Slashdot Mirror
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."
Yup, today your kid is doing algebra, tomorrow he's smoking crack. Just say no.
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 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
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)
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
My goverment has informed me, a patriotic citizen, that I should be especially aware of anything arabic sounding around me. They like to sneak things in to try and destroy our beautiful country, and this could be one of the very plots that brings about 9-11 the sequel!
Why, filling our kids heads with islamic math propaganda is the last thing we need right now. Will it help us build bigger bombs? No, I don't think so. Counting to 10 is enough, and if you forget a few numbers in between, that's alright by me. President Bush himself can't count to 10 without his advisors helping, and I bet none of them know al-jebrah either.
Al jebrah is a tool of the devil! It might help when you're trying to decide how many camels to give away to marry off your daughters, and it might even help to figure out how to build those crazy pointy towered mosque thingies. But as americans, what good does that do us?
Besides, they come right out and say it. It leads to godless science, teaching us that we're the grandchildren of monkeys. Yes, cousin Cletis kinda looks like a chimp, but by god he's a good 85% human. Keep your godless atheist algebraic satanic brainwashings out of my kids skulls!
(stupid lameness filter won't even let you do a *** seperator bar)
Dammit. Spent 20 minutes writing one of my best trolls ever, and I can't bring myself to click 'submit'. It wouldn't be a big deal, but I know people like this... ugh. I'm wimping out.
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~
Interestingly enough, our word Algebra comes from the book Hisab al-jabr w'al-muqabala written by a Middle-Eastern man named Abu Abd-Allah ibn Musa al'Khwarizmi around 830.
Brian Ellenberger
In an effort to overcome our country's mathematics woes, the National Council of Teachers of Mathematics (NCTM) put together a monumental group of standards and principles revolutionizing the way that Mathematics is taught at the High School level.
The NCTM-based curriculum is different. Some teachers and college professors believe it to be weak on mathematics because it doesn't look like the curriculum they grew up with. Traditional curriculum (teacher does a couple examples, students practice solving 30+ problems similar) has not been good enough though.
The new curriculum, based on psychology and education research from the latter half of this century, focuses on understanding in addition to the traditional acquisition of skills. It is mathematics rich with connections to other areas, and deep in content. Students start in 6th grade learning basic algebraic concepts, number theory, geometry, probability, etc. Obviously mastery of all these concepts does not happen in a single year. In fact, the curriculum spirals around the same concepts, building new understanding and making new connections with each pass so that, ideally, when students graduate their skills AND understanding will be better than that of previous generations.
Sometimes this math is called "Fuzzy Math" or the "New 'New Math'". Some educators, professionals, parents, and children feel the curriculum is weak on "real math." My concerns were similar before I started teaching the Interactive Mathematics Program (IMP).
Between 9th and 10th grade, students master basic algebra, learn the basics of the trig functions, work with standard deviation and the chi-squared measure, build and solve and maximum profit linear programming (something most math majors don't do until grad school), derive and prove the pythagorean theorem, work with exponential and logarithmic functions, do all sorts of number-theory related problems, and so much more. Still IMP and other standards-based curricula have their problems. In my opinion, although there's plenty of problem-solving and understanding-based activities, there needs to be more traditional skill work. I supplement my lessons with such work where appropriate. Any teacher worth their stuff would do the same. Additionally, the curricula is very wordy, which is fine for middle-class suburbanites, but when you're teaching in a city where 25% of the students don't speak english as their first language, and 75% are in poverty (typically correlated with smaller vocab and weaker reading/writing skills), a wordy curriculum is just one more thing making it tough to teach/learn math. In sum, there's a lot of hostility from the non-math-teacher world toward this new curriculum because it's so different. But, with the abismal performance of American mathematics when compared internationally, it can't be business as usual. The curriculum is already working well in the classes I've seen. And the research points to positive improvements after curriculum implementation (no large study has been completed as far as I'm aware). NCTM-based curricula is no panacea, but it's a definite improvement over the more archaic traditional curricula.
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.
One of the things that seriously separates humans from other animals is our ability to think, and to think abstractly. Too often the comments are made about algebra- "I'll never use this..." "What is this good for?"
Even if algebra problems per se never occur in whatever "real life" people end up having, the ability to think quantitatively is essential for an reasonable person. Thinking more abstractly about problems of many kinds is essential- for developing efficient code, for having a reasonable business plan, for managing one's person finances, for voting in a responsible way, and basically for being a productive member of society. The evidence for poor critical/mathematical thinking is everywhere- people falling for Ponzi schemes, short-sided economic policy, unwise credit-card debt, bad laws, ridiculous jury decisions, and the list goes on. The proper perspective about mathematical reasoning is that it is fundamental for most productive people, and essential for all citizens.
Unfortunately, this perspective is usually not instilled by our current generation of underpaid, frequently under-qualified (more than half of the math and science teachers in CA have "emergency certification", which can be extended indefinitely since there is no adequate supply of properly trained and willing math and science teachers.) Instead, students are often exposed to math teachers, who, to be honest, don't actually like math or understand its central role as a foundation for science and modern reasoning. Kids are smart- if a teacher doesn't like math and is just going through the motions, they pick up on that. And given the sympathy that students get from parents, teachers, etc for the horror of "word problems" it isn't a surprise that mathematical reasoning skills are a consistent weak point of students at all levels in the US.
Everyone agrees that more resources should be directed at education, but people have been agreeing on that for at least 30 years with much of the same problems enduring. Good education is more expensive an investment than many decision-making bodies are willing to undertake, and that shows in the wide disparity in education between the "haves" and the "have nots". Until there is a significant change in how much energy and money people are willing to invest in education, it seems that these phenomena will continue.
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
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?