More practice with fractions? Screw that! I convert them to decimals and go from there.
Yeah, I couldn’t function without a calculator–and they didn’t become cheap enough for me to buy until AFTER I graduated from college so I was non-functional until I was 22–but after years of converting fractional inches to decimals so I could key them in I can do up through 32nds instantly in my head so I got the answer to your example without entirely knowing how I got there.
However, though this is more suitable for GD than GQ, what is the goal of teaching people math? Simple arithmetic’s value is obvious and as annoying as it may be for a kid he needs to learn his times table and be able to do some calculating mentally to function at all. But what good are algebra and geometry to an adolescent? He learned the difference between a circle and a triangle watching Sesame Street but what good does it do him to go any deeper? And how do you get that across to him (me)? How do you show him (me) the value of learning to solve
-75x[sup]2[/sup] + 290x = 240
by factoring? Without knowing how that relates to his (my) life it’s so much Chinese.
But this is where Calculus is a better subject for younger minds. It does relate to everyday life much more than algebra. Calculus being the math of motion, flows and anything in a changing state. That said, how to make it accessable to someone without a great deal of algebraic and trigonometrical maths background would be a great problem but worthwhiloe if achievable.
I’d rather give Zsofia the benefit of the doubt; after trying to parse the sentence in multiple ways, I eventually arrived at the interpretation “if you take the derivative [correctly], you get a point [on the exam].” Sadly, this interpretation vindicates the critics who deride our school system as overly focused on testing to the extent that real learning rarely happens.
I wish I’d had that when I was in school. I was always interested in the relations between numbers and things like curves and shapes, but got so bored with the arithmetic drills and being set scores of multiple place multiplication each night in the third grade, that I totally turned off math.
Sorry, that’s what I meant - you get a point on the AP exam. Which means I retained exactly no calculus, despite having one of the best math teachers around. Because in my life to date not only have I never used it, but I can’t even imagine how I could possibly use it. (I recall some of the “story problems” on the AP exam - I believe at one point in time I could theoretically determine how tall a given hill was, assuming it had precisely the shape of a sine curve and I knew the equation that graphed it. Yeah, thanks for that and all.)
I know calculus is extremely important to a lot of people, even in their daily work environments - just, you know, not the vast majority of humanity.
Oh, and you don’t have to take it correctly, at least you didn’t. You just had to write “dy/dx”.
And my teacher in no way “taught the test” - she absolutely loathed it when they made her. Boy were we ever shocked when we had to keep doing calculus after the exam!
You could teach calculus a lot earlier, but you have to have algebra first, I’d say. Algebra is just the rules of numbers, including how they combine, the order of operations, all the associativity, commutivity and distrubution rules, etc… well, there’s also graphing and matrices in there somewhere. The x’s and y’s scare kids, but I don’t see why we couldn’t teach algebra concurrently with arithmetic. Then we could reserve the fun math, like calculus, for high school, and some people wouldn’t leave high school thinking that math was just mindless arbitrary rules. Which of course is all that arithmetic and algebra are unless you go further and see why they are that way.
Kind of a hijack, but what’re considered the necessary prerequisites to advanced algebra (the study of groups and rings and such)? Can someone reasonably jump from calculus to that without an intermediate step?
I too am a member of the elistist intellectual club, but I think you haven’t met our lower half in the population.
I’ve taught kids who literally couldn’t draw a graph in 11th grade.
Now, you could argue that they shouldn’t be given a HS diploma, and in California there is now a HS Exit Exam, (CHSEE [KaySee or Kaah See, depending), but I think that understanding calculus is beyond the call of being considered a HS graduate.
As it is, out here in the sticks, a significant portion already can’t pass the algebra component of the CHSEE.
I have not read your reference at this late hour. Is this program just “new math” or the recent trend in “integrated math” in new clothes?
The problem the local districts ran into with “integrated math” was that only the brightest students had enough time to assimilate the new material. Topics were so short-lived that kids couldn’t remember what they had done and the knowledge slipped away because it hadn’t been hammered home enough.
Now, to me, this is evidence of the need to somewhat track kids, meaning that you separate them according to ability, and let the thouroughbreds run to their ability, but in many HS these days, there’s not really any such thing, meaning that the brightest are shortchanged by the limitations of the lower third of the class.
I know that in my chemistry classes, I was pointedly limited by kids that in other districts would never have even been placed in the class, but there I was, needing to go back so that my median score would be a straight “C”, so that I couldn’t be accused of moving to quickly. Any high level class like that will drop a couple people at the semester, but in my district, it was people who couldn’t even deal with covering only the most basic state standards.
Of course, we could say this about any school subject. 90% and more of the population doesn’t use a certain HS subject in real life.
I have a big speech to kids about how HS is not about the subjects, but about how they’ll handle problems and difficulties, and how they’ll charge ahead to “fix the problem”. Their chemistry grade will quickly be forgotten, but if they learn to study when they need to, and to fix the problem because they need to, they’ll move ahead in the real world.
And when someone tells them that the oil companies are repressing the cars that run on water, they can roll their eyes with the rest of us.
I’d rather see primary and secondary education provide the kids with a solid understanding of the basic math through algebra, geometry, and trigonometry. I’m an engineer and I had the adult dose of calculus in college, but unless one is going to be an engineer or a physicist or maybe a chemist, you really aren’t going to use it. Sure, you need calculus to get a solid understanding of physics. But not every student of physics needs to be able to derive the equations of acceleration and velocity and distance in order to make use of them. For the few people that will actually need calculus to succeed in their professions, they will have ample opportunity in college to pick it up. For most college majors and for most careers, calculus isn’t necessary.
There is an incredible amount of sophistication required. to really learn differential and integral calculus. Kids just aren’t ready for it. In fact, if anything is clear from this thread it’s how many 'Dopers don’t understand it.
As it is, we wait until high school or college (with a few exceptional cases) to try to teach it, and still for at least half the people in the class we walk away from them physically exhausted from trying to pound a glimmer of comprehension in. Calculus is not just a new bunch of formal manipulations on algebraic expressions as Stranger On A Train seems to think, but for most students that’s the best we can hope to communicate.
To learn the calculus requires, at the very least, a strong foundation of algebra. Along with this is a basic understanding of the notion of a variable. If you don’t grok what “x” is in an algebraic formula (at least at the high-school algebra level), you’re sunk before you even leave the harbor. Beyond that it needs trigonometry, exponentials, and logarithms to provide more fodder for examples than mere algebra can provide. It also takes an abstract notion of function, which the calculus will refine to notions of continuity and differentiability. Finally, the ability to follow a complicated logical argument is essential, not just because presenting proofs is standard in the pedagogy, but because it’s almost the only way to get to any examples that give a hint of the true scope of the theorems.
Oh, easily. In fact I was playing with groups before I was done with the calculus sequence. The problem is that the prerequisite for abstract algebra is not material, like how you need to know the product rule for derivatives before you can understand integration by parts, but experiential.
Abstract algebra is all about proofs and arguments for this and that proposition. You must be able to read (and understand!) all simple proofs and most moderate ones, and be able to do most of the simple ones yourself before even starting. How do we get students to the point that the can handle proofs? By showing them proofs through calculus and linear algebra and differential equations…
Incidentally, because people have already seen the basic ideas before, “advanced calculus” (an analysis course at the upper undergrad level) is often used as the final ramp into proofs rather than abstract algebra.
Then again, I was born in 1974, so I don’t really know what “new math” is. Or, more properly speaking, I know only “new math” and don’t know what makes it “new”.
This is totally different in pedagogy and methodology, at least. Kids still learn to add, subtract, multiply and divide, but at different rates and with different methods. They introduce functions and statistics in first grade (only they don’t call them that) and things are repeated over and over and over, but not in a rote way. There may be a page of three digit multiplication or fraction division review every few weeks even in the higher grades, where other books assume you remember how to do it - but there aren’t 10 pages to bore you out of your skull. There may be two pages of what looks like algebra, and then a page of geometry, and then two pages of arithmetic. Things are grouped by concept, not by modality. They are arranged logically, so that each lesson inevitably leads to the next, instead of an arbitrary, “We’ll be starting fractions this week!” announcement. Finally, everything is grounded in real world applications. Formerly known as “word problems”. I have, surprisingly enough, never heard “Why do we need to know this stuff?” about math class from my kid, and he’s in eighth grade now. (I *have *heard it about every other subject.)
Slightly off topic, but I think kids should learn double-entry accounting as soon as they can add and subtract well. I learned it in my late 20s, and the whole time I was learning it I was wondering why they didn’t teach me this in 4th grade, when perhaps it might have been absorbed more naturally.
That’s what I did, and it was definitely pretty challenging. I would’ve had a much easier time of it had I waited until after I’d taken analysis and linear algebra.
Slightly off-topic as well, but it kinda fits. My senior year of college I took an intro accounting class to fill a requirement. (It was that or something like intro sociology, and there was no way I was taking something an intro to something that fuzzy. I learned that lesson with an awful intro psych class.) Anyway, I’m taking the class as well as three or four upper-level biology majors who are taking the class for the same reason I was. We wind up all pretty much sitting together in the back of the classroom and none of us have any problems with the material. I’m thinking that this is a piece of cake after calculus and P. Chem. and I assume that the bio majors were thinking about the same, referring to whatever hard classes they had taken.
The class was otherwise pretty much nothing but freshmen who wanted to be business majors and you would not believe how many of them had a hard time understanding double-entry bookkeeping, much less some of the more advanced topics. The class could just completely stall out for five minutes while someone asked the professor, yet again, why that entry goes on the debit side and not the credit side.
If you included Calculus in the same manner that Chicago Math presents all the rest of the math concepts, you would have kids in first grade doing half a differential between their addition and geometry proofs. This latest fad has proven to be a real flop where I live. Kids have to re-learn (learn for the first time, actually) all the basic math concepts when they get to college, unless they have teachers that will stray from the book and teach them along the way.
Interesting. Do you have any statistics or news articles about this? I’d like to know how wide-spread the discontent it, if it’s in kids who were with the program from the start, and what I can do to prevent any problems for my son.
And I’d like to forward anything you have on to my mother, who’s been teaching with the curriculum in another district for years with no complaints from the students or the high schools. Lots of her students return to visit after freshman year in college to thank her - she was the teacher who brought the system to her school district. I wonder what’s going wrong in your district (or what’s going right in hers.)
As I said before, my problems are with my district’s implementation and lack of parental education, not the EM curriculum itself. Whenever I whine to my mother, she says, “But didn’t they give you the handbook? The answer to your question is on page 38!” Which, of course, they didn’t.
