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Representations 2

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Introduction

In a recent post I began a discussion of the mental, physical and mathematical representations of a mathematical object. The discussion continues here. Mathematicians, linguists, cognitive scientists and math educators have investigate some aspects of this topic, but there are many subtle connections between the different ideas which need to be studied.

I don’t have any overall theoretical grasp of these relationships. What I will do here is grope for an overall theory by mentioning a whole bunch of fine points. Some of these have been discussed in the literature and some (as far as I know) have not been discussed.  Many of them (I hope)  can be described as “an obvious fact about representations but no one has pointed it out before”.  Such fine points could be valuable; I think some scholars who have written about mathematical discourse and math in the classroom are not aware of many of these facts.

I am hoping that by thrashing around like this here (for graphs of functions) and for other concepts (set, function, triangle, number …) some theoretical understanding may emerge of what it means to understand math, do math, and talk about math.

The graph of a function

Let’s look at the graph of the function {y=x^3-x}.

What you are looking at is a physical representation of the graph of the function. The graph creates in your brain a mental representation of the graph of the function. These are subtly related to each other and to the mathematical definition of the graph.

Fine points

  1. The mathematical definition [2] of the graph of this function is: The set of ordered pairs of numbers {(x,x^3-x)} for all real numbers {x}.
  2. In the physical representation, each point {(x,x^3-x)} is shown in a location determined by the conventional {x-y} coordinate system, which uses a straight-line representation of the real numbers with labels and ticks.
    • The physical representation makes use of the fact that the function is continuous. It shows the graph as a curving line rather than a bunch of points.
    • The physical representation you are looking at is not the physical representation I am looking at. They are on different computer screens or pieces of paper. We both expect that the representations are very similar, in some sense physically isomorphic.
    • “Location” on the physical representation is a physical idea. The mathematical location on the mathematical graph is essentially the concept of the physical location refined as the accuracy goes to infinity. (This last statement is a metaphor attached to a genuine mathematical construction, for example Cauchy sequences.)
  3. The mathematical definition of “graph” and the physical representation are related by a metaphor. (See Note 1.)
    • The physical curve in blue in the picture corresponds via the metaphor to the graph in the mathematical sense: in this way, each location on the physical curve corresponds to an ordered pair of the form {(x,x^3-x)}.
    • The correspondence between the locations and the pairs is imperfect. You can’t measure with infinite accuracy.
    • The set of ordered pairs {(x,x^3-x)} form a parametrized curve in the mathematical sense. This curve has zero thickness. The curve in the physical representation has positive thickness.
    • Not all the points in the mathematical graph actually occur on the physical curve: The physical curve doesn’t show the left and right infinite tails.
    • The physical curve is drawn to show some salient characteristics of the curve, such as its extrema and inflection points. This is expected by convention in mathematical writing. If the graph had left out a maximum, for example, the author would be constrained (by convention!) to say so.
    • An experienced mathematician or advanced student understands the fine points just listed. A newbie may not, and may draw false conclusions about the function from the graph. (Note 2.)
  4. If you are a mathematician or at least a math student, seeing the physical graph shown above produces a mental image(see Note 3.) of the graph in your mind.
  5. The mathematical definition and the mental image are connected by a metaphor. This is not the same metaphor as the one that connects the physical representation and the mathematical definition.
    • The curve I visualize in my mental representation has an S shape and so does the physical representation. Or does it? Isn’t the S-ness of the shape a fact I construct mentally (without consciously intending to do so!)?
    • Does the curve in the mental rep have thickness? I am not sure this is a meaningful question. However, if you are a sufficiently sophisticated mathematician, your mental image is annotated with the fact that the curve has zero thickness. (See Note 4.)
    • The curve in your mental image of the curve may very well be blue (just because you just looked at my picture) but you must have an annotation to the effect that that is irrelevant! That is the essence of metaphor: Some things are identified with each other and others are emphatically not identified.
    • The coordinate axes do exist in the physical representation and they don’t exist in the mathematical definition of the graph. Of course they are implied by the definition by the properties of the projection functions from a product. But what about your mental image of the graph? My own image does not show the axes, but I do “know” what the coordinates of some of the points are (for example, {(-1,0)}) and I “see” some points (the local maximum and the local minimum) whose coordinates I can figure out.

Notes

1. This is metaphor in the sense lately used by cognitive scientists, for example in [6]. A metaphor can be described roughly as two mental images in which certain parts of one are identified with certain parts of another, in other words a pushout. The rhetorical use of the word “metaphor” requires it to be a figure of speech expressed in a certain way (the identification is direct rather than expressed by “is like” or some such thing.)  In my use in this article a metaphor is something that occurs in your brain.  The form it takes in speech or writing is not relevant.

2. I have noticed, for example, that some students don’t really understand that the left and right tails go off to infinity horizontally as well as vertically.   In fact, the picture above could mislead someone into thinking the curve has vertical asymptotes: The right tail looks like it goes straight up.  How could it get to x equals a billion if it goes straight up?

3. The “mental image” is of course a physical structure in your brain.  So mental representations are physical representations.

4. I presume this “annotation” is some kind of physical connection between neurons or something.  It is clear that a “mental image” is some sort of physical construction or event in the brain, but from what little I know about cognitive science, the scientists themselves are still arguing about the form of the construction.  I would appreciate more information on this. (If the physical representation of mental images is indeed still controversial, this says nothing bad about cognitive science, which is very new.)

References

[1] Mental Representations in Math (previous post).

[2] Definitions (in abstractmath).

[3] Lakoff, G. and R. E. Núñez (2000), Where Mathematics Comes From. Basic Books.

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Sets don't have to be homogeneous?

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Colm Bhandal commented on my article on sets in abstractmath.org.

Let me first of all say that I am impressed with your website. It gave
me a few very good insights into set notation. Now, I’ll get straight
to the point. While reading your page, I came across a section
claiming that:

“Sets do not have to be homogeneous in any sense”

This confused me for a while, as I was of the opinion that all objects in a set were of the same type. After thinking about it for a while, I came to a conclusion:

A set defines a level of abstraction at which all objects are homogeneous, though they may not be so at other levels of abstraction.

Taking the example on your page, the set {PI^2, M, f, 42, -1/e^2} contains two irrational numbers, a matrix, a function, and a whole number. Thus, the elements are not homogeneous from one perspective (level of abstraction as I call it) in that they are spread across four known sets. However, in another sense they are homogeneous, in that they are all mathematical objects. Sure, this is a very high level of abstraction: A mathematical object could be a lot of things,
but it still allows every object in the set to be treated homogeneously i.e. as mathematical objects.

You are right.  I think I had better say “The elements of a set do not have to be ‘all of the same kind’ in the sense of that phrase in everyday speech.”  Of course, a mathematician would say the elements of a set S are “all of the same kind”, the “kind” being elements of S.
 
Apparently, according to the way our brains work, there are natural kinds and artificial kinds.  There is something going on in my students’ minds that cause them to be bothered by sets like that given about or even sets such as {1,3,5,6,7,9,11} (see the Handbook, page 279).   Philosophers talk about “natural kinds” but they seem to be referring to whether they exist in the world.  What I am talking about is a construct in our brain that makes “cat” a natural kind and “blue-eyed OR calico cat” an artificial kind.  Any teacher of abstract math knows that this construct exists and has to be overcome by talking about how sets can be arbitrary, functions can be arbitrary, and so on, and that’s OK.

 This distinction seems to be built into our brains.  A large part of abstractmath.org is devoted to pointing out the clashes between mathematical thinking and everyday thinking. 

Disclaimer:  When I say the distinction is “built into our brains” I am not claiming that it is or is not inborn; it may be a result of cultural conditioning. What seems most likely to me is that our brains are wired to think in terms of natural kinds, but culture may affect which kinds they learn.  Congnitive theorists have studied this; they call them “natural categories” and the study is part of prototype theory.  I seem to remember reading that they have some evidence that babies are born with the tendency to learn natural categories, but I don’t have a reference.

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Mental Representations in Math

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This post is part of the abstractmath article on images and metaphors.  I have had some new insights into the subject of mental representations and have incorporated them in this rewritten version (which omits some examples).  I would welcome comments.

Mathematicians who work with a particular kind of mathematical object have mental representations of that type of object that help them understand it.  These mental representations come in various forms:

  • Visual images,  for example of what a right triangle looks like.
  • Notation, for example visualizing the square root of 2 by the symbol “\sqrt{2}“.  Of course, in a sense notation is also a physical representation of the number.  An important fact:  A mathematical object may be referred to by many different notations. There are examples here and here. (If you think deeply about the role notation plays in your head and on paper you can easily get a headache.)
  • Kinetic understanding, for example thinking of the value of a function as approaching a limit by imagining yourself moving along the graph of the function.
  • Metaphorical understanding, for example thinking of a function such as  as a machine that turns one number into another: for example, when you put in 3 out comes 9.   See also literalism and this post on Gyre&Gimble.

Example

Consider the function h(t)=25-(t-5)^2.   The chapter on images and metaphors for functions describes many ways to think about functions.  A few of them are considered here.

Visual images You can picture this function in terms of its graph, which is a parabola.   You can think of it more physically, as like the Gateway Arch.  The graph visualization suggests that the function has a single maximum point that appears to occur at t = 5.

I personally use visual placement to remember relationships between abstract objects, as well.  For example, if I think of three groups, two of which are isomorphic (for example C_2 and \text{Alt}_3.), I picture them as in different places with a connection between the two isomorphic ones.  I know of no research on this.

Notation You can think of the function as its formula .  The formula tells you that its graph will be a parabola (if you know that quadratics give parabolas) and it tells you instantly without calculus that its maximum will be at (see ratchet effect).

Another formula for the same function is -t^2+10t.   The formula is only a representation of the function.  It is not the same thing as the function.  The functions h(t) and k(t) defined on the real numbers  by h(t)=25-(t-5)^2 and k(t)=-t^2+10t are the same function; in other words, h = k.

Kinetic The function h(t)  could model the height over time of a physical object, perhaps a ball shot vertically upwards on a planet with no atmosphere. You could think of the ball starting at time t = 0 at elevation 0, reaching an elevation of (for example) 16 units at time t = 2, and landing at t = 10.  You are imagining a physical event continuing over time, not just as a picture but as a feeling of going up and down (see mirror neuron).  This feeling of the ball going up and down is attached in your brain to your understanding of the function h(t).

Although h(t) models the height of the ball, it is not the same thing as the height of the ball. A mathematical object may have a relationship in our mind to physical processes or situations but is distinct from them.

According to this report, kinetic understanding can also help with learning math that does not involve pictures.  I know that when I think of evaluating the function  at 3, I visualize 3 moving into the x slot and then the formula  transforming itself into 10.  I remember doing this even before I had ever heard of the Transformers.

Metaphor One metaphor for functions is that it is a machine that turns one number into another.  For example, the function h(t)  turns 0 into 0 (which is therefore a fixed point) and 5 into 25.  It also turns 10 into 0, so once you have the output you can’t necessarily tell what the input was (the function is not injective).

More examples

  • ¨ “Continuous functions don’t have gaps in the graph“. This is a visual image.
  • ¨ You may think of the real numbers as lying along a straight line (the real line) that extends infinitely far in both directions. This is both visual and a metaphor (a real number “is” a place on the real line).
  • ¨ You can think of the set containing 1, 3 and 5 and nothing else in terms of its list notation {1, 3, 5}. But remember that {5, 1,3} is the same set. In other words the list notation has irrelevant features – the order in which the elements are listed in this case.
  • The interior of a closed curve or a sphere is called that because it is like the interior in the everyday sense of a bucket or a house. Note The boundary of a real-life container such as a bucket has thickness, in contrast to the boundary of a closed curve or a sphere. This observation illustrates my description of a metaphor as identifying part of one situation with part of another. One aspect is emphasized; another aspect, where they may differ, is ignored.

Uses of mental representations

Integers and metaphors make up what is arguably the most important part of the mathematician’s understanding of the concept.

  • Mental representations of mathematical objects using metaphors and images are necessary for understanding and communicating about them (especially with types of objects that are new to us).
  • They are necessary for seeing how the theory can be applied. 
  • They are useful for coming up with proofs.

Many representations

Different mental representations of the same kind of object help you understand different aspects of the object.

Every important mathematical object has many representations and skilled mathematicians generally have several of them in mind at once.

New concepts and old ones

We especially depend on metaphors and images to understand a math concept that is new to us.  But if we work with it for awhile, finding lots of examples, and eventually proving theorems and providing counterexamples to conjectures, we begin to understand the concept in its own terms and the images and metaphors tend to fade away from our awareness…

Then, when someone asks us about this concept that we are now experts with, we trundle out our old images and metaphors – and are often surprised at how difficult and misleading our listener finds them!

Some mathematicians retreat from images and metaphors because of this and refuse to do more than state the definition and some theorems about the concept.   They are wrong to do this. That behavior encourages the attitude of many people that

  • mathematicians can’t explain things
  • math concepts are incomprehensible or bizarre
  • you have to have a mathematical mind to understand math

All three of these statements are half-truths. There is no doubt that a lot of abstract math is hard to understand, but understanding is certainly made easier with the use of images and metaphors

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