The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook SolvEq.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

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.