Simple question, I believe. Why wait till the teen years to begin learning calculous or physics and the like?
For one, you should be able to spell “calculus” before learning it. 
But mainly, there’s a lot of foundational stuff you need to learn frist. Calculus doesn’t make sense without first learning basic algebra and some trigonometry. I suppose you don’t really need to learn geometry first, but knowing geometry helps with trig.
For example, taking a differential to find the instantaneous rate of change of a function for a specific input doesn’t make much sense if you’ve never graphed a function before and found the slope the manual way.
I think it’s ok to wait until high school to teach the calculus, but I think high school graduates should be required to have a general understanding of it.
(dang you for pointing out my spelling)
I understand that everything builds upon the basics. But why not move things along a little faster. I remember learning addition and subtraction in kindergarten and first grade, then multiplication and division in third, and then finally algebra in 8th grade. That seems like a lot of down time in between those years.
Don’t the younger kids soak up knowledge and learns things at a faster rate?
There are also changes in the brain that happen at or just before puberty that allows for the more logical/abstract thinking that higher math requires. The human brain goes thru various stages of intellectual development before a person is both phsyically and mentally mature. So, even if you tried to give kids a crash course in math to go from algebra to calculus in grade school, most kids would simply be unable to grasp the concepts.
I suppose there’s no reason you couldn’t move things faster, but the disadvantage of any school system is that you have to cater to the students who aren’t prepared to go so fast.
Remember “word problems” from elementary school? Like, “Billy picked ten apples and gave two to Sam and three to Charlie. How many apples does Billy have left?” This type of thing is really just very simple algebra – they’re trying to prepare you for the thought processes involved in learning formal algebra later.
In grammar school you also learned things like fractions and decimal places, and (if your school was into that “new math” craze) counting in alternative bases. If you learned multiplication and division in third grade, you probably learned how to do those operations on fractions in fourth grade.
Basic algebra requires operating on fractions all the time.
In grammar school you probably also learned about shapes: the difference between a square and a rectangle, the three types of triangles, various polygons, etc. Later on you probably learned how to measure angles. This foundation knowledge helps a lot when you start doing formal geometric proofs in high school.
So, the point is, often you may not even realize how all the basic skills you learn in grammar school are necessary for more formal mathematics later in life. The most important foundation skill, which is often overlooked early on, is simply learning to think in a logical way. That’s one of the harder things to teach and can’t be learned by rote.
The newest, bestest mathematics program out there right now that has math teachers excited is the University of Chicago’s Everday Mathematics program. While it doesn’t include calculus, it’s trying to move educators away from the Mathematics/Geometry/Algebra separation paradigm, and interweaves concepts from all these disciplines to teach math in a totally different way. (I should note that it’s a fantastic program for kids who started it from the beginning. It’s a bear if you move into a district teaching it and you have to jump in the middle somewhere.)
Kids learn fast, true. But a classroom is, of necessity, only going to move as fast as the slowest kid (or maybe the third slowest kid). Plus, there are dozens of other requirements that have to be taught each day. A math class is about 45 minutes per day, of which at least 10 are consumed with finding your homework from yesterday, passing in papers, finding your page, finding a pencil, sharpening it, passing a note to the girl in the next row asking her if she thinks Mike is hot yes or no, brushing your hair, dropping your pencil and shooting Mike dirty looks when he kicks it under the cabinet.
Plus, y’know, all those weeks straight where no one is learning anything at all new, because we’re “preparing” for state testing. Not to mention the weeks of testing themselves.
When walking home from grammar school, I passed the junior high, and could see and hear where they were teaching algebra. I used to sit on the fence outside that classroom and soak up the algebra well before I got to the grade where they were teaching it. I found that grammar school math was boring – I really wanted to move ahead. Had there been avenues for taking calculus early, I would have. But, as others have noted above, public schools have to serve a broad range of students.
It’s true as well that you need the background to take calculus – that algebra class was one of those bases. I ended up taking summer classes to get further ahead, and in high school took a pre-calculus mathy class that was half calculus, then finally a year of calculus in high school. In fact, that (and a summer of study on my own) enablecd me to advance-place out of my first term of calculus in college. At MIT.
So, yeah, you could do it. My high chool wasn’t particularly well-supplied, but I still left it with over a year’s worth of calculus.
Interesting program! I’ve long felt that where math instruction falls down is relating what is being taught to the student’s experience. Word problems, instead of being the source of kids’ nightmares, should be the base upon which the class is built and, if they are taught right, they can be. The abstract should only come later, after the practical basics of all the disciplines are mastered.
There’s really no reason why you can’t teach calculus at a conceptual level in grammar school, as it’s really just a generalization of arithmetic, especially if you start out with the infinitesimal approach and leave the more mindwarping difficult topic of limits to later. The concepts of functions, graph theory, sequences, series, the fundamental theorem of calculus and the mean value theorem can all be demonstrated visually and by experiment without using any “ugly” math involving trigonometry. This would create at least a good foundation for later understanding the details of trig, geometry, and calculus.
This would of course require educators to understand calculus, which I daresay would be an uncommon ability among grammar school teachers. This is probably the fundamental reason why we don’t. In my experience, many grammar school teachers lack a fundamental grounding in many subjects that they teach, including math, science, and history, and resort to teaching by rote from textbook. This may be fine for the average student, but it’s an absolute drug for students who are able to grasp general concepts.
Stranger
I never took calculus. I ran aground on the shoals of advanced algebra in 8th grade, managed to claw my way through geometry and trigonometry, and then dropped math except for a nasty year of statistics in the graduate sociology program many years later. I hate math.
My recollection of math in elementary school is like this:
1st & 2nd grade: Learn addition, subtraction, multiplication, division. Principles taught mostly implicitly.
3rd grade: Drill drill drill drill drill drill drill drill. Monotonous and boring beyond belief. I think they might’ve introduced simple “word problems”. Show your work. Add subtract multiply and divide.
4th grade: “New Math”. More drill drill drill drill drill except now the principles are taught explicitly and given names. Oh, and fractions and decimals, which are still just variations on the same old same old multiply and divide stuff. Pounds and pounds of homework. Monotonous and boring beyond belief.
5th grade: More drill drill drill drill. A few new and fresh ways to make us crank away at the same old addition subtraction multiplication and division, such as “Base 8” and “convert to and from metric”. Some new concepts (exponents and scientific notation; logarithms, probably).
6th grade: More drill drill drill drill drill drill drill. Everything in math is about making us add subtract multiply and divide. 7 + 7 is still 14 and 7 x 7 is still 49, no matter what contrived & convoluted situation they invent to make us add subtract multiply and divide. Life as a human calculator. Boring boring boring boring boring boring boring boring boring boring boring .
7th grade: Lots and lots and lots of yet more ways to make us add, subtract, multiply and divide. Did I mention boring boring boring boring boring boring boring boring boring boring yet? Obviously math exists only to make us brain dead and they will never introduce any genuinely new concepts.
8th grade: What are the factors of 8x[sup]2[/sup] + 11xy - 9y[sup]4[/sup]? Please diagram them on this xy graph. Solve for x for each recursive value of y and find the first and last point where x is a negative number. :eek:
I agree with the OP. There was plenty of room for them to have introduced new concepts instead of making the math part of my brain rust closed. I was totally turned off to math and then when they finally saw fit to actually dump new stuff on us, I could not keep up. (And it was still about getting the right answers more than it was about comprehending what math was and what you could do with it. I don’t like calculating, it makes my brain hurt, but I might have enjoyed the concepts)
What does Grammar School mean to you?
For me (as a Brit) it means a secondary school (ages 13 to 18) with achademic rather than vocational leanings.
(they used to require students to pass a test at age 12 to get in)
Indeed Calculus is taught at Grammar Schools in UK.
In the US grammar school (now more generally referred to as elementary school, primary school, or “grade school”) is the first 6-7 years of education. Classes are typically compulsory with little in the way of individual option or variation. Starting with middle school/junior high school students are more often placed into classes according to ability and/or interest. In the US there are, as a general rule, no separate instututions for vocational versus college prepatory schools at the secondary level, although there are often programs for vocational training and course tracks for college prep (Advanced Placement and Honors classes). This is less uniform from state to state or even between school districts within a state than is the norm for European nations.
Stranger
Let me offer another reason not to teach Calculus in grade school: There are no electives in grade school. Calculus, as an elective, is taught to only a fraction of HSers. How many folks would be able to pass a mandatory calculus class even in HS? Now, imagine the same situation in grade school. It would be a disaster. And offering it as an elective would be almost as bad-- parents with ridiculous expectations of their kids would just make those kids’ lives miserable. Sure, you could do it in the later grades of grade school (5-7), but it would be a terrible idea.
My $0.02 is that geometry and trig are much more massively useful in everyday life that calculus, so it behooves everybody to have geometry and trig taught first, and wait to teach calc when not everyone is still taking math classes.
Of course, if I had my way, some practical statistics would be mandatory, too, but that’s another rant. (Darn kids… grumble…)
It is interesting, isn’t it? It’s been really hard to be a parent (not highly skilled in math) in the program, however. My kid has learned entirely different algorithms than I did - for instance, he multiplies from the left, and doesn’t “carry” numbers. I understand the logic of his system (called a “partial product algorithm”, described here), but it’s not at all intuitive or quick for me. Yet he can beat me, speed-wise, and it makes the underlying concept of place value more explicit than my method.
The only part where my child’s district has really fallen short in the program is by not educating me about it better so I can help him. There are parent books, but they are not routinely distributed in our district.
I never did take calculus, and it’s never been a hindrance to me, honestly. It’s just not important in any other field I’ve studied, or as a homemaker. Fractions and geometry? Heck yeah. I’d never be able to double a recipe or figure out how much paint we need for the bedroom if I couldn’t do that. I did take statistics, which I loved, and I did get as far as trig. But I’ve not even needed functions or graphing that I can think of since college, and calculus has never come up. But I think if you’re bored in math class you should absolutely study what challenges you - your brain needs “exercise” to keep making those neural developments in childhood. Teaching math teaches your brain how to learn, not just math.
Trigonometry/Statistics and Algebra 2 teacher checking in here.
I have seniors in my Trig/Stats class who still can’t do the math they should learned by rote: add/subtract or multiply/divide fractions.
Me: “The sine of alpha is three-fifths, and the cosine of alpha is four-fifths. What is the tangent of alpha?”
Crickets: “Chirp, chirp, chirp.”
Me: “What is the ratio identity for tangent?”
Meek voice: “sine over cosine”
Me: “So what is the tangent?”
2nd Meek voice: “three-fifths over four-fifths.”
Me: “And that simplified?”
Crickets: “Chirp, chirp, chirp.”
Me: “What do we do when we divide by fractions?”
Multitude: “reciprocal… multiply… numerator… denominator…”
Me (interpreting): “multiply the numerator by the reciprocal of the denominator. So that’s 3/5 times?..”
3rd Meek voice: “five-fourths?”
Me: “Right. 3/5 times 5/4 is?”
Loud voice: “Mr. B, can’t you cancel the 5’s?”
Me: “Yes! Canceling the 5’s leaves 3 in the numberator and 4 in the denominator, which is…”
1st Meek voice: “three-fourths?”
Me (sighing): “Yes.”
(continuing)
So if kids aren’t retaining 3rd/4th grade math until 12th (even though they should’ve been practicing it for 8 years), what hope would there be that they’d retain calculus? Better for them to learn it closer to the age when they’d actually use it.
Although, schools could/should start formal algebra in 7th grade. Up through 8th now, they’re taught psuedo-algebra. But if they were taught Algebra in 7th and 8th, then Geometry 9th, Algebra 2 in 10th, then Trig/Stats in 11th, I think concepts would stick better.
Also, a lot more practice with fractions.
(BTW, my students are supposed to be Honors level. Also, I had football players that I gave F’s on weekly interims, but lo and behold, they played on the team.)
Good lord, why would we want to? I had AP Calculus in high school; passed the AP exam with a 4. Know what I remember? If you take the derivative, you get a point. I wish to hell I hadn’t been born in time for “new math”, maybe then I wouldn’t have to look it up every time I need to know how many teaspoons in a tablespoon or how much four cups of broth is in gallons. If they spent more time teaching kids geometry, simple trig, fractions, percentages, and estimations in your head, now that would be useful. Oh, and understanding statistics.
You do realize that’s not correct, right? I’m not trying to make fun of you or anything, but I just want to point out that you can’t remember that from calculus anymore than you can remember learning that the capital of they US is Paris in geography class. A point would only be the derivative in one specific case: f(x) = ax + b, where a and b are constants.