Gyre&Gimbleposts about math, language and other things that may appear in the wabe

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. Mirror neurons (Wikipedia)
9. Representations and models (article in abstractmath)
10. Representations II: dry bones (article in abstractmath)
11. The transition to formal thinking in mathematics, David Tall, 2010
12. What is the object of the encapsulation of a process? Tall et al., 2000.

The Mathematical Definition of Function

2011/05/20 — Charles Wells

Introduction

This post is a completely rewritten version of the abstractmath article on the definition of function. Like every part of abstractmath, the chapter on functions is designed to get you started thinking about functions. It is no way complete. Wikipedia has much more complete coverage of mathematical functions, but be aware that the coverage is scattered over many articles.

The concept of function in mathematics is as important as any mathematical idea. The mathematician’s concept of function includes the kinds of functions you studied in calculus but is much more abstract and general. If you are new to abstract math you need to know:

• The precise meaning of the word “function” and other concepts associated with functions. That’s what this section is about.
• Notation and terminology for functions. (That will be a separate section of abstractmath.org which I will post soon.)
• The many different kinds of functions there are. (See Examples of Functions in abmath).
• The many ways mathematicians think about functions. The abmath article Images and Metaphors for Functions is a stub for this.

I will use two running examples throughout this discussion:

• $latex {F}&fg=000000$ is the function defined on the set $latex {\left\{1,\,2,3,6 \right\}}&fg=000000$ as follows: $latex {F(1)=3,\,\,\,F(2)=3,\,\,\,F(3)=2,\,\,\,F(6)=1}&fg=000000$. This is a function defined on a finite set by explicitly naming each value.
• $latex {G}&fg=000000$ is the real-valued function defined by the formula $latex {G(x)={{x}^{2}}+2x+5}&fg=000000$.

Specification of function

We start by giving a specification of “function”. (See the abstractmath article on specification.) After that, we get into the technicalities of the definitions of the general concept of function.

Specification: A function $latex {f}&fg=000000$ is a mathematical object which determines and is completely determined bythe following data:

• $latex {f}&fg=000000$ has a domain, which is a set. The domain may be denoted by $latex {\text{dom }f}&fg=000000$.
• $latex {f}&fg=000000$ has a codomain, which is also a set and may be denoted by $latex {\text{cod }f}&fg=000000$.
• For each element $latex {a}&fg=000000$ of the domain of $latex {f}&fg=000000$, $latex {f}&fg=000000$ has a value at $latex {a}&fg=000000$, denoted by $latex {f(a)}&fg=000000$.
• The value of $latex {f}&fg=000000$ at $latex {a}&fg=000000$ is completely determined by $latex {a}&fg=000000$ and $latex {f}&fg=000000$ .
• The value of $latex {f}&fg=000000$ at $latex {a}&fg=000000$ must be an element of the codomain of $latex {f}&fg=000000$.

The operation of finding $latex {f(a)}&fg=000000$ given $latex {f}&fg=000000$ and $latex {a}&fg=000000$ is called evaluation.

Examples

• The definition above of the finite function $latex {F}&fg=000000$ specifies that the domain is the set $latex {\left\{1,\,2,\,3,\,6 \right\}}&fg=000000$. The value of $latex {F}&fg=000000$ at each element of the domain is given explicitly. The value at 3, for example, is 2, because the definition says that $latex {F(2) = 3}&fg=000000$. The codomain of $latex {F}&fg=000000$ is not specified, but must include the set $latex {\{1,2,3\}}&fg=000000$.
• The definition of $latex {G}&fg=000000$ above gives the value at each element of the domain by a formula. The value at 3, for example, is $latex {G(3)=3^2+2\cdot3+5=20}&fg=000000$. The definition does not specify the domain or the codomain. The convention in the case of functions defined on the real numbers by a formula is to take the domain to be all real numbers at which the formula is defined. In this case, that is every real number, so the domain is $latex {{\mathbb R}}&fg=000000$. The codomain must include all real numbers greater than or equal to 4. (Why?)

Comment: The formula above that defines the function $latex G$ in fact defines a function of complex numbers (even quaternions).

Definition of function

In the nineteenth century, mathematicians realized that it was necessary for some purposes (particularly harmonic analysis) to give a mathematical definition of the concept of function. A stricter version of this definition turned out to be necessary in algebraic topology and other fields, and that is the one I give here.

To state this definition we need a preliminary idea.

The functional property

A set R of ordered pairs has the functional property if two pairs in R with the same first coordinate have to have the same second coordinate (which means they are the same pair).

Examples

• The set $latex {\{(1,2), (2,4), (3,2), (5,8)\}}&fg=000000$ has the functional property, since no two different pairs have the same first coordinate. It is true that two of them have the same second coordinate, but that is irrelevant.
• The set $latex {\{(1,2), (2,4), (3,2), (2,8)\}}&fg=000000$ does not have the functional property. There are two different pairs with first coordinate 2.
• The graphs of functions in beginning calculus have the functional property.
• The empty set $latex {\emptyset}&fg=000000$ has the functional property .

Example: Graph of a function defined by a formula

The graph of the function $latex {G}&fg=000000$ given above has the functional property. The graph is the set

$latex \displaystyle \left\{ (x,{{x}^{2}}+2x+5)\,\mathsf{|}\,x\in {\mathbb R} \right\}.&fg=000000$

If you repeatedly plug in one real number over and over, you get out the same real number every time. Example:

• if $latex {x = 0}&fg=000000$, then $latex {{{x}^{2}}+2x+5=5}&fg=000000$.  You get 5 every time you plug in 0.
• if $latex {x = 1}&fg=000000$, then $latex {{{x}^{2}}+2x+5=8}&fg=000000$.
• if $latex {x =-2}&fg=000000$, then $latex {{{x}^{2}}+2x+5=5}&fg=000000$.

This set has the functional property because if $latex {x}&fg=000000$ is any real number, the formula $latex {{{x}^{2}}+2x+5}&fg=000000$ defines a specific real number. (This description of the graph implicitly assumes that $latex {\text{dom } G={\mathbb R}}&fg=000000$.)  No other pair whose first coordinate is $latex {-2}&fg=000000$ is in the graph of $latex {G}&fg=000000$, only $latex {(-2, 5)}&fg=000000$. That is because when you plug $latex {-2}&fg=000000$ into the formula $latex {{{x}^{2}}+2x+5}&fg=000000$, you get $latex {5}&fg=000000$ every time. Of course, $latex {(0, 5)}&fg=000000$ is in the graph, but that does not contradict the functional property. $latex {(0, 5)}&fg=000000$ and $latex {(-2, 5)}&fg=000000$ have the same second coordinate, but that is OK.

How to think about the functional property

The point of the functional property is that for any pair in the set of ordered pairs, the first coordinate determines what the second one is. That’s why you can write “$latex {G(x)}&fg=000000$” for any $latex {x }&fg=000000$ in the domain of $latex {G}&fg=000000$ and not be ambiguous.

Mathematical definition of function

A function$latex {f}&fg=000000$ is a mathematical structure consisting of the following objects:

• A set called the domain of $latex {f}&fg=000000$, denoted by $latex {\text{dom } f}&fg=000000$.
• A set called the codomain of $latex {f}&fg=000000$, denoted by $latex {\text{cod } f}&fg=000000$.
• A set of ordered pairs called the graph of $latex { f}&fg=000000$, with the following properties:
• $latex {\text{dom } f}&fg=000000$ is the set of all first coordinates of pairs in the graph of $latex {f}&fg=000000$.
• Every second coordinate of a pair in the graph of $latex {f}&fg=000000$ is in $latex {\text{cod } f}&fg=000000$ (but $latex {\text{cod } f}&fg=000000$ may contain other elements).
• The graph of $latex {f}&fg=000000$ has the functional property. Using arrow notation, this implies that $latex {f:A\rightarrow B}&fg=000000$.

Examples

• Let $latex {F}&fg=000000$ have graph $latex {\{(1,2), (2,4), (3,2), (5,8)\}}&fg=000000$ and define $latex {A = \{1, 2, 3, 5\}}&fg=000000$ and $latex {B = \{2, 4, 8\}}&fg=000000$. Then $latex {F:A\rightarrow B}&fg=000000$ is a function.
• Let $latex {G}&fg=000000$ have graph $latex {\{(1,2), (2,4), (3,2), (5,8)\}}&fg=000000$ (same as above), and define $latex {A = \{1, 2, 3, 5\}}&fg=000000$ and $latex {C = \{2, 4, 8, 9, 11, \pi, 3/2\}}&fg=000000$. Then $latex {G:A\rightarrow C}&fg=000000$ is a (admittedly ridiculous) function. Note that all the second coordinates of the graph are in $latex {C}&fg=000000$, along with a bunch of miscellaneous suspicious characters that are not second coordinates of pairs in the graph.
• Let $latex {H}&fg=000000$ have graph $latex {\{(1,2), (2,4), (3,2), (5,8)\}}&fg=000000$. Then $latex {H:A\rightarrow {\mathbb R}}&fg=000000$ is a function.

According to the definition of function, $latex {F}&fg=000000$, $latex {G}&fg=000000$ and $latex {H}&fg=000000$ are three different functions.

Identity and inclusion

Suppose we have two sets A and B with $latex {A\subseteq B}&fg=000000$.

• The identity function on A is the function $latex {{{\text{id}}_{A}}:A\rightarrow A}&fg=000000$ defined by $latex {{{\text{id}}_{A}}(x)=x}&fg=000000$ for all$latex {x\in A}&fg=000000$. (Many authors call it $latex {{{1}_{A}}}&fg=000000$).
• The inclusion function from A to B is the function $latex {i:A\rightarrow B}&fg=000000$ defined by $latex {i(x)=x}&fg=000000$ for all $latex {x\in A}&fg=000000$. Note that there is a different function for each pair of sets A and B for which $latex {A\subseteq B}&fg=000000$. Some authors call it $latex {{{i}_{A,\,B}}}&fg=000000$ or $latex {\text{in}{{\text{c}}_{A,\,B}}}&fg=000000$.

Remark The identity function and an inclusion function for the same set A have exactly the same graph, namely $latex {\left\{ (a,a)|a\in A \right\}}&fg=000000$.

Graphs and functions

• If $latex {f}&fg=000000$ is a function, the domain of $latex {f}&fg=000000$ is the set of first coordinates of all the pairs in $latex {f}&fg=000000$.
• If $latex {x\in \text{dom } f}&fg=000000$, then $latex {f(x)}&fg=000000$ is the second coordinate of the only ordered pair in $latex {f}&fg=000000$ whose first coordinate is $latex {x}&fg=000000$.

Examples

The set $latex {\{(1,2), (2,4), (3,2), (5,8)\}}&fg=000000$ has the functional property, so it is the graph of a function. Call the function $latex {H}&fg=000000$. Then its domain is $latex {\{1,2,3,5\}}&fg=000000$ and $latex {H(1) = 2}&fg=000000$ and $latex {H(2) = 4}&fg=000000$. $latex {H(4)}&fg=000000$ is not defined because there is no ordered pair in H beginning with $latex {4}&fg=000000$ (hence $latex {4}&fg=000000$ is not in $latex {\text{dom } H}&fg=000000$.)

I showed above that the graph of the function $latex {G}&fg=000000$, ordinarily described as “the function $latex {G(x)={{x}^{2}}+2x+5}&fg=000000$”, has the functional property. The specification of function requires that we say what the domain is and what the value is at each point. These two facts are determined by the graph.

Other definitions of function

Because of the examples above, many authors define a function as a graph with the functional property. Now, the graph of a function $latex {G}&fg=000000$ may be denoted by $latex {\Gamma(G)}&fg=000000$.  This is an older, less strict definition of function that doesn’t work correctly with the concepts of algebraic topology, category theory, and some other branches of mathematics.

For this less strict definition of function, $latex {G=\Gamma(G)}&fg=000000$, which causes a clash of our mental images of “graph” and “function”. In every important way except the less-strict definition, they ARE different!

A definition is a device for making the meaning of math technical terms precise. When a mathematician think of “function” they think of many aspects of functions, such as a map of one shape into another, a graph in the real plane, a computational process, a renaming, and so on. One of the ways of thinking of a function is to think about its graph. That happens to be the best way to define the concept of function.  (It is the less strict definition and it is a necessary concept in the modern definition given here.)

The occurrence of the graph in either definition doesn’t make thinking of a function in terms of its graph the most important way of visualizing  it. I don’t think it is even in the top three.

Templates in mathematical practice

2010/03/11 — Charles Wells

This post is a first pass at what will eventually be a section of abstractmath.org. It’s time to get back to abstractmath; I have been neglecting it for a couple of years.

What I say here is based mainly on my many years of teaching discrete mathematics at Case Western Reserve University in Cleveland and more recently at Metro State University in Saint Paul.

Beginning abstract math

College students typically get into abstract math at the beginning in such courses as linear algebra, discrete math and abstract algebra. Certain problems that come up in those early courses can be grouped together under the notion of (what I call) applying templates [note 0]. These are not the problems people usually think about concerning beginners in abstract math, of which the following is an incomplete list:

• Reasoning from axioms
• Encapsulating processes (Tall et al, 2000).
• Semantic contamination (abstractmath)
• Translating between mathematical English and logic (abstractmath)

The students’ problems discussed here concern understanding what a template is and how to apply it.

Templates can be formulas, rules of inference, or mini-programs. I’ll talk about three examples here.

The template for quadratic equations

The solution of a real quadratic equation of the form $latex {ax^2+bx+c=0}&fg=000000$ is given by the formula

$latex \displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}&fg=000000$

This is a template for finding the roots of the equations. It has subtleties.

For example, the numerator is symmetric in $latex {a}&fg=000000$ and $latex {c}&fg=000000$ but the denominator isn’t. So sometimes I try to trick my students (warning them ahead of time that that’s what I’m trying to do) by asking for a formula for the solution of the equation $latex {a+bx+cx^2=0}&fg=000000$. The answer is

$latex \displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2c}&fg=000000$

I start writing it on the board, asking them to tell me what comes next. When we get to the denominator, often someone says “$latex {2a}&fg=000000$”.

The template is telling you that the denominator is 2 times the coefficient of the square term. It is not telling you it is “$latex {a}&fg=000000$”. Using a template (in the sense I mean here) requires pattern matching, but in this particular example, the quadratic template has a shallow incorrect matching and a deeper correct matching. In detail, the shallow matching says “match the letters” and the deep matching says “match the position of the letters”.

Most of the time the quadratic being matched has particular numbers instead of the same letters that the template has, so the trap I just described seldom occurs. But this makes me want to try a variation of the trick: Find the solution of $latex {3+5x+2x^2=0}&fg=000000$. Would some students match the textual position (getting $latex {a=3}&fg=000000$) instead of the functional position (getting $latex {a=5}&fg=000000$)? [Note [0]). If they did they would get the solutions $latex {(-1,-\frac{2}{3})}&fg=000000$ instead of $latex {(-1,-\frac{3}{2})}&fg=000000$.

Substituting in algebraic expressions have other traps, too. What sorts of mistakes would students have solving $latex {3x^2+b^2x-5=0}&fg=000000$?

Most students on the verge of abstract math don’t make mistakes with the quadratic formula that I have described. The thing about abstract math is that it uses more sophisticated templates

• subject to conditions
• with variations
• with extra levels of abstraction

The template for proof by induction

This template gives a method of proof of a statement of the form $latex {\forall{n}\mathcal{P}(n)}&fg=000000$, where $latex {\mathcal{P}}&fg=000000$ is a predicate (presumably containing $latex {n}&fg=000000$ as a variable) and $latex {n}&fg=000000$ varies over positive integers. The template says:

Goal: Prove $latex {\forall{n}\mathcal{P}(n)}&fg=000000$.

Method:

• Prove $latex {\mathcal{P}(1)}&fg=000000$
• For an arbitrary integer $latex {n>1}&fg=000000$, assume $latex {\mathcal{P}(n)}&fg=000000$ and deduce $latex {\mathcal{P}(n+1)}&fg=000000$.

For example, to prove $latex {\forall n (2^n+1\geq n^2)}&fg=000000$ using the template, you have to prove that $latex {2^2+1\geq 1^1}&fg=000000$, and that for any $latex {n>1}&fg=000000$, if $latex {2^n+1\geq n^2}&fg=000000$, then $latex {2^{n+1}+1\geq (n+1)^2}&fg=000000$. You come up with the need to prove these statements by substituting into the template. This template has several problems that the quadratic formula does not have.

Variables of different types

The variable $latex {n}&fg=000000$ is of type integer and the variable $latex {\mathcal{P}}&fg=000000$ is of type predicate [note 0]. Having to deal with several types of variables comes up already in multivariable calculus (vectors vs. numbers, cross product vs. numerical product, etc) and they multiply like rabbits in beginning abstract math classes. Students sometimes write things like “Let $latex {\mathcal{P}=n+1}&fg=000000$”. Multiple types is a big problem that math ed people don’t seem to discuss much (correct me if I am wrong).

The variable $latex {n}&fg=000000$ occurs as a bound variable in the Goal and a free variable in the Method. This happens in this case because the induction step in the Method originates as the requirement to prove $latex {\forall n(\mathcal{P}(n)\rightarrow\mathcal{P}(n+1))}&fg=000000$, but as I have presented it (which seems to be customary) I have translated this into a requirement based on modus ponens. This causes students problems, if they notice it. (“You are assuming what you want to prove!”) Many of them apparently go ahead and produce competent proofs without noticing the dual role of $latex {n}&fg=000000$. I say more power to them. I think.

The template has variations

• You can start the induction at other places.
• You may have to have two starting points and a double induction hypothesis (for $latex {n-1}&fg=000000$ and $latex {n}&fg=000000$). In fact, you will have to have two starting points, because it seems to be a Fundamental Law of Discrete Math Teaching that you have to talk about the Fibonacci function ad nauseam.
• Then there is strong induction.

It’s like you can go to the store and buy one template for quadratic equations, but you have to by a package of templates for induction, like highway engineers used to buy packages of plastic French curves to draw highway curves without discontinuous curvature.

The template for row reduction

I am running out of time and won’t go into as much detail on this one. Row reduction is an algorithm. If you write it up as a proper computer program there have to be all sorts of if-thens depending on what you are doing it for. For example if want solutions to the simultaneous equations

|         |    |    |
|---------|----|----|
| 2x+4y+z | =  |  1 |
| x+2y    | =  |  0 |
| x+2y+4z | =  |  5 |

you must row reduce the matrix

|    |    |    |    |
|----|----|----|----|
|  2 |  4 |  1 |  1 |
|  1 |  2 |  0 |  0 |
|  1 |  2 |  4 |  5 |

(I haven’t yet figured out how to wrap this in parentheses) which gives you

|    |    |    |    |
|----|----|----|----|
|  1 |  2 |  0 |  0 |
|  0 |  0 |  1 |  0 |
|  0 |  0 |  0 |  1 |

This introduces another problem with templates: They come with conditions. In this case the condition is “a row of three 0s followed by a nonzero number means the equations have no solutions”. (There is another condition when there is a row of all 0′s.)

It is very easy for the new student to get the calculation right but to never sit back and see what they have — which conditions apply or whatever.

When you do math you have to repeatedly lean in and focus on the details and then lean back and see the Big Picture. This is something that has to be learned.

What to do, what to do

I have recently experimented with being explicit about templates, in particular going through examples of the use of a template after explicitly stating the template. It is too early to say how successful this is. But I want to point out that even though it might not help to be explicit with students about templates, the analysis in this post of a phenomenon that occurs in beginning abstract math courses

• may still be accurate (or not), and
• may help teachers teach such things if they are aware of the phenomenon, even if the students are not.

Notes

1. Many years ago, I heard someone use the word “template” in the way I am using it now, but I don’t recollect who it was. Applied mathematicians sometimes use it with a meaning similar to mine to refer to soft algorithms–recipes for computation that are not formal algorithms but close enough to be easily translated into a sufficiently high level computer language.
2. In the formula $latex {ax^2+bx+c}&fg=000000$, the “$latex {a}&fg=000000$” has the first textual position but the functional position as the coefficient of the quadratic term. This name “functional position” has nothing to do with functions. Can someone suggest a different name that won’t confuse people?
3. I am using “variable” the way logicians do. Mathematicians would not normally refer to “$latex {\mathcal{P}}&fg=000000$” as a variable.
4. I didn’t say anything about how templates can involve extra layers of abstract.  That will have to wait.
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