An educated mind is an opened mind. An opened mind is a liberal mind. Teachers don't have to intend to create liberals, it happens naturally. 
On the inside:
 On Beauty
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On Beauty
In an interesting confluence of events, I was thinking about the beauty of mathematics the other day and lo and behold, a discussion about it breaks out in Marion Brady's diary. It's probably still going on, but I had to leave for awhile to get something written for Teacher's Lounge. So why don't I just write a bit about mathematics? I am, after all, an abstract algebraist [If there are others out there, I'm a Frank Anderson student out of the University of Oregon]. I am a studier of rings and their modules. A ring is a nonempty set with two operations, roughly equivalent to addition and multiplication (in that we require the addition to be commutative, associative, and have an identity and inverses and that multiplication be associative and distribute over the addition. Sometimes we also require multiplication to have an identity. Sometimes not. Sometimes we require multiplication to be commutative, but not me. I study what are called noncommutative rings, but what is true when the commutative law is not assumed is still true if we assume it. A left module (handedness matters) over that ring, on the other hand, is like a vector space (think matrices)...sort of one that you left on the radiator overnight. It's an abelian group (oops! I mean it's a nonempty set with an operation roughly akin to addition (commutative, associative, identity and inverses)), and a "scalar multiplication of elements of the ring times elements of the module (in that order) which produces other elements of the module. We require that the distributive law holds...and if there is an identity 1 in the ring, that 1 * x = x. Math is hard if we don't assume that. My specific area of expertise could be said to be in the possible formation of fractions in such rings. The modules are studied to shed light on the problem. It's hard to study fractions when x(1/y) and (1/y)x may not be the same element. Or when 1/y may or may not exist (remember, 1/0 doesn't exist and so if xy = 0 for some element x, which happens more often than you might think, then an element 1/y = x/xy = x/0, which isn't going to exist either. So I studied topologies on rings and torsion theories in the categories of the rings' modules. I won't go further. But the intricasy and beauty of the structures are awesome to behold! And discovering levels of depth to meaning that eventually inform our life philosophies is incredible. Wow! What a screed! I probably lost most of you there. I'm sorry that I tend to do that, caught up in the beauty flowing through my brain. What saddens me is that there are so few people I can talk about this with, this artistic pastiche called mathematics. I did my best for 25 years trying to share the joy with people, as their algebra teacher (I feel the urge to expound on what an algebra is, but I'll save that for another day...other than to note that I said an algebra). I taught in college, remedial level through grad school level. Now I have moved on to programming languages, art and poetry. But I feel the need...I have the duty...to defend my field and promote its wellbeing. No place in my screed up there did I refer to any numbers other than 0 and 1. Mathematics is about words. It's about logically pasting thoughts together to see what is true and what is not. It is assembling building blocks provided by other people and adding our own blocks and the paste to put them together to create new knowledge. We have a very difficult time trying to get students to think that way. You know why? Because in grade school, while their curiosity is being stomped out of them somewhere between grade 2 and grade 6, the students are told that arithmetic is mathematics and mathematics is arithmetic, that mathematics is all about the numbers and if they don't learn the numbers right, they can't go on. Who said so? Where is it written? If we want to improve education, which I believe is the goal, then we have to think about it critically. That requires that we think deeply about all of our assumptions with an open mind. I try to have an open mind about people who have good enough verbal skills to understand and master challenging mathematics being kept out of my subject because they can't master skills they will not need in order to understand that mathematics. I understand that other people don't have my agenda of preserving the beauty of the past and passing it on to future generations, adding to and embellishing that creative structure and presenting its results to the rest of humanity, that they may be applied wherever by whomever. To me it is all about the beauty. My side doesn't get expressed a lot. 
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