Math Education: Encouraging Mediocrity and the Negative Effects of Algebra

[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.

 

[Vorde]

I think one thing that needs to be thought of is that school is part preparation for life, and part just teaching the student.

I could easily make the argument that 90% of what I learn in a history class is useless for any (non-historical) professional career, but I still think we should be taught it.

 

[Drakkith]

I'm with Zooby and his article on this one. I don't have anything to back my opinion up other than my own experiences, but I would expect that most people never use algebra outside of the classroom.

I think one thing that needs to be thought of is that school is part preparation for life, and part just teaching the student.

I could easily make the argument that 90% of what I learn in a history class is useless for any (non-historical) professional career, but I still think we should be taught it.

Absolutely. I believe that knowing basic history is essential to having a basic understanding of the world and how it works. But I dare say that learning the countries of the world or the history of the middle ages is a far cry from struggling through mathematical concepts that you will probably never use and the understanding of is far from necessary to a successful life.

 

[zoobyshoe]

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?

That information comes from a guy who has specifically studied this:

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.”

There may be other 'educational psychologists' with dissenting opinions, but it's not the kind of assertion you can challenge without actually collecting data about what's going on in the U.S. workplace.

 

[zoobyshoe]

I'm with Zooby and his article on this one. I don't have anything to back my opinion up other than my own experiences, but I would expect that most people never use algebra outside of the classroom.

I'm not on the author's side about the solution, necessarily. I posted it because he highlights the problem well.

I, personally, never failed an algebra course, but I also never exceeded a B. What bothers me is that it all evaporated from my mind within a year of getting out of high school, that is: as soon as it wasn't reinforced any more. I learned algebra to pass algebra and it never had the least application to my real life. It was a weird, tedious problem solving exercise that got more and more complex without having any purpose.

 

[Mech_Engineer]

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.

Well I guess I'm trying to make the point that there will always be a "hardest class" that some students may struggle with. I've heard this argument for calculus before and agree that lots of people don't need it, but algebra seems so fundamental to me...

Maybe instead of algebra they (we) could implement a series of "applied mathematics" courses that focuses more on real-world problems in statictics and finance and and less on understanding abstract algebraic math. Still, it seems to me that algebra has so many useful real-world examples its method of teaching doesn't have to be abstract at all...

 

[WannabeNewton]

If the argument being made is let's not teach algebra to students who have no interest in it and/or will never use it later in life then why can't I say the same thing about history or literature classes? I doubt I will ever in my life use what I have learned about the Mesopotamian era or the Ottoman empire or even the Puritans. These things many would not find essential but I find it hard to believe that people would think high school algebra is not an essential job skill for a reasonable amount of jobs. My mom is a biologist who deals mainly with immunology and there is a good amount of math that crops up in the articles she reads but should such math not be important to an aspiring biology student simply because he/she is not "interested" in it?

 

[micromass]

If the argument being made is let's not teach algebra to students who have no interest in it and/or will never use it later in life then why can't I say the same thing about history or literature classes? I doubt I will ever in my life use what I have learned about the Mesopotamian era or the Ottoman empire or even the Puritans.

Right. It's not only about algebra being necessary in later life. I will concede that many people will never use math in their later life (although many people will, which is already a reason to teach it).
But that's not the point here. High school is supposed to give a general education. How can you call somebody well-educated if he never heard of Shakespeare before? How can you call somebody well-educated if he doesn't know who won the US civil war? Likewise, somebody cannot be well-educated if he never learned to do basic algebra.
Algebra is truly one of the triumphs of human reasoning. It is truly terrible that people do not want to learn it.

 

[Vorde]

I think the problem here was highlighted by someone else when they were talking about the apprentice system. In the old days, you became an apprentice to learn the skills needed for your job, and you went to school to be taught many different things. Both aspects of those two things are combined in modern day schools.

It sort of comes down to what you expect a high school diploma to contain. Drakkith seemed to draw a distinction between math and literature (history really) and their relevance to modern day life. I see no such distinction.

If you are going to say school is about learning skills for life then I would say the curriculum should consist of Foreign Language, Engineering, Writing and Economics/Statistics. If you are going to say school is about being taught, then I would say that History, Literature, Science and Mathematics are all on equal footing with regards to their relevance.

From that equality, you can't righteously remove a course because people are struggling in it, you need to examine your way to teaching it to see what's wrong with it.


Edit: Micromass said a very similar thing in a more concise way.

 

[zoobyshoe]

If the argument being made is let's not teach algebra to students who have no interest in it and/or will never use it later in life then why can't I say the same thing about history or literature classes?

If people were being barred from college based on widespread failure in History and Literature, the same question would apply to those subjects. The fact is, it's not that hard to pass History and Literature even if you have little interest in them.

 

[WannabeNewton]

If people were being barred from college based on widespread failure in History and Literature, the same question would apply to those subjects. The fact is, it's not that hard to pass History and Literature even if you have little interest in them.

I can't say this with generality but at my high school the history and literature classes were tougher than math classes simply because of the hours and hours of daily work we would receive from those classes. I loved US history but the daily multiple page written home-works would easily make my LA or DE class look like heaven on earth. It was a pretty controlled experiment at my school because both the US history teacher and the LA \ DE teacher (same one for both) were excellent; it came down to the nature of the classes. I think getting good marks on a math test would be considerably easier than pulling off consistent A's on history essays given that the math teacher is a good one. We need to have better math teachers not discard algebra from being a mandatory high school class. That would be absurd in all honesty.

 

[zoobyshoe]

Algebra is truly one of the triumphs of human reasoning.

This is probably true, but is it an insight you arrived at yourself, or was algebra presented to you in this enthusiastic way?

 

[zoobyshoe]

We need to have better math teachers not discard algebra from being a mandatory high school class. That would be absurd in all honesty.

Word. I think I would have loved algebra if it had only been taught differently.

 

[micromass]

If people were being barred from college based on widespread failure in History and Literature, the same question would apply to those subjects. The fact is, it's not that hard to pass History and Literature even if you have little interest in them.

So since when is it wrong to challenge the children? Should we dumb everything down and make everything easy for them??
The fact is that algebra isn't even that hard. The children should just apply themselves and struggle through it.

 

[micromass]

This is probably true, but is it an insight you arrived at yourself, or was algebra presented to you in this enthusiastic way?

My algebra teacher was very bad. I hated it. But it was important and I studied for it anyway. In the end, I'm glad I know it.

 

[WannabeNewton]

Word. I think I would have loved algebra if it had only been taught differently.

Unfortunately it would be a monumental task getting such teachers in each high school in the country or even a majority of the high schools.

 

[Andy Resnick]

Andrew Hacker is asking the wrong question. While his logic is sound, the underlying question is not "Is Algebra Necessary?", but "Should Schools Teach or Train?" Historically, professional schools (Medical, Law, Graduate science programs, etc.) trained future professionals, while K-12 and liberal-arts undergraduate programs provided a broad education. The rise of standardized testing has introduced the notion of training K-12 and undergraduate students, and as a result, increasing numbers of educators (like Professor Hacker) are rightly questioning the fundamental purpose of education- should schools provide a 'learning environment', or 'train the future workforce'?. That is the correct question to be discussed.

For an excellent critique of the current state of K-12 Mathematics 'training', Paul Lockhart's 2002 essay "Mathematician's Lament" is required reading.

 

[D H]

This is probably true, but is it an insight you arrived at yourself, or was algebra presented to you in this enthusiastic way?

On my own (there was no internet in the stone age), I came up with a proof that 0.999... is 1. I showed it to my 9th grade algebra teacher. He said I was wrong. He was quite adamant. He ranked right up there in teacher quality with my 9th grade history teacher, whose main job was football coach. He made rocks look downright intelligent. At least my high school algebra teacher was smarter than a rock. But not much.

 

[zoobyshoe]

Unfortunately it would be a monumental task getting such teachers in each high school in the country or even a majority of the high schools.

I don't think it's a matter of replacing teachers, but of getting the ones that are in place to understand what they're doing wrong and how to correct it.

However, I'm not sure anyone knows exactly what's being done wrong or what they should be doing instead. All that might be certain at this point is that there's a big problem.

 

[zoobyshoe]

Andrew Hacker is asking the wrong question. While his logic is sound, the underlying question is not "Is Algebra Necessary?", but "Should Schools Teach or Train?" Historically, professional schools (Medical, Law, Graduate science programs, etc.) trained future professionals, while K-12 and liberal-arts undergraduate programs provided a broad education. The rise of standardized testing has introduced the notion of training K-12 and undergraduate students, and as a result, increasing numbers of educators (like Professor Hacker) are rightly questioning the fundamental purpose of education- should schools provide a 'learning environment', or 'train the future workforce'?. That is the correct question to be discussed.

For an excellent critique of the current state of K-12 Mathematics 'training', Paul Lockhart's 2002 essay "Mathematician's Lament" is required reading.

Very good point.

On my own (there was no internet in the stone age), I came up with a proof that 0.999... is 1. I showed it to my 9th grade algebra teacher. He said I was wrong. He was quite adamant. He ranked right up there in teacher quality with my 9th grade history teacher, whose main job was football coach. He made rocks look downright intelligent. At least my high school algebra teacher was smarter than a rock. But not much.

This gives me an idea for another thread with a poll.

 

[Drakkith]

Warning: Read the following at your own risk. Author may not know what he is talking about. You have been warned.

But that's not the point here. High school is supposed to give a general education. How can you call somebody well-educated if he never heard of Shakespeare before? How can you call somebody well-educated if he doesn't know who won the US civil war? Likewise, somebody cannot be well-educated if he never learned to do basic algebra.

I would argue that a knowledge of the basics of history allow us to judge current events and keep bad things from happening. And this doesn't require an advanced understanding of all of history, only the most basic understanding is needed. How does algebra help the average person? If it's JUST so they can be considered "educated", I would say to hell with that. Let's not invent reasons just to have some.

Algebra is truly one of the triumphs of human reasoning. It is truly terrible that people do not want to learn it.

I could argue the same thing about almost anything, such as Quantum Mechanics. But the fact is that we don't teach QM to most people because it's pointless and they don't care about it.

I can't say this with generality but at my high school the history and literature classes were tougher than math classes simply because of the hours and hours of daily work we would receive from those classes. I loved US history but the daily multiple page written home-works would easily make my LA or DE class look like heaven on earth.

Wow, sounds pretty hard. I don't remember having anywhere near this amount of homework.

I think getting good marks on a math test would be considerably easier than pulling off consistent A's on history essays given that the math teacher is a good one. We need to have better math teachers not discard algebra from being a mandatory high school class. That would be absurd in all honesty.

Would you say that writing a paper on U.S. History requires that you know the history of Ancient Greece? I would say no. However, getting A's on an Algebra test DOES require you to know stuff from previous years of math. But, when does anyone actually use math as a kid? Only on schoolwork. Math more advanced than maybe multiplication just isn't commonly used. You don't talk about it, you don't think about it, nothing. You don't need it to understand the world around you for almost all purposes. It doesn't give you insight into every day matters, help you make decisions, or serve as a moral compass. It does NOTHING. That's why it's so hard for people to care about it. Because for them, it does nothing. It's only useful as a tool for doing other things, and if you don't do things that require you to know algebra, then why would you want to learn it?

So since when is it wrong to challenge the children? Should we dumb everything down and make everything easy for them??
The fact is that algebra isn't even that hard. The children should just apply themselves and struggle through it.

I don't understand this common mentality of "X person should just do Y thing". It's a BS statement that means nothing. It is a FACT that algebra IS difficult to many people. If you have to "just apply yourself and struggle through it", you have failed. Even if you passed the class, you have failed. The fact that you struggled through it probably means that you don't really understand it and only memorized enough information to get past the tests and will likely not be able to do a real life problem involving algebra. This contrasts to classed like history where you can forget the exact date that Abe Lincoln got shot and it doesn't matter. You know he got shot, you know the general idea of why and how and the events leading up to it (Civil War), and that's about all that really matters. (My personal opinion of course) And if you don't know those things, THEN you have failed history.

Now, I am NOT saying we should stop teaching algebra. I really have no idea how useful it is to most people. I'm saying that IF it is useless for the great majority of people, THEN we should probably not teach it.