Glad to hear her school is open to her changing tracks. She can take a look at edx.org - these are free online courses, and offer a comprehensive view of the material than individual youtube videos. Especially useful for understanding limits:
https://www.edx.org/course/apr-calculus-challenging-concepts-davidson-next-calapccx-1
I viewed a few of the videos and they looked engaging. The courses are archived and self-paced and also come with self-check questions.
HI all, D started her self study today. The text book is Calculus (graphical, numerical, algebraic) by Finney, Demana, waitz, Kennedy. She likes the book, but really wants more practice problems. Can you recommend a workbook she can use that has solutions?
Thanks!
DS did the AoPS Calc book as independent study in 8th grade. He took AP Calc BC this year as a 9th grader. Interestingly, the AB and BC questions were both a bit hard for him when he first did them. I think he indicated AoPS was more theoretical. Overall though he breezed through the class and thinks/hopes he got a 5 on the recent exam. He has a very high A in the class.
He took all of the AoPS courses though. I would imagine it would be quite difficult if Calculus was your first AoPS experience, not sure.
@gallentjill Schaum’s outline for calculus is a good resource. You can preview some pages on amazon.
Here is a free calculus textbook that has problems that she can try (or look in for alternative explanations if the usual book’s explanation of something is unclear to her).
https://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/textbook/
@gallentjill As a supplement to their textbooks, my kids used “Be Prepared for the AP Calculus Exam” by Howell and Montgomery and “Multiple Choice Questions to Prepare for the AP Calculus BC Exam” by Rita Korsunsky. Both can be found on Amazon. The only downside to Korsunsky’s workbook is that while the answer key is contained at the back of the workbook, if you want the full solutions (which are very detailed) you must purchase a separate CD.
I would not recommend the AoPS books. They are all about competition math. There has been a lot written about how detrimental competition math is in general and how especially bad it is for women and minorities in general. A Fields medalist once said something to the effect that true math is done by a tortoise very carefully not like a race car driver as is required in competition math.
Below is a quote from the AoPS web site on their calculus book. Their philosophy is counter productive and extremely detrimental to the learning of math.
“The book includes hundreds of problems, ranging from routine exercises to extremely challenging problems drawn from major mathematics competitions such as the Putnam Competition and the Harvard-MIT Math Tournament. Many of the problems have full, detailed solutions in the text, and the rest have full solutions in the accompanying Solutions Manual.”
If you want a textbook how about Larson/Edwards. I believe it is the most popular Calculus textbook. There must be a reason!
@collegedad13 I agree with not recommending the AoPS books for this student, but your assessment of the AoPS texts is absolutely false. AoPS’s approach encourages students to understand the theory behind the processes used in mathematical calculations. It is an approach that will not work for many students, but for gifted math students who crave understanding behind formulas and want to prove the theorems for themselves, AoPS is a great approach.
Your contention, “Their philosophy is counter productive and extremely detrimental to the learning of math,” is disproved by all of the grads of their textbooks who go on to excel in math dominated programs.
FWIW, my ds attributes his great success in physics to his math foundation from AoPS.
@Mom2aphysicsgeek I have a DD who is extremely gifted in math itself and is completely opposed to the AoPS approach.
You may want to look at Cathy Oneils writings on this subject. She has a PHd in math from Harvard and taught at MIT.
My contention has entrenched the good old boy network in the field of math and has kept it from being in tune with the twenty first century at many instituitions
@collegedad13 And for every person like your dd and Onelis, you will find others with differing views.
I would suggest my ds is gifted in math since he graduated from high school having completed multivariable, diffEQ, and linear alg 1 and 2. During middle school, he had a math mentor who introduced us to AoPS. His mentor has a PhD in math and has kids who equally gifted in math who used AoPS and went on to elite colleges in math related fields. These kids have all thrived via AoPS’s approach.
My ds was bored to tears with the Larson text (what we originally considered for BC) and ran back to AoPS for cal. He would rather spend hrs working through proofs than plugging and chugging equations.
I have 2 other kids who are Larson type students. They are excellent math students, but AoPS is not their desired approach. For my dd, AoPS’s approach drove her crazy and she refused to use the texts. Just bc she didn’t like the approach does not negate the value of the approach for kids like her brother.
@collegedad13 The only thing that the AoPS books have in common with competition math is problem solving. It just so happens that high-level math competitions involve a lot of problem solving, i.e., the same types of problems.
The AoPS teaching approach may not be everyone’s cup of tea, as its style of teaching involves working through hard problems in order to learn the material, but that is wholly unrelated to competition or gender perceptions.
@gallentjill My senior (as of today - can’t believe I just wrote that) is also planning to get a leg up on calc this summer, to prepare for Physics C and for the BC class (which long had AB as a prereq at her school, dropped the prereq but didn’t really change the teaching to include the AB material, according to other kids). She will be working through the first half of the Strang text along with the Strang videos linked in this thread. On the page of each video, there is also brief PDF of notes summarizing the concepts and some of them include some practice problems. The Strang text, also linked in this thread (free), would have extra practice problems. Schaum’s outline is worth picking up, both for extra problems now and as a resource next school year.
@gallentjill Another video resource is CalcView that is free and used with the Larson’s texts.
My daughter did this this year (as a junior), and it was a mistake. It was very hard for her, has led to an unnecessary level of stress and undermined her self esteem.
One useful resource for one of our children was the online set of questions by albert.io. You can pay for a 3-month or 10-month subscription. She started it just a week or so before her AP test, but we wish in retrospect that she had used it during the school year.
As for the Art of Problem Solving books, I agree that the Calculus book is a little terse (and was written by another author), but I think that the Algebra and Precalculus books are just wonderful. They actually emphasize very basic concepts. For example, I think there is a whole chapter on understanding negative numbers. This conceptual approach, as opposed to recipes or rote formulas, helps students learn how to approach problem solving, as @evergreen5 noted above. The books teach problem solving by posing a difficult problem, and then going step-by-step on how to solve it. The books are filled with warnings about possible misconceptions and common errors made by students. Yes, the problems range from easy to competition level, but you don’t need to do all of the problems. But if you understand the basic concepts and approaches to problem solving that are taught in the AoPS books, that will give you the grounding to solve difficult problems, not just in math but in other fields.
Problem solving skills are very useful in preparation for college. It is true that most students will not do math contests, but in college, they will face tests that are much like math contests. Unlike many high school tests, tests in college won’t ask you to solve problems you have seen before, but rather on your ability to apply basic concepts to new problems.
@NoKillli Every student is different so you just have to decide what is workable for your daughter then advocate for her until she gets it. Our daughter, who is smart but not a “genius” by any stretch, negotiated herself on to an accelerated track in middle school and then again in high school so that she took AP Calc BC as a sophomore. She scored a 5 on the AP exam and a 5 on the AB subtest. She earned an A in the course. For some kids, it can be a disaster. For others, it is do-able. It was no big deal for our kid. AP Calc BC was hard but that just meant she had to work. She took Honors Alg 2 in 8th. Honors Pre-Calc in 9th. Skipped AB. If it seems like the right decision for your child, be persistent with her school until she is accommodated. Schools can’t always look out for the individual child as they administer school wide policies. That is where we parents come in. 
@evergreen5 Physics C will be hard without already having Calc BC fully nailed down. Just know to be ready for that.
@Mom2aphysicsgeek the point I was trying to make was that the hyper competitive approach to math that AoPS seems to advocate may work for non minority boys but it does not work generally for women or minorities(ie blacks or hispanics) They do not thrive in that environment. There is lots of literature on the subject. My DD completed MV and DEQ in the 8th grade and just completed a degree in math. This is not a race by any stretch of the imagination. It is quest for knowledge. I just looked at the AoPS Calculus excerpts that were online. It seems like it has lots of problems but not lots of proofs. At that level that is what I would expect.
As one Fields medalist said the mathematician looks for solutions to problems with unknown solutions and slow math works just fine.
@gallentjill good for your D for wanting to get ahead and willing to put in the time. I know she’s already made up her mind and is moving forward with things, but I just wanted to throw out a little food for thought, which is I’m wondering, if studying math over the summer is this the best use of her time (and it may be!) Here’s a couple of pro/cons that jump to mind:
Last piece of advice, if she can get her common app finished before Senior Year starts, this will be a huge help to her.
I think we are getting off the main topic re:AoPS. But I think it is worth mentioning that AoPS approach to doing math problems is what you need to do advanced work in math - virtually all proof based classes. If I had exposure to stuff like AoPS when I was in high school, I would have been much more ready for the analysis and abstract algebra classes in my undergrad years.
I am not sure whether competitions are the best way to encourage mathematical problem solving. A lot of the top students in the prestigious AMC competitions are children of recent Asian American immigrants . There is a lot of push and extra classes in the community to prep for these contests. So the hyper competitiveness of the contests themselves is a separate issue from the usefulness of the AoPS materials.
I work with my second child using AoPs materials. The competitions are not the goal. A deeper and more nuanced understanding of mathematics is - and that’s true for anyone pursuing advanced work in math -male, female, URM, ORM, whatever.
For OP’s purposes, however, I would go along with the non-AoPS suggestions given here.