Math Education: A Better Understanding of the "Why"

[chrisdimassi]

I like your signature, Astronuc. You would be surprised how many people drop out of math because they lose sight of the "why" to math, it's difficult not to.
Did anybody record this Dr. Phil episode? http://www.drphil.com/shows/show/767"

In it, a young woman's mother mentioned that she was doing very well in math and her teacher did not understand what was wrong. The young lady said that she was just memorizing theorems but had no idea WHAT she was doing!
Dr. Phil's reply was something along the lines of 'maybe you just aren't cut out for math, your gift may be in something else'.
Great advice, Dr. Phil.

My question is this: do teachers really know what's happening with numbers beyond arithmetic or have they just 'memorized theorems' to get through college and get a teaching degree?

 

[Doc Al]

Note: I moved this from the tutorial section to GD, where it may get more attention. (The tutorial section for posting tutorials, not for discussion.)

 

[chroot]

Some teachers are brilliant, while others are more akin to parrots. I don't think it would be fair to make a sweeping generalizations about all teachers, do you?

Furhermore, does this thread really have a point?

- Warren

 

[JasonRox]

It's not fair to make generalizations, but being part of classes who have prospective teachers, I would have to say not very many of them know what's going. In fact, I haven't met one yet.

I'd say they don't know anything. If you don't want to generalize, let's just say that maybe 1 in 1000 know what they are doing.

Sometimes it's not what you know that makes great teachers. A teacher who cares about the individual successes of the students and who will help them explore areas they enjoy is a great teacher.

 

[JasonRox]

I like your signature, Astronuc. You would be surprised how many people drop out of math because they lose sight of the "why" to math, it's difficult not to.

It's funny because I never heard anyone asking the arts teacher...

...why are we drawing pictures? We throw them in the garbage after class anyways.

...why are we playing instruments? It sounds horrible.

...why are we acting out a play? Nobody is watching.

And the list goes on for all kinds of subjects.

Ignorance gets you nowhere.

 

[Chi Meson]

Some teachers are brilliant, while others are more akin to parrots.

- Warren

I've met some brilliant parrots too.

 

[chrisdimassi]

It's funny because I never heard anyone asking the arts teacher...

...why are we drawing pictures? We throw them in the garbage after class anyways.

...why are we playing instruments? It sounds horrible.

...why are we acting out a play? Nobody is watching.

And the list goes on for all kinds of subjects.

Ignorance gets you nowhere.

Er, I wasn't referring to 'why we need math'. As I had mentioned with the young girl telling Dr. Phil that she didn't want to continue with math because she didn't understand it beyond the rote memorization of formulas, people need an intuitive understanding of the learning of math as opposed to an ignorance of its construction.

 

[chrisdimassi]

It's not fair to make generalizations, but being part of classes who have prospective teachers, I would have to say not very many of them know what's going. In fact, I haven't met one yet.

I'd say they don't know anything. If you don't want to generalize, let's just say that maybe 1 in 1000 know what they are doing.

Sometimes it's not what you know that makes great teachers. A teacher who cares about the individual successes of the students and who will help them explore areas they enjoy is a great teacher.

One in one thousand? My worst nightmare has been realized.
Yes, bravo for teachers who can teach the material in whatever way, creative or otherwise, that they can. Teachers who really care about the student's future should receive medals.
Out of over 60+ teachers from 1st-12th (I moved a lot), I can think of only two who knew how to communicate well and they were the only classes I ever made A's in.

 

[chrisdimassi]

Some teachers are brilliant, while others are more akin to parrots. I don't think it would be fair to make a sweeping generalizations about all teachers, do you?

Furhermore, does this thread really have a point?

- Warren

I didn't really say 'all' teachers, I just said 'teachers'. If that means 'all teachers', well, grammar was not my strong suit. Otherwise, it's an assumption of my intentions.
Noting from what the young lady said and from my own experience, it seems to be a common occurrence that many teachers do not have an intuitive understanding of what they teach.
Note I'm not referring to college professors here, necessarily (many I have met seem to be brilliant personally as well as being fully competent in imparting knowledge to students)

A good point would be: what's being done to change this in the educational system?

 

[Cyrus]

I didn't really say 'all' teachers, I just said 'teachers'. If that means 'all teachers', well, grammar was not my strong suit. Otherwise, it's an assumption of my intentions.
Noting from what the young lady said and from my own experience, it seems to be a common occurrence that many teachers do not have an intuitive understanding of what they teach.
Note I'm not referring to college professors here, necessarily (many I have met seem to be brilliant personally as well as being fully competent in imparting knowledge to students)

A good point would be: what's being done to change this in the educational system?

In order to teach in college you must have a PHD. For high school, you don't even have to major or minor in the subject you teach. Plus anyone worth a damn isint going to waste their time teaching to get min. wage, when they can make double the amount doing real work.

Those who can do, those who can't teach.

 

[chroot]

I can think of only two who knew how to communicate well and they were the only classes I ever made A's in.

Your comments are indicative of a sweeping selection bias shared by, well, almost everyone: the only good teachers are the ones who gave me A's.

That's more than a little short-sighted, in my opinion. Education is difficult for the teacher, but, frankly, it must be even more difficult for the student.

- Warren

 

[Gelsamel Epsilon]

Now I want to know if I was just recited too. o_o

I think not though, my math teacher is very good.

~Gelsamel

 

[Moonbear]

There are a lot of students who get through high school by memorizing and regurgitating without processing and understanding, not just in math. These are the students who have the toughest time adjusting to college courses, because they haven't learned to learn in high school and think they'll coast through college the same way they coasted through high school. These are the students who wind up in the instructors' offices after the first exam wondering how they only got a C or D when they've gotten As all their life.

However, a lot of what is taught at the high school level DOES require simple, rote memorization. Until you have some basics in your head, you really can't fully understand the concepts involved. A good teacher will remind the students from time to time that they are asking them to memorize something now because they have not yet been taught enough to understand the reasoning behind it, but will get to that in later courses, so they don't lose sight that there's more to it than just memorization.

One needs to realize that learning is not a linear process. You have to do a lot of doubling back and revisiting of material. High school and introductory college courses are essentially intended as an overview of an entire discipline. It's just skimming the surface to give you the big picture, and then as you get into higher level courses, you'll be able to keep the details in perspective.

It's a shame that the young lady in question has been done a disservice by discouraging her from a subject when she has realized on her own that there's more to learning than memorizing, and has developed a thirst for that higher level of learning.

As for the question of whether teachers themselves know the subject beyond the level of memorization, when you get into high school subject specialties, unless someone is being asked to teach a class that is not their specialization (i.e., the teacher certified to teach high school English being dumped into an algebra class because the school can't get a more qualified teacher), they usually choose their subject because of what interests them.

As with any profession, you will find a range of qualifications among teachers. For every outstanding teacher, there is also one who scraped by and barely met the requirements for certification.

There's a saying:
"A good teacher explains.
A superior teacher demonstrates.
A great teacher inspires."

Unfortunately for the young lady discussed in the OP, her teacher has missed an opportunity to inspire her, as has Dr. Phil, but we already know he's a hack. When a student expresses frustration with memorizing subject material without being enlightened to a higher understanding, that is the time to suggest either a tutorial to delve deeper into some of the concepts, or to do an extra credit assignment that is more challenging and requires linking concepts, or even just to sit down and explain that she's going to need to have these theorems memorized in order to tackle the conceptual part of the subject that is taught at the college level, and to just hang onto that enthusiasm for college.

 

[Chi Meson]

In order to teach in college you must have a PHD. For high school, you don't even have to major or minor in the subject you teach. Plus anyone worth a damn isint going to waste their time teaching to get min. wage, when they can make double the amount doing real work.

Those who can do, those who can't teach.

Grrrrrrrrrrrrrrrrrrrrrrrrrrrrr!

CY! I tusted you! I tusted yooooooooou!

Anyway, it's not minimum wage and we get 13 weeks off every year. I could get more money but less time off if I worked for a local acoustical engineering unit that's nearby. I chose more time.

 

[marlon]

Unfortunately for the young lady discussed in the OP, her teacher has missed an opportunity to inspire her, as has Dr. Phil, but we already know he's a hack.

So is Dr Phil considered to be a fake media psychologist ? Do intelligent people take him seriously ?

We have his shows on TV here but we donnot take that stuff very seriously. The entire concept of that show appears quite strange to me and many others.

marlon

 

[chroot]

I don't think anyone with any sense considers Dr. Phil (or any other TV personality) to be anything more substantive than brain-dead entertainment.

- Warren

 

[JasonRox]

There's a saying:
"A good teacher explains.
A superior teacher demonstrates.
A great teacher inspires."

I like that one.

 

[JasonRox]

I think we are living in perverse times when in the country with the largest economy the majority of high schools are not giving kids a proper education when they are the most well equipped in the world.

I think this is because in North America we take everything for granted.

 

[Cyrus]

Grrrrrrrrrrrrrrrrrrrrrrrrrrrrr!

CY! I tusted you! I tusted yooooooooou!

Anyway, it's not minimum wage and we get 13 weeks off every year. I could get more money but less time off if I worked for a local acoustical engineering unit that's nearby. I chose more time.

I knew when I typed that I was going to get it on the head from you.

I would put you with the small minority of teachers that actually give a damn about what they teach. I had one teacher like that in freshman year. He was my american history teacher. He was in Vietnam and had some of his fingers blown off and told us about how he got hit with a mortar round. And he taught at the Naval Academy, wrote books, and did work at the Arlington Cemetery. He was a really cool guy and we all respected him. Then he died from a heart attack the next year...too bad, we all loved him.


As for the rest of the teachers, they couldn't find their own ass if they tried. Complete and total idiots, I mean where do they find some of these people?

I have met a good handful of africans that came to this country for school while at college (were talking 3rd world countries here). They all tell me about their high school and how they learned calc1, calc2, some of calc3, linear algebra, french, english.

The US school system for K-12 is a joke. I would NEVER put my kid in any lousy public school. It's going to be private school for k-8, and then either private school or community college in place of high school. NO WAY are they going to a public high school.

God, I could go on and on and on about how I hate public schools...

And those lunches! People in prison get better food, seriously...

 

[chrisdimassi]

For high school, you don't even have to major or minor in the subject you teach.

I had no idea.
We really can't blame people for wanting to take higher paying jobs, teaching sounds like a major sacrifice.
One thing that occurred to me is that teachers are the products of this flawed system which is a possible other cause.

 

[chrisdimassi]

Your comments are indicative of a sweeping selection bias shared by, well, almost everyone: the only good teachers are the ones who gave me A's.

That's more than a little short-sighted, in my opinion. Education is difficult for the teacher, but, frankly, it must be even more difficult for the student.

- Warren

Well, I sort of just threw that comment in. It is true that most of the students did very well in those classes and became quite proficient yet after moving, students in that subject did not do as well or seem as enthused about the subject. Same textbook, different teachers.

 

[chrisdimassi]

I don't think anyone with any sense considers Dr. Phil (or any other TV personality) to be anything more substantive than brain-dead entertainment.

- Warren

What? All psychiatrists aren't geniuses? What qualifies them to give IQ tests?

 

[Moonbear]

So is Dr Phil considered to be a fake media psychologist ? Do intelligent people take him seriously ?

We have his shows on TV here but we donnot take that stuff very seriously. The entire concept of that show appears quite strange to me and many others.

marlon

Intelligent people don't take him seriously, but sadly, there are a lot of people who do take him seriously. He does manage to give advice that you want to hear him give on some shows, but that doesn't make it good advice. (Sort of the "Just Get Over It Already!" type comments that get a good laugh, but are totally useless for someone whose problem is they haven't gotten over "it" already, whatever "it" is.)

 

[JasonRox]

Intelligent people don't take him seriously, but sadly, there are a lot of people who do take him seriously. He does manage to give advice that you want to hear him give on some shows, but that doesn't make it good advice. (Sort of the "Just Get Over It Already!" type comments that get a good laugh, but are totally useless for someone whose problem is they haven't gotten over "it" already, whatever "it" is.)

He does give a lot of help to people on the show though. He will have people keep up with you and everything.

Also, we have to remember that his show is for general information only. So, for general problems that viewers might have, the line "Just Get Over It Already!" will suffice. For example, when he had guests that were overweight and trying to lost weight, the first key was to get out of the state of denial. This only applies to the general overweight person though.

 

[chrisdimassi]

His comment to the girl was very surprising. It showed no depth of thought nor understanding of intellectual processes.
It's a shrink's job to give advice so bad advice=incompetence in my book.
So maybe he shows some common sense sometimes but I recall the first thing that ever struck me about him is that he acts like a prick. How is that therapy? Tough love, my arse.

 

[chrisdimassi]

Not that I dislike all shrinks, of course. I've run into a couple good ones on science forums over the years.

 

[chrisdimassi]

I knew when I typed that I was going to get it on the head from you.

I would put you with the small minority of teachers that actually give a damn about what they teach. I had one teacher like that in freshman year. He was my american history teacher. He was in Vietnam and had some of his fingers blown off and told us about how he got hit with a mortar round. And he taught at the Naval Academy, wrote books, and did work at the Arlington Cemetery. He was a really cool guy and we all respected him. Then he died from a heart attack the next year...too bad, we all loved him.


As for the rest of the teachers, they couldn't find their own ass if they tried. Complete and total idiots, I mean where do they find some of these people?

I have met a good handful of africans that came to this country for school while at college (were talking 3rd world countries here). They all tell me about their high school and how they learned calc1, calc2, some of calc3, linear algebra, french, english.

The US school system for K-12 is a joke. I would NEVER put my kid in any lousy public school. It's going to be private school for k-8, and then either private school or community college in place of high school. NO WAY are they going to a public high school.

God, I could go on and on and on about how I hate public schools...

And those lunches! People in prison get better food, seriously...

That's exactly right, they are institutions. Not an environment conducive to learning if you ask me.
The problem is, I assume, that many teachers graduated from the same system and it's hard to break out of a cycle from the environment that you were brought up in (be it family or school).
I think of it like someone who is born in the ghetto. It's nobody's fault where they are born but the environment is set against them.
Sure there are the very few who get out with luck, hard work, meeting the right people (any combination) and we praise those people for persevering under harsh circumstances but that's not the point.
The solution is to change the environment. Why? Because our country has to IMPORT many of our scientists and other intellectuals. Nothing wrong with that but it indicates that many of our students are missing out on what others from other countries are evidently not. So you bet your ass something needs to change in the educational system of this country!

Yes, thank God for the teachers who have been able &/or willing to rise above and break out of the cycle. However, good teaching skills need to be the norm, not the exception!
It seems that one goal of this website is doing a great service to the community. Many of us had parents who were unable to afford tutors or unaware that they should. Thank you to the volunteers for giving your time to make up for what our education system can not do and to the teachers who care.

 

[moose]

A great example is the calculus teacher at my high school.

I took Calculus I with him last year, second semester. I thought that he was a pretty good teacher because my previous math teachers were worse than him. I received an A in the course without much trouble. Over the summer, I took Calculus II taught by a grad student. This grad student was by FAR my favorite math teacher I have ever had. He explained everything in a very logical order, in a way that makes you understand immediately if you are thinking along quickly enough. He proved almost everything that he wrote (theorems and rules and such). The final grade in the course consisted of four tests and ten quizes. I payed very close attention in class and thought along with everything that he said, and he did a very good job convering a plethora of topics in one class thoroughly. Due to this, I glanced over my notes everyday for about 10 minutes, and was ready for the next day. I studied for a few hours before tests, but nothing too major. I spoke with several of the students in the course, and the ones who had trouble were the ones who weren't following along with the grad student as well as others were. In the end, only two people out of the class of 35 received an A, me being one of them. The tests were seriously difficult. The final was insane. I loved how he had problems on there which we were never taught and weren't in the book, but you could figure out if you understood well enough and through it through. After this, I took differential equations. I didn't take the prerequisite, which was multivariable calculus, but some math director dude said it should be fine. Should be... I took differential equations and had a good professor, not as good as the grad student, but good none the less. So I go through the extremely challenging course, studying at least 3 hours a night, and end up with a D (about class average...). I did what I could, but the fact that it was a summer course (5 weeks long), and that it was very heavily based on multivariable calc, just turned out to be too much.

SO where am I getting at?

I now tutor for the Calculus I class at my high school. I never realized how horrible of a teacher the teacher really is. He never explains anything, just states things. When someone asks him a simple question, like how someone came up with the product rule for taking derivatives, he said that mathematicians found a pattern. I proved the damn thing in about 20 seconds on a receipt I had in my pocket. WTF? He has done the same thing for anything anyone asks. He says things without saying anything why, and then people struggle with everything. Like for example, whily doing something about implicit differentiation, when someone asked why they put a d(whatever variable)/dx after whatever variable's derivative, he said "because after x, we put dx/dx which cancels". How does that explain ANYTHING!? Then I looked at the tests, and I never realized how easy he makes them (same with Calc II, which I can actually make a valid comparison with). They are so straight forward, like, take the derivative of this (its a simple function). Or, if the radius of a sphere is increasing at 3cm/s, what is the rate of increase in volume at r=5cm? All of the problems have been previously covered in class.

Also, he fails to even mention sinh, cosh, tanh, etc. When I walked into my first Calc II test at UA, I missed a ****load of points because I didnt know ANYTHING about hyperbolic sines and cosines.

 

[Pythagorean]

...anyone worth a damn isint going to waste their time teaching to get min. wage, when they can make double the amount doing real work.

Those who can do, those who can't teach.

exceptions:

the physics professors at my university who work for DOD and are required to teach as part of the deal for their 300+ k a year.

OR

My political economy teacher is one of the few political scientists to run for a political position. He ran for the board of education, and succeeded three times in a row, made political economy part of the core, and chose to teach the 100 level core class because he thinks its important information for everyone to know. He has a TA from a higher course he teaches too.

Do you really believe there's not a lot of teachers like this elsewhere in the world who actually think quality education is important.. ?

 

[Ivan Seeking]

I think we need to consider what things look like from the teacher's side of the classroom.

http://www.pointlesswasteoftime.com/monkeyhair.jpg