Problem-Solving Skills

Solving problems in this chapter refers to creating any specified mathematically related result.  Some examples of mathematics problem include creating proofs of theorems, discovering theorems, finding counterexamples, doing mathematical experimentation, creating useful axioms and definitions, creating a method to solve a class of problems, etc.  In order to be an effective mathematician, you need to have problem-solving skills to create mathematics and solutions to problems on your own.

According to Schoenfeld [51] there are four main keys to problem-solving as follows:

1. Resources—available truths such as theorems and methods from various branches of mathematics.

2. Heuristics—general strategies used to approach difficult problems in various circumstances.

3. Control—careful control and selection of strategies, paths and methods chosen to attack a problem in order to solve problems as efficiently and effectively as possible.

4. Skeptical Defeasible Reasoning†—reasoning by what "usually happens", what is "usually done", or what "usually works" in similar circumstances.  Defeasible reasoning is not logical and thus the beliefs that you acquire from using this can often lead to serious mistakes and misjudgment.  Therefore beliefs given by defeasible reasoning should always be doubted by its user until the beliefs are carefully verified to at least some degree when possible and it is not too time consuming to verify the conclusions.

† (Schoenfeld uses the word "beliefs" in problem solving; however “defeasible reasoning” is more general.)

Resources are essential to problem-solving because you cannot blindly solve a problem.  Resources are your sight to what is available for use in your problem.  Heuristics are essential because many problems require a complicated method to solve them which calls for the need of general strategies to approach difficult problems in a variety of circumstances.  Control is essential because each path chosen in a problem-solving process leads to many other possible paths that lead to even more possible paths and so on.  The number of possible paths grows so large that improper control by choosing unpromising paths or according to Schoenfeld "taking wild goose chases" can make problem-solving so time consuming that a problem may never be solved.  Defeasible reasoning is essential because using perfect logic in uncertain scenarios is not usually feasible or may be so time consuming that a problem-solver would never finish or have the proper intuitive guidance to solve a problem.

Therefore because each of these 4 keys to problem-solving is essential, a problem-solver must use each of these keys properly and effectively with few exceptions in order to be a skilled problem-solver.  These skills are like a 4-link chain where if one of the links is broken, so is the problem-solving process.  On the other hand, an expert mathematician in a specific branch of mathematics may have so much experience that his/her intuition (defeasible reasoning) may be so strong that they almost always know exactly how to approach a problem in a given field with no need for general heuristics or for control.  This speaks much for experience and expertise in a field and how it greatly enhances problem-solving skills.

Principles of Control

Carefully Monitor and Control the Methods You Choose to Apply to Solve a Problem [41].

Carefully monitoring and controlling the problem-solving process is essential because otherwise you will likely waste much time pursuing information or methods that may have no application to solving your problem.  You have to carefully decide which heuristics to try and use, which sub-problems to try and solve, or any other method of attack to solve a problem.  Without careful monitoring and control of the problem-solving process, problem-solving can be far more time consuming or even impossible due so much time wasted on fruitless attempts.   Thus knowing how to control the problem-solving process to choose more promising problem-solving paths is essential to being an effective problem solver.

In order to control the problem-solving process properly, you should periodically monitor and assess:

1. The plan you follow (or sub-plan of a sub-problem).  Consider the likelihood that the plan will help achieve the goal of the problem.

2. Probable truths you search for.  Consider the likelihood that a truth you search for will be useful in solving the problem.

3. Heuristics you use.  Consider the likelihood of the heuristic being useful for solving your problem.  Consider if you are using the heuristic properly for the given type of problem, and what the most effective way to use the heuristic would be.  For example, in the problem-solving strategy: “Consider specific examples of your problem (See the problem-solving methods catalog at the end of this chapter)”, the specific example of your problem that would be most useful is one that would allow you to use available tools (resources) to solve your problem.  Thus more promising example solutions allow you to create hypothesis about the solution of your general problem.  See pg 78 of Schoenfeld [51] to understand the dilemma of applying heuristics properly to a variety of circumstances.

4. Path you choose to solve the problem.  Consider and search for an easier way to approach the problem that takes less time [41].

5. Make sure a sub-problem you are attempting to solve will be of use to your goal before pursuing its solution.  Avoid solving sub-problems that you have little evidence would help solve your original problem.  For example, solving complicated differential equations that have little evidence of being useful to your problem would waste a lot of precious mind power and time [41].

An important set of control considerations to take before seeking a particular choice path:

6. Never implement difficult or time consuming procedures unless you have checked to see whether other, far simpler procedures work [41].

7. Check to verify that a procedure will likely help you solve a problem before following a procedure such as a heuristic.  If you cannot tell if a procedure helps you to solve the problem, then consider the likelihood the procedure will help you to solve the problem.  Then consider the conditional likelihood that the subsequent steps required after the initial procedure will help you to solve the problem.  Choose the path with the best balance of likelihood of solving the problem in conjunction with expected time to follow the path [41].

8. Choose a representation of the problem that simplifies the mathematics and the problem-solving process.  If you later find that your representation or notation of the problem clutters the problem-solving process, then search for new representations and notations that simplify the problem-solving process [41].

Representation of a problem and the model you use to solve a problem is critical.  For example consider multiplying MCXVI by MCCMIIV in roman numerals without ever resorting to the use of other numeric representations.  This model would be terribly complicated to do  multiplication (Schoenfeld used this example in [41], and Polya in [52]).

Other Control Principles

When a path for solving a problem gives no success, don’t throw away the information you acquired during that process, because the faulty plan could have useful elements to the correct solution [41].  You avoid throwing away information such as known information, elements of an abandoned plan, etc.

In [41], Schoenfeld mentions that there is a difference between an expert in a mathematical subject that has no need of control, and someone intelligent that needs the use of control to solve a problem.  Schoenfeld explains that experts in a mathematical field need little control or problem-solving strategies.  However, intelligent people who are not experts in a mathematical field need to choose resources carefully, and abandon strategies appropriately when careful monitoring of the strategy gives little evidence of having application to the problem.  Intelligent non-experts have to use control strategies [41].

Try to solve your problem as efficiently as possible.  Efficiency is essential in being good at solving problems.  Try to search for many problem-solving paths (plans) first, and quickly (possibly using defeasible reasoning) follow many of them until you feel you can somewhat predict the difficulty of the path.  Follow each path to the point where you feel you can somewhat predict which path would be the least work, and begin your most intense work there.  Later you can try other paths for the sake of learning.

Also, when searching for information (resources) that might help in the problem-solving process, do not search excessively for information if you feel you have already found enough information to construct promising plans for solving the problem.  You can return to searching for resources later if doing so becomes necessary.

Classify problems into types.  Doing this allows you to use methods that usually work for certain classes, or types of problems and to use commonly used tools for those types of problems.

Principles of Defeasible Reasoning

We as humans are hardwired to use defeasible reasoning for practically everything we do.  In fact, use of defeasible reasoning is where our commonly used fallacies come from.  Defeasible reasoning is an excursion into fallacy and false logical reasoning in hopes of discovering something useful.  Being irrational and using fallacies is very useful if used properly.  Otherwise, why would the whole human race in all of its great intelligence always lean towards such use of fallacy if it weren't truly useful in some way?  Those fallacies we use usually do tell the truth even though they are not logical (although we commonly incorrectly refer to use of defeasible reasoning as logical).  So there is no contradiction to what was said in an earlier section on understanding logic, because although you use fallacy to do defeasible reasoning, your skepticism of its results is guided by proper usage of logic.  However, because defeasible reasoning is fallacious, the commonly used fallacies often tell us serious lies as well.  Using fallacies can be a good guide to solving problems and finding potential truth, but it is important to understand that they are not logical and will give you false information on many occasions.

         Although we take such risks of obtaining false information by using defeasible reasoning, it is worthwhile to use this tool due to the level of power it provides for discovering likely truths and likely solutions to problems.  Given that we as humans have defeasible reasoning so hardwired into our minds—being so accustomed to using it for practically everything we do, we usually do not give sufficient doubt to what our defeasible reasoning tells us.  We often fail to investigate further as to whether our defeasible reasoning provided a truth or a falsehood.  For example, when people learn the distributive property: a(b + c) = ab + ac, they often use that experience and by defeasible reasoning assume that everything distributes such as the falsities:  sin(x + y) = sin(x) + sin(y), 1/(x + y) = 1/x + 1/y, (x + y)2 = x2 + y2, etc.  Now it is a good idea for people to use their defeasible reasoning to consider the possibilities of these relationships, but should do so with great skepticism and carefully experiment, check and possibly even try to prove these relationships true or false.

         Because of the great risks involved with using defeasible reasoning, people must still use defeasible reasoning liberally, but with a consistently lingering skepticism of its results until the results are verified using logic.  Many people are so used to taking their defeasible reasoning as absolute truth that it is difficult for them to even begin to doubt its results.  In fact, students’ absolute acceptance of results provided by defeasible reasoning in math classes is the greatest cause of their confusions.  This is because they take the results of their defeasible reasoning as absolute truth.  In fact, because the use of defeasible reasoning is so 2nd nature to our minds, we usually do not even recognize that we are using our defeasible reasoning to even consider the need to be skeptical of its results.  We need to be taught to use this wonderfully powerful tool without misusing it.