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Finishing PreCalc by December. What to do for the rest of the year?

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We school year round, so except for online classes, we start and finish whenever we start and finish.  

 

So it looks like we will finish AoPS PreCalculus by December.  The plan is for dd15 to take AP calculus next year with an online course, starting in the fall.  

 

Since she hasn't had discrete math, I was thinking about starting her on AoPS Intermediate C&P.  But I know she (ever the planner) will want to do what she can to prepare for AP Calc BC.   (She has a good handle on PreCalc, solving all but the Putnam, IMO, and similar problems, lol.)  

 

What are your ideas for math for the rest of the school year and possibly the summer?  

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When DS was between math classes, he would watch MIT OpenCourseware videos (OCW Scholar courses) and work on problems that interested him.

 

She might also enjoy the AoPS Vol 1/ Vol 2 and Beyond books.

Edited by quark
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What about some calculus with ALEKS? Of course a book would also work, but ALEKS is self-graded, does assessments, and is easy to fit into the schedule (anywhere, anytime with internet access). Edited to add that the ALEKS course is called Prep for Calculus. 

 

It's not my ideal for a stand-alone course (even tho' youngest is using it that way right now, lol), but would be great prep and review for an intense class next year.

 

I would probably do that or a book that gives an overview of calculus, one that shows some of the more interesting concepts. Or possibly a Coursera-type class. 

 

Basically, I would plan something low intensity that would make next year easier, probably not for credit. if you have the time and inclination, you could do that plus more fun days with puzzles, challenges, videos. Going that route, it could be a half credit of Math Explorations or similar. 

Edited by katilac
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I'd actually be more inclined to do something else rather than precalc review. I wouldn't want her to get stale, and if she can do all but the imo/putnam problems, she will be excellently prepared. Some focused review right before the class should be all that's necessary. 

 

Discrete or stats would be my vote. 

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How about slow start AP Calc BC along with some stats or discrete math?

 

If you really want solid Calc 1 and Calc 2 equivalents (higher echelon colleges) it may be worth starting sooner and go beyond AP Calc BC by the end of next year. (using MIT or other options).

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Unfortunately, this one looks like it's out of print. I would look for it at the library...

 

The Hitchhiker's Guide to Calculus: Michael Spivak: 9780883858127: Amazon.com: Books

https://www.amazon.com/Hitchhikers-Guide-Calculus-Michael-Spivak/dp/0883858126/ref=la_B000APBL44_1_5?s=books&ie=UTF8&qid=1475153578&sr=1-5

 

She might enjoy this one:

 

Calculus Made Easy: Silvanus P. Thompson, Martin Gardner: 9780312185480: Amazon.com: Books

https://www.amazon.com/dp/0312185480?_encoding=UTF8&isInIframe=0&n=283155&ref_=dp_proddesc_0&s=books&showDetailProductDesc=1#product-description_feature_div

 

And, of course, my knee-jerk-response title for high schoolers who want to know more about math:

 

What Is Mathematics? An Elementary Approach to Ideas and Methods: Richard Courant, Herbert Robbins, Ian Stewart: 9780195105193: Amazon.com: Books

https://www.amazon.com/Mathematics-Elementary-Approach-Ideas-Methods/dp/0195105192/ref=sr_1_1?s=books&ie=UTF8&qid=1475154685&sr=1-1&keywords=what+is+mathematics

 

Peace,

Janice

 

Enjoy your little people

Enjoy your journey

Edited by Janice in NJ
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We might be in this situation next year and I was thinking that would be a good time to take a statistics course.  

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If your daughter is applying to colleges which require a certain number of SAT subject tests, she might consider scheduling one after completing Pre-calculus.

 

Regards,

Kareni

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Thanks, everyone!  These are some really great ideas.  Dd is now considering taking the SAT Level 2 math in a few months.  I had also forgotten that she's already taking AP statistics, and dd is considering taking a CS class, or starting on the AoPS calculus textbook so she can get a bit more AoPS before switching to what will probably be another textbook for AP calc.  I'm also very interested in the books recommended by Janice.  Who can resist a Hitchhiker's Guide?  I'll try to convince her to do discrete math her senior year.  

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She might also be ready for the SAT if she has not taken it yet. DS found prepping for SAT math and SAT 2 Math together to be much easier than doing both months apart. He did the SAT 2 Math in December then SAT in January.

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Thanks, everyone!  These are some really great ideas.  Dd is now considering taking the SAT Level 2 math in a few months.  I had also forgotten that she's already taking AP statistics, and dd is considering taking a CS class, or starting on the AoPS calculus textbook so she can get a bit more AoPS before switching to what will probably be another textbook for AP calc.  I'm also very interested in the books recommended by Janice.  Who can resist a Hitchhiker's Guide?  I'll try to convince her to do discrete math her senior year.  

 

OH MY GAWD

 

I almost missed this recommendation (hitchhikers guide).

 

Now I must buy it.  

 

:ohmy:

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I'm also very interested in the books recommended by Janice.  Who can resist a Hitchhiker's Guide?  I'll try to convince her to do discrete math her senior year.  

 

Please note that according to the page I found on Amazon, the title is out of print. DO NOT spend $100+ on this one. It's a super-thin paperback - MORE than worth the ten bucks I paid for it, but it's definitely not worth $100+. See if your library can find it for you.

 

If you really want to experience the glory that is Spivak, I can whole-heartedly recommend his Calculus text. A beautiful book. And I'm not alone in this. It's hard to chuck a rock in the math world and find someone who doesn't have positive things to say about it. (There are absolutely others; however, Spivak has solutions which is very important for self-study.)

 

Calculus, 4th edition: Michael Spivak: 9780914098911: Amazon.com: Books

https://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918

 

Combined Answer Book For Calculus Third and Fourth Editions: Michael Spivak: 9780914098928: Amazon.com: Books

https://www.amazon.com/Combined-Answer-Calculus-Fourth-Editions/dp/0914098926/ref=pd_sbs_14_t_0?ie=UTF8&psc=1&refRID=0RFFRSSVN2GH6915C62B

 

Here's the rub: Most kids who take the calc sequence in college are majoring in something other than math. They never discover why AOPS and Spivak are so valuable. If they go on to graduate school in physics or a host of other disciplines that require heavy lifting in math OR if they make it to real analysis as undergrads, then they look back and nod in agreement. The foundation was laid, and it served them well! These texts are dropping a TON of bread crumbs along the way! And let's be honest, that's what we look for in an education. It's just good pedagogy. Great teachers (in person and in their books) offer kids boots they can clomp around in now. Sure, they are a bit too big, but that is part of great teaching. Once the students fully grow into them, they look back and say, "Ahhhh! That's what that was all about!" Sure, you could offer them boots that fit just right and make the students feel all warm and fuzzy (i.e. great! As in, I got a 100% on that test even though I hardly studied.); however, there is a certain amount of discomfort associated with not only 3rd/4th year undergrad math but also with graduate level mathematics. It can feel a bit like herding cats; there is so much going on! You have to trust the process and just be content to jump into the deep end and try to swim. It's that process that causes things to coalesce. AND I suspect it's that process that causes a lot of kids to quit. (Can you tell I've been reading and studying when it comes to this rabbit trail?) So these kinds of texts offer more than just the actual standard Calc I/II content. They offer a preview of what's coming, and they generate an appropriate atmosphere. 

 

Along those lines, here's another rec (and a free one!): Gilbert Strang's MIT Highlights of Calculus series is well-done. Gilbert is a linear algebra guy. For many, he is the go-to guru on the subject. And as most mathematicians will tell you, you can never learn enough linear algebra.  :) So it's great to watch these as someone who is aware of what's coming. Love Gilbert!

 

You could watch these all the way through and then start with AOPS and/or Spivak's Calculus. After all, calculus is a fun, fun subject - a foundation that should not be rushed but rather savored and enjoyed. There is much to appreciate! It's so beautiful! Plenty to wallow in. 

 

Highlights of Calculus | MIT OpenCourseWare

https://ocw.mit.edu/resources/res-18-005-highlights-of-calculus-spring-2010/index.htm

 

Peace,

Janice

 

Enjoy your little people

Enjoy your journey

Edited by Janice in NJ
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Please note that according to the page I found on Amazon, the title is out of print. DO NOT spend $100+ on this one. It's a super-thin paperback - MORE than worth the ten bucks I paid for it, but it's definitely not worth $100+. See if your library can find it for you.

 

If you really want to experience the glory that is Spivak, I can whole-heartedly recommend his Calculus text. A beautiful book. And I'm not alone in this. It's hard to chuck a rock in the math world and find someone who doesn't have positive things to say about it. (There are absolutely others; however, Spivak has solutions which is very important for self-study.)

 

older edition online

https://ia801606.us.archive.org/22/items/Calculus_643/Spivak-Calculus.pdf

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Please note that according to the page I found on Amazon, the title is out of print. DO NOT spend $100+ on this one. It's a super-thin paperback - MORE than worth the ten bucks I paid for it, but it's definitely not worth $100+. See if your library can find it for you.

 

 

 

Here's the rub: Most kids who take the calc sequence in college are majoring in something other than math. They never discover why AOPS and Spivak are so valuable. 

 

Lovely post, Janice.  I hadn't noticed the high resale price, since I ordered a copy from the library.  Now I can steal several copies and sell them on Amazon.    :leaving:   (Just kidding.  I was just looking for an excuse to use the sneaky emoji.)

 

The only big revelation I received from calculus was a derivation of the volume of a sphere, which we had been deploying for years without justification.  

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I've heard good things about the Life of Fred Calculus book.   It may be a good way to have an alternate point of view on it.   The more different ways you learn that level of math the more likely you are to understand the core of it.

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Hi Amy Marie,

 

This isn't directed to you personally; however, I would not use programs like LoF with an advanced math student. A student who has successfully completed AoPS PreCalc is likely to be offended/bored by a text that spends a page and a half talking about diapers in order to explain what a function is while relegating the interesting material (the epsilon-delta definition of a limit immediately jumps out at me) to the appendix. 

 

When you read mathematics, you read slowly. It does take 30 minutes to read the page. Or an hour. Or two. Unless you already know the material. In that case, we feel a bit of a thrill in being able to read the page faster. It's like playing a Beethoven Sonata movement that you've already mastered; it reminds you that you actually are making progress. In the Note to the Student in the calc text, the LoF author immediately begins with this. He claims that you can read at, "whatever speed feels good when you’re enjoying the life adventures of Fred." 

 

That's fine if we are worried about what makes the student feel good. Most students who have completed AoPS PreCalc at age 15 probably would prefer just to learn some useful math. For example, here is a very beautiful idea. Click on the link below and scroll down the the section titled "Precise Statement."

 

https://en.wikipedia.org/wiki/(ε,_δ)-definition_of_limit

 

Stop. Think. Re-read. Think some more. 

 

I don't have Fred's appendix discussion. Anyone have it? Care to share his explanation? Does it take longer than 30 minutes to read? And if it takes less, does it really offer the student as much insight as AoPS or Spivak or that very concise sentence that comes after the words Symbolically on the Wikipedia page? Because I just read several pages about diapers and baby talk in LoF, but he failed to explain what a bijection is. In fact, I didn't even see the word mentioned. However, he seems to have moved on (see pg 19); he has asked me to pull out a yellow highlighter and color a duck on the previous page. 

 

No thanks.

 

If you want to learn to write in French, you have to learn to read French. And it takes time. 

 

Peace,

Janice

 

Enjoy your little people

Enjoy your journey

 

 

Edited by Janice in NJ
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Long P.S. to my previous post:

 

To be, or not to be, that is the question:

Quote by some prince...has anyone every heard of him?

 

Now there is one of the most loaded statements in all of literature. You can talk about what it means for pages and pages. I suspect one could write an entire dissertation on the topic.  :001_smile: 

 

The final sentence after the word Symbolically in the article referenced above is like that. It is packed with meaning. And the more you know, the more it means. 

 

A beautiful process. 

 

Students who are ready to study calculus are ready to engage with that process. They don't need a watered-down version. Reminds me of this debate when it comes to the humanities:

 

http://learning.blogs.nytimes.com/2012/01/20/seven-minute-shakespeare/?_r=0

 

This line from the Cliff-version cracks me up:

 

HAMLET: To be or not to be, that’s the question…right? I mean, if you think about it…

 

hmmmm... not exactly the same if our goal is an education...

 

Peace,

Janice

 

Enjoy your little people

Enjoy your journey

 

And I'm sure you've all seen this. But I couldn't resist posting it just in case you haven't...

 

To be or not to be - Shakespeare Live on Vimeo

https://vimeo.com/174434551

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