Math Education: Encouraging Mediocrity and the Negative Effects of Algebra

[zoobyshoe]

"Is Algebra Necessary?"

...The toll mathematics takes begins early. To our nation’s shame, one in four ninth graders fail to finish high school. In South Carolina, 34 percent fell away in 2008-9, according to national data released last year; for Nevada, it was 45 percent. Most of the educators I’ve talked with cite algebra as the major academic reason.

Shirley Bagwell, a longtime Tennessee teacher, warns that “to expect all students to master algebra will cause more students to drop out.” For those who stay in school, there are often “exit exams,” almost all of which contain an algebra component. In Oklahoma, 33 percent failed to pass last year, as did 35 percent in West Virginia.

Algebra is an onerous stumbling block for all kinds of students: disadvantaged and affluent, black and white. In New Mexico, 43 percent of white students fell below “proficient,” along with 39 percent in Tennessee. Even well-endowed schools have otherwise talented students who are impeded by algebra, to say nothing of calculus and trigonometry...

http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html?_r=1&pagewanted=all

 

[arildno]

The great thing about algebra is that it mercilessly weeds out those who do not have the capacity to think in properly abstractways.
That is why algebra should be kept in school.

 

[AlephZero]

I don't think the problem is that 30% or 43% of kids aren't being taught to be "proficient" (whatever that means).

The bigger problem is the mindset that 100% of kids should capable of achieving 100% success at anything they want.

 

[Jimmy Snyder]

I don't tink spellling iz neccasery eether. And neether gramer.

 

[arildno]

The bigger problem is the mindset that 100% of kids should capable of achieving 100% success at anything they want.

Yes.
It is the hatred of reality, and actually, hatred of existing humanity as such which is the biggest problem.
Both extreme liberalists and extreme communists cannot conceive of human beings other than tabulae rasa onto which anything can be written, with no inherent limitations.

 

[arildno]

I don't tink spellling iz neccasery eether.

Dits nott, ju fuul!

 

[chiro]

I remember George Carlin talking about this kind of thing and he mentioned that soon enough, the standards will be lowered until all it takes to get into university is a pencil.

I don't think he was too far off the mark with the comment to be honest (in terms of the general direction of education, not so much the specifics of his predictions).

 

[FredericGos]

I think the problem is in how mathematics is introduced to young kids. We pretty much loose at least half of them from the start. Not because those kids are dumb or anything, but just because they are introduced to the subject without any context or purpose.

We should incorporate mathematics in other subjects. Woodshop, sports, etc comes to mind.

Just my 2 cents.

 

[phyzguy]

I don't tink spellling iz neccasery eether. And neether gramer.

Right. By the logic of the article, why teach anything? Do you use literature on the job? Or history? I had to work to learn lots of dates in my history class in high school, and I never in my job needed to know that Rome fell on 476AD. Why was I forced to go through this "pain"? With this attitude being espoused in a major newspaper, I fear that America's future will not be what any of us want.

 

[arildno]

I think the problem is in how mathematics is introduced to young kids. We pretty much loose at least half of them from the start. Not because those kids are dumb or anything, but just because they are introduced to the subject without any context or purpose.

We should incorporate mathematics in other subjects. Woodshop, sports, etc comes to mind.

Just my 2 cents.

1. Why have a falsified premise regarding humanity? Most humans ARE dumb, with highly limited capacity for abstract thinking (i.e, they are..dumb). Simply because abstract thinking is a non-adaptive, and quite probably, COUNTER-adaptive trait in general.
2. those who need woodshop or sports in order to learn maths won't ever accomplish much in maths.

 

[chiro]

I think the problem is in how mathematics is introduced to young kids. We pretty much loose at least half of them from the start. Not because those kids are dumb or anything, but just because they are introduced to the subject without any context or purpose.

We should incorporate mathematics in other subjects. Woodshop, sports, etc comes to mind.

Just my 2 cents.

That is a great idea: the only thing is that it to get it done, you would need a lot of cross-discipline co-ordination for this kind of thing but it is a good suggestion in so many ways.

The first way is that since people tend to naturally gravitate to their interests (even if it takes a little while or requires a nudge), then obviously the relevance of something, even cross-disciplinary (like your suggestion) will be a lot higher, more meaningful, and something that will not just be appreciated, but retained and more than likely used (isn't this what education is partly about?)

The kind of thing you are suggesting sounds almost like a slightly modified apprentice-ship where cross-disciplinary skills are introduced out of relevance: the normal apprentice-ship is based out of need, but there's probably a very blurry line between the two.

I have a question though for you: The thing about the education system is that measures are taken to attempt to standardize in some way, the levels for demonstration of particular skillsets or knowledge foundations for things like technical colleges, universities, other non-technical colleges, and even in some respect for people that are looking to get entry level work.

My question to you is how you would deal with the standardization in a cross-disciplinary system of learning: If skills were to be recognized how would they be recognized? How would this be affected system-wide for recognition of training for tertiary institutions? How would it be co-ordinated?

Again I think it's a great idea, I'm just curious on your own take to these questions.

 

[FredericGos]

1. Why have a falsified premise regarding humanity? Most humans ARE dumb, with highly limited capacity for abstract thinking (i.e, they are..dumb). Simply because abstract thinking is a non-adaptive, and quite probably, COUNTER-adaptive trait in general.
2. those who need woodshop or sports in order to learn maths won't ever accomplish much in maths.

Complete nonsense.

I'm talking about young kids age 5-6 or whatever. After a while, it can become a subject on it's own.

Most kids get scared and are forever lost to mathematics. Again, not because they are stupid, but because we teach them arithmetic and algebra the wrong way. The human mind is not meant to be so excact and it sure is no fun. My sister is a perfect example, we are both quite intelligent but she says stuff like 'I hate math' etc while I love the stuff. But I only started loving it in my adult life. I had to find a purpose, and that was computer programming and simulations. When I was a kid, I hated it as well.

But whatever, by your logic, I'm dumb as door, allthough my IQ is around 145. Not that It matters to me much...

 

[FredericGos]

That is a great idea: the only thing is that it to get it done, you would need a lot of cross-discipline co-ordination for this kind of thing but it is a good suggestion in so many ways.

The first way is that since people tend to naturally gravitate to their interests (even if it takes a little while or requires a nudge), then obviously the relevance of something, even cross-disciplinary (like your suggestion) will be a lot higher, more meaningful, and something that will not just be appreciated, but retained and more than likely used (isn't this what education is partly about?)

The kind of thing you are suggesting sounds almost like a slightly modified apprentice-ship where cross-disciplinary skills are introduced out of relevance: the normal apprentice-ship is based out of need, but there's probably a very blurry line between the two.

I have a question though for you: The thing about the education system is that measures are taken to attempt to standardize in some way, the levels for demonstration of particular skillsets or knowledge foundations for things like technical colleges, universities, other non-technical colleges, and even in some respect for people that are looking to get entry level work.

My question to you is how you would deal with the standardization in a cross-disciplinary system of learning: If skills were to be recognized how would they be recognized? How would this be affected system-wide for recognition of training for tertiary institutions? How would it be co-ordinated?

Again I think it's a great idea, I'm just curious on your own take to these questions.

Yeah, I didn't say it would be easy to do, and I don't really know the answers to your questions. The reason I'm saying it, is that mathematics can be found in EVERY single subject you might learn about. And I'm saying this as a way to not scare kids off from the start and gently introduce them to math so that they understand it's usefull. How many times have we heard kids ask their teacher: 'Why do I need to learn this' ?

 

[dipole]

1. Why have a falsified premise regarding humanity? Most humans ARE dumb, with highly limited capacity for abstract thinking (i.e, they are..dumb). Simply because abstract thinking is a non-adaptive, and quite probably, COUNTER-adaptive trait in general.
2. those who need woodshop or sports in order to learn maths won't ever accomplish much in maths.

I tend also to take the pessimistic stance on this issue - the fact that most people are bad at math is because most people are in fact quite stupid and unable to mentally adapt to unfamiliar concepts.

There has to be a huge cultural component to this though. If you compare American scores to, say, Chinese scores, the gap can't be genetics. I think that most Americans are stupid because of the culture they're born into, which is I suppose is a compromise to the completely pessimistic viewpoint because at least then you leave open the possibility that the problem can be addressed.

Honestly though there's no reason in my opinion to assume it will ever improve in America, and I think as time goes by the Asian and European countries will simply pass the U.S. in technological superiority and the U.S. will lose its position in the world, and yet another cycle of power will begin.

 

[arildno]

There has to be a huge cultural component to this though. If you compare American scores to, say, Chinese scores, the gap can't be genetics. I think that most Americans are stupid because of the culture they're born into, which is I suppose is a compromise to the completely pessimistic viewpoint because at least then you leave open the possibility that the problem can be addressed.
.

It is not a particularly weighty counter-argument against genetics, although it MIGHT be true.
A reservation:

Each gene is probably involved in many, many phenotypic properties. Thus, some genes of higher prevalence in one ethnic group might well influence human properties we wouldn't have dreamt of was relevant for it.

Local requirements of adaptivity may have had non-adaptive consequences on other areas of life, due to the multifunctionality of a given gene variant, in which only one of those functions is what triggers natural selection to prefer it.

 

[dydxforsn]

Statistics like the ones mentioned in the original post can only be explained by bad teachers, it's impossible that such a large population couldn't master the simple subject of algebra because not everyone is "meant for it" as I've seen a few people already say in this thread.. It's pretty basic really..

 

[DaveC426913]

Only if you eat your corn-on-the-cob horizontally.

 

[Vorde]

Statistics like the ones mentioned in the original post can only be explained by bad teachers, it's impossible that such a large population couldn't master the simple subject of algebra because not everyone is "meant for it" as I've seen a few people already say in this thread.. It's pretty basic really..

I think there is a bit of a catch-22 here, considering we are on a physics forum...

But I agree with your point.

 

[Ivan Seeking]

We should be teaching Intelligent Design and not Algebra. Seems pretty clear to me.

 

[Vorde]

We should be teaching Intelligent Design and not Algebra. Seems pretty clear to me.

This man is a genius.

 

[Integral]

IMHO, a large part of the problem is that many elementray school teachers are math phobes themselves. Since they are the ones that set the tone early, we are fighting a uphill battle.

Rather then no child left behind we should be finding the talented and pushing them ...HARD!

 

[SW VandeCarr]

I think kids develop a mental block right at the start. They understand that a+a=2a. But then you have to tell them about a+b.

 

[miglo]

saw this article today, i was going to post it on PF but i forgot, glad to see it popped up though

i think a lot of kids who are failing at algebra just don't care about the subject, and because of that they would rather just fail than have to put in the time to get a decent grade

but then again there are some bad teachers out there, and i would know since i had an algebra teacher who spent more time playing world of warcraft during class rather than teaching

 

[arildno]

IMHO, a large part of the problem is that many elementray school teachers are math phobes themselves. Since they are the ones that set the tone early, we are fighting a uphill battle.

Rather then no child left behind we should be finding the talented and pushing them ...HARD!

Correct. The no child left behind policy is silly and unrealistic, simply because there aren't enough good teachers who'd bother with teaching 6 and 7-year olds who are too dumb to begin with.
We WILL have enough good teachers to teach the bright ones, though.

 

[arildno]

I think kids develop a mental block right at the start. They understand that a+a=2a. But then you have to tell them about a+b.

Most would understand it a lot better if we simply wrote: a+a=2*a, just like 3+3=2*3, and so on.
It is extremely unpedagogic to use algebraic short hand 2a, but I don't know if we ever manage to get even this idiotically simple thing across to the school system.

 

[SW VandeCarr]

Most would understand it a lot better if we simply wrote: a+a=2*a, just like 3+3=2*3, and so on.
It is extremely unpedagogic to use algebraic short hand 2a, but I don't know if we ever manage to get even this idiotically simple thing across to the school system.

I don't agree. Your suggestion happens to work in this particular example, but in general: a+3a+3a=7a. It's understood that 2a means 2*a and 1*a is equivalent to a. You missed the point of my example. My post was to show the difference between arithmetic and algebra. Arithmetic addition is reducible. That is, a string of numbers can be reduced to one number. However there are additive expressions in algebra that are irreducible (as formulas).

 

[arildno]

3+2*3+3*3=6*3

In short:
Prior to even teaching algebra, kids should learn to do arithmetic in "new" ways, for example:

"Calculate 3*2+4*2-2*2 in two different ways"

 

[arildno]

In the bad, standard way of teaching algebra, you mix together three new "things" from the start, making the required leap of understanding that much bigger:

1. Introduction of letters-as-numbers
2. Suppression of multiplication signs
3. Adding together in a wholly new manner.

2. isn't necessary at all, and just muddles the whole thing, while 3. could be profitably taught at pre-algebra level.

 

[SW VandeCarr]

3+2*3+3*3=6*3

In short:
Prior to even teaching algebra, kids should learn to do arithmetic in "new" ways, for example:

"Calculate 3*2+4*2-2*2 in two different ways"

OK. But you have to remember that multiplication is simply a shorthand for repeated addition.

 

[arildno]

"My post was to show the difference between arithmetic and algebra. Arithmetic addition is reducible. That is, a string of numbers can be reduced to one number"
The weight of this being??
The need to suppress the multiplication sign??
The need to wait utilizing the distribution law in computation until you get to algebra level?

essentially, I don't see your point at all.

 

[arildno]

OK. But you have to remember that multiplication is simply a shorthand for repeated addition.

And why should therefore the multiplication sign be suppressed from the start in learning algebra??

 

[Vorde]

I have no academic or scientific experience with this subject, only what I have learned from middle school tutoring and actually being in an Algebra I class, so take my statements with due skepticism.

But of the many things I have seen causing trouble and halting understanding in math, the shorthand for multiplication is not one of them.

I think VandeCarr's point was that if you are going to respond so negatively to the suppression of the multiplication sign, why aren't you responding equally so to the suppression of repeated addition as multiplication. Sure you can define multiplication as an operation of its own, but I promise all young math students know multiplication only as repeated addition.

 

[arildno]

"why aren't you responding equally so to the suppression of repeated addition as multiplication."
That is supposed to be thoroughly learned when multiplication is introduced to begin with, and is therefore to regarded as "mastered" when you get to algebra.

 

[SW VandeCarr]

I have no academic or scientific experience with this subject, only what I have learned from middle school tutoring and actually being in an Algebra I class, so take my statements with due skepticism.

But of the many things I have seen causing trouble and halting understanding in math, the shorthand for multiplication is not one of them.

I agree. There is a conceptual difference between arithmetic and algebra. An expression like a+b is irreducible because it involves different categories. 'b' things and 'a' things cannot be added unless one reformulates the problem and this involves categories or sets. These formal concepts are usually reserved for higher math, so the kids have deal with this issue in their own way.

 

[Drakkith]

My question is whether algebra actually serves a useful purpose to MOST students. If not, why make it required by those students who who no interest in it?

 

[arildno]

I agree. There is a conceptual difference between arithmetic and algebra. An an expression like a+b is irreducible because it involves different categories. 'b' things and 'a' things cannot be added unless one reformulates the problem and this involves categories or sets. These formal concepts are usually reserved for higher math, so the kids have deal with this issue in their own way.

"a+b" adds "a" and "b"

You can perfectly well, in higher ARITHMETICS introduce the "collection way of adding":

2*3-4*5+1*3+3*5+2*5=3*3+1*5=9+5=14

The reducibility issue and your "conceptual difference" does not become relevant until the penultimate step.

 

[Jimmy Snyder]

My question is whether algebra actually serves a useful purpose to MOST students. If not, why make it required by those students who who no interest in it?

Teach the best, burn the rest?

 

[GregJ]

I found that I used basic algebra after school in regular jobs and everyday activities. Even trying to figure out simple stuff, like finding out how many martial arts classes I could get with my savings/earnings.

I don't think algebra should be optional in school.

 

[dydxforsn]

I keep hearing people say the "students aren't interested in algebra". I hope this is being used loosely, because it's very hard to say whether the student not being interested in algebra is a product of bad teaching or not. Motivation and creating interest is the most understated and key jobs of the teaching profession. Sometimes this motivation needs to come from better teaching (people like what they're good or taught well), but sometimes this motivation needs to come from psychological-type discussion. There's really a number of ways to produce a student with no motivation to learn algebra, the last conclusion I would jump to is that they "can't" or "shouldn't" do it...

While we're on the subject I would like to point out something. Somebody mentioned the fact that there is inevitably going to be a shortage of good algebra teachers. I accept this, but I have a solution. There should be a propensity for algebra teachers in high school to start over from 2 + 2 = 4 all the way to complicated factoring tricks or whatever may be the conclusion of the course these days. This could at least become a "survey" course at the high school level, where teachers are forced to go back and give students that complete picture of algebra so they're not looking through the dark with only a flashlight on the subject. This is like the only cumulative subject in their schooling career and most simply don't have the foundation either because they were unaware of how important their foundations would be, or they simply didn't get them from a particularly terrible teacher (of which this survey course idea is a solution to "gaps" produced by the inevitable terrible teacher.)

 

[DragonPetter]

This is my own personal opinion. I think the biggest problem is that we often try to teach 30+ students with 1 teacher in 1 hour chunks and expect every kid's mind to conform to this one perspective of learning from the 1 selected textbook. Then we place equal importance and homework on a different subject with their next hour chunk, and repeat for the rest of the day, even though these subjects might be relatively insignificant compared to something more important and difficult like algebra. I don't think it is fair to blame the teachers that much, because they have to work within a rigid system, although there will always be bad teachers.

Then, we think we can encourage other students to help each other, as if it is win-win to promote collaboration skills and help bring the stragglers back up. It is unrealistic to expect learning to happen spontaneously by working in partners at that age, and it probably holds back the kid who already knows it and doesn't help the one who is struggling. And when we finally acknowledge a child needs 1 on 1 tutoring, it is usually too late in the process to bring them up to speed or they are already thoroughly confused. The school systems are extremely passive, reactive, and unrealistic when it comes to realizing a kid's potential. The ones excelling and the ones struggling are hurt the most. I just think back to when I had my class in it and realize all of the missed opportunities for myself and others to ask any question they might have. 1 hour and 30 kids leaves no time for all the possible questions and clarification, and at that age it is hard to know how to ask the right questions.

Difficulty is a complete turn off for a lot of people, and the common teaching systems makes it more difficult and confusing than it needs to be. The whole weeding out argument is counter productive, as it dismisses anyone's potential. If every kid had Johann Bernoulli I as their private tutor like Euler did, I think most kids would be able to master the basics of algebra by the time they're 18. Of course there would still be some who aren't capable, but not at the failure percentages that the education stats give.

Tutors for every kid is not affordable, but parents could fill in the gaps. The problem is that parents failed algebra too, and many don't have the skills to teach or the priorities to value their child's education.

 

[SW VandeCarr]

I think the concept of a variable (which is basic to algebra) is important for reasoning in general. It has enormous practical applications. However, many find this concept difficult. Outside of the hard sciences and engineering, one can earn a good living without knowing algebra.

 

[DragonPetter]

so they're not looking through the dark with only a flashlight on the subject.

I think that's a great analogy of why learning and mastering many subjects is difficult for some, especially when the teachers already have the lights on for themselves.

 

[Drakkith]

Teach the best, burn the rest?

I wouldn't put it like that. We don't teach Calculus to everyone. Why? Because most wouldn't get any use out of it.

 

[WannabeNewton]

Stripping algebra from the mandatory elementary school classes would be a catastrophic blow to this country's already failing secondary school system.

 

[Mech_Engineer]

It seems to me that any professional career makes some use of algebra, even looking at things as simple as a credit card offer or news story with graphs require it's use at some base level.

Maybe the argument can be made that not everyone needs calculus in high school, but algebra is so pervasive in society removing its requirement in school can't actually solve anything without adding 10 new problems in it's place IMO. Maybe if failure rates are high we should consider reevaluating teaching methods instead of just lowering the standard...

 

[zoobyshoe]

Stripping algebra from the mandatory elementary school classes would be a catastrophic blow to this country's already failing secondary school system.

The article points to the algebra required in high school as preparation for college entrance as what starts to trip people up.

The toll mathematics takes begins early. To our nation’s shame, one in four ninth graders fail to finish high school. In South Carolina, 34 percent fell away in 2008-9, according to national data released last year; for Nevada, it was 45 percent. Most of the educators I’ve talked with cite algebra as the major academic reason.

Shirley Bagwell, a longtime Tennessee teacher, warns that “to expect all students to master algebra will cause more students to drop out.” For those who stay in school, there are often “exit exams,” almost all of which contain an algebra component. In Oklahoma, 33 percent failed to pass last year, as did 35 percent in West Virginia.

Algebra is an onerous stumbling block for all kinds of students: disadvantaged and affluent, black and white. In New Mexico, 43 percent of white students fell below “proficient,” along with 39 percent in Tennessee. Even well-endowed schools have otherwise talented students who are impeded by algebra, to say nothing of calculus and trigonometry.

 

[zoobyshoe]

It seems to me that any professional career makes some use of algebra, even looking at things as simple as a credit card offer or news story with graphs require it's use at some base level.

Maybe the argument can be made that not everyone needs calculus in high school, but algebra is so pervasive in society removing its requirement in school can't actually solve anything without adding 10 new problems in it's place IMO. Maybe if failure rates are high we should consider reevaluating teaching methods instead of just lowering the standard...

From the article:

Nor is it clear that the math we learn in the classroom has any relation to the quantitative reasoning we need on the job. John P. Smith III, an educational psychologist at Michigan State University who has studied math education, has found that “mathematical reasoning in workplaces differs markedly from the algorithms taught in school.” Even in jobs that rely on so-called STEM credentials — science, technology, engineering, math — considerable training occurs after hiring, including the kinds of computations that will be required. Toyota, for example, recently chose to locate a plant in a remote Mississippi county, even though its schools are far from stellar. It works with a nearby community college, which has tailored classes in “machine tool mathematics.”

That sort of collaboration has long undergirded German apprenticeship programs. I fully concur that high-tech knowledge is needed to sustain an advanced industrial economy. But we’re deluding ourselves if we believe the solution is largely academic.

 

[Mech_Engineer]

I like the idea of a specialization in education rather than everyone taking the exact same course load, but what happens when algebra gets taken out of the curriculum and long division or multiplication tables become the new weed out topic? It seems to me it would be difficult to even separate algebra from the earlier math classes...

 

[GregJ]

Regarding the "considerable training occurs after hiring" part. Can't say I have been in many jobs where that happens to be honest. Most jobs I have worked at require you to have the necessary skills before you start working for them.

But that's just me I guess. Maybe I am the odd one out?

 

[zoobyshoe]

...but what happens when algebra gets taken out of the curriculum and long division or multiplication tables become the new weed out topic?

Why do you think this would happen? I doubt the stuff that gets used and reinforced constantly in real life would ever be questioned as necessary to teach.