AP Calculus - How we (and others) did it...

[Jane Elliot]

I'll add here that I agree with regentrude and Luann. If you want to double up in math, doing algebra 2 and geometry simultaneously makes sense.

 

Whew! I was nervous after I posted that, wondering what Kathy in Richmond would say. :tongue_smilie: Crazy, I know, but your opinion matters to me even though I don't know you IRL. Your advice has been tremendously helpful to us. Thank you!

[8filltheheart]

Thanks, 8. When I suggested doubling on math, I was thinking of actually spending double time, not doing math faster. Over the past few weeks, DD has spent 10+hours per week on math because she is excited and motivated. I was thinking of a workload like this... which, of course, has to be the student's choice.

 

At 10+ hrs/week it will take you far longer than 1/2 a school yr to finish the text.

 

Two things:

 

(1)The AoPS online Precalculus class moves really fast, too fast for what I like to accomplish. I worked through the Precalc text alone with a highly gifted student last school year, taking time to do many of the challenge problems. We started in late September and worked through the end of May, if I remember correctly. Even so, we didn't do everything in the book (skipped the last couple of chapters). I admit that we had a lot of fun in Chs. 7 & 8, trigonometry and complex numbers & the geometry of complex numbers, and that took extra time. Sometimes even the discovery problems within the text would be old AIME problems (look at 7.4 if you have the book already for an example of this); these sections just took lots of time and thought to get through. If you try to get through a section such as 7.4 in a single day, you'll miss out on something good. The scope goes way beyond any other precalculus book out there. Totally worth it, though, for us. I think that learning how to spend a day or two knocking their brains against a really tough problem and finally conquering it was the main benefit my kids took away from their AoPS classes. And the precalculus text is one of their most challenging books.

 

(2) Traditionally, precalculus covers some topics in advanced algebra, also, and those topics are completely missing from the AoPS Precalculus text & course. That's what AoPS covers in its Intermediate Algebra text, though. So I'd recommend that kids using AoPS for precalculus also look at that text in addition to the Precalculus text. In the AoPS online school, that corresponds to their Algebra 3 course.

 

Your daughter will already have finished Intermediate Algebra already, though. Sure, she *could* finish up precalculus by the end of the school year if you want to get done. You can just adjust the time spent by doing the text sections and skipping enough of the Challengers to get through at the speed you choose. I think that you'd be missing something good, though. If I had my choice, I'd cut calculus short rather than precalculus (just look at the relative sizes of those two AoPS texts - 528 pp for precalc and 336 pp for calc - they've got it right:)). I'd rather puzzle over the complex transformations of the plane or linear transformations and how they're represented as matrices wrt different bases than learn yet another integration technique or do yet another contrived min-max problem. Calculus seems like two main ideas held together by a year's worth of tricks. If you really want to learn that sort of math, take a real analysis class instead. Just my two cents :D , but I think in retrospect that these are the decisions that served us well over the long haul.

 

 

:iagree:

 

Regentrude, even if you reduce it to simple comparison in textbook size (which as Kathy has stated is grossly underestimating the content of the pre-cal book), if you compare ds's pace w/ the cal bk vs. the pre-cal bk, 25 weeks at a minimum of 10-12 hrs/wk would only be 2/3 of the pre-cal text.

 

I personally don't think looking at the avg length of their online classes gives anywhere near a realistic representation of getting through their texts w/real appreciation for the content unless you are talking about the kids at the peak of the peak in skill. Even really strong math students are going to require more time to get the most out of it. The only reason ds was able to get as much out of their alg 3 course as he did was that he took it mostly over the summer so he had lots of time to devote strictly to it b/c he didn't have anything else he had to work on.

 

FWIW, when ds gave his estimate for math last yr, he was guesstimating strictly in terms of actually working on problems, not reading the material and learning the concepts.

 

I am not meaning to discourage you, but I would be prepared for the pre-cal book to take more time if your dd needs it.

[regentrude]

Your daughter will already have finished Intermediate Algebra already, though. Sure, she *could* finish up precalculus by the end of the school year if you want to get done. You can just adjust the time spent by doing the text sections and skipping enough of the Challengers to get through at the speed you choose. I think that you'd be missing something good, though. If I had my choice, I'd cut calculus short rather than precalculus (just look at the relative sizes of those two AoPS texts - 528 pp for precalc and 336 pp for calc - they've got it right:)). I'd rather puzzle over the complex transformations of the plane or linear transformations and how they're represented as matrices wrt different bases than learn yet another integration technique or do yet another contrived min-max problem. Calculus seems like two main ideas held together by a year's worth of tricks. If you really want to learn that sort of math, take a real analysis class instead.

 

Thanks for all the advice, Kathy.

I guess my approach to math is colored by being a theoretical physicist :) Sure, calculus is only two main ideas, but the year's worth of tricks is actually several years worth of tricks and then the main bread and butter for some colleagues (I shudder when I see the integrals waltzing across DH's pages, who is a condensed matter theorist). So, I see us doing a lot more calculus than the AoPS text (which does not seem to cover all that much material, judging from the TOC)

My DD is dying to get to calculus based physics so that she can get the prerequisites to study some "cool" physics. So we are tailoring our approach to math according to her plans which involve calc based physics in Spring of 11th grade.

While I consider matrices and transformation interesting, I must admit that I personally only fully appreciated them when I encountered them in a physics context. In my field of physics, I have found them to be less important than calculus (my colleagues in crystallography will be of a different opinion)

 

We will see how it goes. Since DD will take several years of math in college, I am not too worried should we decide to cut something short or postpone certain topics until later (right now we are planning for two years of calculus, which would afford us plenty of time to revisit some topics concurrently with their emergence in physics.)

But thanks a lot for the advice. We are definitely not rushing math; we may, however, based on our perspective, decide to rearrange some material.

[regentrude]

I am not meaning to discourage you, but I would be prepared for the pre-cal book to take more time if your dd needs it.

 

Thanks, 8. We will definitely take whatever time she needs. We are not rushing math!

As I explained in my previous post, our view of math is a bit colored with respect to physics; we may choose to postpone certain topics of AoPS precalc until after calculus 1. Calculus is actually pretty simple - I personally find introductory calculus much simpler than some of the matrix transformations AoPS includes in precalc :)

(Now, three dimensional integrations and vector calculus is another beast....)

[Kathy in Richmond]

Whew! I was nervous after I posted that, wondering what Kathy in Richmond would say. :tongue_smilie: Crazy, I know, but your opinion matters to me even though I don't know you IRL. Your advice has been tremendously helpful to us. Thank you!

 

Oh my goodness, that's too funny!

 

Do you realize that it's moms like you and 8Fill whom I look up to?!

 

You two and all the other moms of many are who are classically educating so many kids so well are my heroes. I don't think that I could do it. I was often maxed out trying to get everything done with only two kids!

 

I've had the honor of meeting 8Fill in real life. Her family is so NICE; they're tops not only in academics but also in character; how she raises so many so well is totally beyond me! I picture you and your crew to be a lot like hers. You've been around these boards forever like I have (I joined when my oldest was a sixth grader)...I feel like we're friends in spirit by now, even if we've never met IRL. :)

Edited November 11, 2011 by Kathy in Richmond

[Kathy in Richmond]

Thanks for all the advice, Kathy.

I guess my approach to math is colored by being a theoretical physicist :) Sure, calculus is only two main ideas, but the year's worth of tricks is actually several years worth of tricks and then the main bread and butter for some colleagues (I shudder when I see the integrals waltzing across DH's pages, who is a condensed matter theorist). So, I see us doing a lot more calculus than the AoPS text (which does not seem to cover all that much material, judging from the TOC)

My DD is dying to get to calculus based physics so that she can get the prerequisites to study some "cool" physics. So we are tailoring our approach to math according to her plans which involve calc based physics in Spring of 11th grade.

While I consider matrices and transformation interesting, I must admit that I personally only fully appreciated them when I encountered them in a physics context. In my field of physics, I have found them to be less important than calculus (my colleagues in crystallography will be of a different opinion)

 

We will see how it goes. Since DD will take several years of math in college, I am not too worried should we decide to cut something short or postpone certain topics until later (right now we are planning for two years of calculus, which would afford us plenty of time to revisit some topics concurrently with their emergence in physics.)

But thanks a lot for the advice. We are definitely not rushing math; we may, however, based on our perspective, decide to rearrange some material.

 

Yeah, that makes sense. We mathematicians like our math straight up with as few numbers as possible, lol. I love stuff like number theory and linear algebra even w/o applications now, though it's always satisfying to see how it's all used by *others*.;)

 

We're all studying math with different goals in mind. And yes, of course, your dd has a lifetime ahead of her to learn more. I didn't come to my love of pure math till later in life. I was on the calculus track as a kid, and I was more or less pushed into studying applied fields in college. I completed my engineering major OK, but really never found my passion till I snuck in extra math classes. Sounds as if your daughter's already found her passion, and it makes great sense to get her ready for calc-based physics as soon as you can.

 

Btw, have you chosen a calculus text for your dd yet? From what you've described about your goals and your dd's abilities, I just have a feeling that you'd like the Stewart calculus text. I tutored two kids from it this year, and it's really perfect for the future engineer/physicist who is after the tough/rigorous approach, but oriented toward applications rather than theoretical math. It's also a good choice if your dd is aiming for the AP Calculus exam (of course, I'd always supplement with specific test prep).

[regentrude]

Btw, have you chosen a calculus text for your dd yet? From what you've described about your goals and your dd's abilities, I just have a feeling that you'd like the Stewart calculus text. I tutored two kids from it this year, and it's really perfect for the future engineer/physicist who is after the tough/rigorous approach, but oriented toward applications rather than theoretical math. It's also a good choice if your dd is aiming for the AP Calculus exam (of course, I'd always supplement with specific test prep).

 

Thanks, Kathy!

No, we have not looked into calculus texts yet. We were going to have a look at AoPS because we love the series so far. I will definitely check out the Stewart.

Do you prefer the Stewart over AoPS? Or use both? I am always grateful for you sharing your expertise.

 

ETA: Just looked it up, seems there are several different ones. Which Stewart is it that you recommend?

[Jane Elliot]

You've been around these boards forever like I have (I joined when my oldest was a sixth grader)...I feel like we're friends in spirit by now, even if we've never met IRL. :)

 

I am honored. :)

[8filltheheart]

Oh my goodness, that's too funny!

 

Do you realize that it's moms like you and 8Fill whom I look up to?!

 

You two and all the other moms of many are who are classically educating so many kids so well are my heroes. I don't think that I could do it. I was often maxed out trying to get everything done with only two kids!

 

I've had the honor of meeting 8Fill in real life. Her family is so NICE; they're tops not only in academics but also in character; how she raises so many so well is totally beyond me! I picture you and your crew to be a lot like hers. You've been around these boards forever like I have (I joined when my oldest was a sixth grader)...I feel like we're friends in spirit by now, even if we've never met IRL. :)

 

:blushing: aw, shucks, Kathy :blushing: Thank you so much for the kind words and compliment. I am very honored.

 

If anyone ever had any doubts, Kathy and her family are even more amazing in person! I don't think I have ever encountered such positive reinforcement and encouragement from anyone else. What a blessing for our family and this forum to have someone so willing to share her experiences and success w/such humility and kindness!

[Kathy in Richmond]

Thanks, Kathy!

No, we have not looked into calculus texts yet. We were going to have a look at AoPS because we love the series so far. I will definitely check out the Stewart.

Do you prefer the Stewart over AoPS? Or use both? I am always grateful for you sharing your expertise.

 

ETA: Just looked it up, seems there are several different ones. Which Stewart is it that you recommend?

 

My son used my old edition of Thomas and Finney calculus from the 70's. I like that well enough, but it was not a good match for my daughter. She would have done well with AoPS, but it wasn't out yet (neither the text nor the online class). So I had her read through Calculus for the Forgetful instead. She learns best by seeing the big picture first, and it's a slim text meant for adults who had calculus in the past and want to relearn it. When she had finished that book, I had her go back through Barron's AP calc review guide to get the details (i.e., work tons of problems and improve her fluency.)

 

I usually don't care for modern math texts, but I really grew to like Stewart last year in my tutoring. The explanations were mathematically correct and didn't gloss over the theory; it had plenty of intelligently designed practice problems covering applications that future scientists and engineers might enjoy. I used this version for my first student who was in Calc 1 at William & Mary, and this version for my second student who was an engineering major in Calc 2 at Vanderbilt. The latter is some crazy version designed specifically for Vanderbilt students (there was very little difference between the two). I'm sure you'd be fine with any of the versions; I'd get a slightly older edition myself to save $$ over a higher-priced current edition.

 

I own and like the AoPS calculus text, also, though I've yet to use it with any student. It's less traditional, more math theory oriented, and set up like all the AoPS texts in format. I don't think you could go wrong either way.

[regentrude]

I usually don't care for modern math texts, but I really grew to like Stewart last year in my tutoring. The explanations were mathematically correct and didn't gloss over the theory; it had plenty of intelligently designed practice problems covering applications that future scientists and engineers might enjoy. I used this version for my first student who was in Calc 1 at William & Mary, and this version for my second student who was an engineering major in Calc 2 at Vanderbilt.

 

Thank you. One more question: what does the title "Single variable essential calculus: Early transcendentals" mean? I understand the single variable calc, but "early transcendentals" sounds, well, a bit like a cult. :)

Seriously: does it just mean that transcendental functions are covered early?

[snowbeltmom]

 

I own and like the AoPS calculus text, also, though I've yet to use it with any student. It's less traditional, more math theory oriented, and set up like all the AoPS texts in format. I don't think you could go wrong either way.

 

I am grateful that I found this group and a big "thank you" to you all for sharing your advice. I have a follow-up question to the the AoPS Calc. discussion. My oldest has taken all of AoPS online classes up through Pre-Calc (As an aside, he is taking pre-calc now, and the pace is extremely fast. On a "light math day" he is spending 2 hours on the class.)

 

He plans on majoring in chemistry in undergrad with hopes of going on to medical school. He does not necessarily enjoy the proofs in math. He would much rather solve a problem that has a numerical answer.

 

I love AoPS's approach. I feel my kids are really understanding math rather than memorizing a bunch of plug and chug types of problems (like I did in high school.)

 

In light of my son's future plans of study, would there be a better Calculus program for him than AoPS, or do you think he would be better served with AoPS. Also, he does enjoy the online classes and does not complain at all about needing to devote hours a day to math.

[memphispeg]

We are using Larson. I like it b/c I can easily match problems to text examples. Dd would prefer NOT to do practical applications. The book has some applied problems and also some puzzlers. Dd will probably have to take Calc. in college so, she just wants some good technique training at this point.

BTW - I really do think some of the more brilliant posters on this board should get together and create a Calc/Physics year-long course. I stumbled around a bit trying to do this last year but, I lack the math/physics expertise. I could see where it would be a really terrific course. Making knowledge less segregated, etc.

[Kathy in Richmond]

Thank you. One more question: what does the title "Single variable essential calculus: Early transcendentals" mean? I understand the single variable calc, but "early transcendentals" sounds, well, a bit like a cult. :)

Seriously: does it just mean that transcendental functions are covered early?

 

Yep, that's all. All these choices over a calculus textbook. Silly, no? Do you want your sines and logs served up earlier or later in the course? I suspect that late transcendentals came into vogue when students started arriving in college calculus classrooms with only the fuzziest notions of trig or exponential functions. It helped the profs to teach the calculus of polynomials first. Only after that was mastered would they reteach the transcendental functions and then incorporate them into calculus.

 

And what has it come to when a 600 page calculus text is advertised as brief and condensed? :lol:

 

Oh well. I did like its no-nonsense approach (no silly sidebars, etc) and challenging problem sets. I also like Resnick & Halliday for calculus-based physics, and it had the same sort of flavor IMO.

 

Here's a link I found this morning that gives the tables of contents of Stewart's various textbooks. The edition that I used is at the very bottom of the list. Note that it only includes material for two semesters of calculus (i.e., the Calculus BC syllabus). He also offers an edition with the third semester (multivariable calculus) included.

[regentrude]

Thanks, Kathy. I definitely prefer early transcendentals then.

Now I am getting all excited about calculus...off to check the TOC.

 

Over breakfast this morning, DH and I felt compelled to derive that cos x is the derivative of sin x ... we fully admit to being dreadfully nerdy at our house.

[memphispeg]

Larson does transcendental right after integration. There's a sort of review of log and trig f(x) behaviors first and then on into the derivatives and integrals. When I took calc. in the old days (we had a backassward syllabus, integration and then derivatives), we spent time with the trig and log functions first before moving to the polynomial stuff. It was a computer-oriented class where we modeled calc. techniques on known curves and functions and then extrapolated the formulae from our programming conclusions.

[Kathy in Richmond]

I have a follow-up question to the the AoPS Calc. discussion. My oldest has taken all of AoPS online classes up through Pre-Calc (As an aside, he is taking pre-calc now, and the pace is extremely fast. On a "light math day" he is spending 2 hours on the class.)

 

He plans on majoring in chemistry in undergrad with hopes of going on to medical school. He does not necessarily enjoy the proofs in math. He would much rather solve a problem that has a numerical answer.

 

I love AoPS's approach. I feel my kids are really understanding math rather than memorizing a bunch of plug and chug types of problems (like I did in high school.)

 

In light of my son's future plans of study, would there be a better Calculus program for him than AoPS, or do you think he would be better served with AoPS. Also, he does enjoy the online classes and does not complain at all about needing to devote hours a day to math.

 

Well, if your son doesn't especially enjoy the proof side of math and wants lots of numerical problems, then maybe he'd also like to use Stewart for calculus. His text includes LOTS of problems, while AoPS takes its usual brief-is-better approach. They're both fine, just different approaches. Stewart's text also includes the theoretical explanations for calculus, and I wouldn't consider it "plug-and-chug". It's just that the emphasis is more on applied problem solving. Stewart is usually considered to be a tough calculus choice, but after AoPS precalculus, your son would be more than prepared.

 

I don't know of any online classes using Stewart. PA Homeschoolers uses the Larson text (a more common choice), I believe.

 

AoPS calculus is terrific, too. I love what I've seen flipping through the book. I've never taught from it, but 8Fill's son is currently using it with the online class. I know that he's very happy with it so far; in fact, I've only heard good reports of the class from different people.

 

In either case, you'd want to supplement with AP prep books. We like using Barrons, this text for FRQ practice (it has very detailed solutions showing exactly what they're looking for), and a released exam or two just before the real test in May.

[Kathy in Richmond]

Thanks, Kathy. I definitely prefer early transcendentals then.

Now I am getting all excited about calculus...off to check the TOC.

 

Over breakfast this morning, DH and I felt compelled to derive that cos x is the derivative of sin x ... we fully admit to being dreadfully nerdy at our house.

 

Heehee...we're like that here, too. I'm all excited because tonight dh and I are going out to hear Dr. Robert Ballard lecture on archaeological oceanography and his discovery of the Titanic. Nerds all around. :D

[Kathy in Richmond]

When I took calc. in the old days (we had a backassward syllabus, integration and then derivatives), we spent time with the trig and log functions first before moving to the polynomial stuff. It was a computer-oriented class where we modeled calc. techniques on known curves and functions and then extrapolated the formulae from our programming conclusions.

 

Wow, that really is a different sort of approach to calculus! Back in my day, Carnegie Mellon taught an integrated calculus/physics course like you mentioned in your other post. Great idea; I wish more schools still did that.

[snowbeltmom]

Kathy, thanks for the info. There are just so many choices it can be overwhelming.

[memphispeg]

Kathy -

When I was planning our physics and calc. curriculum I thought that since the fundamental theorems grew out of consideration of physics calculation difficulties there would be a curriculum that mirrored some of the original classical reasoning. Not to be. It could prove interesting to students, to look at where alg. failed to answer the great physics questions and then to learn why/how calculus does.

[Joan in GE]

This site is for AP Calculus teachers and has a lot of useful information, including a page with links for the different typical textbook choices...

 

http://designatedderiver.wikispaces.com/

 

Joan