@Thucy

What?

statistics being important

Oh, sure. I guess I don't understand why statistics wouldn't be considered math...

I would say that a solid understanding of ratios could only assist in making design choices in art.

Get passionate about FGM. This is one social issue that can be affected massively by a concerted effort by those in education.

"If a kid wants to be an engineer, having a solid understanding of calculus before entering college will help them immensely. Do you disagree? "

First, you suggested it was (more or less) necessary, not just that it would help them immensely.

Second, yes, I do disagree. A majority who take calculus in high school are going to have to take it again in college anyway, and I haven't really seen evidence that they do any better than those who haven't. In fact, often they are vastly underprepared in algebra, and end up doing very poorly in college calculus and having to change their majors, without ever fully understanding how it is that they weren't able to do in college what they thought they could do in high school.

I'm not saying it has to be a DISadvantage to take calculus in high school, if a student is taught it well and really gets it. But I'm saying that I can't really imagine what the "immense[] help" of taking it there is supposed to be, either. There's plenty of time to take it in college.

Essentially, I agree with these articles (which I just now found), based on my own experience both taking and teaching calculus.

http://www.washingtonpost.com/blogs/class-struggle/post/do-the-math-too-much-calculus/2012/08/16/9a35034e-e4fd-11e1-8f62-58260e3940a0_blog.html

http://www.maa.org/the-changing-face-of-calculus-first-semester-calculus-as-a-high-school-course

http://blogs.edweek.org/edweek/curriculum/2009/04/a_rush_to_calculus.html

" There's plenty of time to take it in college."

Let me guess, mommy and daddy paid for your education? Those of us who paid for our own through work and student loans are glad we took Calculus in high school and then too the AP exams so we didn't have to take it in college. One less expensive class to pay for.

And if done right, taking Calculus in HS doesn't have to neglect Algebra. I took Algenbra 1 in junior high, Geometry freshman year, Alg 2 sophomore, Prob&Stats, Trig, and circular Functions junior, and Calc senior year. After which I took the Calc AB Advanced Placement exam.

"First, you suggested it was (more or less) necessary, not just that it would help them immensely."

The difference between "(more or less) necessary" and "immensely helpful" seems like splitting hairs...

"A majority who take calculus in high school are going to have to take it again in college anyway"

Good. Rigorous subjects like calculus should be taken more than once.

"In fact, often they are vastly underprepared in algebra, and end up doing very poorly in college calculus and having to change their majors, without ever fully understanding how it is that they weren't able to do in college what they thought they could do in high school."

I don't understand how this is relevant. Of course a solid understanding of algebra precedes a solid understanding of calculus.

"There's plenty of time to take it in college."

Have you looked at an engineering curriculum recently?


I'll take a look at your articles are respond.

I took calculus at junior college, Draug. Technically, it was on a scholarship, so neither I nor my parents paid for it; but the cost for all my courses would have been about $1000 a semester, total. So calculus would have cost maybe $200 per course. Junior college is an excellent place to take calculus (and, in many scenarios, an excellent way to start college).

In any case, I don't think that abge was referring to financial advantages, so I restricted my remarks to academic considerations.

If people are under sufficient financial pressure that colleges finances enter in a serious way into calculating what courses to take and when, then I would counsel very seriously considering junior college at the outset. Whether, at a four-year university, adding calculus would cost additional money just depends on the specific university and its tuition plan.

@semck

So, it appears your concern is that people are being rushed through math to get them to into calc before finishing HS. If this is true, I agree it is a serious problem, because, as you say, the fundamentals of algebra, trig, and geometry are essential to a successful career in math.

With that being said, I find it hard to believe that a competent math department couldn't teach those subjects in 3 years, leaving the 4th year of HS to teach calculus. If they aren't able to do it in 3 years, will one extra year help that much? It seems to me the issue is larger, in that case, than lack of time.

abge,

"Good. Rigorous subjects like calculus should be taken more than once."

What nonsense. Almost every course in an engineering degree is "rigorous." Should we also take calculus-based E&M twice? Statics and dynamics? Maybe go ahead and throw in thermodynamics a second time?

But let me clearer.

You are claiming that it is an "immense advantage" to take calculus in high school. Given that most students who do so will have to take calculus again in college, skipping college calculus must not be the "immense advantage" you have in mind. (Unless you are restricting your comments to those who pass the AP test).

My remark therefore eliminated one possible interpretation of the advantage. The remainder of my point is that, since the students will take calculus in college (and have time to do so anyway), then no, I don't think it matters much whether they've seen it before.

But maybe I should ask this: just what are you claiming the advantage is?

"The difference between "(more or less) necessary" and "immensely helpful" seems like splitting hairs..."

Well, it's an enormous difference. They mean completely different things. Do I need to explain this, or can you see it now on further thought?

"Have you looked at an engineering curriculum recently?"

Yes. It's fairly impractical to achieve in less than five years. Why do you mention it?

@abge,

That is a concern, yes. But as I said, I'm ultimately not arguing that it has to be a big DISadvantage (though it becomes one, to the extent that that happens, so that is a net negative on the high-school-in-calculus side).

More relevantly, rather, I am focusing on the fact that I simply fail to see an actual advantage to taking calculus that would come *anywhere near* "immense" or "more or less necessary." Your initial claim, recall, was that "by the end of HS they better be able to [integrate] if they want to be an engineer," suggesting they should actually have seen Calculus *II*. I continue to completely fail to see any reason you would attach such urgency to such knowledge at such a point.

So, of course you can't take every course in full twice, although if you had infinite time, that would certainly be ideal. Instead, it's helpful if there's some overlap between the end of one course and the beginning of another. This reinforces ideas and presents them from a different perspective, which can be very helpful.

In my experience, Calc in HS usually comprises of both Differential and Integral Calculus. Taking this in HS accomplishes two things:

1) It gives you a taste of more advanced math, which is helpful for determining your career.

2) It gives you an opportunity to see the subject more than once, which I still maintain is helpful, even if it isn't always practical.

"Yes. It's fairly impractical to achieve in less than five years. Why do you mention it?"

So, agreeing that a 4 year degree is already impractical to obtain in under 5 years, how can you say there's "There's plenty of time to take it in college"?

Is Calculus-based Physics not a standard things for engineering majors to take freshman year? How can you do well in physics, if you're not only learning physics concepts but also integration for the first time?

I actually took Physics in high school the year before Calc and it helped immensely being able to apply practical real world examples (like the relationship of acceleration, speed, and distance).

Yeah, I learned how to integrate in my physics class in HS before I learned in math, as well. But I'm not sure it was the best approach.

"glad we took Calculus in high school and then too the AP exams so we didn't have to take it in college"

I took an absurd amount of AP and Running Start classes in High School.

(Running Start is a state program that offers Community College classes while in High School)

After a quarter in college, I was a senior needing only five Humanities credits to fulfill my General Education Requirements. I will graduate within three years of entering, double-majoring in two of the most course-heavy degree programs offered.

AP is $90 per test (roughly $18 per equivalent college credit for the average AP test).
Running Start was paid by the state for everything but books and transportation.

In short, AP and Running Start are quite literally the best.

What were your majors?

Math and Physics.

That's quite impressive.

@abge,

I agree that the right way to solidify knowledge is to build on it in later courses. I do *not* agree that taking any course a second time would be ideal. It would be enormously boring if you had learned it well the first time.

Let us turn to the two advantages you present for taking calculus in high school:

"1) It gives you a taste of more advanced math, which is helpful for determining your career."

That seems reasonable, for those undecided on a career. It doesn't help your original claim at all, though, which was that, "by the end of HS they better be able to [integrate] if they want to be an engineer."

If we're talking about somebody who already wants to be an engineer, then this point simply doesn't apply to them.

"2) It gives you an opportunity to see the subject more than once, which I still maintain is helpful, even if it isn't always practical."

OK, I'll agree that for some it might be helpful. What I don't see is how this "helpful"ness reaches the level of "if you want to be an engineer, you'd better do it." If you understand math well, you're going to be able to do calculus, whether it's at the college level or at high school. It might be nice to have seen it before, but it just doesn't really matter.

Let me try to explain some more my perspective on this stuff. Over the years, as I've tutored, taught, and graded the exams of many hundreds of calculus students, I have noticed several different types that I see a lot*:

1) The ones who don't really understand algebra or trig, and so simply cannot complete a problem correctly because they haven't internalized the rules or what they mean, and they cannot manipulate expressions. This is quite a large portion of students (including many who have taken calculus).

2) The students who are solid on algebra and pretty solid on trig, but who have conceptual problems with calculus, especially word problems and reasoning about derivatives, integrals, etc. in the real world. This is a sizeable portion.

3) Those who are excellent at math, have internalized what is going on, and quickly are able to use and apply calculus. This is a small minority.

*I'm creating this broad, rough classification by thinking back now. I *of course* do not classify students when I am interacting with them, and there are many personal variations, differences, etc.

Now, the students in group 2 can get there. I am not ultimately concerned about their performance, if they work hard. They are the students who seek help and are able to be helped.

The students in group one, on the other hand, face horrible odds. They can occasionally work so hard that they catch up and make it through well enough that they have some chance at continuing in science or engineering; but the odds are terrible. I have graded hundreds of their exams, and it's a depressing and awful experience. They are arriving at the last days of their hopes of being engineers, and there is little we can do for them. (Of course there are outliers. I know at least one person who struggled mightily with algebra early, had to go to junior college to take developmental math, and ended up with a mathematics major. Gloriously, it does happen, and it should always be our goal).

These experiences greatly impact my view of what is *crucial* for somebody who wants to be an engineer or scientist. It is crucial that you come to college calculus knowing how to do algebra and trig. (Take precalculus if you can't do trig). If you know that, then we really can teach you calculus. Maybe it would be easier if you had seen it before, but you can do it, and it's not so likely to develop into the roadblock to your career. The better you are at algebra, the more unlikely it is. It's very rare to see student wipe out on calculus without bunches of algebra errors, if I've ever seen it at all.

So that drives my perspective on the necessity of taking calculus before. I can't even visualize a student who fails calculus because he hasn't seen *calculus* before. I live every day among students who fail it because they don't really get algebra.

"So, agreeing that a 4 year degree is already impractical to obtain in under 5 years, how can you say there's 'There's plenty of time to take it in college'? "

Yeah, because I don't really think that's mostly because of calculus. Of course there are hundreds of universities and tens of thousands of different students situations, so doubtless there are some for whom taking calculus in high school is the difference of a year in college. That's certainly something to take into account, then; but given your remarks, and the discussion I've given, I take it this can't be the primary motivation for a *necessity* of taking calculus in high school.

So the social issue here would be:
"social and structural barriers to affective learning in mathematics and it's associated subjects?"

I agree with you, PSMongoose. Community colleges are great, and being able to start them in high school (and to test out of college courses) is wonderful.

@abge,

The physics/calculus interaction works differently at different schools. But most that I'm aware of have set it up so that it can be done. Where I went to school, for example, Mechanics was intended to be taken in the spring of the freshman year, so that calculus I would already have been taken in the fall; and E&M would be taken in the autumn the next year (during Calculus III). It varies by school, but I think it's unusual for there not to be at least some way to make it work.

"If we're talking about somebody who already wants to be an engineer, then this point simply doesn't apply to them."

I disagree strongly, since many people think they want to be something before they have any actual experience with it. Knowing how to integrate is importantly not only because it is a skill you need in the field, but because it shows you have some idea of what that field entails.

So, I understand your concerns in regards to students who are struggling with math and feel they have to choose between taking Calc (and very likely failing) or giving up on STEM. That certainly isn't a good position to put kids in. I guess where we disagree is how to remedy this. While more time certainly would help, and a senior year without calculus would give you that, I can't help but feel the problem is more systemic than that. Did the student not try hard enough? Were their algebra/trig teachers bad?

"but given your remarks, and the discussion I've given, I take it this can't be the primary motivation for a *necessity* of taking calculus in high school."

Well, yeah, if you discard each one of my points one by one, I'm not left with much : )

It is hard to finish an engineering degree in 4 years, but it can and should be done if at all possible. Taking 1-3 AP courses in HS can certainly be the difference in being able to complete the degree on time. To me, it seems natural that one of those courses should be Calc.

On the subject of teaching mathematics, this is always a good document to pull out...

http://mysite.science.uottawa.ca/mnewman/LockhartsLament.pdf

But I certainly agree with semck about the variety of Calculus students - I certainly saw that in my calculus class.

30 years on, the pressure to take AP courses has probably increased dramatically. Most of the students in my Calc AP class in HS were top students overall and went on to STEM careers if that was where their long term interests lay.

There are, of course, practical reasons why taking Calc in HS is important. Mainly it increases your chances of getting accepted into an engineering program. And while we both may wish that wasn't the case, that's the reality of it.

Abge,

Well, I feel a little confusion myself whether you're arguing for time or not. On the one hand, you keep bringing it up. On the other hand, you keep fighting for people to take calculus twice.

Anyway, you speak of a "remedy" for putting kids in a position where they either have to take calculus in high school or give up on STEM. I think we disagree slightly -- I am asserting that they simply don't have to take calculus in high school, so there is nothing to remedy. (Except perhaps the false impression that they do, and any pressure along such lines from university admissions departments).

"It is hard to finish an engineering degree in 4 years, but it can and should be done if at all possible. Taking 1-3 AP courses in HS can certainly be the difference in being able to complete the degree on time. To me, it seems natural that one of those courses should be Calc."

Well, for those who can take high school AP calculus and actually pass the exam, I'm certainly not saying there's a *dis*advantage, and in the form of time, at least, there's a clear advantage, so I think we agree there. I'm just disputing that it's at all necessary for any deeper reasons than time and logistics.

I'm certainly a big fan of both AP and CLEP for those in rigorous majors, and would encourage looking to humanities and electives for possible eliminable courses for an engineer, although of course not all engineering students would be in a position to do so). Certainly I was all about cutting out unnecessary courses myself, in college, so I can't at all dispute your advice here. : )