that.
Now with multiplication and division, I'd use a different model. Instead of rotating the beach ball around an axis, left and right (the additive model), I'd talk about the beach ball itself growing and shrinking (down to grapefruit size, up to some really big geodesic sphere, like a Cloud 9 if you're familiar with that invention).
In making multiplication be about "scaling" (resizing), we're getting into exponential curves, 2nd and 3rd powering most immediately (change in surface area, change in volume). To "do and undo" in this environment is less about rotating left and right (additive) and more about expanding and contracting (multiplicative).
Double every edge of the rhombic dodecahedron, and its volume goes from 6 to 48 (6 * 8). To undo, multiply by 1/2 to have every edge shrink to 50% of its previous size. Volume goes to 0.125 of its former self.
Might you still call it a function, when a polyhedron maps to another polyhedron because of x2? Sure.
Therefore I would not "dumb it down" to just being about real numbers for the elementary school teachers. On the contrary, we'd do enough of the group theory to pick up the shop talk, explore a computer language, and then get away from numbers into finite groups with permutations.
I'd probably bill this as Casino Math but the localizing docs will customize however (not saying its up to me how a locale adapts in a mix with plenty of other sources).
> What you might not realize is that we can redefine any function of more than one variable as a function of one variable, by simply letting the value of all but one of the variables be fixed. That is, we can redefine the function f(x,y,z) = P of three variables to be a function of one variable, and by letting y,z be fixed might write something like g(x)_{y,z} = P to denote this, where if we want and if we make things clear enough to the audience, we can drop the subscript notation and just let the actual formula P tell the audience that the original two other variables are still there by virtue of the original formula containing them, just held constant, and I write g(x) = P. >
There's also the option of taking all the inputs and bundling them and calling that "one variable". For example, with A sin (B x) + C, we might accept (A,B,C) as a "single point" in a phase space. (x, y) is likewise a single object. When I listed allproducts and allanswers for totatives of 28, I was listing the same number of math objects in both lists. In that sense, you may make any function a unary function. Just consider the combination of arguments, several objects of different types, as "the argument" (collapse the args to "one thing").
Then yeah, what you're talking about may be called "currying" in a computer science context. Pythons functools module contains partial( ) which is used in just such a manner (some examples elsewhere in this thread).
> Note that if a function of more than one variable is not injective and we redefine it so that we have a function of one variable, that redefined function of one variable may sometimes be injective. And that's what we get when, in a cancellative or rightcancellative multiplicative groupoid, we redefine the binary function (operation) to a unary one: We go from not having an injective function to having one. >
I'm not persuaded a rigorous theory of functions is helping explain much, while we're getting a lot of overhead.
The concept of "function" needs a lot more elaboration, before we start explaining how to think about binary operations in a groupoid, where the only property is closure. The binary operation need not conform to the function idea i.e. might be more random than formal functions would allow.
I'm happy to dabble in group theory with these folks, but not at the expense of a lot more notation, and no handson with computers. Seems like a long shot that this'd prove edifying. But then, as you say, this isn't for elementary math teachers.
> People do this all the time in math. Maybe you haven't seen it, but I have  I was taught it, and used this technique myself in writing the functions I created in my paper I published while an undergrad. All the mathematicians who saw the paper had no trouble understanding and no problem with this redefining of a function of more than one input as another function of one input, even though their formulas are the same. These functions generalize the Euler totient function in a new way. They count the number of arithmetic progressions of k reduced residues modulo n, with and without the first term a, common difference d, and any median term m fixed. Even though there are multiple variables a,d,k,m,n, I defined all these functions to be a function of n only, where all the other variables' values were fixed or held constant. The paper's title is "Arithmetic progressions that consist only of reduced residues" and it's available online free, open access. Just > Google the title with the quotation marks.
I am familiar with this convention, as I've endeavored to make clear, with examples in source code.
> > And so I'm just using the exact same sort of technique in this document I'm writing, where I redefine binary multiplication as unary multiplication by simply holding constant one of the variables. > > I'm doing this move from binary to unary to uncover and make explicit what is actually there in the relationship between multiplication and division, addition and subtraction. >
There seems to be the added assumption that what you're doing should somehow be useful to elementary school teachers, and I'm suggesting that there's more than one way to introduce the idea of doing and undoing. Whether one relates this back to group theory is an option, not mandatory, the way I see it, so we're free to debate your proposal in that light.
On the other hand, if you're saying you just want to see Wiki pages, may I recommend Wikieducator, which is where I store a lot of my own stuff (I eat my own dog food in that sense).
> I again note that this sort of redefining, this redefining of functions of more than one variable into a function of one variable, is at least implicitly done all the time under our very noses, where for instance we see the equivalence of the cancellation property on a set S under a binary function and this binary function when redefined as a unary function being an injective or onetoone function. The Wikipedia article on the cancellation property touches on this equivalence: > > http://en.wikipedia.org/wiki/Cancellation_property > > "Interpretation > To say that an element a in a magma (M,*) is leftcancellative, is to say that the function g: x > a * x is injective." > > THEOREM: Equivalence of injection and cancellation. > > Let S be a multiplicative groupoid, and let all individual variables denote elements in S. Then unary multiplication on S is injective if and only if S is cancellative. Stated another way: For any p, let unary multiplication on S be defined as the functions f(x) = xp and g(x) = px. Then the functions f and g are injective if and only if p is cancelable under multiplication. >
Multiplicative groupoid would be a groupoid with some additional assumptions, I know not which specifically, but probably there's an identity element. I assume this simply from your prepending the adjective "multiplicative" i.e. there's some allusion to properties associated with multiplication rather than, say, addition.
> Proof. Case 1: By the definitions of cancellation and injective functions, we have the implication that f(x_1) = x_1p = x_2p = f(x_2) implies x_1 = x_2, which shows that f being injective is equivalent to p being rightcancelable under multiplication. Case 2: By the definitions of cancellation and injective functions, we have the implication that g(x_1) = px_1 = px_2 = g(x_2) implies x_1 = x_2, which shows that g being injective is equivalent to p being leftcancelable under multiplication. Taking these two cases together shows that unary multiplication on S is injective if and only if S is cancellative. > > And since we can define on the domain and range of any injective function an inverse function (that domain and range form a bijective function), we have an inverse function of f that we can call division and write as f^{1}(f(x)) = f(x)/p = x. >
I'd probably go with "operations" such as edits in a word processor or text editor, and talk about the undo feature. I'd also talk about functions that are useful precisely because they're "one way" i.e. you can do, but undoing is hard. Example: multiply two giant prime numbers together. Will you then be able to factor the product? Likely not, as we don't have an algorithm that is guaranteed to work in any reasonable amount of time (on the contrary, is pretty much guaranteed to take quasiforever).
We want to help future physics students understand about what's "reversible" versus what comes with an "arrow of time". Turning back the clock is not always an option, at least not locally.
Yes, these sound rather like philosophical conversations, and that opens teachers to the charge of not having all the answers, which leaves students uneasy as they just want to hold it together long enough to pass whatever tests, don't want to waste time on any "philosophy" that won't be there (on the AP or IB exam or whatever). Perhaps that's another reason to move into ~M! within the literature curriculum? We want students to develop fluency, logical problem solving abilities, without a discouraging level of abstraction that just serves as dead weight after a certain level. What's the right balance of abstract and concrete? Remember the pun: concrete = CONtinuous + disCRETE (mentioned in the Litvins intro, alluding to titles such as 'Concrete Mathematics' out of Standford, Knuth a coauthor).
>> >> As Clyde was pointing out, the inverse of f(a,b) > c >> would seem to be >> "to factor" i.e. to crack a product into two constituent >> factors. As >> you point out, though, factor pairs are far from unique. >> >> I don't recall anyone ever teaching me that division was >> the inverse >> function of multiplication, in the sense that arccos is the >> inverse of >> cos, such that D(M((a,b)) > (a,b) where (a,b) is the >> input tuple. > > You were never taught that multiplication and division "undo" each other or some similar thing? >
I remember the number line and hopping off to the right in a positive direction (these are invariable links, "positive" and "right"  no one cares to question). Then the opposite direction might be called the negative ("left handed, sinister") direction.
The imagery tends to be less clear around multiplication, which is where that earlier demo comes in (relating to scaling).
10 * F * F + 2 is one example of a function we use. Coxeter found it delightful and wrote a whole paper on formulae of this genre, which it only takes some high school algebra to prove. Prove what? That it gives the number of balls in layer F, packing outwardly from a center. 1, 12, 42, 92... I've used that a lot, including in Martian Math?
Why exactly?
It's a long story if getting into the details, but has to do with keeping a lexicalgraphical bridge that will likely be of utility even in adulthood. Why? Spacefilling tessellations. Why does that matter? Chemistry. Why does that matter? The processes of life.
Anyway, 10 * F * F + 2 may be rewritten as x2(P)(x) +2 where x2 means "times two" or doubling (multiplicative two) and +2 means "add two" (additive two). P is means some product of primes, a fancy way of saying either a prime or composite number, i.e. positive integer. In the case of 10 * F * F + 2, P is of course 5. However, as I mentioned, H.S.M Coxeter introduced other values for P that effectively changed the shape of the packing, meaning instead of a cuboctahedron, you'd have some other expanding shape. Here's a link to said paper, followed by a link to a proof, followed by a literary link (for the ~M! readers):
http://preview.tinyurl.com/ylzcsmj (stops at tinyurl along the way, to allow previews  then goes to Google Books)
http://mybizmo.blogspot.com/2007/01/gnumathmemo.html (proof of 10 * F * F + 2 = sequence of cuboctahedral numbers  topological considerations).
http://worldgame.blogspot.com/2009/11/kickingcandownroad.html (Ernest Hemingway and Python generators)
Again, why I bring this up in some detail is we're talking about do/undo of two different kinds, multiplicative (x2) and additive (+2).
And yes, here's a bridge to chatter about groups, rings and fields (not so sure we need magmas, but maybe, for some teachers who wanna get detailed).
A primary advantage of the group theory approach is we're at last free of real numbers, might use other objects instead, some quite alien, which is why I'd *not* dumb it down by turning back the clock, by making "real numbers" the only math objects we care about (they're just another subclass of the C type after all, not that special).
Rotating polyhedra is already deeply embedded in group theory as you know doubt know (I've published a sequence of pages on that, relating to crypto), so for me to be sprinkling in some simple functions, some algebra, to generate polyhedral numbers, is hardly a major rewrite.
What's different (a market niche, but also a school of thought) is allowing that tetrahedron of four closepacked uniradius balls to have unit volume (seems alien!).
Per the article in Quadrays in Wikipedia, that takes some getting used to and we'll offer elementary school teachers some useful heuristics to help make the transition (if doing OCN brand digital math, it'll be hard to avoid that univolume tetrahedron segment, as we get back to it along many turns of the spiral).
For further reading (for elementary math teachers included):
http://controlroom.blogspot.com/2010/03/connectingdots.html
>> >> > Elementary school children are taught that division is >> the inverse operation of multiplication  and they are >> taught this on the natural numbers. It is not the case that >> division is not taught until fractions and rational numbers >> with multiplicative inverses. >> > >> >> I'd like to see more evidence of this. > > C'mon Kirby, students are taught all the time that subtraction and addition are inverses. Same for multiplication and division. They are taught subtraction as the inverse of addition before negative numbers, and division as the inverse of multiplication before reciprocals. (That review of the South Korean curriculum talked about how they really pushed in early elementary the concept that these operations "undo" each other.) >
But like you say, the word "inverse" is used kind of loosely, to mean something like "undo"  actually not even "something like" as "undo" is a good word for it.
Again, my tendency would *not* be to plow into groupoids if looking for college prep material. I'd prefer to talk about when turning back the clock is an option, when "inverting" makes sense. The "undo" feature in word processing is actually relatively new.
You probably see where I'm going with this: version control. What is it and how is it used? Do we use it enough? What if literature majors were to start using cvs and svn for there PhD dissertations. Perhaps some already are? Remember, it was Gene Fowler the poet who rolled his own XML editor in Delphi  self taught all the way.
My point: if math teachers don't see fit to teach these "how things work" topics (what's the Internet? what's XML? what's programming like? what's SQL?), then I think it's time for maybe English majors (or Russian majors, or...) to take it out of their hands. Literacy is literacy, and that includes numeracy (or how else will you understand history, as anything but a series of mindless wars  which is close, but numeracy makes a difference, e.g. helps us appreciate Baghdad's contributions "above the fray" as it were (I'm reading Edwin Black's 'Banking on Baghdad' these days)).
No, it's too important to "just wait" for the NCTM to finally decide these proposals make sense (MAA to the rescue?). That could take until like 2030 at this rate, and we've already skipped a generation or two, wasted a lot of time (ever since Apollo?  1960s/70s was maybe a high water mark for cultural literacy?).
> It's all over the Internet. The word "inverse" is used. For instance Google > addition subtraction inverse > without the quotation marks. Over 1 million hits. And Google > multiplication division inverse > without the quotation marks. Over 6 million hits. >
OK, it's all over the Internet but what does it mean? Is it struggling to articulate something about groupoids? Why not use the word "reversible" more. What functions are reversible? What *operations* are reversible (more generic than functions).
> > >> >> I'm not persuaded that you've done more than define a set >> of >> functions, such as add2, add3, add4, which may be paired >> with inverse >> functions that undo the results of the first. > > Well, in a multiplicative group, the binary operation f(x,y) = xy is not a bijective function. Surjective, yes, but not injective. But hold the value of y constant, denoting this by replacing it with b, and we now do have a provably surjective and injective function g(x) = xb, which to say, if we replace g(x) with a for convention's sake, we can prove that the equation a = xb has a solution and that it is unique for all a,b. >
I've already laid down some video/audio tracks on how I want to edge into the discrete math stuff. Whether I'll be able to adapt what I find on your Wiki pages to fit my needs remains to be seen.
The current plan is to offer more of my stuff through Saturday Academy or perhaps Math Learning Center or ISEPP (isepp.org), in cahoots with other "rad math" teachers (a brand we've developed). Elementary school teachers who come through may not go back to their schools saying "I need to use Python now". That's not the point. It's more like getting a sense of what the job market is like. Robert Hansen could probably do something similar in Florida, taking teachers around his company, explaining what engineers really do and don't do. MIT's Bucciarelli is also good on this, might be assigned reading (or even better, we might have the video/audio of his talk in our downtown theater, ISEPP hosting).
http://controlroom.blogspot.com/2006/05/isepplectureendofseason.html
> What I was getting at before was that that group theorem's proof we all know about can be restated: Applying the definition of a surjective function gives us that we always have a solution in x for any a,b, and applying the definition of an injective function gives us that we always have a solution in x that is unique for any a,b. >
Keep in mind that one way functions exist in the sense that we have an algorithm for multiplying p, q to get N, but then may have no formula for getting back to p, q from that same N. This is how RSA works in that (p,q) are used to derive d from e, visavis N. N becomes your public key, c the message sent to you using it (raised to the e power modulo N), and d your private key for getting m (plaintext message) out of c (ciphertext).
This same algorithm used at the chip level will encode / decode audio/video signal, which is how DirecTV works.
> In fact, these identities enable us to >> see explicitly how it is that in groups division is the >> inverse of multiplication. >> > >> >> On the contrary, in showing the f(a, b) = r and f(c, d) = >> r, where (a, >> b) and (c, d) are not equivalent in the domain, and where f >> stands for >> "multiplication", you are proving that f does not have an >> inverse >> function. > > I should have said unary division is the inverse of unary multiplication. > > Ultimately, it's about cancellation. When cancellation or at least rightcancellation holds, it will be possible to reduce a function of more than one variable that is not injective to a function of one variable that is injective. This uncovers the equivalence of cancelation and injection that exists "under the covers" for that original function of more than one variable when cancellation or at least rightcancellation holds. > >
I'd say ultimately it's about undoing, where I'd anchor this concept in the murky / messy bed of everyday experience  as this is where mathematics actually has foundations if we're to go with Wittgenstein's picture. To overformalize too early is not necessarily a positive feature, so trying to collapse the notion of do/undo into group theory might not be the right cup of tea for everyone in China. A diversity of approaches is indicated (doc talk). Some teachers might want to try the graphicalgeometry way of distinguishing x2 and +2 the way I've suggested, building off preexisting understanding of scaling (resizing: broadcasting / radiating, contracting) and translation (moving sideways in a lattice, angular momentum, motion in a rail car along some track, play head on a tape).
I prefer to avoid reducing mathematics to anything that's purely lexical or algebraic. Bourbaki insisted this was what one *had* to do in the end, but I'm not seeing it as either/or. The "central dogma" of biochemistry is as surely about algorithms (reversible and not) as the theory of functions. Adding energy to the picture, and still calling it "mathematical" is *not* verboten where radical math teachers are concerned.
Kirby

