User:Devon McCormick/VisualVsSymbolicUnderstanding

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The following examples contrast a visual style of teaching and learning against an intrinsically different symbolic way. This is taken from a five-minute talk given at a Future Salon Meetup in New York City in November, 2008.

Visual versus Symbolic Understanding

Here we illustrate the difference between two dissimilar ways of understanding by providing an example of each. First we look at some crude comics which attempt to explain what an index fund is; this should be a fairly esoteric topic for most people but we try to make it more understandable by picturing things about an index fund, like the market in which it operates and how it is composed of some of the most important companies in a market.

We contrast this visual understanding with a different kind of understanding: symbolic understanding. The example we give for symbolic understanding is again a fairly esoteric topic: we explain what it means to transpose multi-dimensional and how by demonstrating how easy it is to reason about this using symbols in a system with simple grammar and syntax.

Example of Visual Understanding

This is the first page of an example of learning and teaching visually. In this example, I'm attempting to explain some basic ideas about the stock market, starting with an illustration of what an index fund is.

Continuing our explanation, we now relate the idea of an index fund to equity markets and relate how the performance of an asset is driven largely by the over-all direction of the market of which it is part.


Next, we begin to elaborate on some related themes:

1. How market movements reflect the collective judgment of all its participants.

2. The uncertainty of the past in predicting future performance versus

3. the certainty of fees detracting from total performance.


Example of Symbolic Understanding

To contrast with these visual concepts, we'll now consider a concept so irrevocably abstract that we cannot grasp it visually but must rely on symbols to convey the idea.

Here, we attempt to explain the notion of dyadic transpose of a multi-dimensional array. This is the re-arrangement of the axes of a multi-dimensional array while retaining the relative ordering of elements within the array (subject to this re-arrangement).

We start by explaining the simpler case of monadic transpose, which is the simple reversal of the axes of an array, starting with the simplest one for which this makes a difference - a two-dimensional array, often known as a matrix or a table.

For a two-dimensional array, monadic transpose switches the rows and columns as shown here by comparing the original matrix (column 1) to its transpose in column 2: think of it as rotation about an axis on the major diagonal. We use the symbol "|:" to indicate transpose.


Now we demonstrate how dyadic transpose, in this simplest interesting case, either leaves the matrix the same (column 1) or performs the same as monadic transpose (column 2), depending on the axis ordering specified by the left argument to the transpose function |:.


We continue by considering the simplest interesting case for dyadic transpose by applying it to a three-dimensional array. This contrasts the single result available from monadic transpose to all of the five possible new axis configurations available from dyadic transpose.

Note that monadic transpose is equivalent to the 2 1 0 |: transpose (furthest right below) because this reverses the order of the axes. All possible transposes of a 3D array.jpg

Here, we extend the explanation to a four-dimensional array by considering how the axis ordering specified by the left argument affects the shape of this array. We've chosen an array with a different size for each axis to clarify the result. This examination of the affect of dyadic transpose on the shape of the 4-D array refers to the first example shown in the illustration following.


Here, we illustrate two different dyadic transposes on a four-dimensional array.


Note that, although we do use visual examples to explain this concept, it becomes increasingly difficult to illustrate as the number of dimensions increases. An understanding of the symbolic explanation however, is easily extended to much higher dimensions.

The Relation between the two kinds of understanding

These two forms of understanding complement each other insofar as the explanation of each above used both techniques: the section on symbolic understanding used a lot of visual examples to provide a way of visualizing what it means to transpose an array in all possible orientations but the visual examples become increasingly unwieldy for much higher dimensional arrays whereas the symbolic understanding extends well to these higher dimensions.

Similarly, the example of visual understanding relied on written word and some comic conventions for indicating motion and such; these conventions become symbolic as they become well accepted.