A person drops out of High School because the confusion generated by his schooling seems to be increasing exponentially. He feels hopeless about learning. He can’t see any benefit from continuing in school. So he drops out.

What underlie that confusion are the holes in his understanding. He may start out with just a few holes in Kindergarten. But such holes, when not resolved, multiply to become significant by the end of the elementary school. The student tries to overcome the lack of understanding by memorizing his school materials. But such evasion serves only to increase those holes in his understanding. Even memorization fails to get good grades by the end of the middle school. That is when parents scramble to get the student tutored.

Teachers and after-school help focus only on the confusion related to his present grade. Overcoming confusion from earlier grades appears to be such a daunting task that it is not even attempted. But as long as those earlier holes are not filled, they keep on multiplying exponentially during the high school years until the student simply drowns in the resulting confusion.

When you sit down to help this student, you find that he can’t tell you coherently what he doesn’t know. Even when you know that these confusions may depend on just a few holes in student’s understanding from way back, you find the situation to be unsolvable. The student’s attention is so fixed on the present confusion that he cannot trace anything back. This has been the problem all along. One-on-one troubleshooting works but it is a hit and miss affair and not efficient enough.

The subject of this essay is an idea that seems to handle this dilemma. Any subject, whether it is mathematics or language-arts, begins from some important need. As observations follow that need, the subject develops slowly to form a structure based on logic. We may trace the logic of the subject all the way to a beginning premise. The later concepts develop out of the earlier concepts. Obviously, it would be difficult to understand the later concepts if earlier concepts are not understood.

This brings up a parallel between arrangement of holes in the student’s mind and the flow of logic in a subject. It is the filling of earlier holes that makes it possible to resolve the present confusion. Similarly, it is the understanding of earlier concepts that make it possible to comprehend the later ones.

**The simple idea is that follow the logic of a subject to discover and resolve the earliest holes in the understanding of the student. Then come forward in a logical sequence resolving rest of the holes. **

We seem to have found a simple and straight forward method to resolve confusion in any student quite rapidly and efficiently. When we applied this idea to a group of school dropouts, it worked like a charm. The students soon discovered the earliest difficulty they had with the subject. They knew the source of their difficulty now, and were eager to resolve it. This quickly established a two-way dialog and eager participation.

Getting that two-way dialog going was the first breakthrough. We knew how to approach the confusion; but we soon discovered that we had a tiger by the tail. We needed better planning to handle that confusion.

It was like putting an accident victim through a planned rehab.

## Comments

I find that many if not all my education contained irrational conclusions which I continually had to rehash. I remember my teacher trying to convince me that 0 was a number which could not be expressed with her fingers. She hid her hands behind her back. Today I find it extremely difficult to convince a mathematician that 0 can not possibly be a rational number. Yet, it surfaces in the failure of most mathematics!?(

Something that worked for me was an unusual professor who allowed us (the students) to write the book in our numerical analysis class. The technique of just asking the right questions of the student allows fundamental understanding to be acquired.

I now do this for myself and ask why is it the case. Often it is not!

Most have in place beliefs in which they do not want to give up. The want machines are not very logical:?) They have to want to know.

0 (nothing) and 1 (something) are two extremes of a duality. There is a continuum of rational and irrational numbers between them. 1 may be taken as infinity but that doesn’t much change the argument.

The two extremes of a duality are not discrete elements. There exists a continuous scale between them. That is the essential logic.

Science uses the perspective of “particles in void” This perspective looks at the duality of matter and space and regards these two elements as discrete. This is the fundamental illogic at the base of science.

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