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In my last few weeks of the school term, I had many discussions in class about the education system in Singapore and countries elsewhere.
I'm not too sure when it was that I developed the passion to learn. I just happened to realize that every bit of knowledge that was available to me was the embodiment of what it was to be a human being. Everything was part of this giant cosmic web that I've only seen a small part of. I got hungry, I got famished. I needed to consume information and have my many questions answered. Since then, I've been an extremely eager student, and my teachers have noticed this.
I just found everything so fascinating, and it didn't even matter what it was. I used to hate Maths and Science because I didn't see the relevance of them and didn't have the passion for them. I made up excuses for my bad results in Math and in Biology and Chemistry, saying that I was "right-brained" or was naturally inclined towards the arts. Then I stopped, because I suddenly realized that what I was being stupid.
I took a tremendous amount of interest into everything, and I stopped thinking thoughts that went like this, "Why do I even need to learn trigonometry? I'm not going to be walking along the road one day and suddenly find myself with the urge to calculate the angle of inclination of some lamp." or other puerile nonsense.
Nothing was redundant, and as I've mentioned - everything was linked to something else. It was like an infinite dance of information, and I was in the front row. I could reach out and grab whatever I wanted.
Of course, I took precautions: I avoided learning how schools wanted me to. I made sure that the drive I had to learn was intrinsic – I didn't absorb facts for the sake of absorbing them. I became self-possessed and passionate. I believed that everything I could learn was beneficial to me as a person. I kept trying to see the beauty in things, and now being awed by numerals and mathematical concepts and the majesty of the Sciences comes naturally. The wonder which I have when I approach these subjects is genuine and real.
It's been observed that most of the students in Singapore are simply masters of absorbing facts and statistics. They do not see the relevance of anything and everything they learn in school. They've been labelled as sponges, they mindlessly absorb whatever is presented to them. When they're wrung - when they sit for an examination, facts are forced out of them.
I learn because I enjoy learning about what it is to be human, and everything that entails being one.
+ Show Spoiler +
I'm not entirely confident about where I was going with this blog. I guess I just wanted to share the wonderment of a starry-eyed schoolboy with regards to learning.
  
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Reading about the history of math is a fun way of seeing its relevance. ET Bell's "Men of Mathematics" gives short, entertaining biographies of famous mathematicians and traces the development of math. It is pretty amazing when you think about it all. Philosophy and literature have fascinating histories as well.
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I'm not sure what the proper way to respond to this blog is, but I approve its contents. Also, you always have nice music; I already have you to thank for discoving Rebecca Brandt and Bonobo, add Emancipator to that list 
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This is good! But why not question the primacy of utility to begin with? Where did our assumption about the value of utility come from anyways? Why must everything that is of value be useful, and everything useful be valueable?
The Master said: "The Superior Man is aware of Goodness, the inferior man is aware of advantage."
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Thanks, I didn't realize his new album came out.
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Keep in mind all science and math is philosophy, and you'll go far with learning.
PS Wasn't the whole right-brain left-brain thing debunked?
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Also it probably matters what you're studying. I don't know if anyone can say that every subject was interesting to them, so it probably makes more sense for you to say that you only have a passion to learn certain subjects (unless you really like *all* of them?). But maybe there are some people out there who genuinely enjoy learning about everything; I haven't seen many people like that. People usually like specializing in one area, and would be thoroughly depressed working as a shoe salesman (for example).
That's why you get the "puerile nonsense" like how am I going to use trigonometry in my daily life. But I don't think its childish or silly at all. I think you would probably do well to recognize that not everyone is like you, and not everyone has a passionate interest in learning math or the infinite connections associated with it. There are a lot of aspects of mathematics that they probably won't use for the rest of their life, so they shouldn't be forced to learn something that holds no innate interest to them. Of course there may be other arguments for learning the math that's taught at that level.
Incidentally, math gets 100x more interesting in university if you decide to specialize in math (I'm taking one such course now, the textbook is excellent, by Michael Spivak), but you better be a semi-genius or you'll have trouble!
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On March 17 2013 22:54 Iranon wrote:I'm not sure what the proper way to respond to this blog is, but I approve its contents. Also, you always have nice music; I already have you to thank for discoving Rebecca Brandt and Bonobo, add Emancipator to that list
I'm glad
On March 17 2013 23:46 Roe wrote:
Keep in mind all science and math is philosophy, and you'll go far with learning.
PS Wasn't the whole right-brain left-brain thing debunked?
I have no idea. I didn't bother to look up more info on the matter.
On March 18 2013 00:09 radscorpion9 wrote:
Also it probably matters what you're studying. I don't know if anyone can say that every subject was interesting to them, so it probably makes more sense for you to say that you only have a passion to learn certain subjects (unless you really like *all* of them?). But maybe there are some people out there who genuinely enjoy learning about everything; I haven't seen many people like that. People usually like specializing in one area, and would be thoroughly depressed working as a shoe salesman (for example).
That's why you get the "puerile nonsense" like how am I going to use trigonometry in my daily life. But I don't think its childish or silly at all. I think you would probably do well to recognize that not everyone is like you, and not everyone has a passionate interest in learning math or the infinite connections associated with it. There are a lot of aspects of mathematics that they probably won't use for the rest of their life, so they shouldn't be forced to learn something that holds no innate interest to them. Of course there may be other arguments for learning the math that's taught at that level.
Incidentally, math gets 100x more interesting in university if you decide to specialize in math (I'm taking one such course now, the textbook is excellent, by Michael Spivak), but you better be a semi-genius or you'll have trouble!
To be honest, I think it's because I only take a few subjects in school. That is:
-English
-Elementary Math
-Additional Math
-Chemistry
-Biology
-History/Social Studies
And because I'm only studying them all at a very basic level. I suppose that once I have to learn everything at a higher level, my interest for everything might wane. If we're talking about true passion, then mine lies in Literature. For this, I have sam!zdat and farvacola to thank ^_^
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Seeking education and knowledge for intrinsic purposes- because you truly want to obtain information and have those interesting and diverse conversations that only a well-rounded and educated person could have- is an incredibly worthwhile project, and one that educators wish all their students would believe in. Unfortunately, most motivators for education (at least, until students become adults) tend to be external (e.g., learn about this topic because you want a good grade, or do your homework because your parents and teachers want you to). And while those aren't terrible per se, they're not very helpful in the long run if they don't eventually create some level of inherent desire to learn as well. You'll tend to stop caring once you complete the task that's being asked of you, especially if educators are more focused on you regurgitating information than actually generating a high level of interest in the subject. But if you honestly have a thirst for knowledge, even if the purpose of it isn't immediately practical, you welcome debates and discussions and research and reading, even after you pass your class. And that's rare for many students, because many systems of education are more concerned with tangible results like grades than how students feel about their education. (One can make an argument that they're linked, although students have learned how to play the game of school and to do well in classes without truly learning or remembering anything. Because it's not their priority, especially if it doesn't seem like their teachers' priority. And it's much harder to quantify a love of learning than it is to look at one's grades.)
Thanks for calling my attention to your blog post It makes me happy- both as an educator and as a student- that some people can develop a thirst for knowledge on their own. I only hope it will be reinforced and even strengthened by those you meet in the future.
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Seeking education and knowledge for intrinsic purposes- because you truly want to obtain information and have those interesting and diverse conversations that only a well-rounded and educated person could have- is an incredibly worthwhile project, and one that educators wish all their students would believe in.
So how are educators actively trying to accomplish this?
Unfortunately, most motivators for education (at least, until students become adults) tend to be external (e.g., learn about this topic because you want a good grade, or do your homework because your parents and teachers want you to).
many systems of education are more concerned with tangible results like grades than how students feel about their education.
What do you think should be done so things can improve?
I'm glad you enjoyed my blog, and thanks for the well wishes
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The main reason I learn is because it's challenging and everything else is boring.
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On March 23 2013 18:47 Recognizable wrote:
The main reason I learn is because it's challenging and everything else is boring.
I'm only theorizing here, but isn't everything you do learning, in a sense? When you practice StarCraft, it's still falls under learning, since you're learning how to play better after all. When you listen to music, you're learning how to appreciate music, etc., so on and so forth. I hope you get the idea.
I think you only stop learning if your mind is stagnant and it's literally a blank.
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Yes. However, all those things you mentioned, games/music/movies just can't keep my mind occupied for long enough.
They do not see the relevance of anything and everything they learn in school. They've been labelled as sponges, they mindlessly absorb whatever is presented to them. When they're wrung - when they sit for an examination, facts are forced out of them.
Very true, there is this weird dichotomy in my school. Everyone that has chosen a science curriculum generally get's high grades and likes to actually learn, everyone that did not, doesn't. It's very understandable as well. Because something like History, at least on a high school level, requires almost no creativity whatsoever and it's just memorizing facts. Very boring and dull. The amount of times I've gotten, "so what can you do with Math/Physics?" Make me sick; I don't know and I don't care.
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On March 23 2013 19:20 Recognizable wrote:Yes. However, all those things you mentioned, games/music/movies just can't keep my mind occupied for long enough. Show nested quote +They do not see the relevance of anything and everything they learn in school. They've been labelled as sponges, they mindlessly absorb whatever is presented to them. When they're wrung - when they sit for an examination, facts are forced out of them. Very true, there is this weird dichotomy in my school. Everyone that has chosen a science curriculum generally get's high grades and likes to actually learn everyone that did not, doesn't. It's very understandable as well. Because something like History, at least on a high school level, requires almost no creativity whatsoever and it's just memorizing facts. Very boring and dull. The amount of times I've gotten, "so what can you do with Math/Physics?" Make me sick; I don't know and I don't care.
I am confident that I definitely understand what you're talking about. Soft subjects like History can be infinitely easier to do better in when you take it at a high school level. I'm not sure how the Sciences are tested in the Netherlands, but I would say that in Singapore, it's much harder to do well in the humanities. The reason for this is that how History is tested isn't just memorizing facts at all, although you still do it to a certain extent. The History tests I take here tests how well you can interpret a political cartoon/picture or whatever. With the inference you make from said picture, you then have to back up it up with the knowledge you have on the subject.
For example, there's a picture of Hitler shoving a piece of meat labelled Czechoslovakia into a meat grinder, and the product goes into a bowl labelled "the Reich". So you have to explain what you think of Hitler from that cartoon, by saying that he was expansionistic and so on. Then you have to reinforce this inference based on your knowledge on Hitler. You have to bring up the ideals that he expressed in Mein Kampf, or talk about the Czechoslovakian crisis in 1938.
Now let's consider how the Sciences are taught and examined in Singapore (in general). I learn a few things in school, exothermic and endothermic reactions, electrolysis, the calculation of moles, the whole works. But the approach which the teachers take to teaching these things are largely inspired abut what is going to come out on the test. I need to learn the first 20 elements in the periodic table because those are the elements that are going to appear in the test. The rest of the periodic table? Fuck `em. So what if they're interesting. You must memorize what happens when you add hydrochloric acid to Potassium. You must know how to prepare potassium nitrate from potassium hydroxide and nitric acid via titration.
Do you see what I'm trying to get across? Everything that I am taught that concerns the Sciences is just confined to this pre-defined box. I get sick of it sometimes, I really do. But then I realize that it's just what Science is. You're merely taking down observations of nature, that's why it seems like there is no creativity involved when learning it in school. I believe Neil deGrasse Tyson even said something like this once. Van Gogh painted starry night, and if he didn't, nobody would have painted it. But if Newton didn't discover the law of optics, somebody else would have discovered it. Because the study of science is already pre-defined, and art is not, it's one of the reasons why art fascinates me much more than science sometimes.
Of course, I understand that there is so much more to Science, and that innovative creativity is integral to scientific development, but still - I remain exponentially more interested in literature. It's because the laws of science is universal and unchanging. I guess what I'm trying to say is this - it takes different kinds of people. There are people that are thrilled by studying how the world works sans humans, and there are people that are interested in studying the world that humans create.
I have a feeling that I'm just spewing shit.
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I am confident that I definitely understand what you're talking about. Soft subjects like History can be infinitely easier to do better in when you take it at a high school level. I'm not sure how the Sciences are tested in the Netherlands, but I would say that in Singapore, it's much harder to do well in the humanities. The reason for this is that how History is tested isn't just memorizing facts at all, although you still do it to a certain extent. The History tests I take here tests how well you can interpret a political cartoon/picture or whatever. With the inference you make from said picture, you then have to back up it up with the knowledge you have on the subject.
Yes we have cartoons as well. Our last one on our exam was about Churchill and FDR sitting on a fishing boat, maybe you've seen it? They definitely strive to test critical thinking skills, however the solutions are very apparent and you just have to link a couple of things to what you've learnt. You aren't thinking across boundaries, the answer was somewhere exactly in the book, and if it was not, the reasoning is always so apparent it's hard not to stumble over the answer.
Now let's consider how the Sciences are taught and examined in Singapore (in general). I learn a few things in school, exothermic and endothermic reactions, electrolysis, the calculation of moles, the whole works. But the approach which the teachers take to teaching these things are largely inspired abut what is going to come out on the test. I need to learn the first 20 elements in the periodic table because those are the elements that are going to appear in the test. The rest of the periodic table? Fuck `em. So what if they're interesting.
This is beyond dumb. We get a book with basically all the information including the periodic table and we can use it at every exam.
You must memorize what happens when you add hydrochloric acid to Potassium. You must know how to prepare potassium nitrate from potassium hydroxide and nitric acid via titration.
Somewhere, probably beneath a layer of Mathematics and years of study, there are principles which explain why the things happen as they happen when you add hydrochloric acid to Potassium.
You're merely taking down observations of nature
I believe this to be a common misconception of science. Many of science wasn't done by observing nature and forming a theoretical model around the observations but by forming a theoretical model, which is where the creativity comes in, and confirming your theoretical model with nature.
Because the study of science is already pre-defined, and art is not, it's one of the reasons why art fascinates me much more than science sometimes.
Seeing as how art is made by humans whom are bounded by the same physical laws you could in theory form a theoretical model which explains why we humans enjoy art and what art will be succesful and whatnot. I'm just messing around :p
What is your opinion about Math? Seeing as how Math is the pinnacle of reasoning and creativity unbounded by physical laws. Well, you probably wouldn't know because they don't teach mathematical reasoning in high schools. Which is incredibly sad because it's what Mathematics is all about. I hope I can get good at it someday, because it's incredibly hard.
I should get back to studying.
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Then can't we just apply the same logic that you used for the humanities to the sciences? You said that you just get a book with all the information, so isn't it essentially the same thing - the answers are all found in the book, one way or another.
On March 23 2013 19:59 Recognizable wrote:I believe this to be a common misconception of science. Many of science wasn't done by observing nature and forming a theoretical model around the observations but by forming a theoretical model, which is where the creativity comes in, and confirming your theoretical model with nature.
But aren't they just observations of nature after all? Not too bastardize the whole scientific theory, but isn't it much like saying,"I think the sky is purple.". Then I go outside, check it out, and go, "Guess I'm wrong!".
On March 23 2013 19:59 Recognizable wrote:Show nested quote +Because the study of science is already pre-defined, and art is not, it's one of the reasons why art fascinates me much more than science sometimes. Seeing as how art is made by humans whom are bounded by the same physical laws you could in theory form a theoretical model which explains why we humans enjoy art and what art will be succesful and whatnot. I'm just messing around :p What is your opinion about Math? Seeing as how Math is the pinnacle of reasoning and creativity unbounded by physical laws. Well, you probably wouldn't know because they don't teach mathematical reasoning in high schools. Which is incredibly sad because it's what Mathematics is all about. I hope I can get good at it someday, because it's incredibly hard. I should get back to studying.
As you've mentioned, I don't learn mathematical reasoning, and I haven't been exposed that much to it. But I know there's something much more beneath and beyond everything I'm taught in school. So, I'm sad that I really don't have an opinion on Mathematics. However, I do know that Plexa is doing a PhD in Mathematics, pluripotential theory, if my memory serves me correctly.
I should get back to studying too
Edit:
If there's an easy way to introduce myself to mathematical reasoning, do tell.
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On March 23 2013 16:31 Azera wrote:Show nested quote +
Seeking education and knowledge for intrinsic purposes- because you truly want to obtain information and have those interesting and diverse conversations that only a well-rounded and educated person could have- is an incredibly worthwhile project, and one that educators wish all their students would believe in.
So how are educators actively trying to accomplish this?
I think that much of it can come from how teachers can get students actively engaged in the course material, motivate learning, and generate interest. This can occur through instructional methods (e.g., open class discussion and conversations as opposed to teachers just lecturing to students, boring them), as well as the specific projects and ideas that are organized by the teacher (either as a grading policy or just general philosophies that help students grow intellectually).
Unfortunately, I think many teachers feel that they need to focus more on testing and abiding by specific curricula, and are therefore not able to explore the variety of ideas they may have. Teachers often have to worry more about time restraints and the sheer quantity (not necessarily quality) of material that needs to be taught by the time class tests, course exams, and standardized testing occurs. As with any profession, feeling rushed tends to decrease performance, and a subject as important as education is something that truly should not be half-assed.
Show nested quote +
Unfortunately, most motivators for education (at least, until students become adults) tend to be external (e.g., learn about this topic because you want a good grade, or do your homework because your parents and teachers want you to).
Show nested quote +
many systems of education are more concerned with tangible results like grades than how students feel about their education.
What do you think should be done so things can improve? I'm glad you enjoyed my blog, and thanks for the well wishes
We can take a few pages out of other countries' books when it comes to educational policy. We can't copy everything that Finland or China or Singapore does, but we can work towards a culture that promotes the respect of educators and education (e.g., less hand-holding of students, making sure parents work together with teachers, and less political bullshit and walking on eggshells), as well as shifting the focus of standardized tests from an obsession that decides everything in life, to occasional benchmarks and general statistics. Granted, there are plenty of other educational problems that can't be solved at the teacher-level (e.g., federal funding and how that trickles down), but teachers can make sure that students are being mentally stimulated by the material, and not just in it for the grades.
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On March 23 2013 20:32 Azera wrote:Show nested quote +On March 23 2013 19:59 Recognizable wrote:Because the study of science is already pre-defined, and art is not, it's one of the reasons why art fascinates me much more than science sometimes. Seeing as how art is made by humans whom are bounded by the same physical laws you could in theory form a theoretical model which explains why we humans enjoy art and what art will be succesful and whatnot. I'm just messing around :p What is your opinion about Math? Seeing as how Math is the pinnacle of reasoning and creativity unbounded by physical laws. Well, you probably wouldn't know because they don't teach mathematical reasoning in high schools. Which is incredibly sad because it's what Mathematics is all about. I hope I can get good at it someday, because it's incredibly hard. I should get back to studying. As you've mentioned, I don't learn mathematical reasoning, and I haven't been exposed that much to it. But I know there's something much more beneath and beyond everything I'm taught in school. So, I'm sad that I really don't have an opinion on Mathematics. However, I do know that Plexa is doing a PhD in Mathematics, pluripotential theory, if my memory serves me correctly. I should get back to studying too Edit: If there's an easy way to introduce myself to mathematical reasoning, do tell.
I was raised on brain teasers and logic puzzles, and so my math classes usually seemed enjoyable, as I perceived the information through the lens of riddles and problem solving, rather than the regurgitation of theorems and facts. My math teachers weren't absolutely terrible either, which tends to be a huge determining factor in whether or not students get turned away from a specific subject. When I was a student teacher, one of the courses I taught was a high school geometry class, and I was forced to teach the topic that most students hate with a passion: proofs. Granted, there are some really interesting proofs out there, but the logic and reasoning portion of geometry does nothing to further the passion of mathematics. Students find out the hard way (read as: the wrong way) that mathematics prides itself on proof and logical consistency, but the proofs they have to perform are proving that triangles are congruent (SSS, SAS, ASA, blah blah blah) and then eventually again in trigonometry (using identities to prove that two trigonometric quantities are equal). These sections of the curricula seem almost purposely designed to turn people away from mathematics (or at least make it seem boring), when in reality we can be focusing on how important logic and reasoning and rational, consistent steps and the need for defending our ideas are.
Thinking mathematically often comes with experience. Students aren't accustomed to proving things (at least, past their usual extent of needing to defend their answers and claims, which doesn't happen nearly as much as it should), and they become uncomfortable with the fact that you can't just follow a simple mindless procedure (e.g., the order of operations) and have every math question solved for you. I was able to create some logic puzzles and brain teasers for my geometry students to give them in between the boring-but-necessary-for-the-test geometry proofs, and most students found the overall instruction to be entertaining and successful (I constantly asked for their feedback and input). So I salvaged some of geometry's reputation in my class, but only by introducing my own ideas and being flexible with my own pedagogy... and I feel that that's something that teachers in general should do more of, but are frequently unable to do because they're bound by time constraints and curricula.
As far as thinking mathematically is concerned, I find it to be very hard to do so if you're accustomed to thinking in other ways. There tends to be one right answer in mathematics, but you can take several paths to get there, and not all of them force you to think of things in terms of black and white, or boring, or "follow the procedure and you'll do it right".
+ Show Spoiler +In college, my bachelor's degree was in mathematics, and then I got my master's in math education. Currently, I'm doing a PhD in math education too
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On March 24 2013 01:25 DarkPlasmaBall wrote:Show nested quote +On March 23 2013 20:32 Azera wrote:On March 23 2013 19:59 Recognizable wrote:Because the study of science is already pre-defined, and art is not, it's one of the reasons why art fascinates me much more than science sometimes. Seeing as how art is made by humans whom are bounded by the same physical laws you could in theory form a theoretical model which explains why we humans enjoy art and what art will be succesful and whatnot. I'm just messing around :p What is your opinion about Math? Seeing as how Math is the pinnacle of reasoning and creativity unbounded by physical laws. Well, you probably wouldn't know because they don't teach mathematical reasoning in high schools. Which is incredibly sad because it's what Mathematics is all about. I hope I can get good at it someday, because it's incredibly hard. I should get back to studying. As you've mentioned, I don't learn mathematical reasoning, and I haven't been exposed that much to it. But I know there's something much more beneath and beyond everything I'm taught in school. So, I'm sad that I really don't have an opinion on Mathematics. However, I do know that Plexa is doing a PhD in Mathematics, pluripotential theory, if my memory serves me correctly. I should get back to studying too Edit: If there's an easy way to introduce myself to mathematical reasoning, do tell. I was raised on brain teasers and logic puzzles, and so my math classes usually seemed enjoyable, as I perceived the information through the lens of riddles and problem solving, rather than the regurgitation of theorems and facts. My math teachers weren't absolutely terrible either, which tends to be a huge determining factor in whether or not students get turned away from a specific subject. When I was a student teacher, one of the courses I taught was a high school geometry class, and I was forced to teach the topic that most students hate with a passion: proofs. Granted, there are some really interesting proofs out there, but the logic and reasoning portion of geometry does nothing to further the passion of mathematics. Students find out the hard way (read as: the wrong way) that mathematics prides itself on proof and logical consistency, but the proofs they have to perform are proving that triangles are congruent (SSS, SAS, ASA, blah blah blah) and then eventually again in trigonometry (using identities to prove that two trigonometric quantities are equal). These sections of the curricula seem almost purposely designed to turn people away from mathematics (or at least make it seem boring), when in reality we can be focusing on how important logic and reasoning and rational, consistent steps and the need for defending our ideas are. Thinking mathematically often comes with experience. Students aren't accustomed to proving things (at least, past their usual extent of needing to defend their answers and claims, which doesn't happen nearly as much as it should), and they become uncomfortable with the fact that you can't just follow a simple mindless procedure (e.g., the order of operations) and have every math question solved for you. I was able to create some logic puzzles and brain teasers for my geometry students to give them in between the boring-but-necessary-for-the-test geometry proofs, and most students found the overall instruction to be entertaining and successful (I constantly asked for their feedback and input). So I salvaged some of geometry's reputation in my class, but only by introducing my own ideas and being flexible with my own pedagogy... and I feel that that's something that teachers in general should do more of, but are frequently unable to do because they're bound by time constraints and curricula. As far as thinking mathematically is concerned, I find it to be very hard to do so if you're accustomed to thinking in other ways. There tends to be one right answer in mathematics, but you can take several paths to get there, and not all of them force you to think of things in terms of black and white, or boring, or "follow the procedure and you'll do it right". + Show Spoiler +In college, my bachelor's degree was in mathematics, and then I got my master's in math education. Currently, I'm doing a PhD in math education too
I've been wondering. What is different about the reasoning you learn in like Chemistry and Physics in High School and the reasoning you do in Mathematical proofs? I know they are different because I suck at the latter, but I can´t put my finger on it. I know almost no one in my class is comfortable with proofs, probably because we don't practice this nearly enough. Anyway, just look at this question: There is an object which sends out a wave under water with (whatever) speed which pulsates for 100ns, the wave hits a wall 12cm further away and bounces back. When the end of the wave hit's the pulsating object it starts to pulse again. Show that the frequency of the pulsating object is hearable. Why is this so different than: Here are a couple of triangles and circles, proof this angle is 3 times the other angle. Both exercices you reason based on information you have gotten. I have a great Math teacher by the way. He is just so happy to teach Math. I also had a great Physics teacher. He would always ask: Do you guys understand it? We would nod. "Yes, yes..... you always nod"
Yelling: "BUT DO YOU REALLY UNDERSTAND IT? Kudos for becoming a teacher
We can take a few pages out of other countries' books when it comes to educational policy. We can't copy everything that Finland or China or Singapore does, but we can work towards a culture that promotes the respect of educators and education
This is very important imo. I'm in class full of bright kids. None of them want to work in such an underappreciated profession.
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On March 24 2013 05:31 Recognizable wrote:Show nested quote +On March 24 2013 01:25 DarkPlasmaBall wrote:On March 23 2013 20:32 Azera wrote:On March 23 2013 19:59 Recognizable wrote:Because the study of science is already pre-defined, and art is not, it's one of the reasons why art fascinates me much more than science sometimes. Seeing as how art is made by humans whom are bounded by the same physical laws you could in theory form a theoretical model which explains why we humans enjoy art and what art will be succesful and whatnot. I'm just messing around :p What is your opinion about Math? Seeing as how Math is the pinnacle of reasoning and creativity unbounded by physical laws. Well, you probably wouldn't know because they don't teach mathematical reasoning in high schools. Which is incredibly sad because it's what Mathematics is all about. I hope I can get good at it someday, because it's incredibly hard. I should get back to studying. As you've mentioned, I don't learn mathematical reasoning, and I haven't been exposed that much to it. But I know there's something much more beneath and beyond everything I'm taught in school. So, I'm sad that I really don't have an opinion on Mathematics. However, I do know that Plexa is doing a PhD in Mathematics, pluripotential theory, if my memory serves me correctly. I should get back to studying too Edit: If there's an easy way to introduce myself to mathematical reasoning, do tell. I was raised on brain teasers and logic puzzles, and so my math classes usually seemed enjoyable, as I perceived the information through the lens of riddles and problem solving, rather than the regurgitation of theorems and facts. My math teachers weren't absolutely terrible either, which tends to be a huge determining factor in whether or not students get turned away from a specific subject. When I was a student teacher, one of the courses I taught was a high school geometry class, and I was forced to teach the topic that most students hate with a passion: proofs. Granted, there are some really interesting proofs out there, but the logic and reasoning portion of geometry does nothing to further the passion of mathematics. Students find out the hard way (read as: the wrong way) that mathematics prides itself on proof and logical consistency, but the proofs they have to perform are proving that triangles are congruent (SSS, SAS, ASA, blah blah blah) and then eventually again in trigonometry (using identities to prove that two trigonometric quantities are equal). These sections of the curricula seem almost purposely designed to turn people away from mathematics (or at least make it seem boring), when in reality we can be focusing on how important logic and reasoning and rational, consistent steps and the need for defending our ideas are. Thinking mathematically often comes with experience. Students aren't accustomed to proving things (at least, past their usual extent of needing to defend their answers and claims, which doesn't happen nearly as much as it should), and they become uncomfortable with the fact that you can't just follow a simple mindless procedure (e.g., the order of operations) and have every math question solved for you. I was able to create some logic puzzles and brain teasers for my geometry students to give them in between the boring-but-necessary-for-the-test geometry proofs, and most students found the overall instruction to be entertaining and successful (I constantly asked for their feedback and input). So I salvaged some of geometry's reputation in my class, but only by introducing my own ideas and being flexible with my own pedagogy... and I feel that that's something that teachers in general should do more of, but are frequently unable to do because they're bound by time constraints and curricula. As far as thinking mathematically is concerned, I find it to be very hard to do so if you're accustomed to thinking in other ways. There tends to be one right answer in mathematics, but you can take several paths to get there, and not all of them force you to think of things in terms of black and white, or boring, or "follow the procedure and you'll do it right". + Show Spoiler +In college, my bachelor's degree was in mathematics, and then I got my master's in math education. Currently, I'm doing a PhD in math education too I've been wondering. What is different about the reasoning you learn in like Chemistry and Physics in High School and the reasoning you do in Mathematical proofs? I know they are different because I suck at the latter, but I can´t put my finger on it. I know almost no one in my class is comfortable with proofs, probably because we don't practice this nearly enough. Anyway, just look at this question: There is an object which sends out a wave under water with (whatever) speed which pulsates for 100ns, the wave hits a wall 12cm further away and bounces back. When the end of the wave hit's the pulsating object it starts to pulse again. Show that the frequency of the pulsating object is hearable. Why is this so different than: Here are a couple of triangles and circles, proof this angle is 3 times the other angle. Both exercices you reason based on information you have gotten.
Well it's definitely true that you can use your previous experiences with similar problems and relevant formulas to make headway with either math or science questions. However, mathematics (especially regarding proofs) tends to be far more abstract, and so sometimes it's harder to recognize what steps are needed to even complete the proof. Often times, you merely know possible theorems or strategies that may help because you've built an association between similar problems and those ideas in the past, but they may not be completely useful in every situation. Sometimes you need to just make a decision and run with it, just to get some ideas fleshed out, and then when you hit a dead end, backtrack a bit or start over. Sometimes there's trial and error involved in math proofs, especially when you're unfamiliar with the steps of any similar problems.
Do you have any general tips concering proof based Mathematics?
With mathematical proofs in particular, it helps to know what kinds of strategies you can use to prove something (e.g., proof by induction, indirect/ proof by contradiction, proof by cases, etc.). The more of these you're comfortable with, the more ways you can approach a proof. That's not to say that every method will work for every proof, but it gives you more options to solve something.
After that, you reflect on what kinds of identities or theorems might be useful. If you're proving triangle congruences, you need to know all the different ways (arrangements of sides and angles) that you can prove such a thing, and then think about how you could prove that specific sides or angles are congruent to complete the proof (e.g., reflexive property if the triangles share a side).
With trigonometric proofs, reciprocal and Pythagorean identities tend to be useful to manipulate certain functions (especially if they're squared terms). And as you do more of these, you'll become quicker at recognizing which changes you'll need to make to get closer to your end result. If you're proving that one group of trigonometric functions is mathematically equal to another set, maybe messing with both sides at the same time can help you uncover the missing parts. Moving forwards a few steps with one side of the equation, and moving "backwards" with the other side may make it more clear as to how to fill in the missing pieces.
Any proof (especially algebraic) where the claim is that X or Y or Z can occur given an assumption implies that you should be looking at how you can break your domain (set of integers, all real numbers, etc.) into that many groups (what's special about the numbers that make X true, as opposed to Y or Z?). Often times, using actual numbers or representations will provide some clarity on this issue (for three possibilities and a domain of integers, does that imply that the groups are "negatives, zero, and positives" or maybe "numbers that give either a remainder of 0 or 1 or 2 when dividing by 3" (relating to modular arithmetic) or some other grouping? Plug in some arbitrary numbers and make sure you understand what the proof is explaining, and why there needs to be more than one kind of occurrence.
Here's one of my favorite mathematical statements: Any perfect square (0, 1, 4, 9, 16, etc.) can be written as either 4k or 8k+1, where k is some integer. For example, 16 = 4(4), 100 = 4(25), 1 = 8(0)+1, and 81 = 8(10)+1. Why is this the case (i.e., prove it)?
+ Show Spoiler +As I previously stated, you need to figure out why there are two different possible results (either 4k or 8k+1). What are the groups? + Show Spoiler +4k works for all even perfect squares, and 8k+1 works for all odd perfect squares. That's nice and all, but you need to be able to write out what it means to be even or odd in algebraic terms before you can actually make headway with this concept. + Show Spoiler +Even means a number can be written as twice another number, or all evens = 2m (arbitrary variable) where m is an integer.
Odd means a number can be written as twice another number and then plus one, or all odds = 2m+1 (arbitrary variable) where m is an integer.
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A perfect square is the product of the same integer with the result being an integer as well. You have odd and even integers. Which can be written as 2m and 2m+1. If the square is a product of an even integer you get 2*2k=4k. If the square is a product of an uneven integer you would assume you get 2*2k+1. But this doesn't work because you don't get a perfect square(5). So it's 8K+1. haha.
I have no idea... 
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On March 24 2013 06:53 Recognizable wrote:A perfect square is the product of the same integer with the result being an integer as well. You have odd and even integers. Which can be written as 2m and 2m+1. If the square is a product of an even integer you get 2*2k=4k. If the square is a product of an uneven integer you would assume you get 2*2k+1. But this doesn't work because you don't get a perfect square(5). So it's 8K+1. haha. I have no idea...
I'm not sure if... + Show Spoiler +you've skipped the steps proving this, but it appears (based on your generalization to odds) that you might be messing up the few steps that go from "all evens can be represented as 2m, where m is an integer", to "all even squares can be written as 4k, where k is an integer". I only think this because you didn't show the fact that m^2 can be written as k:
-Even = 2m
--An even squared = (2m)^2 = (2m)(2m) = 4m^2
---m is defined as an integer, and an integer squared is also an integer. We can call m^2 "k", and therefore use substitution to call squared evens 4k instead of 4m^2, thus proving the first case.
The substitution of variables and pulling out the coefficient of 4 are what's key here, and it's what you'll need to do for the odds squared equaling 8k+1 (although the odd case is significantly harder than the even case). Regardless, you still start by multiplying out (2k+1)^2 to show what an odd number squared is.
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On March 24 2013 07:14 DarkPlasmaBall wrote:Show nested quote +On March 24 2013 06:53 Recognizable wrote:A perfect square is the product of the same integer with the result being an integer as well. You have odd and even integers. Which can be written as 2m and 2m+1. If the square is a product of an even integer you get 2*2k=4k. If the square is a product of an uneven integer you would assume you get 2*2k+1. But this doesn't work because you don't get a perfect square(5). So it's 8K+1. haha. I have no idea... I'm not sure if... + Show Spoiler +you've skipped the steps proving this, but it appears (based on your generalization to odds) that you might be messing up the few steps that go from "all evens can be represented as 2m, where m is an integer", to "all even squares can be written as 4k, where k is an integer". I only think this because you didn't show the fact that m^2 can be written as k:
-Even = 2m
--An even squared = (2m)^2 = (2m)(2m) = 4m^2
---m is defined as an integer, and an integer squared is also an integer. We can call m^2 "k", and therefore use substitution to call squared evens 4k instead of 4m^2, thus proving the first case.
The substitution of variables and pulling out the coefficient of 4 are what's key here, and it's what you'll need to do for the odds squared equaling 8k+1 (although the odd case is significantly harder than the even case). Regardless, you still start by multiplying out (2k+1)^2 to show what an odd number squared is.
Huh, oh. That was actually what I wanted to do in the beginning but got m^2 and immediately dismissed it :/ I'll try the odd one.
Edit:
(2k+1)^2=4k^2+4k+1=k(4k+4)+1
-->To make the equations equal k must be larger than 0. The next possible integer is 1. So we can substitute k inside the bracket with 1 keeping the equations equal which gives-->8k+1.
It's probably wrong :/
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On March 24 2013 07:18 Recognizable wrote:Show nested quote +On March 24 2013 07:14 DarkPlasmaBall wrote:On March 24 2013 06:53 Recognizable wrote:A perfect square is the product of the same integer with the result being an integer as well. You have odd and even integers. Which can be written as 2m and 2m+1. If the square is a product of an even integer you get 2*2k=4k. If the square is a product of an uneven integer you would assume you get 2*2k+1. But this doesn't work because you don't get a perfect square(5). So it's 8K+1. haha. I have no idea... I'm not sure if... + Show Spoiler +you've skipped the steps proving this, but it appears (based on your generalization to odds) that you might be messing up the few steps that go from "all evens can be represented as 2m, where m is an integer", to "all even squares can be written as 4k, where k is an integer". I only think this because you didn't show the fact that m^2 can be written as k:
-Even = 2m
--An even squared = (2m)^2 = (2m)(2m) = 4m^2
---m is defined as an integer, and an integer squared is also an integer. We can call m^2 "k", and therefore use substitution to call squared evens 4k instead of 4m^2, thus proving the first case.
The substitution of variables and pulling out the coefficient of 4 are what's key here, and it's what you'll need to do for the odds squared equaling 8k+1 (although the odd case is significantly harder than the even case). Regardless, you still start by multiplying out (2k+1)^2 to show what an odd number squared is. Huh, oh. That was actually what I wanted to do in the beginning but got m^2 and immediately dismissed it :/ I'll try the odd one. Edit: (2k+1)^2=4k^2+4k+1=k(4k+4)+1 -->To make the equations equal k must be larger than 0. The next possible integer is 1. So we can substitute k inside the bracket with 1 keeping the equations equal which gives-->8k+1. It's probably wrong :/
You have the right idea by just focusing on grouping the first two terms, as leaving the +1 alone will allow it to cancel with the +1 in the statement 8k+1. Showing that k(4k+4) = 8k might not be ideal though, because that doesn't allow you to directly cancel out anything else (although you may stumble upon a way to eventually get it into a form that is more manageable). Instead, I'm going to pull out the common 4, rather than k, and stick to using m instead of k for now because I'll still need to do substitution and manipulation later on:
-Odd = 2m+1
--An odd squared = (2m+1)(2m+1) = 4m^2 + 4m +1, which must equal 8k+1 according to the statement.
---4m^2 + 4m = 8k
----4(m^2 +m) = 8k
-----m^2 +m = 2k ...but why/ how?
+ Show Spoiler +What that means is that for all integers m, m^2 +m must be even (since it can be written as 2k). That being said, m can be odd or even, as you can take either of those types of numbers, multiply by 2, add 1, and then come up with an odd. For example, 13 = 2(6)+1 and 15 = 2(7)+1... 6 and 7 are the m terms, which means that you need to prove that m^2 +m must be even regardless of whether m is odd or m is even. So we do another proof by cases inside of this larger proof by cases. (A side proof of this nature, often used to achieve a larger goal or proof, is called a lemma): 1. If m is even, then m^2 is even. m^2 +m is an even plus an even, which is even. + Show Spoiler +(2m)*(2m) is 4m^2, which can be written as 2(2m^2) or 2k, making an even squared always even. 2. If m is odd, then m^2 is odd. m^2 +m is an odd plus an odd, which is even. + Show Spoiler +(2m+1)*(2m+1) is 4m^2 + 4m +1, which can be- again- grouped as 2k+1, where k= 2m^2 +2m; therefore, an odd squared is always odd. >Therefore, regardless of the parity of m (i.e., whether m is even or odd), m^2 +m will be even. >>Therefore, we can pull out that final multiple of 2 to get 8k instead of 4k, finishing off with an odd squared being equal to 8k+1, where k is some integer. QED.
Now granted, after seeing it written out, you can probably follow each step. And you may very well think to yourself "Okay I'm (eventually) convinced, but there's no way I'd ever be able to do this problem on my own, all the way through."
To which I say: I know, I don't blame you, and it takes a lot of experience. This was one of the first problems I proved in my abstract algebra class (junior year of undergraduate), and as we did more of these, we started to see some helpful patterns and wordings that suggested to us to go down the proof by cases route, and then specifically look at odds vs. evens, and then we learned how to manipulate the algebra accordingly. Often times, proofs don't even start out easy; they take a lot of getting used to. I was once where you were, and determination and grinding through these proofs (especially looking at finished proofs to compare) go a long way.
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Just a simple outline of a proof I would make for your question and some of my thought processes for getting there less-mathematically inclined people:
First thing to notice is that even perfect squares are what is divisible by by 4 and odd ones are 8k+1. Took me a few seconds to come up with that. Tried a few shits before that like trying Euler's Theorem or if it was possible to do it by induction.
So we break up the problem into two cases. Notice that when the perfect squares are even, their square root is also even, and when they are odd, square root is odd.
For even ones, perfect square can be broken down into its prime factorization for example:
2^2 = 2^2
4^2=2^4
6^2=2^2*3^2
Now notice that all even perfect numbers can be written down as multiples of 2^2 or 4. This is the easy part. How you would write that out is something like if x is even, then it's divisible by 2:
x = 2*m where m is some integer
x^2 = 2^2*m^2 = 4n where n = m^2 (by closeness of multiplication n is also an integer)
For odd ones:
3^2 = 3^2
5^2 = 5^2
But that doesn't help you, go back to the same idea as before
x = 2m +1 (as x is odd in this case) where m is some integer
x^2 = (2m+1)^2
= 4m^2 +4m +1
Now we try and figure out how to write 4m^2 + 4m in terms of 8n. This had me stuck for a little while, but you can probably prove this mechanically by plugging it into the proof by induction(but in general I hate mechanical proofs where i don't understand where the formula came from).
After thinking for a while, I came up with this alternative:
x^2 = 4(m^2+m) + 1
so you just need (m^2 + m) to be divisible by 2.
Then you break it down into further two cases once again, even and odd:
m is even:
m = 2y for some integer y
m^2 = 4y^2
m + m^2 = 2y + 4y^2 = 2(y+ 2y^2)
Then just replace that other thing again by some integer and voila you got some even number.
m is odd:
m = 2y+1
m^2 = 4y^2 +4y+1
m+m^2 = 4y^2 + 6y + 2
which is even
Combining it together
x = 4(m^2+m) +1
where m^2 + m can be written down as 2n
x = 4(2n) +1
and magic tada
This took longer than I thought to write out all my thought processes. Would also be a lot easier to write out using modulus operation.
But yeah it's pretty much just trial and error of a bunch of different methods and getting creative when you hit a roadblock.
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Nemesis,
Now notice that all even prime numbers can be written down as multiples of 2^2 or 4.
Do you mean even perfect squares? Surely you don't mean all even primes
For odd ones:
3^2 = 3^2
5^2 = 5^2
Can you elaborate on what you're trying to do with this part? What were you checking or testing out?
Overall though, your explanation and series of steps seem to be similar to the ones I previously wrote down in the earlier posts, but it's always nice to see other people's thought processes
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Aaah yes that was a typo, that was perfect squares.
Writing out my thought process was rather messy and disorganized lol.
I was just trying to see if prime factorization would help me find any pattern to it just like with how I found a pattern to the even ones using prime factorization. Just a trial and error failure attempt.
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On March 24 2013 09:14 DarkPlasmaBall wrote:Show nested quote +On March 24 2013 07:18 Recognizable wrote:On March 24 2013 07:14 DarkPlasmaBall wrote:On March 24 2013 06:53 Recognizable wrote:A perfect square is the product of the same integer with the result being an integer as well. You have odd and even integers. Which can be written as 2m and 2m+1. If the square is a product of an even integer you get 2*2k=4k. If the square is a product of an uneven integer you would assume you get 2*2k+1. But this doesn't work because you don't get a perfect square(5). So it's 8K+1. haha. I have no idea... I'm not sure if... + Show Spoiler +you've skipped the steps proving this, but it appears (based on your generalization to odds) that you might be messing up the few steps that go from "all evens can be represented as 2m, where m is an integer", to "all even squares can be written as 4k, where k is an integer". I only think this because you didn't show the fact that m^2 can be written as k:
-Even = 2m
--An even squared = (2m)^2 = (2m)(2m) = 4m^2
---m is defined as an integer, and an integer squared is also an integer. We can call m^2 "k", and therefore use substitution to call squared evens 4k instead of 4m^2, thus proving the first case.
The substitution of variables and pulling out the coefficient of 4 are what's key here, and it's what you'll need to do for the odds squared equaling 8k+1 (although the odd case is significantly harder than the even case). Regardless, you still start by multiplying out (2k+1)^2 to show what an odd number squared is. Huh, oh. That was actually what I wanted to do in the beginning but got m^2 and immediately dismissed it :/ I'll try the odd one. Edit: (2k+1)^2=4k^2+4k+1=k(4k+4)+1 -->To make the equations equal k must be larger than 0. The next possible integer is 1. So we can substitute k inside the bracket with 1 keeping the equations equal which gives-->8k+1. It's probably wrong :/ You have the right idea by just focusing on grouping the first two terms, as leaving the +1 alone will allow it to cancel with the +1 in the statement 8k+1. Showing that k(4k+4) = 8k might not be ideal though, because that doesn't allow you to directly cancel out anything else (although you may stumble upon a way to eventually get it into a form that is more manageable). Instead, I'm going to pull out the common 4, rather than k, and stick to using m instead of k for now because I'll still need to do substitution and manipulation later on: -Odd = 2m+1 --An odd squared = (2m+1)(2m+1) = 4m^2 + 4m +1, which must equal 8k+1 according to the statement. ---4m^2 + 4m = 8k ----4(m^2 +m) = 8k -----m^2 +m = 2k ...but why/ how? + Show Spoiler +What that means is that for all integers m, m^2 +m must be even (since it can be written as 2k). That being said, m can be odd or even, as you can take either of those types of numbers, multiply by 2, add 1, and then come up with an odd. For example, 13 = 2(6)+1 and 15 = 2(7)+1... 6 and 7 are the m terms, which means that you need to prove that m^2 +m must be even regardless of whether m is odd or m is even. So we do another proof by cases inside of this larger proof by cases. (A side proof of this nature, often used to achieve a larger goal or proof, is called a lemma): 1. If m is even, then m^2 is even. m^2 +m is an even plus an even, which is even. + Show Spoiler +(2m)*(2m) is 4m^2, which can be written as 2(2m^2) or 2k, making an even squared always even. 2. If m is odd, then m^2 is odd. m^2 +m is an odd plus an odd, which is even. + Show Spoiler +(2m+1)*(2m+1) is 4m^2 + 4m +1, which can be- again- grouped as 2k+1, where k= 2m^2 +2m; therefore, an odd squared is always odd. >Therefore, regardless of the parity of m (i.e., whether m is even or odd), m^2 +m will be even. >>Therefore, we can pull out that final multiple of 2 to get 8k instead of 4k, finishing off with an odd squared being equal to 8k+1, where k is some integer. QED. Now granted, after seeing it written out, you can probably follow each step. And you may very well think to yourself "Okay I'm (eventually) convinced, but there's no way I'd ever be able to do this problem on my own, all the way through." To which I say: I know, I don't blame you, and it takes a lot of experience. This was one of the first problems I proved in my abstract algebra class (junior year of undergraduate), and as we did more of these, we started to see some helpful patterns and wordings that suggested to us to go down the proof by cases route, and then specifically look at odds vs. evens, and then we learned how to manipulate the algebra accordingly. Often times, proofs don't even start out easy; they take a lot of getting used to. I was once where you were, and determination and grinding through these proofs (especially looking at finished proofs to compare) go a long way.
This part I hate the most. It's always so obvious when you see the proof /sigh.
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On March 24 2013 17:55 Recognizable wrote:Show nested quote +On March 24 2013 09:14 DarkPlasmaBall wrote:On March 24 2013 07:18 Recognizable wrote:On March 24 2013 07:14 DarkPlasmaBall wrote:On March 24 2013 06:53 Recognizable wrote:A perfect square is the product of the same integer with the result being an integer as well. You have odd and even integers. Which can be written as 2m and 2m+1. If the square is a product of an even integer you get 2*2k=4k. If the square is a product of an uneven integer you would assume you get 2*2k+1. But this doesn't work because you don't get a perfect square(5). So it's 8K+1. haha. I have no idea... I'm not sure if... + Show Spoiler +you've skipped the steps proving this, but it appears (based on your generalization to odds) that you might be messing up the few steps that go from "all evens can be represented as 2m, where m is an integer", to "all even squares can be written as 4k, where k is an integer". I only think this because you didn't show the fact that m^2 can be written as k:
-Even = 2m
--An even squared = (2m)^2 = (2m)(2m) = 4m^2
---m is defined as an integer, and an integer squared is also an integer. We can call m^2 "k", and therefore use substitution to call squared evens 4k instead of 4m^2, thus proving the first case.
The substitution of variables and pulling out the coefficient of 4 are what's key here, and it's what you'll need to do for the odds squared equaling 8k+1 (although the odd case is significantly harder than the even case). Regardless, you still start by multiplying out (2k+1)^2 to show what an odd number squared is. Huh, oh. That was actually what I wanted to do in the beginning but got m^2 and immediately dismissed it :/ I'll try the odd one. Edit: (2k+1)^2=4k^2+4k+1=k(4k+4)+1 -->To make the equations equal k must be larger than 0. The next possible integer is 1. So we can substitute k inside the bracket with 1 keeping the equations equal which gives-->8k+1. It's probably wrong :/ You have the right idea by just focusing on grouping the first two terms, as leaving the +1 alone will allow it to cancel with the +1 in the statement 8k+1. Showing that k(4k+4) = 8k might not be ideal though, because that doesn't allow you to directly cancel out anything else (although you may stumble upon a way to eventually get it into a form that is more manageable). Instead, I'm going to pull out the common 4, rather than k, and stick to using m instead of k for now because I'll still need to do substitution and manipulation later on: -Odd = 2m+1 --An odd squared = (2m+1)(2m+1) = 4m^2 + 4m +1, which must equal 8k+1 according to the statement. ---4m^2 + 4m = 8k ----4(m^2 +m) = 8k -----m^2 +m = 2k ...but why/ how? + Show Spoiler +What that means is that for all integers m, m^2 +m must be even (since it can be written as 2k). That being said, m can be odd or even, as you can take either of those types of numbers, multiply by 2, add 1, and then come up with an odd. For example, 13 = 2(6)+1 and 15 = 2(7)+1... 6 and 7 are the m terms, which means that you need to prove that m^2 +m must be even regardless of whether m is odd or m is even. So we do another proof by cases inside of this larger proof by cases. (A side proof of this nature, often used to achieve a larger goal or proof, is called a lemma): 1. If m is even, then m^2 is even. m^2 +m is an even plus an even, which is even. + Show Spoiler +(2m)*(2m) is 4m^2, which can be written as 2(2m^2) or 2k, making an even squared always even. 2. If m is odd, then m^2 is odd. m^2 +m is an odd plus an odd, which is even. + Show Spoiler +(2m+1)*(2m+1) is 4m^2 + 4m +1, which can be- again- grouped as 2k+1, where k= 2m^2 +2m; therefore, an odd squared is always odd. >Therefore, regardless of the parity of m (i.e., whether m is even or odd), m^2 +m will be even. >>Therefore, we can pull out that final multiple of 2 to get 8k instead of 4k, finishing off with an odd squared being equal to 8k+1, where k is some integer. QED. Now granted, after seeing it written out, you can probably follow each step. And you may very well think to yourself "Okay I'm (eventually) convinced, but there's no way I'd ever be able to do this problem on my own, all the way through." To which I say: I know, I don't blame you, and it takes a lot of experience. This was one of the first problems I proved in my abstract algebra class (junior year of undergraduate), and as we did more of these, we started to see some helpful patterns and wordings that suggested to us to go down the proof by cases route, and then specifically look at odds vs. evens, and then we learned how to manipulate the algebra accordingly. Often times, proofs don't even start out easy; they take a lot of getting used to. I was once where you were, and determination and grinding through these proofs (especially looking at finished proofs to compare) go a long way. This part I hate the most. It's always so obvious when you see the proof /sigh.
If that's really the case, then it should give you inspiration that you can do similar proofs like these in the future
Even if you were incapable of generating the proof on your own, think about what kinds of strategies and procedures you can now add to your arsenal of proof methods in the future (whether the proof is extremely similar to the odd and even squares one, or quite different). The more proofs you look at, the bigger your arsenal can be for the future
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The two most important things I got from this is
1: 4m^2 + 4m +1, which must equal 8k+1 according to the statement.
2. What that means is that for all integers m, m^2 +m must be even
Basically, I tried to algebraically get from this 4m^2 + 4m to this 8k. But that's not necesarry/possible. You know those two are equal. You just have to proof why they are equal. And when are they equal? When they are both even(or uneven). Because you know 2k is even, you have to proof m^2 + m is even. The thing about proofs is that you really have to understand what is going on.
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On March 24 2013 21:02 Recognizable wrote:The two most important things I got from this is 1: 2. Basically, I tried to algebraically get from this 4m^2 + 4m to this 8k. But that's not necesarry/possible. You know those two are equal. You just have to proof why they are equal. And when are they equal? When they are both even(or uneven). Because you know 2k is even, you have to proof m^2 + m is even. The thing about proofs is that you really have to understand what is going on.
Agreed, and that's really the point; it's significantly harder than just throwing a problem into a skeletal equation and then filling in the blanks with specific numbers. The procedures are less fixed.
I also think you can get more out of this proof than just the fact that the statement is true. Look at the different strategies and ideas we've used, and know that in the future, you may very well be utilizing some of these again in your proofs:
-Now you've seen the form that a proof by cases takes, in case you want to use it
-Now you know that two groups of integers may imply odds and evens, and you know how to write them out algebraically
-Now you know about lemmas (sub-proofs)
-Now you've seen that grouping items can help further the proof, and that grouping isn't arbitrary and one factoring may help more than another
-Now you have some algebraic and abstract experience dealing with perfect squares and parity (odd vs. even) in general
And even if you've seen some of these before, the organization and form that they take on in this proof will be helpful to you in the future. As you complete (or even just read) more of these, you'll end up with fewer moments where you have no idea what to do next, or you end up rushing to a dead end because you're completely unfamiliar with the territory.
With regular science or math (non-proof) problems, this would be the point where you then are introduced to a few more similar questions and extensions, to cement your understanding of the procedures and strategies used (e.g., repeated math problems on order of operations, emphasizing the structure and tiers used in PEMDAS). However, proofs sometimes vary to the point where you can only use broad strokes and a few ideas from previous proofs to make headway with new ones... and I unfortunately don't have other similar proofs to give you off the top of my head.
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This is much more interesting than the Calculus stuff I'm learning right now ^.^ Altough some of the optimization you can do with Calculus is pretty cool, so far it's been most of the same over and over again :/
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DarkPlasmaBall,
Thanks for the replies!
I agree that teachers are bound by the set curriculum set by the ministry of education (or whatever board you guys have over there ), and they just end up rushing through the syllabus because of the time constraints. How do you propose to fix this? Change how school funding works?
I'm also not too sure about why you think educators in Singapore have a lot of respect, and that teaching is something prestigious - because they're not. That's just my views anyway. Singapore has it's own problems with it's education system, regarding the extremely rigid curriculum and the over-emphasis on grades. One thing however, I would say that all the schools are adequately funded. I guess the reason for this is that Singapore is a much smaller country that the US, and as such, is much easier to divide resources amongst the schools.
On March 24 2013 01:25 DarkPlasmaBall wrote:Show nested quote +On March 23 2013 20:32 Azera wrote:On March 23 2013 19:59 Recognizable wrote:Because the study of science is already pre-defined, and art is not, it's one of the reasons why art fascinates me much more than science sometimes. Seeing as how art is made by humans whom are bounded by the same physical laws you could in theory form a theoretical model which explains why we humans enjoy art and what art will be succesful and whatnot. I'm just messing around :p What is your opinion about Math? Seeing as how Math is the pinnacle of reasoning and creativity unbounded by physical laws. Well, you probably wouldn't know because they don't teach mathematical reasoning in high schools. Which is incredibly sad because it's what Mathematics is all about. I hope I can get good at it someday, because it's incredibly hard. I should get back to studying. As you've mentioned, I don't learn mathematical reasoning, and I haven't been exposed that much to it. But I know there's something much more beneath and beyond everything I'm taught in school. So, I'm sad that I really don't have an opinion on Mathematics. However, I do know that Plexa is doing a PhD in Mathematics, pluripotential theory, if my memory serves me correctly. I should get back to studying too Edit: If there's an easy way to introduce myself to mathematical reasoning, do tell. I was raised on brain teasers and logic puzzles, and so my math classes usually seemed enjoyable, as I perceived the information through the lens of riddles and problem solving, rather than the regurgitation of theorems and facts. My math teachers weren't absolutely terrible either, which tends to be a huge determining factor in whether or not students get turned away from a specific subject. When I was a student teacher, one of the courses I taught was a high school geometry class, and I was forced to teach the topic that most students hate with a passion: proofs. Granted, there are some really interesting proofs out there, but the logic and reasoning portion of geometry does nothing to further the passion of mathematics. Students find out the hard way (read as: the wrong way) that mathematics prides itself on proof and logical consistency, but the proofs they have to perform are proving that triangles are congruent (SSS, SAS, ASA, blah blah blah) and then eventually again in trigonometry (using identities to prove that two trigonometric quantities are equal). These sections of the curricula seem almost purposely designed to turn people away from mathematics (or at least make it seem boring), when in reality we can be focusing on how important logic and reasoning and rational, consistent steps and the need for defending our ideas are. Thinking mathematically often comes with experience. Students aren't accustomed to proving things (at least, past their usual extent of needing to defend their answers and claims, which doesn't happen nearly as much as it should), and they become uncomfortable with the fact that you can't just follow a simple mindless procedure (e.g., the order of operations) and have every math question solved for you. I was able to create some logic puzzles and brain teasers for my geometry students to give them in between the boring-but-necessary-for-the-test geometry proofs, and most students found the overall instruction to be entertaining and successful (I constantly asked for their feedback and input). So I salvaged some of geometry's reputation in my class, but only by introducing my own ideas and being flexible with my own pedagogy... and I feel that that's something that teachers in general should do more of, but are frequently unable to do because they're bound by time constraints and curricula. As far as thinking mathematically is concerned, I find it to be very hard to do so if you're accustomed to thinking in other ways. There tends to be one right answer in mathematics, but you can take several paths to get there, and not all of them force you to think of things in terms of black and white, or boring, or "follow the procedure and you'll do it right". + Show Spoiler +In college, my bachelor's degree was in mathematics, and then I got my master's in math education. Currently, I'm doing a PhD in math education too
I can't agree more with the bolded part!
Personally, trigonometry and trigonometric proofs are the bane of my existence. I can never properly visualise the triangles and angles. Do you have any suggestions and advice on how I should practice my proofs (at a high school level) and skills in trigonometry?
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My pleasure, Azera
I agree that teachers are bound by the set curriculum set by the ministry of education (or whatever board you guys have over there ), and they just end up rushing through the syllabus because of the time constraints. How do you propose to fix this? Change how school funding works?
Educators can't really change the funding system, and the way America gives money to schools (trickling down from the federal government to states to districts, etc.) is conditioned upon results (e.g., standardized test scores). This harms education, as teachers and administrators need to focus more on improving those specific scores, hence teachers "teaching to the test" and education being too focused on grades rather than knowledge. Many teachers even prefer to work at private schools instead of public schools, exchanging their smaller salaries for more freedom from national standards and restrictive curricula (the zero tolerance policies aren't so bad either!). More successful countries focus far less on standardized testing as a metric for how much money schools deserve, and instead trust teachers more and continue to provide adequate funding.
I'm also not too sure about why you think educators in Singapore have a lot of respect, and that teaching is something prestigious - because they're not.
I mentioned Singapore along with other countries because they're notorious for having incredibly good science and math results compared to the rest of the world. There's an international standardized test for mathematics and science that compares students (and therefore countries) from all over the world: It's called Trends in International Mathematics and Science Study (TIMSS). The TIMSS shows how poorly America is doing in comparison to many other countries (who you'd think we'd beat or at least measure up to), which has helped provide the recent push for more American students going into the STEM (science, technology, engineering, mathematics) fields. You can check out the TIMSS website here: http://nces.ed.gov/Timss/ We can look at the structure and atmosphere of countries who are performing better than us, and possibly modify our approaches to education accordingly.
Personally, trigonometry and trigonometric proofs are the bane of my existence. I can never properly visualise the triangles and angles. Do you have any suggestions and advice on how I should practice my proofs (at a high school level) and skills in trigonometry?
As with any proof topic in general, it's about becoming familiar with the relevant (or possibly relevant) concepts and formulas so that you can implement them whenever possible, and then just grinding through a ton of problems so that you start to realize situations where certain identities or steps are more likely to occur. Get comfortable with manipulating and transforming trigonometric functions (tan = sin/cos, sin^2 + cos^2 = 1, etc.) to push towards the final answer (or a benchmark that might give you part of the final answer). You're going to hit dead ends along the way, and you'll need to backtrack or restart the proof. Don't just view that as failure; instead, remember that it didn't work for the next time you end up in the same situation (and also remember what does eventually work!). Proofs just really take a lot of practice, so that you learn the patterns.
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Once again, thanks for the answers!
It seems like in order to really get education down correctly, everything has to be changed - from ground up. How schools are funded, the expectations of educating students, the 'correct' way to prepare students for society, etc. All these are very interesting stuff and too deep for me to talk about before reading up on it.
I've enjoyed this conversation, and I've learned a lot. Thanks. Thank you for the advice too. I'll get cracking on the proofs!
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On March 18 2013 00:09 radscorpion9 wrote:
Also it probably matters what you're studying. I don't know if anyone can say that every subject was interesting to them, so it probably makes more sense for you to say that you only have a passion to learn certain subjects (unless you really like *all* of them?). But maybe there are some people out there who genuinely enjoy learning about everything; I haven't seen many people like that. People usually like specializing in one area, and would be thoroughly depressed working as a shoe salesman (for example).
That's why you get the "puerile nonsense" like how am I going to use trigonometry in my daily life. But I don't think its childish or silly at all. I think you would probably do well to recognize that not everyone is like you, and not everyone has a passionate interest in learning math or the infinite connections associated with it. There are a lot of aspects of mathematics that they probably won't use for the rest of their life, so they shouldn't be forced to learn something that holds no innate interest to them. Of course there may be other arguments for learning the math that's taught at that level.
Incidentally, math gets 100x more interesting in university if you decide to specialize in math (I'm taking one such course now, the textbook is excellent, by Michael Spivak), but you better be a semi-genius or you'll have trouble!
I enjoyed every class I took in HS and Uni. I hated learning for them but I loved listening to the teachers, trying to navigate my way to understanding.
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On March 27 2013 20:17 Azera wrote:
Once again, thanks for the answers!
It seems like in order to really get education down correctly, everything has to be changed - from ground up. How schools are funded, the expectations of educating students, the 'correct' way to prepare students for society, etc. All these are very interesting stuff and too deep for me to talk about before reading up on it.
I've enjoyed this conversation, and I've learned a lot. Thanks. Thank you for the advice too. I'll get cracking on the proofs!
Anytime
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EPT
