Mathematics Education: Taking a Clue
From the Recent Technological Revolution
Is mathematics object oriented?
We already saw that a class hierarchy came in handy for problem solving. Additional examples will show that object orientation has significance in mathematics.
- Mathematics distinguishes between the abstract and the concrete.
- Mathematics has hierarchy and inheritance
Everything that holds in a superclass is inherited by a subclass
- Mathematics has polymorphism
Drawing a line depends on the geometry at hand.
It's easy to see that encapsulation makes mathematics human. Whitehead and Russell aside, do mathematicians care to see proofs that start with axioms?Encapsulation along with abstraction (conceptualization) make mathematics pluggable. (It is pluggable - there is no question about that.)
So, what do you think?
We may also note that mathematics has an inside and an outside. Paraphrasing E.T.Bell who said "The place of rigor in mathematics is in mathematics." (The Place of Rigor in Mathematics (Monthly (41)1934)), I can say that the inside of mathematics is mathematics. The outside of mathematics is not mathematics.
So, what is outside of mathematics? I say: objects without classes.
I. Kant said
A concept without a percept is empty
a percept without a concept is blind.
2 + 3 = 5 in itself is just a fact which one may believe in, be sure of, or even know for sure. Memorization, gadget manipulation without conceptualization is blind - not pluggable, and is not mathematics.
And, of course, if there is no proofs, there may hardly be mathematics. Sorry, can't talk of anything and not mention its unique features.
I would also like to say that mathematics is not dull and, therefore, anything dull is not mathematics. This sounds so uplifting. But enthusiasm devoid of tangible manifistation is grotesque, not just empty. Mathematics is not dull for those who do it. Technolgy has a potential to make mathematics not dull for those who learn it. Technology has a potential to make the invisible visible. It also shows the way how to do that.