<p>Agree with you that AoPS was one of the best resources my kids had when they were in school. If you buy Rusczyk’s two part book, it’s really interesting in that each chapter is about a different area of math, so you get a broad look at the subject, and done in a bit of a conversational way that keeps your attention.</p>
<p>In my opinion, the mistake is to look at Calculus as the ultimate pinnacle in math, so if you get to it, you must have mastered everything else. There’s a lot more that can be learned in Algebra, trig, or the sometimes-non-HS subjects like linear algebra or discrete math as others have commented and as is evident in the AoPS site. But the question is whether learning any of these areas in greater depth gives an edge over going to calc. (I do agree heartily with an earlier poster that stats gives a of bang for the buck for the average non-STEMer, and I don’t even see if calculus belongs as a must-have for this population). </p>
<p>One of the best benefits I see in getting to Calc before 11th grade is that I found Physics more effective when taught to students who were familiar with Calculus. For the student who wants both AP Phy B (or Phy SAT II) and C, the first for the breadth - optics atomic/nuc/thermodynamics, etc also, and the second for the depth, it made sense to have the basics of Calc done by 10th grade. In this regard I see a benefit for someone likely to go Engineering/Sci in college in getting to calculus earlier than is normally planned, and perhaps doing other non-calculus math in their Jr and Sr years.</p>
<p>I could not agree more. I think calculus was the easiest part of the math education in my country (before I came to the US). They taught geometry every year from 7 to 12th grade. Plane (3D) geometry was taught in 9th grade and 11th grade. Calculus was taught in 12th grade but it was only 1/3 or 1/4 of the entire math curriculum for 12th graders. And they did not use thick books like “Calculus for Engineers” for highschool students. The calculus curriculum was equivalent to Cal AB in the US, but the problems were harder.</p>
<p>QM,
Thanks for the lengthy posts.
A thought:
""Mathematics is a very weird subject in the sense that once one understands something, it is very hard to reconstruct why it ever caused problems. “”
I feel that THIS is why teaching Math is so difficult. A person who has had no problems understanding Math may be the most likely one end up being a teacher, but I feel that such a person usually does not “get” why others may not understand Math, and also may not be able to explain and show things in a variety of ways, to help learners with different levels and styles “get” it. In effect, a gifted mathematician is probably NOT the best person to teach Math. Same for Physics and Chemistry, IMO.
Most especially, a person gifted with numbers, quantification, visual representation, concrete into abstract thinking may NOT be able to VERBALIZE these things very well.
And we all know that there are many ways to get to an answer, but many Math teachers do not honor or explore this.</p>
<p>An equation is a elegant way of representing a question. Then one needs to know the techniques for solving the equation, i.e working with the numbers- that can be learned and remembered by many.
Some have a very hard time translating a verbal or concrete problem into the right number and operations and formulas. This is different for “critical thinking”, “logical thinking” and “abstract thinking”. It may be a combination of all three!</p>
<p>I am most definitely the non-Math type, as are my children and my H.
I regret for my children’s sake NOT using numbers and quantitative points of view as a matter of course every day in real life… Without thinking, we emphasized reading, language, writing, speaking, ideas about people and the world, and not enough practical stuff about measuring things, asking why the physical world works the way it does…</p>
<p>As a so-called “traditionalist”, I can’t agree more. When did we decide that only things which can directly be connected to adult life are useful to learn? For example, trigonometry was directly useful for me to know as a physicist but it is likely that 80%-90% of high-school graduates will never need to use a sine or cosine again. But yet we should teach trig, IMO, along with division of polynomials and other when-will-I-ever-use-this math topics, to (hopefully) develop the analytical, problem-solving skills that every adult needs in a productive life. There are probably way too many topics in the typical US math curriculum, but that’s another story.</p>
<p>No one uses Shakespeare’s English anymore. So why do we teach, read, and act the plays? I imagine that CC people can give many reasons.</p>
<p>By that logic I could have stopped english after maybe 8th or 9th grade (I couldn’t agree more). I am not ever going to need to find the theme of some fictional story in my career. I don’t see the point of 90% of the english curriculum. Might as well just be replaced with Tech Comm in my opinion.</p>
<p>With out wasting so much time in English, we could expand the math and science curriculum more. I don’t think anyone is being pushed too far. I took the “two years ahead” track too. I think Alg 1 in 7th -> Geometry -> Alg 2/ Trig -> Pre Calc -> BC Calc -> and then I took Lin Alg and Calc 3 through a post-secondary program. But I never felt pushed. Thank God we have this or I would have been bored out of my mind. My HS allowed kids to take something like “Sr. Math Topics” their Sr. year or some other easy math course. If anything we don’t push students enough in math and we don’t have enough students going into STEM and the like, and other countries are passing us up.</p>
<p>It’s good to see more kids taking higher level math…our country is lacking in math and science to begin with. But I agree with OP that some kids try to rush through too fast and might not fully absorb all the concepts. I personally wish I learned Algebra a little bit better when I was in grade school. My differential equations course now is very heavy on the algebra and it took me a little time to get refreshed with some of the intricacies.</p>
<p>I never needed to solve a quadratic equation in any of my four careers. So by your logic, all the math I took from Algebra II on was a waste of time. </p>
<p>Except that it wasn’t. Part of being an educated person is having a broad understanding of the world one lives in. That includes some theoretical math (I’m using “theoretical” in a broad sense here).</p>
<p>I went out of my way to avoid taking calculus and have never “needed” to know about integrals and differentials. I regret not knowing calculus, because IMO, a knowledge of the most important area of advanced math is part of being an educated person. </p>
<p>As, IMO, is at least a passing familiarity with the greatest works of our language, and an understanding of how to read them intelligently. </p>
<p>Life and living are about much more than what you do to make money. Although given the culture we live in, I’m not surprised that is a foreign concept to so many young people.</p>
<p>Maybe some kids are being pushed too far ahead, which would be too many, but I think as a whole, American students are not being pushed far enough. As everyone knows, we do terribly in math compared internationally. I think part of the issue is that Americans seem to have mathphobia and that becomes a self-fulfilling prophecy.</p>
<p>Your are obviously the student for whom being two grades ahead is perfectly appropriate. And any good school district will allow moving ahead for any student for whom the regular sequence (or less accelerated sequence) will result in useless boredom.</p>
<p>But the question really comes up because there are apparently many students who are two grades ahead but are afraid to take calculus BC immediately after precalculus, or who attend schools that force them to take it over two years instead of one.</p>
<p>In the first case, it may be that the students were pushed beyond their ability and are not confident in math (when I went to high school, a two year ahead student was someone who easily got A grades in all high school honors math courses including calculus BC as well as college sophomore math, and one year ahead students were generally A students in honors math). Students two grades ahead but not confident in taking calculus BC do not fit this model, which likely points to them being accelerated beyond their actual math preparation and ability.</p>
<p>In the second case, the high school may believe that it has to spread calculus over two years because of poor performance in one year calculus courses due to students with poor preparation or insufficient math ability being pushed too far ahead of their actual math preparation and ability.</p>
<p>Is said student who has been pushed beyond their ability doing well in the classes they have taken leading up to AP Calc BC? If so, then yes, maybe they have been pushed in their abilities but they clearly have adapted, adjusted, and been able to succeed, as their good grades would illustrate. In this case, there is no legitimate reason to fear AP Calc BC for any reason based on personal ability or readiness. Pushed ahead or not, AP Calc seems to be intimidating to ANYBODY, regardless of whether you’re on track and about to take it or are ahead of the game and about to take it. Some kids at my school are on completely regular track (Alg 1, Geo, Precalc, AP Calc senior year) but AP Calc is intimidating to them even though they did perfectly fine in the previous three courses… so they opted to take college algebra at the community college (piece of cake) instead (dual enrollment).</p>
<p>Now, if the student was pushed beyond their ability and didn’t do so hot in the classes previous to AP Calc, then they have legitimate reason to fear AP Calc. That doesn’t mean they can’t or shouldn’t do it, but letter grades and GPAs aside, why would they take a class when they can’t master its prerequisites?</p>
<p>In our school district and many surrounding, there is no option of skipping Calc AB. All accelerated math students must take Calc AB, then BC, if they want to take BC. These are top students who are not intimidated or perform poorly. I do not know the reason for this and did not think there was a problem with it. I am even under the impression that some area schools do not offer Calc BC. So, in answer to your question, maybe some students are not pushed enough.</p>
<p>Meh kids in Germany and Korea are busy doing differential equations and linear algebra in high school. Meanwhile kids here get 3 months off for summer vacations, forget everything they learned, and then think pre-calc is hard that many of them take in the 11th or 12th grade. </p>
<p>Kids should stop getting so long off for summer vacation here, then they’d be able to catch up to their world counterparts. We pay teachers full year salaries anyway.</p>
<p>I don’t think summer vacation is the issue. In many areas, maybe most, it is no longer a full 3 month break. When I lived in Ca, it was 8 weeks (an extra week was given in fall) and where I live now, it’s about 9 1/2.</p>
<p>I love long summer breaks for my D, time to catch up to grow up. Otherwise, I agree other countries are teaching at least as advanced math to their HS students. US gave up steel industry when cheap labor became available, gave up auto industry, textiles long gone. The tech industry that is expanding and hiring will need well educated in math and sciences. How are we going to compete if we just coddle our kids? We spend the most money in educating students, yet it doesn’t look like we have the best prepared students by any measure. Most of the time we hear how stressed our kids are. How is that possible? We spend more money in education than others, our kids are not doing better and yet they are so stressed out that they have to use the most drugs in the world to relieve their stress? Does this make sense to anyone?</p>
<p>Your high school’s regular track appears to be compressed, with what is normally four years of high school math compressed into three. Regular math students do not reach calculus in high school; they take it, if needed, in college. Of course, college calculus courses are full speed, similar to high school calculus BC over one year.</p>
<p>Is “college algebra” just repeating high school math?</p>
<p>The calculus over two years requirement appears to be holding back the top achievers who could easily handle calculus BC over one year, probably in order to accommodate those who were pushed ahead beyond their math preparation and ability.</p>
<p>If students who were on the regular track (completing precalculus as high school seniors) are expected to take college freshman calculus (which calculus BC over one year approximates), why are the students who are better at math not allowed to take calculus at an equivalent pace?</p>
<p>Are you suggesting giving the advanced students the option of Calc AB, then BC or taking Calc BC, then multivariable calculus? Our school does not offer the latter (multivariable calculus) at this point in time. I do not really know why they teach Calc AB/BC over 2 years. It seems to be standard in the area.</p>
<p>UCB - there is some disconnect between what the schools are programmed to teach vs what the engineering state schools within the state are expecting.</p>
<p>In our school district (I think the biggest in texas), if someone got to calculus ab or bc in senior year, they are considered an year ahead in Math, having completed algebra I in 8th. However, UT engineering school expects people to have completed an AP exam in AB or BC by the time they apply and to have scored a 3 or else it is a conditional admission until the score shows up in July after graduation. So one has to wonder how they would come up such a rule knowing that it does not match the public school’s curriculum.</p>
<p>Pizzagirl - I agree Statistics is important. The main problem seems to be that a lot of colleges don’t accept it for credit and expect you to retake it as a required class. Every master’s program I was in needed a Stats class as a requirement. Since my undergrad engineering school did not need it, I had to retake it and when I did an MBA used my other master’s credit to get a waiver.</p>
<p>Igloo - what you say would make sense if the primary place for learning and growing for our top students were our local schools. But that’s not the case. Long summers provide an opportunity to remix kids geographically and instead group them by talents and interests. The various math camps bring together that high school talent and teach these willing and avid learners more real math in 5-8 weeks than they get all year in school while making friends with those other “oddballs” who are just as excited about learning as they are. It’s not for test scores or grades or credits. It’s just to learn math because it’s beautiful, not because it’s useful or necessary. </p>
<p>I’m sure this goes on in other arenas too. Our schools pull from local populations and can only take kids so far, even when they have the best of intentions, teaching ability and funding. So school is for the basics --the floor of learning. Summer is for flying high with no ceilings. </p>
<p>Take away the summer, and all you have is floors.</p>