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. 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 mixed with this . 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?