Geometry and Euclid
You know, some math history can help spark interest in whatever you're teaching, and enliven the math (make it 'live math' so to speak). So today I want to educate you just a little bit about Euclid and geometry.
Did you know that a typical high school geometry course today with axioms, definitions, and theorems follows after the way Euclid presented geometry in his book Elements... and that this happened around 300 B.C. in Alexandria!
So the theorems your student is learning date back 2300 years!
Euclid was great - not because he found many great theorems back in his time (he didn't), but because he organized all then known mathematics into a logical presentation in his book Elements.
The geometry parts of Elements guided geometrical research for a long time after Euclid, and your school course of geometry is basically still based on this book.
So Euclid gives his name to Euclidean geometry - also called plane geometry.
In the beginning of his book, Euclid stated some definitions and postulates (we would call them axioms). Then he went on to prove various theorems, using only the postulates, and previously proven theorems.
A system organized this way is called an axiomatic system, and that's what your high school geometry book does too.
You might be interested in checking out a website that makes Euclid's Elements 'live' with interactive Java applets.
You might ask, "So why do we study geometry in such an ancient way?"
Because it is NOT an ancient way! Euclid started this 'business' of organizing math into axiomatic systems, and mathematicians have been doing that ever since. In other words, mathematicians organize everything in that manner, not just geometry.
Actually, in school mathematics, high school geometry is the ONLY place where you encounter an axiomatic system: axioms that are assumed as true without proving, and theorems logically proved from those.
Of course people argue whether that is good or bad. Mathematicians say it is good; it gives youngsters the only opportunity to encounter proofs in school mathematics. Some say it's not good because it's just too difficult for today's teenagers.
I feel it is VERY good and needful that school mathematics involves some proving and justification of the math facts instead of mathematics that is plain 'announced'. But I also feel there could be other ways to do this than the current high school geometry course. It would be better, in my opinion, to involve proving in other levels and other math topics, too - but not necessarily in this rigid 'two-column proof' way.
But while things stand as they do, you might be interested in reading my previous articles: Why high school geometry is difficult and What to do about it?
I wish everyone a prosperous year 2006!
Categories: history, geometry
Did you know that a typical high school geometry course today with axioms, definitions, and theorems follows after the way Euclid presented geometry in his book Elements... and that this happened around 300 B.C. in Alexandria!
So the theorems your student is learning date back 2300 years!
Euclid was great - not because he found many great theorems back in his time (he didn't), but because he organized all then known mathematics into a logical presentation in his book Elements.
The geometry parts of Elements guided geometrical research for a long time after Euclid, and your school course of geometry is basically still based on this book.
So Euclid gives his name to Euclidean geometry - also called plane geometry.
In the beginning of his book, Euclid stated some definitions and postulates (we would call them axioms). Then he went on to prove various theorems, using only the postulates, and previously proven theorems.
A system organized this way is called an axiomatic system, and that's what your high school geometry book does too.
You might be interested in checking out a website that makes Euclid's Elements 'live' with interactive Java applets.
You might ask, "So why do we study geometry in such an ancient way?"
Because it is NOT an ancient way! Euclid started this 'business' of organizing math into axiomatic systems, and mathematicians have been doing that ever since. In other words, mathematicians organize everything in that manner, not just geometry.
Actually, in school mathematics, high school geometry is the ONLY place where you encounter an axiomatic system: axioms that are assumed as true without proving, and theorems logically proved from those.
Of course people argue whether that is good or bad. Mathematicians say it is good; it gives youngsters the only opportunity to encounter proofs in school mathematics. Some say it's not good because it's just too difficult for today's teenagers.
I feel it is VERY good and needful that school mathematics involves some proving and justification of the math facts instead of mathematics that is plain 'announced'. But I also feel there could be other ways to do this than the current high school geometry course. It would be better, in my opinion, to involve proving in other levels and other math topics, too - but not necessarily in this rigid 'two-column proof' way.
But while things stand as they do, you might be interested in reading my previous articles: Why high school geometry is difficult and What to do about it?
I wish everyone a prosperous year 2006!
Categories: history, geometry
Comments
I've used that web site you linked to for years in my college junior-level course in geometry for math and math education majors.
I think highly motivated homeschooling parents would do well to read through the whole thing and even think about using the Elements itself as a geometry textbook.
Also, you bring up a good point about proofs -- namely that junior high and high school level students can and should study them, and that there aren't enough of them in a standard curriculum. Just keep in mind that a proof is nothing more than a clear and mathematically sounds explanation of why something is true; no more and no less. Don't we want students to be able to explain why something is true and not just believe it?
But I personally do like Geometry, A Guided Inquiry by G. D. Chakerian, Clavin D Crabill, Sherman K Stein.