Modern Mathematics Education a Failure?

[pawnwarp]

Hello,

Thought I would share this link to "Lockharts Lament"

It basically talks about how the way math is being taught in schools is a terrible way to teach it and that it takes all of the real mathematics out of it.

http://www.maa.org/devlin/LockhartsLament.pdf

 

[chiro]

Hey pawnwarp and welcome to the forums.

You might be interested to know that this has been talked about before in these forums quite a few times (if not that particular article which has been talked about several times, something that is very similar).

If you go to the search feature and type in 'Lockhart' for the thread (search by thread and not by post), you will see quite a few threads on this very topic.

 

[Deveno]

i used to day-dream about teaching math to kids.

here was one of my favorites:

it's the first day of class. the trepidation is palpable. what we will do first? areas? percentages? fractions?

no, here is my idea:

math is has committed some terrible crime. it MUST have, because we hate it so. well, what do we do with criminals? we put them on trial. so, i would announce we are putting math on trial. we'll choose a prosecutor, a defense attorney (maybe more than one of each..teamwork is cool), and some jurors. either side gets to argue however they like (within reason, illegal and/or rather loud actions could attract the attention of undesriables like: the principal), and can call witnesses from the class to testify.

the point? the get people to think about how they feel about something, and about what it is, we are going to study. and...because it's fun to play make-believe.


idea number 2:

i've always wondered...what should be the first number? 0? 1? -∞? nothing (blank)? well, this is america! and what do we do in america when we can't decide on things? we vote! yes, i propose to make the first number an elected office. let the ardent defenders of these candidates speak their minds, and sway the masses!

i read about a teacher who taught students about "quantity" by going out and measuring things. they climbed trees, dug holes, wound their way alongside river-banks. that's the way to go in my opinion!

 

[chiro]

Deveno, thanks for the interesting perspective but making the first number an elected office? Maybe you should run for president and test out your theory :)

 

[Deveno]

Deveno, thanks for the interesting perspective but making the first number an elected office? Maybe you should run for president and test out your theory :)

what do YOU think should be the first number? why?

 

[chiro]

what do YOU think should be the first number? why?

Well if you are a Sci-Fi fan, the answer would be 42:

http://en.wikipedia.org/wiki/Phrases_from_The_Hitchhiker's_Guide_to_the_Galaxy

If you were a Bill and Ted fan (or extremely fascinated with sexual innuendo) the answer would be 69:

If were some obscure Indian mathematician you answer would be 1729:

http://en.wikipedia.org/wiki/1729_(number)

But since I'm none of those people I am going to go with the advice of Leopold Kronecker which states: "God made the integers, all else is the work of man". We can create anything else with the integers (even if we can't actually evaluate it), so based on this the first number should be 0 or nothing at all.

 

[Deveno]

Well if you are a Sci-Fi fan, the answer would be 42:

http://en.wikipedia.org/wiki/Phrases_from_The_Hitchhiker's_Guide_to_the_Galaxy

If you were a Bill and Ted fan (or extremely fascinated with sexual innuendo) the answer would be 69:

If were some obscure Indian mathematician you answer would be 1729:

http://en.wikipedia.org/wiki/1729_(number)

But since I'm none of those people I am going to go with the advice of Leopold Kronecker which states: "God made the integers, all else is the work of man". We can create anything else with the integers (even if we can't actually evaluate it), so based on this the first number should be 0 or nothing at all.

1729, that famous taxi-cab number. it's not a very interesting number, you know :)

my point isn't to "re-invent" or "re-define" what the first number should be.

my point is...ok, there was another thread somewhere else on these forums, about whether the complex plane was "all the numbers". before one answers that question, one would do well to think about what a number might be. my answer would be that what a number is, depends on what you plan to do with it. I'm a big believer in "contextual truth" not absolutes. use the right tools for the job at hand, whether it be accounting, slicing and dicing, patterning, or just having fun.

i believe the best way for people to learn anything, is to think about what they are learning. this requires striking out blind, sometimes. some things work better than others, isn't it better to see for yourself, than to just take someone else's word for it?

i saw a video once about a guy who said that a textbook is just about the worst thing you can use to teach math. the best way is to try to solve a problem, and use what little you know, to find out more. often, there are several paths to solving a problem. the person who knows that, from experience, is going to be the more skilled problem-solver. the same guy also said homework was a bad idea. if you don't teach when you teach, what are you there for? the book will always be around, but...it's just words on paper. ideas...those things are alive, if one doesn't have the skill to make the ideas BEHIND the words dance, perhaps another line of employment would be better for all concerned.

there is a quote spivak used in his calculus book which i adore:

"a man ought to read just as his inclination leads him. for what he learns as a task will do him little good."

 

[chiro]

my point isn't to "re-invent" or "re-define" what the first number should be.

my point is...ok, there was another thread somewhere else on these forums, about whether the complex plane was "all the numbers". before one answers that question, one would do well to think about what a number might be. my answer would be that what a number is, depends on what you plan to do with it. I'm a big believer in "contextual truth" not absolutes. use the right tools for the job at hand, whether it be accounting, slicing and dicing, patterning, or just having fun.

I agree with what you're saying, but when you incorporate a wide range of audiences you are always going to get a lot of different viewpoints and ideas of what the answer to this might be.

Most people associate numbers to something they can count or measure. Some people might even consider numbers as labels in the way that a particular index of a set would give some kind of object. This seems to be what you are talking about in the above quote.

If you are trying to solve the problem of getting everyone to understand (or make an attempt to) each other and appreciate the context that the other person sees, then I wish you a lot of luck.

The unfortunate thing is that it takes a lot of effort to do that very thing. Many people just don't want to go to the effort to do that kind of task. The ones that do are often very very good teachers whether they do it in the classroom, behind the computer, in the bar, with others in front a campfire or otherwise.

Also if you tried to do that kind of thing in your standard classroom, I think you would get so much outrage from many of the different groups because they might end up dismissing it as it gives the opportunity for students to see it in a different way to what they see it as. Also many people might say its not pragmatic and would cause more chaos and confusion that it would solve.

I think your ideas though are good ideas.

 

[Deveno]

I agree with what you're saying, but when you incorporate a wide range of audiences you are always going to get a lot of different viewpoints and ideas of what the answer to this might be.

Most people associate numbers to something they can count or measure. Some people might even consider numbers as labels in the way that a particular index of a set would give some kind of object. This seems to be what you are talking about in the above quote.

If you are trying to solve the problem of getting everyone to understand (or make an attempt to) each other and appreciate the context that the other person sees, then I wish you a lot of luck.

The unfortunate thing is that it takes a lot of effort to do that very thing. Many people just don't want to go to the effort to do that kind of task. The ones that do are often very very good teachers whether they do it in the classroom, behind the computer, in the bar, with others in front a campfire or otherwise.

Also if you tried to do that kind of thing in your standard classroom, I think you would get so much outrage from many of the different groups because they might end up dismissing it as it gives the opportunity for students to see it in a different way to what they see it as. Also many people might say its not pragmatic and would cause more chaos and confusion that it would solve.

I think your ideas though are good ideas.

indeed, like the english professor in "dead poet's society" got fired for his trouble. i really dislike the whole standardized test thing that is all the rage these days. people aren't machines, and the whole idea that equality/democracy means we ought to treat everyone as interchangeable feels every kind of wrong i know.

but i also believe that if people were taught properly, they'd do better on the standardized tests that those given "the standardized test training". we (humans) are curious creatures, we like being able to do stuff, it's satisfying. our educational system is increasingly competitive, rather than cooperative, and in the long run, everybody is the worse for it.

you really want "no child left behind'? better work on the ones you're dooming to failure, then. the self-motivated, over-achievers are going to succeed no matter what.

 

[chiro]

Interestingly enough sometimes the best teachers are the ones that have struggled a lot themselves because they are able to relate to the people they are teaching that are having a hard time as well.

I hear a lot on other forums about teachers being put down because they 'got crap marks and couldn't do anything else' or things like 'I don't want my kids taught by someone who got a 60 ATAR' (ATAR is a ranking that you get when you complete the Higher School Certificate much like your SAT with the note that it is ranked not raw).

Those people don't realize that the ones who really come through for the struggling ones are those who often struggled themselves.

It's not surprising that in my experience that sometimes the people that are super smart are just really impatient with the slow learners. I am lucky however to go to a small sized university that has great teachers and I can imagine that at some universities (especially big ones known for high research output) that this would probably not be a common thing (although comments like those by twofish-quant who has come from MIT have been enlightening).

 

[alan2]

I had significant experience teaching mathematics at the college level, mostly calc, linear algebra, ode's. I decided that teaching was my passion so I applied to, and was accepted by, what is thought to be one of the top 5 math ed Ph.D. programs in the U.S. I left after less than one year because I found that those involved in research to improve math ed really didn't know anything about math. They were heavily involved in why kids make mistakes in symbol manipulation. My point that they weren't being taught to understand math was not well received. I recall a long seminar about why kids couldn't "cancel" common factors correctly in a quotient. I pointed out that there wasn't really any such operation as "canceling", but rather it was factorization based upon the distributive law and resultant multiplication by one, and maybe kids would understand if they were properly taught what they were doing. I was met with dumb stares. I had to teach a class called Math for Elementary Ed which was a joke. As I saw it, the people charged with teaching future teachers how to teach had little to no understanding of the fundamentals themselves. Mathematicians with an interest in education were basically ignored or ridiculed for not understanding education. My personal opinion is that nothing will change until people who love and understand mathematics are allowed to interact with students and future teachers.

 

[Cenote]

As someone who has just been through basic maths education in Germany, I think my biggest gripe is that my teachers did very little to actually encourage me to try again if you fail at some problem, or to continuously keep trying until you get an answer. I think therein lies one of the great strengths of mathematics, the fact that getting to an answer autonomously is very gratifying.
Maybe maths has has a kind of bad reputation, because of it's preciseness. In other subjects, e.g. most languages, you can always blame part of the bad marks on the teacher because it will always be, to a certain degree, subjective, whereas in maths you know that something is either right or wrong and if it's wrong then it is your own fault.
I also heard, or read, a good analogy on it recently, which went something along the lines of "Being able to do maths is like trying to play a piano, you hear someone else play and are intrigued, but then immediately disheartened when you realize that when you try it, it sounds terrible, so you think you will always be unable to do maths, that it is an innate thing, that some people are just able to do maths or play a piano, but they always forget that like the person playing the piano, the person who can do maths had to acquire that skill through long periods of continuous studying as well."

Edit: Oh, and one should always try to challenge the students. If one feels too comfortable in a maths class, one might get the feeling that you already know everything, when in reality your teacher just tries to "do you a favor" by giving you easy problems, which might be very counterproductive in the end. My favorite maths teacher was a grumpy old man who never gave anyone the satisfaction of being lazy and always tried to push them forward.

 

[Deveno]

the best thing a teacher can do for a student, is teach them how to teach themselves.

a favored method for this, is to "push" students until they realize that if they push themselves, they don't get pushed.

the best teachers do this in such a way, that the students never realize they were pushed in the first place, they were "too involved" to think it was anthing else but "all them".

but this is a skill, and its not a skill one can easily teach (if teaching is hard, then it stands to reason that "teaching to teach" is doubly hard).

yes, math is precise (or can be). and this precison is a stumbling block for many people (they see there's "only one right answer" so no "wiggle room" for getting the "gist" of it). as one author put it, you have a choice:

a) easy definitions and hard theorems
b) hard definitions and easy theorems

i prefer (b). i think if you are going to talk about "anything", one ought to clearly "define your terms". this helps organize our thoughts. i think that, further, what's necessary for communication (and teaching is a form of communication), is "agreement upon definitions". so the actual different viewpoints possible on "what a number is" isn't all that important...what is important, is that if you and i are talking about numbers, we have some sort of agreement on what we are talking about.

now often, a teacher has some clear definition that he/she wants the class to accept. and often the reason behind this is, other teachers will expect the students in the future to be familiar with the previous teacher's definition. and to be fair, a teacher needs to prepare his/her students for this future event.

but...there's no reason why, at the same time, the teacher can't also make a definition (contract) with the students based on their ideas, too. perhpas it might happen, that some of the students come to see why "their definition" isn't as useful as they thought it might be. what a beautiful thing that is, that's true knowledge...because now they are motivated to find a "better definition".

sure, left unchecked, a math class might devolve into esoteric philosophy, or sheer non-sense. so, sometimes a teacher has to stop being "nice" and steer a little harder. hopefully, though, not too often.