deductive MATHEMATICS:  an example

You can probably remember a textbook like this from your own school days.  The lessons are unclear and it is expected that by the repeated examples that follow, the student will get used to seeing this sort of problem and then apply the lesson in the future.  Believe it or not, the text above was included in a parent's guide to help those who lack expertise in math! 

The biggest problem here is that the student does not learn WHY any of this is or is not true.  Due to this lack of understanding the student cannot use the knowledge for any problem except those that resemble the examples previously studied--if the student actually did them!  This method of teaching math does not work. Students never gain a sense of mastery and generally feel that their chances of doing well are dependent upon "what's on the test" and whether it will be familiar or not.  Bad scores are normally followed with complaints that the teacher put questions on the test that weren't done in class. 

It's okay when most of the kids don't get it and the average test scores are very low.  Schools simply "curve" the grades and everybody wins!  Yea! 

Axiom V.  A whole is equal to the sum of all of its parts.

Axiom XIX.  If equal quantities are multiplied into equal quantities, the products will be equal.

Both of these propositions are obvious to anyone after some reflection--and every student loves the challenge of thinking whether they are really true or not.  You can go ahead and ask your own child whether they are true and see for yourself.

After learning a number of axioms and definitions, the student is presented with this Theorem:

Theorem I.  The product of two quantities is equal to the sum of the products of the parts of one quantity multiplied into the other.

Of course the language is challenging if you haven't studied Grammar recently, but don't let that distract you.  The problem is using what we know to "invent" new knowledge, or solutions to new problems. The student must think of what he has learned and how it can possibly help him to prove this theorem.  Let's walk through it.

First, we can see that in this theorem is a statement about a whole and its parts.  If each quantity is a whole, then each is equal to the sum of its parts (Axiom 5).

Having settled that, we see that this theorem is combining what we know about a whole and its parts with what we know about the products of equal quantities (Axiom XIX).  After all, a whole and the sum of its parts are equal quantities. 

So, if we had a quantity (a+b) and another (c+d), the theorem says that the product of (a+b) multiplied by (c+d) would be the same as the sum of a(c+d) and b(c+d).  Well, that looks fine, but how can we prove it?  We don't want to take a theorem for granted because a theorem is, by definition, a proposition that requires proof--it is not self-evident.

Therefore, using our axioms we may prove this theorem:

PROOF OF THEOREM I:

1.  If equal quantities are multiplied into equal quantities, then their products must be equal (Axiom XIX).
2.  Every whole is equal to the sum of all of its parts. (Axiom V).
3.  Therefore, if a whole and sum of its parts are multiplied into equal quantities then the products will be equal. Q.E.D..

(Note: Q.E.D. stands for Quod erat demonstrandum., which means "Which was to be demonstrated.")

On the other hand, the modern student is asked to accept a theorem without proof, which contradicts the very idea of a theorem and the scientific method which the modern schools claim to be their standard.  The modern textbook isn't so scientific after all!  Worse, because he never gains an understanding of the concept through what is certain, the student will never use this principle confidently or creatively to solve new problems.  Ultimately, the modern student is not taught the art of Arithmetic, but merely satisfies a checklist of exercises that will be forgotten even more quickly than were learned.  This is the difference between inductive and deductive mathematics.