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#1 lewelma

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Posted Today, 12:41 AM

Based on some lovely advice from you guys last month, ds and I have decided to have him do a run through of Calc 1 on his own and start Calc 2 at the University in July (school year here is Feb - Nov).  We decided to do this for a few reasons.  1) Calc 1 is a repeat (at a slightly higher level) of highschool calc and you are allowed to skip it if you earned an 'excellence' (so top 15% of kids). So the kids in the class will not be as mathy as mine. 2) He has already placed out of all the other 100 level courses, so if he does not take Calc 2 there, he will be dumped into the deep end at 15 in 200 level classes. So a major goal is to get him to be able to do timed tests at a 100 level to prep him for the 200 level (he has never taken a timed math test except for the Squad exam for the IMO team.  3) AoPS calc runs Oct - April and he does not want to wait until 2016 to take Calc 3 because he wants some human interaction....  The downside is that Calc 2 is reasonably algorithmic and is using the Anton, Bivens, and Davis book. Also, he will be taking AoPS PreCalculus *concurrently* with this self study of Calculus.  A bit unusual, but there you go.

 

Ok, with that summary of where we are at, here is the problem.  DS started reading the book yesterday.  He got through the first 2 pages which are about 'what is calculus' and give a brief description of the 2 problems that historically motivated people to invent it.  In those first 2 pages, they showed a gradient line, and a curve with the area broken into boxes.  Very vague stuff.

 

I come in 30 minutes later and he has his notebook out already.  I'm thinking 'cool, he is doing some practice problems, but I didn't expect him to want to.'  So I ask him what he is working on.  He tells me 'I'm trying to figure out the area under the curve, and I'm struggling to decide where to put the rectangles.'  Ok, cool, I really have no idea what he is talking about -- 30 years pass before my eyes and I'm trying to remember the chain rule or something.  I don't really remember why it matters what you do with any rectangles.  So about 20 minutes later he comes out very excited.  The answer is 1/3.  Um. :confused1:  1/3 of what?  Yes, it has been 30 years.  So I ask him to explain it.  Well, apparently he has set himself the job to prove this area under the curve thing (this is NOT suggested in the textbook on page 2! He just asked a question and went about solving it), and after a whole bunch of manipulation , he gets 1/3.  So I ask him the obvious question.  Are you right?  Well, he does not know, and I don't know what he has just proved. So I look in the index and on page 356 is a proof that looks *exactly* like his, and his answer is right.  He is ecstatic. and kind of looks like this :hurray: mixed with this :willy_nilly:  . And then he gets all excited about the discussion on page 356 about why you should use the rectangles in the middle, because he decided on one side. He tells me that he *loves* calculus.

 

So here is the problem.  Is this book the right book? Seems like no. It is very algorithmic with some proofs scattered throughout, and the way he thinks does not seem to match the way this book teaches.  But he needs to use it so he is prepared for the way the calc2 class will run (same textbook for calc1, 2 and 3).  What would be a good format for the next 6 months of self study?  How can we make this book work?  What can I suggest to him? How am *I* going to help him? or even assess him because he needs to practice timed assessments? He does not need a tutor exactly, but he might need someone with some purpose that I am unclear about.

 

I am open to suggestions.  

 

Ruth in NZ 

 

ETA: now he has just coded geogebra to draw the equation of the tangent line at any point for any function of any power. He's got the variables on a slider so it is pretty cool to watch as he changes the variable which changes both the function and then the location of the tangent line. He does not know what a derivative is yet.  So it appears that he is solving this by first principles or something. He told me he figured it out at martial arts tonight (-: then he came home and coded it.  I think he needs some sort of program that just guides him through exploration of these principles.  Does this exist?



#2 Mike in SA

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Posted Today, 07:20 AM

Have you taken a look at MIT OCW?  It might provide the theory he craves, and the video lectures can be quite good...

 

Other than some of the calculator-heavy "AP" books, most Calculus books are roughly the same.  Leithold and Anton are a bit more theoretical, but what I've seen of the rest (AoPS included), they follow the same approximate arc.  Additional theory needs to be provided by the instructor.  A rigorous course is your best bet for theory.

 

You might need to bear through calculus, as it is something of a "basic math" to the rest of analysis and applied math.  When you get to calculus 3, it will be more fun, but until then, you might just need to take what you can get.



#3 Kathy in Richmond

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Posted Today, 07:49 AM

Yes, MIT OCW's Single Variable Calculus lectures would be a great addition. Try the OCW Scholar edition.

 

Maybe have him read some history of calculus? See how others came up with these ideas?

 

Get the AoPS calculus textbook if you can. It would be more appropriate for him than Anton. He could read it on the side. If you can't do that, then milk Anton for all you can by making sure to identify & do the more challenging problems in each chapter. Try the MIT assignments & exams in the OCW course for a bigger challenge. They come with complete solutions, so he can check himself.

 

Keep messing around with Geogebra for explorations. I googled "geogebra and calculus" and found lots of sites with applets. It might be fun to explore some of them.

 

What he was working on to find area under a curve are called Riemann Sums. Wonderful that he thought the rectangle placement through himself!

 

If he ever gets stuck or wants a solution checked, ask here or send it my way!



#4 Mike in SA

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Posted Today, 08:48 AM

Yes, MIT OCW's Single Variable Calculus lectures would be a great addition. Try the OCW Scholar edition.

 

Maybe have him read some history of calculus? See how others came up with these ideas?

 

Get the AoPS calculus textbook if you can. It would be more appropriate for him than Anton. He could read it on the side. If you can't do that, then milk Anton for all you can by making sure to identify & do the more challenging problems in each chapter. Try the MIT assignments & exams in the OCW course for a bigger challenge. They come with complete solutions, so he can check himself.

 

Keep messing around with Geogebra for explorations. I googled "geogebra and calculus" and found lots of sites with applets. It might be fun to explore some of them.

 

What he was working on to find area under a curve are called Riemann Sums. Wonderful that he thought the rectangle placement through himself!

 

If he ever gets stuck or wants a solution checked, ask here or send it my way!

 

Yes, Anton is not the friendliest of texts.  :)

 

I absolutely LOVE Leithold, though.  Imho the best calculus text you can get.



#5 quark

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Posted Today, 01:09 PM

DS's few weeks with MIT OCW Single Variable Calc really "spoiled" first semester calculus 1 for him at the CC (in a totally good way). :) I was going to recommend OCW Scholar too but with the caveat that I couldn't help him with it so I don't really know what he did. It was purely self study with the videos and some selected problems. He was totally bored through the first 10 weeks of calculus 1 at the CC and is only now starting to find it interesting. So cool that your DS figured it out, Ruth.



#6 kiana

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Posted Today, 01:41 PM

Yeah, he sounds like he'd be more interested in Spivak or Apostol than that calculus text. I'd recommend getting one of those for fun. Book Depository has Spivak for a reasonable price and Apostol for an unreasonable price.

 

Apostol is what MIT uses for calculus with theory, Spivak is what Ohio State uses for the same course (or did last time I looked) and both are excellent books for someone who's very mathematically mature but hasn't been exposed to calculus. 

 

I really don't think he needs to follow Anton, but more that he needs to be able to do the problem sets in Anton, if that makes sense? 



#7 lewelma

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Posted Today, 02:36 PM

Wow!  You guys are great!

 

Yes, Anton is not the friendliest of texts.   :)

I'm noticing this.  He'll have 3 course with this book.  Sigh.

 

I really don't think he needs to follow Anton, but more that he needs to be able to do the problem sets in Anton.

 

This is a really good point.  Thanks for spelling it out for me.  So he can learn with the MIT OCW and the Spivak book and geogebra apps, and then go through the problem sets in Anton with a focus on the challengers.  I think that will work!  That sounds so much better than working through the Anton book linearly.  It looks pretty soul sucking.

 

Kathy, we do have the AoPS calculus book, but at first peak he said it was pretty hard.  Do you think this is because he has not done AoPS PreCalc yet?  We looked through the PreCalc text and it only looked like he needed the 2 trig chapters for Calculus. Is this true? DS thought that AoPS PreCalc looked like a ragbag of topics that need to be covered for the IMO.  Is this right? 

 

What he was working on to find area under a curve are called Riemann Sums. Wonderful that he thought the rectangle placement through himself!

He was pretty excited!  Especially because he came up with his own question and then used his mathematical knowledge to figure it out.  I'm not sure he has done that before.  He has always solved problems that someone else came up with.

 

Quark, I'll tell ds your ds's experience.  He loves hearing about his 'arch nemesis.' :001_smile:

 

Thanks everyone!

 

 



#8 Kathy in Richmond

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Posted Today, 02:51 PM

Kathy, we do have the AoPS calc book, but at first peak he said it was pretty hard.  Do you think this is because he has not done AoPS PreCalc yet?  We looked through the PreCalc text and it only looked like he needed the 2 trig chapters for Calc. Is this true? DS thought that AoPS PreCalc looked like a ragbag of topics that need to be covered for the IMO.  Is this right?


Oh, I didn't realize that he hasn't seen trig yet! Yes, I'd have him do the first 5 chapters of Precalculus before moving onto AoPS calculus. Ch1 is probably review, Chs 2-4 cover trig, and Ch 5 covers parameterization, polar coords, and more trig. He'll need all of those for AoPS calc.

The remainder of the Precalc book covers a very nice introduction to complex numbers and linear algebra. Complex numbers will help him with olympiad math, but both topics will also lay a foundation for his future math major theoretical studies. I actually thought that the intermediate alg text had more of the ragbag contest math stuff.

#9 lewelma

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Posted Today, 04:14 PM

Got it.  We decided to have him take Olympiad Geometry over our summer rather than PreCalculus because the Geo class is less work and we weren't sure when it would be taught again.  We also had not talked to the university yet when he signed up for AoPS classes, so did not know how he would need to slot into university.  Kind of a muddle now.  sigh.

 

However, he just told me that he knows trig.  Um how?  Apparently when he did his month long preparation for Olympiad Geometry, he had to cover a lot of trig to make sense of the questions.  So after discussing it this morning, he has decided to read through the AoPS PreCalc trig chapters and focus on what he does not know, but he thinks it will only take about 2 weeks if he does not do the challengers.

 

Here is timing, just so you know what in the world we are doing:

Nov - Jan  AoPS Olympiad Geometry

Jan - June AoPS PreCalc

July - Nov Calc2 at university using Anton

 

So basically, he has to do Anton-level Calc1 sometime between now and July, which was reasonable because it is very algorithmic.  The problem is that he does not appear to be able to only study Calc at an algorithmic level, and by nature studies more theoretically, so now things are a bit more muddled.

 

The other problem is that Calc2 is not ever taught Feb-June, so if he wants to do a 100 level calc class it is either July or a full year later, by which point he will really want to be in 200 level courses.

 

Hope that makes sense.  I know it is a mess.

 

 



#10 kiana

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Posted Today, 04:25 PM

I think he'd be fine studying theoretical single-variable calc AND aops precalc if he's got the time to do it after his brief run through trig.

 

If not, quite honestly, given how much he'd prefer to do it theoretically, I'd postpone the aops precalc and study calc theoretically. Given his mathematical maturity, if he has solid algebra and trig skills he'll be fine without it.



#11 Kathy in Richmond

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Posted Today, 04:42 PM

I like kiana's suggestion. If he self teaches trig & has already completed AoPS intermediate algebra, then he's completed the basics of most precalculus classes anyway. Sounds as if he'd enjoy theoretical calculus now, so let him! He could safely leave the rest of AoPS precalc (complex & linear algebra) for a later date if there's not time to do both.

#12 lewelma

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Posted Today, 04:48 PM

The problem is the human interaction -- he must have it.  AoPS Calculus is not taught until October, so PreCalc is the only class left.  Also, we are only talking Calc 1 that he has to get through, and I think that the AoPS Calc book is 1 and 2, isn't it?  So I'm thinking that he has 7 months, including 8 weeks of summer with nothing but math, so I think he would have time to do half of the AoPS calculus book while concurrently doing both Oly Geometry and PreCalc.  Or is this unrealistic?



#13 lewelma

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Posted Today, 04:51 PM

But we also really really need to pound on test taking skills, so I do have to make sure that he practices timed math tests.  So he actually would only have about 5 months, so we have 2 months to practice. 

 

I could also get him to do Anton problem sets at the end of each topic that he studies with the AoPS book, so that he is not stuck at the end with tons of Anton to do.

 

 



#14 kiana

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Posted Today, 05:05 PM

I really doubt doing the whole problem set from Anton is going to be necessary. I think unless he's struggling on something (one topic that's notorious for requiring immense amounts of practice is integration by substitution) I'd choose problems very judiciously. For example -- I really, really doubt he's going to require 20 practice problems on differentiating polynomials.

 

I think his school has assignments posted for calc 1, with solutions, as well as an old mid-term -- those may be helpful.



#15 lewelma

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Posted Today, 05:09 PM

Oh heavens, not a *whole* problem set!  hee hee.  He would kill me

 

I think his school has assignments posted for calc 1, with solutions, as well as an old mid-term -- those may be helpful.

 

oooh.  What a great idea.  That would be perfect if they are available!



#16 kiana

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Posted Today, 05:10 PM

FTR, calculus at an Anton-level really isn't that difficult for a student with a solid algebra/geometry/trigonometry foundation who already thinks mathematically. The big issue is with students who are weak in prerequisites or do not think well mathematically and try to learn by rote. When I was about your son's age (and much less well-prepared -- there was NO aops! alas!) I went through calc 1 in 3 weeks and was perfectly well-prepared for calc 2 at the state university in the fall.

 

I mention this not as a boast (please don't think that) but simply as reassurance.



#17 Kathy in Richmond

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Posted Today, 05:20 PM

The problem is the human interaction -- he must have it.  AoPS Calculus is not taught until October, so PreCalc is the only class left.  Also, we are only talking Calc 1 that he has to get through, and I think that the AoPS Calc book is 1 and 2, isn't it?  So I'm thinking that he has 7 months, including 8 weeks of summer with nothing but math, so I think he would have time to do half of the AoPS calculus book while concurrently doing both Oly Geometry and PreCalc.  Or is this unrealistic?


Sure, if he need the interaction of a real class, then do it that way. My guess is that the Calc 1 material is going to be pretty simple for him. Especially for a boy who already 'invented' Riemann sums on his own! Do what kiana suggested and be judicious in selecting problems from Anton.

You're correct that AoPS calculus covers both 1 and 2. Most of the time-consuming problems are in the Calc 2 half (infinite series, methods of integration, calculus in polar coordinate, etc).

There are some released & practice AP calc AB exams on the College Board site, both multiple choice & free response, with grading rubrics.  They might come in handy if you want to work on timed testing using Calc 1 material.



#18 lewelma

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Posted Today, 05:25 PM

FTR, calculus at an Anton-level really isn't that difficult for a student with a solid algebra/geometry/trigonometry foundation who already thinks mathematically. The big issue is with students who are weak in prerequisites or do not think well mathematically and try to learn by rote. When I was about your son's age (and much less well-prepared -- there was NO aops! alas!) I went through calc 1 in 3 weeks and was perfectly well-prepared for calc 2 at the state university in the fall.

 

I mention this not as a boast (please don't think that) but simply as reassurance.

 

Thanks for that. The head of the department, after looking at ds's portfolio of AoPS proofs, told him he should just self study Calc1 and go straight to calc2.  He thought exactly what you are thinking.  So I went and bought some high school work books, one for him and one for me, that went through algorithmic Calc 1.  I've been looooving them.  They are super fun, but really just algebraic manipulation over and over again.  They make it messy with fractional exponents and with tricky simplifications, but end the end it is just rote learning.  My ds looked at them for about 3 seconds and has never gone back.  All I really need him to do is get through the 2 workbooks (about 80 pages each with room for the practice) and then practice taking the high school math exam under time pressure.  A simple, very straightforward goal.  The problem is, this is just not my son.  It would be nice in some ways if it were, but it is not.  So sounds like we will need to do a more theoretical Calculus and then have him practice timed tests at a lower level.  I keep harping on the timed test aspect because the only timed math test my son has ever take is the squad exam for the IMO and he had 4.5 hours for 5 problems.  I really don't know if he has the ability to accurately plod through mechanical calculations under time pressure.  I seriously don't know.



#19 lewelma

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Posted Today, 05:29 PM

There are some released & practice AP calc AB exams on the College Board site, both multiple choice & free response, with grading rubrics.  They might come in handy if you want to work on timed testing using Calc 1 material.

 

cool.  Thanks!



#20 kiana

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Posted Today, 05:37 PM

I googled "calc 1 final exam" and found a bunch with solutions:

 

http://cims.nyu.edu/...inal Exams.html

http://www.math.pitt...220-CALCULUS-I/

http://ocw.mit.edu/c...010/final-exam/

 

I'm sure you could find more if you need them.




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