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
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 terms that need 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 is obvious without the convoluted mess that is outlined in the OP.
But what does this help them with? For your average HS math student, what does this contribute in terms of real benefit? I love math, its something I pursue as a career, and I love telling people about it and teaching it. But at a certain point you have to realize how difficult and unnatural it is for many people. You're not going to be better equipping any but a small subset of likeminded individuals, and for the majority you're sacrificing giving them at least a fundamental understanding in favor of mathematical correctness.
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?
When high school kids think of multiplying vectors (1,2)*(4,5), their not thinking of the dot product (as this isn't introduced until university), their thinking of (1*4,2*5).
It was drilled into me in high school that you can't multiply 2 vectors.
After all (1,2)+(4,5) = (1+4,2+5), so why shouldn't (1,2)+(4,5) = (1+4,2+5)? The answer is because the definition of a vector is far more general than what is taught in high school -- vector spaces are defined to have only scalar multiplication and not vector multiplication, and that vector multiplication is not needed for the physics application of vectors in high school.
Fun fact: the algebraic structure that is a vector space with vector multiplication is called an "algebra".
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.
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 think something to keep in mind of when thinking of teaching stuff to high school students.
How long did it take before one can understand how to understand this vector space concept? Well it takes a substantial amount of effort in the discipline of mathematics... one will need to learn how to prove things, have an understanding of Fields, mastery of applying "basic" principles.... the list goes on and on and on.... and that is the pre-req for understanding what a vector space is....
Clearly, high school students have absolutely none of that. Simply explaining what a vector space with absolutely no context to build off with is guaranteeing the average high school student to fail (... maybe with the exception of the odd genius here and there). So... what is the next best thing to teach.... AN EXAMPLE OF A VECTOR SPACE R^2 to start off with.
University math in general attempts to "generalize" what one has learned. Technically speaking, the way high school presents vectors isn't wrong... it is an example of what a vector is. However, in higher level math, one will attempt to say X, Y and Z is considered to be vectors because it follows these generalized principles.
On February 05 2012 00:57 unifo wrote: I think something to keep in mind of when thinking of teaching stuff to high school students.
How long did it take before one can understand how to understand this vector space concept? Well it takes a substantial amount of effort in the discipline of mathematics... one will need to learn how to prove things, have an understanding of Fields, mastery of applying "basic" principles.... the list goes on and on and on.... and that is the pre-req for understanding what a vector space is....
Clearly, high school students have absolutely none of that. Simply explaining what a vector space with absolutely no context to build off with is guaranteeing the average high school student to fail (... maybe with the exception of the odd genius here and there). So... what is the next best thing to teach.... AN EXAMPLE OF A VECTOR SPACE R^2 to start off with.
University math in general attempts to "generalize" what one has learned. Technically speaking, the way high school presents vectors isn't wrong... it is an example of what a vector is. However, in higher level math, one will attempt to say X, Y and Z is considered to be vectors because it follows these generalized principles.
This.
@OP, I think it's clear you definitely are predisposed to a mathematical way of thinking and to you it may be a truly natural way to look at it, which is absolutely fantastic. But to me at least, the difference between being a math teacher and being a good math teacher is being able to understand that not everyone shares this quality, and for a real majority this requires thinking in a way that is difficult and intimidating. I agree that almost everyone has the innate ability to understand this level of mathematical thought, and also agree that math education, particularly in lower levels is in desperate need of improvement. But this isn't the right way to go about it. You have a gift at understanding this, and that's great, because it means you are better suited to understanding the intricacies of the subject and, with thought, understanding where other people might have trouble. And god knows we could use more people with your talent out here. But don't fall into the trap of thinking everyone can do this naturally; anyone can talk at a student, but only someone with a deep, intuitive understanding can explain something well.
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 ... ?
On the whole, I agree with your assessment that the way vectors are taught at an introductory level is usually 'wrong' in the rigorous mathematical sense, but I am not convinced that your solution would fix the problem.
I am a TA for an introductory physics course at the university level and the #1 problem that I see with students' exams and homework is a lack of understanding of how to manipulate vectors. So, I agree that this is a problem.
Your basis in linear algebra is of course correct and completely rigorous and is how I eventually learned about vector spaces for my advanced mathematics. The problem is that most of the students that I teach are not in a position to understand the linear algebra any more than they understand the pragmatic definitions like the ones used in the video you linked to. The concept of a vector space makes the notion of a vector very abstract and for students who aren't majoring in mathematics; it runs counter to the way they have learned math for their entire lives. Most students (at least in the US) learn mathematics only in small chunks that they need to use and not starting from a rigorous foundation. Although most university students could take the derivative of a polynomial function or the limit of some rational function as it approaches zero, they would have a hard time justifying their responses using the formal definitions of a derivative and a limit(I cant even remember this definition most days.....its something like for every epsilon there exists a delta such that......).
Note that we actually do this all the time in mathematics education. We teach a restricted special case in a sometimes inconsistent way to get students to have some basic intuition about the objects and perform basic tasks, then in higher level classes we generalize the notions. I'll list some examples here: 1) The imaginary number i: why assume that there is only one such number? What happens when a function tries to take i as an input, such as Sin(i*x) or Log(2*i*x)? Most students couldn't answer these question after they learn about i to solve the quadratic formula in high school. You need an entire class on complex analysis for that (and even then they might not cover what happens if you assume there is more than one imaginary number (Quaternions). 2) The Dirac delta 'function': here is an idea which even I don't actually understand. Physicists and engineers use and abuse the delta function every day without ever thinking about the fact that it isn't actually a function at all, it is a functional or distribution. It is normally defined as an object with is zero everywhere except a single point and has total integral 1. This definition is patently incorrect if you consider certain sets of series which converge to the delta function, even though their values do not converge to zero almost everywhere, but for most everyone the definition is good enough. More importantly, it helps gives students the intuition of what happens when you use the delta function. I have lots of nagging questions about the delta function because I don't know the theory of distributions, but I get along okay.
I think the crux of the issue is that teaching students the abstract linear algebra version of vectors does not give them a strong physical intuition about how vectors work in physics, and this is the reason that we don't teach it that way. The whole point of our physics course is to develop intuition, not teach specific skills. Teaching vectors as arrows is a much more physically relevant approach given how we deal with objects like velocity. We need concepts like decomposition along a basis vector (which is hard to do/explain with functions in a Hilbert space) and direction (which is nearly impossible to do in that same case) to understand that physics. It is not so important to us if the math is rigorous. I'll note here that quantum mechanics existed for more than 40 years and was used all over the world before its basis was made mathematically rigorous (the idea of a rigged Hilbert space). Ultimately, we teach the way we do because it is best for physics, and I'll let the math teachers speak for themselves as to why they don't delve more deeply into the idea of a vector space.
EDIT: I wanted to add after reading previous responses that I learned about vectors first in high school in a mathematics class and we definitely discussed both dot and cross products. They are important concepts that we mostly expect students to understand when they step into our physics courses.
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 terms that need 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 is obvious without the convoluted mess that is outlined in the OP.
But what does this help them with? For your average HS math student, what does this contribute in terms of real benefit? I love math, its something I pursue as a career, and I love telling people about it and teaching it. But at a certain point you have to realize how difficult and unnatural it is for many people. You're not going to be better equipping any but a small subset of likeminded individuals, and for the majority you're sacrificing giving them at least a fundamental understanding in favor of mathematical correctness.
I understand the need to balance understandability with rigor. For example, I have no problems with the loose definition of a limit that is given in high school calculus. I certainly won't be advocating teaching the delta-epsilon definition in high school. But that's because the high school definition of a limit makes sense and is sufficient at that level. However, the way that vectors are taught does not make sense, questions like those in the OP are very hard to answer or understand, and when teachers are hammering home the point that a vector is a quantity with magnitude and direction, as the video does, and as my high school physics teachers did, it is also completely wrong.
Personally, I think a lot of my grudge came from the seemingly conflicting ways vectors are taught in high school physics courses as arrows, and in high school math courses as all of the above or whichever is more convenient at the moment.
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.
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?
Yes, but then again those are only special cases od the more general inner product as well. I.e. a product <.,.> thats satisfies: a.) <u+v,w> = <u,w> + <v,w>, b.) <av,w> = a<v,w>, c.) <v,w> = <w,v>, d.) <v,v> >= 0 while <v,v> = 0 if and only if v = 0.
The problem is where do you stop explaining things. Because now you should also start explaining about inner product spaces, metric spaces, linear operators etc.
The biggest problem I find with comming from highschool, is that you often don't even know you were being tought incorrect / incomplete knowledge and that you often lack understanding of the concepts behind what it is you are doing. Especialy the latter is super important in really understanding math. IMO, even without going into all the details, teachers could definately put more effort into explaigning the connections between topics and concept in maths.
They could at least mention that the way they are teaching about vectors is incomplete and that actually the vectors they are showing are only a specific type of vectors in a broader framework.
I'm also familiar with vectors and don't understand what you're trying to explain at all. It's just a bunch of facts - you're showing what the result of teaching vectors properly would be, but there's absolutely no way there as far as I can see.
They could at least mention that the way they are teaching about vectors is incomplete and that actually the vectors they are showing are only a specific type of vectors in a broader framework.
This is a key point I think. It was never 'hammered' in to me that vectors are arrows and nothing else by my math/physics teachers. It was always a much more pragmatic approach of: "here's what we need to learn." Teachers definitely do need to make an effort to not give students incorrect information.
On February 05 2012 01:01 Anytus wrote: 2) The Dirac delta 'function': here is an idea which even I don't actually understand. Physicists and engineers use and abuse the delta function every day without ever thinking about the fact that it isn't actually a function at all, it is a functional or distribution. It is normally defined as an object with is zero everywhere except a single point and has total integral 1. This definition is patently incorrect if you consider certain sets of series which converge to the delta function, even though their values do not converge to zero almost everywhere, but for most everyone the definition is good enough. More importantly, it helps gives students the intuition of what happens when you use the delta function. I have lots of nagging questions about the delta function because I don't know the theory of distributions, but I get along okay.
Agreed with this. The delta function is the single most abused piece of mathematics that I know of. Although, if I recall my funational analysis paper correctly, when treated as a linear functional \delta_x represents the functional which takes f to f(x) which captures a lot of the "zero everywhere except at a single point" idea.
On February 05 2012 01:01 Anytus wrote: 2) The Dirac delta 'function': here is an idea which even I don't actually understand. Physicists and engineers use and abuse the delta function every day without ever thinking about the fact that it isn't actually a function at all, it is a functional or distribution. It is normally defined as an object with is zero everywhere except a single point and has total integral 1. This definition is patently incorrect if you consider certain sets of series which converge to the delta function, even though their values do not converge to zero almost everywhere, but for most everyone the definition is good enough. More importantly, it helps gives students the intuition of what happens when you use the delta function. I have lots of nagging questions about the delta function because I don't know the theory of distributions, but I get along okay.
Agreed with this. The delta function is the single most abused piece of mathematics that I know of.
It's a measure. You need to learn measure theory to understand it.
On February 05 2012 01:01 Anytus wrote: 2) The Dirac delta 'function': here is an idea which even I don't actually understand. Physicists and engineers use and abuse the delta function every day without ever thinking about the fact that it isn't actually a function at all, it is a functional or distribution. It is normally defined as an object with is zero everywhere except a single point and has total integral 1. This definition is patently incorrect if you consider certain sets of series which converge to the delta function, even though their values do not converge to zero almost everywhere, but for most everyone the definition is good enough. More importantly, it helps gives students the intuition of what happens when you use the delta function. I have lots of nagging questions about the delta function because I don't know the theory of distributions, but I get along okay.
Agreed with this. The delta function is the single most abused piece of mathematics that I know of.
It's a measure. You need to learn measure theory to understand it.
Okay, Riesz representation theorem gg. But understanding it in that context isn't as useful as calling it a linear functional which is much more description or even as a distribution. But if you recall was RRT says is that "every linear functional can be represented by a Radon measure integrated" (loosely speaking) so in that respect it is more a linear functional. I call it a function as a force of habit as it was grind into me in applied math.
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?
The issue with what you're saying about vectors is that those definitions aren't obviously motivated unless you've already internalized the idea of a basis.
If you have internalized the idea of a basis, then those definitions are completely obvious and can easily be worked with. If you haven't, then those definitions seem to be completely random.
IMO, the point of the current system is more to teach people about a basis on a vector space than about vectors.
For an analogous situation that might or might not be more familiar to you: Topological definitions don't make sense unless you understand neighborhoods.
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.
[B]On February 05 2012 01:22 paralleluniverse wrote: It's a measure. You need to learn measure theory to understand it.
Absolutely, and yet every physicist I have ever talked to says, "Don't take a class on measure theory, it won't help you use the Dirac delta function any better than you already do." Not to mention that we certainly can't require every physics major to take: Calculus 1-3, Linear Algebra, Differential Equations, Partial Differential Equations, AND Measure Theory(plus pre-requisites) before they take their first course in quantum mechanics. There simply aren't enough semesters. We need a working definition of the delta function so that we can use it, even if it is actually incorrect.
On February 05 2012 01:01 Anytus wrote: On the whole, I agree with your assessment that the way vectors are taught at an introductory level is usually 'wrong' in the rigorous mathematical sense, but I am not convinced that your solution would fix the problem.
I am a TA for an introductory physics course at the university level and the #1 problem that I see with students' exams and homework is a lack of understanding of how to manipulate vectors. So, I agree that this is a problem.
Your basis in linear algebra is of course correct and completely rigorous and is how I eventually learned about vector spaces for my advanced mathematics. The problem is that most of the students that I teach are not in a position to understand the linear algebra any more than they understand the pragmatic definitions like the ones used in the video you linked to. The concept of a vector space makes the notion of a vector very abstract and for students who aren't majoring in mathematics; it runs counter to the way they have learned math for their entire lives. Most students (at least in the US) learn mathematics only in small chunks that they need to use and not starting from a rigorous foundation. Although most university students could take the derivative of a polynomial function or the limit of some rational function as it approaches zero, they would have a hard time justifying their responses using the formal definitions of a derivative and a limit(I cant even remember this definition most days.....its something like for every epsilon there exists a delta such that......).
Note that we actually do this all the time in mathematics education. We teach a restricted special case in a sometimes inconsistent way to get students to have some basic intuition about the objects and perform basic tasks, then in higher level classes we generalize the notions. I'll list some examples here: 1) The imaginary number i: why assume that there is only one such number? What happens when a function tries to take i as an input, such as Sin(i*x) or Log(2*i*x)? Most students couldn't answer these question after they learn about i to solve the quadratic formula in high school. You need an entire class on complex analysis for that (and even then they might not cover what happens if you assume there is more than one imaginary number (Quaternions). 2) The Dirac delta 'function': here is an idea which even I don't actually understand. Physicists and engineers use and abuse the delta function every day without ever thinking about the fact that it isn't actually a function at all, it is a functional or distribution. It is normally defined as an object with is zero everywhere except a single point and has total integral 1. This definition is patently incorrect if you consider certain sets of series which converge to the delta function, even though their values do not converge to zero almost everywhere, but for most everyone the definition is good enough. More importantly, it helps gives students the intuition of what happens when you use the delta function. I have lots of nagging questions about the delta function because I don't know the theory of distributions, but I get along okay.
I think the crux of the issue is that teaching students the abstract linear algebra version of vectors does not give them a strong physical intuition about how vectors work in physics, and this is the reason that we don't teach it that way. The whole point of our physics course is to develop intuition, not teach specific skills. Teaching vectors as arrows is a much more physically relevant approach given how we deal with objects like velocity. We need concepts like decomposition along a basis vector (which is hard to do/explain with functions in a Hilbert space) and direction (which is nearly impossible to do in that same case) to understand that physics. It is not so important to us if the math is rigorous. I'll note here that quantum mechanics existed for more than 40 years and was used all over the world before its basis was made mathematically rigorous (the idea of a rigged Hilbert space). Ultimately, we teach the way we do because it is best for physics, and I'll let the math teachers speak for themselves as to why they don't delve more deeply into the idea of a vector space.
EDIT: I wanted to add after reading previous responses that I learned about vectors first in high school in a mathematics class and we definitely discussed both dot and cross products. They are important concepts that we mostly expect students to understand when they step into our physics courses.
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'm probably in the ideal demographic for this, as I am educated in programming/probability-related mathematics but have never encountered a vector in my life. I have to say, the first definition left me with a few questions, but I grasped the overall idea of a vector and it mostly made sense to me. The definition you gave made no sense to me whatsoever, and seemed overly complicated. It may be a truer definition, and it may make more sense to someone who already understands what a vector is, but as a method for teaching, I do not think it is superior by any stretch.
It saddens me your need to show off pushed you into creating this topic. If you look closely to khansacademy, you'll notice there are 40+ lessons on vectors only, some even 10 or more minutes long. So you're basically picking on (the first) 6 minutes out of 4-6 hours of content.
There's a threshold where theory needs to take a pause and practicality kick in. We each have it, I for example was taught vectors the way you described them and found it rather easy, while during uni-level statistics i had to slow down as i reached mine.
On February 05 2012 01:01 Anytus wrote: On the whole, I agree with your assessment that the way vectors are taught at an introductory level is usually 'wrong' in the rigorous mathematical sense, but I am not convinced that your solution would fix the problem.
I am a TA for an introductory physics course at the university level and the #1 problem that I see with students' exams and homework is a lack of understanding of how to manipulate vectors. So, I agree that this is a problem.
Your basis in linear algebra is of course correct and completely rigorous and is how I eventually learned about vector spaces for my advanced mathematics. The problem is that most of the students that I teach are not in a position to understand the linear algebra any more than they understand the pragmatic definitions like the ones used in the video you linked to. The concept of a vector space makes the notion of a vector very abstract and for students who aren't majoring in mathematics; it runs counter to the way they have learned math for their entire lives. Most students (at least in the US) learn mathematics only in small chunks that they need to use and not starting from a rigorous foundation. Although most university students could take the derivative of a polynomial function or the limit of some rational function as it approaches zero, they would have a hard time justifying their responses using the formal definitions of a derivative and a limit(I cant even remember this definition most days.....its something like for every epsilon there exists a delta such that......).
Note that we actually do this all the time in mathematics education. We teach a restricted special case in a sometimes inconsistent way to get students to have some basic intuition about the objects and perform basic tasks, then in higher level classes we generalize the notions. I'll list some examples here: 1) The imaginary number i: why assume that there is only one such number? What happens when a function tries to take i as an input, such as Sin(i*x) or Log(2*i*x)? Most students couldn't answer these question after they learn about i to solve the quadratic formula in high school. You need an entire class on complex analysis for that (and even then they might not cover what happens if you assume there is more than one imaginary number (Quaternions). 2) The Dirac delta 'function': here is an idea which even I don't actually understand. Physicists and engineers use and abuse the delta function every day without ever thinking about the fact that it isn't actually a function at all, it is a functional or distribution. It is normally defined as an object with is zero everywhere except a single point and has total integral 1. This definition is patently incorrect if you consider certain sets of series which converge to the delta function, even though their values do not converge to zero almost everywhere, but for most everyone the definition is good enough. More importantly, it helps gives students the intuition of what happens when you use the delta function. I have lots of nagging questions about the delta function because I don't know the theory of distributions, but I get along okay.
I think the crux of the issue is that teaching students the abstract linear algebra version of vectors does not give them a strong physical intuition about how vectors work in physics, and this is the reason that we don't teach it that way. The whole point of our physics course is to develop intuition, not teach specific skills. Teaching vectors as arrows is a much more physically relevant approach given how we deal with objects like velocity. We need concepts like decomposition along a basis vector (which is hard to do/explain with functions in a Hilbert space) and direction (which is nearly impossible to do in that same case) to understand that physics. It is not so important to us if the math is rigorous. I'll note here that quantum mechanics existed for more than 40 years and was used all over the world before its basis was made mathematically rigorous (the idea of a rigged Hilbert space). Ultimately, we teach the way we do because it is best for physics, and I'll let the math teachers speak for themselves as to why they don't delve more deeply into the idea of a vector space.
EDIT: I wanted to add after reading previous responses that I learned about vectors first in high school in a mathematics class and we definitely discussed both dot and cross products. They are important concepts that we mostly expect students to understand when they step into our physics courses.
A few points:
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 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 you 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 just because we when to school in different countries.
But then how do you answer the question "What's the point of defining vector spaces?" The reason it's taught as it is ATM is because the concept of a vector space as "something with a basis" is much more intuitive and initially useful than the standard definition is.
On February 05 2012 01:01 Anytus wrote: On the whole, I agree with your assessment that the way vectors are taught at an introductory level is usually 'wrong' in the rigorous mathematical sense, but I am not convinced that your solution would fix the problem.
I am a TA for an introductory physics course at the university level and the #1 problem that I see with students' exams and homework is a lack of understanding of how to manipulate vectors. So, I agree that this is a problem.
Your basis in linear algebra is of course correct and completely rigorous and is how I eventually learned about vector spaces for my advanced mathematics. The problem is that most of the students that I teach are not in a position to understand the linear algebra any more than they understand the pragmatic definitions like the ones used in the video you linked to. The concept of a vector space makes the notion of a vector very abstract and for students who aren't majoring in mathematics; it runs counter to the way they have learned math for their entire lives. Most students (at least in the US) learn mathematics only in small chunks that they need to use and not starting from a rigorous foundation. Although most university students could take the derivative of a polynomial function or the limit of some rational function as it approaches zero, they would have a hard time justifying their responses using the formal definitions of a derivative and a limit(I cant even remember this definition most days.....its something like for every epsilon there exists a delta such that......).
Note that we actually do this all the time in mathematics education. We teach a restricted special case in a sometimes inconsistent way to get students to have some basic intuition about the objects and perform basic tasks, then in higher level classes we generalize the notions. I'll list some examples here: 1) The imaginary number i: why assume that there is only one such number? What happens when a function tries to take i as an input, such as Sin(i*x) or Log(2*i*x)? Most students couldn't answer these question after they learn about i to solve the quadratic formula in high school. You need an entire class on complex analysis for that (and even then they might not cover what happens if you assume there is more than one imaginary number (Quaternions). 2) The Dirac delta 'function': here is an idea which even I don't actually understand. Physicists and engineers use and abuse the delta function every day without ever thinking about the fact that it isn't actually a function at all, it is a functional or distribution. It is normally defined as an object with is zero everywhere except a single point and has total integral 1. This definition is patently incorrect if you consider certain sets of series which converge to the delta function, even though their values do not converge to zero almost everywhere, but for most everyone the definition is good enough. More importantly, it helps gives students the intuition of what happens when you use the delta function. I have lots of nagging questions about the delta function because I don't know the theory of distributions, but I get along okay.
I think the crux of the issue is that teaching students the abstract linear algebra version of vectors does not give them a strong physical intuition about how vectors work in physics, and this is the reason that we don't teach it that way. The whole point of our physics course is to develop intuition, not teach specific skills. Teaching vectors as arrows is a much more physically relevant approach given how we deal with objects like velocity. We need concepts like decomposition along a basis vector (which is hard to do/explain with functions in a Hilbert space) and direction (which is nearly impossible to do in that same case) to understand that physics. It is not so important to us if the math is rigorous. I'll note here that quantum mechanics existed for more than 40 years and was used all over the world before its basis was made mathematically rigorous (the idea of a rigged Hilbert space). Ultimately, we teach the way we do because it is best for physics, and I'll let the math teachers speak for themselves as to why they don't delve more deeply into the idea of a vector space.
EDIT: I wanted to add after reading previous responses that I learned about vectors first in high school in a mathematics class and we definitely discussed both dot and cross products. They are important concepts that we mostly expect students to understand when they step into our physics courses.
A few points:
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 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 you 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 just because we when to school in different countries.
There's no question what you're saying is correct in terms of the math, but your post is on teaching. You have yet to provide any sort of pedagogical benefit from teaching kids this perspective other than "its right mathematically, and if I can do it they should be able to too". If we taught kids in HS math in full rigor, we'd have to accept 20% as an A+.
The most important thing when educating students of the high school level is to make explanations simple and intuitive. Most high school students don't have the foundation in math required to understand your way of describing it. Keep in mind that education caters towards the lowest common denominator. Can you honestly say that the average student would be able to actually understand (not just remember the axioms) this?
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.
On February 05 2012 01:26 sukarestu wrote: 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?
I don't want to confuse you, so I'll try and answer your question while still maintaining what you have already learned. In thsi thread, we are talking about a large number of objects (be they numbers, matricies, functions, etc.) and when given certain operations (like addition) and a certain set of formal rules (the ones in the OP), these objects are said the be 'elements of a vector space' ie they are vectors. This is a much more general definition than what you will be using.
To answer your question, technically yes: the set of all real numbers along with addition and multiplication (that you learned in primary school) forms a vector space. So, as long as you define addition and multiplication in the way that you are familiar with, then yes the number 45 is an element of that vector space and is then a vector itself. You can check that this is true by taking ordinary numbers and checking that the rules listed in the OP work for those ordinary numbers.
There are 2 problems here. The first was noted by parallel universe: speed doesn't consider ALL the real numbers, just the positive ones so that messes things up. Also, The definitions listed here don't tell you about the qualitative differences between scalars like 45 and the vectors you will use in physics class like <45,0,0>. Yes, technically the set of all real numbers with addition and multiplication forms a vector space, and yes objects like velocity are also part of a (different) vector space, but this doesn't really help you understand that there are HUGE differences in how we manipulate ordinary numbers and vectors like velocity. Just because they are both 'vectors' in certain contexts doesn't meant that we can treat them the same way.
As a undergraduate currently double majoring in mathematics and physics, and going for a minor in education, I must almost completely disagree with this post :S
From a "who is correct" standpoint, yes, your definitions of vectors are better, but when you look at the practicality of teaching this concept to high school students, none of this is required. Students do not want nor need to have knowledge about spaces, space elements, fields, or what ever else is associated with introductory university-level linear algebra.
On February 05 2012 00:41 Djabanete wrote: 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.
This is exactly what I was thinking as well. Practically speaking, using a graphical representation is often the easiest way to begin demonstrating a complex concept. When you use an arrow to describe vectors, it shows students why some of its properties make sense, and why others don't. If a student asks what it means to add two vectors, you show them that adding arrows can be visualized through the "tail-to-head" method. If they wonder why you can't multiply vectors, you refer back to the arrows, since it doesn't really make sense to "multiply arrows".
Yes, this definition would be crude and perhaps unsatisfying to some students,but you would present more formal definitions for these students on request. But for most students who are just looking to make basic sense of concepts in class so they don't fall behind or get frustrated, this would be the way to go.
High school (which for me was not long ago) was where I developed my love of physics and decided that it would be my area of concentration in undergraduate studies. This was because of my teachers who made the concepts easy to understand and knew who to convey lessons at the high school level. This also fostered my desire to pursue a career in education. As long as your target audience is below the undergraduate level, you must seek the simplest and easiest definitions that accommodate their aptitude and motivation.
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?
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
I didn't use vectors that much in high school, so I can't really complain about the way they were explained. At university (computer science) I had to use vectors a loooooot during my graphics course, there I learned things like dot/cross product. Vectors are still a bit trickt for me though. (had to use them in combination with matrices).
To me, this sounds a little ridiculous, rather like teaching Dedekind cuts in ~6th grade when real numbers are introduced. (After all, what is a number? A distance on a line? An amount of things? We are told many seemingly contradictory "definitions" of numbers, just like vectors)
Vectors are usually taught first in physics, where the arrow interpretation is useful. I'm not sure many high school physics students would bother paying attention to the entire general definition of a vector space, since it would be quite hard to see how it relates to physics.
I do agree that if our aim was to get students to high-level, proof-oriented mathematics as quickly and seamlessly as possible, then a drastic re-ordering of subjects taught in math would be necessary. We teach things like trig functions, logarithms, and indeed vectors at the earliest opportunity that they are useful, often at a level where students must simply memorize properties without understanding.
On February 05 2012 00:36 Excludos wrote: 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 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?
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.
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?
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 01:26 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.
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.
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.
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 00:27 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)
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 01:26 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 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.
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.
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 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.
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.
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.
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.
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.
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.
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).
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.
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 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.
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.
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
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.
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.
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.
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.
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.
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...
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.
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.
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.
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 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 02:12 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.
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 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 02:22 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.
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 02:29 ~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?
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.
On February 05 2012 00:36 Excludos wrote: 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.
They do in my final high school. It's really not complex at all. Most difficulties people have with school-level maths are a result of being taught by people who don't realise how staggeringly easy to understand the concepts they're teaching are. Teachers who believe material is difficult pass their assumptions on to students, teachers who are completely confident that all the students will succeed have more students succeed. If you believe material is easy to understand using only basic concepts students already get, you explain the new material in terms of the old more easily and focus the examples better on what the students know. Confidence makes both student and teacher persevere for longer, and makes them more creative in their attempts to understand/teach - they expect that some explanation will work really well for the student, so they search for it.
(Integration can be introduced using playdoh* as soon as area is covered. The basic idea of it doesn't even require functions to be understood, just that there you can work out the area between a line and the edge of the playmat the same way you work out the area of a shape. If a kid knows what integration and differentiation are before they know what a function is they can learn to integrate, differentiate and graph functions all at the same time.)
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.
Exactly this. Even though now in university I've been taught these correct definions of vectors and vector spaces I'm glad they don't teach it like this in high school. There's a reason they teach stuff in easier forms in school and then build on it later. Reminds me of a story a math teacher told me once about how in the 70's (i think) someone decided that instead of teaching first graders that 1+1=2 they should teach set theory first instead. Teachers would draw something like venn diagrams with stuff like apples and cats and try to teach the first grade kids about unions and intersections. These kids are now spoken of as the lost generation.
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.
Exactly this. Even though now in university I've been taught these correct definions of vectors and vector spaces I'm glad they don't teach it like this in high school. There's a reason they teach stuff in easier forms in school and then build on it later. Reminds me of a story a math teacher told me once about how in the 70's (i think) someone decided that instead of teaching first graders that 1+1=2 they should teach set theory first instead. Teachers would draw something like venn diagrams with stuff like apples and cats and try to teach the first grade kids about unions and intersections. These kids are now spoken of as the lost generation.
And I think this really illustrates the issue. In my personal experience of trying to teach people maths, you have to start by convincing them that the things you teach them are actually useful, which often comes down to tying it closely to the real world. Once the simplified, reality-connected presentation of a subject is finished, then the student will hopefully know enough to at least suspect that they haven't seen the whole picture yet. If they want to, then is the time to give a more rigorous presentation. If not, at least they learnt something.
And to me, it was a great experience once I got that A+B doesn't have to equal B+A, and that we should appreciate commutativity when we get it. When giving seemingly trivial definitions, I always found it most instructive to look at examples which didn't fall inside of those definitions.
Meh, it's common to teach a certain example of mathematical object before the general case. It's a pedagogically sound technique. Even at university level you'd see neighbourhoods defined on real numbers instead of metric spaces or topological spaces. People who know the Euclidiean algorithm for natural numbers will usually get it faster for arbitrary rings etc.
I agree with the point about (0,0)->(2,3)=(1,1)->(3,4). I had found that part confusing. I can see how this wouldn't happen in the more abstract case but that doesn't justify losing 90% of your class in the first 3 minutes.
It's also nice to keep in mind that words like "vector" are just labels. We want students to understand the underlying concepts. For high school this is the set of vectors over R^2 and R^3 or directed line segments on 2 or 3 dimensional space. Those vectors do have a "length" and a "direction" (in a certain sense).
We can worry about relabeling when we teach or learn the more general concept.
I disagree with you OP with your view that the formal generalisation should be taught before the concrete example, if only because historically the concrete examples were manipulated first. People were dealing with points in R^n before the notion of infinite dimensional vector spaces.There was the study of permutations of the roots of a polynomial before the notion of a group (and a group action). Can you imagine being first taught about categories before having examples of concrete categories?
Lets look at what yor point of view would give to the generalization of vector spaces: ok we have magma -> monoids -> groups -> rings -> fields (each time adding more structures). The (compatible) action of a field on a commutative groupe is a vector space, the one of a ring is a module. On the other hand we can talk about topology: norm -> distance -> topology as open and closed spaces (here relaxing the structure). Ok, but we have a notion of "sets over a topological space" which is a sheaf, and then the particular notions of groups/rings/ over a topological space according to wether the sheaf takes values in a group or a ring... And then we have vector spaces over a topological space which are vector bundles. Ok so then we can combine algebra and topology which leads to the notion of scheme. But the notion of sheaf also allows to further relax the definition of a topological space which leads to Grothendieck topologies and topos. Applying that to scheme give (algebraic) stacks (so at this part the "topological part" is in the notion of gluing, and the "algebraic" part is that we can glue automorphisms roughly speaking). So in this way, we get the following generalization of vectors spaces: gerbes (which are "vector spaces that are stacks"), since they are the 2-category generalisation of principal bundles.
So with your point of view we should first introduce gerbes as in Wikipedia: A gerbe on a topological space X is a stack G of groupoids over X which is locally non-empty and transitive. I am pretty sure the only way to understand a gerbe is to first understand a vector space (and then a vector bundle) than the other way around...
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?
What difference does it make if it's 9th grade or 12th grade? Teaching the intuition before the formalism is more effective in either case.
This is kind of stupid. You say that you need university level knowledge to understand the "traditional" way of teaching it and then you give the definitions that are used in any university level linear algebra lecture, wtf?
While I prefer the formal definition there is no way that it is more didactic and it will confuse people more.
The only thing I don't like about how they teach math is the fact that they say you CAN'T do something, only to next year, teach you that 'thing'. I don't understand why they cannot just say, you will learn that later as you need to build you fundamentals up before you can tackle that.
I think it necessary to teach in small steps, that are basically right, but somewhat wrong, as they are slowly building up the the whole picture.
I didn't pick up vectors from memorizing the definition of a vector or the slew of operations on vectors, but rather by just doing a ton of problems, which slowly refined my understanding of vectors. As previously stated, I don't think your approach is any better (I think its more difficult honestly) than the traditional approach. But in all honesty you will learn vectors best by doing tons of problems and looking through tons of examples, or at least that's how I learn almost everything.
Slightly off topic (still about vectors tho), anyone know of a simple/reasonable/cute way to remember the Laplace operator for cylindrical and spherical coordinates?
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?
Didn't have much of a problem answering those 4 with only the mental picture of "scalar is a number, vector is a number with a direction" (or whatever boiled down approach you hate that teachers teach).
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?
5 is pretty easy when you think about vectors being just magnitude & direction. Same magnitude? Check. Same direction? Check. Same with 6. Now, if you want to delve into what exactly are imaginary numbers, and how to teach them, then 8 can be easy or difficult.
I'm not saying the old way is perfect, but your new way appears overly complex. Abstract concepts such as vector space and Field? Defining what a vector is using the term "vector space?" You'll get your high school student scratching his head to start. Oh yeah, I've always known what a vector space is and only wanted to know how that fits in with a vector ...
Stick to the old way, in my opinion. Your new idea, "proper" idea ... just as hard for newcomers to the concept.
I was taught vectors in less than a class in AP physics and I've never seen anyone that has any problems with it. I think vectors are a lot easier to understand if they are presented as components of distance or force. Everyone that I have seen that learned vectors in physics understands them easily, while people that waited untill maybe Calc 3 have a hard time. I'm a physics major though, so maybe I'm biased.
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.
OK, so everyone agrees that noone cares if physics is wrong in a mathematically rigorous sense. Do you have anything to back your claim that it makes no sense to students? Because the only thing I find confusing is that "vectors" and scalars are both vectors and you think teaching it as such would somehow clear things up.
Yeah I don't think you are in the right place to denounce Khan Academy or teachers in general... In physics (in particular) and chemistry we are taught certain things that just are not true, or just are not nearly in depth enough to be the full truth and this is simply because not everyone is planning on going to Uni to take Math, but you may need physics 12 or Chem 12 to graduate and go into a certain program...
The link you posted, is a good easy to learn way of thinking for specific vectors (ones normally used) and is great for a student in high school because it's right up his park.
A good example though, of how good Khan academy is would be... Well you know that guy, who made microsoft and windows? What's his name, slips my mind sadly. He not only fully endorses it, but states he uses it quite frequently.
I taught chemistry to 6th grade class for a while. Introduced atoms as being weighed in protons. AKA Hydrogen weighs 1 proton and Carbon weighs 6 protons.
Now Carbon actually weighs 12amu.
But it's easier to understand the periodic table counting 1,2,3,4 etc. I introduced neutrons later.
@Plexa - I think the OP's qualifications are a 1st or 2nd year undergrad who's just discovered higher level mathematics.
On February 05 2012 14:14 Sufficiency wrote: So basically, OP is a mathematician who does not really understand the plight of the common people who have difficulties with abstract mathematics.
@Plexa - I think the OP's qualifications are a 1st or 2nd year undergrad who's just discovered higher level mathematics.
Really? Maybe I gave him too much credit then.
This is a problem, however, since in my experience, a lot of teachers understand their subject well, but they don't know how to bring it down to the students' level. Being smart or learned does not make you qualified to be an educator, something that I don't think the OP understands at all.
On February 05 2012 14:14 Sufficiency wrote: So basically, OP is a mathematician who does not really understand the plight of the common people who have difficulties with abstract mathematics.
On February 05 2012 14:11 husniack wrote:
@Plexa - I think the OP's qualifications are a 1st or 2nd year undergrad who's just discovered higher level mathematics.
Really? Maybe I gave him too much credit then.
This is a problem, however, since in my experience, a lot of teachers understand their subject well, but they don't know how to bring it down to the students' level. Being smart or learned does not make you qualified to be an educator, something that I don't think the OP understands at all.
Abstract mathematics is simply out of the realm of most people in the world, and expecting average high school students to think abstractly, especially with vector spaces, is quite absurd.
Building from the previous author's message, I like to add that "experts in the field often forget what it feels like to be a beginner again". Something that is intuitive for an expert in a subject domain, is not necessarily intuitive for a beginner learner. Thus experts often don't remember what it is like to be a beginner... they can only guess. Thus, sometimes the "smart know-it-all" cannot teach the subject well because everything is so intuitive to the teacher, but neglect what the learner is going through.
Why bother stopping at introducing vector spaces? The conceptual difficulty in moving on to modules over a ring versus vector spaces over a field (as you described it anyway) is nil and they might as well learn the general version if they are going to learn anything. Now that I think about it, it's not that much harder to just jump right into category theory and handle additive functors from preadditive categories into the category of abelian groups instead. Of course at that point we should just handle morphisms, objects, functors, and the rest of abstract nonsense and let the students figure out how to apply it themselves. Life is so much easier when all you have to do is find the right commutative diagram!
Vectors are taught they way they are taught for a very good reason. If you don't believe me and you haven't taken category theory yet, pick up MacLane and see if you can reconstruct vector space theory in the framework of categories without appealing to your intuition from the "easier" case of modules over a field. I don't know any mathematicians who jump right into an abstract field without trying to master the examples first (that is I don't know Grothendieck) and I don't see why we should expect students to do that either.
e: As an aside, magnitude and direction generalize perfectly well to normed vector spaces. We even have a notion of angles between vectors in inner product spaces.
On February 05 2012 15:06 Lpspace wrote: Why bother stopping at introducing vector spaces? The conceptual difficulty in moving on to modules over a ring versus vector spaces over a field (as you described it anyway) is nil and they might as well learn the general version if they are going to learn anything. Now that I think about it, it's not that much harder to just jump right into category theory and handle additive functors from preadditive categories into the category of abelian groups instead. Of course at that point we should just handle morphisms, objects, functors, and the rest of abstract nonsense and let the students figure out how to apply it themselves. Life is so much easier when all you have to do is find the right commutative diagram!
I would LOVE that! But let me guess: OP hates it because it's too abstract.
On February 05 2012 15:06 Lpspace wrote: Why bother stopping at introducing vector spaces? The conceptual difficulty in moving on to modules over a ring versus vector spaces over a field (as you described it anyway) is nil and they might as well learn the general version if they are going to learn anything. Now that I think about it, it's not that much harder to just jump right into category theory and handle additive functors from preadditive categories into the category of abelian groups instead. Of course at that point we should just handle morphisms, objects, functors, and the rest of abstract nonsense and let the students figure out how to apply it themselves. Life is so much easier when all you have to do is find the right commutative diagram!
Vectors are taught they way they are taught for a very good reason. If you don't believe me and you haven't taken category theory yet, pick up MacLane and see if you can reconstruct vector space theory in the framework of categories without appealing to your intuition from the "easier" case of modules over a field. I don't know any mathematicians who jump right into an abstract field without trying to master the examples first (that is I don't know Grothendieck) and I don't see why we should expect students to do that either.
e: As an aside, magnitude and direction generalize perfectly well to normed vector spaces. We even have a notion of angles between vectors in inner product spaces.
You should only need to explain enough to understand the content and not teach something that is wrong, for that you only need to go to vector spaces. Groups, rings, modules, etc aren't necessary.
On February 05 2012 15:06 Lpspace wrote: Why bother stopping at introducing vector spaces? The conceptual difficulty in moving on to modules over a ring versus vector spaces over a field (as you described it anyway) is nil and they might as well learn the general version if they are going to learn anything. Now that I think about it, it's not that much harder to just jump right into category theory and handle additive functors from preadditive categories into the category of abelian groups instead. Of course at that point we should just handle morphisms, objects, functors, and the rest of abstract nonsense and let the students figure out how to apply it themselves. Life is so much easier when all you have to do is find the right commutative diagram!
Vectors are taught they way they are taught for a very good reason. If you don't believe me and you haven't taken category theory yet, pick up MacLane and see if you can reconstruct vector space theory in the framework of categories without appealing to your intuition from the "easier" case of modules over a field. I don't know any mathematicians who jump right into an abstract field without trying to master the examples first (that is I don't know Grothendieck) and I don't see why we should expect students to do that either.
e: As an aside, magnitude and direction generalize perfectly well to normed vector spaces. We even have a notion of angles between vectors in inner product spaces.
You should only need to explain enough to understand the content and not teach something that is wrong,
Will you stop this nonsense already? You also took for granted some old man dressed in a red suit is flying in a sledge pulled by reindeers. Did that hinder your progress in physics class later?
Agree with op The highest level classes in highschool should learn how things realy work and not learn the most easy way to get a general understanding of something, That can be left for classes at a bit lower level. There is nothing in highschool for the people who go study math and they basicly have to start all over again wich is a huge waste.. unless they tought themselves in their spare time somehow. Its good to make a program suited for the average student but the best students should be tought the right way right from the start, already at highschool.
thx for the kahn academy btw, i didnt knew that site and already loving it
On February 05 2012 15:06 Lpspace wrote: Why bother stopping at introducing vector spaces? The conceptual difficulty in moving on to modules over a ring versus vector spaces over a field (as you described it anyway) is nil and they might as well learn the general version if they are going to learn anything. Now that I think about it, it's not that much harder to just jump right into category theory and handle additive functors from preadditive categories into the category of abelian groups instead. Of course at that point we should just handle morphisms, objects, functors, and the rest of abstract nonsense and let the students figure out how to apply it themselves. Life is so much easier when all you have to do is find the right commutative diagram!
Vectors are taught they way they are taught for a very good reason. If you don't believe me and you haven't taken category theory yet, pick up MacLane and see if you can reconstruct vector space theory in the framework of categories without appealing to your intuition from the "easier" case of modules over a field. I don't know any mathematicians who jump right into an abstract field without trying to master the examples first (that is I don't know Grothendieck) and I don't see why we should expect students to do that either.
e: As an aside, magnitude and direction generalize perfectly well to normed vector spaces. We even have a notion of angles between vectors in inner product spaces.
You should only need to explain enough to understand the content and not teach something that is wrong,
LOL the irony is killing me...but seriously, none of this discussion on vector spaces are required for the application of vectors on the high school level, nor is it something that the average high school student will readily understand. You seem to just be set on your opinion despite better educators and mathematicians in this thread who show you otherwise. If you really want to teach high school math in this way, go start your own private school. But don't suggest that teachers on the whole should teach this way (who are employed in public tax-funded schools)
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.
agreed.
i just finished my engineering degree and i think that if vectors were taught by their formal mathematical definition in highschool
1) it would have confused more at the time [at least i would have resisted it. i still resist mathematical definitions.] 2) wouldn't really have applied it in later years either
[…] - A vector is a complex number, like 2+3i. […] - You can't multiply 2 vectors together, you can only multiply scalars with vectors. […]
This is one example why I think you shouldnt try to teach. If you want to explain fundamentals you have to get it right otherwise you are hurting the understanding more than you help it in the long run. I dont think any teacher would have said that.
Visualizing is for most pupils far more important than strict rules. Sure if you are good in math these rules are clearer and dont leave room for interpretation but if you abandon 80% of your class its not worth it.
Also that "but that's not the right way to teach it." is a pretty bold statement.
[…] - A vector is a complex number, like 2+3i. […] - You can't multiply 2 vectors together, you can only multiply scalars with vectors. […]
This is one example why I think you shouldnt try to teach. If you want to explain fundamentals you have to get it right otherwise you are hurting the understanding more than you help it in the long run.
No shit. That's the whole point of this post.
I dont think any teacher would have said that.
I know teachers who would say that. Complex numbers are offten introduced as vectors.
The very first picture depicts a complex number as a vector.
Yet, students get it drilled into them that you cannot multiply 2 vectors (which is correct), but then why can you multiply two complex numbers if they are vectors? The answer requires an understanding of vector spaces, and that when complex numbers are presented as a vector over the reals, there is no vector multiplication, so complex multiplication isn't vector multiplication. But when complex numbers are presented as a field, the field multiplication is complex multiplication.
Visualizing is for most pupils far more important than strict rules. Sure if you are good in math these rules are more clear and dont leave room for interpretation but if you abandon 80% of your class its not worth it.
As I said, the visual representation of vectors naturally arises from the abstract definition. Define a vector space, then show that R^2 is a vector space, next show that points in R^2 are vectors by definition, finally show that points in a number plane can be represented as arrows.
On February 05 2012 00:36 Excludos wrote: 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.
I giggled at that, whether the pun was intended or not.
hi, did a phd in theoretical physics, and taught 4 years at 2:nd year physics where they learned the "proper" definition of a linear space. I largely agree with See.Blue and the others (did read every post really, just first few pages, sorry) that high school student just are not ready for the formal definition.
Tbh, most of my 2:nd year physics student were already struggling to truly understand this more abstract definition... I would write down the definitions and show how they make sense in the "arrow representation" they were used to understand linear spaces in, and most were fine with that. But I know many of the students never really understood how a function is a vector, or how an integral of a product of two functions is a scalar product. I don't think that is because I am a bad teacher, or because most of my student were stupid, but because this is a damn hard concept to understand when you are not used to this kind of abstract thinking.
I agree that much can be done better in math (and physics) education at all levels (micro probably knows more about that), but it is not as easy as starting to use very formal mathematics already in high school.
The underlying question is: should math at (high) school be taught more akin to what is currently taught in the university or should it be more like what it is now. Personally, I'd give the first a try, for there is no beauty in whatever is currently done.
On February 05 2012 14:11 husniack wrote: I taught chemistry to 6th grade class for a while. Introduced atoms as being weighed in protons. AKA Hydrogen weighs 1 proton and Carbon weighs 6 protons.
Now Carbon actually weighs 12amu.
But it's easier to understand the periodic table counting 1,2,3,4 etc. I introduced neutrons later.
@Plexa - I think the OP's qualifications are a 1st or 2nd year undergrad who's just discovered higher level mathematics.
I've seen him post stuff about Bayesian interference so I assume he has done more than that
I am currently taking Calc B in my college freshman year, and I've taken regular math in highschool with only introduction to vectors. So I am what you would call a stereotypical -meh- math student that would be learning vectors.
I read through your explanation of vectors 3 times and I have not a single fucking clue what its saying.
When I was taught the arrow definition I understood the general concept pretty fast... however I have no idea what your explanation of vectors is saying at all.
On February 05 2012 19:11 Jombozeus wrote: I am currently taking Calc B in my college freshman year, and I've taken regular math in highschool with only introduction to vectors. So I am what you would call a stereotypical -meh- math student that would be learning vectors.
I read through your explanation of vectors 3 times and I have not a single fucking clue what its saying.
Which I guess proves you wrong.
Physics vectors and real numbers 17, (1,2) or (3,4,5) are just a special case of vectors called n-tuples with n being the dimension. They are the vectors of vector spaces R, R^2, R^3 which are called euclidean vector spaces after the famous mathematician Euclid who developed the geometry in these spaces in his book "Elements".
Vector spaces are just sets of elements like the real numbers with additional structure, like how to combine them etc(that big list of things he wrote).
These elements can also be functions, complex numbers and polynomials etc so long as they obey those properties he wrote. And because they do obey those property's it gives rise to a lot of new property's that can be derived like the inner product etc.
And just like sets you can have subsets(smaller sets that are contained in the original set) which when they obey these property's they are subspaces
Linear algebra is just the study of these Vector spaces and their subspaces and mappings between vector spaces which preserve linearity(maps a line to a line).
But Im curious if vectors are just a further special case of tensors?, only In first year so I havent come across them properly
when i see average high school students being taught the way as outlined as OP and the majority of them understanding it quickly, i'll agree with the OP. otherwise not.
This is one example why I think you shouldnt try to teach. If you want to explain fundamentals you have to get it right otherwise you are hurting the understanding more than you help it in the long run.
No shit. That's the whole point of this post.
The very first picture depicts a complex number as a vector.
Yet, students get it drilled into them that you cannot multiply 2 vectors (which is correct), but then why can you multiply two complex numbers if they are vectors? The answer requires an understanding of vector spaces, and that when complex numbers are presented as a vector over the reals, there is no vector multiplication, so complex multiplication isn't vector multiplication. But when complex numbers are presented as a field, the field multiplication is complex multiplication.
Well if that's the point, here's an example issue from your posts. The reason you can multiply two complex "vectors" together is not that the complex numbers form a field. It is because the complex numbers form an algebra over the reals. There are plenty of other interesting vector spaces which you can extend to algebras but do not have a field structure. For example, look at the multiplication defined for lie algebras which gives rise to a whole lot of "zero divisors".
You should only need to explain enough to understand the content and not teach something that is wrong
Physics teaches the difference between scalars and vectors. That can't be done by using a mathematically rigorous definition of a vector.
Complex numbers are often introduced as vectors.
Introduced as the same special case R^2 vector used in physics. It is almost like teachers got together and agreed upon how vectors should be taught outside of university level linear algebra courses.
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)
