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Back to top of Understanding Math chapterLast edited 8/9/2008 10:08:00 AM
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.
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Contents
Mental representations are imperfect
Mental representations in math
Images and
metaphors on this website
Two levels of images and
metaphors
Metaphors and images are dangerous
Mental
representations are physical representations
Mental 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.
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.
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.
Mental representations 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 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 “ ”. 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. See a function example here and set examples
here and here.
¨ Kinetic understanding, for example thinking of the value of a function as approaching a limit by imagining yourself moving along it.
¨
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
Consider the function . The chapter
on images and metaphors for functions describes many ways to think about
functions. Only 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 looks like it occurs
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 a 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 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.