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Number Theory vs. Counting & Probability in AoPS


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Has anyone had experience with the Introduction to Number Theory book in the Art of Problem Solving curriculum?  Judging from the short list of topics, it seems that much of the same information is covered in Introduction to Number Theory that are covered in the Number Theory chapter of the PreAlgebra book.  Is the depth much more sufficient?

We do not have enough years to cover both Introduction to Number Theory and Counting & Probability.  My son does not want to go into computer sciences, or extensive mathematics as a profession, so we will not be covering the Intermediate Counting & Probability.  It is more a choice of which introductory text to use for foundational knowledge and later basis to pull from in more complex mathematics.

 

The other option is to do both books in the year, instead of just one.  Number Theory is an area my son is fairly strong in and the number theory chapter in the PreAlgebra book has been mostly review.  Would it be too much to go through both books simultaneously?

 

Any experiences or thoughts here would be welcomed.

 

Thanks!

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NT is barely touched in the PreAlgebra book.

 

Dd has worked through both the C&P and the NT books while working through other AoPS books. I don't see why your ds couldn't do both C&P and NT simultaneously in one year, or he could do one and then the other. The online classes take 12 weeks per book.

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You could also start with C&P while doing another text to see how it goes. If your student is not going into math or computer science, I'd start with one or the other in case you get bogged down along the way. I think you want to be sure you've got a couple of algebra courses, geometry, and, from some source, a good class or two in data analysis and statistics done before finishing high school -- of course, more is better, but I think those are the basic needs nowadays for kids without learning challenges. If you've got plenty of time, including possibly extra time for a slow class (due to difficulty or extra interest), then doing both C&P and NT is a good option.

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My son finished both books this year.  Kathy in Richmond warned me that some students take quite some time to get their heads around number theory, as it is just a different way of thinking.  I have found this true for both number theory and counting for my older son. My son did the books sequentially, and I found that he needed to review the material every week for months. 

 

So if I were to do it again, I would have a student work each week on *both* books, so 2 days number theory and 3 days counting, then reverse the next week.  In this manner, the material is stretched out in time, and the student's mind has a chance to absorb it. 

 

Ruth in NT

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Ds is currently doing NT intro text. He loves it, says there is plenty to learn, and is moving through it rather quickly. He has a solid algebra background fwiw. Interesting idea about switching off with C&P. From the level of enthusiasm (working on it on thanksgiving) I would say it is very worth trying. Happy Thanksgiving.

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Thanks for the info everyone.  Your experience is very helpful.  The switching off sounds like a good place to start.  It allows for more ruminating time in his brain, but also lets him spend more time on one if he is having trouble.  Much appreication to all!  Happy Holidays!

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For review add in Alcumus at the appropriate difficulty level and with review frequency set to "high." There are a few problem types in Alcumus not covered in the text (sum of divisors problems come to mind), but there is sufficient teaching in the solutions this isn't much of an issue. There are also a few twists that go beyond the text. DD almost met her nemesis in the Euclidean Algorithm section, but persevered and came out on top.

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Has anyone had experience with the Introduction to Number Theory book in the Art of Problem Solving curriculum?  Judging from the short list of topics, it seems that much of the same information is covered in Introduction to Number Theory that are covered in the Number Theory chapter of the PreAlgebra book.  Is the depth much more sufficient?

 

 

 

 

The PreAlgebra book pretty much covers the first 4-5 chapters of the NT book.  You can probably skip those if he's familiar with it, GCM, LCD, prime factorizations, etc.  

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The PreAlgebra book pretty much covers the first 4-5 chapters of the NT book. You can probably skip those if he's familiar with it, GCM, LCD, prime factorizations, etc.

Having him do the review (or just the challenge) problems might be a better option than just skipping totally. There could be one small segment of a topic that might need further study.

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The PreAlgebra book pretty much covers the first 4-5 chapters of the NT book.  You can probably skip those if he's familiar with it, GCM, LCD, prime factorizations, etc.  

 

It's hard to tell from the TOC, but I would be surprised if the Intro to NT text did not include greater depth on these topics than the single NT chapter in the Prealgebra text.  So, I agree w/Luckymama that I'd do the review and challenge sections to make sure all bases are covered and nothing juicy and worthwhile is missed.

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Has anyone had experience with the Introduction to Number Theory book in the Art of Problem Solving curriculum?  Judging from the short list of topics, it seems that much of the same information is covered in Introduction to Number Theory that are covered in the Number Theory chapter of the PreAlgebra book.  Is the depth much more sufficient?

 

We do not have enough years to cover both Introduction to Number Theory and Counting & Probability.  My son does not want to go into computer sciences, or extensive mathematics as a profession, so we will not be covering the Intermediate Counting & Probability.  It is more a choice of which introductory text to use for foundational knowledge and later basis to pull from in more complex mathematics.

 

The other option is to do both books in the year, instead of just one.  Number Theory is an area my son is fairly strong in and the number theory chapter in the PreAlgebra book has been mostly review.  Would it be too much to go through both books simultaneously?

 

Any experiences or thoughts here would be welcomed.

 

Thanks!

 

My daughter has done both of the books you asked about, and if I were to pick only one I'd choose the Counting and Probability.  She found the Number Theory book to be mostly review.  Also, she thought what she learned in the C&P was more useful to her on the AMC 8 exam she recently took, but she has a pretty strong base in Number Theory.

 

There is nothing wrong with doing both, of course, but if I had to pick one it'd be C&P. 

 

Good luck!

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  Number Theory is an area my son is fairly strong in and the number theory chapter in the PreAlgebra book has been mostly review.  Would it be too much to go through both books simultaneously?

My boy who likes cryptography and did the Elements of Mathematics Foundations free course (some time ago) found the Number Theory book mostly review.  It wouldn't be too much to do both books. The C&P book is the harder of the two.

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We didn't use AoPS for Pre-Alg, so I can't comment on that.  My eldest did the first part of Intro of Alg as Alg I, then Intro to Number Theory, and is now finishing up Intro to Alg as Alg II.  He found Intro to Number Theory much easier than Alg I, but it was a nice break for him.  He loved the content too!  So much so, that he would talk about the topics to anyone who listened - usually 9 and 10 year old public school kids. 

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Where would these fall in the preA to algebra sequence? Especially if you were not using AoPS for PreA?

 

AFAIK, it is best to have some or all of algebra 1 under one's belt first.  Eta, there are pre-tests on the website.

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Where would these fall in the preA to algebra sequence? Especially if you were not using AoPS for PreA?

 

I taught high school math in U.S. public schools for either advanced or remedial learners (I worked with at-risk kids so it was the extremes of the spectrum).  If your child is into math, or going on an academic track, I would say that everything I have seen in Number Theory is a standard Pre-Algebra course.  The combinitorics course (Counting/Prob) is more of a the end of Algebra 1, though binomial theorum falls in many different places for different schools.  Currently we are doing the Pre-Algebra book and it is exactly that, Pre-Algebra fairly solidly.

 

I think the difference in the curriculum is not the content, but what the text is asking you to do with the content.  Instead of showing you a solution based method of study (a process is given to the student, the student completes the process over and over, the answer is derived), AoPS is process based  method of study.  No process is given to the student and the answer isn't the important part.  Each section only has about 10-15 problems for the student to independently work out rather than hundreds.  However, in order to answer those 10-15 problems, the student has to completely understand the topic and have been able to develop a process for which to apply it.  In this way, it is a critical thinking book surrounded in math, and not a traditional math textbook.

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