Sunday, October 16, 2011

A Few Hints How to Introduce Mathematical Concepts

Teacher should postpone introduction of strange, exotic mathematical names and labels to, otherwise, most likely, easy to explain and easy to understand concepts in mathematics. Teacher should  first explain mathematical concepts in terms of required sequence of mathematical operations on numbers and on sets involved and then introduce labels or historically accepted names for them. Most of us will agree that many of those names are there for historical reasons, and often they are confusing, misleading, and even intimidating (if Borel, Hausdorff needed so much work to prove that theorem or formulate it, what chances do I have?). To me, it is sometimes better to refer to a theorem by a number first, like Theorem 1, Theorem 2, … and only later assign historical names or other labels to reference them.

Math lectures sometimes look to students like putting misleading, cluttered, over-detailed traffic signs on an easy to use roads and highways.

But, what are we really doing or what we want to do when we say "introducing math concepts through A or B or C real world examples"? Do math concepts need to be introduced through real world examples at all? No, they don't. They can be derived or formulated directly from axioms.  So, what would be the goal of "introducing math" through real world examples? To show that math concepts can be motivated by real life examples, but, at the same time the same math concepts can be derived inside math only, without referencing any real world domain. To me, the mandatory step of introducing some math concepts from real world example is to mandatory show that the same concept can be defined or derived from ZFC axioms. This would clearly show the border and connections, if you wish, at the same time, between applied math and pure math developed from ZFC axioms.

[introducing math, math ], math concepts, math education, mathematics, theorems ]