Math STRATEGY needed for Graphing

[TubbaBlubba]

I'm with SezMe here. It is more useful to do it the other way. Start off with real life experiences where they read graphs and gradually introduce the jargon as required.

Ah, I'm not saying you should start by teaching the actual definition, rather that when working out what to teach you should keep the strict definition in mind. For example, there appears to be ambiguities in whether the discussion concerns actual functional graphs or charts.

 

[Earthborn]

What you're saying is generally true up to the level of say, first-year college calculus, and in applied mathematics.

So what I said was true up to a level quite a bit beyond what Truethat's students (or pretty much anyone) has to know? I can live with that.
Was I oversimplifying things to the point of snarkiness? Sure.

Beyond that, the emphasis is typically on the need for strict and rigorous definitions, and on the drawn graph as nothing more than a mental aid at best.

Everything mathematicians squiggle on paper or white/blackboards is a "mental aid".

Try drawing a nice picture representing the tensor product - it's not exactly easy.

When Truethat's students get to that chapter, I am sure she'll appreciate your help.

 

[TubbaBlubba]

So what I said was true up to a level quite a bit beyond what Truethat's students (or pretty much anyone) has to know? I can live with that.
Was I oversimplifying things to the point of snarkiness? Sure.

I just objected to the notion that advanced mathematicians were ubiquitously reliant of graphs to make sense of mathematics. Graphsof some sort still do appear in advanced mathemathics, but they almost always serve a completely pedagogic purposes of limited effectiveness. Plus, they can even cause trouble. Possibly a legend, but Euclid supposedly only understood a product to be an area or a volume, rather than a relationship between numbers, preventing him from computing them with more than three factors.

Everything mathematicians squiggle on paper or white/blackboards is a "mental aid".

No, in a proper axiomatic system, numerals, variables and operators are more significant than that. They summon mathematical entities; they have a life they live, and play by a set of rules. They may end up playing in a way that you could never imagine, never foresaw and may ever understand. When figures are written, theorems are proved, lemmas are shown... the is more that happens then than some scruffy mathematician writing down things to keep in mind and thinking his way to the answer. The true mathematics happen on the paper, like a film playing out on the silver screen.

Yes, a symbol is far more than a memo. It is the often unambiguous representation of a mathematical creature.

When Truethat's students get to that chapter, I am sure she'll appreciate your help.

I'm speaking to you right now, about the things you wrote. Other people will read and consider your statements in other contexts and may come to misunderstanding about the complexity of advanced mathematics. That is why I made that point.

You said for example

If some mathematician invents/discovers some new superformula, the first thing s/he is going to do is draw it into a graph and say "Aha! So that's what it does!".

That new superformula will probably involve an infinite-dimension tensor or a tripple integral with five differental forms. That's not somethhing you just scetch up. Not that many mathematicians are concerned with new ways to put together various nice elementary functions of a single variable that make a nice Cartesian graph.

 

[TheGnome]

I'm still not sure what is meant by graph in this thread. This might be due to a language barrier. I'm not very fluent in math but I know some Mathematik.

So let me list some possibilities I see:

• A visualization of some mathematical relation or data, like a chart or a diagramm.
• A plot of a real valued function which confusingly enough is not its graph
• What comes to my mind first when I hear graph is the graph as the subject of graph theorie, a thing with vertices and edges that you could use to think abaout e.g. the Seven Bridges of Königsberg.

As I don't know much about how math is taught in truethat's setting or about the vocabulary that is used there I may just be a little clueless.


While I probably don't have much advice to offer, I would like to follow the discussion. So, any help?

 

[TubbaBlubba]

I'm still not sure what is meant by graph in this thread. This might be due to a language barrier. I'm not very fluent in math but I know some Mathematik.

So let me list some possibilities I see:

• A visualization of some mathematical relation or data, like a chart or a diagramm.
• A plot of a real valued function which confusingly enough is not its graph
• What comes to my mind first when I hear graph is the graph as the subject of graph theorie, a thing with vertices and edges that you could use to think abaout e.g. the Seven Bridges of Königsberg.

As I don't know much about how math is taught in truethat's setting or about the vocabulary that is used there I may just be a little clueless.


While I probably don't have much advice to offer, I would like to follow the discussion. So, any help?

I believe truethat has boht the first and second one in mind, but not the third; I've restrited myself to the second.

 

[caveman1917]

Given a (real-valued) function f(x), a graph is the set of all ordered pairs (x, f(x)), and this is often represented by a figure in a Cartesian coordinate system (x,y). Thus, by locating some x in the coordinate system, we can find f(x), and vice versa, provided the function is well-behaved.

Aaaaah, I was reading the thread thinking a graph being the vertices & edges thing, and the thread wasn't making much sense

Just one nitpick then, the function f is the set of ordered pairs (a function is a subset of the Cartesian product of its domain and codomain).

 

[TubbaBlubba]

Aaaaah, I was reading the thread thinking a graph being the vertices & edges thing, and the thread wasn't making much sense

Just one nitpick then, the function f is the set of ordered pairs (a function is a subset of the Cartesian product of its domain and codomain).

As I said: A function f is a set of ordered pairs (a,b) where you have the pairs (a,b) and (a,c) iff (or only if, whichever) b = c. And yes, by this definition, the (x, f(x)) definition of a graph becomes all but the same as the function itself, although the graph is dependent on the existence of said function. You couldn't define a function as the pairs (x, f(x)) since that would be circular.

 

[caveman1917]

As I said: A function f is a set of ordered pairs (a,b) where you have the pairs (a,b) and (a,c) iff (or only if, whichever) b = c.

I didn't see you say that, perhaps I missed it, I'm not going to go looking. My response was under the assumption that you left the function undefined but used the set of pairs as definition of the graph.

ETA: I see I missed your later post before writing my response to the earlier one, apologies.

And yes, by this definition, the (x, f(x)) definition of a graph becomes all but the same as the function itself, although the graph is dependent on the existence of said function.

Does that help anything, though? Irrespective of whether you want to denote the set as (x, f(x)) or (x, y) or {(a, b),(c,d),...} or f, we're still talking about the same set.

You couldn't define a function as the pairs (x, f(x)) since that would be circular.

Depends on how exactly you introduce the notation "f(x)", I think. It seems you could do it non-circularly if you really wanted to, not that I see much reason why one would want to.

 

[TubbaBlubba]

Does that help anything, though? Irrespective of whether you want to denote the set as (x, f(x)) or (x, y) or {(a, b),(c,d),...} or f, we're still talking about the same set.

Not if you already establish that definition of a function, no (although there may be some obscure axiomatic motivation for it), but it's nice to have a definition of graph that's independent of how you define a function, and you might want to have different restrictions on the elements (e.g you might accept pairs of vectors as a function, but not as a graph).

Depends on how exactly you introduce the notation "f(x)", I think. It seems you could do it non-circularly if you really wanted to, not that I see much reason why one would want to.

Yeah, function notation is a whole can of worms in itself.

 

[caveman1917]

but it's nice to have a definition of graph that's independent of how you define a function

There could be many ways to go about defining these, but end of the day they will define the same mathematical object (namely a set of pairs), the difference is only notational. So why not just sidestep the issue and consider a graph a certain notation for a function? We are not restricted to just a couple of boring squiggles as in "f" for what we consider a notational symbol for this set of pairs.

and you might want to have different restrictions on the elements (e.g you might accept pairs of vectors as a function, but not as a graph).

This leads to another issue, such as with 3-dimensional graphs (that are 2-dimensional projections). If g(f) is the graph of a function f, then the following could be true: f \subset R^3, g(f) \subset R^2.

You'd have to do something like mapping f to E^3, projecting to E^2 and then mapping that to R^2.

 

[TubbaBlubba]

There could be many ways to go about defining these, but end of the day they will define the same mathematical object (namely a set of pairs), the difference is only notational. So why not just sidestep the issue and consider a graph a certain notation for a function? We are not restricted to just a couple of boring squiggles as in "f" for what we consider a notational symbol for this set of pairs.

I don't know exactly, but I do know even rigorous and axiomatic mathematical textbooks (e.g. Spivak's) keep graphs and functions separate. I suspect it is partly historical (although Spivak happily does away with historical notation he personally doesn't like), but probably also because of the many subtleties in defining functions in various spaces can be ignored when speaking about graphs.

This leads to another issue, such as with 3-dimensional graphs (that are 2-dimensional projections). If g(f) is the graph of a function f, then the following could be true: f \subset R^3, g(f) \subset R^2.

You'd have to do something like mapping f to E^3, projecting to E^2 and then mapping that to R^2.

Ah, yes, to connect this to what I wrote above: graphs derived from higher-dimensional functions (i.e. projections) might not be functions in their own space! Consider a three-dimensional paraboloid surface (x,y,z) defined by some function z = f(x,y). You can construct another graph, a level surface, in R^2 "all (x,y) such that f(x,y) = some constant". This, of course, is a circle. Which, obviously, is not a function (for almost all x in the set, we have both (x,y) and (x,-y) in the set).

So I really think it's about not having the graphs be restricted in the same way functions are (and yes, this would require an extension of my original definition of a graph above).

 

[caveman1917]

I don't know exactly, but I do know even rigorous and axiomatic mathematical textbooks (e.g. Spivak's) keep graphs and functions separate.

Indeed, I just never really thought about it again.

Ah, yes, to connect this to what I wrote above: graphs derived from higher-dimensional functions (i.e. projections) might not be functions in their own space!

Good point.

So I really think it's about not having the graphs be restricted in the same way functions are (and yes, this would require an extension of my original definition of a graph above).

The extension might get problematic. If for every subset g of R^2 there exists an n such that a function R^n->R f exists that has g as its graph, then you have to accommodate the entire powerset of R^2 in your definition. It's not clear how you're going to specify "graph" as differently from "any g in P(R^2)".

 

[TubbaBlubba]

The extension might get problematic. If for every subset g of R^2 there exists an n such that a function R^n->R f exists that has g as its graph, then you have to accommodate the entire powerset of R^2 in your definition. It's not clear how you're going to specify "graph" as differently from "any g in P(R^2)".

It's probably not necessary to define the term "graph" in a painstakingly specific way anyway since they're hardly used in formal calculations - you just speak of "the graph of a real-valued function" as one set, the "graph of a level surface" as another, etc.

 

[AlaskaBushPilot]

As I don't know much about how math is taught in truethat's setting or about the vocabulary that is used there I may just be a little clueless.

It isn't very difficult to discern. You have students who don't know what a graph is, meaning the examples she gave: bar charts, a line, the four quadrants - there should be no question about what she is doing and the only confusion arises because these are graduate students not competent at grade school level expectations concerning math.

I know what exams are being prepared for, have studied the sample questions, and the exams use the proper terminology. So rejecting the exam terminology for terms like "squiggle" isn't a good strategy.

For one thing, obviously you want to get exam-takers familiar with the terminology they are using on the exam itself. Secondly, this term clearly buys into an anti-math belief system where we trivialize math by choosing pejorative terms specifically implying they are meaningless doodles. Were I allowed to say who these people are, then it would be immediately apparent why it is pretty urgent to extinguish an anti-math attitude.

With the first and second grade graphing my four and five year old encountered, they use pictograms like this (We like Khan Academy):

So I understand the point about gently sliding people into graphing by using the familiar (the apples themselves) instead of symbolic representations to start with. That is how we teach graphing to little boys and girls.

But we are actually teaching them math. We are not coaching them on how to eliminate obviously wrong answers on an exam so they can guess between the remainder. That is the nature of the OP, and we can see the rationalization that despite concealing who these people are, we are assured that math has nothing to do with their jobs and how unfair it is to expect any understanding of grade-school level math out of them. Therefore, under this rationalization, it is appropriate that we coach them on how to make higher-probability guesses instead of understanding math.

These are six year olds, who do not need to have an explanation for what a graph is and why we do it.

And yes, I would show them this instead of making excuses for them. In my teaching career I ended up being thanked by people for cracking the whip and expecting more out of them. This is a very low expectation threshold for adults.

 

[TheGnome]

It isn't very difficult to discern. You have students who don't know what a graph is, meaning the examples she gave: bar charts, a line, the four quadrants - there should be no question about what she is doing and the only confusion arises because these are graduate students not competent at grade school level expectations concerning math.

I know what exams are being prepared for, have studied the sample questions, and the exams use the proper terminology. So rejecting the exam terminology for terms like "squiggle" isn't a good strategy.

For one thing, obviously you want to get exam-takers familiar with the terminology they are using on the exam itself. Secondly, this term clearly buys into an anti-math belief system where we trivialize math by choosing pejorative terms specifically implying they are meaningless doodles. Were I allowed to say who these people are, then it would be immediately apparent why it is pretty urgent to extinguish an anti-math attitude.

With the first and second grade graphing my four and five year old encountered, they use pictograms like this (We like Khan Academy):

So I understand the point about gently sliding people into graphing by using the familiar (the apples themselves) instead of symbolic representations to start with. That is how we teach graphing to little boys and girls.

But we are actually teaching them math. We are not coaching them on how to eliminate obviously wrong answers on an exam so they can guess between the remainder. That is the nature of the OP, and we can see the rationalization that despite concealing who these people are, we are assured that math has nothing to do with their jobs and how unfair it is to expect any understanding of grade-school level math out of them. Therefore, under this rationalization, it is appropriate that we coach them on how to make higher-probability guesses instead of understanding math.

These are six year olds, who do not need to have an explanation for what a graph is and why we do it.

And yes, I would show them this instead of making excuses for them. In my teaching career I ended up being thanked by people for cracking the whip and expecting more out of them. This is a very low expectation threshold for adults.

Ah yes thank you. That gave me some insight into the process. I mean it's probably the same here in Switzerland but it's a long time since I experienced this first hand.

Actually I was thinking along these lines until TubbaBlubba introduced the notion of the graph of a function which definitely is not a pictorial presentation of a function. I think this is cleared up in my mind now.

 

[caveman1917]

It's probably not necessary to define the term "graph" in a painstakingly specific way anyway since they're hardly used in formal calculations

Indeed, which brings me back to my earlier point about why even bother attempting to define it rather than just saying it's a symbolic representation of a function and be done with it.

 

[TubbaBlubba]

Indeed, which brings me back to my earlier point about why even bother attempting to define it rather than just saying it's a symbolic representation of a function and be done with it.

I guess because it's nice to be able to speak of the graph of, say, f(x) = sin(1/x) despite the fact that it is impossible to make anything resembling a symbolic representation of it around the y-axis. And to keep different graphs distinct.

 

[TubbaBlubba]

I guess because it's nice to be able to speak of the graph of, say, f(x) = sin(1/x) despite the fact that it is impossible to make anything resembling a symbolic representation of it around the y-axis. And to keep different graphs distinct.

Ah, I forgot that Spivak actually dedicates a whole chapter to graphs in Calculus.

Here's a few tidbits:

... we ca draw a function by drawing each of the pairs in the function. The drawing obtained in this way is called the graph of the function. In other words, the graph of f contains all the points corresponding to pairs (x,f(x)).

[He proceeds to an informal demonstration of the fact that functions of the form f(x) = cx are straight lines - TB]

This demonstration has neither been labeled nor treated as a formal proof. ... The rigorous proof of any statement connecting geometric and algebraic concepts would first require a real proof (or a precisely stated assumption) that the points on a straight line correspond in an exact way to the real numbers. ... We shall use geometric pictures only as an aid to intuition; for our purposes (and for most of mathematics) it is perfectly satisfactory to define the plane to be the set of all pairs of real numbers and to define straight lines as certain collections of pairs, including, among others, the collections {(x, cx) : x a real number}.

[More notions on the properties of graphs and functions follow. He considers circles, ellipses etc to be composed of graphs, not graphs themselves]

The gist of it is: We want the pictures as mental aids, but we also want to be able to speak of graphs in formal settings and to distinguish "graphs of a function" from other geometric figures. Proofs connecting geometry and analysis are incredibly difficult, so we're better off having a set-theoretic, rather than a geometric notion of a graph.

So essentially, I think you've got the right idea. It is worth noting that nowhere in the chapter does Spivak invoke his formal notation for definition, as he does for functions, limits, continuitiy, etc.

 

[caveman1917]

Ah, I forgot that Spivak actually dedicates a whole chapter to graphs in Calculus.

I checked the one we used, Calculus: A Complete Course (Adams & Essex), and it essentially has just one sentence on it

A graph of an equation (inequality) is the set of points P(x,y) that satisfy that equation (inequality).

It also has a bit on coordinate systems in the plane to get to the notion of "point", but it does all this informally.

 

[TubbaBlubba]

My single variable calculus coursebook didn't even have an epsilon-delta definition of the limit in it, and also claimed that 3.1415... was a "limit of pi". So... yeah. Lots if variation there... it had good exercises.

Spivak's text (as well as his Calculus on Manifolds) is often considered the single best calculus textbook there is, and also provides extensive guidance in how to actually do real math, define things, prove theorems, etc. Whenever I do assignments, usually in my physics courses, I've noticed that I find it extremely easy to chunk out a precise definition of whatever vague object the assignment wants me to treat, a skill I think I picked up from that book.