Model building is far more important than memorizing facts or procedures. First, there are so many facts and procedures that even a certified genius cannot memorize a significant fraction of them. Second, unorganized facts do not lead to further directed thoughts.
A good scientist requires an excellent model and a well-developed ability to, by thinking, organize new facts into additional models or meld them with existing models.
This is the reason that problem solving is a substantial part of a scientific education. From the simplest problem in arithmetic to the most complicated problems in physics, problem solving is the organization of facts into useful models.
Most teachers and students do not understand this. They think that problem solving consists of “getting the answer.” They seek solutions manuals that show the student how to work each problem. Newtonian mechanics, the principal subject of first-year high school and college physics, affords an excellent example.
Most high school texts provide a formula that allows a student to calculate the length of time required for an apple to fall from a specified position in a tree to the ground .The student plugs numbers into the formula and computes the answer.A properly educated 8-Year-old child should be able to do this just fine -which is about the level of math skills of many tax-financed high school students.
This is not problem-solving. Since there are essentially uncountable variations of this problem that require different equations, it is not even useful .The student will never be able to memorize all of these different equations.
In an appropriate course,the student is presented with the problem and required to apply Newton’s laws of motion to derive the necessary equation. If a numerical solution is required, he will, indeed, plug the numbers into his equation, but this is a final and essentially trivial part of the exercise. The student must apply the Newtonian model to the necessary facts in his derivation.
This derivation, however, requires calculus. Since most tax-financed high school students do not know calculus, they cannot learn to solve physics problems. So, the school provides a non-calculus “physics” course. The school pretends to teach physics, and the students pretend to learn.
Newtonian mechanics involves primarily objects that can be seen and that move in ways that are consistent with ordinary experience. This makes it easier to comprehend the scientific model that Newton invented.
Most of modern science, however, involves the submicroscopic molecular world that cannot be seen by the unaided eye and obeys rules that are often not consistent with ordinary experience.In this world, therefore, models are especially important. It is essentially impossible to work productively in such a field without well-developed models.
To be sure, the simplest models are often eventually reduced to automatic memory. After understanding the concepts, a six-year-old should commit the simple arithmetic operations to memory. “Seven times eight equals fifty six” should be memorized by unthinking instant rote memory. This result will be used so often that its repeated derivation would slow the student down in later work. It should not be memorized, however, until the student knows that he can derive the result from his logical model and has done so.
The acquisition of problem-solving ability – the skill to apply logic and an appropriate model to determine the correct solution of a problem – requires many, many years of practice involving the unaided solution of tens of thousands of problems. Gradually the student builds both the requisite ability and also self-confidence in his ability. Both are required for successful problem solving. He is unlikely to solve difficult problems unless he is confident that he can do so.
Tax-financed schools are uninterested in ability, but they have noticed the part about self-confidence. The students and parents must “believe”that the students are doing well, so “self-esteem” is promoted. Misplaced self-esteem without real ability is, however, a cruel fraud. It holds negative value for the student.
The teacher can aid the student by selecting a good course of study and by example in demonstrating good study habits and methodical intellectual techniques. When, however, the teacher actually works a problem for the student he deprives the student of the problem. Its value is gone. The student will only develop skills with those problems he actually solves. Teacher solutions also harm self-confidence and discipline because the idea that, if the problem is difficult, he can refer it to the teacher will always be lurking in the student’s mind.
The school must provide 1) a good study environment, 2) good study habits, and 3) a superb course of study that is appropriate for all students and also for the best of students. Not even one of these three requirements is met in today’s tax-financed schools.
Moreover, problem-solving ability – the ability to move with facility back and forth between a mental model and the facts that relate to it and to apply the model and the facts to the discovery of new facts and new models – is an essential part of virtually all human higher mental activity. It is best taught with introductory mathematics, physics, and chemistry, but it is applicable to all endeavors.
Economists have a model. If parts of that model are omitted, odd conclusions are the result. It is intuitively obvious, for example, that tax revenues will be zero if the tax rate is 0 % and also if it is 100%. At 100%, there is zero incentive to produce. Fear of death or torture can, of course, elicit some work from slaves, but this is not taxation. So, the boundary conditions of a graph of governmental income from taxes require that the curve pass through zero income at 0% and at 100% taxation. In between, income rises from zero at 0%, reaches a maximum, and then declines to zero at 100%.
This is commonly known as the Laffer curve, but it should be trivially obvious to any good problem solver even with no economics training. Yet, with taxation well beyond the maximum of the Laffer curve, politicians and the media continually claim to the public that raising taxes will “soak the rich” and result in more money for distribution to the public. The public supports this for two reasons. First, they have adopted the unethical view that stealing is right if the majority votes for the government to do it. Second, most of them have almost no problem-solving ability and cannot therefore realize that higher taxes are not even in the economic self-interests of the thieves.