This post isn't only for those who know a lot of mathematics, but also for those who known none.
/soapbox.
I came across this video on YouTube and it makes me want to puke.
This video is basically how I was taught vectors in high school, particularly in physics. It’s wrong and it makes no sense to students.
For those who know better, the "definition" given in the video is not the right definition of a vector, it’s merely an example of a vector. In fact, it's incorrect. The function cos(x) is a vector. So what's it's magnitude and direction? Can it even have one?
Here's what is usually taught: - A vector is a quantity with magnitude and direction, a scalar is just a magnitude. - "45 m/s in the NW direction" is a vector. "45 m/s" is a scalar. - A vector is a point, like (2,3). - A vector is an arrow.
- A vector is a complex number, like 2+3i. - You can add vectors like this: (5,1) + (2,3) = (7,4). - But you can also add vectors together by adding arrows (see below). - 2 arrows with the same magnitude and direction, but different positions are the same vectors, so the 2 red arrows are both vector a.
- You can't multiply 2 vectors together, you can only multiply scalars with vectors.
What the fuck...
Does this make sense to anyone who hasn't studied university level mathematics? If you have not studied university level mathematics, I encourage you to reply to tell me how much sense you can make of this.
In particular, you might have the following questions: 1. How do you harmonize the above 3 "definitions"? 2. If a vector has magnitude and direction, then what is the direction of (2,3)? 3. If it’s the direction the arrow from (0,0) to (2,3) makes, why (0,0)? 4. Why should 2 arrows with the same magnitude and directions, as shown in the picture below, be the same vector? 5. You might get an answer from a teacher like "because (1,2) - (0,0) = (1,2) and (4,3) - (3,1) = (1,2), so they are the same", but why should this mean they are the same vector? 6. You might get the answer because they have the same magnitude and direction, but the tip of the arrows are at different points (1,2) and (4,3), so why shouldn't we call them different? 7. If you can't multiply 2 vectors together, but 3+i and 2+2i are vectors, why can I do (3+i)*(2-2i) = 8-4i, which is also a vector? 8. If an arrow v is a vector, 1+i is a vector and (-1,0) is a vector, what is v+(1+i)+(-1,0), and does this make sense? 9. Is 5 a vector? 10. Is f(x) = sin(x) a vector? 11. Is (-2,4,5) a vector? 12. Is a student learning vectors by being taught the above facts and definitions in a position to answer these questions?
If a student having only been taught with the pedagogically incorrect way outlined above, can answer more than, say, 5 of those questions, he/she would be a genius.
This list of questions shows that vectors are taught nonsensically at school, without rigor, and in fact, it is outright wrong. I had many similar questions, that were never satisfactorily resolved until I correctly learned what a vector really is.
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 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.
We call the above-mentioned addition, vector addition, and the above-mentioned multiplication, scalar multiplication.
What is the field F? Basically any set of numbers you know is a field, such as the rational numbers, the real numbers, or the complex numbers. A full definition of a field is given below. + Show Spoiler +
A field F is a set with 2 operations, addition and multiplication, satisfying the following rules: 1. a+b is in F and ab is in F 2. a + (b + c) = (a + b) + c and a(bc) = (ab)c 3. a + b = b + a and ab = ba 4. There exists an element 0 in F, such that a + 0 = a for all a in F 5. There exists an element 1 in F, such that a1 = a for all a in F 6. For every a in F, there exists an element −a in F, such that a + (−a) = 0 7. For every a in F, there exists an element 1/a in F, such that a (1/a) = 1 8. a(b + c) = (ab) + (ac) where a, b, c are any elements in F.
To specify a vector space, you need to specify the set F (which is usually taken to be the real numbers, R) and the set V (which is usually taken to be the points in the number plane, R^2). A vector is simply an element of a vector space, and a scalar is simply an element of the field which the vector space is over.
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.
How does this help students understand vectors?
Firstly, this correctly defines a vector. The definition of a quantity with magnitude and direction is simply wrong. It is merely an example of a vector. 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 over the field of real numbers, so is also a vector.
The vector space which is used for high school physics is the set of points in the number plane R^2 over the field of real numbers. It then follows that points on the plane, such as (1,2) can be represented by arrows, such as the arrow from (0,0) to (1,2), or the arrow from (5,1) to (6,3), or complex numbers, such as 1+2i, because all these vector spaces are isomorphic to each other. Loosely, we say that 2 vector spaces are isomorphic if you can rename all elements of one set with the elements of the other, while keeping addition and multiplication consistent with how it was before the renaming. But isomorphism isn’t something that is taught until 1st or 2nd year university mathematics, which is why all these different things that are called vectors in high school shouldn’t be introduced as vectors. Either teach it the way I outlined or only call one of these things as an example of a vector.
From this the answer to all the 12 questions that I raised are mostly self-evident. And the reason high school teachers insist on the magnitude and direction requirement is because of application to high school physics, but that's not the right way to teach it. Looking at the vector space of points in the number plane R^2 over the field of real numbers, it is then obvious why we can introduce the geometric representation of this particular example of vectors as arrows.
Sal (the guy who made khanacademy.org and this video) generally explains things really well. I've revised a lot of math with his videos and it's been a lot of fun.
I don't know about this one, but don't knock Sal in general. He's a great guy.
my math teacher used to employ the "scarabäus algebraicus" (scarab beetle, it sounds kinda latin-ish when pronounced in german) who would walk along vectors. So to get from point B to point A when you only have vectors to A and B from a third point C, he had to walk along the vector C->B backwards, so you take that negative, and then C->A forwards, so you take that positive. Adding up, you get vector B->A = - C->B + C->A.
still helps me sometimes, although usually i don't need to think about this a lot anymore. At some point it starts coming naturally. But that beetle analogy can really help when you have to figure out formulas to get the formula for a path through some geometric shape with only a few known vectors in it
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.
I was taught the normal way. Then while looking at stuff for high-school physics (relation between force, b-field and current in electomagnetism) I saw KahnAcademy tutorials of him using dot product, to do PHYSICS (OMG!). We only used dot product for math, we always had a cox(theta) somewhere for our equations, so how could he be using vectors for physics? Note the lack of understanding. I still don't get how to use the math we've learnt in mechanics \:
Fast forwards to university, doing a semester in linear algebra first year. You've run through about 1/3 the course-work. Still doesn't teach me how to use vectors in physics tho ):
The problem with teaching linear algebra in high-school is kids that age aren't mature enough to learn it. I was in the most rigorous math class, but still people were confused as fuck and bored out of their minds when we did simple definitions of groups and sup-groups. Imagine what teaching simplistic ~vectors~ must be like to the people who are anything less than 'I wanna do math in uni' ? But then there's a huge gap between the physics you do and the math you need to understand it.
If anyone is interested in learning 'Linear Algebra', which is what the OP is talking about, there are great resources on the web for free. I recommend this one http://linear.ups.edu/download.html it is 3106 pages long, completely free, and absolutely in-depth and clear. Linear algebra is the sort of thing you teach yourself in a day before your exam, after realizing you didn't listen/understand the entire semester - true story bro.
And I agree with you, OP. I was so used to the the high school definition of a vector that when I learned the proper definition of a vector I was completely bamboozled.
You seem to know a little bit about this stuff. Tell us about bases (if that's the proper English term).
While what you're saying is mathematically correct your definition is completely impractical to teach to someone not already predisposed to mathematics. Speaking as someone with a degree with mathematics who has also logged several hundred hours tutoring and teaching math classes, your approach is fine with someone who is in an honors class (not because they are smarter but because they're more predisposed to a more general mathematical style of thought). For anyone else, you get in to terms and concepts that are well beyond anything they will ever need or use or see again, and frankly, you're not going to teach them anything useful. Part of teaching math is knowing when to sacrifice mathematical rigor and generality in favor of understandability. It is the teacher's job to equip students to operate in the real world, for most student's education, an algebraic treatment of vectors is useless and frankly, they won't have the understanding to utilize the additional power and generality.
Most people's first exposure to many other fields in math is similarly ill-founded, look at statistics. Most (read: almost all) intro stats classes look at counting, then move right in to these god-given distribution functions and random variables, then to regression analysis and hypothesis testing. Sure you could give it a proper set/measure theoretic treatment, even at sophmore level courses you could easily give an informal treatment of measure theory and explain notions like "almost surely," but what does this add? Yes, this is correct formally but who cares? How are you better preparing all but the most theoretically inclined students? At some point you just want to help people get to where they want to be, and while for people like you and I this may be a point of passion, you have to recognize that not everyone shares that view, and at a certain point you have to reassess your teaching priorities if you're serious about it.
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."
I don't see any problem teaching the first way to highschool kids. Its the same reason they start off saying you can't square root a number below zero (which you can). Because its complex as hell and highschool students don't need it.
On February 05 2012 00:29 See.Blue wrote: While what you're saying is mathematically correct your definition is completely impractical to teach to someone not already predisposed to mathematics. Speaking as someone with a degree with mathematics who has also logged several hundred hours tutoring and teaching math classes, your approach is fine with someone who is in an honors class (not because they are smarter but because they're more predisposed to a more general mathematical style of thought). For anyone else, you get in to terms and concepts that are well beyond anything they will ever need or use or see again, and frankly, you're not going to teach them anything useful. Part of teaching math is knowing when to sacrifice mathematical rigor and generality in favor of understandability. It is the teacher's job to equip students to operate in the real world, for most student's education, an algebraic treatment of vectors is useless and frankly, they won't have the understanding to utilize the additional power and generality.
The only term that needs to be defined is vector space. You can completely ignore the field part, and substitute in R, which is "well-understood" by high school kids. R^2 (or C) can be introduced as an example of a vector space, and the geometric interpretation would then be obvious without the convoluted mess that is outlined in the OP.
In fact, there might not even be a need to introduce the term "vector" in high school. R^2 is easy enough to understand.
On February 05 2012 00:10 paralleluniverse wrote: - You can't multiply 2 vectors together, you can only multiply scalars with vectors.
I did skim through the rest but didn't find anything else correcting this. Vectors can be multiplied, there's a dot product and a cross product method for vectors?
On February 05 2012 00:29 See.Blue wrote: While what you're saying is mathematically correct your definition is completely impractical to teach to someone not already predisposed to mathematics. Speaking as someone with a degree with mathematics who has also logged several hundred hours tutoring and teaching math classes, your approach is fine with someone who is in an honors class (not because they are smarter but because they're more predisposed to a more general mathematical style of thought). For anyone else, you get in to terms and concepts that are well beyond anything they will ever need or use or see again, and frankly, you're not going to teach them anything useful. Part of teaching math is knowing when to sacrifice mathematical rigor and generality in favor of understandability. It is the teacher's job to equip students to operate in the real world, for most student's education, an algebraic treatment of vectors is useless and frankly, they won't have the understanding to utilize the additional power and generality.
Well put.
I first encountered vectors in a physics class, where they were explained as something having magnitude and direction, an "arrow," if you will. This immediately let me grasp how to use them to calculate net forces and velocities and things like that, using trigonometry. When I later studied quantum mechanics and linear algebra, I got a more general picture. It wasn't a big leap to see that you could think of a point or a function as a vector if you're doing the appropriate things with it. I think I'd have been able to cope with an algebraic definition up front (instead of an intuitive definition), but it would have been a waste of time in the physics class. So I think your teaching method needs to be tailored to the students and the class. If it's a physics class, then an "arrow" is a very seviceable definition that will let you fit the most physics teaching into your lecture. If it's a math class, especially an honors class or something where the students are expecting to be using a lot more of this kind of math in the future, it might be a good idea to use your formal definition right up front.
In some classrooms you'd just scare people off.
The most important parts of teaching are presenting the material clearly, making the material useful to students, and passing on to the students your passion for the subject. If you do those things then you can take any number of specific approaches and have them all work.
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
EPT![[image loading]](http://upload.wikimedia.org/wikipedia/commons/5/5d/Position_vector.svg)
![[image loading]](http://upload.wikimedia.org/wikipedia/commons/2/28/Vector_addition.svg)
![[image loading]](http://i.imgur.com/suoO8.png)
