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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."
I was blessed here in Tallahassee, FL, with some really great math teachers, as well as the option to take a "real" algebra course as "early" as the 7th grade. And we're not talking "algebraic concepts" here - I was required to derive the quadratic formula w/o completing the square, which is TOUGH when you're 12 or 13.
It disappoints me to see schools lowering their standards to raise average test scores. I'm one of the minority who believes that D should be passing, but that a C truly should be an "average" grade (just like it says on the report card). My H.S. has an average GPA of something like 3.4! That's just silly - there's nothing differentiating the truly exceptional from those who could either kiss a lot of arse or slough through it and do all the extra credit.
I also see a very disturbing trend of schools offering classes that, in essence, "teach the test", be it the SAT, ACT, or the FCAT (in FL's case). Doesn't this skew the results? I'd like to hear some others' opinions on this...
Just my $0.02 worth of incoherent rambling...
Brandon
Well I can't imagine what school's must be teaching these days cause the younger generation I do run into seems to be completely clueless for many things.
//.
Math, science. But also literature, geography, world events. But no couth is one of the biggest problems.
I admin from home. Sit in my underwear, drink beer, do not shave. See me in public like that? Hell no. I go on an interview for a possible client and I look like the man from IBM in the 80's. The orginal Men in Black
I am 32 and not that old(or at least I dont think so). Here is what I know.
Late 80's schools had gotten so horrid they had to administer tests that had to be taken before graduation. Basic skills tests. You might have passed your exams but still had to take this one. I never took it but I saw one and it was frightfully easy. Along the lines of the ASVAB for the military.
Schools dropped physics and trig to go to things like Alebgra 1,2 and geometry and that was it in math.
Anyone have that physics teacher who used the overhead for the notes? And he had written the notes originally back in 63 and over the years had made corrections to them? But sill used them. Probably still teaching.
TENURE - stay here long enough and we will give you a free cushion for your ass.
I went to a boarding school for my formative years and while I did recieve a fair amount of ass whuppins I did get some great teachers who really got me into science and math and literature. We built a Heathkit Hero in the dorm and fiddled with ham radios, and even had a unix box in '83. A DEC. And I owned your ass playing miner 2049er and Lode Runner on the Apple
I then switched to a local school and bam. I saw the wonders of a regular high school. Sure I got girls and booze and had quite a bit of fun, but I did not learn near as much or the teachers did nothing to generate my interest in things. Well, methane soap bubble torches were fun.
Teachers aren't paid enough. Private schools do tend to get the better ones. I graduated in the end from a public school, and had good teachers, but my private school experience was by far superior. And when I choose to lay my eggs I will make the sacrifice and send my little geeks to a private school. For them.
Teachers also need to be recertified every couple of years, just like people in the tech industry. "I had a TRS-80 back in the day so I don't ever need to take a computer class". Teachers get complacent, light a fire under their asses.
Bit of a rant here, but we do need to do something about it. Our world ya know.
And I do not care if you are 18 and can write a script that will control the weather, make Bill Gates give it all to charity, or even make Slashdotters a more level-headed bunch. Education is the the real fucking deal.
Take the time. I had to do it at 32 and it sucks.
Puto
The Revolution Will Not Be Televised
Thats rather silly. You're basically implicitly conceding that there is no intrinsic importance to mathematics, and it is only "useful" as a means for solving scientific problems. Of course, as far as most students in middle/high school are concerned, what is being taught in science classes is "useless" also, and simply saying "learning X is important so you can do Y" is not a sound argument in the view of a student who sees no reason to understand either X or Y.
I would suggest that this attitude is the main problem, and based on my own experience, it is something that the educational system in general seems to promote. After all, instructors are not necessarily encouraged to promote a real appreciation for and understanding of a given subject, but rather meeting various "standards", increasingly codified very strictly in terms of various new state standardized tests. This environment leaves a student no goal but passing these tests, which whether they reject it or accept it does not enhance their long-term understanding.
Personally, I would rather have seen the intrinsic logic and beauty first, and the "real-world" applications later.
"(Man) tries to live his own life as if he were telling a story. But you have to choose: live or tell." --Sartre
Well, no. What she needed was to retake math, beginning in about 6th grade. Not that she was dumb, but it is almost impossible to do well on a timed test if you're using your calculator to divide by two. All the basic tools the above post talks about were completely foreign to her, although - to her credit - she could FOIL up a storm. Unfortunately, I don't think she knew what factoring actually meant. Forget deriving formulas by common sense or making intelligent guesses to narrow the range of choices. She was convinced that math was difficult and "other", something to be crammed before tests... but nothing she would ever understand. And understanding should be the goal of instruction in any subject (says a future teacher, with her fingers crossed).
Personally, I'm in favor of combining math with anything - science, as above, or music (as one of y'all suggested) - that will help students like mine think of algebra as a helpful tool, or even a "fun puzzle" (our local slang for calc), as opposed to some kind of senseless ordeal.
To be fair, I got an excellent education in public schools (please, Lord, may there be no typos in this boast... err, post), but then, I watch "Square One" and _Donald in Mathmagic Land_ for fun. A good nerdy environment will do wonders.
I think the social interaction issue is a false argument. How much valid social interaction is there hanging around all day with a bunch of dopey kids your same age? I'd much rather see kids grow up interacting with people of all ages in all walks of life. Public school is an unnatural system created to churn out docile factory workers. Any well adjusted, educated people turned out at the end of twelve years is an accident and failure of the designed system, not a success. Of course the schools are turning out poorly educated people. Thats what they are designed to do.
Pardon if this is a repeat.
I remember reading somewhere, and, after much thinking, agreeing with it, that science is currently taught in backward order.
That is, instead of biology-chemistry-physics, we should teach physics-chemistry-biology.
The reason for this is that to really get chemistry, you need a strong grounding in why all those little particles do what they do. To really understand biology, you need to have a strong grasp of chem.
Students today have a very hard time with math - and that's crazy. They shouldn't.
One way to make math more "real" to students is to apply it to science - perhaps if they aren't math-nuts, they'll be science nerds, and the connection will draw them into both.
The problem with this, of course, is that physics is classically taught as a calculus-based course, (although it's perfectly possible to do it with trig and algebra - my AP test 5 can vouch for that)
Chemistry "needs" algebra - at least it works a lot better with it.
Biology (at least at the high- and middle-school level) needs very little math at all.
Therefore, we teach them in reverse order.
As to not teaching algebra, there is no excuse.
I explained the basic principle behind algebra to a bunch of fifth-graders and had them doing "x+59 = 226" in about fifteen minutes.
Everything else is derivitive of that - if the textbooks can't get that across, blame them.
(Note - I would not suggest blaming teachers in the slightest - teaching from books works, even bad books, and teachers, at least in my district, are required to teach from a book - they were good teachers with bad material)
So damn the torpedoes and shut down Houghton-Mifflin!
~Mac~
I was lucky.
My father has a masters in math and physics education. Taught me almost everything, mostly how to think about the problem. That's they key. That's what most schools really lack. They teach how to get it done quick, hard, and dirty. Not to really "think" about it.
Then, to top things off I was accepted to a special mathematics program from the University of Minnesota and took Algebra I&II in 8th grade, geometry and Pre Calculus in 9th grade and calculus I & II (essentially) in 10th.
Had I gone further I would have completed most of my required college math courses by the time I graduated high school.
I can certainly sympathize with you there. By any chance, did you take "Math for Engineers"? I was required to (thank goodness I'm done with that my first year) and I suddenly found myself horribly mediocre in a subject at which I have been the best as long as I can remember. Why? Because the prof said "here is theorem 1. We are interested in these applications.... Ok, here is theorem 2..." while the kids in the Arts and Sciences school were being taught as "now suppose you have delta and epsilon... What can we derive from this? Thus we arrive at theorem 1. Now suppose.... Thus we arrive at theorem 2." They assumed the students were bright enough to figure out "how" with knowing "why," the way math ought to be taught. Ugh. At least I can take real math now that bogus engineering math is over with.
Comment removed based on user account deletion
and nothing else taught in school is boring.
all through middle school and high school i had a ~40% homework average in algebra through calc 2. i also had a ~97% test average to make my B- or C+... i got the whole "you aren't applying yourself speil" but i DID know the material. i was just a lazy fuck and didn't do homework. then i go on to college... homework isn't graded and suddenly i have straight A's.
can not grading homework WORK for a middle school student? or will they all just not do homework and fail?
i have always hated the learning process in math for that very reason...
MARIJUANA, SHROOMS, X: ONLINE?! - E
Back when I was at Uni (not that long ago really) our Faculty held seminars for all the Maths teachers in the local School Region (SE Queensland). The Minister of Education gave an address one year and made the following statement:
None of you should ever expect to meet a student who can apply a quardratic equation to a appplied problem
This boggled those of us who had been teaching awhile (17 years teaching Maths in my case).
What I've seen is that very few mathematicians seem to go into teaching. After all, if you have a decent Maths Degree (whether Arts or Science) you will get a much better job in the Computing/Finance/Actuarial industries. So the vocational teacher who is also a mathematician is a rare bird indeed.
So we have all these ppl who are teaching because they had to do something with their degrees. They don't really understand maths, they don't appreciate the beauty or the simplicity of it. These people's approach to algebra is to 'transliterate' things from one side of an equation to the other! Now how we can take anything balanced and transfer bits from one side of it to the other and still have something balanced escapes me. I know that if I always perform the same operations on both sides of an equation it stays balanced though...
It's the same as "the square root of 4 is plus or minus 2" - absolute crap. Square root of 4 is 2. Roots of the equation x^2-4=0 are plus or minus 2.
Until the pecunary rewards for teaching can attract people who feel passionate about the subject as teachers we will continue to see the ongoing slide in Maths Education.
Note that I'm not teaching anymore - I can't afford to. I work in IT now.
Well, here's my story.
Seventh grade, sitting in the back of a pre-algebra course. Early in the year. The teacher puts a problem on the board and expects everyone to come up with the answer. The goal is to teach order of operations, I think, which is old news to me. The problem is something like (5 * 7 + 3 * 4 - 9 / 3 + 8 - 2 ^ 4) / 2 = ?
So I sit there. As the teacher is walking around, she stops at my desk and asks, "Aren't you going to do this?" I said, "Yeah, it's 18." "You didn't write anything down." And I reply, "No, I did it in my head."
So they move me up to algebra proper. The only thing I remember missing was the idea of the difference of two squares, which took about 3 minutes to explain. They were still using FOIL enough that they didn't need to stop to tell me about that. After taking algebra in 7th grade, I had to visit the high school first thing in the morning to take geometry in 8th grade.
This was in Cary, IL, sometime around 1991. Over ten years later, I'm living outside of Boston, MA. I sure hope they're still as astute as ever.
The result of students doing arithmetic on that $2 calculator is that they have no number sense. Their brains can't make connections between this concept and that because they have no clue what their neighborhood is like. They aren't familiar at all with the "world" of numbers. They don't immediately see the connection between points (1,2) and (3,6) and equation y=2x. They don't see what numbers to use when factoring x^2+4x-21. They have to run the numbers through their calculator.
You say that students already know arithmetic, and really, they don't. They can add and subtract well. Multiplying if they are lucky. Long division is anxiety producing. Fractions are downright scary. Decimals are ok as far as addition, subtraction and multiplication goes, but division is again tough. They get mixed up with negatives and positives -- is -3 - 2 = -5 or -1 or 1...or is it 5, since -3 and -2 have the same sign?
Indeed, the problem is that the dominant philosophy in mathematics education in the past decade has been a bastardization of NCTM Standards. Standards can be boiled down to one [very astute] statement: Most of a student's work should be related to what you are teaching. In other words, (for example) students doing graphing shouldn't be bogged down in long division. This statement got twisted to become "You're in 4th grade now. You can use that calculator for your all of your arithmetic. Look! It even does fractions for you!"
This is what we call a strawman -- misrepresenting a statement so that you can knock it down. If students' arithmetic was all calculator based (which is pretty much the size of it now), and if students' really didn't get substitution (which is pretty much the size of it now), I would be spending 2-3 weeks on both mechanics and meaning.
THAT all said, you're right in that students' (not necessarily numerical) problem solving skills do need to be stretched AND celebrated in math as well (read my previous post about the dyslexic student). You can't go wrong getting a kid to exercise those parts of their brain. Though, you do have finite time and you do have to teach stuff that is explicitly math too, so you have to balance your time that way.
Last year I did a week long project at the end of each quarter. This year I would like to put a smaller logic problem at the end of each chapter to replace half of the larger problems (I have a couple of *REALLY* good projects that I don't want to give up...my group theory project is one of those!)
As an aside: Everything I remember of myself and my friends, from before prolonged exposure to education, suggests to me that children in their "natural" state really do enjoy learning.
It seems humans are hardwired for feeling this way -- until puberty. Children must put forth immense work in learning language, but they do it "automatically" because of this hardwiring. However, at puberty, for biological reasons, students become much more interested in their developing sexuality, which overrides the older paradigm. Because of the great reduction in mean age of puberty since the US general curriculum was developed (during the Great Depression), there is more distraction than was expected by those who crafted this pedagogical system.
Religion is the opiate of the masses. The wealthy smoke the real stuff.
I'll second that. I am very angered by the poor mathematics and physics education I received in high school. For a long time, I felt I never had the patience for math and that it was a topic I only dealt with because I'd been programming since my first "Hello, World!" at the age of four. It was only in first year of college that I recognized that the problem had been not with me, but with my education. I finally had the need to develop more robust mathematics skill to ease other parts of my college education, and so I retaught myself trigonometry, analytic geometry, and differential and integral calculus. Now, I absolutely love the topic and I'm teaching myself differential equations and linear algebra to help further my understanding of the mathematical mechanics I'm sure to encounter in my graduate studies.
It took a while, but I now feel I'm a reasoably decent math student, and I also feel like maybe I can learn what I need. What do I attribute this to, mostly? My high school math classes. It wasn't possible for me to get the learning I needed. I need to get the following information up front: the general idea, the implications of that idea, and the way it works. Once I have that, I can generally teach myself the process of using a mathematical tool. I had the same teacher for my advanced algebra, trig, and calculus class, and she taught only one way- assign homework, let the students blunder on their own overnight, then give credit for the assignment and spend the next day's class discussing what students had done wrong.
Then I'd stroll over to my physics class...my physics teacher avoided math at all opportunities. Often, she would throw a formula up on the blackboard, then replace all the variables with numbers to do a "sample problem" and then simply refuse to complete the problem. She'd throw any random number in at any point in the process of solving the formula. When I asked her in class why the numbers literally weren't adding up, her response was that she was just putting in numbers to "show the class what the formula looks like when it's used." I offered to give her my calculator if she'd keep the numbers mathematically correct, and suddenly I'm facing a disciplinary action for "insubordination."
When I got a C in my high school calculus class, the teacher called me in after school and told me "You'll never be able to get a degree in computer science if you can't do better than this in my class." I can only snicker now, as I not only have the degree she said I couldn't get, but I'm also going back for a Masters. In many ways, I did this despite her "education," since I re-taught myself the subject.
So, I really feel my secondary school educational process failed me. I learned mostly what not to do in alegbra and calculus, and any hope of interdisciplinary use of math was dashed by teachers who were simply too lazy to bother. I shudder to think how it is five years later- from what I hear, public education in the US is only getting worse.
And, in a lot of ways college-level math in state colleges is just as bad. My discrete mathematics professor was nearly fired for gross incompetence while she was teaching my class. My class was the one who filed the complaint of incompetence, actually. Between her inept teaching and the time in class that was lost due to the investigation, I never got much of an education of finite state automata or languages. I taught myself a survival-level amount of automata for a different undergraduate course, but I know so little about languages and how automata relate to them that I am now having to "reteach" that to myself.
Just goes to remind you...you shouldn't let school get in the way of your education.