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On February 05 2012 01:53 Plexa wrote:He's actually incorrect in saying 45 is an element of a vector space because he didn't appreciate that speed is a function which gives values in a positive reals only (which aren't a field). Note that 45 is also an element of the integers, also not a field. The underlying space is important to appreciate in this context  see if he were talking about velocity, which outputs onto all of the reals, then it is a vector despite the fact the number "45" hasn't changed. OP is correct when he says vectors don't necessarily have magnitude and direction (hello topological vector spaces!!) but as far as high school physics is concerned, every vector will have a size and a direction and indeed this should be true for any higher level engineering (except maybe engineering science). In essence, saying a vector has size and direction is a meaningful way to distinguish them from a number, or a pair of numbers.
Another unintuitive but important consequences of vector spaces. The numeral 45 doesn't always represent the same thing. Whether it represents a number in the field of all real numbers, or an object in the positive real numbers is important. This, I think, is the fundamental problem with teaching all these things from the fundamental idea of "they're all elements of a vector space!" In the mind of the student you lose important distinctions between vector spaces.
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Aotearoa39261 Posts
On February 05 2012 02:10 paralleluniverse wrote:Show nested quote +On February 05 2012 01:59 Plexa wrote:On February 05 2012 01:55 paralleluniverse wrote:On February 05 2012 01:29 Plexa wrote:On February 05 2012 01:26 sukarestu wrote:I'm one week through year 11 just started my physics class, had 3 lessons (obviously pretty clueless) Hearing vectors can be a point.. "TRIPPING OUT MAN!!!".. functions "meh.. fine w/e" In fact, claiming the speed 45 m/s is a vector would lose you marks at school because a teacher would claim it has no direction, but 45 is an element of a the vector space of real numbers or the field of real numbers, so is also a vector.
Whut? ok.. let me think here, Vector = compound, 45 = atom 45 exists in Vector therefore is vector? So everything in the field of real numbers is considered a vector? In fact, claiming the speed 45 m/s is a vector would lose you marks at school because a teacher would claim it has no direction, but 45 is an element of a the vector space of real numbers or the field of real numbers, so is also a vector.
Speed has no direction btw. That is why the answer is wrong. If the velocity was 45 m/s then the answer is right. speed = length(velocity), speed is an element of R+ (all real numbers greater or equal to zero) - not a field. Define the vector space of all real numbers over the field of real numbers. The nonnegative numbers in this vector space can represent speed, and 45 is a element of this vector space, so is a vector by definition. No, you are wrong. You are embedding your speed in the reals, which in reality it's embedded in the positive reals. Counter example using your argument. Consider the vector space over the reals, the intergers {...,-1,0,1,...} are elements in this vector space so are a vector by definition. Clearly incorrect. No, I'm not confusing speed and velocity. Velocity is modeled in physics as an arrow, that is a vector in the vector space of points in R^2 over the field R. So the point (1,2) corresponds to the velocity "sqrt(5) m/s 1.11 radians from the x-axis". But the set R over the field R, is a vector space, so 45 is a vector, whether it is 45 apples, 45 m, 45 m/s, 45 degrees, whatever.
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speed: R->R+, t----> |v(t)|
R+ is not a vector space.
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Aotearoa39261 Posts
On February 05 2012 02:13 Anytus wrote:Show nested quote +On February 05 2012 01:53 Plexa wrote:He's actually incorrect in saying 45 is an element of a vector space because he didn't appreciate that speed is a function which gives values in a positive reals only (which aren't a field). Note that 45 is also an element of the integers, also not a field. The underlying space is important to appreciate in this context see if he were talking about velocity, which outputs onto all of the reals, then it is a vector despite the fact the number "45" hasn't changed. OP is correct when he says vectors don't necessarily have magnitude and direction (hello topological vector spaces!!) but as far as high school physics is concerned, every vector will have a size and a direction and indeed this should be true for any higher level engineering (except maybe engineering science). In essence, saying a vector has size and direction is a meaningful way to distinguish them from a number, or a pair of numbers. Another unintuitive but important consequences of vector spaces. The numeral 45 doesn't always represent the same thing. Whether it represents a number in the field of all real numbers, or an object in the positive real numbers is important. This, I think, is the fundamental problem with teaching all these things from the fundamental idea of "they're all elements of a vector space!" In the mind of the student you lose important distinctions between vector spaces.
I think that is something the OP doesn't grasp, personally.
@OP, what exactly are your qualifications?
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United States24805 Posts
On February 05 2012 02:12 paralleluniverse wrote:Show nested quote +On February 05 2012 02:10 perestain wrote:On February 05 2012 00:10 paralleluniverse wrote:
So how should vectors correctly be taught? Like this:
What is a vector? A vector is an element of a vector space.
A vector space V over a field F is a set with 2 operations, addition between elements of V, and multiplication of elements of V with elements of F, satisfying the following rules:
1. u + (v + w) = (u + v) + w
2. u + v = v + u
...
You will already have lost the attention of 95% of the students in a typical school situation at that time. While I understand your need for complete and correct definitions, you just cannot introduce the concept of vectors like this to someone who has not heard of it before and is not used to approach mathematics in a purely symbolic and formal way yet. From the point of view of a student, the definition can not make sense, because he has no grasp what a mathematical space is, let alone a vector space. If you develop a strategy how to introduce vectors in a way that enables people to understand a formal definition faster, then power to you, and please let us know. But as is, the average joe at school will have a better understanding of vectors after watching the video, since the definition you cite will simply not make any sense to him. There is no shortcut to enlightenment. Also it appears a bit elitist wanting to burn the ladders which you used yourself to climb upwards. A 5 year old would not only be able to make sense of those axioms, but be asking why we have bothered listing such obvious facts. And I never really gained any real understanding of vectors until a proper definition was taught to me, so I don't think the elitist complaint applies.
I don't know how you are attached to the way you presented those axioms... but the presentation alone is enough to turn off a student. I think most students would realize that 2 + 3 gives you the same answer as 3 + 2. However, if you show a student u + v = v + u you will turn a lot of them off right there.
Of course there are ways to present the same axioms that would be less of a turn off to the average student.
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On February 05 2012 00:41 Mordanis wrote:
The "new" way is confusing to me, and I'm majoring in Physics so I'm fairly familiar with vectors. There are a few things specifically that could use some clearing up.
1.) what are "u","v","a","b"? are they an arbitrary vector, an arbitrary scalar, or what?
2.) I don't really get what a vector space over a field means.
If you could clear this up, I'd be very grateful :D
1. I presume they are vectors. He should have bolded them to indicate they are vectors but he went away from standard notation.
2. You'll understand this concept after you take a Vector and Multivariable Calculus class. Actually, you will understand it after the first or second chapter. You will also learn this in a Linear Algebra class.
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So basically instead of teaching a vector as something with a magnitude and direction (which makes intuitive sense), you want to completely formalize it and just teach people abstract algebra from day 1.
Yeah... good luck with that.
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On February 05 2012 02:12 paralleluniverse wrote:Show nested quote +On February 05 2012 02:10 perestain wrote:On February 05 2012 00:10 paralleluniverse wrote:
So how should vectors correctly be taught? Like this:
What is a vector? A vector is an element of a vector space.
A vector space V over a field F is a set with 2 operations, addition between elements of V, and multiplication of elements of V with elements of F, satisfying the following rules:
1. u + (v + w) = (u + v) + w
2. u + v = v + u
...
You will already have lost the attention of 95% of the students in a typical school situation at that time. While I understand your need for complete and correct definitions, you just cannot introduce the concept of vectors like this to someone who has not heard of it before and is not used to approach mathematics in a purely symbolic and formal way yet. From the point of view of a student, the definition can not make sense, because he has no grasp what a mathematical space is, let alone a vector space. If you develop a strategy how to introduce vectors in a way that enables people to understand a formal definition faster, then power to you, and please let us know. But as is, the average joe at school will have a better understanding of vectors after watching the video, since the definition you cite will simply not make any sense to him. There is no shortcut to enlightenment. Also it appears a bit elitist wanting to burn the ladders which you used yourself to climb upwards. A 5 year old would not only be able to make sense of those axioms, but be asking why we have bothered listing such obvious facts.
I'm going to assume you're just using hyperbole here, and not being ridiculously illogical and biased. In your efforts to introduce a new concept, you brought in 3-4 more new concepts that are far more abstract and useless at the high-school level. Though you may be right about most of your ideas on this mathematical concept, I sincerely hope you never have an influence in the world of education.
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As an undergraduate Physics major receiving my degree in April, I have to admit I think the OP is silly, with apologies. Mathematically speaking everything you're preaching is completely accurate, and a much stronger definition. Yes it is like cheating to teach vectors the "old/example" way that you mention, but I 100% believe that if someone is already having difficulties grasping simple physics that trying to teach them your purely mathematical sense of vectors, especially throwing around terms like vector space and field, will not make the subject any more tangible. It's simply the nature of all math, and while it's fun to rant about from an "educated" perspective, in a satirical sense perhaps, it's completely unrealistic to teach such concepts when students are first introduced to them. Those who need to know the proper definitions and rules will learn after they've been indoctrinated and are used to the math work/terminology, but those who never need to know the details, only the practicality (such as engineers) shouldn't be expected to learn it because it's simply unnecessary complication. This of course isn't to say that they COULDN'T, or that it wouldn't HELP, but from the practical standpoint, in terms of curriculum and pacing, it's just obnoxious and unnecessary.
It's the same reason why you learn about orbitals in [high school / intro Uni] chemistry, but you aren't introduced to the wavefunction at this stage or even remotely the nature/math of why they are the way they are. It's the same as expecting an electrical engineer to have an extremely advanced knowledge of E&M, or better yet START with advanced E&M and then get to the "realistic, practical example" of electronics, pieces in a circuit. Sure it can't hurt to be learning/teaching it the "right way" first, but it's almost never a realistic goal.
Edit: past few posts said the same I have, just very succinctly. However they did use a word I love and should have included, FORMALISM. Yes learning the formalism of this kind of math is "the right way", OF COURSE IT IS NO ONE CAN ARGUE, but it's absolutely silly, from a practical sense and also in a curricular sense, to expect people to learn in this way as soon as they are first introduced to it.
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Every science and maths course taught at high school level are either "wrong" or "incomplete". Most of the time, these are done purposefully in order to gradually introduced the ideas to the students and not to overload them with too much information. So usually the students are just given some of the basic rules ( with simplified or no explanation) with some simple examples which they can understand.
Is this the right way to teach kids science? Maybe, maybe not. On one hand, the students will get a general idea of what the topics are about. On the other hand, what they learn are simplified or not exactly true. In many branch of science, knowing the ideas are usually more important, and the students can learn the "details" later when they are ready or if they really need it, usually at university. The system probably works fine for the average high school students.
For more dedicated and curious students, I will say they should not just rely on what they learn in the classroom, but seek out the "truth" for themselves. In an ideal world, you should be able to ask your teachers and they can give you the answer but in reality, they are probably not train to a high enough degree to teach you anything outside the high school syllabus (or even the high school syllabus itself).
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parallel, I really respect your intentions, but your goals are immature. You've been imprecise a few times and when you're dealing with increased abstraction that's a dangerous thing. Yeah, the function sin(x) is an element of the vector space of (continuous) functions over the reals. And yeah, that's a useful thing, especially for stating the existence theorem for solutions to ODEs. But historically a vector was an element of honest-to-God R^3, or a specific R^2 inside R^3, or maybe R^n if you were really abstract. The definition you gave of a vector space as an F-module came in the late 1800s, long after the intuition of Descartes and Newton (on which classical physics is based and with which it still works).
If you want to know why all of the less formal definitions are actually good ones, here's the mantra: the abstract vector space R^n acts effectively on affine n-space. Once you choose an origin, you really can identify points in R^n with elements of the abstract vector space. That's why your definition is a useful abstraction. But it's not a necessary one, and it occludes a lot of the nicest facts about R^3 -- that it has a norm, that it has an inner product, that it's connected (heck, contractible), that R is complete, etc. -- all of which are amenable to abstractions too.
And that's my point. As mathematicians we abstract not because it lets us be more formal with our arguments. Working mathematicians know rigor when we see it. If you showed me a symbolic proof that, say, the surface of a coffee cup is homeomorphic to a torus, it would be ugly and incomprehensible. We abstract because we find certain properties of certain objects useful, and we want to know what it is about those objects that makes them useful. Until you've spent years working with the reals, and R^3, by themselves, it's very difficult to see what's so great about them that we would want to pull out the abstract properties that make them work the way that they do.
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On February 05 2012 02:05 gyth wrote:Show nested quote +In fact, claiming the speed 45 m/s is a vector would lose you marks at school because a teacher would claim it has no direction, but 45 is an element of a the vector space of real numbers or the field of real numbers, so is also a vector. Physics teaches Euclidean Vectors, and since it is the only type of vector taught (at least in introductory courses) they are just referred to as vectors. Show nested quote +The rules listed in the definition of a vector space and a field would be obviously true and well-known to anyone who has studied beyond sixth grade mathematics. I'm gonna call BS on that one. http://www.australiancurriculum.edu.au/Mathematics/Curriculum/F-10Where is a mathematically rigorous definition of field or vector space gone over at a gradeschool level?
You didn't know in high school primary 3 + 4 = 4 + 3?
What about 2*(3*5) = (2*3)*5? In what grade did this dawn on you?
Did you know that if 3 and 4 are numbers then 3 + 4 is also a number?
For 5, there's a number -5 and 5 + -5 = ...?
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I am waiting for the followup thread where we learn how first grade arithmetic is taught all wrong and it would be so much easier for kids to understand it with a proper definition of ℤ and just a few simple concepts from set theory.
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United States24805 Posts
On February 05 2012 02:22 dementrio wrote:
I am waiting for the followup thread where we learn how first grade arithmetic is taught all wrong and it would be so much easier for kids to understand it with a proper definition of ℤ and just a few simple concepts from set theory.
Well my understanding is that second grade math is wrong because instead of having kids memorize the multiplication table (1-10) we have them use a calculator now!
There actually are many problems with how math is taught, and most of them are not what people online complain about.
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On February 05 2012 02:12 paralleluniverse wrote:Show nested quote +On February 05 2012 02:10 perestain wrote:On February 05 2012 00:10 paralleluniverse wrote:
So how should vectors correctly be taught? Like this:
What is a vector? A vector is an element of a vector space.
A vector space V over a field F is a set with 2 operations, addition between elements of V, and multiplication of elements of V with elements of F, satisfying the following rules:
1. u + (v + w) = (u + v) + w
2. u + v = v + u
...
You will already have lost the attention of 95% of the students in a typical school situation at that time. While I understand your need for complete and correct definitions, you just cannot introduce the concept of vectors like this to someone who has not heard of it before and is not used to approach mathematics in a purely symbolic and formal way yet. From the point of view of a student, the definition can not make sense, because he has no grasp what a mathematical space is, let alone a vector space. If you develop a strategy how to introduce vectors in a way that enables people to understand a formal definition faster, then power to you, and please let us know. But as is, the average joe at school will have a better understanding of vectors after watching the video, since the definition you cite will simply not make any sense to him. There is no shortcut to enlightenment. Also it appears a bit elitist wanting to burn the ladders which you used yourself to climb upwards. A 5 year old would not only be able to make sense of those axioms, but be asking why we have bothered listing such obvious facts.
A 5 year old wouldn't normally know what "+" "(" ")" "=" "vector space" "V" "field F" "2 operations" "elements of V", "multiplication" means. So you probably would have to start there ...
In other words, I think you are an idiot to stand by this claim
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On February 05 2012 02:22 paralleluniverse wrote:Show nested quote +On February 05 2012 02:05 gyth wrote:In fact, claiming the speed 45 m/s is a vector would lose you marks at school because a teacher would claim it has no direction, but 45 is an element of a the vector space of real numbers or the field of real numbers, so is also a vector. Physics teaches Euclidean Vectors, and since it is the only type of vector taught (at least in introductory courses) they are just referred to as vectors. The rules listed in the definition of a vector space and a field would be obviously true and well-known to anyone who has studied beyond sixth grade mathematics. I'm gonna call BS on that one. http://www.australiancurriculum.edu.au/Mathematics/Curriculum/F-10Where is a mathematically rigorous definition of field or vector space gone over at a gradeschool level? You didn't know in high school primary 3 + 4 = 4 + 3? What about 2*(3*5) = (2*3)*5? In what grade did this dawn on you? Did you know that if 3 and 4 are numbers then 3 + 4 is also a number? For 5, there's a number -5 and 5 + -5 = ...?
How can any fact about how vectors behave possibly be obvious before you know what a vector is?
Suppose a and b are members of a group and + is the group operation. Is it obvious that a+b=b+a? I can't imagine how this could be any less obvious than commutativity for vectors when you know neither what a vector nor an element of a group is.
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I'm going to para-phrase what my Real Analysis lecturer said at almost every lecture: the problem isn't with your mind not being able to comprehend what we're doing, the problem is with your mind throwing up a mental block saying "Oh my god, it's so complicated and hard, how could we possibly understand that?". I think it's much more important to address that issue if you wish to improve mathematical literacy (which is a noble pursuit), and your approach doesn't help in the slightest. The majority of my friends that didn't take maths past high school get irked at the first sight of a formula and throw up their hands saying "Maths, I hate maths. It's so complicated". I think your proposal takes things in the opposite direction.
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The problem is that the "proper" way you describe at the end is at a level of abstraction that will take most people years to come to terms with. And most people will only ever do the very basic math required of them by society.
As indicated by some posters; even for people who've done a lot of math this is fairly hard. Abstract thinking IS hard. Most people like when things are made intuitive. Most people prefer intuitive arguments/persuasions over rock solid mathematical theory.
But I do agree that there is a problem...
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On February 05 2012 02:14 Plexa wrote:Show nested quote +On February 05 2012 02:10 paralleluniverse wrote:On February 05 2012 01:59 Plexa wrote:On February 05 2012 01:55 paralleluniverse wrote:On February 05 2012 01:29 Plexa wrote:On February 05 2012 01:26 sukarestu wrote:I'm one week through year 11 just started my physics class, had 3 lessons (obviously pretty clueless) Hearing vectors can be a point.. "TRIPPING OUT MAN!!!".. functions "meh.. fine w/e" In fact, claiming the speed 45 m/s is a vector would lose you marks at school because a teacher would claim it has no direction, but 45 is an element of a the vector space of real numbers or the field of real numbers, so is also a vector.
Whut? ok.. let me think here, Vector = compound, 45 = atom 45 exists in Vector therefore is vector? So everything in the field of real numbers is considered a vector? In fact, claiming the speed 45 m/s is a vector would lose you marks at school because a teacher would claim it has no direction, but 45 is an element of a the vector space of real numbers or the field of real numbers, so is also a vector.
Speed has no direction btw. That is why the answer is wrong. If the velocity was 45 m/s then the answer is right. speed = length(velocity), speed is an element of R+ (all real numbers greater or equal to zero) - not a field. Define the vector space of all real numbers over the field of real numbers. The nonnegative numbers in this vector space can represent speed, and 45 is a element of this vector space, so is a vector by definition. No, you are wrong. You are embedding your speed in the reals, which in reality it's embedded in the positive reals. Counter example using your argument. Consider the vector space over the reals, the intergers {...,-1,0,1,...} are elements in this vector space so are a vector by definition. Clearly incorrect. No, I'm not confusing speed and velocity. Velocity is modeled in physics as an arrow, that is a vector in the vector space of points in R^2 over the field R. So the point (1,2) corresponds to the velocity "sqrt(5) m/s 1.11 radians from the x-axis". But the set R over the field R, is a vector space, so 45 is a vector, whether it is 45 apples, 45 m, 45 m/s, 45 degrees, whatever. ... speed: R->R+, t----> |v(t)| R+ is not a vector space.
That's one way to define speed, but you can also simply change the codomain (but not the range) to R.
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I understood the OP's defintion....And I've never even had the term vector thrown at me before...-_-....idk why everyone's arguing that it'd be hard to teach either...
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On February 05 2012 02:22 dementrio wrote:
I am waiting for the followup thread where we learn how first grade arithmetic is taught all wrong and it would be so much easier for kids to understand it with a proper definition of ℤ and just a few simple concepts from set theory.
As Day9, points out, the world of mathematics education is constantly mind-blowing, in that you break old assumptions with every year that passes.
We tell children that you can't subtract larger numbers from smaller ones because they don't need knowledge about negative numbers to understand subtraction until later. Trying to bring university mathematics to high school students is similarly fruitless, and causes problems that they never needed or cared about.
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EPT
