Mathematics: Principles and Formulae

I gather that when most people think about mathematics, they think about two things, and in this order: numbers and formulas. In doing mathematics, most students are quick to add something else, though not by name: recipes – usually grossly oversimplified recipes. It is this tendency that runs them into so much trouble and many are the able teachers of mathematics that have tried to slap them (figuratively) into using something better, namely, principles.

A famous work of Sir Isaac Newton was titled, in Latin, Philosophiæ Naturalis Principia Mathematica ("Mathematical Principles of Natural Philosophy"). But you don't need to be as smart as Sir Isaac Newton to understand the importance of principles as superior to recipes. First of all, what's the difference?

A recipe is a sequence of steps or components with no (particular) reference to reasons. Computers do recipes - algorithms. Students that try to be computers (and nothing more) short-change themselves because there is already a much more efficient tool for that work than themselves. (And thus it is little wonder they complain about the pointlessness of the learning they are engaged in. Often of their own accord, they are not learning the really important lessons from mathematics that they should be.) Principles help me to justify, or at least partly understand, the steps of the recipe and apply it usefully. The ability Principles are one of the key advantages you have over your calculator.

People try to simplify the setting up and solving of proportion problems. They do it this way: "cross-multiply and divide." They don't really know why. It is only a recipe to them. How do you know which numbers to multiply and divide? It increases the number of things to memorize and (I have observed) it distracts people from the real principles. Forget about cross-multiplying and dividing. If I know a/b = c/d, I do what I always do with equations: do "opposites" and do the same thing to both sides (subject to the constraint that the operation must be permissible on the values in the equation). If I want to isolate a, I have a divided by b, so I do the opposite of divide by b, namely, I multiply by b, and I do it to both sides. The end. "Cross-multiple and divide" doesn't bridge well into solving other forms of equations, is unnecessary, gives little clue as to why it is supposed to work, and increases the number of things to memorize. How do you solve an equation?

1. Do "opposites."
2. Do the same thing to both sides.
3. Eliminate any apparent solutions which do not satisfy the original equations or the constraints of the problem (which may be implicit, such as, "negative area is not allowed").

Throw "cross-multiply and divide" and its ilk into the garbage bin.

Here are a few very generic principles to apply to problem solving:

1. Don't ask "What should I do?", ask "What is true?"  You won't make very much ground figuring out what to do if you haven't established a few things that you know about the situation. If you want something to do, then do this: find out more true things and organize them. This often starts with writing down some given information. Perhaps you print a few formulas and consider which variables you know and which you don't. Ask yourself, "Do I know any other equations that involve these unknowns? Are they valid in this situation?" Don't worry too much about whether an equation is going to give you the final answer, but concern yourself with whether it actually holds true in the situation. Thomas Edison didn't invent so many things by only trying things he knew would work. You also will write down formulas and find that they do not help you solve the problem – even though they are true and valid in the situation, they might not be useful to you. It isn't a mistake (yet), to write it down as a tool in the toolbox.
2. Don't ask "Why can't I do this?", ask, "Can this method or statement be proven correct?"  People want to apply rules that are applicable to one type of operation to a completely different operation. They see that they are allowed to make the statement a (b + c) = a b + a c and want to know why they can't say log(b + c) = log(b) + log(c).  This is one or both of two fundamental misunderstandings. Either it is assuming that the parentheses have the same meaning in both cases (which means they don't know what a function is and certainly not what a log is) or they misunderstand the importance of proof. This later misunderstanding in particularly important. Often the student assumes that a similar appearance means a similar treatment is permitted. The fact is, the statement a (b + c) = a b + a c, where a, b, and c are numbers can be proven. Our acceptance of the statement is based on proof from simpler principles. We don't "just know" that that's true (although this example is among the more intuitive that I could have chosen). Similarly, we don't "just know" that the statement  log(b + c) = log(b) + log(c) is true. If we can prove somehow that it is true, then we know we're allowed to make this expansion. On the other hand, if we can find a counter-example, we know it is not true. So here's a counter-example to the above (very, very silly) suggestion: log(10 + 10) = log 20 < 2, but log(10) + log(10) = 1 + 1 = 2; so they are clearly not equal. 
1. This doesn't mean you need to do proofs, but you should be satisfied that a proof exists for what you are doing and that you are not assuming something which cannot be established.
1. (Did you notice yourself proving your own work while you applied that principle? Tricky, huh?)
3. Look for patterns. Patterns have some kind of rule behind them. If you can determine the rule that makes the pattern, it may reduce work and memorization required. The rule will be more generic than the sampling that you noticed the pattern in – otherwise, it is a false pattern, because a counter-example exists. Don't assume the pattern is real. Test it, prove it, disprove it, as may be necessary.
4. Make analogies. The main reason for teaching the principle of proportional triangles is not to teach students how to solve triangle problems – although there's good uses for that. Proportions are so common place that everyone should get comfortable with the archetypical example: proportional triangles. There are analogies between force and momentum, between torque and force, momentum and rotational inertia, fluid pressure and electricity. Some things are more intuitive to us than others and if an analogy exists between something you are familiar with and something you are not, it can help you with a "working understanding" of the unfamiliar concept.

Understanding principles helps you to evaluate proposed solutions or solution methods. Recipes can't do that very effectively. Recipes also don't work well on a "new to you" problem. Knowing principles is a key point of difference between people who know how to implement a given method of solution and someone who can actually develop a method of solution to a problem they have not previously seen. This is true problem solving. Perhaps the most important thing anyone can do to improve their problem solving ability is to prize principles like gold and recipes as mere silver, maybe only bronze.