In my previous post I wrote about the idea of offloading abstraction, the sort of things we do with geometric figures, diagrams (that post emphasized manipulable diagrams), drawing the tree of an algebraic expression, and so on.  This post describes a way to offload chunking.

### Chunking

I am talking about chunking in the sense of encapsulation, as some math ed. people use it.  I wrote about it briefly in [1], and [2] describes the general idea.  I don't have a good math ed reference for it, but I will include references if readers supply them.

Chunking for some educators means breaking a complicated problem down into pieces and concentrating on them one by one.  That is not really the same thing as what I am writing about.  Chunking as I mean it enables you to think more coherently and efficiently about a complicated mathematical structure by objectifying some of the data in the structure.

### Project

This project an example of how chunking could be made visible in interactive diagrams, so that the reader grasps the idea of chunking.  I guess I am chunking chunking.

Here is a short version of an example of chunking worked out in ridiculous detail in reference [1].

Let $f(x)=.0002{{\left( \frac{{{x}^{3}}-10}{3{{e}^{-x}}+1} \right)}^{6}}$  How do I know it is never negative?  Well, because it has the form (a positive number)(times)(something)$^6$.    Now (something)$^6$ is ((something)$^3)^2$ and a square is always nonnegative, so the function is (positive)(times)(nonnegative), so it has to be nonnegative.

I recognized a salient fact about .0002, namely that it was positive: I grayed out (in my mind) its exact value, which is irrelevant.  I also noticed a salient fact about ${{\left( \frac{{{x}^{3}}-10}{3{{e}^{-x}}+1} \right)}^{6}}$ namely that it was (a big mess that I grayed out)(to the 6th power).  And proceeded from there.  (And my chunking was inefficient; for example, it is more to the point that .0002 is nonnegative).

I believe you could make a movie of chunking like this using Mathematica CDF.  You would start with the formula, and then as the voiceover said "what's really important is that .0002 is nonnegative" the number would turn into a gray cloud with a thought balloon aimed at it saying "nonnegative".  The other part would turn into a gray cloud to the sixth, then the six would break into 3 times 2 as the voice comments on what is happening.

It would take a considerable amount of work to carry this out.  Lots of decisions would need to be made.

One problem is that Mathematica doesn't provide a way to do voiceovers directly (as far as I know).  Perhaps you could make a screen movie using screenshot software in real time while you talked and (offscreen) pushed buttons that made the various changes happen.

You could also do it with print instead of voiceover, as I did in the example in this post. In this case you need to arrange to have the printed part and the diagram simultaneously visible.

I may someday try my hand at this.  But I would encourage others to attack this project if it interests them.  This whole blog is covered by the Creative Commons Attribution – ShareAlike 3.0 License", which means you may use, adapt and distribute the work freely provided you follow the requirements of the license.

I have other projects in mind that I will post separately.

### References

1. Abstractmath article on chunking.
2. Wikipedia on chunking

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.)

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.

### Examples

• 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.

## Prechunking

The emerging theory of how the brain works gives us a new language to us for discussing how we teach, learn and communicate math.

### Modules

Our minds have many functionalities.  They are implemented by what I called modules in Math and modules of the mind because I don’t understand very much about what cognitive scientists have learned about how these functionalities are carried out.  They talk about a particular neuron, a collection of neurons, electrical charges flowing back and forth, and so on, and it appears there is no complete agreement about these ideas.

The functions the modules implement are physical structures or activities in the brain.  At a certain level of abstraction we can ignore the mechanism.

Most modules carry out functionalities that are hidden from our consciousness.

• When we walk, the walking is carried out by a module that operates without our paying (much) attention to it.
• When we recognize someone, the identity of the person pops into our consciousness without us knowing how it got there.  Indeed, we cannot introspect to see how the process was carried out; it is completely hidden.

Reasoning, for example if you add 56 and 49 in your head, has part of the process visible to your introspection, but not all of it.  It uses modules such as the sum of 9 and 6 which feel like random access memory.  When you carry the addition out, you (or at least I) are conscious of the carry: you are aware of it and aware of adding it to 9 to get 10.

Good places to find detailed discussion of this hiddenness are references [2] and [4] below.

### Chunking

Math ed people have talked for years about the technique of chunking in doing math.

• You see an algebraic expression, you worry about how it might be undefined, you gray out all of it except the denominator and inspect that, and so on.  (This should be the subject of a Mathematica demo.)
• You look at a diagram in the category of topological spaces.  Each object in the diagram stands for a whole, even uncountably infinite, space with lots of open and closed subsets and so on, but you think of it just as a little pinpoint in the diagram to discover facts about its relationship with other spaces.  You don’t look inside the space unless you have to to verify something.

Students have a hard time doing that.  When an experienced mathematician does this, they are very likely to chunk subconsciously; they don’t think, “Now I am chunking”.  Nevertheless, you can call it to their attention and they will be aware of the process.

There are modules that perform chunking whose operation you cannot be aware of even if you think about it.  Here are two examples.

Example 1. Consider these two sentences from [2], p. 137:

• “I splashed next to the bank.”
• “There was a run on the bank.”

When you read the first one you visualize a river bank.  When you read the second one you visualize a bank as an institution that handles money.  If these two sentences were separated by a couple of paragraphs, or even a few words, in a text you are likely not to notice that you have processed the same word in two different ways.  (When they are together as above it is kind of blatant.)

The point is the when you read each sentence your brain directly presents you with the proper image in each case (different ones as appropriate).  You cannot recover the process that did that (by introspection, anyway).

Example 2. I discussed the sentence below in the Handbook.  The sentence appears in references [3].

…Richard Darst and Gerald Taylor investigated the
differentiability of functions $latex f^p$ (which for our
purposes we will restrict to $latex (0,1)$) defined for
each $latex p\geq1$ by

In this sentence, the identical syntax $latex (a,b)$ appears twice; the first occurrence refers to the open interval from 0 to 1 and the second refers to the GCD of integers m and n.  When I first inserted it into the Handbook’s citation list, I did not notice that (I was using it for another phenomenon, although now I have forgotten what it was).  Later I noticed it.  My mind preprocessed the two occurrences of the syntax and threw up two different meanings without my noticing it.

Of course, “restricting to (0, 1)” doesn’t make sense if (0, 1) means the GCD of 0 and 1, and saying “(m, n) = 1doesn’t make sense if (m, n) is an interval.  This preprocessing no doubted came to its two different conclusions based on such clues, but I claim that this preprocessing operated at a much deeper level of the brain than the preprocessing that results in your thinking (for example) of a topological space as a single unstructured object in a category.

This phenomenon could be called prechunking.  It is clearly a different phenomenon that zooming in on a denominator and then zooming out on the whole expression as I described in [1].

### This century’s metaphor

In the nineteenth century we came up with a machine metaphor for how we think.  In the twentieth century the big metaphor was our brain is a computer.  This century’s metaphor is that of a bunch a processes in our brain and in our body all working simultaneously, mostly out of our awareness, to enable us to live our life, learn things, and just as important (as Davidson [4] points out) to unlearn things.  But don’t think we have Finally Discovered The Last Metaphor.

### References

1. Zooming and chunking in abstractmath.org.
2. Mark Changizi, The vision revolution.  Benbella Books, 2009.
3. Mark Frantz, “Two functions whose powers make fractals”.  American Mathematical Monthly, v 105, pp 609–617 (1998).
4. Cathy N. Davidson, Now you see it.  Viking Penguin, 2011.  Chapters 1 and 2.
5. Math and modules of the mind (previous post).
6. Cognitive science in Wikipedia.
7. Charles Wells, The handbook of mathematical discourse, Infinity Publishing Company, 2003.

## Playing with Riemann Sums

I had a satori [Note 2].  I felt like the guy in the ads who sits in front of his new ultrafast computer with the wind blowing his hair back and bracing himself by holding onto the desk.  (My hair was dark then but I certainly was not wearing a tie.)

That convergence theorem was talking about something BIG.

I visualized a Cloud of Riemann Sums floating around and swerving closer to the Right Answer as their meshes decreased.

A Riemann Sum has a lot of parameters:

• Its mesh.  This can be any positive real number.
• Its choice of subintervals. Any positive integer!  There can be billions of subintervals.
• And, ye gods, the individual choice of each evaluation point for each subinterval in each Riemann Sum

Those are three independent parameters, except for the constraint imposed by the mesh on each choice of subintervals.  [Note 3].

I tell my students that we have to zoom in and zoom out [Reference 2] from a problem.  When we zoom out a complicated structure is thought of as a point in a certain relationship with other structures-as-points.  Then to understand something we zoom in (selectively) to see the details that make it work.  What I remember from my satori is that I didn’t visualize them as points but rather as little blurs, sort of like the blurs in Mumford’s red book [Reference 3], which I think was the first non-constipated math text I had ever seen.

### Riemann Sums in Mathematica

In the nineties, I was on a grant to create Mathematica programs for students, and one of the notebooks I created allowed you to easily exhibit Riemann sums with various parameters.  I also included code that would show a cloud.

Below is a cloud.  It is a plot of the values of 300 Riemann sums for $latex \int_0^{\pi} \sin x \,dx$.  They have randomly chosen meshes from $latex 0$ to $latex \pi/2$ and the subintervals and individual evaluation points for each subinterval are also chosen randomly.

The cloud below is a plot of the values of 300 Riemann sums for the area of the upper right quarter circle of radius 2 with center at origin.  Its meshes range from 0 to 1, and other properties are similar to the one above.  The vertical spread of the points is considerably bigger,  presumably because of the vertical tangent line at the right hand end of the integral.

When you click on the code for either of these you get a different cloud with the same parameters.

You can access the notebook containing the code for this via Abmath Gate.    Be sure to read the ReadMe file.

### Notes

[1] This was 1961.  Of course the book didn’t say things such as “with any choice of points-to-evaluate-at”.  It said what it had to say in stilted academic prose which required reading it two or three times before understanding it.  Academic prose is much better these days.  See Reference [1].

I was quite good at reading complicated prose. My ACT scores were a tad higher in English or Language or whatever it is called that they were in Math.  With the Internet, math exposition should do much more with pictures, interactive things, and lots of examples (which don’t waste paper now).  But that is another diatribe…

[2] This is a snooty word for lightbulb flashing over your head.  Every once in awhile I give in to the temptation to use some obscure word to impress people as to the variety of things I know about.  Teachers, don’t do this to your students.  Other professors are fair game.

[3] The same choice of subinterval can correspond to many different meshes, if your definition of mesh requires only that each subinterval be narrower than the mesh, rather than requiring that the mesh be the size of the biggest subinterval.  (I had never thought about that until I wrote this.)

[4] The Mathematica Demonstrations website has several other notebooks that exhibit Riemann Sums.

### References

[1]  The Revolution in Technical Exposition II, post on this blog.

[2]  Zooming and Chunking in abmath.

[3] D. Mumford, The Red Book of Varieties and Schemes (second expanded ed.), Springer Lecture Notes in Math 1358, Springer-Verlag, Berlin, 1999.   (I have not seen this edition.  What I remember is the Red Book as it was in the 1967 Algebraic Geometry Summer School at Bowdoin.  I hope the smudges survive in the new version.  As I remember the smudges were bigger for points that were more generic or something like that.  Those smudges caused me a kind of sartori, too.)