OK, I know this is a stupid question so please go easy. What I would like to know is if the differentiation operator is usually the first operator that students encounter, or is there one (or more) they are already familiar with?
Thanks.  Timotha  End of Forwarded Message 
The inverse of a function is an operator.
Bob Hansen
On 5/22/15, 9:59 AM, "Timotha Trigg" <[hidden email]> wrote:

In reply to this post by Timotha Trigg
On Fri, 22 May 2015 07:59:59 0600, Timotha Trigg <[hidden email]>
wrote: > OK, I know this is a stupid question so please go easy. What I would > like to know is if the differentiation operator is usually the first > operator that students encounter, or is there one (or more) they are > already familiar with? > > Thanks. > >  Timotha > >  End of Forwarded Message > If you mean the first operator that they are encouraged to think of as something that transforms functions into other functions, it probably is. But it isn't the first such that they encounter. For just one example, there are the translation operators, which transform the function f: x > f)x) into the function f_h: x > f(x  h). While they encounter such operators, they're rarely asked to think of them as "operators". Given the difficulty the notion of "operator" seems to cause even calculus students, that's probably wise. Part of the trouble probably arises from our premature (in the sense of placement in curriculum) formalization of realvalued functions of a single real variable as being certain sets of ordered pairs of numbers.  Louis A. Talman Department of Mathematical and Computer Sciences Metropolitan State University of Denver <http://rowdy.msudenver.edu/~talmanl>  End of Forwarded Message 
In reply to this post by Timotha Trigg
On Fri, May 22, 2015 at 6:59 AM, Timotha Trigg <[hidden email]> wrote: OK, I know this is a stupid question so please go easy. What I would like to know is if the differentiation operator is usually the first operator that students encounter, or is there one (or more) they are already familiar with? I think you might be implying something similar to what Joe did: that there might be a "right answer" to this question. Whereas in fact we have no global meanings, only "language games" (Wittgenstein), partially overlapping, with local meanings. [ What you mean by "pawn" depends on the context and if I immediately leap to the conclusion that I under stand you, I may be setting myself up for some amusing misunderstandings when I come to your "pawn shop" and there isn't any, hah hah  you have to buy whole chess sets, not individual pawns, at WalMart ] Maths do a good job of propagating broad agreement, but in Python = is an operator, the assignment operator, used for binding a name to an object. In most math textbooks, = is not so specifically defined and == for strict Boolean equality, a comparator, like <, >, =< and =>, is for us also an operator. All of these would be familiar to a 3rd grader in many STEM programs. That's six operators already. I turned my reply to Joe's post into a blog post, re "game": http://worldgame.blogspot.com/2015/05/beunusualitsok.html As for differentiation, it's cool that Spivak in 'Calculus on Manifolds' (one of my textbooks at Princeton, Dr. Thurston my prof) is clear that D(f) is eating a function and returning a function. Functions are objects that may be fed to other functions. We do that in Pythonic Math (part of Gnu Math) too: use functions as arguments to other functions, or to operators. That can be somewhat mindblowing to the average poor slob USAer in Lower48, where such maths are still verboten (except in Michigan?). Signing up for maths via Distance Learning will be mindblowing then. Welcome to 'Intro to Lambda Calc' (not your grandfather's high school Calc), which will help you with Differential Calc as well. Lambda Calc and Delta Calc I call 'em, sometimes using the Greek letters. But I don't expect this usage to gain much currency outside of my exclusive Quaker channels, at least not at first. Kirby 
In reply to this post by Robert Hansen
On Fri, May 22, 2015 at 8:49 AM, Robert Hansen <[hidden email]> wrote:
Here you get the standard sage and pious nonsense from our sageinresidence. Every list should have at least one of these clowns. Kirby 
In reply to this post by Louis Talman
Given the difficulty the notion of "operator" seems to cause even calculus students, that's probably wise. Part of the trouble probably arises from our premature (in the sense of placement in curriculum) formalization of realvalued functions of a single real variable as being certain sets of ordered pairs of numbers. Here I'd say Lou has already bought into a specific namespace, as known to him as water to a fish. He's a native here. But in Alien Math, we say: . A = 10 is using "the assignment operator". +  / * are the "four operators" used in typical arithmetic. Another operator, %, is usually not taught in Lower48 which is another reason the colleges are mostly what we'd call "remediation academies" in translation. Typical REPL (dialog with an interpreter  "like Eliza" says Wayne): . 150 % 75 0 . 48394579717912 % 4884 208 Kirby 
In reply to this post by Timotha Trigg
Lou Talman says:
>Part of the trouble probably arises from our premature (in the sense of placement in curriculum) formalization of realvalued functions of a single real variable as being certain sets of ordered pairs of numbers. How so? I've heard you say similar things (all directed at functions as ordered pairs) but I don't appreciate the nature of the problem you see. In fact, I would think the abstraction would actually help, but not having learned it say, from the perspective you would prefer, that's just a guess. Cheers, Joe N  End of Forwarded Message 
In reply to this post by kirby urner4
On 5/22/15, 2:07 PM, "kirby urner" <[hidden email]> wrote:
On Fri, May 22, 2015 at 8:49 AM, Robert Hansen <[hidden email]> wrote: What’s wrong? I didn’t use python?
Bob Hansen

In reply to this post by Timotha Trigg
Robert Hansen wrote:
http://mathforum.org/kb/message.jspa?messageID=9779516 > The inverse of a function is an operator. You might be thinking of "relation". The typical usage of the term "operator" is that it's just a function, except that its domain (and often its range as well) are a little more exotic than the real numbers (or ndimensional Euclidean space). Thus, there are operators defined on matrices (e.g. the adjoint operator, the exponential operator, the transpose operator, etc.), operators defined on appropriate sets of functions (e.g. various differentiation operators, various integration operators, the translation operator that Louis Talman mentioned, etc.), operators defined on subsets of a topological space (the closure operator, the interior operator, the derived set operator, the frontier operator, etc.), and other kinds, but in all cases we're still talking about relations that are functions. That is, each of these operators associates to each object in its domain exactly one object in its range. So why say "operator" instead of "function"? I think it's just convention, which began before the word "function" began to take on the abstract meaning it now has (in higher level math, at least), back when "function" meant the kinds of things you study in calculus (single variable and multivariable). Incidentally, people even work with operators whose domain and range are various sets of operators. See my comments here: http://math.stackexchange.com/questions/372429/isthereasuchthingasanoperatorofoperatorsinmathematics Dave L. Renfro  End of Forwarded Message 
In reply to this post by Timotha Trigg
Kirby says:
>I think you might be implying something similar to what Joe did: that there might be a "right answer" to this question. Talkin' about me? I don't follow. How/when did I do that? Cheers, Joe N  End of Forwarded Message 
In reply to this post by Joe Niederberger
On Fri, 22 May 2015 12:49:29 0600, Joe Niederberger
<[hidden email]> wrote: > Lou Talman says: >> Part of the trouble probably arises from our premature (in the sense of >> placement in curriculum) formalization of > realvalued functions of a single real variable as being certain sets of > ordered pairs of numbers. > > How so? I've heard you say similar things (all directed at functions as > ordered pairs) but I don't appreciate the nature of the problem you see. > In fact, I would think the abstraction would actually help, but not > having learned it say, from the perspective you would prefer, that's > just a guess. > > Cheers, > Joe N > >  End of Forwarded Message > The issue is unnecessary formalization that tends, for the beginner, to obscure rather than illuminate. At what point of, say, the calculus curriculum, is any use made of the nature of a function as a set of ordered pairs? If there is none, then why introduce the definition at that level or before?  Louis A. Talman Department of Mathematical and Computer Sciences Metropolitan State University of Denver <http://rowdy.msudenver.edu/~talmanl>  End of Forwarded Message 
In reply to this post by Dave L. Renfro
I was thinking of “a function from a function” to distinguish from “a value from a value”.
Bob Hansen
On 5/22/15, 4:21 PM, "Dave L. Renfro" <[hidden email]> wrote:

In reply to this post by Timotha Trigg
Lou Talman says:
>The issue is unnecessary formalization that tends, for the beginner, to obscure rather than illuminate. At what point of, say, the calculus curriculum, is any use made of the nature of a function as a set of ordered pairs? If there is none, then why introduce the definition at that level or before? Oh, reconsidering, we're probably not so far apart. My 60s "New Math" showed little cartoons of "function machines" and whatnot, but didn't really delve into formal definitions of ordered pairs. The emphasis was always on the "toone" requirement, and the fact that one has "sets" (informal) for the domain and range (or codomain). Never really thought much about the formalities of it all till Halmos' "Naive Set Theory". I liked the way he treaded carefully around the issue of some people insisting their be some kind of "active agent" (not his words) to functions. I know your favorite formulation is to speak of a "rule" that gives one element of the range for each element of the domain; I like the word "associate" better, but its of little import I think. I also like simple diagrams employing single headed arrows. Cheers, Joe N  End of Forwarded Message 
In reply to this post by Dave L. Renfro
On Fri, May 22, 2015 at 1:21 PM, Dave L. Renfro <[hidden email]> wrote: << ... >> So why say "operator" instead of "function"? I think it's just Yes, just a convention. Python the math notation (runnable) clearly indicates the equivalence of operators with functions in the sense that A + B is equivalently A.__add__(B), i.e. if A, B are members of domains where "+" makes sense between them, then feed one to the other, and if they're of the same type, which method is picked (__add__) will make no difference. Ditto for *, /,  which are __mul__, __truediv__ and __sub__ respectively. We can also negate. The coder of the type gets to design behavior for each of these key operators, setting the stage for Abstract Algebra concepts later. That's a segue to another usual next topic: "binary versus unary" operators, with  being a paradigm example of one that works in both roles, as 3 and as 03, which are equal ( 3 == 03 ). But then some will argue 3 is not an operator in action but the static member of integer set i.e. 3 is a name, not a verb. These fine points dog the beginner. Kirby

In reply to this post by Louis Talman
I would speculate that the introduction of the notion of function is to distinguish from 'relation'  so that differentiation is manageable at an early stage.
Gary Tupper
On Saturday, May 23, 2015, Louis Talman <[hidden email]> wrote: On Fri, 22 May 2015 12:49:29 0600, Joe Niederberger <[hidden email]> wrote: 
In reply to this post by Joe Niederberger
On 5/22/15, 2:49 PM, "Joe Niederberger" <[hidden email]> wrote:
How so? I've heard you say similar things (all directed at functions as ordered pairs) but I don't appreciate the nature of the problem you see. In fact, I would think the abstraction would actually help, but not having learned it say, from the perspective you would prefer, that's just a guess. I’ve never seen a curriculum in high school where abstraction came out strong. I’ve seen abstraction in high school math, but not where it blossomed like it does in upper or graduate level college classes. I’ve always taken that to mean something about
cognition and the stages of coming of age in such subjects.
Bob Hansen

In reply to this post by Louis Talman
At 08:04 AM 5/23/2015, Louis Talman wrote:
>On Fri, 22 May 2015 12:49:29 0600, Joe Niederberger ><[hidden email]> wrote: > >>Lou Talman says: >>>Part of the trouble probably arises from our premature (in the >>>sense of placement in curriculum) formalization of realvalued >>>functions of a single real variable as being certain sets of >>>ordered pairs of numbers. >> >>How so? I've heard you say similar things (all directed at >>functions as ordered pairs) but I don't appreciate the nature of >>the problem you see.In fact, I would think the abstraction would >>actually help, but not having learned it say, from the perspective >>you would prefer, that's just a guess. >The corrected she is unnecessary formalization that tends, for the >beginner, to obscure rather than illuminate. At what point of, >say, the calculus curriculum, is any use made of the nature of a >function as a set of ordered pairs? If there is none, then why >introduce the definition at that level or before? Because the New Math gurus of SMSG so dictated. Inappropriate formalism where their seminal meeting was held in 1958 at Stanford and Ed Begle did his best to persuade Stanford's universally respected mathematics problemsolving guru, George Polya, to participate in the group. He refused because he viewed it as a completely misguided enterprise and doomed for appropriate eventual failure in most schools with most students. As with the "Moore Method" for college math, he was never addressing all math students at the precollegiate level (even at the collegiate level) much less all students in all programs led by poorly prepared teachers with little interest or ability in mathematics. As with Moore at UT Austin, he was mostly interested in graduatelevel students of Stanford quality but extended that interest to students with exceptional mathematics ability and interest at any level. He recognized full well that being a powerful problem solver requires an extensive "toolbox" of concepts and algorithms. In attacking a new situation, the more parts that can be seen as a special case of something wellknown, the more one's brain can focus attention on genuinely new aspects Think Newton's famous quote on his ability to see further and that was Newton, not ordinary students with ordinary teachers much less worse than ordinary on both counts. Bringing things uptodate, Common CoreMath advocates are misusing Polya to support their "discovery" pedagogy although some recognize the inappropriateness: http://truthinamericaneducation.com/commoncorestatestandards/ccsscontentissuesreviews/commoncorestandardsformathematicalpracticeparti/ Lone voices crying in the wilderness aside, Santayana is alive and well. Wayne  End of Forwarded Message 
In reply to this post by gatupper
I don't think this is something that necessitates the orderedpairs definition at the level of calculus or before. On Sat, 23 May 2015 15:06:30 0600, Gary Tupper <[hidden email]> wrote: I would speculate that the introduction of the notion of function is to distinguish from 'relation'  so that differentiation is manageable at an early stage.  Louis A. Talman Department of Mathematical and Computer Sciences Metropolitan State University of Denver <http://rowdy.msudenver.edu/~talmanl> 
In reply to this post by Timotha Trigg
R Hansen says:
>I was thinking of ?a function from a function? to distinguish from ?a value from a value?. For some sets of functions whose inverse relations are functions (any way you want to say that) then indeed you have an "function inversion operator" as long as the domain (invertible functions of some type) is appropriately specified. I strongly doubt that "all invertible functions" would be a proper set. Sounds too big. Anyone? Cheers, Joe N  End of Forwarded Message 
In reply to this post by Joe Niederberger
On Sat, 23 May 2015 14:45:39 0600, Joe Niederberger
<[hidden email]> wrote: > I know your favorite formulation is to speak of a "rule" that gives one > element of the range for each element of the domain; I like the word > "associate" better, but its of little import I think. I also like simple > diagrams employing single headed arrows. All good, and just fine through most, if not all, of the undergraduate curriculum. But there are still textbooks that insist on more formal definitions that, in my opinion, achieve little more than separate most students from their entirely adequate intuitions. The function concept is, for most students, a difficult one: Witness the centuries it took mathematicians to develop the concept. I think it's a mistake to cloud it with formalization in the first place, and much worse to cloud it with formalization that isn't needed for the work they're doing.  Louis A. Talman Department of Mathematical and Computer Sciences Metropolitan State University of Denver <http://rowdy.msudenver.edu/~talmanl>  End of Forwarded Message 
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