One of the major obstacles to teaching a topic in mathematics is that the student has little or no idea why they are learning it.
When I was learning the C Language, I never understood pointers because I never knew why they were there. What problem do they solve? Evidently the guy who invented them did it to make programming easier but I didn't see why.
I struggled with epsilon-delta process until we got into 20th Century mathematics; when I saw what the process was used for the concept became trivial.
As a rule, mathematics spring topics on students like a magician pulling a rabbit out of a hat. The rabbit is obscure because why would a person keep a rabbit in a hat.
The part of a method or a theorem that the student sees is the result of lots of thought with the thought removed. It is one thing to see that the steps of a proof lead to a correct result, it is quite another see where the steps came from and why you are proving the result in the first place.
And now we come to graphs.
What are graphs for? They allow a person to use their eyes to help in the understanding of functions. Graphs are of no use to a person blind from birth. Graphs were invented by people who could see.
There are several ways to look at functions visually. Some graphs use the magnitude of the distance between two points, as can be seen, to visually represent a numerical quantity. You see that points A and B are farther apart than points C and D. Another method is to represent a number as a color. This is the method used to tell when steel is hot enough to pour. Film comes in different color temperatures.
So, the point of the graph is to see a function and the basic idea is to represent numbers in the domain and in the range by the size of the separation between two points.
But to understand graphs, the student has to understand functions. If they don't understand functions they can't see a reason for graphs.
The next time you are trying explain graphs, see how many of your listeners can tell you what a function is.
Supposing that the student understands the function, then they are ready for graphs. I suggest Chapters 3,4 and 5 of The Calculus: An Opinion by J. Davis
I think that paper and pencil is the best way to start. I think that the student gets a feeling for drawing the picture of a function. If I want to really look at a face, I will sketch it, not take a Polaroid.
The graphing calculator is of use when you want a more accurate picture of the function like when you want a more exact number you use a calculator.
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