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On February 05 2012 00:37 paralleluniverse wrote:Show nested quote +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.
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On February 05 2012 00:41 NecroSaint wrote:Show nested quote +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".
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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.
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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 : Show nested quote +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.
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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.
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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.
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On February 05 2012 00:56 paralleluniverse wrote:Show nested quote +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 ... ?
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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.
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On February 05 2012 00:41 See.Blue wrote:Show nested quote +On February 05 2012 00:37 paralleluniverse wrote: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.
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Vector reminds me of Despicable me and that reminds me of my undying question:
Why the fuck aren't there any movie about Minions?
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Aotearoa39261 Posts
The statements about vectors made in HS are consistent with the definition presented - what is your problem?
EDIT: oh and the vectors = arrows thing is mostly for engineering
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On February 05 2012 01:01 Apom wrote:Show nested quote +On February 05 2012 00:56 paralleluniverse wrote:On February 05 2012 00:51 Apom wrote:I agree with the person who said that your definitions are vastly overcomplicated for high school needs. Actually I don't recall ever hearing a definition of a vector before reading this thread, which didn't prevent me from being somewhat good at math. Good enough to notice the following, at least : What is the field F? Basically any set of numbers you know is a field, such as the integers, the rational numbers, the real numbers, or the complex numbers. A full definition of a field is given below. Integers are not a field, they are a ring. The definition you give in spoilers even proves it. That is a sloppy mistake, and not how you want to start your explanation. Yeah you're right. But I find it hard to believe that you know about rings before knowing about vector spaces. I learned them in the order monoïd > ring > field > vector space > algebra. In fact I don't see how any other order would make sense, since each definition builds on top of the previous one ... ?
So you took an abstract algebra course before a linear algebra course? Not that there's anything wrong with it, it just seems a bit rare.
While vector spaces can be defined in terms of groups or rings, it's possible to correctly define a vector spaces without it, while doing so adds almost nothing to the content of a standard 1st or 2nd year linear algebra course. This is different from defining vectors without the notion of a vector space, because of it's incorrectness and the confusion it gives high school students.
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On February 05 2012 00:41 NecroSaint wrote:Show nested quote +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.
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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.
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United States24700 Posts
Just to spite you OP I'm showing this video to my physics class.
Teaching something in its most general form isn't always practical and often only helps a small portion of a group of students.
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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.
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Aotearoa39261 Posts
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.
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On February 05 2012 01:19 Plexa wrote:Show nested quote +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.
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TBH you can't expect high school students (most of which are NOT going into eng/math/etc) to eat down formal definition of vector spaces.
Countries want their students to pass high school.
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Aotearoa39261 Posts
On February 05 2012 01:22 paralleluniverse wrote:Show nested quote +On February 05 2012 01:19 Plexa wrote: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.
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EPT
