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## Main Question or Discussion Point

I have posted this here for two reasons, one I think you all will really appreciate this

as a teaching tool to hopefully increase your students understanding (and hopefully

grades) & second because you all might find/make more concept maps on more

advanced topics or with focus shifted in a different direction (i.e. a concept map

of theorems for example!). Please if you ever come across or make anything of the sort

please share it here & we'll hopefully amass quite a collection. References to books

containing concept maps more than welcome too!

I have become convinced of the power of concept maps as a way to

truly understand a subject. Please allow me to convince you with this

opening post then as payment for this miraculous gift I am giving you

I merely request you make/give/find me concept maps on areas of

mathematics such as:

Here's an elementary image of a concept map in real analysis:

Not very convincing is it? Not very powerful or illuminating, I mean you

can look in the cover of your book & see this & it's probably the image of

a concept map a lot of us would think of, something not very informative.

However, look how powerful a concept map can become!

That is the kind of knowledge it would take over 100 pages, and some

serious thought, to acquire & to be honest diff eq's aren't fun because

this is precisely the knowledge you gain after all that hard work! The

fun should be devoted to the theorems holding differential equations

together (You know when you read a book & some author will put in brackets the word

"proof?" after a statement sometimes, I do it with "concept map?"). So to repeat, the

fun should be devoted to the theorems holding differential equations

together (concept map?).

If that one didn't convince you it's probably because there was some

detail missing in it, here's a more complete version:

approach the subject knowing that this is the general form you're striving

for with 1st order ode's that are solvable without recourse to numerical

methods you'd be pretty confident!

I took those pictures from this pdf file, there are more concept maps in it

on differential equations. Looks just fantastic!

Still, these concept maps illustrate the complexity of the subjects you're

trying to master, it's bad enough meeting all these new concepts in one

go but you also have to make sense of how they all fit together & how

they interconnect. Here is a drastic illustration of the complex web of

interconnectedness a subject (Linear Algebra) can contain:

(Download that picture or use firefox zoom or something to examine

that beast! taken from here). Still, I've actually learned from just looking at that

picture! Made sense instantly!

Still, a theorem-interconnectedness concept map would be even more

useful & revealing! For example in my naivety I didn't fully understand

how a subspace of a vector space can be validated just by using λa + b,

I mean there's a nice way to build up to that by proving, I think, two or

three previous theorems.

Now, I want to emphasize the main reason why I think this is so important is because

it illuminates why a book is structured the way it is. In electromagnetism, a subject I

have yet to study properly, I have been slowly finding out about it but a lot of it is in

the dark & I don't have the time to do it properly. Still looking at the following concept

map

and more of it makes sense now (that Gauss & Faraday are for electrostatics &

Amere is for magnetostatics, taken from here). This is tiny but that's because the concept map here

contains nothing compared to the differential equations map above! I'd make more

connections if the map was better! It would just clarify why the book is structured the

way it is, I know nothing substantial about E&M, nothing, but if I can get some grasp

on where I'm going as regards concepts it would help. This is universally true, in

advanced mathematics especially! Something like this:

contains info on where you're going in elementary mechanics, but I mean concept maps

on moment of inertia in various situations, or friction situations etc... could be made

too! That's the stuff people have trouble with & for which a map would help make

sense out of!

Hopefully I've made my case, I'd be interested in opinions on this, here are a few of

the links I've found that contain useful info & maps that some people might appreciate:

http://www.merga.net.au/documents/RP562006.pdf

http://cmc.ihmc.us/papers/cmc2004-271.pdf

link

http://www.informaworld.com/smpp/section?content=a916574853&fulltext=713240928#references

http://www.mtedu.tmue.edu.tw/leeys_teaching/%E7%A0%94%E8%A8%8E%E6%9C%83/PME/PME32%282008%29/Volumen%204/Research_Reports,_vol._4-36.pdf [Broken]

The idea of concept maps within concept maps within concept maps

within a locker in the film Men in Black isn't such a weird thought for a

person to have anymore, is it?

as a teaching tool to hopefully increase your students understanding (and hopefully

grades) & second because you all might find/make more concept maps on more

advanced topics or with focus shifted in a different direction (i.e. a concept map

of theorems for example!). Please if you ever come across or make anything of the sort

please share it here & we'll hopefully amass quite a collection. References to books

containing concept maps more than welcome too!

I have become convinced of the power of concept maps as a way to

truly understand a subject. Please allow me to convince you with this

opening post then as payment for this miraculous gift I am giving you

I merely request you make/give/find me concept maps on areas of

mathematics such as:

------------------metric spaces, elementary fixed point theory, partial differential equations, elementary differential geometry, modern diff geom, measure theory & Lebesgue integration, modern geometry, elementary probability & statistics, modern probability & statistics, lie theory, abstract algebra, topology,

differential topology, algebraic topology, algebraic geometry, whatever else!!!!

Here's an elementary image of a concept map in real analysis:

Not very convincing is it? Not very powerful or illuminating, I mean you

can look in the cover of your book & see this & it's probably the image of

a concept map a lot of us would think of, something not very informative.

However, look how powerful a concept map can become!

That is the kind of knowledge it would take over 100 pages, and some

serious thought, to acquire & to be honest diff eq's aren't fun because

this is precisely the knowledge you gain after all that hard work! The

fun should be devoted to the theorems holding differential equations

together (You know when you read a book & some author will put in brackets the word

"proof?" after a statement sometimes, I do it with "concept map?"). So to repeat, the

fun should be devoted to the theorems holding differential equations

together (concept map?).

If that one didn't convince you it's probably because there was some

detail missing in it, here's a more complete version:

Now, there may be a few bits and bobs missing, still if you were able to

approach the subject knowing that this is the general form you're striving

for with 1st order ode's that are solvable without recourse to numerical

methods you'd be pretty confident!

I took those pictures from this pdf file, there are more concept maps in it

on differential equations. Looks just fantastic!

Still, these concept maps illustrate the complexity of the subjects you're

trying to master, it's bad enough meeting all these new concepts in one

go but you also have to make sense of how they all fit together & how

they interconnect. Here is a drastic illustration of the complex web of

interconnectedness a subject (Linear Algebra) can contain:

(Download that picture or use firefox zoom or something to examine

that beast! taken from here). Still, I've actually learned from just looking at that

picture! Made sense instantly!

Still, a theorem-interconnectedness concept map would be even more

useful & revealing! For example in my naivety I didn't fully understand

how a subspace of a vector space can be validated just by using λa + b,

I mean there's a nice way to build up to that by proving, I think, two or

three previous theorems.

Now, I want to emphasize the main reason why I think this is so important is because

it illuminates why a book is structured the way it is. In electromagnetism, a subject I

have yet to study properly, I have been slowly finding out about it but a lot of it is in

the dark & I don't have the time to do it properly. Still looking at the following concept

map

and more of it makes sense now (that Gauss & Faraday are for electrostatics &

Amere is for magnetostatics, taken from here). This is tiny but that's because the concept map here

contains nothing compared to the differential equations map above! I'd make more

connections if the map was better! It would just clarify why the book is structured the

way it is, I know nothing substantial about E&M, nothing, but if I can get some grasp

on where I'm going as regards concepts it would help. This is universally true, in

advanced mathematics especially! Something like this:

contains info on where you're going in elementary mechanics, but I mean concept maps

on moment of inertia in various situations, or friction situations etc... could be made

too! That's the stuff people have trouble with & for which a map would help make

sense out of!

Hopefully I've made my case, I'd be interested in opinions on this, here are a few of

the links I've found that contain useful info & maps that some people might appreciate:

http://www.merga.net.au/documents/RP562006.pdf

http://cmc.ihmc.us/papers/cmc2004-271.pdf

link

http://www.informaworld.com/smpp/section?content=a916574853&fulltext=713240928#references

http://www.mtedu.tmue.edu.tw/leeys_teaching/%E7%A0%94%E8%A8%8E%E6%9C%83/PME/PME32%282008%29/Volumen%204/Research_Reports,_vol._4-36.pdf [Broken]

The idea of concept maps within concept maps within concept maps

within a locker in the film Men in Black isn't such a weird thought for a

person to have anymore, is it?

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