Explaining math

2013/03/26 — SixWingedSeraph

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 source for the demos is the Mathematica notebook SolvEq.nb.

This post explains some basic distinctions that need to be made about the process of writing and explaining math.  Everyone who teaches math knows subconsciously what is happening here; I am trying to raise your consciousness.  For simplicity, I have chosen a technique used in elementary algebra, but much of what I suggest also applies to more abstract college level math.

An algebra problem

Solve the equation "$ax=b$" ($a\neq0$).

Understanding the statement of this problem requires a lot of Secret Knowledge (the language of ninth grade algebra) that most people don't have.

  • The expression "$ax$" means that $a$ and $x$ are numbers and $ax$ is their product. It is not the word "ax". You have to know that writing two symbols next to each other means multiply them, except when it doesn't mean multiply them as in "$\sin\,x$".

  • The whole expression "$ax=b$" ostensibly says that the number $ax$ is the same number as $b$.  In fact, it means more than that. The phrase "solve the equation" tells you that in fact you are supposed to find the value of $x$ that makes $ax$ the same number as $b$.

  • How do you know that "solve the equation" doesn't mean find the value of $a$ that makes $ax$ the same number as $b$? Answer: The word "solve" triggers a convention that $x$, $y$ and $z$ are numbers you are trying to find and $a$, $b$, $c$ stand for numbers that you are allowed to plug in to the equation.

  • The conventions of symbolic math require that you give a solution for any nonzero value of $a$ and any value of $b$.  You specifically are not allowed to pick $a=1$ and $b=33$ and find the value just for those numbers.  (Some college calculus students do this with problems involving literal coefficients.)

  • The little thingy "$(a\neq0)$" must be read as a constraint on $a$.  It does not mean that $a\neq0$ is a fact that you ought to know. ( I've seen college math students make this mistake, admittedly in more complex situations). Nor does it mean that you can't solve the problem if $a=0$ (you can if $b$ is also zero!).

So understanding what this problem asks, as given, requires (fairly sophisticated in some cases) pattern recognition both to understand the symbolic language it uses, and also to understand the special conventions of the mathematical English that it uses.

Explicit descriptions

This problem could be reworded so that it gives an explicit description of the problem, not requiring pattern recognition.  (Warning: "Not requiring pattern recognition" is a fuzzy concept.)  Something like this:  

You have two numbers $a$ and $b$.  Find a number $c$ for which if you multiply $a$ by $c$ you get $b$.

This version is not completely explicit.  It still requires understanding the idea of referring to a number by a letter, and it still requires pattern recognition to catch on that the two occurrences of each letter means that their meanings have to match. Also, I know from experience that some American first year college students have trouble with the syntax of the sentence ("for which…", "if…").

The following version is more explicit, but it cheats by creating an ad hoc way to distinguish the numbers.

Alice and Bob each give you a number.  How do you find a number with the property that Alice's number times your number is equal to Bob's number? 

If the problem had a couple more variables it would be so difficult to understand in an explicit form that most people would have to draw a picture of the relationships between them.  That is why algebraic notation was invented.

Visual descriptions

Algebra is a difficult foreign language.  Showing the problem visually makes it easier to understand for most people. Our brain's visual processing unit is the most powerful tool the brain has to understand things.  There are various ways to do this.  

Visualization can help someone understand algebraic notation better.  

You can state the problem by producing examples such as

  • $\boxed{3}\times\boxed{\text{??}}=\boxed{6}$ 
  • $\boxed{5}\times\boxed{\text{??}}=\boxed{2}$ 
  • $\boxed{42}\times\boxed{\text{??}}=\boxed{24}$

where the reader has to know the multiplication symbol and, one hopes, will recognize "$\boxed{\text{??}}$" as "What's the value?". But the reader does not have to understand what it means to use letters for numbers, or that "$x$ means you are suppose to discover what it is".  This way of writing an algebra problem is used in some software aimed at K-12 students.  Some of them use a blank box instead of "$\boxed{\text{??}}$".

Such software often shows the algorithm for solving the problem visually, using algebraic notation like this:

I have put in some buttons to show numbers as well as $a$ and $b$.  If you have access to Mathematica instead of just to CDF player, you can load SolvEq.nb and put in any numbers you want, but CDF's don't allow input data. 

You can also illustrate the algorithm using the tree notation for algebra I used in Monads for high school I  (and other posts). The demo below shows how to depict the value-preserving transformation given by the algorithm.  (In this case the value is the truth since the root operation is equals.)

This demo is not as visually satisfactory as the one illustrating the use of the associative law in Monads for high school I.  For one thing, I had to cheat by reversing the placement of $a$ and $x$.  Note that I put labels for the numerator and denominator legs, a practice I have been using in demos for a while for noncommutative operations.  I await a new inspiration for a better presentation of this and other equation-solving algorithms.

Another advantage of using pictures is that you can often avoid having to code things as letters which then has to be remembered.  In Monads for high school I, I used drawings of the four functions from a two-element set to itself instead of assigning them letters.  Even mnemonic letters such as $s$ for "switch" and $\text{id}$ for the identity element carry a burden that the picture dispenses with.

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

Syntax Trees in Mathematicians’ Brains

2010/03/29 — Charles Wells

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 $latex {ax^2+bx+c=0}&fg=000000$ is given by the formula

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

When you ask students to solve $latex {a+bx+cx^2=0}&fg=000000$ some may write

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

instead of

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

That’s because they have memorized the template in terms of the letters $latex {a}&fg=000000$, $latex {b}&fg=000000$ and $latex {c}&fg=000000$ instead of in terms of their structural meaning — $latex {a}&fg=000000$ is the coefficient of the quadratic term, $latex {c}&fg=000000$ 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 $latex {3+5x+2x^2=0}&fg=000000$ after having learned the quadratic formula in terms of $latex {ax^2+bx+c=0}&fg=000000$? Some may make the same kind of mistake, getting $latex {x=-1}&fg=000000$ and $latex {x=-\frac{2}{3}}&fg=000000$ instead of $latex {x=-1}&fg=000000$ and $latex {x=-\frac{3}{2}}&fg=000000$. 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 $latex {3+5x+2x^2}&fg=000000$. 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 $latex {3+5x+2x^2}&fg=000000$, the branching is wrong in (1). I think of $latex {3+5x+2x^2}&fg=000000$ as “Take 3 and add $latex {5x}&fg=000000$ to it and then add $latex {2x^2}&fg=000000$ 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!

Posted in abstractmath.org, Handbook, language, language of math, math, Proposals for research, representations, understanding math. Tags: , , , , , , , , , , , . 8 Comments »

Learning by osmosis

2010/01/15 — Charles Wells

In the Handbook, I said:

The osmosis theory of teaching is this attitude: We should not have to teach students to understand the way mathematics is written, or the finer points of logic (for example how quantifiers are negated). They should be able to figure these things on their own —”learn it by osmosis”. If they cannot do that they are not qualified to major in mathematics.

We learned our native language(s) as children by osmosis.  That does not imply that college students can or should learn mathematical reasoning that way. It does not even mean that college students should learn a foreign language that way.

I have been meaning to write a section of Understanding Mathematics that describes the osmosis theory and gives lots of examples.  There are already three links from other places in abstractmath.org that point to it.  Too bad it doesn’t exist…

Lately I have been teaching the Gauss-Jordan method using elementary row operations and found a good example.   The textbook uses the notation [m] +a[n] to mean “add a times row n to row m”.  In particular, [m] +[n] means “add row n to row m”, not “add row m to row n”. So in this notation “ [m] +[n] ” is not an expression, but a command, and in that command the plus sign is not commutative.   Similarly, “3[2]” (for example) does not mean “3 times row 2″, it means “change row 2 to 3 times row 2″.

The explanation is given in parentheses in the middle of an example:

…we add three times the first equation to the second equation.  (Abbreviation: [2] + 3[1].  The [2] means we are changing equation [2].  The expression [2] + 3[1] means that we are replacing equation 2 by the original equation plus three times equation 1.)

This explanation, in my opinion, would be incomprehensible to many students, who would understand the meaning only once it was demonstrated at the board using a couple of examples.  The phrase “The [2] means we are changing equation [2]” should have said something like “the left number, [2] in this case, denotes the equation we are changing.”  The last sentence refers to “the original equation”, meaning equation [2].  How many readers would guess that is what they mean?

In any case, better notation would be something like “[2] ← 3[1]“. I have found several websites that use this notation, sometimes written in the opposite direction. It is familiar to computer science students, which most of the students in my classes are.

Putting the definition of the notation in a parenthetical remark is also undesirable.  It should be in a separate paragraph marked “Notation”.

There is another point here:  No verbal definition of this notation, however well written, can be understood as well as seeing it carried out in an example.  This is also true of matrix multiplication, whose definition in terms of symbols such as $latex a_ib_j$ is difficult to understand (if a student can figure out how you do it from this definition they should be encouraged to be a math major), whereas the process becomes immediately clear when you see someone pointing with one hand at successive entries in a row of one matrix while pointing with the other hand at successive entries in the other matrix’s columns.  This is an example of the superiority (in many cases) of pattern recognition over definitions in terms of strings of symbols to be interpreted.  I did write about pattern recognition, here.

Posted in abstractmath.org, exposition, Handbook, language of math. Tags: , , , , . 1 Comment »

Constraints on the Philosophy of Mathematics

2009/03/18 — Charles Wells

In a recent blog post I described a specific way in which neuroscience should constrain the philosophy of math. For example, many mathematicians who produce a new kind of mathematical object feel they have discovered something new, so they may believe that mathematical objects are created rather than eternally existing. But identifying something as newly created is presumably the result of a physical process in the brain. So the feeling that an object is new is only indirectly evidence that the object is new.  (Our pattern recognition devices work pretty well with respect to physical objects so that feeling is indeed indirect evidence.)

This constraint on philosophy is not based on any discovery that there really is a process in the brain devoted to recognizing new things. (Déjà vu is probably the result of the opposite process.) It’s just that neuroscience has uncovered very strong evidence that mental events like that are based on physical processes in the brain. Because of that work on other processes, if someone claims that recognizing newness is not based on a physical process in the brain, the burden of proof is on them.  In particular, they have to provide evidence that recognizing that a mathematical object is newly discovered says something about math other than what happened in your brain.

Of course, it will be worthwhile to investigate how the feeling of finding something new arises in the brain in connection with mathematical objects. Understanding the physical basis for how the brain does math has the potential of improving math education, although that may be years down the road.

Posted in math, understanding math. Tags: , , , . No Comments »
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