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 teaching. Show all posts
Showing posts with label teaching. 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.
Friday, July 31, 2009
I loled
Jen sent me a comic today and I laughed really hard. The strip comes from a blog called Coelacanth Diaries written by the talented Stephen Collins.
Enjoy!
Enjoy!
Monday, July 27, 2009
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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