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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