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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.)
*Aimed at lower-mid years in elementary school.
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Hey how can I get in on this math tournament? What's first prize?
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I love Khan academy, dat guy man, day guy.
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Are vectors not just a certain rank of tensor also, anyone mind explaining dual spaces to me?
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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.
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On February 05 2012 03:43 Juisson wrote:Show nested quote +On February 05 2012 00:27 Flameberger wrote:
I've taken university level math, but I didn't have any trouble with vectors in high school. In fact as far as I could tell the entire class understood and applied them without any difficulty.
Yes the way it is taught is not strictly accurate math-wise, but do you honestly think
1. u + (v + w) = (u + v) + w
2. u + v = v + u
3. There exists an element 0 in V, such that v + 0 = v for all v in V.
4. For every v in V, there exists an element −v in V, such that v + (−v) = 0
5. a(u + v) = au + av
6. (a + b)v = av + bv
7. a(bv) = (ab)v
8. 1v = v,
9. u+v is in V
10. av is in V
where v, u, w are any elements in V, and a is any element in F.
is something highschool students are going to look at and think: "oh yeah, that makes a lot more sense", what are you even supposed to do with that? memorize it?
The way I learned vectors has served me well in my various physics courses since high school, when I needed a proper mathmatic understanding of them in Calculus III it was very easy for me to adapt my current understanding to the proper definitions. 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.
http://en.wikipedia.org/wiki/New_Math
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.
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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.
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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...
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On February 05 2012 03:01 ~OpZ~ wrote:Show nested quote +On February 05 2012 02:36 iSTime wrote: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.
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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.
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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.
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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?
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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.
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On February 05 2012 00:17 mechanix wrote:
what do you get when you cross a mosquito with a mountain climber?
nothing, you cant cross a scalar with a vector
:o
LMAO. Nomination for Best TL.net Post of 2012.
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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.
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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.
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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.
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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.
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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.
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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. Show nested quote +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.
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EPT
