|
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" 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.
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?
That isn't something you should take too seriously. It was just a cute example of high school teachers saying something that is technically wrong.
When high school teachers think of vectors, they usually mean arrows.
In that case 45 is not a vector, in fact 45 isn't even an arrow.
But the correct definition of a vector is more general than the set of all arrows. In a different setting, in the vector space of real numbers, 45 is a vector. But that's not what you're teacher means when he/she says vector. So don't repeat any of this in class or you might lose marks.
|
I mean, everything you said is correct, but is basically the beginning of one's Linear Algebra course, yes? My knowledge of "Physics" vectors helps me with Physics but it usually has very little to do with concepts on my Linear Algebra homework. Likewise, what I'm learning in Linear Algebra has helped me in terms of its theory of vector spaces and bases to explain aspects of my signals course rather than knowledge of basic vector calculations and such. So although they could, and perhaps should be taught together, they aren't definitively conflicting ideas. One needs practice with both for very different reasons.
|
Aotearoa39261 Posts
On February 05 2012 01:45 sukarestu wrote:Show nested quote +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. Right thanks. But didn't he state that 45 is an element of a vector because it occupies a space in the field of real numbers, therefore it is a vector? Show nested quote +but 45 is an element of a the vector space of real numbers or the field of real numbers, so is also a vector Or was he referring to the number generally and not the speed mentioned beforehand That's what I was confused about AND Didn't the OP say that vectors don't necessarily have to have speed and direction? Especially considering how it was said "A vector is a point, like (2,3)" that would be stationary.. it definitely does not have direction.. or speed 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.
|
On February 05 2012 01:29 Plexa wrote:Show nested quote +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? 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.
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.
|
your trying too muddle lots of similar things into the same idea imo.
(2,3) has a direction of 2,3 ;p ... or you can solve it into a magnitude and angle using trig.
The problem i had with vectors (and then matricies) is that you get to the point where you cannot intuitivley understand things .. you end up employing rules (eg dot product and cross product) to work.
The point of complex numbers as a vector is that i guess you could think of it as a 4 dimensional number (3 real and 1 complex) but all the real dimensions have been collapsed down into 1 so then you are left with the 2 dimensions of real and complex which is what you are talking about.
What you are running into is that maths becomes more about symbolic manipulation the deeper you go as it rapidly spills out side of the 2d that you can intuit. The point is that when you talk of dot and cross products you are already talking about an abstraction of a whole set of calculations that you are going to do under it. Its far easier to think of them functionally as finding the magnitude in parrallel and finding a vector to the perpendicular.
I am really out of practisde but i found maths got hard but really rewarding when i hit 1st year physics at uni
|
On February 05 2012 01:40 BrickTop wrote:
You could criticize a large majority of high school math material the exact same way. I'm not really sure why you singled out vectors. The high school 'definitions' are usually incorrect, and you could present the real definitions in contrast. But what's the point? I agree with many others in this thread: in my opinion teaching university level definitions to a general high school audience would not be productive.
I singled out vectors, because out of everything in high school, I don't feel anything is as wrongly and confusingly taught as vectors.
|
Aotearoa39261 Posts
On February 05 2012 01:55 paralleluniverse wrote:Show nested quote +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 should be the positive reals. Counter example using your argument. Consider the vector space over the reals, the naturals {1, 2,...} are elements in this vector space so are a vector by definition. Clearly incorrect.
|
There's a reason they do it this way in high school ... it's so those that don't really care about math or physics may actually have a shot at understanding it.
|
On February 05 2012 01:56 MrTortoise wrote:
your trying too muddle lots of similar things into the same idea imo.
No.
I'm trying to unmuddle this mess that high school teaching has made.
(2,3) has a direction of 2,3 ;p ... or you can solve it into a magnitude and angle using trig.
The problem i had with vectors (and then matricies) is that you get to the point where you cannot intuitivley understand things .. you end up employing rules (eg dot product and cross product) to work.
The point of complex numbers as a vector is that i guess you could think of it as a 4 dimensional number (3 real and 1 complex) but all the real dimensions have been collapsed down into 1 so then you are left with the 2 dimensions of real and complex which is what you are talking about.
What you are running into is that maths becomes more about symbolic manipulation the deeper you go as it rapidly spills out side of the 2d that you can intuit. The point is that when you talk of dot and cross products you are already talking about an abstraction of a whole set of calculations that you are going to do under it. Its far easier to think of them functionally as finding the magnitude in parrallel and finding a vector to the perpendicular.
I am really out of practisde but i found maths got hard but really rewarding when i hit 1st year physics at uni
When I talked about complex numbers, I'm referring to the teaching that a complex number a+bi is represented as the arrow from (0,0) to (a,b), that's 2 dimensions, not 4.
Mathematical computations should also be done symbolically, but a geometric interpretation is very valuable for understanding theory. And I suggest a teaching vectors in a way that starts with the former and leads to the latter.
|
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-10
Where is a mathematically rigorous definition of field or vector space gone over at a gradeschool level?
|
As far as the discussion goes on saying that "a number is a vector", it just goes to show again why the OP's suggestion on how to teach vectors is flawed. If you were to find a high school teacher who teaches that 45 is a vector, his/her students would be MUCH more confused. Going back to what I said before, the arrow representation worked well for me, and it justifies why there is a difference between speed and velocity.
Breaking these rules down is NOT the job of a high-school mathematics/physics teacher. It's the job of a university professor whose students are genuinely interested in purifying their knowledge in the field.
|
On February 05 2012 01:34 paralleluniverse wrote:
A few points:
Your point on some things not being property explained, particularly in engineering or physics course is quite right, sometimes it's necessary. But there are a few differences. Not having a complete understanding is different from having a wrong understanding. University level mathematics like analytic continuation and the Dirac delta function is harder to teach than vectors. It's not as bad for a engineering class to teach wrong or incomplete math and it is a math class.
The geometric interpretation of vectors as arrows falls out quite naturally in the teaching method I suggest: Define a vector as an element of a vector space, show that R^2 is a vector space, it follows that the points in R^2 are vectors, then the representation as arrows is obvious.
Clearly, this would require more work on the teachers and students part, but I think the much greater clarity this provides is well worth it.
From the vector space R^n, a geometric intuition of decomposing vectors into basis vectors is also natural (this is university level math, and it's usually done in this correct way anyway). I'm not sure what your point on Hilbert spaces is about. The typical example of a Hilbert space is the space of continuous functions, and the basis vectors being the sin and cos function is graphically obvious if you watch an animation of a Fourier series converging.
I wasn't taught the dot product or cross product in high school, but that's just because we when to school in different countries.
I totally see your point, the representation as arrows definitely does fall out of this approach (as it should). The thing I am worried about is this: we spend a lot of time trying to teach students that vectors are NOT scalars and that scalars are NOT vectors. We do this because we need them to manipulate the two objects in a fundamentally different way (at least from their perspective). The problem I think we will have (overlooking that the rules for what makes something a vector space are abstract) is that we lose the distinction between speed and velocity, distance and displacement, etc that we need in introductory physics. Do we actually lose it from a mathematical point of view? No, of course not but we may lose the distinction in the students mind.
This is what I imagine their though process would be. "These are the rules for a vector space. R^2 is a vector space, those are the arrows. R is also a vector space those are just ordinary numbers. Well, the rules are the same for both and they're both vectors, so I must be able to treat them the same way." We all know that there are a lot of faux pas in this reasoning, but I think that its the kind of thought process that we would have to spend a lot of time correcting if we used this approach. Going through the process of defining a vector space and then continually stressing that objects in different vector spaces can not be treated the same way will probably leave most of them wondering, "What is the point then of these vector spaces?"
Ultimately, I want to say that the formal idea of a vector space is very important for understanding the difference between things which are elements of a vector space and things which are not. For example, the distinction between linear and non-linear operators is very clear because from the vector space formalism they have very different properties. However, it is not sufficient for understanding the difference between 2 objects which are both elements of their respective vector space, but belong to different spaces. The key point about vectors/scalars that we stress is that they are different and can not be treated the same way and teaching formalism about vector spaces doesn't seem to help accomplish that goal, and it might even hurt it.
My point about Hilbert spaces was this. Most students that I teach barely know what sine and cosine are. Thinking of them as basis vectors for the space of all continuous 2p periodic functions is an idea well beyond their level. They don't know anything about Fourier analysis, many don't even know how to handle infinite series. Functions as elements of a vector space kind of opens up a 'can of worms' so to speak. I think we'd almost certainly have to leave this part out.
|
On February 05 2012 00:31 paralleluniverse wrote:Show nested quote +On February 05 2012 00:27 Flameberger wrote:
I've taken university level math, but I didn't have any trouble with vectors in high school. In fact as far as I could tell the entire class understood and applied them without any difficulty.
Yes the way it is taught is not strictly accurate math-wise, but do you honestly think
1. u + (v + w) = (u + v) + w
2. u + v = v + u
3. There exists an element 0 in V, such that v + 0 = v for all v in V.
4. For every v in V, there exists an element −v in V, such that v + (−v) = 0
5. a(u + v) = au + av
6. (a + b)v = av + bv
7. a(bv) = (ab)v
8. 1v = v,
9. u+v is in V
10. av is in V
where v, u, w are any elements in V, and a is any element in F.
is something highschool students are going to look at and think: "oh yeah, that makes a lot more sense", what are you even supposed to do with that? memorize it?
The way I learned vectors has served me well in my various physics courses since high school, when I needed a proper mathmatic understanding of them in Calculus III it was very easy for me to adapt my current understanding to the proper definitions. When I saw that list for the first time, do you know what I was thinking? "Why are we listing facts so obvious that a 5 year old would understand."
That list being obvious how? That's what defines V. They aren't really obvious, what if V would instead be defined as Nul (M), I mean, it's not really obvious considering it's a definition.
|
I have been TA fro 4 courses in Economics and agree with OP that the oversimplifications make it much harder to explain and in the end the kids get confused on things that are obvious once you know the bigger picture. And the other disadvantage is that student learn by heart and don't try to understand.
The approach the OP suggests puts emphasis on understanding rather than memorizing. I had problems with vectors in high school. No problems any more once I took Linear Algebra. It's one of the simplest concept in mathematics and et is made so obscure that it's really difficult to understand.
Students taking classes that at higher level use sophisticated mathematics should be required to take math classes. I can't understand why in US to be a major in economics you don't need to take even multivariate calculus. I have to spend always lots of time to explain the concept of marginal increase while it is just simple derivative.
The correct approach would require students to take calculus up to vector calc, linear algebra, advanced linear algebra, real analysis and a course in optimization. It would be also helpful to take measure theory. You know that currently having only Econ major makes you illegible to apply for PhD in economics? That's weird isn't it? You better have at least a minor in maths, or better major in maths. Schools will care more about the math courses you took than the econ. A math major without any econ has a higher chance (actually quite decent depending on the grades of course) of getting into a PhD than Econ major (who has no chances at all)
|
Sigh, I was actually hoping for a discussion on math education, not a math theory penis measuring contest 
|
Khan ftw! Gogo you should all watch his Youtube starting vid on his page <3
|
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.
|
On February 05 2012 01:59 Plexa wrote:Show nested quote +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.
|
On February 05 2012 02:10 perestain wrote:Show nested quote +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.
|
Aotearoa39261 Posts
|
|
|
|
|
|
EPT
