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Algebra As A Gateway Subject
Spock the Baptist writes: "The Washington Post started a two article series Sunday, and Monday August 18 and 19 2002. The articles deal with something that the math, engineering, and physics faculties at colleges, and universities have long known. Algebra is a 'gateway subject' for math, science, and technology, and secondary schools in general are not doing a good job teaching algebra."
Currently algebra is taught as a "You'll need to know this eventually" kind of a subject. Most of it is forgotten in a few days. Instead of teaching algebra, and then a few years later using it, math classes should be integrated with the science classes in which math skills are usefull.
A skill without a use is going to be forgotten quickly.
I agree that public schools can't do the job. The teachers are told to crank the kids through as fast as they can with little to no support from the board or, more importantly, the parents. It's not their fault. They are among the lowest paid professionals doing a thankless job.
Solution, home school. My wife stays at home and raises our two kids. My 3 year old can count to 20 in English and Spanish (no, I'm not bilingual), do simple sums, and knows her alphabet. I plan on testing her knowledge of the Pythagorean theorem before she hits 10. She will not be rushed, pressured, bullied, or pampered. But we can give her a far better education than some underpaid, overworked teacher afraid to discipline her class for fear of losing her job or his life.
I thought it was an "I'll never need this or see it again" when I was in HS. Problem is, I became an Instructor Pilot. Algebra was life and used every day.
I read in the Washington Post that the Maryland schools are putting BS into the standardized tests and calling it "algebra" and then they wonder why Johnny cannot do anything in real life.
Perhaps we can get back to basic R, R, and R one day and not be as worried about people getting their feelings hurt when they need help in the subjects.
DanH
Cav Pilot's Reference Page
UNIX - Not just for Vestal Virgins anymore
One of my biggest problems teaching algebra is that my students were never given a firm foundation in basics throughout middle school. The philosophy described by the article is accurate as to what I am seeing in middle school math education, but results in a bunch of students who can only solve linear equations in a "trained monkey" kind of way. They have no real cognition as to what their actions mean (ie. When you add to both sides of an equation, you aren't REALLY changing it.) I was halfway through last year (my first year in a new district) before I realized that most of my [otherwise intelligent] students really didn't understand basic concepts like substitution, the difference between an expression and an equation, why you do things to both sides of an equation, etc etc etc.
Let me tell you how much of a nightmare solving solutions were.
I also think that algebra is pushed on students before they are cognitively ready. The average middle school student should go as far as evaluating expressions, variable substitutions, (MAYBE) 1 step equations and (MOST importantly) reading an expression (ie. 3x + 4 means three times x plus 4). The rest of their time should be spent brushing up and applying their ARITHMETIC skills, such as working with/reducing fractions. Give me a class of students who know how to substitute and know their arithmetic, and I'll give you a class of all stars.
In this upcoming year, I'm dedicating the first 2-3 weeks to an intensive review of arithmetic and bare bones algebra. Hopefully that will smooth things over as we go on.
I really like the suggestion of merging science with math. I would love to see those two subjects team taught over a double period.
This seemed to be pointed more towards the middle-school level math courses, but I never had algebra that low. I took algebra I, II and precalculus in highschool, and IMHO (this being two years after i graduated) the problem is that algebra classes have to cater to the lowest commmon denominator, since they're almost universally required for graduation. Even in college calc, our teacher had to spend a few minutes refreshing everyone's memory on basic algebra (factoring, synthetic division, etc)because we never really learned it.
Of course, one approach would be to fail the fuckwits that can't hack it, but apparently teachers catch more flak for failing lazy students than passing smart ones.
If you can't see the value in jet powered ants you should turn in your nerd card. - Dunbal (464142)
Since when is Algebra advanced math? That sort of attitude doesn't help this country at all. I was going to write Ann a reply letter, but since she was already dead I didn't bother.
Disclaimer: I'm currently working on a Ph.D. in applied mathematics
Your code without algebra:
10 print "I never learned algebra"
20 goto 10
Your code with algebra:
for (i=0; i<10; i++) {
printf("I learned my algebra!!!\n");
}
-tpg
I taught math for exactly one year. My biggest problem with teaching was not teaching algebra but fractions!! They were never taught how to add and multiply fractions, except by using a calculator. Some of these kids were quite intelligent and had no problems with
x^2 +6x +8 =0 but (x+1)/2 = 4 and they were lost. All the blame can't be laid on the jr/sr high some of it also falls before they get there.
Some may point to Special Education and/or Gifted programs as alleviating this, but they are typically under funded, help only the lower/upper 3-10%, and don't have any set way to help, instead focusing on the main weaknesses/strengths of the bottom/top 2-3 individuals.
Example: my HS gifted program was essentially a quiz bowl team. Why? It wasn't because we learned a lot(we didn't), but because we had 3 people who were really good. Everyone else was perfectly happy, because going to the events meant they could hang out with their friends and usually get free food. For them, it was just a bonus to watch the top 3 do so well sometimes.
Why hasn't a solution been found and used? Quite simple: parents don't want their kids labeled negatively, and quite often kids don't want to be labeled positively by teachers because it leads to more negative labels from their peers. Having multiple classes, each for a certain level of performer, and you will have complaints, and lots of them.
In other words, don't necessarily blame the teachers or the buereaucrats for the problems of the system--blame our culture for being too Politically Correct.
Moreso than any other subject, mathematics has more of a linear structure- meaning dependence upon previous material.
If you have a bad teacher for 7th grade English, you may never quite be the greatest at diagramming sentence grammar, but the chances are high that you can overcome that shortcoming and still learn to compose good essays, read literature for more than just content, and so on. Other subjects also have the potential to recover from a bad teacher or missed material.
But mathematics has much more of a reliance on prerequisite material. If you have a bad instructor and don't develop good algebra skills, you will struggle and have a great deal of difficulty in algebra 2, trig, etc. When people find out that I do research in mathematics, (a casual conversation-killer if there ever was one) they often have a story, something like "I was always good at math until Mrs. Crabapple in 10th grade" or something like that. One bad experience leads to poor understanding in that subject, and, unfortunately, is rarely overcome and years of struggle result.
I've seen people get derailed at all levels and it really is a problem that needs addressing. At the undergraduate level, sometimes it is particularly painful to witness when a student passes a class (such as first-semester calculus) without learning the material. This can put them into a hopeless limbo- they have no chance of passing the next class, and will probably fail it a few times, but they cannot take the preceding class since they already passed it (sometimes even with a reasonable grade.)
There is a unfortunate stigma to taking something a second time, and that stigma undermines healthy mathematical learning. Sometimes it takes seeing things more than once, or from more than one teacher, before it makes sense. Passing students who just barely have a grasp of the material does them little good and may doom them to years of floundering.
Until there is more recognition of this fundamental aspect of mathematical learning, there will be way too many people who grow up dreading "story problems" and "meaningless algebra"
It's psychosomatic. You need a lobotomy. I'll get a saw.
For this spectacular collapse of education, we have the renowned professor John Dewey of Columbia to thank. Yes, the same amazing mind behind the Dewey Decimal system also flagrantly defied centuries of knowledge about the way humans learned and decided that in fact, humans do not learn by experience, but learn by rote.
Men used to learn as apprenctices, learning while doing for years at a time. The educated labored over Socratic dialogues written over two thousand years before, learning that wisdom and knowledge comes only in knowing to ask the right question.
Many students used to take great pleasure in practicing Socrates' dark art by befuddling others into realizing their own ignorance.
But then, the powers that be at the great school of Columbia looked at the masses of the great unwashed in the masses of tenaments of the South Bronx and decided that man was in fact a machine, ready to be programmed at any time. One must merely sit, listen, and learn from those more knowledgeable than he.
And that is when the transformation took place. Instead of teaching children to ask the right questions, it was the teacher who asked the questions and the student who answered them. Critical thinking was no longer a necessary aspect of learning. One could merely develop the inhuman ability to memorize on end without any care as to its purpose. And then succeed. Some can do this, no doubt. Most likely, the abundance of Cocaine in numerous remedies for uncooperative children in the 1890's probably led some to believe humans could practice such tasks better than they otherwise could. Those complaining of stimulant use by children today are sadly ignorant of a tradition going back 120 years.
But there is a limit, all the stimulant drugs in the world can't teach a child to think critically.
The human being is different than other creatures in that we solve problems creatively, by using our heads, not our bodies. The dog when attacked, knows it will fight back. It cannnot imagine any other way to do this than by using its teeth. When it is hungry, it cannot imagine any other way to get food unless that food is right in front of it.
Humans possess the spark of imagination that is wonderous in its abilities to do and create like never before. It is unfortunate when I see anyone creating the false dichotomy of beauty, art, and science, for they are all the same. We must teach children from the beginning to solve problems, to create what has never existed before, and help them along the way. Algreba should not be a subject in and of itself, it is the most basic form of deductive logic that should be a part of a simple logic class. Math in general should not be a stand alone subject, but taught as a tool in the course of study.
We have followed John Dewey's advice for nearly one hundred years, that a child's brain should be poured full of knowledge. It is false, and destructive. We now have a nation of zombies, unable to question anything or solve any problems. They are hardly human, other than form. is it any wonder they merely stuff their faces with food and vicariously live out there sexual fantasies on television? They know nothing of humanity, they feel only the urges of animals. Eat and fuck, eat and fuck. Is this all life is? Of course, they cannot even ask THAT question...
I don't read or respond to AC posts
I could not help seeing quite a few postings about how "boring" algebra is. Algebra is boring in proportion to how boring the person is who is teaching it. It IS important, and it IS required for all sorts of mathematical subjects later on.
For anyone who has object orientation sussed out, think about containers... what are they? They are (abstract) algebraic structures with certain operations that can be performed on them. You can only get a deeper understanding once you know algebra!
This might seem like a trivial example, but there are many similar examples where a sound mathematical basis helps considerably in software design (because you can approach the problems differently)
Oh and of course I have not even started on the importance of math in engineering (but I am not an engineer, so I'll leave it at that).
Moral of the story - the moment that the standard of mathematics drops, the entire IT industry will feel the blow of ineptitude.
"I hate people who fabricate unintelligent quotes to add to their work seemingly by some 'anon' sage" -- anon
Ah, so in other words we should go back to the old days of apprenticeship and merely allow the curious to move forward.
Sure. Go for it. After all, the last 10000 years of human society clearly had a far better education level and standard of living than we do today.
Or, hell, we don't even have to go back that far. Go look at some of the areas of the world that don't have mandatory schooling. They're top notch. Just last week I was thinking of moving to sub Saharan Africa because they have the best quality of life in the world.
The reality is that you're completely wrong. Even as far back as Socrates and Plato the teacher posed questions to the student. Did students ask questions too? Sure. And *gasp* -- they can now too. If you want to bitch about the (US) educational system, bitch about the funding. Teachers work harder than just about any other profession (hrm, an 8 hour day with no breaks plus another 4-8 hours of planning and grading after school hours), pay them relatively little, make them pay for class supplies out of their own budget, and expect them to educate and morally instruct our children at the same time. With little or no parental backup.
The other minor fact you forgot to mention is the expansion of knowledge in the past 150 years. The concept of a Renaissance Man is dead -- because there is no way for one person to hold the sum of human knowledge now. You can (and should) have a broad base of education, but "jack of all trades, master of none" is becoming increasingly true. Without modern schooling it's impossible to tutor our youth in even a small amount of the knowledge base. Do you know what literacy rates were prior to mandatory education? How many of the illiterate learned basic math, much less algebra?
Sorry, but the arguement that rote learning is evil and useless is bullshit. Rote learning isn't good for EVERYTHING, but in some subjects, it's neccessary, especially for young minds. It's got it's place. In most of the countries where schoolchildren regularly beat the piss out of US children in math and science scores, rote learning is the preffered method of teaching, at least in most of the math classes. All learning is NOT going to be fun and fascinating. There are neccessary things to learn in ANY education that are going to be just plain boring and tedious. We've gotten this idea that all classroom instruction should be creative and "personaly fullfilling", when a lot of the bedrock knowledge neccessary for things like theoretical physics must come from hard, repetitive memorization. I had both kinds of instruction, and it seems the class always did better when we had to memorize the principles first, then "drill till it kills". Once that solid foundation is laid, THEN you can better understand the theoretical.
Life is hard, and the world is cruel
First of all, teachers can't serve as the sole source of motivation for students. Parents and communities have to do that too. The transition for fractions to algebra is one of the hardest on young people. As noted above, one problem is that students that did not have a good grasp of fractions just become more lost in algebra. A second problem is the motivation to learn this new, hard subject.
Students need to understand that "the future is now." This is part of a runup to calculus in college (if not sooner), and that what you can or cannot do in math can and will shape your future. If you do not know algebra II and trigonometry, you are going nowhere in Physics I. No Physics I, no engineering, no chemistry, likely no computer science, etc.
Second, we have to face the fact that many students in math want to get through the class with a decent grade, but have no ambitions to actual understanding. They WANT to be trained monkeys. Their parents often have uncritical aspirations too, and will be happy with trained monkeys.
Thus, they do not want to understand the associative and distributive properties. A trained monkey type of student can solve problems while not fully grasping the properties. A student who understands these properties will have an important intellectual tool available. The idea that certain types things can or can't be related in certain well-defined ways is an important idea.
To those who want to teach math only in the context of solving science problems I say: foo. Mathematical training needs to be broader than the known scientific problems to be solved or you encourage inside-the-box thinking. Where in a physic experiment does someone like Godel become relevant? What about Fermat's last theorem?
Gear the teaching to allow the best to be the best. The crank-churners who don't want to excel will find a way to get a B or C on the test. That's why they call average grades "mediocre." The system has to tolerate the mediocre accepting their lot, but it doesn't have to discourage virtuosity in doing so.
I think that for young students, a good connection with the real world is a requirement for coming to an understanding of the math. I've taught CS at the graduate level, and was always utterly disappointed in the degree of math understanding of the average student. My wife researches grade school math education, is us utterly disappointed with the manner in which it's taught.
Algebra isn't that hard, really. I like to claim that any sixth grader who can figure out what he can get for lunch with the money in his packet has a basic grasp of it already. Part of the problem is that students are encouraged, from a very early age, to believe that they won't really understand math. "Just do it this way", and you'll get the right answer. They aren't usually taught why that way works, or what's going on. They just push numbers around the right way, and write down what you get. There's a definite lack of connection between the "real world" that the students live in and the way it gets talked about in math class.
I also agree that there's far too much repitition in the math curriculum. From my own experience, we learned "fractions" in third grade, did them again in 4th, reviewed them in 5th, went back to them in 6th, etc. By this time I had already dropped out, and started doing algebra and trig as a way to keep myself occupied. Many other students just stopped paying attention, not because it was hard or they didn't get it, but because it was clear that it wasn't ever going anywhere. What a sham!
--tsw
Linear is not the word you want. If math were linear, after learning one concept, there'd be only one direction to go, only one choice for the next concept. Nothing could be further from the truth. The correct word for the concept you're using is cumulative, meaning you have to understand the last concept before you can understand the next. Cumulativity and linearity, in this context, are sort of chronological reverses of one another: linearity dictates the next concept, cumulativity dictates the previous.
The original Howling Frog is a fictional character and has no UID.