Well, RTFA! 😉 I think David was not very careful in his post, so “teaching of the facts” should really be “teaching only the facts”. Some people think that if you teach students their multiplication tables, you have taught them mathematics. No, you haven’t. You have taught them facts about mathematics, nothing more. If you learn the dates of historical events, some people will say you have studied history. No, you haven’t.
The way technology factors in is that I no longer need to spend them learning the dates of historical events, or what 8 times 9 is. These bits of knowledge are obselete. You need to go a step above. You have to move into “using the facts” because the facts alone have no value. This implies you should be “familiar” with the facts, but you don’t want to spend them “memorizing” the facts.
I really don’t give a damn when America was discovered. I can find out in seconds using Google and so can any kid. Don’t waste their time with it. But do teach them why and how it was discovered, and what “discover” means, and why it wasn’t “discovered” before. Better yet, teach them how they can figure out this knowledge from Google.
Bartsays:
Warning: I did not RTFA. I am only reacting to the statement that one would no longer have to learn(=getting to know) facts.
A good reason for doubting it the way it is formulated: do you think calculus students are equally good at solving problems if they have to use their calculator every time they need to graph of exp(x) then they are when they just know the basic shape of this function?
So they need to know the facts about the shape of exp(x).
Conclusion: you need to be a lot more careful about which facts you no longer have to know, now that you can look it up easy.
(a second, maybe more difficult problem, how do you know certain facts are out there without having an idea what they are. How do you get this idea without “learning” the facts?)
dsays:
Bart’s comment about students not knowing the shapes of certain curves, reminded me of of a talk by V I Arnold on the necessity of keeping physics and math tied together. I nshort the tralk addresses the issue of students who don’t know how to interpret or represent a problem geometrically. My favorite part :” To the question “what is 2 + 3” a French primary school pupil replied: “3 + 2, since addition is commutative”. He did not know what the sum was equal to and could not even understand what he was asked about!”
Bartsays:
I have now RTFA. In response to your comment:
I agree that memorizing the facts about when America was discovered doesn’t really seem to do a lot of good. There is a problem with reliance on the internet for this information though. Have you tried the google search? I did, trying not to use any of the facts I know, and what I got would be difficult to interpret without knowing the things I do. (*)
I agree dealing with this only requires familiarity with the facts, but how do you ensure familiarity with the facts that lasts? I only know one way to ensure this — memorize at some point in time, and familiarity will last until some other point in time. (**) (***)
It seems that no longer having to learn these facts at any point in your live requires something _much_ more than having the network be very quick and always be at your finger tips. I am not sure what this much more is exactly, but I don’t believe we have seen any of it yet.
One of the forms this could take are maybe advanced information literacy skills (mentioned in the article). But being able to decide who and when discovered America after some research by using these new skills (which I don’t really know what they are if not a general knowledge of facts) negates the whole argument that the network makes not knowing the facts ok. Yes you can find “facts” quickly, but then deciding which of the “facts” are facts takes time.
Concluding: I agree that technology like the web has a distinct and direct influence on what people have to know and how they have to know this (and that this influence is big). Stating this in any way close to “there is (or will be) no more need to know/learn a lot of facts” is silly. Much more is needed.
(*)
I don’t know much about what historians think of who discovered America when (how debatable this is, if the recieved opinions on this changed since I went to high-school. But I think I have a good idea), but from the google search it would seem there is barely consensus, and it certainly is not clear what the recieved opinion on the matter is.
(**)
This is from a teachers point of view, you have to ensure familiarity with facts until late in life. Memorizing (which is something which can also be tested) is a proven technique, I don’t know of others.
(***)
In the case about discovering America. It is the case you don’t need to remember the year Columbus discovered the us. But knowing Columbus did and having some idea of when, do seem to be important facts. Realizing Columbus (or whomever for that matter) discovered America before the discovery of phones and cars seems something anyone should know and feel.
Bartsays:
In your first paragraph you are confusing two issues:
1) not getting the truth while learning something.
2) needing to learn something.
And part of this is exactly what I was getting at in an earlier comment.
In the second you are changing the subject. The initial reasoning/statement I objected to was: with the pervasiveness and speed of the web there is no more need to know facts b/c you can find them quickly. This was also what you were defending in your first reply to my first comment. There you said one should be familiar with the facts but don’t need to memorize them.
I agree with this to a large extend, but this does not mean that memorization does not have to happen. I think memorization is the only good and reliable way to become familiar with the facts (and for teachers to ensure this familiarity).
In this second paragraph of yours you are reasoning from properties about the fact about when America was discovered — a fact where I agree with you is unlikely to be of tremendous importance in most peoples lives — to properties of all facts. There are many facts, possibly a different set of facts for different people, that you really need quick access to. I take this to be clear. For those facts you can not wave off the difficulties with finding them out through the internet in a similar way you can with the date of discovery of America.
But still I think there are good arguments that it is good to know this date, together will all the things you mention there which I agree are more important.
Now back to the first paragraph. Think of a fact as I mentioned above, one of the ones you require quick access to. How do you get to this state of having quick access to it? Well, you need to learn the fact at some point. This time of learning is the right time to wonder about what discover (as in discovering America) means, and how reliable the information you get is.
So what you get by learning the fact is a separation of issues: at the time of learning you have the time to wonder about the fact, to what amount it is a fact, what the fact means, etc. And then when you need it you can use it. So you get to separate finding which “facts” are facts from using the facts (see my reply numbered 4 paragraph just before the concluding one). One of the main points of education I think.
I think memorization is the only good and reliable way to become familiar with the facts (and for teachers to ensure this familiarity).
This is the entire foundation of our current schools. You’ve been trained this way and naturally, you will say that this is only good and reliable process.
Well, I disagree. I never, ever, memorize anything. If you don’t believe me, ask people who know me. I don’t know my phone number. In fact, in kindergarden, we had to learn our phone numbers and I refused to learn it (I was 5 years old). I really do not know my phone number, I don’t know the number of my office at the university, and so on. Let me be clear: there is no evidence that my brain is damaged in some way (though surely, I’m insane in many ways), I simply always refused to “memorize” because I think it is degrading to be trained like a computer. Of course, I didn’t think this through when I was 5 years old… but now, looking back at my young self, I think I was onto something important.
Strangely enough, despite the fact that memorisation is the only reliable way of learning, in almost all things at school, I got very good grades and my academic life has been mostly without trouble. In my work life, I have really no problem being productive. Everything I need to know can be found on the Web or my PDA. I keep nothing in memory except my general understanding and some problems I work on. Everything else is in some form of electronic memory.
Asking students to memorize things is easy and convenient for the teachers, I still argue, despite your arguments, that we need to move beyond it… no, I’ll go further and say we’ll have to move beyond it… because soon, students will be accutely aware that finding the “capital of Uganda” can be found in a less than a second using Google, the “date of birth of Alexander Graham Bell” (try searching the quoted text, and you’ll see, you get the answer).
Bartsays:
I agree that my formulation “memorization is the only good and reliable way to become familiar with the facts” is poor and does not clearly describe what I mean.
Let me give an example of what I am thinking about. Imagine you are a logician working in Set Theory. One of the subjects you are working on is a special type of group. During your studies of this group you find that there is a certain freeness in the construction. Suddenly the fact that subgroups of a free group are free becomes very important.
I believe I had this fact available to me b/c at some point it was firmly embedded into my brain (memorized). What was actually memorized was the fact that there are transformation on a generating set that keep the set generating and allow to transform it to a basis.
I would like to believe that this was a product of my general understanding (deep insight) into free groups, but really it was just a fact lying in my brain. I don’t think I could work without these type of facts available to me.
This fact and its use are of different character than the capital of Uganda of the date of birth of Alexander Graham Bell. These seem not very required in the first place, but I think you would be missing something if you didn’t know which continent Uganda lies on, or which century Bell lived in.
(
Does one really not have to know that Japan and China are countries that are close to each-other and that France and Germany border? Still, there are the facts (like the subgroup theorem) which are of different nature and have to be known.
I believe there are in fact more important facts to know, but I have difficulty figuring out which they are as I think these are very deeply embedded.
)
Now if I had learned the fact about subgroups in a course I would expect the teacher to have me do the exam where I could need this fact without a book. That would mean that I would have to have had this fact memorized at the time of the exam. This memorization (having readily available of certain facts) I think is essential. Reason being that the teacher is trying to help me do my work later on in live. So he/she has to ensure that if I am in the situation I am in now, I have the fact available to me.
This I believe is exactly the same as the fact about the shape of exp(x) I mentioned in post number 1.
BTW: I am not in disagreement with you on the fact that there are serious issues with current education. And that facts of certain types play too big a role also seems somewhat clear. I hope the above makes what I meant with my “Concluding” in post number 4 clearer.
Yes, it does make things clearer, but allow me to still disagree (though, of course, I don’t claim to be right, I only argue my point because I have yet to see a killer argument on the other side of the fence).
I’m a mathematician, but I don’t recall what a free group is… hell, I can’t even remember the axioms of a group… I honestly think I never learned what a free group is.
Ok, so I go to planetmath.org/encyclopedia/FreeGroup.html. Ah you say! You knew about planetmath.org… well, no, I didn’t… planetmath.org is mentionned on my blog so I researched it (see the “search” textbox on the right?) So far, no memorisation.
Ok, then I read toward the end of the page that “The Nielsen-Schreier Theorem states that every subgroup of a free group is itself free.” I actually remember now that I once studied what free groups are.
In any case, from your post, and in a few seconds (not minutes!!!) I was able to access probably as much information as you have in your brain now about free groups (less if you are an expert).
This doesn’t make me an expert on free groups, not by a long shot. But I now “know” all of the basic facts about free groups, right there.
Are you suggesting I now memorize these facts? For what purpose? I will surely forget them soon after I memorize them because in my daily life, I don’t have to deal with them.
Oh! BTW, I have never memorized what a group is… just wait a second while I click on the link there on the page… associativity, closure, identity, and inverses over one operation… ah ah, ok.
You know what I think is really important??? What actually matters? Is that now that I have these facts handy, I could actually solve problems on free groups. If I have the facts, but not the competence to solve problems with them, the facts are useless because anyone can get to them nearly as quickly as I can.
So, what is important? Problem solving skills? Creative thinking? Or memorizing facts?
The basic problem here is: if you are taking a course on group theory and you are to solve a problem you know that the facts you need are in the book somewhere. In real live you never know which fact you’ll need when. Just being able to look things up doesn’t seem to work out in recognizing when something should be looked up (I think the killer argument you are looking for is in this sentence).
Doesn’t ring like a killer argument to me. I took group theory several times, I never memorized the axioms of a group, and they are already forgotten as we speak. I argue it is not a weakness. The fact is that there is just too much to know out there… Just waaaayyyy too much. So, if you actually need to memorize things so that later you can come back to them if you need them, you’ll be memorizing and memorizing and memorizing… I’m sorry, but nobody has enough memory to be able to absorb all of this.
I argue that the brain mostly work by pattern recognition. You become familiar with a pattern, then another one, then another one… then one day, you encounter a pattern that seems familiar… you’ll naturally go back, dig a bit, and voilà! you’ll find the missing pattern…
I think that’s how it works. You don’t actually need to “know” what exp(1) is. You need to “know” the pattern of exponential growth… but you don’t acquire this sort of intuition by memorizing things, but by “working it out”. If you spend enough time in a pattern, your brain will become familiar with it.
I’ve done enough algebra to know how to prove algebra results and if I need to teach a group theory course tomorrow, I’ll do just fine. I’ll do fine because I know most important patterns because I did a lot of algebra problems, some quite hard.
If you live in algebra land, then yes, you need always have in mind the exact axioms and definitions, and if in 6 months, I start working in algebra, I’ll know these axioms because I’ll use them often.
But I argue that at no point should you memorize these axioms… do the problem solving part… solve very hard problems in the field… that’s what you should spend you time doing, and without knowing it, you’ll go beyond the axioms…
So I want the students to go beyond “what exp(x) looks” like and move into “why should I care it looks like this”…
It seems that a solution to this problem is to at some point spend the time studying free groups and proofs of theorems about them so that you know their truth, then to have them in memory, and then when you need them you don’t need to spend the time verifying.
One word: not doable. You can do it with algebra, and if algebra is all you do, you’ll be fine… but most people are not so narrow. The guy who build bridges can’t also know the axioms of group theory at all time. If he knows this, he better also know how to speak Chinese, Spanish and basic aerospace engineering.
But if he has learned how we solve algebra problem, he might, one day, have a clever geometry problem and he might know a way to solve it that nobody else will… and if he needs to look up the math., he will do it without a problem. He merely needs to have gained intuition, to have internalized the patterns….
Bartsays:
Off course I allow you to disagree, more strongly, I like it better that you do. This exchange has made me think about this much more careful than I otherwise would have. And it seems to me we are converging to the place in my mind where the action is. (b.t.w. as answer to your question, I’d say problem solving skills, creative thinking and memorizing facts (by which I mean what was mentioned in earlier posts)).
Reason I think some facts need to be readily accessible in memory is that without the notion of free group, knowing that a lot is known about them and knowing some of those things, I would never have gotten to the point of using planetmath.org or something similar looking more facts up.
This is similar (if not identical) to a calculus student having to solve a word problem and not getting anywhere b/c they don’t realize that exp(x) has a certain shape that suggests what to do next.
The basic problem here is: if you are taking a course on group theory and you are to solve a problem you know that the facts you need are in the book somewhere. In real live you never know which fact you’ll need when. Just being able to look things up doesn’t seem to work out in recognizing when something should be looked up (I think the killer argument you are looking for is in this sentence).
A second point I want to bring up here (easier to solve admittedly): some random person put up the item about free groups at planetmath. How do you know this is reliable? It seems that a solution to this problem is to at some point spend the time studying free groups and proofs of theorems about them so that you know their truth, then to have them in memory, and then when you need them you don’t need to spend the time verifying.
(easy but as of yet nonexistent solution: verified database of such facts. Still far away though. And doesn’t solve the first problem.)
Well, RTFA! 😉 I think David was not very careful in his post, so “teaching of the facts” should really be “teaching only the facts”. Some people think that if you teach students their multiplication tables, you have taught them mathematics. No, you haven’t. You have taught them facts about mathematics, nothing more. If you learn the dates of historical events, some people will say you have studied history. No, you haven’t.
The way technology factors in is that I no longer need to spend them learning the dates of historical events, or what 8 times 9 is. These bits of knowledge are obselete. You need to go a step above. You have to move into “using the facts” because the facts alone have no value. This implies you should be “familiar” with the facts, but you don’t want to spend them “memorizing” the facts.
I really don’t give a damn when America was discovered. I can find out in seconds using Google and so can any kid. Don’t waste their time with it. But do teach them why and how it was discovered, and what “discover” means, and why it wasn’t “discovered” before. Better yet, teach them how they can figure out this knowledge from Google.
Warning: I did not RTFA. I am only reacting to the statement that one would no longer have to learn(=getting to know) facts.
A good reason for doubting it the way it is formulated: do you think calculus students are equally good at solving problems if they have to use their calculator every time they need to graph of exp(x) then they are when they just know the basic shape of this function?
So they need to know the facts about the shape of exp(x).
Conclusion: you need to be a lot more careful about which facts you no longer have to know, now that you can look it up easy.
(a second, maybe more difficult problem, how do you know certain facts are out there without having an idea what they are. How do you get this idea without “learning” the facts?)
Bart’s comment about students not knowing the shapes of certain curves, reminded me of of a talk by V I Arnold on the necessity of keeping physics and math tied together. I nshort the tralk addresses the issue of students who don’t know how to interpret or represent a problem geometrically. My favorite part :” To the question “what is 2 + 3” a French primary school pupil replied: “3 + 2, since addition is commutative”. He did not know what the sum was equal to and could not even understand what he was asked about!”
I have now RTFA. In response to your comment:
I agree that memorizing the facts about when America was discovered doesn’t really seem to do a lot of good. There is a problem with reliance on the internet for this information though. Have you tried the google search? I did, trying not to use any of the facts I know, and what I got would be difficult to interpret without knowing the things I do. (*)
I agree dealing with this only requires familiarity with the facts, but how do you ensure familiarity with the facts that lasts? I only know one way to ensure this — memorize at some point in time, and familiarity will last until some other point in time. (**) (***)
It seems that no longer having to learn these facts at any point in your live requires something _much_ more than having the network be very quick and always be at your finger tips. I am not sure what this much more is exactly, but I don’t believe we have seen any of it yet.
One of the forms this could take are maybe advanced information literacy skills (mentioned in the article). But being able to decide who and when discovered America after some research by using these new skills (which I don’t really know what they are if not a general knowledge of facts) negates the whole argument that the network makes not knowing the facts ok. Yes you can find “facts” quickly, but then deciding which of the “facts” are facts takes time.
Concluding: I agree that technology like the web has a distinct and direct influence on what people have to know and how they have to know this (and that this influence is big). Stating this in any way close to “there is (or will be) no more need to know/learn a lot of facts” is silly. Much more is needed.
(*)
I don’t know much about what historians think of who discovered America when (how debatable this is, if the recieved opinions on this changed since I went to high-school. But I think I have a good idea), but from the google search it would seem there is barely consensus, and it certainly is not clear what the recieved opinion on the matter is.
(**)
This is from a teachers point of view, you have to ensure familiarity with facts until late in life. Memorizing (which is something which can also be tested) is a proven technique, I don’t know of others.
(***)
In the case about discovering America. It is the case you don’t need to remember the year Columbus discovered the us. But knowing Columbus did and having some idea of when, do seem to be important facts. Realizing Columbus (or whomever for that matter) discovered America before the discovery of phones and cars seems something anyone should know and feel.
In your first paragraph you are confusing two issues:
1) not getting the truth while learning something.
2) needing to learn something.
And part of this is exactly what I was getting at in an earlier comment.
In the second you are changing the subject. The initial reasoning/statement I objected to was: with the pervasiveness and speed of the web there is no more need to know facts b/c you can find them quickly. This was also what you were defending in your first reply to my first comment. There you said one should be familiar with the facts but don’t need to memorize them.
I agree with this to a large extend, but this does not mean that memorization does not have to happen. I think memorization is the only good and reliable way to become familiar with the facts (and for teachers to ensure this familiarity).
In this second paragraph of yours you are reasoning from properties about the fact about when America was discovered — a fact where I agree with you is unlikely to be of tremendous importance in most peoples lives — to properties of all facts. There are many facts, possibly a different set of facts for different people, that you really need quick access to. I take this to be clear. For those facts you can not wave off the difficulties with finding them out through the internet in a similar way you can with the date of discovery of America.
But still I think there are good arguments that it is good to know this date, together will all the things you mention there which I agree are more important.
Now back to the first paragraph. Think of a fact as I mentioned above, one of the ones you require quick access to. How do you get to this state of having quick access to it? Well, you need to learn the fact at some point. This time of learning is the right time to wonder about what discover (as in discovering America) means, and how reliable the information you get is.
So what you get by learning the fact is a separation of issues: at the time of learning you have the time to wonder about the fact, to what amount it is a fact, what the fact means, etc. And then when you need it you can use it. So you get to separate finding which “facts” are facts from using the facts (see my reply numbered 4 paragraph just before the concluding one). One of the main points of education I think.
I’ll get at the core of your argument:
This is the entire foundation of our current schools. You’ve been trained this way and naturally, you will say that this is only good and reliable process.
Well, I disagree. I never, ever, memorize anything. If you don’t believe me, ask people who know me. I don’t know my phone number. In fact, in kindergarden, we had to learn our phone numbers and I refused to learn it (I was 5 years old). I really do not know my phone number, I don’t know the number of my office at the university, and so on. Let me be clear: there is no evidence that my brain is damaged in some way (though surely, I’m insane in many ways), I simply always refused to “memorize” because I think it is degrading to be trained like a computer. Of course, I didn’t think this through when I was 5 years old… but now, looking back at my young self, I think I was onto something important.
Strangely enough, despite the fact that memorisation is the only reliable way of learning, in almost all things at school, I got very good grades and my academic life has been mostly without trouble. In my work life, I have really no problem being productive. Everything I need to know can be found on the Web or my PDA. I keep nothing in memory except my general understanding and some problems I work on. Everything else is in some form of electronic memory.
Asking students to memorize things is easy and convenient for the teachers, I still argue, despite your arguments, that we need to move beyond it… no, I’ll go further and say we’ll have to move beyond it… because soon, students will be accutely aware that finding the “capital of Uganda” can be found in a less than a second using Google, the “date of birth of Alexander Graham Bell” (try searching the quoted text, and you’ll see, you get the answer).
I agree that my formulation “memorization is the only good and reliable way to become familiar with the facts” is poor and does not clearly describe what I mean.
Let me give an example of what I am thinking about. Imagine you are a logician working in Set Theory. One of the subjects you are working on is a special type of group. During your studies of this group you find that there is a certain freeness in the construction. Suddenly the fact that subgroups of a free group are free becomes very important.
I believe I had this fact available to me b/c at some point it was firmly embedded into my brain (memorized). What was actually memorized was the fact that there are transformation on a generating set that keep the set generating and allow to transform it to a basis.
I would like to believe that this was a product of my general understanding (deep insight) into free groups, but really it was just a fact lying in my brain. I don’t think I could work without these type of facts available to me.
This fact and its use are of different character than the capital of Uganda of the date of birth of Alexander Graham Bell. These seem not very required in the first place, but I think you would be missing something if you didn’t know which continent Uganda lies on, or which century Bell lived in.
(
Does one really not have to know that Japan and China are countries that are close to each-other and that France and Germany border? Still, there are the facts (like the subgroup theorem) which are of different nature and have to be known.
I believe there are in fact more important facts to know, but I have difficulty figuring out which they are as I think these are very deeply embedded.
)
Now if I had learned the fact about subgroups in a course I would expect the teacher to have me do the exam where I could need this fact without a book. That would mean that I would have to have had this fact memorized at the time of the exam. This memorization (having readily available of certain facts) I think is essential. Reason being that the teacher is trying to help me do my work later on in live. So he/she has to ensure that if I am in the situation I am in now, I have the fact available to me.
This I believe is exactly the same as the fact about the shape of exp(x) I mentioned in post number 1.
BTW: I am not in disagreement with you on the fact that there are serious issues with current education. And that facts of certain types play too big a role also seems somewhat clear. I hope the above makes what I meant with my “Concluding” in post number 4 clearer.
Yes, it does make things clearer, but allow me to still disagree (though, of course, I don’t claim to be right, I only argue my point because I have yet to see a killer argument on the other side of the fence).
I’m a mathematician, but I don’t recall what a free group is… hell, I can’t even remember the axioms of a group… I honestly think I never learned what a free group is.
Ok, so I go to planetmath.org/encyclopedia/FreeGroup.html. Ah you say! You knew about planetmath.org… well, no, I didn’t… planetmath.org is mentionned on my blog so I researched it (see the “search” textbox on the right?) So far, no memorisation.
Ok, then I read toward the end of the page that “The Nielsen-Schreier Theorem states that every subgroup of a free group is itself free.” I actually remember now that I once studied what free groups are.
In any case, from your post, and in a few seconds (not minutes!!!) I was able to access probably as much information as you have in your brain now about free groups (less if you are an expert).
This doesn’t make me an expert on free groups, not by a long shot. But I now “know” all of the basic facts about free groups, right there.
Are you suggesting I now memorize these facts? For what purpose? I will surely forget them soon after I memorize them because in my daily life, I don’t have to deal with them.
Oh! BTW, I have never memorized what a group is… just wait a second while I click on the link there on the page… associativity, closure, identity, and inverses over one operation… ah ah, ok.
You know what I think is really important??? What actually matters? Is that now that I have these facts handy, I could actually solve problems on free groups. If I have the facts, but not the competence to solve problems with them, the facts are useless because anyone can get to them nearly as quickly as I can.
So, what is important? Problem solving skills? Creative thinking? Or memorizing facts?
Doesn’t ring like a killer argument to me. I took group theory several times, I never memorized the axioms of a group, and they are already forgotten as we speak. I argue it is not a weakness. The fact is that there is just too much to know out there… Just waaaayyyy too much. So, if you actually need to memorize things so that later you can come back to them if you need them, you’ll be memorizing and memorizing and memorizing… I’m sorry, but nobody has enough memory to be able to absorb all of this.
I argue that the brain mostly work by pattern recognition. You become familiar with a pattern, then another one, then another one… then one day, you encounter a pattern that seems familiar… you’ll naturally go back, dig a bit, and voilà! you’ll find the missing pattern…
I think that’s how it works. You don’t actually need to “know” what exp(1) is. You need to “know” the pattern of exponential growth… but you don’t acquire this sort of intuition by memorizing things, but by “working it out”. If you spend enough time in a pattern, your brain will become familiar with it.
I’ve done enough algebra to know how to prove algebra results and if I need to teach a group theory course tomorrow, I’ll do just fine. I’ll do fine because I know most important patterns because I did a lot of algebra problems, some quite hard.
If you live in algebra land, then yes, you need always have in mind the exact axioms and definitions, and if in 6 months, I start working in algebra, I’ll know these axioms because I’ll use them often.
But I argue that at no point should you memorize these axioms… do the problem solving part… solve very hard problems in the field… that’s what you should spend you time doing, and without knowing it, you’ll go beyond the axioms…
So I want the students to go beyond “what exp(x) looks” like and move into “why should I care it looks like this”…
One word: not doable. You can do it with algebra, and if algebra is all you do, you’ll be fine… but most people are not so narrow. The guy who build bridges can’t also know the axioms of group theory at all time. If he knows this, he better also know how to speak Chinese, Spanish and basic aerospace engineering.
But if he has learned how we solve algebra problem, he might, one day, have a clever geometry problem and he might know a way to solve it that nobody else will… and if he needs to look up the math., he will do it without a problem. He merely needs to have gained intuition, to have internalized the patterns….
Off course I allow you to disagree, more strongly, I like it better that you do. This exchange has made me think about this much more careful than I otherwise would have. And it seems to me we are converging to the place in my mind where the action is. (b.t.w. as answer to your question, I’d say problem solving skills, creative thinking and memorizing facts (by which I mean what was mentioned in earlier posts)).
Reason I think some facts need to be readily accessible in memory is that without the notion of free group, knowing that a lot is known about them and knowing some of those things, I would never have gotten to the point of using planetmath.org or something similar looking more facts up.
This is similar (if not identical) to a calculus student having to solve a word problem and not getting anywhere b/c they don’t realize that exp(x) has a certain shape that suggests what to do next.
The basic problem here is: if you are taking a course on group theory and you are to solve a problem you know that the facts you need are in the book somewhere. In real live you never know which fact you’ll need when. Just being able to look things up doesn’t seem to work out in recognizing when something should be looked up (I think the killer argument you are looking for is in this sentence).
A second point I want to bring up here (easier to solve admittedly): some random person put up the item about free groups at planetmath. How do you know this is reliable? It seems that a solution to this problem is to at some point spend the time studying free groups and proofs of theorems about them so that you know their truth, then to have them in memory, and then when you need them you don’t need to spend the time verifying.
(easy but as of yet nonexistent solution: verified database of such facts. Still far away though. And doesn’t solve the first problem.)