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On February 05 2012 14:44 jaerak wrote:Show nested quote +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.
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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.
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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.
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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.
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On February 05 2012 15:43 paralleluniverse wrote:Show nested quote +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?
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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
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On February 05 2012 15:43 paralleluniverse wrote:Show nested quote +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)
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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
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[…]
- 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.
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On February 05 2012 17:27 Namenlos wrote:Show nested quote +[…]
- 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.
http://en.wikipedia.org/wiki/Complex_numbers
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.
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TL;DR OP did a course in linear algebra.
All the high school definitions are enough for a basic understanding of vectors.
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I love this guy, SO helpful.
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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.
I giggled at that, whether the pun was intended or not.
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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.
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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.
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Aotearoa39261 Posts
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 
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PatrickJMT is a much better math teacher than Sal form KhanAcademy in my opinion.
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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.
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I'm glad I'm doing aerospace engineering so I mostly have to worry about how to apply the math and not all theory behind it .
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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.
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EPT
