Note: To manipulate the diagrams in this post and in most of the files it links to, you must have Wolfram CDF Player installed on your computer. It is available free from the Wolfram website.
The diagram above shows you the tangent line to the curve $y=x^3-x$ at a specific point. The slider allows you to move the point around, and the tangent line moves with it. You can click on one of the plus signs for options about things you can do with the slider. (Note: This is not new. Many other people have produced diagrams like this one.)
I have some comments to make about this very simple diagram. I hope they raise your consciousness about what is going on when you use a manipulable demonstration.
Farming out your abstraction load
A diagram showing a tangent line drawn on the board or in a paper book requires you visualize how the tangent line would look at other points. This imposes a burden of visualization on you. Even if you are a new student you won't find that terribly hard (am I wrong?) but you might miss some things at first:
- There are places where the tangent line is horizontal.
- There are places where some of the tangent lines cross the curve at another point. Many calculus students believe in the myth that the tangent line crosses the curve at only one point. (It is not really a myth, it is a lie. Any decent myth contains illuminating stories and metaphors.)
- You may not envision (until you have some experience anyway) how when you move the tangent line around it sort of rocks like a seesaw.
You see these things immediately when you manipulate the slider.
Manipulating the slider reduces the load of abstract thinking in your learning process. You have less to keep in your memory; some of the abstract thinking is offloaded onto the diagram. This could be described as contracting out (from your head to the picture) part of the visualization process. (Visualizing something in your head is a form of abstraction.)
Of course, reading and writing does that, too. And even a static graph of a function lowers your visualization load. What interactive diagrams give the student is a new tool for offloading abstraction.
You can also think of it as providing external chunking. (I'll have to think about that more…)
Simple manipulative diagrams vs. complicated ones
The diagram above is very simple with no bells and whistles. People have come up with much more complicated diagrams to illustrate a mathematical point. Such diagrams:
- May give you buttons that give you a choice of several curves that show the tangent line.
- May give a numerical table that shows things like the slope or intercept of the current tangent line.
- May also show the graph of the derivative, enabling you to see that it is in fact giving the value of the slope.
Such complicated diagrams are better suited for the student to play with at home, or to play with in class with a partner (much better than doing it by yourself). When the teacher first explains a concept, the diagrams ought to be simple.
- The Definition of derivative demo (from the Wolfram Demonstration Project) is an example that provides a table that shows the current values of some parameters that depend on the position of the slider.
- The Wolfram demo Graphs of Taylor Polynomials is a good example of a demo to take home and experiment extensively with. It gives buttons to choose different functions, a slider to choose the expansion point, another one to choose the number of Taylor polynomials, and other things.
- On the other hand, the Wolfram demo Tangent to a Curve is very simple and differs from the one above in one respect: It shows only a finite piece of the tangent line. That actually has a very different philosophical basis: it is representing for you the stalk of the tangent space at that point (the infinitesimal vector that contains the essence of the tangent line).
- Brian Hayes wrote an article in American Scientist containing a moving graph (it moves only on the website, not in the paper version!) that shows the changes of the population of the world by bars representing age groups. This makes it much easier to visualize what happens over time. Each age group moves up the graph — and shrinks until it disappears around age 100 — step by step. If you have only the printed version, you have to imagine that happening. The printed version requires more abstract visualization than the moving version.
- Evaluating an algebraic expression requires seeing the abstract structure of the expression, which can be shown as a tree. I would expect that if the students could automatically generate the tree (as you can in Mathematica) they would retain the picture when working with an expression. In my post computable algebraic expressions in tree form I show how you could turn the tree into an evaluation aid. See also my post Syntax trees.
This blog has a category "Mathematica" which contains all the graphs (many of the interactive) that are designed as an aid to offloading abstraction.