I'm not sure if "abstract" is exactly the right word, since all those algebra stuff high school students have to learn is also pretty much abstract. What was learning math like for you in high school? I was helping my nephew with his homework (8th grade) and he was working on functions. It basically has tables labeled "x" and "f(x)" for him to put in the numbers according to an equation, which is the function. Similar to this way is the machine approach, where a number enters a machines, something happens, and out comes the function. wtf? Why don't schools just get to the meat of it by stating the definiton of the function?! If students thorougly understand what a function is, then they do not have to memorize "the vertical line test", know what an onto, 1-1, bijections are. From knowing these things, they can move on to better understand continuous functions. Eventually, learning calculus will not be hard, since they are not intimidated by the many theorems and not have to resort to rote memory of the formulas. I'm not an educator, but I believe teaching students how to do proofs is much more useful than shoving a bunch of formulas down their throat. You can ask, "well, when will they use these proofs anyway?" My answer is, "when will students use such things as the quadratic formula and trigonometric identities?" Of course, those who go into engineering and science will benefit most from the current math programs. But with the skill to break things down through proofs, they are more likely to absorb what is taught in college mathematics, engineering, and science courses. So here is my idea: 9th grade: Teach students basics in math, such as properties in numbers (associative, commutative, etc.), inequalities, and absolute numbers. They should be able to show why if a>b, then -a<-b...without having to memorize "oh, change the sign whenever you multiply both sides by negative". 10th grade: Teach basics in set theory (union, intersection, etc.), basic methods in proof writing (by contradiction, contrapositive, etc.), and division... except proof by induction. Basics of algebra is also taught (like 2x+3=4, find x). 11th grade: Mappings, functions, proof by induction. Word problems in algebra are also taught. 12th grade: Continuous functions, limits. What do you think?
If you are a parent and your kids are in high school, ask them these simple math questions and see if they can answer you. Why is it -(-x) = x ? Why does the calculator say "undefined" when you punch in 3/0 ? Why is x > y, but -x < -y? If |x| > 3, where on the number line does x cover? If they answer to the tone of "because the teacher said so", then you should talk to the teacher. :grin:
You're probably correct. It's a public school. So I'm guessing rich private schools teach these things? I wouldn't know because I too went to a poor school.
I can't really say what your nephew's school is like, because I went to Bellaire. It may not be a private school, but in terms of math and science it's quite good. Took Calculus in Sophmore, and Calculus 2 in my junior year. In terms of what we should teach to students, my opinion is that it mostly depends on who the students are. Abstract math is almost worthless in some fields. But vitality important in others. I honestly believe that for some, there's no reason to understand math beyond normal algebra and geometry. Seriously, who sees f(x) in everyday life anyway? In China, many high schools are divided, where students pick either a math/science heavy curriculum or a fine arts curriculum. So that's their method of dealing with what you mention. Obviously it's not applicable in the US, since we teach to the lowest common denominator to make sure everyone passes...
I totally understand your point. But the knowledge of abstract math leads to better absorption of the material taught. Similar to what you said, who sees those calculus functions in everyday life anyway? The main thing to learn from proofs is the skill to break a problem down, an important ingredient to success in college. It's good that you learned calculus in high school. But if you had a better understanding of the theorems, maybe you might have pursued a scholarly career in math or science. Just a guess. I never made it to calculus in high school, only to pre-calc. But it was because I didn't understand what the heck I was being taught. It was just a bunch of applications of the formulas. About US teaching the lowest common denominator so everyone passes. Yes, it's true. But what is the requirement to pass high school? That they complete a course of geometry and algebra? Well, instead of learning all those geometric figures, students could be taught to think mathematically through some basic proof techniques. My point is that it's not the usefulness of the abstract math that most students will use later in life. It is the skill gained from learning it and the motivation that it leads students to.
What do you think is the reason why many college students these days take remedial math classes? Even the "college algebra" classes are mainly remedial stuff. I think this is because students did not understand the first time around in high school. And since algebra is taught in college anyway, why do we need to cover it in high school? The same goes with precalculus and calculus.
Another question would be what percentage of people who take classes such as trig, pre-cal, cal, etc. in high school actually benefit from those classes during life. In other words, how many people actually use much of anything they learn from those classes once they get out of high school/college?
I made it up to Calculus 2. I went to CFISD. I met a lot of people who I couldn't believe passed Algebra 1 in my high school. How far they make it into math really depends on how much they like math/numbers.
I agree that presenting formulas and theorems without proofs is basically worthless. Even now at college some lecturers do that. I usually look up the proofs in literature if that is the case. Even if I can't reproduce the proof or derivation after a while, I still know the basic gist of it and, as you say, it helps remember everything. But teaching basics of algebra in the 11th grade seems late. I was taught these things in 5th or 6th grade. And then we went deeper in it in high-school.
I personally can't remember something like this ever happening. The heaviest memorization work I had to do was for trigonometric identities, since my teacher didn't prove 'em, but that was nowhere as hard as memorizing dates for History. Requiring students to prove is overkill. It's an application of logic which, by itself, is an ordeal and requires a level of maturity that only a very few have at a young age. I would like math to be taught on a somewhat historical basis like how Pythagoras discovered that the hypotenuse of a right triangle is the square-root of the sum of the squares of the two other sides. It gives credence to the material which, normally, students feel alienated by, by letting them realize how so and so was discovered.
You can reduced any K-12 (or K-16) education this way, omitting tons of stuff "you never used." And I have never ever read Edgar Poe again! I have never ever done an experiment involving dry ice again! I have never talked about the assassin Princip again! It's now pretty hard and fast neuroscience that the mind improves with exercise and this is especially important in our youth. The more kinds of circuitry you try to wire, while wiring is actively happening, the better. I wish I had pushed myself even harder when young, and I especially wish I'd taken on more foreign language study. I'm a pro-learning-for-learning's-sake, a Stanley Fish kind of guy. We didn't get to be great in this country by just saying "what's the minimum we need here?" BUT even on a pragmatic level, the dudes who found out that you could build solid state circuits out of the quantum mechanical properties of a semiconductor slab... they used a lot of their math and science learning. Think maybe it was worth the shotgun approach of everyone getting exposed to it, just so these key people were able to change the planet? Opinionated educator is opinionated! :grin:
So did anyone read post #3 where I asked parents to ask their high school students those questions? Try it and you'll be surprised that they don't know.
As someone now teaching in a public middle school... the problem starts way before 8th grade or high school math. The way they are teaching elementary kids basic things like subtraction and multiplication is hindering their learning all the way up. They teach lattice instead of regular multiplication, they tell them you can't subtract 5 from 2, etc etc. Plus they are forced to teach only material that is going to be on the standardized tests because it affects the amount of funding the schools get. There are a lot of problems with the public school system in the US and simply stating teach the abstract isn't going to get it done. Let me give you an example. This week we are starting our Algebra chapter in 6th grade. We reviewed exponents first, which they should have learned in elementary, and they did okay with that. Then we taught substitution, if a=3 and b=4 what does ab = ? They struggled with this a little because they couldn't keep the signs/operations the correct way. So we worked through those problems and moved to substitution with formulas. You would have thought we were speaking a foreign language with the word problems we chose. Each of the problems GAVE the formula... all the kids had to do was match the values in the question to the variables in the formulas.... and they struggled HUGE! Educating is a lesson is containing frustration, that's for sure.
Yeah, I see college freshmen who struggle with those concepts for sure. It's sobering, and what Hayesfan says in containing frustration is spot on. For the record, Hayesfan, what you are doing is amazing. You are really in the trenches. By the time someone like me sees students, they have already tipped one way or the other in terms of loving/hating learning/school.
Wow. That's what some people don't get, that learning the current algebra taught in school is just as abstract as abstract math. I decided to see if what I was talking about is true, so I asked a few (a small sample, I know) college freshmen to correctly do some problems involving absolute numbers. Like I predicted, they didn't even have the definition down. After I showed them the definition...they still couldn't do it. This was what I did. I asked, "what does |x| mean?" Their answer was "x and -x" Then I asked "what is |-4|?" they wrote, "4"...without knowing why. So I had to show them the definition of absolute numbers and showed them |-4|= |-(-4)|=|4|= 4 , since |x|= -x if x< 0. Yes...they did not realize that -4 replaces the "x" in "|x|" so that |-x|=|-(-4)|. Shocking. But back to your point that teaching abstract math isn't going to get it done. What about maybe the schools slow down on the math taught (as if it isn't slow already) and teach just ways to think mathematically? Teach students to hammer down on the basics in math before moving on to algebra. If it takes so long that they barely cover algebra when they reach college, then so be it cause they'll learn algebra again in college anyway. Go to a community college and see how many students are in remedial math, so obviously the current way of teaching is not working either. But I'm surprised after you mentioned the problem starts way before high school.
I hate proofs. Proofs are pretty pointless.The people who come up with them are probably some of the smartest human being to ever live. Understand the concepts and then you can write a program to solve it.
