Templates in mathematical practice

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 ${ax^2+bx+c=0}$ is given by the formula

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

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

For example, the numerator is symmetric in ${a}$ and ${c}$ 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 ${a+bx+cx^2=0}$ . The answer is

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

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

The template is telling you that the denominator is 2 times the coefficient of the square term. It is not telling you it is “ ${a}$ ”. 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 ${3+5x+2x^2=0}$ . Would some students match the textual position (getting ${a=3}$ ) instead of the functional position (getting ${a=5}$ )? [Note [0]). If they did they would get the solutions ${(-1,-\frac{2}{3})}$ instead of ${(-1,-\frac{3}{2})}$ .

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

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 ${\forall{n}\mathcal{P}(n)}$ , where ${\mathcal{P}}$ is a predicate (presumably containing ${n}$ as a variable) and ${n}$ varies over positive integers. The template says:

Goal: Prove ${\forall{n}\mathcal{P}(n)}$ .

Method:

Prove ${\mathcal{P}(1)}$
For an arbitrary integer ${n>1}$ , assume ${\mathcal{P}(n)}$ and deduce ${\mathcal{P}(n+1)}$ .

For example, to prove ${\forall n (2^n+1\geq n^2)}$ using the template, you have to prove that ${2^2+1\geq 1^1}$ , and that for any ${n>1}$ , if ${2^n+1\geq n^2}$ , then ${2^{n+1}+1\geq (n+1)^2}$ . 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 ${n}$ is of type integer and the variable ${\mathcal{P}}$ 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 ${\mathcal{P}=n+1}$ ”. Multiple types is a big problem that math ed people don’t seem to discuss much (correct me if I am wrong).

Free and bound

The variable ${n}$ 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 ${\forall n(\mathcal{P}(n)\rightarrow\mathcal{P}(n+1))}$ , 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 ${n}$ . 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 ${n-1}$ and ${n}$ ). 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

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.
In the formula ${ax^2+bx+c}$ , the “ ${a}$ ” 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?
I am using “variable” the way logicians do. Mathematicians would not normally refer to “ ${\mathcal{P}}$ ” as a variable.
I didn’t say anything about how templates can involve extra layers of abstract. That will have to wait.

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4 thoughts on “Templates in mathematical practice”

Edwin Montufar says:

2010/03/13 at 1:07 pm

I’m glad to know you’re focusing on abstractmath again. It’s been really useful to me these past couple semesters, given that I’m a math major. I think the website is great, and if you’d like some constructive feedback, I’d be more than happy to contribute. As for the notion of a template, I myself have always thought of formulas and algorithms as templates associated with a form of pattern matching. In this pattern matching, I never thought of looking for textual positions, instead I focused on functional positions that met certain criteria and corresponded with a position of the template in mind.

In response to this:
“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”

I think it’s a matter of establishing the notion of equivalent forms. That is, we can think of certain templates with variations as being equivalent to each other, but of course being presented differently. I don’t know if this sounds right, but it’s kind of like saying how the well-ordering principle is equivalent to mathematical induction. Or how mathematical induction is equivalent to the principle of induction for well-formed formulas in mathematical logic. Because of this, not all equivalent forms may seem to have an easily identifiable template as well.

some guy on the street says:

2010/03/17 at 3:53 am

Thinking about ax2+bx+c, and what to call the role that a plays — if you’re not content to say “a is the coefficient of the quadratic term”, wanting an expression that gives notional priority to a, you might say that a‘s cofactor is x2. That would agree nicely with the usage of the word cofactor in matrix algebra — the determinant of a matrix being a (normalized or restricted) sum of coefficients-times-cofactors.

Alternatively, (it’s ugly!) one might write the quadritic formula as x=(-f'(0) +/- sqrt( f'(0)^2 – 2 f(0) f”(0)))/(2f(0)), where f is the quadratic function whose roots are wanted. I also think this fully qualifies as precision at the expense of clarity.

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Templates in mathematical practice

4 thoughts on “Templates in mathematical practice”

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