<p>I think that the better way is to use the Chinese language schools that you can find near large metro areas in the US. Our kids did these for eight years each and I think that learning it over a much longer period of time is easier than doing it in college. Learning chinese is tough, tough, tough though.</p>
<p>There are Chinese language schools in most areas, but quality of instruction can be dubious and many are unprepared to teach non-heritage students (e.g. until about 5 years ago, my local Chinese school taught all classes–from kindergarten onward–IN CHINESE).</p>
<p>Yes, it doesn’t work well if you don’t have a native speaker at home. I’m assuming that the poster was Chinese given the username and the interest in the student wanting to learn Chinese.</p>
<p>Back in the seventies, IIRC, lots of schools used Apostol for their honors calc classes, for students who scored above 700 on old SAT and got A’s in high school math. Most of these students would also have already taken high school calculus.</p>
<p>I remember 2/3 of my own honors calc class at a small LAC dropping after the first semester, though. Pre-meds (except for a very few) did not want to work that hard, and many others who would have done fine with the standard text (Thomas?) decided that they just weren’t suited to learn math, in spite of previous success and high SAT scores, once they got a B or C. The second semester did seem easier. (Perhaps we had gotten used to the instructor and text?) But, nearly all of the several students my year who ended up majoring in math had passed on the honors intro class and taken the regular calc classes.</p>
<p>I do remember Thomas in high-school. </p>
<p>It sounds like math was harder way back when. There’s an interesting review at Amazon for Spivak’s book on where math is today. I don’t think that that many schools use something like Apostol or Spivak for their honors classes anymore. Most of the common textbooks today are heavily application-oriented. There is no assumption that students know how to do proofs when they get to college.</p>
<p>I believe Chicago’s 160s Honors Calc uses Spivak. It’s such a tough course because they break the calc down to the basics and proof it all. At many colleges, proofs happen in Analysis (and some schools have an Intro to Proof after calc before Analysis).</p>
<p>S1 was lucky because his HS calc teachers used materials from top colleges’ Calc 1-3 courses and real problem sets. He learned proofs at HCSSiM. Was terrific preparation. He has a copy of Spivak for fun and reference, too.</p>
<p>S1 did IBL Analysis and I’m not even sure they had a text!</p>
<p>The first Amazon Spivak entry mentions that the text is used at Chicago.</p>
<p>We used Salas and Hille way back when at BC for Honors Multivariable and it was almost all theory. Very little in the way of applications. Does this mean that the high school geometry course doesn’t cover proofs anymore? Do high-school students even take geometry anymore?</p>
<p>BCEagle, where I go there is an intro to calculus series which is completely proof based. It’s not the standard one which everyone takes, but it is an option for anyone who wants to take it. Do most schools (large schools) not have a series like that?</p>
<p>We did proofs in our Geometry class in high school (this was just regular Geometry, not honors) but then never again until Calc. And from what I understand most high school Calc classes do not do proofs or delta-epsilon definitions. The proofs we did in Geometry were very easy, just straight forward problems that required you explain your steps. Not sure if the honors Geometry was harder, but I doubt it was much harder.</p>
<p>Am I being a “bad” parent here? All I know is DS has taken 4 classes. I know nothing as what were they. I asked - doing o.k. at school he answered - ya. That was it.</p>
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<p>I don’t know. I believe that there are a lot of high-schools that don’t offer calculus at all. It seems that most schools that do offer calculus offer something compatible with AP calculus as that seems to be the major goal. It’s the enlightened schools that offer an honors version that is proof-based.</p>
<p>I’m familiar with Jacobs Geometry which does have proofs that can take a while. It also has constructions with compass and straightedge.</p>
<p>Art. S2 was waffling between art and soft sciences all through high school and he does have raw talent. He said the pre-req art class kicked his butt (but he got an A-). So funny to hear that from him. I sent him off telling him that could be his breather class. Hah.</p>
<p>“> Do most schools (large schools) not have a series like that?”</p>
<p>No, I was talking about universities. Do most universities not have a proof based introduction to calculus series?</p>
<p>“No, I was talking about universities. Do most universities not have a proof based introduction to calculus series?”</p>
<p>Most Calculus classes aren’t proof based. Of course if the professor is good at teaching the subject they will often show the proof. However, I didn’t have any real proof questions on any of my calculus classes, with the minor exception of a “prove XXX does or does not equal YYY”, which really isn’t a proof, in the formal sense.</p>
<p>I’ve had some other math classes that were heavily proof based (Discrete Structures, Probability Theory).</p>
<p>I think the proofs really kick in with Advanced Calculus/Real Analysis, which is the next level beyond Calc 1 to 3 (single to multivariate calculus).</p>
<p>I don’t think I’ve explained well.</p>
<p>At my university (and I believe most large universities - though perhaps a different exact number) there are 5 different introduction to calculus series. One of them is entirely proof based. Is this common? Do most large universities have multiple introduction to calculus series, with one being based on proofs?</p>
<p>I know that at both of my kid’s schools they offer two calculus series. One for life science majors and one for math, engineering and physical science majors.</p>
<p>I think that most large universities offer a Calc for Science and Engineering and a Calc for social science/business majors. I had a look at the University of Michigan Math department and they do offer a proof-based course.</p>
<p>I also had a look at the University of Vermont and they do offer a variety of calc courses but I didn’t see anything involving proofs. My offhand guess is that most universities don’t offer the proof-based courses. If you want that, you get it later courses.</p>
<p>S is still in hs but took his first cc calculus class. Took Calc BC as a junior(A+) and got a 5 on exam. CC professor (a PhD and really good teacher) looked over the curriculum for BC and said there were some significants gaps between that and calc III. He gave S some work to do over the summer to get on track for his class and warned him that the emphasis he teaches is much different than the AP emphasis.</p>
<p>Well S procrastinated over the summer(didn’t do any extra work) and had a rude awakening in September! He seemed to struggle (for him) all semester. Was able to figure it out eventually and got an A on the final, but still a B in the class. He wants to retake the class when he gets to his 4 year college as he still feels he does not have this at the level he should.</p>
<p>So–I wonder if it might be that Calc BC does not quite line up just right with Calc III and this causes a lot of problems for many kids. Or maybe the courses just emphasize different things or ways of thinking and the students need time to adjust.</p>
<p>Any math profs out there that are familiar with the emphasis in Calc BC vs. Calc III?</p>
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I think there may be a confusion in understanding here. Introductory Calculus at the college level is typically either geared towards engineering/physics/hard sciences, or towards others/business/soft sciences. The non-science calculus (I termed it Calculus Lite when I taught these courses) has virtually zero theory. No proofs to speak of. You lay out the basic concepts, get the techniques down, and work on applying them to various problems. Sometimes a school has a special sequence “Calculus for Business Majors” which focuses the applications on Revenue - cost - profit type applications. </p>
<p>The calculus for engineering and science majors has quite a bit of theory and proofs, and is much more in depth across the spectrum of functions (trig, logs, etc.). In my courses we used Leithold for 3 semesters. </p>
<p>Virtually every college teaching this stuffs will have a similar Lite/Not breakdown, although the non-science path typically is 1 course or maybe 2, while the engineer/science path is at least 4 courses. Neither of these sequences gets anywhere near real analysis, which is at a whole 'nuther level, being solely theories and proofs. You have to have had the intro courses before you can even get near the real analysis level.</p>
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<p>I don’t think that we have much disagreement here but there are
colleges that use Anton, Osterbee, and other for calculus for science
and engineering and those books don’t cover much theory. We could even
take a look at Gilbert Strangs’s calulus textbook where we find this:</p>
<p>“I want to say clearly: Mathematics is not formulas, or computations,
or even proofs, but ideas. The symbols and pictures are the language.
The book and the professor and the computer can join in teaching it.
The computer should be non-threatening (like this book and your
professor) - you can work at your own pace. Your part is to learn by
doing.”</p>
<p>I’m not familiar with Leithold - is it like Apostol or Spivak? I have
about 13 calculus texts at home dating back to the 1940s so I have a
little variety to look at.</p>
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<p>What do you think of teaching the first chapter of Spivak to high-school
or even middle-school kids?</p>
<p>
Probably closer to Apostol. IMHO it was more rigorous than the one I taught from one semester (one of Stewart’s). The full 3 semester course text is over 1000 pages. There was quite a bit of theory and proofs, but not near the level that we got later in Real Analysis. Our RA I text was Bartle and Sherbert, which I didn’t care for.</p>
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I’m all for introducing more of the interesting features and the beauty and subtlety of mathematics at the middle and high school levels. Even some abstract algebra, topology, etc. Too many students are turned off by the system we have now, or they think they are not good at math and abandon all hope before they even get to the actual subject (i.e. while they are still doing arithmetic). I was one of them, and I hate that fact.</p>