Tuesday, May 12, 2009

Second Foundation

Second Foundation

A Picture is Worth a Thousand Words


Functions are most often given by formulae, or at least by some method of giving numbers. Functions at this point are symbolic and they don’t have to get very complicated before they become a blur of symbols that can yield a number but very little insight as to what is going on with the function. I want to see what’s “going on”; I want to see what the function is “doing”. I use the word “doing” because it is short and gives an expression of what I mean; how the graph goes up and down as you follow the graph to the right. I suppose I should talk about what I mean by using a lot more words but I’m not going to; I’m going to say that the function or its graph is “doing” something, like increasing or decreasing or neither.

I can spend time describing a dog biologically or I can draw a picture. (I think about Fonzie’s picture of his lost dog.) Graphs were developed because people needed to see functions; graphs were developed to satisfy a need.
I introduce graphs as pictures that give some insight into what the function is “doing”. The graph is secondary to the function and is just one of several ways to give a visual representation of a function.
I point out the obvious fact that graphs were developed by the sighted.



If I have a function given by a formula and I draw a graph of the function by plotting points, I have missed the point of making a graph, so to speak. Plotting points is a static process; my pencil is still as it plots a point. I am just thinking about what is happening at that point. I want to know what the function is “doing”. I want to know how its graph passes through a point.
The robotic plotting of points is not thinking about what might be happening between the points. I had a student who swore his graph was correct until I showed him that his picture didn’t show what the function was doing, which was going to infinity and back between two of the points he had plotted.
Graphing by plotting points (and later by finding where the derivative is zero etc.) is like getting information through torture, you get answers but they are not reliable. Instructors like the plotting points technique because it is easy to teach and they don’t have to understand graphing all that well themselves. Instructors like rote methods, they like “step one, step two”, techniques.
There should be a Teaching Mathematics Oath: Do no harm.
Graphing by plotting points and not by trying to see what the function is “doing” is harmful.

There aren’t a whole lot of things a function can do. It can increase, decrease, neither increase nor decrease or it can change from increasing to decreasing or vice-versa. And there are only three ways a function can increase and three ways it can decrease. Each of these behavior options is clear from the function’s graph.
You sketch the graph of the function before you take derivatives of the function. From a sketch of the graph it is clear where the derivative is positive, where it’s zero and so on. If you want to find the exact point where something happens, say attain a local maximum, then you set things equal to zero. A graph has a “what it’s doing” aspect and a “computational” aspect.

Can Graphs be Skipped?

I went to a lot of grammar schools and hit the third grade knowing only how to print. My school required cursive. I told the teacher that I could communicate quite well printing and could I skip cursive. I was told that I couldn’t skip cursive.
I suppose that graphs could be skipped. I know a lot of students try to get by on cranking out numbers but they don’t make it, well, they shouldn’t make it.
On the other hand there are blind mathematicians and I have heard of a chess master who played 50 games simultaneously while blindfolded.
But unless the student is exceptional, she shouldn’t skip graphs, nor should the instructor.


Testimonials for Graphs
I was reading the proof of a theorem in complex function theory and there was a point that the author claimed was clear. Being a conscientious student, I thought I’d better go through the proof so that it was, indeed, clear. Several hours later the fog had yet to lift.
The author evidently felt that pictures were for wimps and my attempts at leaping my hurdle were technical. At last I drew a graph and the author was right, the proof was clear. Not only was it clear why the theorem was true but the graph pointed the way to a formal proof.

I was reading the famous book on functional analysis by Riesz and Nagy as a way to practice my mathematical French. I came upon a lemma that I could go through and see that it was step by step true but I had no idea why a person would want to know this arcane result.
A few years later I was taking a course in Harmonic Analysis and the instructor proved the Sun Rising in the Mountains Lemma. Lo and behold it was the lemma in Riesz-Nagy. But now the instructor drew a picture of the sun rising in the mountains and not only was the meaning of the lemma clear but so was the proof.

Teaching graphs
Perhaps functions and their graphs should be taught more in parallel than I do and I have considered this. I chose the more linear approach of first functions and then graphs for several reasons. It seems more natural to do functions first. While representative art may have appeared earlier than numerical descriptions of nature, I am trying to describe nature using numbers. As this description becomes more complex it becomes evident that a picture would be helpful.
I am a big believer in presenting things when there is a perceived need.

Once both concepts are on the table, they can be viewed in parallel, a stereopticon effect. The advantage of sight in the communication of information is that sight can receive information in parallel. The expression, “I see”, implies understanding something all at once, understanding a totality.
A blackboard can display several ideas simultaneously. Books are traditionally linear but can have pictures, for example, graphs or photographs. Speech is linear; I can’t envisage saying two different words at the same time. Movies give information linearly and the flash-back is a device used to overcome this limitation.
Since most of the information that the brain processes comes to it linearly, through the ears as speech, much of what comes through the eyes including the written word and television, it isn’t practiced in parallel processing. I would expect that how well the brain can process two streams of information simultaneously is minimal at best.

I put the expression that gives the rule of the function on the blackboard and a graph of the function next to it. The student now has two concepts to process and it is important that the student get a strong connection between the two. When the student sees an algebraic or trigonometric expression, its graph should immediately come to mind; well, maybe not immediately. And if she sees a drawing that looks like a graph she should wonder what kind of function would have a graph that looks like that. If I see the word ‘dog’ I immediately have a mental picture of a dog. If I see y=2sint, I immediately have a mental picture of a wavy curve moving up and down along the x-axis. I also have a mental picture of a rock bouncing up and down on the end of a spring. The more mental pictures of a function I have, the better.
In my calculus book (mathematicsteacher.org) I give a chapter to functions and then a chapter to graphs. Then I have functions and at least one visualization of them.

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