What is the earliest age that calculus is commonly taught?

Factual Questions
41

I agree. In my experience, calculus in high school is a function of the school being on the ball and offering the opportunity. The path towards Calculus in High School must begin around 7th or 8th grade at a minimum, 6th grade if you want to take Calculus as a freshman or sophomore.

My sister often reminds me that the fact that I took two years of Calculus in high school is largely the product of her class protesting and complaining to the administration that by 6th grade, math had become repetitious and boring for many of the students (6th Grade math is essentially a review of 5th grade math, no new concepts are taught, and 7th grade math was to be more of the same!). The ‘old way’ was Algebra in 9th, Geometry in 10th, Algebra 2 in 11th, and Trigonometry/Analysis in 12th. Failure to start Algebra early locks one out of calculus. My understanding is that schools are trending towards allowing kids to take math according to their level, but there is still resistance to ‘outliers.’ My sister was warned that if the school advanced my nephew to the appropriate level, he would run out of math in high school and wouldn’t be able to graduate. :roll eyes:

Anyway, things are much better than they were in the 80’s and 90’s, but for the kids who probably could handle calculus at a young age, there really isn’t the infrastructure to handle that. One reason I was able to take two years of calculus in high school was because there were about 25 students in that class. 3 of my classmates took Calc2 as Juniors. Senior year they took no math at all. It wasn’t feasible to offer another year of calculus for only 3 people.

42

I agree that it should. There’s no reason for me that algebra couldn’t have been introduced to the advanced students as early as 6th grade. That said, I don’t know anyone in my high school who took algebra before 9th grade, but they got to calculus by 12th without a problem. I’m glad to hear that the curriculum appears to be speeding up. I agree with your sister–I don’t recall learning anything new in math from the 6th grade to my first year of high school.

43

I decided to look up what courses my high school now offers:

http://cory-rawson.k12.oh.us/high/academic/guidance/curriculum%20guide1112.pdf

As you can see, it still doesn’t offer calculus.

When I was there, most of the fathers of my classmates were farmers or factory workers (or both, like my father). Some were office workers or store clerks (with only a high school degree). Most of the mothers were housewives. A very few of the parents had gone to college and taught elementary or high school. I suspect that there is less working in factories and less farming these days and more working at office jobs and in stores.

Certainly less than a majority of the graduates went to college. Those who did took what was known as “college prep” courses. The majority of the students though took what were then called “vocational agriculture,” “industrial arts,” “home economics,” or “business” courses. Even among the students who went to college, most were not able to make it to the last science course (physics) and the last math course (solid geometry/trigonometry). Vocational agriculture was for those students who intended to become farmers, home economics was for those who intended to become housewives, industrial arts was (vaguely) for those working in a factory or some other sort of making things, and business was (vaguely) for those working in some low-level office job.

I looked through the catalogue linked to above to figure out what’s new since I graduated. There are some more art courses. There is a video production course. The business courses are more computer-based. Instead of two years of Latin and two of French, there are three years of Spanish and a course on Spanish culture. The math courses are re-organized, but there’s still no calculus. There are some more science courses, but they are only slightly more difficult. There is actually an A.P. course in American history (the only A.P. course offered). There are a few more agriculture courses, somewhat modernized. The business and industrial arts courses are offered at a consolidated center.

As I said, it’s not the worst high school in the U.S., but it’s not very good. There are a lot of such high schools around, mostly in rural areas and in inner cities. The assumption is that at best you might be able to get into a local state university or small college, assuming that they aren’t too selective. As for getting into a really selective place, forget it.

44

Not so much. You were an odd duck if you hadn’t taken at least 1 year of calculus in high school.

For those concerned about students running out of math classes if they’re too advanced, that’s not really an issue now that there is this thing called the internet. I’m sure the online courses today are higher quality than the ones I took in the 90s.

45

OK, the latest data I could find is 2005, and it’s 13.6% overall for any type of calculus courses. (And 9.2% for AP calculus.) There is a definite upward trend, so I’m guessing it’s probably closer to 15% or so now.

46

ETA: That is, of course, not limited to college-bound seniors, so my estimate probably is way too low. That said, I’d be surprised (but encouraged) if it’s really true that college-bound students would usually take calculus.

47

IME whether you reached calculus in HS was more a function of your general mathematical aptitude than a function of whether you were college bound or not. There were just too many students who never made it past Euclidean geometry but still did really well in everything else. These people–I know from experience–generally don’t plan to major in science or engineering, and in college generally take a much less rigorous version of calculus–calculus lite, if you will. It’s the best they can do because they don’t have the background in trig or second year algebra to derive the fundamentals of calculus–differentiation, integration, limits, etc.

For those who did take calculus in high school, does that generally mean you can skip first year calculus in college?

48

At my university, it was dependent on your AP score or a self-placement exam as to what math courses you should register for. Of course, most non-science/engineering majors didn’t have a calculus requirement, so a good portion of the student body never had to deal with it.

49

Just to amplify on the UK perspective - I attended a good private school in the UK from 13-18. The year was streamed 40:60 into an advanced and a normal maths stream from 13-16. I was in the advanced stream.

We covered basic differentiation (finding maxima for parabola) as an optional enrichment topic at 13, but no calculus was required for the national maths exam (General Certificate of Secondary Education, or GCSE) which was normally taken at 16; the advanced stream took it at 15, and then went on to study calculus in more depth (just integration and differentiation) for a separate ‘Advanced Maths’ exam at 16. That wouldn’t be unusual for other private schools or for well-performing state schools, so I think it’s fair to say that at least 10% of UK students will have covered calculus by 16.

As discussed above, UK secondary students specialize into a few subjects for the final two years of school, leading up to Advanced Level or A-Level exams in those subjects. As I’m a geek (twelve years later I’m the proud owner of a first-class physics degree and a Ph.D. in atomic and laser physics) I took as much math as the school let me, which was two A-levels worth (of four in total).

In that we covered a lot of calculus: integration and differentiation, including trig and hyperbolic functions and the chain, product and coefficient rules, area and volume integrals in Cartesian and polar coordinates, and first and second order differential equations. It wasn’t solely focussed on calc, though: other topics in the pure maths section (which was half the course in total) we also covered imaginary numbers including Euler’s formula, long division with polynomials, and a basic introduction to group theory.

The other half of the course was applied maths - mechanics and statistics. Mechanics was largely statics - ladders against walls, weights suspended by strings, etc - but also included application of ordinary differential equations, rotational motion/moment of inertia, etc. Stats was largely focussed on hypothesis testing with various probability distributions, but also covered some more basic probability work.

By contrast, A-level physics was designed nationally so that you did not need to have studied any calculus beforehand. Basic differentiation and integration were taught as part of the course to anyone who hadn’t already done it. My teachers thought this was a terrible idea, so anyone who wanted to study A-level physics had to separately take enough maths classes to cover the basics of calculus.

50

This is what I was looking for. Common educational tracks for the above average student.

I received just enough HS calculus that I placed into freshman calculus (for scientists and engineers) at university. Oddly enough, my HS math teacher had suggested I take college pre-calc as he had never taught calc before. But had I taken pre-calc instead I would have been a semester behind my peers and would have had a hard time keeping up with freshman physics as it was calc based, which is the only way to teach physics IMO.

51

I went to state school in the UK and also did separate O levels for maths & advanced maths. We did differentiation for the Ordinary O level at age 14 and integration the year after. I’d guess about a quarter of the kids in my school did the advanced maths O level. For A level maths, there was a ton of calculus - significantly beyond what my son is learning in AP Calc now in the US.

As an aside, my son is doing AP calc as a Junior (aged 16) now in California. I find that my son’s classes focus much more on the formal aspects than we did when I learned the same material. They introduce the more nitpicky aspects of maths like restricting the domain and proving that functions are continuous over the domain. The stuff we did was conceptually more difficult (differential equations and the like) but it wasn’t until college that we circled back to the more formal, tedious aspects of maths. As a result, his classes are much more dry and boring, whereas ours were both more interesting and more challenging even if we were allowed to be a bit more sloppy when we were first introduced to the ideas.

52

I went to school in the US. My high school offers pre-calc through calculus and statistics now. They went up to pre-calc when I was there, and there was one kid I knew who was getting some help studying on his own for an AP test in calculus. I was in advanced courses from middle school, so I had algebra in 7th grade instead of 9th when it’s usually introduced. Most kids topped out at algebra II. There was only one pre-calc class of 30 or so when I was in high school.

While I know that teachers say that you need a “solid grounding in the basics” and other sentiments, for me algebra, trig, and geometry didn’t make much sense until calculus. When I finally had calculus in my first year at college, I thought, “Why the FUCK didn’t they teach me this stuff earlier?! Now all that seemingly unrelated boring bullshit is actually useful for something.” My math skills tested lower than language-related ones, but I wonder how things would have been if they’d been teaching math differently when I was in school.

53

That’s how my daughter’s school does it. She took 6th grade math in 5th grade, and then was one of 15 students skipped to Algebra I in 6th grade. High school credit starts with Algebra I, so no real worry (other than potential loss of skill) of running out of math.

54

We had calculus in grades 11 & 12 back in 80s.

I checked it online, and current syllabus also has it in grades 11 & 12.

55

I don’t know what is the requirement now, but I first took differentials in 10th grade; this was the schoolyear 1983-4, in Spain, and every college-track HS student (tradeschool is separate) had to take it. The following year involved integrals but only for students who had chosen the Life Sciences, Pure Sciences, Applied Sciences or Mixed tracks; students from Social Sciences or Pure Humanities didn’t take math at all in the last two years of HS, so no integrals for them.

My father took Calculus (like us, differentials before integrals) as part of his Bachillerato Superior (three years, age-equivalent to the first three of US HS), so more or less at the same age; my mother had only her Bachillerato Elemental (age-equivalent to US middle school) and no Calculus. He went on to become a Profesor Mercantil (the closest thing to a Bachelor’s in Business Administration at that time) and that included applied Calculus; she was a Teacher, the coursework included making sure that the future teachers really were qualified to teach up to Bachillerato Elemental, but logically no further education in Math was needed.

56

The US method of splitting up Grade 8 to 12 math classes into completely separate subjects (i.e. Algebra, Geometry, etc…) is interesting to me, because that isn’t how they do it where I went to school (Alberta, Canada). Here, the math classes in junior high (grade 7 to 9) and high school (grade 10 to 12) are called Math and each class has a number of sub-units.

They’ve since changed the curriculum and class progression slightly, but when I went to high school there were a few different streams of high school classes, depending on your school.

[ul]
[li]The 10-20-30 stream was for average students, or above average students whose schools didn’t have advanced courses[/li][li]The 13-23-33 stream was for below average to average students who would get a high school diploma but almost certainly would never go to university[/li][li]A 14-24 stream, which was basically “life skills” type classes for students who probably had some kind of developmental disability and were mainstreamed into a regular school[/li][li]An AP (advanced placement) or IB (international baccalaureate) steam for advanced students. My school had IB, and those courses were designated 10X-20X-30X.[/ul][/li]All high schools offered Math 31, which was calculus; Math 31 had Math 30 as a co-requisite, so it was taken at the same time as or following Math 30 (or Math 30 IB or AP if your school offered that). I’d say that pretty much all students at my school who took Math IB also took Math 31, and probably less than half of those who took Math 30 also took 31. FWIW, at my high school our graduation rate was almost 100% and probably 85-90% of students went on to some kind of higher education (university, community college or technical school).

So the streams for math at my school when I attended it were:[ul][li]Math 13 - Math 23 - Math 33 or[/li][li]Math 10 - Math 20 - Math 30 - Math 31 or[/li][li]Math 10X - Math 20X - Math 30X - Math 31X[/ul][/li]A couple years after I graduated they changed the curriculum - see link herefor a good flow-chart of the course sequences, and here for a link to a description of what type of students are in each stream. They split the Math 10-20-30 stream into two streams called Pure (Math 20-1 and 30-1) and Applied (Math 20-2 and 20-3). Generally speaking, students who want to go to university will take Math Pure, or those who aren’t planning on university or who are definitely humanities-focused take Math Applied. Only those who take the Pure stream can take Math 31 (calculus). Many schools also offer AP or IB courses, which are more advanced than the Pure stream.

As I said, it seems strange to me to split up math classes into separate courses like most schools in the US do. Instead of taking one class for a whole school year on algebra, or geometry, or trigonometry, the curriculum here covers a little bit of each topic spread over a number of years. So you do a bit of trigonometry in grade 10, 11 and 12, with new concepts and difficulty each year. I would take a WAG that introducing algebra in junior high and then adding more difficult concepts each year in high school might lead to a better understanding than if you just do algebra in grade 9 and then don’t use if for a few years. Also, in our system the curriculum is designed so that the multiple sub-units in each year are interrelated and you get a better idea of how different mathematical concepts and areas are linked together.

The provincial website didn’t have a list of units in AP or IB courses, but the topics/units currently covered in the 10C, 20-1, 30-1, 31 (Pure) stream include:[ul][li]10C: measurement, trigonometry, polynomial factoring and operations, systems of equations, linear relations and functions[/li][li]20-1: quadratic functions and equations, radical and rational expressions and equations, trigonometry, systems of equations, sequences and series[/li][li]30-1: statistics of the normal curve, algebraic transformations, permutations, combinations, probability, circular functions, exponential and logarithmic functions, conic sections[/li][*]Math 31: precalculus and limits, derivative and derivative theorems, application of derivatives, integrals, integral theorems and integral applications[/ul]

57

I’ve always found that weird too. The boundaries seem arbitrary to me and, like you say, I imagine it makes it harder to remember when you do, say, a bit of trig and don’t come back to it for a couple of years.

In England, circa 1980, we had single subject maths up to O level (aged 16). For A levels, there was an option to take separate subject pure and applied maths where pure maths was more theoretical stuff and applied maths was more about solving real world problems from physics and engineering. A typical science-bound student might take 3 A levels (say, pure maths, applied maths and physics), so the last two years of school were entirely about maths and science undiluted by literature, art, history or foreign languages.

58

I agree, the formulas are easy enough to memorize, but that’s the best you can do if you don’t have the background to understand the derivations.

59

Curiously, without being particularly good at math, much less an engineer, I always enjoyed word problems and the extra step of analysis they require.

60

Also, the better students of a given class (or year) tend to have gotten their general education requirements out of the way by junior (penultimate) year, so they are more free to specialize. Universities impose their own course requirements on HS students planning to attend them after graduation, but the key difference there is generally in the grades (marks) they require. In a way this all results in a de facto specialization from around 16 y.o.a. on.

Until he answers the question himself, I assume he means a more senior HS student who is doing mostly or only mathematics courses, in much the same way an upper division university student majoring in math does. From what’s been explained here about the UK’s system of secondary schools, it seems possible.