General Problem-Solving Habits
After learning problem-solving skills, begin to practice solving problems using those skills. Do this in order to make the problem-solving methods and common tools in particular mathematical fields 2nd nature to you. This will make problem solving easier for you.
Whenever you solve a problem, look back to review how you solved the problem [52]. Consider what was ineffective and inefficient in your process to find a method of solution and think about what you could have done to be more effective and efficient in finding a solution. Doing this should help you be more effective next time you attack a problem, and become better at problem solving in general.
Practice solving problems that challenge you because you learn more from working on harder problems than you do by solving simpler ones. Without a challenge, you learn little or nothing.
Problem solving can require lots of ideas and strategy on paper. Use paper liberally in order to be able to solve problems efficiently and effectively.
Create a personal catalog of commonly used methods or techniques to solve different types of problems. Do this as you read math books and as you discover new methods and techniques that may be common tools to solving problems in various branches of mathematics.
Address the emotional aspects that affect the problem-solving process.
-Remembering that by trying to solve the problem, you will likely become better at solving problems in general.
-Varying or changing the details of the problem when you start to get tired of the problem [52]. Additionally, your variation of the problem may give insight into how to solve your original problem.
Be certain that the mathematical resources you use to solve mathematical problems are correct because using incorrect mathematics to solve a step of a problem will almost surely give you an incorrect result. In mathematics we learn many principles and methods. Often when we are in a situation that is similar in some way to other mathematical situations that we have learned to deal with one way, we try to alter the methods to our new similar problem or step and apply them. This is a good and useful approach, but we must always be unwilling to accept the altered method or principle at face value until we prove that the altered method or principle used is correct. Otherwise you will be very prone to making dangerously common and terrible mistakes. For example, when people use the distributive property they apply a(b + c) = ab + ac which is good. However, in similar problems they may apply a/(b + c) = a/b + a/c which is an alteration to the similar correct tool of the distributive property. If you were new to algebra, it may be a good idea to consider this as a possible similar principle that may be useful. But you should not trust such a generalization of a property until you are sure it is true. In fact after considering the example 1/(1 + 1) ≠ 1/1 + 1/1, it becomes obvious how terribly wrong a problem will go by using this falsely generalized property.
Most people who claim they are terrible at math suffer from readily accepting altered methods or principles from similar mathematical situations without investigating those altered principles fully before applying them [49]. Another common example of an incorrect generalization creating resource errors is for example taking the principle that allows you to say (x - 1)(x - 2) = 0 implies x - 2 = 0, x - 1 = 0 which is correct, and then thinking you can apply (x - 5)(x - 3) = 3 then x - 5 = 3, x - 3 = 3 which is false [49].
Then after practicing many incorrectly altered methods and principles, these false principles can become second nature and become engrained in the mind as true [49]. It requires a lot of work to “debug” or reprogram these mathematical misconceptions. The reason for such error patters is that in most common sense applications it is good to take a method or principle that worked in a previous situation and apply that principle to similar circumstances that arise. Although this approach is useful in mathematics as well, you must take far more care not to accept alterations to those principles at face value until you have fully investigated their truthfulness. The mathematician viewpoint is “The similar principle may be worth considering, but I refuse to accept it as true until proof or strong evidence of its truth is provided.” The common sense viewpoint is “The principle worked before, so a minor alteration of the principle should work in my similar problem”. Common sense has its place, but it does not always work, and can be dangerous in a mathematics environment.
Learn to think for yourself. Many people become accustomed to having to rely too heavily on what other people or the authority on the subject thinks in order to verify truth. You should not trust everything you hear or read in science. Even if a person with a PhD makes a comment, you should seek for evidence that backs up their claims. Something should make sense to you first from sound logic or solid evidence before fully trusting a scientific or mathematical statement. Question everything in science and mathematics and think for yourself in order to become a good independent mind. This is important in order for you to be a good problem-solver that has learned to use your own mind to discover, learn, and create instead of having to excessively depend on others to tell you the truth of how to think about everything. If a person does not think for themselves, then it is difficult to understand or create mathematics.
You might just want to know the “steps” to solve a type of problem (like factoring), and may be disinterested in understanding the process by which those steps were created and discovered. You may ask: “What does the teacher say are the steps to get the right answer?”. It is important that you use your skills and your mind instead of having to lean completely on what others think or say is the “method”. You are far better off if you consider what makes sense to you by considering what axioms, theorems, problem-solving skills, etc. relate to your problem to help find the steps of solution you may have been given. Doing this teaches you far more about problem-solving skills and mathematical processes than if you merely trust the steps to solving a problem that your teacher gives you. It is not necessarily bad to learn the steps to solving a problem if you understand how problem-solving skills were used to find those steps.
Use logic properly. Know what consists of a fallacy, and what consists of sound logical argument. Logic will be discussed in more detail in chapter XI.
Formal mathematics and logic are used frequently during the problem-solving process. Problems cannot be successfully solved if false resources obtained from fallacious arguments are used to justify steps of solving a problem. For example, often when people witness that event A and event B happen together, they may think A causes B, or that B causes A whereas for now there is no logical proof of either relationship. Witnessing such an event should be evidence that some relationship might exist, but does not consist of proof.
Common fallacies people use to prove mathematical statements include but are not limited to:
-Appeal to Ignorance: We can’t prove it, so it must be false.
-Argument from Omniscience: Most everyone else believes it, so it must be true.
-Appeal to Tradition: Most other people do it, so it must be okay.
-Argument of Authority: The teacher said it’s true, therefore it is true.
-Circular Reasoning: Using a statement to prove that same statement.
-Confusion of Correlation and Causation: For example, seatbelts and living correlate therefore wearing seatbelts saves lives. Notice that it is obvious that seatbelts often do save lives, but the argument is fallacious. Sometimes a fallacious argument argues something that is true but still is an improper method of argument. An obvious false statement that uses this fallacy is: Every time I see the sky turn blue, I see the sun come out. Therefore, the sky turning blue causes the sun to come out. Correlation is useful evidence that may show a sign that there may be causality, but does not consist of proof.
-Excluded Middle: Considering only extremes to prove something. For example, showing that a statement is true for extreme mathematical entities gives useful evidence, but does not consist of a proof.
-Post Hoc, Ergo Propter Hoc: A result happened after an event; therefore the result was caused by the event.
-Proving Non-Existence: A counter-example can’t be found for a statement; therefore the statement must be true. This gives evidence or a sign that might lead you to believe a statement might be possible to prove, but does not provide proof in itself.
-Statistics of Small Numbers: For example: My uncle smoked all his life and he lived to be 90. Simply because someone can point to a few favorable numbers says nothing about the overall chances.
-Straw Man: Creating a false scenario and then attacking the false scenario to prove a statement. Note that this does not include proof by contradiction.
Pre-Problem-Solving Steps
a) Make the problem and all of its details second nature to your understanding and memory. If you have difficulty understanding parts of the problem, then try to restate the problem in your own words.
You cannot effectively start solving a problem or creatively think about what you are trying to do if you do not understand your problem well, and have each part of the problem 2nd nature to your memory.
b) Concentrate on each part of the unknown, each part of the condition and what you want. If you understand the problem, can rattle off the problem quickly, and see it all at a glance in your mind, then you are ready to for the next pre-problem-solving step.