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Operators

Timotha Trigg
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

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Re: Operators

Robert Hansen
The inverse of a function is an operator.

Bob Hansen

On 5/22/15, 9:59 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?

Thanks.

- -Timotha

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Re: Operators

Louis Talman
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  
real-valued 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>

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Re: Operators

kirby urner-4
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?

Thanks.

- -Timotha


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/be-unusual-its-ok.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 mind-blowing 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
mind-blowing 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


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Re: Operators

kirby urner-4
In reply to this post by Robert Hansen


On Fri, May 22, 2015 at 8:49 AM, Robert Hansen <[hidden email]> wrote:
The inverse of a function is an operator.

Bob Hansen

Here you get the standard sage and pious nonsense from our sage-in-residence. 

Every list should have at least one of these clowns.

Kirby


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Re: Operators

kirby urner-4
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 real-valued 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


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



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Re: Operators

Joe Niederberger
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
real-valued 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

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Re: Operators

Robert Hansen
In reply to this post by kirby urner-4
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:
The inverse of a function is an operator.

Bob Hansen

Here you get the standard sage and pious nonsense from our sage-in-residence.  

Every list should have at least one of these clowns.

What’s wrong? I didn’t use python?

Bob Hansen
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Re: Operators

Dave L. Renfro
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 n-dimensional 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 multi-variable).

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/is-there-a-such-thing-as-an-operator-of-operators-in-mathematics

Dave L. Renfro

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Re: Operators

Joe Niederberger
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

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Re: Operators

Louis Talman
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
> real-valued 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>

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Re: Operators

Robert Hansen
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:

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 n-dimensional 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 multi-variable).

Incidentally, people even work with operators whose domain and
range are various sets of operators. See my comments here:
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Re: Operators

Joe Niederberger
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 "to-one" 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

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Re: Operators

kirby urner-4
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
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 multi-variable).

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 0-3, which are equal ( -3 == 0-3 ).  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
 

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/is-there-a-such-thing-as-an-operator-of-operators-in-mathematics

Dave L. Renfro

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Re: Operators

gatupper
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:

Lou Talman says:
Part of the trouble probably arises from our premature (in the sense of placement in curriculum) formalization of
real-valued 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

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Re: Operators

Robert Hansen
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
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Re: Operators

Bishop, Wayne
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 real-valued
>>>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 problem-solving 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 graduate-level 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 well-known, 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 up-to-date, Common Core-Math advocates are misusing
Polya to support their "discovery" pedagogy although some recognize
the inappropriateness:
http://truthinamericaneducation.com/common-core-state-standards/ccss-content-issues-reviews/common-core-standards-for-mathematical-practice-part-i/

Lone voices crying in the wilderness aside, Santayana is alive and well.

Wayne

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Re: Operators

Louis Talman
In reply to this post by gatupper
I don't think this is something that necessitates the ordered-pairs 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.

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:

Lou Talman says:
Part of the trouble probably arises from our premature (in the sense of placement in curriculum) formalization of
real-valued 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>
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Re: Operators

Joe Niederberger
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

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Re: Operators

Louis Talman
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>

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