Representations of mathematical objects

2012/11/02 — SixWingedSeraph

This is a long post. Notes on viewing.

About this post

A mathematical object, or a type of math object, is represented in practice in a great variety of ways, including some that mathematicians rarely think of as "representations".  

In this post you will find examples and comments about many different types of representations as well as references to the literature. I am not aware that anyone has considered all these different ideas of representation in one place before. Reading through this post should raise your consciousness about what is going on when you do math.  

This is also an experiment in exposition.  The examples are discussed in a style similar to the way a Mathematica command is discussed in the Documentation Center, using mostly nonhierarchical bulleted lists. I find it easy to discover what I want to know when it is written in that way.  (What is hard is discovering the name of a command that will do what I want.)

Types of representations

Using language

  • Language can be used to define a type of object.
  • A definition is intended to be precise enough to determine all the properties that objects of that type all have.  (Pay attention to the two uses of the word "all" in that sentence; they are both significant, in very different ways.)
  • Language can be used to describe an object, exhibiting properties without determining all properties.
  • It can also provide metaphors, making use of one of the basic tools of our brain to understand the world. 
  • The language used is most commonly mathematical English, a special dialect of English.
  • The symbolic language of mathematics (distinct from mathematical English) is used widely in calculations. Phrases from the symbolic language are often embedded in a statement in math English. The symbolic language includes among others algebraic notation and logical notation. 
  • The language may also be a formal language, a language that is mathematically defined and is thus itself a mathematical object. Logic texts generally present the first order predicate calculus as a formal language. 
  • Neither mathematical English nor the symbolic language is a formal language. Both allow irregularities and ambiguities.

Mathematical objects

The representation itself may be a mathematical object, such as:

  • A linear representation of a group. Not only are the groups mathematical objects, so is the representation.
  • An embedding of a manifold into Euclidean space. A definition given in a formal language of the first order predicate calculus of the property of commutativity of binary operations. (Thus a property can be represented as a math object.)

Visual representations

A math object can be represented visually using a physical object such as a picture, graph (in several senses), or diagram.  

  • The visual processing of our brain is our major source of knowledge of the world and takes about a fifth of the brain's processing power.  We can learn many things using our vision that would take much longer to learn using verbal descriptions.  (Proofs are a different matter.)
  • When you look at a graph (for example) your brain creates a mental representation of the graph (see below).

Mental representations

If you are a mathematician, a math object such as "$42$", "the real numbers" or "continuity" has a mental representation in your brain.  

  • In the math ed literature, such a representation is called "mental image", "concept image", "procept", or "schema".   (The word "image" in these names is not thought of as necessarily visual.) 
  • The procept or schema describe all the things that come to mind when you think about a particular math object: The definition, important theorems, visual images, important examples, and various metaphors that help you understand it. 
  • The visual images occuring in a mental schema for an object may themselves be mental representations of physical objects. The examples and theorems may be mental representations of ideas you learned from language or pictures, and so on.  The relationships between different kinds of representations get quite convoluted.

Metaphors

Conceptual metaphors are a particular kind of mental representation of an object which involve mentally associating some aspects of the objects with some aspects of something else — a physical object, an image, an action or another abstract object.

  • A conceptual metaphor may give you new insight into the object.
  • It may also mislead you because you think of properties of the other object that the math object doesn't have.
  • A graph of a function is a conceptual metaphor.
  • When you say that a point on a graph "rises as it goes from left to right" your metaphor is an action. 
  • When you say that the cosets of a normal subgroup of a group "get along" with the group multiplication, your metaphor identifies a property they have with an aspect of human behavior.

Properties of representations

A representation of a math object may or may not

  • determine it completely
  • exhibit some of its properties
  • suggest easy proofs of some theorems
  • provide a useful way of thinking about it
  • mislead you about the object's properties
  • mislead you about what is significant about the object

Examples of representations

This list shows many of the possibilities of representation.  In each case I discuss the example in terms of the two bulleted lists above. Some of the examples are reused from my previous publications.

Functions

Example (F1) "Let $f(x)$ be the function defined by $f(x)=x^3-x$."

  • This is an expression in mathematical English that a fluent reader of mathematical English will recognize gives a definition of a specific function.
  • (F1) is therefore a representation of that function.  
  • The word "representation" is not usually used in this way in math.  My intention is that it should be recognized as the same kind of object as many other representations.
  • The expression contains the formula $x^3-x$.  This is an encapsulated computation in the symbolic language of math. It allows someone who knows basic algebra and calculus to perform calculations that find the roots, extrema and inflection points of the function $f$.  
  • The word "let" suggests to the fluent reader of mathematical English that (F1) is a definition which is probably going to hold for the next chunk of text, but probably not for the whole article or book.
  • Statements in mathematical English are generally subject to conventions.  In a calculus text (F1) would automatically mean that the function had the real numbers as domain and codomain.
  • The last two remarks show that a beginner has to learn to read mathematical English. 
  • Another convention is discussed in the following diatribe.

Diatribe 

You would expect $f(x)$ by itself to mean the value of $f$ at $x$, but in (F1) the $x$ has the property of a bound variable.  In mathematical English, "let" binds variables. However, after the definition, in the text the "$x$" in the expression "$f(x)$" will be free, but the $f$ will be bound to the specific meaning.  It is reasonable to say that the term "$f(x)$" represents the expression "$x^3-x$" and that $f$ is the (temporary) name of the function. Nevertheless, it is very common to say "the function $f(x)$" to mean $f$.  

A fluent reader of mathematical English knows all this, but probably no one has ever said it explicitly to them.  Mathematical English and the symbolic language should be taught explicitly, including its peculiarities such as "the function $f(x)$".  (You may want to deprecate this usage when you teach it, but students deserve to understand its meaning.)

The positive integers

You have a mental representation of the positive integers $1,2,3,\ldots$.  In this discussion I will assume that "you" know a certain amount of math.  Non-mathematicians may have very different mental representations of the integers.

  • You have a concept of "an integer" in some operational way as an abstract object.
  • "Abstract object" needs a post of its own. Meanwhile see Mathematical Objects (abstractmath) and the Wikipedia articles on Mathematical objects and Abstract objects.
  • You have a connection in your brain between the concept of integer and the concept of listing things in order, numbering them by $1,2,3,\ldots$.
  • You have a connection in your brain between the concept of an integer and the concept of counting a finite number of objects.  But then you need zero!
  • You understand how to represent an integer using the decimal representation, and perhaps representations to other bases as well. 
  • Your mental image has the integer "$42"$ connected to but not the same as the decimal representation "42". This is not true of many students.
  • The decimal rep has a picture of the string "42" associated to it, and of course the picture of the string may come up when you think of the integer $42$ as well (it does for me — it is a an icon for the number $42$.)
  • You have a concept of the set of integers. 
  • Students need to be told that by convention "the set of integers" means the set of all integers.  This particularly applies to students whose native language does not have articles, but American students have trouble with this, too.
  • Your concept of  "the set of integers" may have the icon "$\mathbb{N}$" associated with it.  If you are a mathematician, the icon and the concept of the set of integers are associated with each other but not identified with each other.
  • For me, at least, the concept "set of integers" is mentally connected to each integer by the "element of" relation. (See third bullet below.)
  • You have a mental representation of the fact that the set of integers is infinite.  
  • This does not mean that your brain contains an infinite number of objects, but that you have a representation of infinity as a concept, it is brain-connected to the concept of the set of integers, and also perhaps to a proof of the fact that $\mathbb{N}$ is infinite.
  • In particular, the idea that the set of integers is mentally connected to each integer does not mean that the whole infinite number of integers is attached in your brain to the concept of the set of integers.  Rather, the idea is a predicate in your brain.  When it is connected to "$42$", it says "yes".  To "$\pi$" it says "No".
  • Philosophers worry about the concept of completed infinity.  It exists as a concept in your brain that interacts as a meme with concepts in other mathematicians' brains. In that way, and in that way only (as far as I am concerned) it is a physical object, in particular an object that exists in scattered physical form in a social network.

Graph of a function

This is a graph of the function $y=x^3-x$:

Graph of a cubic function

  • The graph is a physical object, either on a screen or on paper
  • It is processed by your visual system, the most powerful sensory management system in your brain
  • It also represents the graph in the mathematical sense (set of ordered pairs) of the function $y=x^3-x$
  • Both the mathematical graph and the physical graph are represented by modules in your brain, which associates the two of them with each other by a conceptual metaphor. 
  • The graph shows some properties of the function: inflection point, going off to infinity in a specific way, and so on.
  • These properties are made apparent (if you are knowledgeable) by means of the powerful pattern recognition system in your brain. You see them much more quickly than you can discover them by calculation.
  • These properties are not proved by the graph. Nevertheless, the graph communicates information: for example, it suggests that you can prove that there is an inflection point near $(0,0)$.
  • The graph does not determine or define the function: It is inaccurate and it does not (cannot) show all of the graph.
  • More subtle details about this graph are discussed in my post Representations 2.

Continuity

Example (C1) The $\epsilon-\delta$ definition of the continuity of a function $f:\mathbb{R}\to\mathbb{R}$ may be given in the symbolic language of math:

A function $f$ is continuous at a number $c$ if \[\forall\epsilon(\epsilon\gt0\implies(\forall x(\exists\delta(|x-c|\lt\delta\implies|f(x)-f(c)|\lt\epsilon)))\]

  • To understand (C1), you must be familiar with the notation of first order logic.  For most students, getting the notation right is quite a bit of work.  
  • You must also understand  the concepts, rules and semantics of first order logic.  
  • Even if you are familiar with all that, continuity is still a difficult concept to understand.
  • This statement does show that the concept is logically complicated. I don't see how it gives any other intuition about the concept. 

Example (C2) The definition of continuity can also be represented in mathematical English like this:

A function $f$ is continuous at a number $c$ if for any $\epsilon\gt0$ and for any $x$ there is a $\delta$ such that if $|x-c|\lt\delta$, then $|f(x)-f(c)|\lt\epsilon$. 

  • This definition doesn't give any more intuition that (C1) does.
  • It is easier to read that (C1) for most math students, but it still requires intimate familiarity with the quirks of math English.
  • The fact that "continuous" is in boldface signals that this is a definition.  This is a convention.
  • The phrase "For any $\epsilon\gt0$" contains an unmarked parenthetic insertion that makes it grammatically incoherent.  It could be translated as: "For any $\epsilon$ that is greater than $0$".  Most math majors eventually understand such things subconsciously.  This usage is very common.
  • Unless it is explicitly pointed out, most students won't notice that  if you change the phrase "for any $x$ there is a $\delta$"  to "there is a $\delta$ for any $x$" the result means something quite different.  Cauchy never caught onto this.
  • In both (C1) and (C2), the "if" in the phrase "A function $f$ is continuous at a number $c$ if…" means "if and only if" because it is in a definition.  Students rarely see this pointed out explicitly.  

Example (C3) The definition of continuity can be given in a formally defined first order logical theory. 

  • The theory would have to contain function symbols and axioms expressing the algebra of real numbers as an ordered field. 
  • I don't know that such a definition has ever been given, but there are various semi-automated and automated theorem-proving systems (which I know little about) that might be able to state such a definition.  I would appreciate information about this.
  • Such a definition would make the property of continuity a mathematical object.
  • An automated theorem-proving system might be able to prove that $x^3-x$ is continuous, but I wonder if the resulting proof would aid your intuition much.

Example (C4) A function from one topological space to another is continuous if the inverse of every open set in the codomain is an open set in the domain.

  • This definition is stated in mathematical English.
  • All definitions start with primitive data. 
  • In definitions (C1) – (C3), the primitive data are real numbers and the statement uses properties of an ordered field.
  • In (C4), the data are real numbers and the arithmetic operations of a topological field, along with the open sets of the field. The ordering is not mentioned.
  • This shows that a definition need not mention some important aspects of the structure. 
  • One marvelous example of this is that  a partition of a set and an equivalence relation on a set are based on essentially disjoint sets of data, but they define exactly the same type of structure.

Example (C4) "The graph of a continuous function can be drawn without picking up the chalk".

  • This is a metaphor that associates an action with the graph.
  • It is incorrect: The graphs of some continuous functions cannot be drawn.  For example, the function $x\mapsto x^2\sin(1/x)$ is continuous on the interval $[-1,1]$ but cannot be drawn at $x=0$. 
  • Generally speaking, if the function can be drawn then it can be drawn without picking up the chalk, so the metaphor provides a useful insight, and it provides an entry into consciousness-raising examples like the one in the preceding bullet.

References

  1. 1.000… and .999… (post)
  3. Conceptual blending (post)
  4. Conceptual blending (Wikipedia)
  5. Conceptual metaphors (Wikipedia)
  6. Convention (abstractmath)
  7. Definitions (abstractmath)
  8. Embodied cognition (Wikipedia)
  9. Handbook of mathematical discourse (see articles on conceptual blend, mental representation, representation, metaphor, parenthetic assertion)
  10. Images and Metaphors (abstractmath).
  11. The interplay of text, symbols and graphics in math education, Lin Hammill
  12. Math and the modules of the mind (post)
  13. Mathematical discourse: Language, symbolism and visual images, K. L. O’Halloran.
  14. Mathematical objects (abmath)
  15. Mathematical objects (Wikipedia)
  16. Mathematical objects are “out there?” (post)
  17. Metaphors in computing science ​(post)
  18. Procept (Wikipedia)
  19. Representations 2 (post)     
  20. Representations and models (abstractmath)
  21. Representations II: dry bones (post)
  22. Representation theorems (Wikipedia) Concrete representations of abstractly defined objects.
  23. Representation theory (Wikipedia) Linear representations of algebraic structures.
  24. Semiotics, symbols and mathematical visualization, Norma Presmeg, 2006.
  25. The transition to formal thinking in mathematics, David Tall, 2010
  26. Theory in mathematical logic (Wikipedia)
  27. What is the object of the encapsulation of a process? Tall et al., 2000.
  28. Where mathematics comes from, by George Lakoff and Rafael Núñez, Basic Books, 2000. 
  29. Where mathematics comes from (Wikipedia) This is a review of the preceding book.  It is a permanent link to the version of 04:23, 25 October 2012.  The review is opinionated, partly wrong, not well written and does not fit the requirements of a Wikipedia entry.  I recommend it anyway; it is well worth reading.  It contains links to three other reviews.

Notes on Viewing  

This post uses MathJax. If you see mathematical expressions with dollar signs around them, or badly formatted formulas, try refreshing the screen. Sometimes you have to do it two or three times.

Posted in exposition, math, representations, understanding math. Tags: , , , , , , , , , , , , , , , , , , , , , , . 2 Comments »

Explaining “higher” math to beginners

2012/09/21 — SixWingedSeraph


Notes on viewing

Explaining math

I am in the process of writing an explanation of monads for people with not much math background.  In that article, I began to explain my ideas about exposition for readers at that level and after I had written several paragraphs decided I needed a separate article about exposition.  This is that article. It is mostly about language.

Who is it written for?

Interested laypeople

There are many recent books explaining some aspect of math for people who are not happy with high school algebra; some of them are listed in the references.  They must be smart readers who know how to concentrate, but for whom algebra and logic and definition-theorem-proof do not communicate.  They could be called interested laypeople, but that is a lousy name and I would appreciate suggestions for a better name. 

Math newbies

My post on monads is aimed at people who have some math, and who are interested in "understanding" some aspect of "higher math"; not understanding in the sense of being able to prove things about monads, but merely how to think about them.   I will call them math newbies.  Of course, I am including math majors, but I want to make it available to other people who are willing to tackle mathematical explanations and who are interested in knowing more about advanced stuff. 

These "other people" may include people (students and practitioners) in other science & technology areas as well as liberal-artsy people.  There are such people, I have met them.  I recall one theologian who asked me about what was the big deal about ruler-and-compass construction and who seemed to feel enlightened when I told him that those constructions preserve exactly the ideal nature of geometric objects.  (I later found out he was a famous theologian I had never heard of, just like Ngô Bảo Châu is a famous mathematician nonmathematicians have never heard of.)

Algebra and other foreign languages

If you are aiming at interested laypeople you absolutely must avoid algebra.  It is a foreign language that simply does not communicate to most of the educated people in the world.  Learning a foreign language is difficult. 

So how do you avoid algebra?  Well, you have to be clever and insightful.  The book by Matthew Watkins (below) has absolutely wonderful tricks for doing that, and I think anyone interested in math exposition ought to read it.  He uses metaphors, pictures and saying the same thing in different words. When you finish reading his book, you won't know how to prove statements related to the prime number theorem (unless you already knew how) but you have a good chance of understanding the statement of some theorem in that subject. See my review of the book for more details.

If your article is for math newbies, you don't have to avoid algebra completely.  But remember they are newbies and not as fluent as you are — they do things analogous to "Throw Mama from the train a kiss" and "I can haz cheeseburger?".  But if you are trying to give them some way of thinking about a concept, you need many other things (metaphors, illustrative applications, diagrams…)  You don't need the definition-theorem-proof style too common in "exposition".  (You do need that for math majors who want to become professional mathematicians.) 

Unfamiliar notation

In writing expositions for interested laypeople or math newbies, you should not introduce an unfamiliar notation system (which is like a minilanguage).  I expect to write the monad article without commutative diagrams.  Now, commutative diagrams are a wonderful invention, the best way of writing about categories, and they are widely used by other than category theorists.  But to explain monads to a newbie by introducing and then using commutative diagrams is like incorporating a short grammar of Spanish which you will then use in an explanation of Sancho Panza's relationship with Don Quixote. 

The abstractmath article on and, or and not does not use any of the several symbolic notations for logic that are in use.  The explanations simply use "and", "or" and "not".  I did introduce the notation, but didn't use it in the explanations.  When I rewrite the article I expect to put the notation at the end of the article instead of in the middle.  I expect to rewrite the other articles on mathematical reasoning to follow that practice, too.

Technical terminology

This is about the technical terminology used in math.  Technical terminology belongs to the math dialect (or register) of English, which is not a foreign language in the same sense as algebra.  One big problem is changing the meaning of ordinary English words to a technical meaning.  This requires a definition, and definitions are not something most people take seriously until they have been thoroughly brainwashed into using mathematical methodology.  Math majors have to be brainwashed in this way, but if you are writing for laypeople or newbies you cannot use the technology of formal definition.

Groups, simple groups

"You say the Monster Group is SIMPLE???  You must be a GENIUS!"  So Mark Ronan in his book (below) referred to simple groups as atoms.  Marcus du Sautoy calls them building blocks.  The mathematical meaning of "simple group" is not a transparent consequence of the meanings of "simple" and "group". Du Sautoy usually writes "group of symmetries" instead of just "group", which gives you an image of what he is talking about without having to go into the abstract definition of group. So in that usage, "group" just means "collection", which is what some students continue to think well after you give the definition.  

A better, but ugly, name for "group" might be "symmetroid". It sounds technical, but that might be an advantage, not a disadvantage. "Group" obviously means any collection, as I've known since childhood. "Symmetroid" I've never heard of so maybe I'd better find out what it means.

In beginning abstract math courses my students fervently (but subconsciously) believe that they can figure out what a word means by what it means already, never mind the "definition" which causes their eyes to glaze over. You have to be really persuasive to change their minds.

Prime factorization

Matthew Watkins referred to the prime factorization of an integer as a cluster. I am not sure why Watkins doesn't like "prime factorization", which usually refers to an expression such as  $p^{n_1}_1p^{n_2}_2\ldots p^{n_k}_k$.  This (as he says) has a spurious ordering that makes you have to worry about what the uniqueness of factorization means. The prime factorization is really a multiset of primes, where the order does not matter. 

Watkins illustrates a cluster of primes as a bunch of pingpong balls stuck together with glue, so the prime factorization of 90 would be four smushed together balls marked 2, 3, 3 and 5. Below is another way of illustrating the prime factorization of 90. Yes, the random movement programming could be improved, but Mathematica seduces you into infinite playing around and I want to finish this post. (Actually, I am beginning to think I like smushed pingpong balls better. Even better would be a smushed pingpong picture that I could click on and look at it from different angles.)

Metaphors, pictures, graphs, animation

Any exposition of math should use metaphors, pictures and graphs, especially manipulable pictures (like the one above) and graphs.  This applies to expositions for math majors as well as laypeople and newbies.  Calculus and other texts nowadays have begun doing this, more with pictures than with metaphors. 

I was turned on to these ideas as far back as 1967 (date not certain) when I found an early version of David Mumford's "Red Book", which I think evolved into the book The Red Book of Varieties and Schemes.  The early version was a revelation to me both about schemes and about exposition. I have lost the early book and only looked at the published version briefly when it appeared (1999).  I remember (not necessarily correctly) that he illustrated the spectrum as a graph whose coordinates were primes, and generic points were smudges.  Writing this post has motivated me to go to the University of Minnesota math library and look at the published version again.

References

Expositions for educated non-mathematicians

Previous posts in G&G

Relevant abmath articles

Notes on Viewing  

  • This post uses MathJax. If you see mathematical formulas with dollar signs around them, or badly formatted formulas, try refreshing the screen. Sometimes you have to do it two or three times.
  • To manipulate the demos in this post, you must have Wolfram CDF Player installed on your computer. It is available free from the Wolfram website. The code for the demos is in the Mathematica notebook algebra2.nb.
Posted in exposition, language of math, math, representations, understanding math. Tags: , , , , , , , , , , , , . No Comments »

Conceptual blending

2012/06/18 — SixWingedSeraph

This post uses MathJax.  If you see formulas in unrendered TeX, try refreshing the screen.

A conceptual blend is a structure in your brain that connects two concepts by associating part of one with part of another.  Conceptual blending is a major tool used by our brain to understand the world.

The concept of conceptual blend includes special cases, such as representations, images and conceptual metaphors, that math educators have used for years to understand how mathematics is communicated and how it is learned.  The Wikipedia article is a good starting place for understanding conceptual blending. 

In this post I will illustrate some of the ways conceptual blending is used to understand a function of the sort you meet with in freshman calculus.  I omit the connections with programs, which I will discuss in a separate post.

A particular function

Consider the function $h(t)=4-(t-2)^2$. You may think of this function in many ways.

FORMULA:

$h(t)$ is defined by the formula $4-(t-2)^2$.

  • The formula encapsulates a particular computation of the value of $h$ at a given value $t$.
  • The formula defines the function, which is a stronger statement than saying it represents the function.
  • The formula is in standard algebraic notation. (See Note 1)
  • To use the formula requires one of these:
    • Understand and use the rules of algebra
    • Use a calculator
    • Use an algebraic programming language. 
  • Other formulas could be used, for example $4t-t^2$.
    • That formula encapsulates a different computation of the value of $h$.

TREE: 

$h(t)$ is also defined by this tree (right).
  • The tree makes explicit the computation needed to evaluate the function.
  • The form of the tree is based on a convention, almost universal in computing science, that the last operation performed (the root) is placed at the top and that evaluation is done from bottom to top.
  • Both formula and tree require knowledge of conventions.
  • The blending of formula and tree matches some of the symbols in the formula with nodes in the tree, but the parentheses do not appear in the tree because they are not necessary by the bottom-up convention.
  • Other formulas correspond to other trees.  In other words, conceptually, each tree captures not only everything about the function, but everything about a particular computation of the function.
  • More about trees in these posts:

GRAPH:

$h(t)$ is represented by its graph (right). (See note 2.)

  • This is the graph as visual image, not the graph as a set of ordered pairs.
  • The blending of graph and formula associates each point on the (blue) graph with the value of the formula at the number on the x-axis directly underneath the point.
  • In contrast to the formula, the graph does not define the function because it is a physical picture that is only approximate.
  • But the formula does represent the function.  (This is "represents" in the sense of cognitive psychology, but not in the mathematical sense.)
  • The blending requires familiarity with the conventions concerning graphs of functions. 
  • It sets into operation the vision machinery of your brain, which is remarkably elaborate and powerful.
    • Your visual machinery allows you to see instantly that the maximum of the curve occurs at about $t=2$. 
  • The blending leaves out many things.
    • For one, the graph does not show the whole function.  (That's another reason why the graph does not define the function.)
    • Nor does it make it obvious that the rest of the graph goes off to negative infinity in both directions, whereas that formula does make that obvious (if you understand algebraic notation).  

GEOMETRIC

The graph of $h(t)$ is the parabola with vertex $(2,4)$, directrix $x=2$, and focus $(2,\frac{3}{4})$. 

  • The blending with the graph makes the parabola identical with the graph.
  • This tells you immediately (if you know enough about parabolas!) that the maximum is at $(2,4)$ (because the directrix is vertical).
  • Knowing where the focus and directrix are enables you to mechanically construct a drawing of the parabola using a pins, string, T-square and pencil.  (In the age of computers, do you care?)

HEIGHT:

$h(t)$ gives the height of a certain projectile going straight up and down over time.

  • The blending of height and graph lets you see instantly (using your visual machinery) how high the projectile goes. 
  • The blending of formula and height allows you to determing the projectile's velocity at any point by taking the derivative of the function.
  • A student may easily be confused into thinking that the path of the projectile is a parabola like the graph shown.  Such a student has misunderstood the blending.

KINETIC:

You may understand $h(t)$ kinetically in various ways.

  • You can visualize moving along the graph from left to right, going, reaching the maximum, then starting down.
    • This calls on your experience of going over a hill. 
    • You are feeling this with the help of mirror neurons.
  • As you imagine traversing the graph, you feel it getting less and less steep until it is briefly level at the maximum, then it gets steeper and steeper going down.
    • This gives you a physical understanding of how the derivative represents the slope.
    • You may have seen teachers swooping with their hand up one side and down the other to illustrate this.
  • You can kinetically blend the movement of the projectile (see height above) with the graph of the function.
    • As it goes up (with $t$ increasing) the projectile starts fast but begins to slow down.
    • Then it is briefly stationery at $t=2$ and then starts to go down.
    • You can associate these feelings with riding in an elevator.
      • Yes, the elevator is not a projectile, so this blending is inaccurate in detail.
    • This gives you a kinetic understanding of how the derivative gives the velocity and the second derivative gives the acceleration.

OBJECT:

The function $h(t)$ is a mathematical object.

  • Usually the mental picture of function-as-object consists of thinking of the function as a set of ordered pairs $\Gamma(h):=\{(t,4-(t-2)^2)|t\in\mathbb{R}\}$. 
  • Sometimes you have to specify domain and codomain, but not usually in calculus problems, where conventions tell you they are both the set of real numbers.
  • The blend object and graph identifies each point on the graph with an element of $\Gamma(h)$.
  • When you give a formal proof, you usually revert to a dry-bones mode and think of math objects as inert and timeless, so that the proof does not mention change or causation.
    • The mathematical object $h(t)$ is a particular set of ordered pairs. 
    • It just sits there.
    • When reasoning about something like this, implication statements work like they are supposed to in math: no causation, just picking apart a bunch of dead things. (See Note 3).
    • I did not say that math objects are inert and timeless, I said you think of them that way.  This post is not about Platonism or formalism. What math objects "really are" is irrelevant to understanding understanding math [sic].

DEFINITION

A definition of the concept of function provides a way of thinking about the function.

  • One definition is simply to specify a mathematical object corresponding to a function: A set of ordered pairs satisfying the property that no two distinct ordered pairs have the same second coordinate, along with a specification of the codomain if that is necessary.
  • A concept can have many different definitions.
    • A group is usually defined as a set with a binary operation, an inverse operation, and an identity with specific properties.  But it can be defined as a set with a ternary operation, as well.
    • A partition of a set is a set of subsets of a set with certain properties. An equivalence relation is a relation on a set with certain properties.  But a partition is an equivalence relation and an equivalence relation is a partition.  You have just picked different primitives to spell out the definition. 
    • If you are a beginner at doing proofs, you may focus on the particular primitive objects in the definition to the exclusion of other objects and properties that may be more important for your current purposes.
      • For example, the definition of $h(t)$ does not mention continuity, differentiability, parabola, and other such things.
      • The definition of group doesn't mention that it has linear representations.

SPECIFICATION

A function can be given as a specification, such as this:

If $t$ is a real number, then $h(t)$ is a real number, whose value is obtained by subtracting $2$ from $t$, squaring the result, and then subtracting that result from $4$.

  • This tells you everything you need to know to use the function $h$.
  • It does not tell you what it is as a mathematical object: It is only a description of how to use the notation $h(t)$.

Notes

1. Formulas can be give in other notations, in particular Polish and Reverse Polish notation. Some forms of these notations don't need parentheses.

2. There are various ways to give a pictorial image of the function.  The usual way to do this is presenting the graph as shown above.  But you can also show its cograph and its endograph, which are other ways of representing a function pictorially.  They  are particularly useful for finite and discrete functions. You can find lots of detail in these posts and Mathematica notebooks:

3. See How to understand conditionals in the abstractmath article on conditionals.

References

  1. Conceptual blending (Wikipedia)
  2. Conceptual metaphors (Wikipedia)
  3. Definitions (abstractmath)
  4. Embodied cognition (Wikipedia)
  5. Handbook of mathematical discourse (see articles on conceptual blend, mental representation, representation, and metaphor)
  6. Images and Metaphors (article in abstractmath)
  7. Links to G&G posts on representations
  8. Metaphors in Computing Science (previous post)
  9. Mirror neurons (Wikipedia)
  10. Representations and models (article in abstractmath)
  11. Representations II: dry bones (article in abstractmath)
  12. The transition to formal thinking in mathematics, David Tall, 2010
  13. What is the object of the encapsulation of a process? Tall et al., 2000.

 

Posted in exposition, Handbook, language of math, math, representations, understanding math. Tags: , , , , , , , , , , , , , , . No Comments »

More about defining “category”

2011/12/05 — SixWingedSeraph


In a recent post, I wrote about defining “category” in a way that (I hope) makes it accessible to undergraduate math majors at an early stage.  I have several more things to say about this.

Early intro to categories

The idea is to define a category as a directed graph equipped with an additional structure of composition of paths subject to some axioms.  By giving several small finite examples of categories drawn in that way that gives you an understanding of “category” that has several desirable properties:

  • You get the idea of what a category is in one lecture.
  • With the right choice of examples you get several fine points cleared up:
    • The composition is added structure.
    • A loop doesn’t have to be an identity.
    • Associativity is a genuine requirement –  it is not automatic.
  • You get immediate access to what is by far the most common notation used to work with a category — objects (nodes) and arrows.
  • You don’t have to cope with the difficult chunking required when the first examples given are sets-with-structure and structure-preserving functions.  It’s quite hard to focus on a couple of dots on the paper each representing a group or a topological space and arrows each representing a whole function (not the value of the function!).

Introduce more examples

Then the teacher can go on with the examples that motivated categories in the first place: the big deal categories such as sets, groups and topological spaces.   But they can be introduced using special cases so they don’t require much background.

  • Draw some finite sets and functions between them.  (As an exercise, get the students to find some finite sets and functions that make the picture a category with $f=kh$ as the composite and $f\neq g$.)
  • If the students have had calculus,  introduce them to the category whose objects are real finite nonempty intervals with continuous or differentiable mappings between them.  (Later you can prove that this category is a groupoid!)
  • Find all the groups on a two element set and figure out which maps preserve group multiplication.  (You don’t have to use the word “group” — you can simply show both of them and work out which maps preserve multiplication — and discover isomorphism!.)  This introduces the idea of the arrows being structure-preserving mape. You can get more complicated and use semigroups as well.  If the students know Mathematica you could even do magmas.  Well, maybe not.

All this sounds like a project you could do with high school students.

Large and small

If all this were just a high school (or intro-to-math-for-math-majors) project you wouldn’t have to talk about large vs. small.  However, I have some ideas about approaching this topic.

In the first place, you can define category, or any other mathematical object that might involve a proper class, using the syntactic approach I described in Just-in-time foundations.  You don’t say “A category consists of a set of objects and a set of arrows such that …”.  Instead you say something like “A category $\mathcal{C}$ has objects $A,\,B,\,C\ldots$ such that…”.

This can be understood as meaning “For any $A$, the statement $A$ is an object of  $\mathcal{C}$ is either true or false”, and so on.

This approach is used in the Wikibook on category theory.  (Note: this is a permanent link to the November 28 version of the section defining categories, which is mostly my work.  As always with Wikimedia things it may be entirely different when you read this.)

If I were dictator of the math world (not the same thing as dictator of MathWorld) I would want definitions written in this syntactic style.  The trouble is that mathematicians are now so used to mathematical objects having to be sets-with-structure that wording the definition as I did above may leave them feeling unmoored.  Yet the technique avoids having to mention large vs. small until a problem comes up. (In category theory it sometimes comes up when you want to quantify over all objects.)

The ideas outlined in this subsection could be a project for math majors.  You would have to introduce Russell’s Paradox.  But for an early-on intro to categories you could just use the syntactic wording and avoid large vs. small altogether.

 

http://en.wikibooks.org/w/index.php?title=Category_Theory/Categories&stableid=2221684

Posted in category theory, exposition, math, understanding math. Tags: , , , , , , , , . No Comments »

Defining “category”

2011/11/28 — SixWingedSeraph

The concept of category is typically taught later in undergrad math than the concept of group is.  It is supposedly a more advanced concept.  Indeed, the typical examples of categories used in applications are more advanced than some of those in group theory (for example, symmetries of geometric shapes and operations on numbers).

Here are some thoughts on how categories could be taught as early as groups, if not earlier.

Nodes and arrows

Small finite categories can be pictured as a graph using nodes and arrows, together with a specification of the identity arrows and a definition of the composition.  (I am using the word “graph” the way category people use it:  a directed graph with possible multiple edges and loops.)

An example is the category pictured below with three objects and seven arrows. The composition is forced except for $kh$, which I hereby define to be $f$.

This way of picturing a category is  easy to grasp. The composite $kh$ visibly has to be either $f$ or $g$.  There is only one choice for the composite of any other composable pair.  Still, the choice of composite is not deducible directly by looking at the graph.

A first class in category theory using graphs as examples could start with this example, or the example in Note 1 below.  This example is nontrivial (never start any subject with trivial examples!) and easy to grasp, in this case using the extraordinary preprocessing your brain does with the input from your eyes.  The definition of category is complicated enough that you should probably present the graph and then give the definition while pointing to what each clause says about the graph.

Most abstract structures have several different ways of representing them. In contrast, when you discuss categorial concepts the standard object-and-arrow notation is the overwhelming favorite.  It reveals domains and codomains and composable pairs, in fact almost everything except which of several possible arrows the composite actually is.  If for example you try to define category using sets and functions as your running example, the student has to do a lot of on-the-go chunking — thinking of a set as a single object, of a set function (which may involve lots of complicated data) as a single chunk with a domain and a codomain, and so on.  But an example shown as a graph comes already chunked and in a picture that is guaranteed to be the most common kind of display they will see in discussions of categories.

After you do these examples, you can introduce trivial and simple graph examples in which the composition is entirely induced; for example these three:

(In case you are wondering, one of them is the empty category.)  I expect that you should also introduce another graph non-example in which associativity fails.

Multiplication tables

The multiplication table for a group is easy to understand, too, in the sense that it gives you a simple method of calculating the product of any two elements.  But it doesn’t provide a visual way to see the product as a category-as-graph does.  Of course, the graph representation works only for finite categories, just as the multiplication table works only for finite groups.

You can give a multiplication table for a small finite category, too, like the one below for the category above.  (“iA” means the identity arrow on A and composition, as usual in category theory, is right to left.) This is certainly more abstract than the graph picture, but it does hit you in the face with the fact that the multiplication is partial.

Notes

1. My suggested example of a category given as a graph shows clearly that you can define two different categorial structures on the graph.  One problem is that the two different structures are isomorphic categories.  In fact, if you engage the students in a discussion about these examples someone may notice that!  So you should probably also use the graph below,where you can define several different category structures that are not all isomorphic. 

2. Multiplication tables and categories-as-graphs-with-composition are extensional presentations.  This means they are presented with all their parts laid out in front of you.  Most groups and categories are given by definitions as accumulations of properties (see concept in the Handbook of Mathematical Discourse).  These definitions tend to make some requirements such as associativity obvious.

Students are sometimes bothered by extensional definitions.  “What are h and k (in the category above)?  What are a, b and c?” (in a group given as a set of letters and a multiplication table).

Posted in category theory, exposition, math, understanding math. Tags: , , , , , . 2 Comments »

Case Study in Exposition: Secant

2011/10/06 — SixWingedSeraph

Note: To manipulate the diagrams in this post, you must have Wolfram CDF Player installed on your computer. It is available free from the Wolfram website. The Mathematica notebooks used here are listed in the references below.

Pictures, metaphors and etymology

Math texts and too many math teachers do not provide enough pictures and metaphors to help students understand a concept.  I suspect that the etymology of the technical terms might also be useful. This post is an experimental exposition of the math concept of “secant” that use pictures, metaphors and etymology to describe the concept.

The exposition is interlarded with comments about what I am doing and why.  An exposition directly aimed at students would be slimmer — but some explanations of why you are doing such and such in an exposition are not necessarily out of place every time!

Secant Line

The word “secant” is used in various related ways in math.  To start with, a secant line on a curve is the unique line determined by two distinct points on the curve, like this:

The word “secant” comes from the Latin word for “cut”, which came from the Indo-European root “sek”, meaning “cut”.  The IE root also came directly into English via various Germanic sound changes to give us “saw” and “sedge”.

The picture

Showing pictures of mathematical objects that the reader can fiddle with may make it much easier to understand a new concept.  The static picture you get above by keeping your mitts off the sliders requires imagining similar lines going through other pairs of points. When you wiggle the picture you see similar lines going through other pairs of points.  You also get a very strong understanding of how the secant line is a function of the two given points.  I don’t think that is obvious to someone without some experience with such things.

This belief contains the hidden claim that individuals vary a lot on how they can see the possibilities in a still picture that stands as an example of a lot of similar mathematical objects.  (Math books are full of such pictures.)  So people who have not had much practice learning about possible variation in abstract structures by looking at one motionless one will benefit from using movable parametrized pictures of various kinds.  This is the sort of claim that is amenable to field testing.

The metaphor

Most metaphors are based on a physical phenomenon.  The mathematical meanings of “secant” use the metaphor of cutting.  When the word “secant” was first introduced by a European writer (see its etymology) in the 16th century, the word really was a metaphor.   In those days essentially every European scholar read Latin. To them “secant” would transparently mean “cutting”.  This is not transparent to many of us these days, so the metaphor may be hidden.

If you examine the metaphor you realize that (like all metaphors) it involves making some remarkably subtle connections in your brain.

  • The straight line does not really cut the curve.  Indeed, the curve itself is both an abstract object that is not physical, so can’t be cut, and also the picture you see on the screen, which is physical, but what would it mean to cut it?  Cut the screen?  The line can’t do that.
  • You can make up a story that (for example) the use was suggested by the mental image of a mark made by a knife edge crossing the plane at points a and b that looks like it is severing the curve.
  • The metaphor is restricted further by saying that it is determined by two points on the curve.   This restriction turns the general idea of secant line into a (not necessarily faithful!) two-parameter family of straight lines.  You could define such a family by using one point on the curve and a slope, for example.  This particular way of doing it with two points on the curve leads directly to the concept of tangent line as limit.

Secant on circle

Another use of the word “secant” is the red line in this picture:


This is the secant line on the unit circle determined by the origin and one point on the circle, with one difference: The secant of the angle is the line segment between the origin and the point on the curve.  This means it corresponds to a number, and that number is what we mean by “secant” in trigonometry.

The Definition

The secant of an angle $latex \theta$ is usually defined as $latex \frac{1}{\cos\theta}$, which you can see by similar triangles is the length of the red line in the picture above.

This illustrates important facts about definitions:

  • Different equivalent definitions all make the same theorems true.
  • Different equivalent definitions can give you a very different understanding of the concept.

The red-line-segment-in-picture definition gives you a majorly important visual understanding of the concept of “secant”.  You can tell a lot from its behavior right off (it goes to infinity near $latex \pi/2$, for example).

The definition $latex \sec\theta=\frac{1}{\cos\theta}$ gives you a way of computing $latex \sec\theta$.  It also reduces the definition of $latex \sec\theta$ to a previously known concept.

It used to be common to give only the $latex \frac{1}{\cos\theta}$ definition of secant, with no mention of the geometric idea behind it.  That is a crime.  Yes, I know many students don’t want to “understand” stuff, they only want to know how to do the problems.  Teachers need to talk them out of that attitude.  One way to do that in this case is to test them on the geometric definition.

Etymology

This idea was known to the Arabs, and brought into European view in the 16th century by Danish mathematician Thomas Fincke in “Geometria Rotundi” (1583), where the first known use of the word “secant” occurs.  I have not checked, but I suspect from the title of the book that the geometric definition was the one he used in the book.

It wold be interesting to know the original Arabic name for secant, and what physical metaphor it is based on.  A cursory search of the internet gave me the current name in Arabic for secant but nothing else.

Graph of the secant function

The familiar graph of the secant function can be seen as generated by the angle sweeping around the curve, as in the picture below. The two red line segments always have the same length.

References

Mathematica notebooks used in this post:

 

Posted in exposition, language, language of math, math, Mathematica, understanding math. Tags: , , , , , , , , , , , . No Comments »

Picturing derivatives

2011/08/19 — Charles Wells

This is my first experiment at posting an active Mathematica CDF document on my blog. To manipulate the graph below, you must have Wolfram CDF Player installed on your computer. It is available free from their website.

This is a new presentation of old work. It is a graph of a certain fifth degree polynomial and its first four derivatives.

The buttons allow you to choose how many derivatives to show and the slider allows you to show the graphs from $latex x=-4$ up to a certain point.

How graphs like this could be used for teaching purposes

You could show this in class, but the best way to learn from it would be to make it part of a discussion in which each student had access to a private copy of the graph.  (But you may have other ideas about how to use a graph like this.  Share them!)

Some possible discussion questions:

  1. Click button 1. Now you see the function and the derivative. Move the slider all the way to the left and then slowly move it to the right.  When the function goes up the derivative is positive.  What other things do you notice when you do this?
  2. If you were told only that one of the functions is the derivative of the other, how would you rule out the wrong possibility?
  3. What can you tell about the zeroes of the function by looking at the derivative?
  4. Look at the interval between $latex x=1.5$ and $latex x=1.75$.  Does the function have one or two zeroes in that interval?  On my screen it looks as if the curve just barely  gets above the $latex x$ axis in that interval.  What does that say about it having one or two zeroes?  How could you verify your answer?
  5. Click button 2.  Now you have the function and first and second derivatives.  What can you say about maxima, minima and concavity of the function?
  6. Find relationships between the first and second derivatives.
  7. Now click button 4.  Evidently the 4th derivative is a straight line with positive slope.  Assume that it is.  What does that tell you about the graph of the third derivative?
  8. What characteristics of the graph of the function can you tell from knowing that the fourth derivative is a straight line of positive slope?
  9. What can you say about the formula for the function knowing that the fourth derivative is a straight line of positive slope?
  10. Suppose you were given this graph and told that it was a graph of a function and its first four derivatives and nothing else.  Specifically, you do not know that the fourth derivative is a straight line.  Give a detailed explanation of how to tell which curve is the function and which curve is each specific derivative.

Making this manipulable graph

I posted this graph and a lot of others several years ago on abstractmath.org.  (It is the ninth graph down).  I fiddled with this polynomial until I got the function and all four derivatives to be separated from each other.  All the roots of the function and all its derivatives are real and all are shown.  Isn’t this gorgeous?

To get it to show up properly on the abmath site I had to thicken the graph line.  Otherwise it still showed up on the screen but when I printed it on my inkjet printer the curves disappeared. That seems to be unnecessary now.

Mathematica 8.0 has default colors for graphs, but I kept the old colors because they are easier to distinguish, for me anyway (and I am not color blind).

Inserting CDF documents into html

A Wolfram document explains how to do this.  I used the CDF plugin for WordPress.  WordPress requires that, to use the plugin, you operate your blog from your own server, not from WordPress.com.  That is the main reason for the recent change of site.

The Mathematica files are New5thDegreePolynomial.nb and New5thDegreePolynomial.cdf on my public folder of Mathematica files.  You may download the .cdf file directly and view it using CDF player if you have trouble with the embedded version. To see the code you need to download the .nb file and open all cells.

Here are some notes and questions on the process.  When I find learn more about any of these points I will post the information.

  1. At the moment I don’t know how to get rid of the extra space at the top of the graph.
  2. I was surprised that I could not click on the picture and shrink or expand it.
  3. It might be annoying for a student to read the questions above and have to go up and down the screen to see the graph.  I had envisioned that the teacher would ask the questions and have the students play with the graph and erupt with questions and opinions.  But you could open two copies of the .cdf file (or this blog) and keep one window showing the graph while the other window showed the questions.
  4. Which raises a question:  Could it be possible to program the graph with a button that when pushed would make the graph (only) appear in another window?

Other approaches

  1. I have experimented with Khan Academy type videos using CDF files.  I made a screen shot and at a certain point I pressed a button and the graph appropriately changed.   I expect to produce an example video which I can make appear on this blog (which supposedly can show videos, but I haven’t tried that yet.)
  2. It should be possible to have a CDF in which the student saw the graph with instructional text underneath it equipped with next and back buttons.  The next button would trigger changes in the picture and replace the text with another sentence or two.  This could be instead of spoken stuff or additional to it (which would be a lot of work).  Has anyone tried this?

Note

My reaction to Khan Academy was mostly positive.  One thing that struck me that no one seems to have commented on is that the lectures are short. They cover one aspect (one definition or one example or what one theorem says) in what felt to me like ten or fifteen minutes.  This means that you can watch it and easily go back and forth using the controls on the video display.  If it were a 50-minute lecture it would be much harder to find your way around.

I think most students are grasshoppers:  When reading text, they jump back and forth, getting the gist of some idea, looking ahead to see where it goes, looking back to read something again, and so on.  Short videos allow you to do this with spoken lectures. That seems to me remarkably useful.

Posted in exposition, math, Mathematica, representations, understanding math. Tags: , , , , , , . No Comments »

Comparing graph and cograph (Version 2)

2011/06/30 — Charles Wells

New Version 6 July 2011

This is a new version of a post originally created on 30 June, 2011. –Charles Wells

When you put the graph and cograph of a function with parameters side by side, interesting things may happen.  I have created the files CographExample.nb and CographExample.cdf to illustrate this.

The .nb form is a Mathematica Notebook, which requires Mathematica to run and allows you to manipulate the objects and change the code in the notebook as you wish.  The .cdf file contains the same material and can be viewed using Mathematica CDF Player, which is available free here.  The CDF Player allows you to change the parameters with the slidebars, so that you can experience the phenomena discusses in the example, but you cannot otherwise modify the file.

You cannot include a Mathematica computable document directly into a Word Press document, so here are screenshots of the cograph example with several different settings of its parameters.

Some Screenshots and sample questions follow.  These not in the original version.  I added these thanks to encouragement by Sam Alexander.

Screenshots


Questions for discussion

My idea is that students will manipulate the slider to see what happens and do some algebra on the relations between a, b and x to explain the phenomena that occur.

Questions about the graph (left figure).

G1. Prove that the two straight lines are parallel for any choice of a and b.

G2. When do the red and blue straight lines in the graph coincide? Answer: The lines coincide when $latex a=1 &fg=000000$ or $latex b=0 &fg=000000$.

  • “When do the…” translates into “for what values of a and b do the…”. I predict that which way you ask the question will make a big difference for some students. To answer this question, you are not solving for x but for a and b. If you ask “When…” they have to discover this for themselves.
  • The answer is disjunctive: This may be a new idea for some of the students.

G3. Find a formula in terms of a and b for the distance between the two straight lines in the graph. Answer: $latex |b-ab|$.

Questions about the cograph.

C1. When do the red and blue arrows in the cograph coincide? Answer: Same as answer to Question G1.

C2. In the cograph, for what aand b are the arrow targets for a given choice of x closer together than the arrow sources? Answer: $latex  |a|>1$. b is irrelevant.

C3. Manipulate a and b. For some values the blue arrows all cross each other at the same point. (Same question for red arrows.) When does this happen? Answer: When a is negative.

  • The abstract setting of the cograph is shifted by this question (and the next) as follows: The arrows originally provided a visual pointer from the input to the output of the function. All of a sudden we are treating them as mathematical objects(straight line segments in the plane).
  • The abstraction is also broken in another way: The space between the source line and the target line is just a visual separation, but after this question both lines lie on the xy plane. The question turns a visual illustration into a mathematical object.

C4.  Describe the common point where all the red arrows cross when a is negative. (Same question for blue arrows.) Answer: The point is $latex \left(\frac{b}{1-a},\frac{3 a}{a-1}\right)&fg=000000$.

Posted in exposition, math, Mathematica, representations, understanding math. Tags: , , , , , . 1 Comment »

Endograph and cograph of real functions

2011/06/09 — Charles Wells

This post is covered by the Creative Commons Attribution – ShareAlike 3.0 License, which means you may use, adapt and distribute the work provided you follow the requirements of the license.

Introduction

In the article Functions: Images and Metaphors in abstractmath I list a bunch of different images or metaphors for thinking about functions. Some of these metaphors have realizations in pictures, such as a graph or a surface shown by level curves. Others have typographical representations, as formulas, algorithms or flowcharts (which are also pictorial). There are kinetic metaphors — the graph of $latex {y=x^2}&fg=000000$ swoops up to the right.

Many of these same metaphors have realizations in actual mathematical representations.

Two images (not mentioned in the abstractmath article) are the cograph and the endograph of a real function of one variable. Both of these are visualizations that correspond to mathematical representations. These representations have been used occasionally in texts, but are not used as much as the usual graph of a continuous function. I think they would be useful in teaching and perhaps even sometimes in research.

A rough and unfinished Mathematica notebook is available that contains code that generate graphs and cographs of real-valued functions. I used it to generate most of the examples in this post, and it contains many other examples. (Note [1].)

The endograph of a function

In principle, the endograph (Note [2]) of a function $latex {f}&fg=000000$ has a dot for each element of the domain and of the codomain, and an arrow from $latex {x}&fg=000000$ to $latex {f(x)}&fg=000000$ for each $latex {x}&fg=000000$ in the domain. For example, this is the endograph of the function $latex {n\mapsto n^2+1 \pmod 11}&fg=000000$ from the set $latex {\{0,1,\ldots,10\}}&fg=000000$ to itself:


“In principle” means that the entire endograph can be shown only for small finite functions. This is analogous to the way calculus books refer to a graph as “the graph of the squaring function” when in fact the infinite tails are cut off.

Real endographs

I expect to discuss finite endographs in another post. Here I will concentrate on endographs of continuous functions with domain and codomain that are connected subsets of the real numbers. I believe that they could be used to good effect in teaching math at the college level.

Here is the endograph of the function $latex {y=x^2}&fg=000000$ on the reals:

I have displayed this endograph with the real line drawn in the usual way, with tick marks showing the location of the points on the part shown.

The distance function on the reals gives us a way of interpreting the spacing and location of the arrowheads. This means that information can be gleaned from the graph even though only a finite number of arrows are shown. For example you see immediately that the function has only nonnegative values and that its increase grows with $latex {x}&fg=000000$.(See note [3]).

I think it would be useful to show students endographs such as this and ask them specific questions about why the arrows do what they do.

For the one shown, you could ask these questions, probably for class discussion rather that on homework.

  • Explain why most of the arrows go to the right. (They go left only between 0 and 1 — and this graph has such a coarse point selection that it shows only two arrows doing that!)
  • Why do the arrows cross over each other? (Tricky question — they wouldn’t cross over if you drew the arrows with negative input below the line instead of above.)
  • What does it say about the function that every arrowhead except two has two curves going into it?

Real Cographs

The cograph (Note [4] of a real function has an arrow from input to output just as the endograph does, but the graph represents the domain and codomain as their disjoint union. In this post the domain is a horizontal representation of the real line and the codomain is another such representation below the domain. You may also represent them in other configurations (Note [5]).

Here is the cograph representation of the function $latex {y=x^2}&fg=000000$. Compare it with the endograph representation above.

Besides the question of most arrows going to the right, you could also ask what is the envelope curve on the left.

More examples

Absolute value function

Arctangent function

Notes

[1] This website contains other notebooks you might find useful. They are in Mathematica .nb, .nbp, or .cdf formats, and can be read, evaluated and modified if you have Mathematica 8.0. They can also be made to appear in your browser with Wolfram CDF Player, downloadable free from Wolfram site. The CDF player allows you to operate any interactive demos contained in the file, but you can’t evaluate or modify the file without Mathematica.

The notebooks are mostly raw code with few comments. They are covered by the Creative Commons Attribution – ShareAlike 3.0 License, which means you may use, adapt and distribute the code following the requirements of the license. I am making the files available because I doubt that I will refine them into respectable CDF files any time soon.

[2] I call them “endographs” to avoid confusion with the usual graphs of functions — – drawings of (some of) the set of ordered pairs $latex {x,f(x)}&fg=000000$ of the function.

[3] This is in contrast to a function defined on a discrete set, where the elements of the domain and codomain can be arranged in any old way. Then the significance of the resulting arrangement of the arrows lies entirely in which two dots they connect. Even then, some things can be seen immediately: Whether the function is a cycle, permutation, an involution, idempotent, and so on.

Of course, the placement of the arrows may tell you more if the finite sets are ordered in a natural way, as for example a function on the integers modulo some integer.

[4] The text [1] uses the cograph representation extensively. The word “cograph” is being used with its standard meaning in category theory. It is used by graph theorists with an entirely different meaning.

[5] It would also be possible to show the domain codomain in the usual $latex {x-y}&fg=000000$ plane arrangement, with the domain the $latex {x}&fg=000000$ axis and the codomain the $latex {y}&fg=000000$ axis. I have not written the code for this yet.

References

[1] Sets for Mathematics, by F. William Lawvere and Robert Rosebrugh. Cambridge University Press, 2003.

Posted in abstractmath.org, exposition, language of math, math, Mathematica, representations, understanding math. Tags: , , , , , , , , , , . No Comments »

Representations 2

2011/05/25 — Charles Wells

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 $latex {y=x^3-x}&fg=000000$.

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 $latex {(x,x^3-x)}&fg=000000$ for all real numbers $latex {x}&fg=000000$.
  2. In the physical representation, each point $latex {(x,x^3-x)}&fg=000000$ is shown in a location determined by the conventional $latex {x-y}&fg=000000$ 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 $latex {(x,x^3-x)}&fg=000000$.
    • The correspondence between the locations and the pairs is imperfect. You can’t measure with infinite accuracy.
    • The set of ordered pairs $latex {(x,x^3-x)}&fg=000000$ 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, $latex {(-1,0)}&fg=000000$) 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.

Posted in abstractmath.org, exposition, language of math, math, representations, understanding math. Tags: , , , , , , , , . No Comments »
« Older posts «
»