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In this chapter, I say something about mental representations (metaphors
and images) in general, and provide examples of how metaphors and images help
us understand math and how they can confuse us.
Pay special attention to the section
called two levels! The distinction made there is
vital but is often not made explicit.
There are other kinds of representations used in math, discussed here.
Contents
Mental
representations are imperfect
Uses of mental representations
Images and
metaphors on this website
Two levels of images and metaphors
Metaphors
and images are dangerous
Mental
representations are physical representations
Images
and metaphors
We think and talk about our experiences of the
world in terms of images and metaphors based on immediate physical
experience. These are our mental representations of our
experiences.
Example of images
We know what a pyramid looks like. But when we refer to the government’s food
pyramid we are not talking about actual food constituting a pyramidal
pile. We are talking about a visual image of the pyramid. In the study of literature, the word “image”
is used in a more general way, to refer to an evocation of a sensory experience
(more).
Examples
of metaphors
We know by direct
physical experience what it means to be warm or cold.
We use these words as metaphors in many
ways:
¨ We refer to a person as having a warm or cold personality. This has nothing to do with their body temperature.
¨ When someone is on a treasure hunt we may tell them they are “getting warm”, even if they are hunting outside in the snow.
Terminology
The metaphors mentioned above involving “warm” and “cold” evoke a sensory experience, and so could be called an image as well. In math education, the phrase “concept image” means the mental structure associated with a concept, so there may be no direct connection with sensory experience. In abstractmath.org, I will usually use the phrase “metaphors and images” to talk about all our mental representations, without trying for fine distinctions.
Mental representations are imperfect
One basic fact
about metaphors and images is that they apply only to certain aspects of the situation.
When someone is getting physically warm we would expect them to start
sweating. But if they are getting warm
in a treasure hunt we don’t expect
them to start sweating. We don’t expect
the food pyramid to have a pharaoh buried underneath it, either.
Our brains handle
these aspects of mental representations easily and usually without our being
conscious of them. They are one of the primary ways we
understand the world.
Images and
metaphors in math
Types of representations
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 many forms. Some of them are listed here, but don’t take
the classification too seriously.
¨ Visual images, for example of what a right triangle looks like.
¨
Notation, for example visualizing the square root of 2
by the symbol
“ ”.
¨ 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
Example
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 .
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 . The formula
is only another representation of the function.
It is not the same thing as the function. The functions h(t) and k(t) defined on
by
and
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 =
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.
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 about types of representations
Visual images
The graph of a function is a visual image, and so is a picture of a rectangle.
I personally use visual images to remember relationships
between abstract objects, as well. For example,
if I think of three groups, two of which are isomorphic (for example and
), I picture them as in three different places
with a connection between the two isomorphic ones. I know of no published research on this kind
of imagery.
Notation
It is not wrong to think of a particular notation of a math
object when you think of the object.
When I think of the square root of 2,
I think of the symbol “ ”. But
it is a very bad idea to think of the symbol as the same thing as the object. A mathematical object may be referred
to by many different
notations. For example the square root of 2 can be represented as
. See a function
example here and set examples here and here
Of course, in a sense notation is both something you visualize in your head and also a physical representation of the object. In fact notation can also be thought of as a mathematical object in itself (common in mathematical logic and in theoretical computing science.) If you think about what notation “really is” a lot, you can easily get a headache…
Kinetic
Our example above
involved a picture (graph of a function).
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
Metaphors are treated in detail below and in the chapter on images and metaphors for functions. See also literalism and this post on Gyre&Gimble.
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
Mental representations 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. (See example
below.)
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
But images and metaphors are also dangerous (see below).
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.
Images and metaphors on this website
This website has many examples of useful mental
representations. Usually, when a chapter
discusses a particular type of mathematical object, say rational numbers, there
will be a subhead entitled Images
and metaphors for rational numbers. This will suggest ways of thinking about them that many
have found useful.
Two levels of images and metaphors
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Images and metaphors have to be used at two different levels, depending on your purpose.
¨ The rich view is for understanding, applications, and coming up with proofs.
¨ You must limit yourself to the rigorous view when constructing and reading proofs.
Math teachers and texts typically do not make an explicit distinction between these views, and you have to learn about it by osmosis. Teachers and texts do make the distinction implicitly in practice. They will say things like, “You can think about this theorem as …” and later saying, “Now we give a rigorous proof of the theorem.” Abstractmath.org makes this distinction explicit in many places throughout the site.
The rich view
The kind of metaphors and images discussed in the mental
representations section above make math rich, colorful and intriguing to think about.
This is the rich view of math. The rich view is vitally
important.
¨ It is what makes math useful and interesting.
¨ It helps us to understand the math we are working with. More about this.
¨ It suggests applications.
¨ It suggests approaches to proving
things about math.
Example
You expect the ball whose trajectory is modeled by the
function h(t) above
to
slow down as it
rises, so the derivative of h
must be smaller at t = 4 than it is at t = 2. A mathematician might
even say that that is an “informal proof” that . A rigorous proof is given below.
The
rigorous view: inertness
When we are constructing a definition or proof we cannot trust all those wonderful
images and metaphors. Definitions must not use metaphors and proofs must use only logical reasoning based on definitions and previously proved theorems.
For the point of view of doing proofs, math objects must be thought of as inert (or static), like your pet rock. This means they
¨ don’t move or change over time, and
¨ don't interact with other objects, even other mathematical objects.
(See also abstract object).
Rigorous
When mathematicians say things like, “Now we give a rigorous proof…”, part of what they mean is that they have to forget about all the color and excitement of the rich view and think of math objects as totally inert. Like, put the object under an anesthetic when you are proving something about it.
Example
The ball function h(t) does not move or change.
When in rigorous mode, a mathematician
will think of h as a complete mathematical object all
at once, not changing
over time. The
function is the total
relationship of the
input values of the parameter t to the output values h(t). It
consists of a bunch
of interrelated information, but it doesn’t
do anything and it doesn’t change.
Formal proof that : Above, I gave an
informal argument for this. The rigorous way to see that
for
the ball function h(t) is to calculate the derivative
and plug in 4 and 2 to get ,
which is less than
. (Note the embedded
phrases). This argument picks
out particular data about the function that prove the statement. It says nothing about anything slowing down
as t increases. It says nothing about anything changing.
Other examples
¨
The rigorous way to say that
“Integers go to infinity in both directions” is something like this: “For every integer n there is an integer k
such that k < n and an integer m such that n < m.”
¨
The rigorous way to say
that continuous functions don’t have gaps in their graph is to use the definition of continuity.
¨ Conditional assertions are one important aspect of mathematical reasoning in which this concept of unchanging inertness clears up a lot of misunderstanding. “If… then…” in our intuition contains an idea of causation and of one thing happening before another (see also here). But if objects are inert they don’t cause anything and if they are unchanging then “when” is meaningless.
The rigorous view does not apply to all abstract objects, but only to mathematical objects. See abstract objects for examples.
Metaphors and images are dangerous
Every mental representation has flaws. Each one provides a way of thinking about an A as a kind of B in some respects. But the representation can have irrelevant
features. People new
to the subject will be tempted to think
about A as a kind of B in inappropriate respects as well. This is a form of cognitive
dissonance.
It may be that most difficulties
students have with abstract math are based on not knowing which aspects of a given representation are applicable
in a given situation. Indeed,
on not being consciously aware that one must restrict the applicability of the mental pictures
that come with a representation.
In abstractmath.org you will sometimes see this statement: “What is wrong with this metaphor:” (or image, or representation) to warn you about the flaws of that particular representation.
Example
The graph of the function h(t) makes it look like the
two arms going downward become so nearly vertical that the curve has vertical asymptotes.
But it deos not have asymptotes.
The arms going down spread out underneath every point of the x-axis. For example, there is a point on the curve underneath
the point (999,0), namely (999, -988,011).
Example
A
set is sometimes described as a container. But consider: the integer 3 is in the set of all odd
integers, and it is also in the set . How could something be in two containers at
once? (More about this here.)
Example
Mathematicians think of the real numbers as constituting a line infinitely long in both directions, with each number as a point on the line. But this does not mean that you can think of the line as a row of points. See real line.
Example
We commonly think of functions as machines that turn one
number into another. But this does not
mean that, given any such function, we can construct a machine (or a program)
that can calculate it. For many
functions, it is not only impractical to do, it
is theoretically impossible to do it.
In other words, the
machine picture of a function does not apply to all functions.
I have been told that the book
by Cutland is a good
starting place to learn about this, but I have not yet examined the book.
The images and metaphors you use to think about a mathematical object
are limited in how they apply
Mental representations can suggest proofs, but they cannot be used in proofs. Only definitions and previously proved theorems can be used in a proof.
The images and metaphors you use to think about
the subject
cannot be directly used in a proof.
Appendices
Mental
representations are physical representations
It seems likely that cognitive phenomena such as images and metaphors are physically represented in the brain as collections of neurons connected in specific ways. Research on this topic is proceeding rapidly. Perhaps someday we will learn things about how we think physically that actually help us learn things about math.
In any case, thinking about mathematical objects as physically represented in your brain (not necessarily completely or correctly!) wipes out a lot of the dualistic talk about ideas and physical objects as separate kinds of things. Ideas, in particular math objects, are emergent constructs in the physical brain.
About metaphors
“Metaphor” is used in
abstractmath.org to describe a type of
thought configuration. It is an implicit
conceptual identification of part of one type of situation with part of another.
Metaphors are a fundamental way we understand
the world, and they are
a fundamental
way we understand math.
The word
“metaphor”
The word “metaphor” is also used
in rhetoric as the name of a type of figure of speech. Authors often refer to metaphor in the meaning
of thought configuration as a conceptual
metaphor. Other figures of speech,
such as simile and synecdoche, correspond to
conceptual metaphors as well.
References for metaphors in general cognition:
Center for the cognitive science of metaphor
Lakoff, G., Women, Fire, and Dangerous Things. The
Lakoff, G. and Mark Johnson, Metaphors We Live By. The
Turner, M., Reading
Minds: The Study of English in the Age of Cognitive Science.
References for metaphors in math:
Byers, W., How
Mathematicians Think.
Lakoff, G. and R. E. Núñez, Where Mathematics Comes
From. Basic Books, 2000.
Núñez, R. E., “Do Real Numbers Really Move?” Chapter in 18 Unconventional Essays on the Nature of Mathematics, Reuben Hersh, Ed. Springer, 2006.
Instantly?
is a square, so its smallest possible value is 0, which
happens when
. This means the
largest
possible value
of
occurs at
. Return