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Syntax Trees in Mathematicians’ Brains

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Understanding the quadratic formula

In my last post I wrote about how a student’s pattern recognition mechanism can go awry in applying the quadratic formula.

The template for the quadratic formula says that the solution of a quadratic equation of the form ${ax^2+bx+c=0}$ is given by the formula

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

When you ask students to solve ${a+bx+cx^2=0}$ some may write

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

instead of

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

That’s because they have memorized the template in terms of the letters ${a}$, ${b}$ and ${c}$ instead of in terms of their structural meaning — $ {a}$ is the coefficient of the quadratic term, ${c}$ is the constant term, etc.

The problem occurs because there is a clash between the occurrences of the letters “a”, “b”, and “c” in the template and in the equation to solve. But maybe the confusion would occur anyway, just because of the ordering of the coefficients. As I asked in the previous post, what happens if students are asked to solve $ {3+5x+2x^2=0}$ after having learned the quadratic formula in terms of ${ax^2+bx+c=0}$? Some may make the same kind of mistake, getting ${x=-1}$ and ${x=-\frac{2}{3}}$ instead of $ {x=-1}$ and $ {x=-\frac{3}{2}}$. Has anyone ever investigated this sort of thing?

People do pattern recognition remarkably well, but how they do it is mysterious. Just as mistakes in speech may give the linguist a clue as to how the brain processes language, students’ mistakes may tell us something about how pattern recognition works in parsing symbolic statements as well as perhaps suggesting ways to teach them the correct understanding of the quadratic formula.

Syntactic Structure

“Structural meaning” refers to the syntactic structure of a mathematical expression such as ${3+5x+2x^2}$. It can be represented as a tree:

(1)

This is more or less the way a program compiler or interpreter for some language would represent the polynomial. I believe it corresponds pretty well to the organization of the quadratic-polynomial parser in a mathematician’s brain. This is not surprising: The compiler writer would have to have in mind the correct understanding of how polynomials are evaluated in order to write a correct compiler.

Linguists represent English sentences with syntax trees, too. This is a deep and complicated subject, but the kind of tree they would use to represent a sentence such as “My cousin saw a large ship” would look like this:

Parsing by mathematicians

Presumably a mathematician has constructed a parser that builds a structure in their brain corresponding to a quadratic polynomial using the same mechanisms that as a child they learned to parse sentences in their native language. The mathematician learned this mostly unconsciously, just as a child learns a language. In any case it shouldn’t be surprising that the mathematicians’s syntax tree for the polynomial is similar to the compiler’s.

Students who are not yet skilled in algebra have presumably constructed incorrect syntax trees, just as young children do for their native language.

Lots of theoretical work has been done on human parsing of natural language. Parsing mathematical symbolism to be compiled into a computer program is well understood. You can get a start on both of these by reading the Wikipedia articles on parsing and on syntax trees.

There are papers on students’ misunderstandings of mathematical notation. Two articles I recently turned up in a Google search are:

Both of these papers talk specifically about the syntax of mathematical expressions. I know I have read other such papers in the past, as well.

What I have not found is any study of how the trained mathematician parses mathematical expression.

For one thing, for my parsing of the expression $ {3+5x+2x^2}$, the branching is wrong in (1). I think of ${3+5x+2x^2}$ as “Take 3 and add $ {5x}$ to it and then add ${2x^2}$ to that”, which would require the shape of the tree to be like this:

I am saying this from introspection, which is dangerous!

Of course, a compiler may group it that way, too, although my dim recollection of the little bit I understand about compilers is that they tend to group it as in (1) because they read the expression from left to right.

This difference in compiling is well-understood.  Another difference is that the expression could be compiled using addition as an operator on a list, in this case a list of length 3.  I don’t visualize quadratics that way but I certainly understand that it is equivalent to the tree in Diagram (1).  Maybe some mathematicians do think that way.

But these observations indicate what might be learned about mathematicians’ understanding of mathematical expressions if linguists and mathematicians got together to study human parsing of expressions by trained mathematicians.

Some educational constructivists argue against the idea that there is only one correct way to understand a mathematical expression.  To have many metaphors for thinking about math is great, but I believe we want uniformity of understanding of the symbolism, at least in the narrow sense of parsing, so that we can communicate dependably.  It would be really neat if we discovered deep differences in parsing among mathematicians.  It would also be neat if we discovered that mathematicians parsed in generally the same way!

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Templates in mathematical practice

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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:

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

  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 {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?
  3. I am using “variable” the way logicians do. Mathematicians would not normally refer to “{\mathcal{P}}” 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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