Calculus in high school over two years -- why?

Parents Forum
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<p>This is part of a question posted in this</a> thread about students being two or more grades ahead in math, but it seems to be ignored there.</p>

<p>Based on posts in these forums, there is an increasing tendency for schools to require that students who reach calculus take it over two years, typically AB one year followed by BC the next year, with no option to take a one year course that covers the entire BC syllabus. (AB and BC refer to the AP syllabi, where AB is commonly given one quarter to one semester of calculus credit in university, and BC is commonly given two quarters to two semester of calculus credit in university.)</p>

<p>Obviously, the existence of such courses and policy implies that there are significant numbers of students who are at least two grades ahead in math, reaching calculus by junior year in high school.</p>

<p>This does not seem to make too much sense -- the students who are two or more grades ahead in math should theoretically be the best students in math, and fully capable of handling a one year BC course. They would probably be bored at the slow pace of taking AB over a year and then the remaining part of BC over another year.</p>

<p>When I was in high school, there was only one class worth of students to teach calculus to (less than 10% of the students, almost all of which were one grade ahead in math, with the rare student every few years who was two grades ahead in math). That course taught the BC syllabus over one year, and some students did get scores of 5 on the BC test (though some of the students not doing that well in the course took the AB test). Later, when the number of students taking calculus senior year grew to two classes' worth, they added an AB course in addition to the existing BC course.</p>

<p>So it is mystifying why some schools require all students who reach calculus to take it over two years, instead of allowing the best students, who reach calculus by junior year, to take a one year BC course.</p>

<p>Any thoughts on this question?</p>

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<p>Some schools have always taught AB and then BC over 2 years, even back in the dark ages when I was in high school. Other schools have never done this. I think it says more about adult expectations (either way) that the ability of the students.</p>

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<p>This is what I ended up doing, but I’m the only person I know who did.</p>

<p>i was intimidated to take BC, but looking back i wish I had taken BC junior year and multivariable senior year.</p>

<p>BC is maybe 20% more material than AB…30% max!</p>

<p>On the other hand, many people have trouble remembering math concepts. Calculus is really helpful to have working knowledge of. Taking 2 years gave me a solid foundation</p>

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<p>It’s not like that at every school. Around where I live, you do all of Calculus BC in one year. You take either Calc AB or BC, never both.</p>

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<p>Calculus A is supposed to be the equivalent of Calc 1 at college. Calculus B is supposed to be the equivalent of Calc 2 at college. So Calc AB is the equivalent of a year of most college’s calculus course.</p>

<p>The problem is that the integration taught in the Calc AB course is not as thorough and is lacking theory that a lot of the college courses require. It’s just the way the course is set up. </p>

<p>There are high schools where the Honors PreCalc course pretty much covers Calc A and those kids can make the jump into Calc BC but this is usually not recommended, because it is a tough course and one where kids do not always do that well, and we are talking about the top kids here. </p>

<p>Calc BC is a review of the Calc B and then segues into Advanced Calc or Calc 3 at many colleges. But most kids will repeat that Calc 3 or Vector Calculus even if they complete BC Calc and/or do well on the AP exam. </p>

<p>So for kids to do well on the Calc BC exam, it’s usually smart to have them take the Calc AG first. It is a heavy duty course for most kids, and grades are important too. Historically, even top kids have a tough time with the course.</p>

<p>As Hahalolk says, there are school where you do take one or the other. That is when there is a Precalc course that covers the Calc A portion so the kids can go directly iinto Calc BC It all depends on how many kids would fit what track.</p>

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<p>^We actually don’t cover any Calc A in precalc, but that’s besides the point. :)</p>

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<p>DS did AB, BC, and in 12th grade, Calc 3, to get 4 years of math. I personally think kids not going in the math/engg track are better served taking Statistics, but very few in his school took it because there wasn’t an AP weighted Stat course at that time. </p>

<p>CALCULUS III Credit Value: 1 Credit (AP weight)
Prerequisite:―“B” or higher grade in Calculus BC
Duration: Full Year
Grade Level(s): 11 and 12
Required: N
Elective: Y
Qualifies for:
Computer Requirement: N
Fine/Practical Art Requirement: N
Humanities Requirement: N
Contains a Culminating Project: Y
Calculus III is designed for students who have completed AP Calculus BC. Topics included: vectors and geometry of space, vector-valued functions, functions of several variables, multiple integration, vector analysis, and differential equations. Additional components include an informal review of BC Calculus, interdisciplinary mathematical activities, and problem-solving using tools such as Mathematica and graphing calculators.</p>

<p><a href=“http://www.franklinregional.k12.pa.us/highschool/stuff/1112_program_of_studies.pdf[/url]”>http://www.franklinregional.k12.pa.us/highschool/stuff/1112_program_of_studies.pdf&lt;/a&gt;
page 37</p>

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<p>I remember taking BC in high school (then the only calculus course available there) and multivariable calculus (generally considered university sophomore level math) at a university. The latter was not a repeat of any significant amount of the BC material, although it certainly made use of the BC material.</p>

<p>I do not know of any college or university from community college to the top public and private universities that gives more than a year of freshman calculus credit for a 5 on the BC.</p>

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<p>The calculus AB often taught as a one year course is often the only calculus course taught in many high schools. It can be considered equivalent to a ONE semester average, not top, college course (many UW-Madison students who pass AP calculus struggle with the second semester course and are best off starting over with the UW first semester course- it covers more than the AP course). Only some high schools offer a second semester course equivalent. Typically a 4-5 college credit AP course takes an entire year for a semester’s worth of material (foreign languages as well as math). </p>

<p>Students gifted in mathematics can of course go through the material more quickly. However, schools (even at the college level) need to teach at a pace that accomodates ALL of the students capable of taking the course. There aren’t usually enough resources to accomodate the gifted learners in anything. In the ideal world students could advance at their pace instead of a classroom’s pace. Your school district, or even most private schools, don’t have the money to do this. One reason you choose a college that suits your academic level- as well as your major. One size doesn’t fit all but it is not cost effective to have all of the possible sizes available.</p>

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<p>The local school offered a one-year, stand-alone AP Calc BC course. Students could take it without taking Calc AB first. They could also take Calc AB first for a year, and then take Calc BC. This involved repeating quite a lot of material.</p>

<p>With this configuration, to me it made no sense for good math students to take Calc AB and then take BC, and many just took Calc BC.</p>

<p>As to why schools would require a two-year Calc sequence–i.e., the students must take AB followed by BC, here are a few thoughts:

  1. To maximize the number of 5’s and the number of passes from the school, on the AP exams.
  2. To avoid repetition of material between the two courses. The students who took Calc BC directly wound up in the same class with students who had covered much of the material before, in Calc AB. In fact, this worked out all right, even though with the local calc teacher, some of the actual exams during the year were repeated from the year before!
  3. To avoid having to pay for college courses for the students who take BC and are still in high school afterwards. Our state mandates that the local school district pay for the college credits, for a student who has exhausted options at the high school.</p>

<p>In terms of your other question: Why there are many more students who are advanced two or more years in math? Reasons vary from student to student and district to district, but here are a few:</p>

<p>Why would a student who is not truly talented in math wind up several years ahead of the old schedule?

  1. The elimination of proof-based geometry in many school districts removed a gate-keeper course that otherwise would have meant that a student had to have a reasonably high level of mathematical maturity by the time he/she took geometry.
  2. Watering down of algebra and reduction in the total material covered makes it easier. Spherical trig, anyone?
  3. The availability of AP Calc AB followed by AP Calc BC, with AP Stats in there somewhere means that a student can load up on AP (and potentially gain GPA weighting) by following this route, without needing to be especially mathematically gifted.</p>

<p>And some reasons that students who actually are quite talented in math are now further ahead than traditionally:

  1. A number of school districts have removed the artificial barriers to acceleration.
  2. There are some special programs, such as the CTY or other talent-search based programs, EPGY, and some universities (e.g. UC, Berkeley) that offer 4 years of high school math compressed into two, and available to middle schoolers.</p>

<p>Interestingly, the people on the Art of Problem Solving web site have some discussion devoted to “the calculus trap.” They suggest that a student may not be well-served by progressing as rapidly as possible toward calculus. There is a lot of interesting pre-calculus mathematics, that can enrich a student’s understanding.</p>

<p>I think that the applicability of the suggestions on the AoPS website depends on the resources available to the student. For example, can the student take post-calculus courses with a similarly talented group, or are the courses not available, or are the students in the courses older, but not as strong mathematically? How good are the high school math teachers, really? Can the student get the non-calculus enrichment that AoPS envisions? Probably only at a really good school.</p>

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<p>I agree with you. At my kids’ school (a suburban public), the top kids take AP Calculus BC, and the kids at the next level down take AP Calculus AB. Both are year-long courses, and no one takes both. If a student takes Calc BC as a junior, then he can take Multivariable Calculus one semester senior year and an honors math topics course (content of which I’m not sure) the other semester. Both are offered at the high school - they’re not dual enrollment or dual credit courses.</p>

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<p>Beats me. The kids who are taking calc as juniors at our school are strong students who wouldn’t dream of taking AB. It’s strictly either or. All the best math students take BC. No one takes first one then the other. (Mind you Honors Pre-Calc always starts on the Calculus book before the end of the year.)</p>

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<p>That makes sense. I would expect the AB course to be filled with one grade ahead seniors who are good, but not great, at math, and/or not planning on studying anything math-intensive in university. I would expect the BC course to be filled with the better at math students, including those great enough at math to be two grades ahead and taking it as juniors (and who would think that an AB course is too slow).</p>

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<p>How commonly do universities accept a high school multivariable calculus course, compared to a community college multivariable calculus course? (Let’s assume a university other than a super-elite science and engineering school which may not accept any other course because its own courses have lots of extra theory.)</p>

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<p>at my school the general pathway is AB–>BC, but kids can skip it.</p>

<p>i take AB now, and i think it benefits me more. a lot of kids skip, but i like having the extra time to really learn the material.</p>

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<p>In my school most people take either or. About half of the juniors who are two years ahead are taking AB and half are taking BC. Most of the people taking AB as juniors will take BC as seniors. They are in the minority though. In my 20 kid BC class, only one kid took AB because he was one year ahead in math from his old school but not strong enough in math to take BC.</p>

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My high school had just switched from the BC curriculum to the AB curriculum when I took the course. They decided that they were doing students a disfavor by pushing them through Calc BC in 80% of a school year. Classes ended in late June, but the AP exams happen in early May.</p>

<p>I for my part did not feel bored in Calc AB at all. My teacher added some topics to the AB curriculum: I got my first exposure to rigorous mathematics (we spent 3 weeks on limits and continuity instead of the 1 week suggested by Collegeboard, to work through the epsilon-delta proofs), I learned how to program (I wrote my own slope field plotter and Riemann sum calculator) and we also set a bit of class time aside for math competitions. I am convinced that I learned more in our enhanced Calc AB class than I would have gotten out of regular Calc BC!</p>

<p>I do hope that schools teaching Calc BC over two years enhance the Collegeboard curriculum in one way or another. It’s hard to imagine how the extra BC topics might fill two full semesters all by themselves.</p>

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<p>You might find this article interesting, from Chronicle of Higher Education</p>

<p>The Rocky Transition From High-School Calculus
By David M. Bressoud
Amid the many signs of underperformance in American education, the growth and geographic spread of Advanced Placement courses in high schools is a widely admired success story. For several decades, AP calculus has been a standard-bearer for this movement, representing a singular mark of academic accomplishment for the ablest students, their teachers, and their schools.</p>

<p>Unfortunately, for the majority of those students, AP calculus has become not a steppingstone but a stumbling block. Each year hundreds of thousands of our best first-year college students are precluded from careers in science, technology, engineering, and mathematics because they fail to advance beyond AP calculus. We urgently need to fix the problems with how AP calculus fits into the college curriculum.</p>

<p>In 1965, the Mathematical Association of America’s Committee on the Undergraduate Program in Mathematics published “A General Curriculum in Mathematics for Colleges,” establishing what would become the standard undergraduate curriculum. The committee noted how well its curriculum fit with the AP curriculum. It observed, however, that students who took calculus in high school were still a tiny minority. Nine thousand students took the AP calculus exam that spring, of the 1.4 million who would start college that fall.</p>

<p>Of the 2.7 million students who matriculated in the fall term of 2008, 300,000 had taken the AP calculus exam, and at least 200,000 more either took the course but chose not to take the exam or participated in another high-school calculus program. Those students—almost one in five of those entering college—constitute the elephant that too many colleges and universities continue to ignore. While some issues must be dealt with in high schools, we in higher education have a responsibility to meet the needs of those students.</p>

<p>The clearest evidence that we have a serious problem comes from the number of students taking Calculus II, the second term of college calculus, in the fall. In 1990, about 50,000 students earned scores of 3 or higher on the AP calculus exam, enabling most of them to begin college mathematics with Calculus II or higher. That fall, according to the Conference Board of the Mathematical Sciences, 110,000 students enrolled in Calculus II. In 2005, 160,000 students earned scores of 3 or higher on the AP calculus exam. That fall, total enrollment in Calculus II was down to 104,000. That drop is even worse than it appears; tens of thousands of students who did not take AP calculus instead earned college credit for calculus via the International Baccalaureate program or programs that allow high-school students to also enroll in college courses. Those students, too, could have taken Calculus II but did not.</p>

<p>The decline in enrollment in Calculus II is also more significant than the numbers suggest. The Conference Board of the Mathematical Sciences breaks down its data by the type of institution. Between 1995 and 2005, fall-term Calculus II enrollments dropped by 20 percent at two-year colleges, undergraduate colleges, and comprehensive universities. At research universities, a 29-percent increase kept the overall total from falling too far. But fall-term enrollments in Calculus II at research universities are almost perfectly correlated with the number of incoming students who intend to major in engineering, which has increased by 31 percent from 1995 to 2005.</p>

<p>We have very little evidence of what has caused this decline or the general decline in enrollments in mathematics at the level of calculus and above. Professors point to what they consider to be poor mathematics instruction in the high schools. High-school teachers point to outdated pedagogical practices in many colleges, especially professors’ reliance on lectures and failure to engage students in active learning.</p>

<p>There is some indication that the pressure to offer calculus in high school is pushing underprepared students prematurely into the subject. The National Education Longitudinal Study of the high-school class of 1992 reported that of every 10 students who studied calculus in high school, three had to take precalculus when they got to college. Three others did not continue with any calculus in college. No more-recent data are available, but the phenomenal growth of students taking calculus in high school suggests that the problem has not abated.</p>

<p>An even bigger problem is the imperfect alignment of AP-calculus course content with the college curriculum. The Committee on the Undergraduate Program in Mathematics originally envisioned Calculus I as an overview, covering most of the common techniques and applications of both differentiation and integration. Calculus II would revisit those ideas in greater depth. Calculus AB, the AP-calculus program designed to correspond to Calculus I, still provides such an overview. But the way calculus is now taught at most colleges, most of the techniques and applications of integration are postponed until the second semester. Calculus I is now a more theoretical and sophisticated treatment of differential calculus. For the best students in the best schools—perhaps a third of those who study calculus in high school—that is not a problem. They are able to follow the expanded syllabus at the level required to be ready, even overprepared, for Calculus II.</p>

<p>But the majority of high-school calculus students earn a satisfactory grade by focusing on algorithms and procedures rather than understanding. Given the breadth of the material to be mastered, that is a common coping mechanism. Many students who retake Calculus I in college think they already know the material, but then get slammed midsemester when the level of sophistication required turns out to be higher than expected. Few of those students recover to complete the course or continue studies in mathematics.</p>

<p>Three things need to happen if we are to meet the challenges created by the explosive growth of high-school calculus:</p>

<ol>
<li><p>Get more information about what happens to students who study calculus in high school. How many of them are deemed unready for calculus when they get to college? How many retake Calculus I, and how successful are they? How many never take another calculus course? What are the factors that affect their decisions? How important is it to future mathematical success to study calculus while in high school?</p></li>
<li><p>Establish and enforce guidelines for high-school programs offering calculus. There is nothing inherently wrong with the growth of AP or other high-school calculus courses. In fact, now that those programs have become the norm, students at schools that do not offer calculus may be disadvantaged. But calculus offered in high school must be designed to facilitate success in college mathematics for all students who take it, rather than creating obstacles for all but the very best.</p></li>
<li><p>Re-examine first-year college mathematics. Colleges and universities must ensure that there is an appropriate next course for students who studied calculus in high school but are not yet ready for college-level calculus—a course that acknowledges and builds on what they have learned while preparing them for further mathematics.</p></li>
</ol>

<p>The time has come for colleges to stop assuming that Calculus I and II constitute two halves of a single course. We need to return to the original vision of Calculus I as a general overview of the themes and tools of calculus. Most of the students who take Calculus I never get to Calculus II. The traditional college Calculus I course serves those students very poorly, giving little indication of why calculus is important or what one might do with it outside the classroom. Moreover, if we try to explain calculus at a high level of sophistication before students understand its fundamental concepts and what it can accomplish, then we open the door to misleading perceptions and unnecessary frustration.</p>

<p>Some students take calculus in high school with the intention of avoiding mathematics in college. Departments of mathematics need courses that entice, engage, and encourage those students. At most colleges, students are expected to pass calculus before moving on to higher mathematics. But there is no reason why statistics, linear algebra, geometry, or discrete mathematics cannot be used instead of calculus as a bridge to higher-level mathematics. Eventually, all mathematics majors will need to continue with calculus and the courses that build directly upon it. But because it has become a stumbling block, we need more options that allow students to pursue challenging mathematics while postponing calculus.</p>

<p>If we want to facilitate a smooth transition from high-school calculus into college mathematics, then we must get more information about the true difficulties students face, ensure that students are ready before they begin calculus, and rethink the college mathematics curriculum. We need engaging, intellectually satisfying courses that will inspire students to continue their study of mathematics.</p>

<p>David M. Bressoud is president of the Mathematical Association of America and a professor of mathematics at Macalester College.</p>

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<p>Here is another:</p>

<p>The Rush to Take More AP Courses Hurts Students, High Schools, and Colleges
By DAVID W. OXTOBY
An entering student at Pomona College last fall submitted the results of 14 Advanced Placement tests, all but one with the top score of 5. In all, 20 members of the entering class each reported the results of 10 or more such exams. Obviously, these are highly talented students who will benefit from the broad range of advanced courses that Pomona offers. But it is far from clear that this proliferation of AP courses — along with the accompanying pressures — truly makes for the best high-school education, or, for that matter, prepares students to get the most out of their college years.</p>

<p>When I was a high-school student in the 1960s, students in good schools might have taken several AP courses, all during their senior year. Now, however, in order to accumulate 10 or more AP exams, it is necessary to begin far earlier. At some high schools, a 10th-grade chemistry course (the first chemistry course a student takes) is now designated as “advanced placement” so that introductory as well as college-level material can be compressed into a single year of work. In a few subjects, AP courses are now available as early as ninth grade. Can a ninth grader truly be said to be doing “college level” work in European history?</p>

<p>Although I applaud the effort to make challenging courses available to growing numbers of high-school students across the country, I worry that advanced-placement programs are rapidly becoming the latest way in which schools are “teaching to the test,” rather than using creativity to excite and challenge students. Too much of the high-school curriculum is turning into a pale imitation of college courses instead of providing the solid foundation that students need to build on in the future, and the new pressures associated with these courses are distorting both the high-school experience and the nature of the courses being taught. In high school, teachers of AP courses must frequently race through a yearlong college syllabus, saving several weeks at the end to coach students for the test. They have little opportunity to innovate or bring their own best ideas to the subject because of the sheer volume of information to be presented. Indeed, some AP courses attempt to cover more material than their college analogs do.</p>

<p>Just as troubling is what happens once these students arrive at college. Too many students now enter with advanced courses on their r</p>

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<p>Regarding:</p>

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<p>It is worth noting that [the</a> Mathematics department at UC Berkeley does not express much confidence in AP calculus scores of less than 5](<a href=“http://math.berkeley.edu/courses_AP.html]the”>http://math.berkeley.edu/courses_AP.html):</p>

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<p>Regarding:</p>

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<p>The proliferation of AP tests of dubious value in terms of university subject credit probably is not helping matters. For example, UC Berkeley’s policies on subject credit for some AP tests:</p>

<p>Chemistry: Nothing for Chemistry or Chemical Engineering majors, but gives one semester of chemistry course credit for other majors.
Physics B: Nothing.
Physics C: Nothing for Physics majors.
Statistics: Non-calculus statistics course. But many majors which require statistics, including Economics and Business Administration, require a calculus-based statistics course.
History (any): Nothing for History majors.
Economics: Need both micro and macro to count for an introductory economics course (minimum score of 5 for Business Administration major, 4 for Economics major).</p>

<p>For practical purposes, it may be better for students and parents to think of many of these AP courses as more advanced than usual high school courses, rather than university level courses. Although, as Oxtoby points out, they may need some changes to function optimally as high school courses giving the best preparation for university level courses.</p>

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<p>There was an article I couldn’t find but had been looking for when I found those, that indicated math professors were concerned that students are now cramming a lot more math courses under their highschool belt, but are actually ended up less prepared because their knowledge is not as deep and internalized (but merely exposed to a more surface level survey understanding of more topics). </p>

<p>It made me wonder about this trade off I hadn’t thought about before. For some students, of course, more would be better. But it could be for many (most?) student, they would actually become more prepared for college math by spending more time on topics, and sinking into them at a deeper and more fundamental level of real understanding, than covering quickly a lot of material memorized for the test. It is an interesting issue.</p>