|
On February 05 2012 02:25 frogrubdown wrote:Show nested quote +On February 05 2012 02:22 paralleluniverse wrote: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.
It would be obvious to high school students, who haven't been introduced to algebraic structures like abstract groups. The only algebraic structure they know are numbers.
|
On February 05 2012 02:12 paralleluniverse wrote:
And I never really gained any real understanding of vectors until a proper definition was taught to me
At what grade level did you come to this understanding?
Do you think you could teach a 5 year old the same understanding???
Because you seem to want physics to stop time and teach all of math before getting to the practicality of solving a problem.
Sometimes you just want to make a pie.
|
Send this to khanacademy and let Sal redo the video if it really is such a wrong explanation.
|
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
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.
While I don't agree entirely with the "bad" way of explaining vectors, I'm pretty sure that this is worse.
This is a formal mathematical representation of a vector. Which is not what most students in middle and high-school who are introduced to vectors need to know.
Just look at your second paragraph. I have a degree in Chemistry and a minor in CS. I've written videogame code professionally that uses vectors a lot. And even I can barely follow that.
Giving the formal, rigorous, mathematical definition of a vector does not "help students understand vectors"; it confuses the shit out of them. That's all it does.
To even begin to understand this explanation, you have to know what a `set` and a `field` are. And your definition of these concepts is... lackluster at best. It's certainly formal and mathematically correct.
But teaching is about a lot more than being formal and mathematically correct. Teaching is first and foremost about communication. And that requires two things: understanding what you want to communciate (which you do) and understanding who you're talking to. The last part is where you fail. Without understanding how your audience thinks, you will be unable to communicate with them.
Set and field theory are not things that middle school and high school students need to know. They aren't ready to know them (certainly not without specific instruction). Giving them an intuitive idea of vectors is a lot more effective overall than just regurgitating a formal definition at them and expecting them to just sort of figure out how it makes sense.
Now, there are times when using intuitive ideas can be problematic. This happens when the intuitive idea is actually contrary to the formal definition to the point that, when the time comes to understand the formal math, understanding things correctly is made more difficult. I don't believe this is the case for vectors. When introduced to the forma; math, people tend to understand vectors better. But they don't have trouble transitioning from the old way of looking at it to the new way.
Basically, you're suggesting the equivalent of not teaching that silly Newtonian mechanics to middle-schoolers and instead teach full-on relativity. Screw algebra and geometry: straight to calculus!
No. Just no.
The function sin(x) is a vector, because it’s an element of the vector space of all continuous real functions over the field of real numbers. Yet it has no magnitude, nor direction. 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.
And... how exactly is that useful information when trying to understand vectors and how they differ from scalars?
Saying that everything is a vector is basically saying that nothing is a vector. If every number they've ever used is a vector, then... what the hell is the point?
A good teacher knows when to refrain from saying things, even true things, if it would confuse the student. Tell them when they have a better grasp of the subject matter, but not before.
|
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.
No he wouldn't.
If you want to teach someone, you have to talk to them in concepts they can understand. Otherwise you're not teaching but just bragging.
|
On February 05 2012 02:29 paralleluniverse wrote:
It would be obvious to high school students, who haven't been introduced to algebraic structures like abstract groups. The only algebraic structure they know are numbers.
But that's exactly the problem you're proposing to correct. Your definition is only "obviously true" because of previously-made assumptions. Not everything is a vector space, not all rings are fields, not all groups are abelian (commutative). Heck, if you start with the sphere (that is, the 2-dimensional sphere in R^3) it's actually provably impossible to come up with a way to put a group structure on its elements that respects its topology.
|
On February 05 2012 02:30 gyth wrote:Show nested quote +On February 05 2012 02:12 paralleluniverse wrote:
And I never really gained any real understanding of vectors until a proper definition was taught to me At what grade level did you come to this understanding? Do you think you could teach a 5 year old the same understanding??? Because you seem to want physics to stop time and teach all of math before getting to the practicality of solving a problem. Sometimes you just want to make a pie.
It was first year university, when I felt I really understood vectors.
The 5 year old comment was a bit of a joke. I was talking about the axioms of a vector space (e.g. 2+3=3+2, 4*(1+2)=4*1+4*2) being obvious to people in, say , grade 6. And that was honestly how I felt when I was first introduced to them.
|
Aotearoa39261 Posts
On February 05 2012 02:27 paralleluniverse wrote:Show nested quote +On February 05 2012 02:14 Plexa wrote: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.
As I said before, you can do that to anything that is a subset of a field (but not a field) and conclude that the subset is also a field.
|
On February 05 2012 02:29 ~OpZ~ wrote:
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...
The point is that expecting 9th graders to gain any intuition from such a formalized concept is wrongheaded and shows a lack of understanding of how students learn. It's much better to teach the intuition and then later to formalize it when we need the formalism to derive something.
For example, we do not teach students geometry by first teaching them abstract algebra and then teaching them how to describe any geometric object as an algebraic object. This would be ridiculous and practically none of the students would know what the hell is going on. It is better to teach them geometry using more familiar things and then only much later teaching them the formal correspondence seen in algebraic topology and algebraic geometry when we need to attack problems that are too difficult with geometry alone.
|
On February 05 2012 02:31 XiGua wrote:
Send this to khanacademy and let Sal redo the video if it really is such a wrong explanation.
There are already 40 lessons on vectors there, that's what makes this topic so useless.
|
On February 05 2012 01:13 paralleluniverse wrote:Show nested quote +On February 05 2012 01:01 Apom wrote:On February 05 2012 00:56 paralleluniverse wrote:On February 05 2012 00:51 Apom wrote:I agree with the person who said that your definitions are vastly overcomplicated for high school needs. Actually I don't recall ever hearing a definition of a vector before reading this thread, which didn't prevent me from being somewhat good at math. Good enough to notice the following, at least : What is the field F? Basically any set of numbers you know is a field, such as the integers, the rational numbers, the real numbers, or the complex numbers. A full definition of a field is given below. Integers are not a field, they are a ring. The definition you give in spoilers even proves it. That is a sloppy mistake, and not how you want to start your explanation. Yeah you're right. But I find it hard to believe that you know about rings before knowing about vector spaces. I learned them in the order monoïd > ring > field > vector space > algebra. In fact I don't see how any other order would make sense, since each definition builds on top of the previous one ... ? So you took an abstract algebra course before a linear algebra course? Not that there's anything wrong with it, it just seems a bit rare. While vector spaces can be defined in terms of groups or rings, it's possible to correctly define a vector spaces without it, while doing so adds almost nothing to the content of a standard 1st or 2nd year linear algebra course. This is different from defining vectors without the notion of a vector space, because of it's incorrectness and the confusion it gives high school students.
I never explicitly took a course in either linear or abstract algebra. I just had a two-year-long math course to start uni with ^_^
Although I did use vectors in high school (which probably counts as a linear algebra course ?), I never encountered the concept of vector space before somewhere around half of my first university year, at which point I had already done copious amounts of abstract algebra. Which, to be honest, was quite helpful in understanding the constructs at the heart of linear algebra, even it didn't make inverting matrices any easier.
|
On February 05 2012 02:25 micronesia wrote:Show nested quote +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.
So true... I wish it got more attention.
|
I'm surprized no one has mentioned the urgent need to teach high school students differential geometry so they can know what a tangent space is and they can learn newtonian physics the proper way.
|
I don't think that you should say the definition taught in schools is false, it's just not complete but includes everything you need until later stages of university...
|
Two university courses of Algebra in and I don't see vectors the same way as I did before...But don't forget that in high school they just rigourously teach you the interpretation of a vector in R2 so that people can picture this. The problem is not necessarily that vectors aren't taught properly, more that people can only function in R1, R2 or R3...nothing else exists at that level of mathematics. There should definitely be an introduction course on abstract algebra. Just the basic number/set/group theory, maybe the basics of vector spaces and fields...I don't know, anything to make them realise that there is more to 'Algebra' than the act of solving equations in R1 or R2, and that Calculus is the the only other branch of mathematics.
THEN you could introduce vectors rigorously. Their definition just makes it self-evident for students.
|
On February 05 2012 02:51 HallBregg wrote:
I'm surprized no one has mentioned the urgent need to teach high school students differential geometry so they can know what a tangent space is and they can learn newtonian physics the proper way.
Why would you want them to learn newtonian physics anyways ? Newtonian physics don't take relativity or strong nuclear force into account, they are just wrong.
// edit : well nevermind, someone already made this point five posts ago.
|
I think the OP is a little elitist about the math.
Yes, the "correct" definition is great for higher level math. But chances are you are learning about vectors before anything past Calculus, and the original definition simply is a better visual for high school students. It's explaining what a vector is in a fairly straightforward way that is useful to kids.
The definition you presented doesn't help kids in a high school physics class or calc AB, it belongs in a higher level math class, and usually that's not until senior year or college, and at this point the original definition wouldn't hurt them very much.
|
On February 05 2012 02:36 iSTime wrote:Show nested quote +On February 05 2012 02:29 ~OpZ~ wrote:
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... The point is that expecting 9th graders to gain any intuition from such a formalized concept is wrongheaded and shows a lack of understanding of how students learn. It's much better to teach the intuition and then later to formalize it when we need the formalism to derive something. For example, we do not teach students geometry by first teaching them abstract algebra and then teaching them how to describe any geometric object as an algebraic object. This would be ridiculous and practically none of the students would know what the hell is going on. It is better to teach them geometry using more familiar things and then only much later teaching them the formal correspondence seen in algebraic topology and algebraic geometry when we need to attack problems that are too difficult with geometry alone.
What about 10th, 11th, or 12th? Where are you most likely to find vectors? Physics was an 11th/12th grade class where I am, and precal and cal were 11th/12th, along with alg 3 and trig where I'm from. Algebra 1/2 and geometry were 9th 10th and 11th grade here....So I'm confused. Did you do vectors in ninth grade?
|
Post removed due to having second thoughts about my decision to enter into a useless discussion.
|
The 5 year old comment was a bit of a joke. I was talking about the axioms of a vector space (e.g. 2+3=3+2, 4*(1+2)=4*1+4*2) being obvious to people in, say , grade 6. And that was honestly how I felt when I was first introduced to them.
But there is a difference between something being obvious and it being true, another step to understanding it, and then another to being able to teach it.
If you could make a *10* minute youtube video of your explanation of vectors I'm sure many of us would be doubly impressed if it offered easier understanding.
|
|
|
|
|
|
EPT
