By now, everyone should be sick and tired about my constant ranting and raving about critical thinking, thinking for yourself, logic, et cetera et cetera et cetera...
I'm a maths guy. It's what I harp on and how I survive. And as a teacher, even more so because I am charged with the task of teaching OTHERS how to think critically.
YES. Your children will be in my hands. Fear for the future.
A friend of mine recently sent me this story written in the style of a Zen lesson. I think it sums my views on critical thinking up very nicely.
I hope you enjoy it!
Showing posts with label math. Show all posts
Showing posts with label math. Show all posts
Tuesday, August 4, 2009
Proving Infinity
I'm going to put on my pedant hat again today and talk about a concept that we all know and love...or maybe not so much love depending on your math skills.
Let's have some personal background to this logical exercise first:
A year ago this September 19th, the Society of Non-Theists at Purdue UniversitySONTAPU, lol held a sort of mock-evangelical rally for the Flying Spaghetti Monster* to try to convey a message. This message was: "Unverifiable claims aren't true just because you cannot disprove them.
Throughout the day, we drew larger and larger crowds. The various people were eliciting emotions ranging the gamut from "lol" to "Alright!" to "What on Earth are they doing?" to "OH LAWD BABEH JEEBUS HELP ME!!!1!1!eleven"
In the corner, however stood two people. One was holding a video camera and the other was talking, perhaps if narrating.
After I read a chapter of the Gospel of the Flying Spaghetti Monster in my best "Preacher Voice" I went to go get some water and was stopped by these guys to talk.
It became clear very quickly that these people were not here to get the poop on what we were doing and why we were doing it, but I talked to them civilly. They asked me how I felt about God and what I, myself believed.
I told them. As a scientist and, much more specifically, a Math major, education classes be damned, I had very specific ideas of what I would need to be persuaded to the side of the believers. That is to say, PROOF. Logic. In order for me to stand up and say "There exists a higher power." I need to see a written proof with QED at the end (although now, I think I'd also accept the heavens parting and having God himself send me on a quest a-la Monty Python).
"How about infinity?" was the response. "Can you prove infinity?"
"Well...prove infinity itself? Hmm...I'm not sure I, myself know how to do that. I suppose you could go through the route proving that the integers have no upper bound and are therefore infinite."
This, of course, didn't help. Eventually, they moved away from Math onto subjects that they were more properly coached on and that I didn't have enough real training in to properly bullshit mine their arguments. I made eye contact with other people from our group in the traditional, "Shit, shit, help me! They won't stop throwing bullshit at me!" fashion, allowed someone else to get caught up in the argument and then bowed out claiming various excuses.
But, it got to me. Infinity. What does infinity mean? We use it in math all the time, don't we? Calculus is basically built around the concept of infinity, isn't it? Differentials...integrals, Reimann sums...The infinite and the infinitesmal are all around...well...sort of.
Infinity is a somewhat wooly concept...and by somewhat wooly, I mean completely incomprehensible. To have an infinite quantity of something is physically impossible. It's a contradiction of terms, really. If you have a quantity, you have quantified it. How can you quantify something that is, by definition, inquantifiable? Well, that's precisely it. You can't.
Infinity is not a thing. It is not a measurement. It is not, really, even a state of mind.
Infinity is, in all senses, the impossible we can never and WERE NEVER MEANT TO reach simply by the very nature of the concept! It's not even a benchmark that is merely set too high.
"But, Mark," You say. "There are so many other concepts that we can't actually see that we use all the time, too!"
"Well, yeah. Sorta."
"I mean, you have imaginary numbers, transendental numbers, even, perhaps, NEGATIVE numbers are also abstract concepts that we are surrounded by in math that we don't actually argue with."
This is very true. You probably couldn't find -1 apple, or 2i dollars in your wallet...and I'd love to see someone come up with exactly pi of something.
However, in each of these situations, regardless of their abstractness, we use them because they appear in nature. Even imaginary numbers have a very useful practical application that translates into something tangible. Just ask your friendly neighborhood electrical engineer. I'm sure he'd be glad to point you in the right direction.**
The point is that infinity is really the only one of these that doesn't get a real, practical analog because it doesn't exist on its own.
Infinity is a tool, certainly, but not something that can be proven.
*Yes, they're bowing down to me. It seems that, on the same day, there was a flash mob. They were going through and taking showers and brushing their teeth in all the fountains on campus. There is a small fountain right by where we were holding our event.
**Just make sure he's bathed recently.***
***I'm just kidding. I love you all. Please don't kill me with your trebuchets.
Let's have some personal background to this logical exercise first:
A year ago this September 19th, the Society of Non-Theists at Purdue University
Throughout the day, we drew larger and larger crowds. The various people were eliciting emotions ranging the gamut from "lol" to "Alright!" to "What on Earth are they doing?" to "OH LAWD BABEH JEEBUS HELP ME!!!1!1!eleven"
In the corner, however stood two people. One was holding a video camera and the other was talking, perhaps if narrating.
After I read a chapter of the Gospel of the Flying Spaghetti Monster in my best "Preacher Voice" I went to go get some water and was stopped by these guys to talk.
It became clear very quickly that these people were not here to get the poop on what we were doing and why we were doing it, but I talked to them civilly. They asked me how I felt about God and what I, myself believed.
I told them. As a scientist and, much more specifically, a Math major, education classes be damned, I had very specific ideas of what I would need to be persuaded to the side of the believers. That is to say, PROOF. Logic. In order for me to stand up and say "There exists a higher power." I need to see a written proof with QED at the end (although now, I think I'd also accept the heavens parting and having God himself send me on a quest a-la Monty Python).
"How about infinity?" was the response. "Can you prove infinity?"
"Well...prove infinity itself? Hmm...I'm not sure I, myself know how to do that. I suppose you could go through the route proving that the integers have no upper bound and are therefore infinite."
This, of course, didn't help. Eventually, they moved away from Math onto subjects that they were more properly coached on and that I didn't have enough real training in to properly bullshit mine their arguments. I made eye contact with other people from our group in the traditional, "Shit, shit, help me! They won't stop throwing bullshit at me!" fashion, allowed someone else to get caught up in the argument and then bowed out claiming various excuses.
But, it got to me. Infinity. What does infinity mean? We use it in math all the time, don't we? Calculus is basically built around the concept of infinity, isn't it? Differentials...integrals, Reimann sums...The infinite and the infinitesmal are all around...well...sort of.
Infinity is a somewhat wooly concept...and by somewhat wooly, I mean completely incomprehensible. To have an infinite quantity of something is physically impossible. It's a contradiction of terms, really. If you have a quantity, you have quantified it. How can you quantify something that is, by definition, inquantifiable? Well, that's precisely it. You can't.
Infinity is not a thing. It is not a measurement. It is not, really, even a state of mind.
Infinity is, in all senses, the impossible we can never and WERE NEVER MEANT TO reach simply by the very nature of the concept! It's not even a benchmark that is merely set too high.
"But, Mark," You say. "There are so many other concepts that we can't actually see that we use all the time, too!"
"Well, yeah. Sorta."
"I mean, you have imaginary numbers, transendental numbers, even, perhaps, NEGATIVE numbers are also abstract concepts that we are surrounded by in math that we don't actually argue with."
This is very true. You probably couldn't find -1 apple, or 2i dollars in your wallet...and I'd love to see someone come up with exactly pi of something.
However, in each of these situations, regardless of their abstractness, we use them because they appear in nature. Even imaginary numbers have a very useful practical application that translates into something tangible. Just ask your friendly neighborhood electrical engineer. I'm sure he'd be glad to point you in the right direction.**
The point is that infinity is really the only one of these that doesn't get a real, practical analog because it doesn't exist on its own.
Infinity is a tool, certainly, but not something that can be proven.
*Yes, they're bowing down to me. It seems that, on the same day, there was a flash mob. They were going through and taking showers and brushing their teeth in all the fountains on campus. There is a small fountain right by where we were holding our event.
**Just make sure he's bathed recently.***
***I'm just kidding. I love you all. Please don't kill me with your trebuchets.
Wednesday, July 29, 2009
Math Time
Apropos of nothing, I want to talk about the Brouwer fixed point theorem.
This theory states, basically, there is at least one point on that function that sends the point to itself. That is, there is always at least one point x such that, for a function f, f(x) = x.
I find this property of life so unequivocally awesome that I have to talk about it.
What this theorem shows is that regardless of how you manipulate the world around you, something will end up back where it started.
Let's start simple...
1. Imagine you had a flat map of your country sitting in front of you...or, indeed, dear reader, go get one now. Were you to drop it on the ground, there would be, without a doubt, one point on that map that was directly above the point it represented no matter the scale of the map as long as one sat inside the other.
2. Alternatively, take two pieces of paper with identical pictures on them, crumble one up and place it on top of the other piece of paper. At least one point on the crumbled up piece of paper would be sitting above its corresponding point.
3. Let's say you had a cup of coffee and a spoon. After stirring your coffee, there would always be at least ONE atom of coffee in that cup that was in the exact same place as where it began.
4. This property is also the reason why you could never have a tie in a game of Hex.
As for the proof itself, it'somfg a head tripsomewhat difficult to understand if you don't know the jargon*.
The proof is normally done by contradiction--that is, we try to prove the opposite (i.e. that there are no fixed points) in the hopes that we find a result that shouldn't exist.
Effectively, we begin by saying "Assume there is no fixed point" and then try to "break" the proof using what we already know. If we can show that saying there is no fixed point is absurd, then there can only be one recourse...that there is at least one fixed point somewhere.
Interestingly enough, later Brouwer rejected this proof because he felt that all proofs must be "constructed." That is to say, he felt that proof by contradiction was cheating and that if you are going to prove something, you should do it directly--which he did end up doing for some proofs, including, eventually, the Fixed Point Theorem**.
I hope you feel more enlightened now.
*I know the feeling quite well. My first time being given this proof was in a special guest lecture about the history of proof. It whizzed by me so fast that when I copied it down, even now it makes no sense...which is a terrible shame.
**Which was, I'm sure, awesome for him. Even I don't want to touch that one.
This theory states, basically, there is at least one point on that function that sends the point to itself. That is, there is always at least one point x such that, for a function f, f(x) = x.
I find this property of life so unequivocally awesome that I have to talk about it.
What this theorem shows is that regardless of how you manipulate the world around you, something will end up back where it started.
Let's start simple...
1. Imagine you had a flat map of your country sitting in front of you...or, indeed, dear reader, go get one now. Were you to drop it on the ground, there would be, without a doubt, one point on that map that was directly above the point it represented no matter the scale of the map as long as one sat inside the other.
2. Alternatively, take two pieces of paper with identical pictures on them, crumble one up and place it on top of the other piece of paper. At least one point on the crumbled up piece of paper would be sitting above its corresponding point.
3. Let's say you had a cup of coffee and a spoon. After stirring your coffee, there would always be at least ONE atom of coffee in that cup that was in the exact same place as where it began.
4. This property is also the reason why you could never have a tie in a game of Hex.
As for the proof itself, it's
The proof is normally done by contradiction--that is, we try to prove the opposite (i.e. that there are no fixed points) in the hopes that we find a result that shouldn't exist.
Effectively, we begin by saying "Assume there is no fixed point" and then try to "break" the proof using what we already know. If we can show that saying there is no fixed point is absurd, then there can only be one recourse...that there is at least one fixed point somewhere.
Interestingly enough, later Brouwer rejected this proof because he felt that all proofs must be "constructed." That is to say, he felt that proof by contradiction was cheating and that if you are going to prove something, you should do it directly--which he did end up doing for some proofs, including, eventually, the Fixed Point Theorem**.
I hope you feel more enlightened now.
*I know the feeling quite well. My first time being given this proof was in a special guest lecture about the history of proof. It whizzed by me so fast that when I copied it down, even now it makes no sense...which is a terrible shame.
**Which was, I'm sure, awesome for him. Even I don't want to touch that one.
Monday, July 27, 2009
Guest Blog Repost: Atheism is not a religion
Here's another one I did that became pretty controversial. I would like to thank Frank for bringing up some fantastic points. Perhaps in a few days I'll comment on them myself. In the meantime, here's the post.
Maybe we can drum up some controversy over here too!
Note: I was not particularly sober at this point, so I'm not going to vouch for the watertightness of my proofs here.
Hey everybody!
Post #4 from Mark!
Beer tends to make me more introspective (Being that it is Blue Moon, I'm also incredibly happy.), so I'm going to dust off an old topic that SHOULD have been laid to rest years ago; but, unfortunately, still pops up around occasionally.
Comparing Atheism to Religion:
Let's begin with a very cliché opening statement:
re*li*gion
–noun
1.
a set of beliefs concerning the cause, nature, and purpose of the
universe, esp. when considered as the creation of a superhuman
agency or agencies, usually involving devotional and ritual
observances, and often containing a moral code governing the
conduct of human affairs.
2.
a specific fundamental set of beliefs and practices generally
agreed upon by a number of persons or sects: the Christian
religion; the Buddhist religion.
3.
the body of persons adhering to a particular set of beliefs and
practices: a world council of religions.
4.
the life or state of a monk, nun, etc.: to enter religion.
5.
the practice of religious beliefs; ritual observance of faith.
6.
something one believes in and follows devotedly; a point or matter
of ethics or conscience: to make a religion of fighting prejudice.
Neglecting 3, 4, and 5 because they are incidental to the argument, I want to go through and explain the rest of these. Surely you, dear reader, will agree with me that, assuming these are the only definitions of religion, if I can show Atheism does not fall into any of these categories (each statement, therefore, is conjoined by an “or”), I will have proved Atheism not a religion. Hooray Analysis classes! I wonder if I can re-write some of these definitions as actual mathematical statements.
Also, this is taken from Random House Dictionary. Credible source if I say so myself.
1. Let's start with “a set of beliefs concerning the cause, nature, and purpose of the universe.”
Let X and Y be sets such that X = {x| x = a belief concerning the cause nature or purpose of the universe} and Y = {y| y = a common beliefs of Atheists regarding the nature of the universe} Then if Atheism is a religion, X ^ Y =/= emptyset
I think we can all agree that Atheism has only one actual concept associated with it: The disbelief that there exists such a thing as God. There is no universal belief as to how the universe was created, what it looks like beyond what we can see, and, especially, the purpose of said universe.
The rebuttal usually comes in the form of the following: “What about the Big Bang? It is generally assumed that if a person does not agree that a god created the universe, it began with 'The Big Bang.'”
Certainly. This is a commonly held theorem by many people. The concept of The Big Bang Theory (which is also a REALLY awesome show, by the way) is, indeed the best we have so far. Years and years of testing, measuring, and pondering have been done and this is the only theory that has stood the test of time. Also, this theory was first hypothesized by a priest. So, the church SHOULD be with us on this one. More importantly, Atheism has nothing to do with guessing at the origins of the universe. I'm sure there is at least one Atheist somewhere who is convinced that Aliens are responsible for some reason. Atheism and scientific thought are not necessarily synonymous.
i.e. Assume that X^Y=/= empty set.
But the infinite intersection of Ya, where a is a subset of A where a is contained in A= {All the atheists in the world} (A is the spanning set of Y where A is all the atheists in the world and Ya is the set of commonly held beliefs of all atheists regarding the nature of the universe)
Ya = {empty set} Therefore, X^Y = empty set.
CONTRADICTION.
“esp. when considered as the creation of a superhuman agency or agencies, usually involving devotional and ritual observances, and often containing a moral code governing the conduct of
human affairs.”
I'm sure we can leave this as an exercise.
2. “a specific fundamental set of beliefs and practices generally agreed upon by a number of persons or sects: the Christian religion; the Buddhist religion.”
Let X and Y be sets such that X = {x| x = a belief} and Y = {y| y = a common beliefs of Atheists} Then if Atheism is a religion X ^ Y =/= emptyset
Again, because Atheism has no particular collection of beliefs, there is no set of beliefs to agree on.
Don't pull the kind of crap with me that says, “It takes FAITH not to believe in God.”
Pointing out that religions have no real case to prove that God exists is NOT a belief. It's merely an observation of a logic flaw.
The proof for #2 is nearly identical to #1.
6. “Something one believes in and follows devotedly; a point or matter of ethics or conscience.”
Let me break this up into two sections starting with the latter first.
“A point or matter of ethics or conscience”
Let X and Y be sets such that X = {x| x = a statement regarding ethics} and Y = {God does not exist} Then if Atheism is a religion X ^ Y =/= emptyset
Atheism says the following: GOD DOES NOT EXIST.
This is not, and I repeat, NOT a statement regarding ethics in any sense.
i.e. God does not exist is not contained in X. Therefore X^Y = empty set.
Part 2:
Something one believes in and follows devotedly
I have never met an Atheist who has spent their life devoted to the thought that God Does Not Exist.
Our thoughts on the existence of a god does not rule our lives. It does not even, normally, play anything more than a tangential part in who we are. I am Mark and, yes, I am indeed an Atheist. HOWEVER, more importantly, I am a teacher, a musician, I have brown hair, I was born in September and I like long walks on the beach. I am devoted only to living my life as I feel it needs to be lived. The only difference in the way my life will be lived compared to if I weren't an atheist, is I'd be spending more time in Synagogue. Given the amount of free time I now have on Saturdays, I can live my life 3 hours more every single week.
Q.E.D.
Maybe we can drum up some controversy over here too!
Note: I was not particularly sober at this point, so I'm not going to vouch for the watertightness of my proofs here.
Hey everybody!
Post #4 from Mark!
Beer tends to make me more introspective (Being that it is Blue Moon, I'm also incredibly happy.), so I'm going to dust off an old topic that SHOULD have been laid to rest years ago; but, unfortunately, still pops up around occasionally.
Comparing Atheism to Religion:
Let's begin with a very cliché opening statement:
re*li*gion
–noun
1.
a set of beliefs concerning the cause, nature, and purpose of the
universe, esp. when considered as the creation of a superhuman
agency or agencies, usually involving devotional and ritual
observances, and often containing a moral code governing the
conduct of human affairs.
2.
a specific fundamental set of beliefs and practices generally
agreed upon by a number of persons or sects: the Christian
religion; the Buddhist religion.
3.
the body of persons adhering to a particular set of beliefs and
practices: a world council of religions.
4.
the life or state of a monk, nun, etc.: to enter religion.
5.
the practice of religious beliefs; ritual observance of faith.
6.
something one believes in and follows devotedly; a point or matter
of ethics or conscience: to make a religion of fighting prejudice.
Neglecting 3, 4, and 5 because they are incidental to the argument, I want to go through and explain the rest of these. Surely you, dear reader, will agree with me that, assuming these are the only definitions of religion, if I can show Atheism does not fall into any of these categories (each statement, therefore, is conjoined by an “or”), I will have proved Atheism not a religion. Hooray Analysis classes! I wonder if I can re-write some of these definitions as actual mathematical statements.
Also, this is taken from Random House Dictionary. Credible source if I say so myself.
1. Let's start with “a set of beliefs concerning the cause, nature, and purpose of the universe.”
Let X and Y be sets such that X = {x| x = a belief concerning the cause nature or purpose of the universe} and Y = {y| y = a common beliefs of Atheists regarding the nature of the universe} Then if Atheism is a religion, X ^ Y =/= emptyset
I think we can all agree that Atheism has only one actual concept associated with it: The disbelief that there exists such a thing as God. There is no universal belief as to how the universe was created, what it looks like beyond what we can see, and, especially, the purpose of said universe.
The rebuttal usually comes in the form of the following: “What about the Big Bang? It is generally assumed that if a person does not agree that a god created the universe, it began with 'The Big Bang.'”
Certainly. This is a commonly held theorem by many people. The concept of The Big Bang Theory (which is also a REALLY awesome show, by the way) is, indeed the best we have so far. Years and years of testing, measuring, and pondering have been done and this is the only theory that has stood the test of time. Also, this theory was first hypothesized by a priest. So, the church SHOULD be with us on this one. More importantly, Atheism has nothing to do with guessing at the origins of the universe. I'm sure there is at least one Atheist somewhere who is convinced that Aliens are responsible for some reason. Atheism and scientific thought are not necessarily synonymous.
i.e. Assume that X^Y=/= empty set.
But the infinite intersection of Ya, where a is a subset of A where a is contained in A= {All the atheists in the world} (A is the spanning set of Y where A is all the atheists in the world and Ya is the set of commonly held beliefs of all atheists regarding the nature of the universe)
Ya = {empty set} Therefore, X^Y = empty set.
CONTRADICTION.
“esp. when considered as the creation of a superhuman agency or agencies, usually involving devotional and ritual observances, and often containing a moral code governing the conduct of
human affairs.”
I'm sure we can leave this as an exercise.
2. “a specific fundamental set of beliefs and practices generally agreed upon by a number of persons or sects: the Christian religion; the Buddhist religion.”
Let X and Y be sets such that X = {x| x = a belief} and Y = {y| y = a common beliefs of Atheists} Then if Atheism is a religion X ^ Y =/= emptyset
Again, because Atheism has no particular collection of beliefs, there is no set of beliefs to agree on.
Don't pull the kind of crap with me that says, “It takes FAITH not to believe in God.”
Pointing out that religions have no real case to prove that God exists is NOT a belief. It's merely an observation of a logic flaw.
The proof for #2 is nearly identical to #1.
6. “Something one believes in and follows devotedly; a point or matter of ethics or conscience.”
Let me break this up into two sections starting with the latter first.
“A point or matter of ethics or conscience”
Let X and Y be sets such that X = {x| x = a statement regarding ethics} and Y = {God does not exist} Then if Atheism is a religion X ^ Y =/= emptyset
Atheism says the following: GOD DOES NOT EXIST.
This is not, and I repeat, NOT a statement regarding ethics in any sense.
i.e. God does not exist is not contained in X. Therefore X^Y = empty set.
Part 2:
Something one believes in and follows devotedly
I have never met an Atheist who has spent their life devoted to the thought that God Does Not Exist.
Our thoughts on the existence of a god does not rule our lives. It does not even, normally, play anything more than a tangential part in who we are. I am Mark and, yes, I am indeed an Atheist. HOWEVER, more importantly, I am a teacher, a musician, I have brown hair, I was born in September and I like long walks on the beach. I am devoted only to living my life as I feel it needs to be lived. The only difference in the way my life will be lived compared to if I weren't an atheist, is I'd be spending more time in Synagogue. Given the amount of free time I now have on Saturdays, I can live my life 3 hours more every single week.
Q.E.D.
Guest Blog Repost: Mark on Math
As many of you don't know...what with most of you just meeting me for the first time and all...I am currently an Undergrad in the school of Math at Purdue who is pursuing a degree in Math Education (because, frankly, it's running away from me and I really want to catch it.)
What this means, for those of you who are not both studying at Purdue OR in a secondary education major, is that I am a Math student who is forced to take six relatively perfunctory education classes in addition to nearly ALL the math classes.
As a result of this particularly rigorous number of math classes (and a few awesome ones I've taken just for the lulz), I've been given a very good understanding of what is necessary to come into these classes and not leave the room crying every day. Let's just say, I didn't have a very excellent background in Math before I came to Purdue and started off on my path to become a math teacher (after, of course, a year and a half detour through the Chemical Engineering department. *sadface*).
Granted, my Calculus and Trig. skills are fantastic, my Algebra skills are awesome, and my Geometry skills are...well, not awesome but I made it through the class and, by the end, had totally made up for the terrible beginning.
“But...but Mark!” You say. “Isn't that Math?”
Well...sort of.
Those things are the sum total of Math in the same way that taking baking soda and vinegar and mixing them together is chemistry.
Sure...these are things you do IN math and things that require math but what is missing is the theoretical aspect.
WHY do these things do what they do? Why does the Calculus do what it is supposed to?
This part of math is called “Analysis.” It mostly consists of “Proofs.” That is to say, the mathematic reasoning behind a given theorem.
The problem is that back in high school (and it seems most high schools nowadays) provide little to no actual analysis backing...specifically because of how state standards are set up. In order to continue functioning as a school, its students must score at certain levels on their standardized tests. As a result, teachers don't always have the option of including logical reasoning and proof as a part of their curriculum.
This is really freaking sad.
To me, this strips Math of all of its science! There is no inquiry. It's just become history with numbers.
This next semester, I will be teaching a class here at Purdue. MA 153 for those in the know and Algebra and Trigonometry I for those who aren't.
I fully intend to sneak in as much logic and reasoning as I possibly can. My students will not just know WHAT they're doing, but I'll actually explain to them WHY they're doing what they're doing and WHERE it comes from so they can understand HOW to do it on a higher level than they might were they just to get equations and algorithms thrown at them.
Until later, this is Mark signing off!
What this means, for those of you who are not both studying at Purdue OR in a secondary education major, is that I am a Math student who is forced to take six relatively perfunctory education classes in addition to nearly ALL the math classes.
As a result of this particularly rigorous number of math classes (and a few awesome ones I've taken just for the lulz), I've been given a very good understanding of what is necessary to come into these classes and not leave the room crying every day. Let's just say, I didn't have a very excellent background in Math before I came to Purdue and started off on my path to become a math teacher (after, of course, a year and a half detour through the Chemical Engineering department. *sadface*).
Granted, my Calculus and Trig. skills are fantastic, my Algebra skills are awesome, and my Geometry skills are...well, not awesome but I made it through the class and, by the end, had totally made up for the terrible beginning.
“But...but Mark!” You say. “Isn't that Math?”
Well...sort of.
Those things are the sum total of Math in the same way that taking baking soda and vinegar and mixing them together is chemistry.
Sure...these are things you do IN math and things that require math but what is missing is the theoretical aspect.
WHY do these things do what they do? Why does the Calculus do what it is supposed to?
This part of math is called “Analysis.” It mostly consists of “Proofs.” That is to say, the mathematic reasoning behind a given theorem.
The problem is that back in high school (and it seems most high schools nowadays) provide little to no actual analysis backing...specifically because of how state standards are set up. In order to continue functioning as a school, its students must score at certain levels on their standardized tests. As a result, teachers don't always have the option of including logical reasoning and proof as a part of their curriculum.
This is really freaking sad.
To me, this strips Math of all of its science! There is no inquiry. It's just become history with numbers.
This next semester, I will be teaching a class here at Purdue. MA 153 for those in the know and Algebra and Trigonometry I for those who aren't.
I fully intend to sneak in as much logic and reasoning as I possibly can. My students will not just know WHAT they're doing, but I'll actually explain to them WHY they're doing what they're doing and WHERE it comes from so they can understand HOW to do it on a higher level than they might were they just to get equations and algorithms thrown at them.
Until later, this is Mark signing off!
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