Help me think this through - math related

[lewelma]

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?

[Mike in SA]

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.

[Kathy in Richmond]

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!

[Mike in SA]

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.

[quark]

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.

[kiana]

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? 

[lewelma]

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!

 

 

[Kathy in Richmond]

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.

[lewelma]

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.

 

 

[kiana]

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.

[Kathy in Richmond]

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.

[lewelma]

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?

[lewelma]

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.

 

 

[kiana]

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.

[lewelma]

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!

[kiana]

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.

[Kathy in Richmond]

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.

[lewelma]

 

 

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.

[lewelma]

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!

[kiana]

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

 

http://cims.nyu.edu/~kiryl/Calculus/Tests/Old%20Final%20Exams.html

http://www.math.pitt.edu/~athanas/MATH-0220-CALCULUS-I/

http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/final-exam/

 

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

[lewelma]

We just watched the first MIT lecture where he derived the derivative using the x and x+changex (where changex will go to 0); and then used the binomial theorem to expand, and then simplified.

 

DS took a different approach while at martial arts, and we are wondering the implication of the difference. He used the function y=x^a He took a stationary point n and a moving point m, where m approached n. 

 

so the slope of the tangent was (n^a - m^a)/(n-m)

 

he then factored the numerator into (n-m)(n^a-1.....m^n-1) where there are 'a' terms and each has the degree a-1.  given he was doing it in his head, he did not write it all out and instead realised that the middle terms were all the same degree so that as m approaches n all of them would be n^a-1.

 

So the (n-m) in the numerator and denominator cancel and he is left with an^a-1 when m approaches n. This is the slope of the tangent.

 

So he put the function y=x^a and the tangent of y=an^a-1 x -(a-1)n^a  into geogebra and then allowed a and n to vary.

 

To me, this seems shorter than what we just watched with the binomial expansion.  Is there a reason to have changex approach 0 rather than m approach n?

 

 

By the way, he loved the MIT OCW, so he might work through that and their problem sets, and then do the problem sets and previous exams at the local university for Calc1 (thanks Kiana).  So he might delay the AoPS Calc book until he does the class in October (kind of as a repeat of Calculus but at a higher level).  We will see.  Thanks for helping me think this through!

[kiana]

Nope. An alternative definition of the derivative that is computationally easier in some cases is the limit as x goes to a of the fraction (f(x) - f(a))/(x - a). The only issue is that strictly speaking he should perform the cancellation *before* he thinks about m approaching n -- if m approaches n before the cancellation he's cancelling out 0/0.

 

Most students get boggled enough by one definition of the derivative that introducing two really throws them for a loop. But they are absolutely the same as your son has noted.

 

If you can afford it I'd really recommend getting Spivak's calc book for your son as well -- I saw it on bookdepository which iirc has free shipping worldwide. I think it would be 100% up his alley.

[lewelma]

Thanks Kiana!  I'll get the Spivak book, and I'll wrap it for x-mas!  I feel like I wasted $180 on Anton, but he will need it for 2 more classes, so perhaps not. 

[1053]

.

[Kathy in Richmond]

Ruth, here's another idea. I was too busy prepping Thanksgiving food today to even recall what I did with my own daughter. :p  We'd had a tough time finding a calculus text that worked for her, too. She's very much a big picture thinker, and she absolutely couldn't stand the various calc books on my shelf. Too tedious, too long, & too much practice on easy stuff, especially since she had the main concepts of calculus down from exposure at math camps. She zones out with repetitious practice & is very independent (ie, didn't want me to teach her directly). For various reasons, I wasn't ready to start her on Apostol yet; she was already stretched pretty thin that year with tough courses and lots of extracurriculars. The AoPS calc text & class were not yet on the market, and I didn't own Spivak.

 

Anyway, we were visiting ds that fall & happened to stop in the MIT bookstore. She spotted Calculus for the Forgetful: How to Understand More & Memorize Less by Wojciech Kosek on a display, looked it over, and asked me whether she could switch to that instead. It was written as a refresher course for adults, but I thought that it was worth a try. This book is 150 pages instead the usual 700+ page tome, covers lots of theory & proofs, and has a few good problems (some toughies) in each section like AoPS instead of hundreds like other texts. If you've ever seen any of Gelfand's volumes, it's similar in size & style IMO. It worked very well for her, and she loved it. She took her time, did all the exercises, and she still finished it in a couple of months. Looking back at my records, I see that she did it concurrently with the AoPS Intermediate Number Theory course online, so it was doable for her to double up like you're proposing. Then she did some focused prep for the AP calc BC exam, which she aced. So...it can also work to do something quite nontraditional. Just another idea!

[kiwik]

I know the professor said self study calc101 but if he did it at uni he would meet some of his social needs at the same time and you could focus more on the other stuff.

[1053]

.

[lewelma]

kiwik, I brought up your idea with ds.  He feels like taking AoPS precalc concurrently with his first University class might be too much.  Plus, it still puts him into a university class having never taken a timed test. But thanks for the idea; it helped to kick around different options.

 

Kathy and 1053, Calculus for the Forgetful is in the NZ National Library, so I might be able to get it shipped down to me.  When I went to buy it on Amazon, it said delivery date between 1Jan and 1Feb, which would take out the entire summer break.  Sigh.  I too think it looks like a good option.  Plus Kiana found the entire course schedule including assignments, solutions, midterm, and final.  So one option we are considering is studying more algorithmically and then using the local university's course materials. 

 

When we looked at the calendar last night, (with christmas, tramping, grandparents visiting, math camp, and 2 weeks where Oly Geo and PreCalc overlap), he is down to only 4 months to study (with another math class running that entire time), so I am thinking that a deep theoretical approach might need to wait.  Perhaps I could buy the Spivak book, and he could use it a long side the Anton book when he takes Calc2, because he will have no other classes running so will have time to dig deep.

 

Given my parents are coming in February and can put a math book in their suitcase, is there a reason to pick Spivak or Apostol? On book depository, Spivak is $90 and Apostol is $375, so the choice was clear.  But if I can order in the USA and ship with parents for free, I can choose either.  I would also want some sort of solutions manual.

 

 

[kiana]

Both texts are good -- Apostol will also cover multivariable calculus and linear algebra in the second volume. I've never seen an answer key for Apostol, but I know there's a solutions guide for Spivak. But I also fairly regularly see people who are self-studying with Apostol asking for help on a site like physicsforums and getting help, so if there's something that's genuinely difficult that's a possibility.

[kiwik]

kiwik, I brought up your idea with ds. He feels like taking AoPS precalc concurrently with his first University class might be too much. Plus, it still puts him into a university class having never taken a timed test. But thanks for the idea; it helped to kick around different options.

 

Kathy and 1053, Calculus for the Forgetful is in the NZ National Library, so I might be able to get it shipped down to me. When I went to buy it on Amazon, it said delivery date between 1Jan and 1Feb, which would take out the entire summer break. Sigh. I too think it looks like a good option. Plus Kiana found the entire course schedule including assignments, solutions, midterm, and final. So one option we are considering is studying more algorithmically and then using the local university's course materials.

 

When we looked at the calendar last night, (with christmas, tramping, grandparents visiting, math camp, and 2 weeks where Oly Geo and PreCalc overlap), he is down to only 4 months to study (with another math class running that entire time), so I am thinking that a deep theoretical approach might need to wait. Perhaps I could buy the Spivak book, and he could use it a long side the Anton book when he takes Calc2, because he will have no other classes running so will have time to dig deep.

 

Given my parents are coming in February and can put a math book in their suitcase, is there a reason to pick Spivak or Apostol? On book depository, Spivak is $90 and Apostol is $375, so the choice was clear. But if I can order in the USA and ship with parents for free, I can choose either. I would also want some sort of solutions manual.

I can see his point. Hope it goes well.

[EndOfOrdinary]

We are directly making my son take tests just to overcome the hurdle of traditional schooling needing test skills. I have found working up to the hour or hour and a half is much better for my son. Giving three problems (incredibly easy ones) and fifteen minutes was where we started. It ended the initial anxiety. Even with anxiety he was more than capable. From there it was four problems that were a smudge harder. Then five. At three minutes a problem he was beginning to see that he had to create a rather definitive process of writing out the equation necessary, plugging in and simplifying, then crunching everything down. I do not think if I just told him that process he would have taken it as seriously as putting him in a place to discover it.

 

As much as rote is icky, stacking up a handful of rote problems and asking your son to solve them in a rather limited time might help him to see that it is part of the silly political junk of some schooling. Unfortunately, no one did that early enough for me and I spent far too many emotional and physical resources banging my ideological head against a wall. It does not matter how brilliant you are if sometimes you cannot smile pretty and be socially acceptable.

[lewelma]

We are directly making my son take tests just to overcome the hurdle of traditional schooling needing test skills. I have found working up to the hour or hour and a half is much better for my son. Giving three problems (incredibly easy ones) and fifteen minutes was where we started. It ended the initial anxiety. Even with anxiety he was more than capable. From there it was four problems that were a smudge harder. Then five. At three minutes a problem he was beginning to see that he had to create a rather definitive process of writing out the equation necessary, plugging in and simplifying, then crunching everything down. I do not think if I just told him that process he would have taken it as seriously as putting him in a place to discover it.

 

As much as rote is icky, stacking up a handful of rote problems and asking your son to solve them in a rather limited time might help him to see that it is part of the silly political junk of some schooling. Unfortunately, no one did that early enough for me and I spent far too many emotional and physical resources banging my ideological head against a wall. It does not matter how brilliant you are if sometimes you cannot smile pretty and be socially acceptable.

 

Thanks for this EndOO. Luckily, ds has no test anxiety because of all the music performance exams he has taken.  But boy am I getting push back for having to study rote problems. Basically, he won't do it, at least not yet.  I think I have given him some idyllic life for the mathematical mind, but left him with a complete disregard for hoop jumping.  But even with all the pushback I have gotten this past week, I refuse to regret taking the approach that I have taken - to allow him to study math as he sees fit rather than how the schools see fit.  But I am definitely reaping the consequences. Sigh.

 

Today, I tried to run him through all the derivative material in 20 minutes, thinking we would just pound it out. But he would not have it.  He told me he had to prove it all.  So I handed him the proofs.  "No.  *I* have to prove it!"  So we made a list of what he will prove today.  He refused to accept that the polynomial rule was distributive, so we started with that, and he proved it using "an infinite sum of an infinite binomial series." I kind of went  :blink:  Next up is:

 

product rule

quotient rule

chain rule

e^x

natural log

sinx

all other trig functions.

Did I forget any?

 

I grabbed the Anton book and found some proofs in the appendix including proofs of basic limit theorems.  "Oh, you might need these." "No!  don't show me, I need to prove those too!"  Um ok.

 

So now I'm thinking a couple of things:  1) This is definitely the blind leading the blind (except of course *he* does seem to know what he wants to do, but is looking for guidance. It is just me who cannot hope to run him through a theoretical calculus course).  2) This is all AoPS's fault! (In a good way of course :001_smile: )

 

Do I need to find a tutor or a mentor?   Or is this exploratory process going to work out because he is motivated and knowledgeable?

 

Also, just ordered Calculus for the forgetful from the National Library and it will be here by Friday.

[quark]

Today, I tried to run him through all the derivative material in 20 minutes, thinking we would just pound it out. But he would not have it.  He told me he had to prove it all.  So I handed him the proofs.  "No.  *I* have to prove it!"  So we made a list of what he will prove today. 

 

[...]

 

Do I need to find a tutor or a mentor?   Or is this exploratory process going to work out because he is motivated and knowledgeable?

 

One time, when DS wanted to prove something, DS's mentor, due to time-crunch, said "just trust me". Suffice to say, I received a tirade of frustrated feelings after that class. Calmed him down, asked him to work on it and present to his mentor at the next class but the mentor, again on a time crunch, didn't get back to him on it. Not long after, DS decided to take a break from the mentor. Yes, the mentor we have used and loved for so long! Lovely gentleman though he is, his schedule just doesn't permit the long drawn out, proof-based discussions they used to have any more. So all this to suggest that if you are using a mentor, keep expectations in check unless you stumble upon one of those gem of a mentors who will take time to work your DS to the extent that he wants things proved and hashed out. This was also a good lesson for my DS I think. How to balance the need to prove everything with also the need to just move along in some ways and get things done kwim?

 

ETA: I think it's absolutely lovely that these boys like exploring and discovering and working their hearts out on proofs etc. I'm just trying to figure out at what point one needs to be able to balance this need for exploration with also accepting that some things need to be done first and maybe later you can go back and work on proving them? I'm in no hurry for this to happen now but just wondering how and when to help him when he needs to work on things at a faster speed vs just proving and proving all day long.

[kiana]

I would let him go until it doesn't work out. The chain rule is a real pain in the butt to prove in detail though -- the proofs in most calculus books (I'm not sure about Anton) skip details.

[Kathy in Richmond]

Yes, let him go to town! He'll learn more at this stage of the game by proving things for himself. It should be doable - there aren't *that* many concepts in Calc 1 anyway.

 

After your list, it's common to learn the derivative of an inverse function & implicit & logarithmic differentiation. Then some theorems like Rolle's Theorem & the Mean Value Theorem, and then applications of derivatives like optimization problems. Then integration with Riemann Sums and the Fundamental Theorem of Calculus, along with applications of integration to areas and volumes.

 

You can always double check topic coverage against the AoPS calculus text's table of contents. Calc 1 coverage would be the first 4 chapters & parts of chapter 5. Here in the US, techniques of integration like integration by parts or partial fractions are usually at the beginning of Calc2.

 

If he gets stuck on any proof, you should be able to find them in the AoPS text...if he'll let you give him hints, that is. :laugh: And yes, the Chain Rule proof is full of technical details, & I wouldn't expect him to necessarily come up with that one on his own. Even AoPS relegates the full details of that proof to an appendix (see the end of Ch 3).

 

Fluency & speed are another thing once the concepts are down. I think it does help in the long run to have that automaticity. Mine worked on it in order to do well on the AP exam. Not too pleasant for them, but not too painful, either. It took maybe a month or so of dedicated practice? We kept a strand of fun math going at the same time (think a fun AoPS course or olympiad problems) so that it wasn't TOO painful. :)

[lewelma]

Thanks guys.  You are so reassuring!

[kiana]

If he gets stuck on any proof, you should be able to find them in the AoPS text...if he'll let you give him hints, that is. :laugh: And yes, the Chain Rule proof is full of technical details, & I wouldn't expect him to necessarily come up with that one on his own. Even AoPS relegates the full details of that proof to an appendix (see the end of Ch 3).

 

Fluency & speed are another thing once the concepts are down. I think it does help in the long run to have that automaticity. Mine worked on it in order to do well on the AP exam. Not too pleasant for them, but not too painful, either. It took maybe a month or so of dedicated practice? We kept a strand of fun math going at the same time (think a fun AoPS course or olympiad problems) so that it wasn't TOO painful. :)

 

Haha, yeah. If AOPS puts it in an appendix ...

 

The chain rule is one that I routinely say "just trust me" when I teach calculus. I feel the proof is so complex that it's not going to be of any use to anyone other than the top 1%, who are usually not in my class because they took it in high school and passed it.

 

I agree with what Kathy has said about fluency and speed. I think that a few weeks to a month before the test would be plenty of time to start practicing, though. I think there the chapter tests from Anton would be ideal both for getting him used to taking tests in a timed manner and for finding any places where additional drill is needed.

 

The drilling for fluency and speed is especially important when it comes to taking derivatives of functions like {e^2x sin (x^5) } / {ln (cos (sqrt(x))} or other horrible monstrosities.

[1053]

.

[lewelma]

He was actually willing to sit down and do some problems yesterday after proving all the things on the list.  Mixed practice!  I don't think he has done mixed practice since he was about 8. 

 

I've been slowly picking at his perception that a speed test of straightforward problems is insulting. I've been reading parts of your responses to him, and we had a breakthrough yesterday using the word 'fluency.' Seems quite funny to me that a single word could break the stalemate, but it did.  I defined fluency as having an intuitive sense of what to do, that you understood it so well that you did not need to think it through, that is was not rote which is learning to work algorithmically without understanding.  All this is quite good in his eyes, and makes fluency a worthwhile goal.  Then he made the leap "Oh, so they make the test under time pressure to test your fluency."  Ah, yes!  Can't believe this has taken me a month to beat this through his head.  Sigh.  Thanks for the word, Kathy!

 

1053, thanks for the brief proof, he enjoyed comparing it to the AoPS one and finding the flaw.  As he has not taken trig yet, yesterday he got hung up with proving that the derivative of sinx is cosx.  He needed to prove sinx/x=1 as x=>0.  I told him it was just an identity, so memorize it, but no.  So off he went drawing a  huge circle, and doing some geometric something with limits.  When he finished, he said "that was a cool proof!"  He just loves this stuff. 

 

He really appreciates that I have internet friends who are mathematicians and mothers of mathematicians.  So thank you all for your advice!  I am learning very quickly that I am a scientist, NOT a mathematician, and I am definitely feeling like I am just doggy paddling in the deep end !  On another note, ds thinks he scored between a 20 and 30 on the BMO1 which would be enough to qualify him for the BMO2 given his age.  We will see.

 

[1053]

.

[lewelma]

I just asked him, and apparently he did use the squeeze theorem.  So I asked when he learned about it, and he said 'I got the diagram off the internet, and solved it on my own from there.  It was obvious that you needed to make the area here squeezed and that area also squeezed.'  When he went to check the proof is when he learned the name of what he did.  I'm just like :blink:

 

 

[raptor_dad]

FYI, archive.org has a 1st ed of Spivak. A quick google will turn it up. It has been up for a couple of years. I assume its copyright status is ambiguous. It is before Spivak started PoP Press so he doesn't own rights... but Addison-Wesley also doesn't have any interest in enforcing their rights... Could be useful for review until you buy a 3rd or 4th ed copy.

[kiana]

FYI, archive.org has a 1st ed of Spivak. A quick google will turn it up. It has been up for a couple of years. I assume its copyright status is ambiguous. It is before Spivak started PoP Press so he doesn't own rights... but Addison-Wesley also doesn't have any interest in enforcing their rights... Could be useful for review until you buy a 3rd or 4th ed copy.

 

I didn't realize that! Thanks!

[lewelma]

two questions:

 

1) how is Spivak different from AoPS calc?

 

2) What is a 'first year analysis class'?  Some people seem to say that is what Spivak is kind of equivalent to.

 

Is this class at our local uni, an "analysis" class? Doesn't sound like what I am looking at with the Spivak 1st edition book (Thanks Raptor_dad!):

 

Foundations of Algebra, analysis and Topology: An introduction to some fundamental structures and spaces, and their study by the axiomatic method. One half of the course will discuss groups, including permutation groups, groups of matrices, and symmetry groups. The other half will study concepts of continuity and analysis in Euclidean spaces and metric spaces.

[Kathy in Richmond]

FYI, archive.org has a 1st ed of Spivak. A quick google will turn it up. It has been up for a couple of years. I assume its copyright status is ambiguous. It is before Spivak started PoP Press so he doesn't own rights... but Addison-Wesley also doesn't have any interest in enforcing their rights... Could be useful for review until you buy a 3rd or 4th ed copy.

 

Count me as another who wasn't aware of this. Thanks so much!

 

[Kathy in Richmond]

He was actually willing to sit down and do some problems yesterday after proving all the things on the list.  Mixed practice!  I don't think he has done mixed practice since he was about 8. 

 

I've been slowly picking at his perception that a speed test of straightforward problems is insulting. I've been reading parts of your responses to him, and we had a breakthrough yesterday using the word 'fluency.' Seems quite funny to me that a single word could break the stalemate, but it did.  I defined fluency as having an intuitive sense of what to do, that you understood it so well that you did not need to think it through, that is was not rote which is learning to work algorithmically without understanding.  All this is quite good in his eyes, and makes fluency a worthwhile goal.  Then he made the leap "Oh, so they make the test under time pressure to test your fluency."  Ah, yes!  Can't believe this has taken me a month to beat this through his head.  Sigh.  Thanks for the word, Kathy!

 

 Oh, how funny! Glad it worked for him, Ruth. I'm with him. Fluency makes me feel that I've mastered a foreign language instead of taken part in boring speed drills.

 

On another note, ds thinks he scored between a 20 and 30 on the BMO1 which would be enough to qualify him for the BMO2 given his age.

And good for him on the BMO1! I didn't realize that it proceeded in stages. Good luck on the next round, too!

 

two questions:

 

1) how is Spivak different from AoPS calc?

 

2) What is a 'first year analysis class'?  Some people seem to say that is what Spivak is kind of equivalent to.

 

Is this class at our local uni, an "analysis" class? Doesn't sound like what I am looking at with the Spivak 1st edition book (Thanks Raptor_dad!):

 

Foundations of Algebra, analysis and Topology: An introduction to some fundamental structures and spaces, and their study by the axiomatic method. One half of the course will discuss groups, including permutation groups, groups of matrices, and symmetry groups. The other half will study concepts of continuity and analysis in Euclidean spaces and metric spaces.

I'm not familiar enough with Spivak to compare it to AoPS Calculus. Apostol user here!

 

Your local uni course description appears to be a class that transitions students from the first couple of years of calc, multi, & diff eqn to higher level, more abstract mathematics. It's a combo of abstract algebra (1st half) and real analysis (2nd half). Then you'd go on to full fledged courses in those individual subjects. I bet that your son would enjoy this class now. :)

[lewelma]

And good for him on the BMO1! I didn't realize that it proceeded in stages. Good luck on the next round, too!

1200 kids are invited to take the BMO1 (it is proof based, so I'm guessing they don't want to grade more than that!) Then 100 kids are invited to take the BMO2. The number I saw was a score above a 17/60 for 10th graders to get an invite (ds is in 9th in the UK school year). However, I don't believe that NZ kids take the BMO2. It is more that it would give us a feel for his real competitiveness. NZ is just so small, and so few kids are interested in math, that he is a big fish in a very small pond. But if he makes it into the top 100 kids in the UK that would be super cool. We won't know until January.

 

Your local uni course description appears to be a class that transitions students from the first couple of years of calc, multi, & diff eqn to higher level, more abstract mathematics. It's a combo of abstract algebra (1st half) and real analysis (2nd half). Then you'd go on to full fledged courses in those individual subjects. I bet that your son would enjoy this class now. :)

They said he could take it after the Calc2 class, because he has placed out of the other 100 level classes based on his AoPS proofs. So in 1 year.