art of problem solving writing

Evan Chen《陳誼廷》

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Math olympiad beginner's page

This page is meant for people who don’t have much past olympiad/proof experience and are looking to get started. If you aren’t interested in proof-based problems yet, then this page is not for you. Try checking FAQ C-0 if you are totally new to math contests.

Before all else, welcome to the olympiad scene! It’s going to be hard as heck, but in my private opinion this is where all the coolest stuff is (as far as math contests go, anyway 1 ). Stay around for long enough, and you will get to see a lot of really amazing problems.

You may also read math contest FAQs for some more philosophical (and less concrete) advice on studying for math contests in general.

0. Syllabus #

I wrote an unofficial syllabus for math olympiads (also linked on handouts ) giving some guidance on what topics appear on math olympiads.

1. First reading: the welcome letter #

For the USA Math Olympiad in 2020, the board of the USAMO prepared an invitation letter 2 for all the qualifiers, congratulating them on their achievement and giving them some suggestions on where to begin. This letter contains:

• A few pretty carefully chosen problems (not necessarily easy!), to give people a sense of what to expect on the contest
• Fully written solutions to those chosen problems, so that you can see what a correct and complete solution is expected to look like.
• Some advice for actually taking the contest: the format of the exam, planning your time, common mistakes, etc.

You can download the letters here:

• JMO welcome letter , and solutions to examples
• USAMO welcome letter , and solutions to examples

I suggest starting by reading through this letter, trying the example problems (you will probably not solve them all; we chose examples from the entire difficulty spectrum), and then comparing your work to the provided solutions. That will give you a taste of what you are getting in to.

2. Writing proofs #

If you don’t have experience with proof-based problems, the first thing I should say is that it is not as hard as you might think . Solving the problems completely is difficult, but if you really have a completely correct solution to a problem, it is actually pretty hard to not get full credit. I would say at least 90% of the time, when a student loses points on the USA(J)MO unexpectedly, it’s because their proof is actually incomplete, not (just) badly-written.

Of course, you should still try to write your solutions as clearly as possible. To that end, here are some links to advice:

• How to write a solution , from Art of Problem Solving.
• Remarks on English , written by me.
• How to write proofs , by Larry W. Cusick.

You don’t need to get too caught up in these links; proof-writing will become more natural over time as you solve more problems. So I would encourage you to continue doing practice problems or reading books at the same time as you are getting used to writing; these go hand-in-hand and I actually suspect it’s counterproductive to try to practice writing in isolation.

If possible, the best way is to have a friend or coach who can check your work and provide suggestions. But the supply of people willing to do this is admittedly very low, so most people are not so lucky to have access to feedback. Almost everyone gets by instead with something like the following algorithm:

• Write up your solution neatly.
• Look up the problem on AoPS contest index 4 ; and compare your solution to those by reputable users.
• Edit your solution and post it on the thread. By Cunningham’s Law , wrong solutions are often exposed quite rapidly.

Together these three steps should catch “most” substantial errors. See Section B.1 of my English handout for more details about this procedure.

If you want a book to follow, the one I grew up with was Joseph J Rotman’s Journey into Mathematics: An Introduction to Proofs . 3

If you like excessive information, you might also read my handout Intro to Proofs for the Morbidly Curious .

3. (For USA) United States Mathematical Talent Search and USEMO #

If you are in the United States, there is a nice proof-based contest called the USAMTS which is a great way for beginners to get started. Things that make the USAMTS special:

• It is a free, individual, online contest open to any students in the USA.
• The problems are chosen to be quite beginner-friendly, though with a spectrum of difficulty each round.
• This contest gives you a full month to work on the problems rather than having a short time limit.
• You get some feedback on your proofs as well, not just a score.

I also run a contest called the USEMO in the fall which is also free and offers feedback, but it is more difficult since is intended to mimic the USAMO and IMO in format and difficulty. One could try using it as “practice USAMO” in the fall.

4. Books to read #

There is some more material you have to learn as well, since there are some new classes of problems (such as olympiad geometry, functional equations, orders mod p, etc.) that you will likely not have seen before from just working on short-answer contests.

Some possible suggestions for introductory books:

• General: Art and Craft of Problem-Solving by Paul Zeitz is a good “first book” for all the fields.
• Geometry: My book E.G.M.O. ; and more alternatives are linked at the bottom of that page.
• Number theory: Modern Olympiad Number Theory is the most comprehensive text I know of now.

The OTIS Excerpts has beginner introductions for several topics:

• Inequalities (chapters 1-2)
• Functional equations (chapters 3-4)
• Combinatorics (chapters 6-9)

More possibilities (including intermediate-advanced texts not listed here) are on the links page . You might also check Geoff Smith’s advice and links .

I really want to stress these are mere suggestions . Just because you have done X does not mean you will achieve your goals, and conversely, there are surely many fantastic resources that I have not even heard of. If you are looking for a list of materials which are “guaranteed to be enough” for solving IMO #1 and #4, then unfortunately I can’t help you.

5. Problem sources #

At some point (sooner rather than later), you also need to start just working through some past problems from recent years of contests. You can think of this as encountering problems in the wild. 5

In case you didn’t know already, on Art of Problem-Solving there is an extensive archive of past problems from basically every competition under the sun, together with community-contributed solutions. The supply of problems here is inexhaustible.

Here are some particular contests I like (alphabetical):

• Canada national olympiad
• European Girls Math Olympiad
• IMO and IMO Shortlist
• NICE , open to anyone
• USA Team Selection Tests

The bottom of the recommendations page has some more suggestions for problems if this list isn’t sufficient.

Good luck and happy solving!

Insomuch as contest problems have intrinsic artistic value, proof-based exams are a more versatile medium than the short-answer exams, much like how videos are more versatile than static photos. In this analogy, videos don’t make stills worthless or obsolete; they aren’t automatically better, either. But as a medium, they expand the space of ideas an artist could express, at the cost of being proportionately more work to create.  ↩

These were written in early January 2020 before COVID-19 wreaked havoc on everything, so the contests still go by their typical name and don’t mention anything specific to the belated administration that year.  ↩

It shares my philosophy that teaching proof-based classes by force-feeding set theory notation is not particularly helpful, and instead develops proof-writing by discussing real mathematical content from geometry, number theory, etc. rather than being overly focused on bookkeeping and formalism.  ↩

I do NOT recommend using the AoPS Wiki in place of the Contest Index. The solution quality in the wiki is generally much poorer than the forum.  ↩

I used to carry a binder with printouts of the IMO shortlist and check them off as I solved them.  ↩

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Review by Pamela Brittain , Common Sense Education | Updated August 2019

Art of Problem Solving

Variety of math content supports struggling and advanced learners

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Community rating, privacy rating, subjects & topics, take a look inside 8 images.

Pros : Presents information in multiple ways; courses are organized specifically by grade level.

Cons : The forums are a bit hard to navigate; continued use of the program could get a bit pricy over time.

Bottom Line : Separate sections for younger and older students help to teach kids at all levels, from foundational skills up to Math Olympiad training.

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The Art of Problem Solving allows students to focus on math concepts that they're struggling with or (for advanced students) that offer a challenge. The site presents the materials as self-paced for younger students or as online, interactive courses for older students. There's also a large collection of additional resources available on the site, including videos and an extensive forum (over eight million posts and counting) where students discuss all sorts of math concepts. 

The site is best designed for students wishing to work independently on their math knowledge, although some of the materials presented -- such as the videos -- may be good for a teacher to use with a whole class to introduce a topic or give a different perspective on it. If you're a Math Olympiad coach or have students preparing for math contests, you'll find the videos of sample test questions very helpful; many of the resources found here are especially geared toward kids who are preparing to compete at a national level.

The Art of Problem Solving has added a lot of math content over the past few years and now includes videos, interactive courses, and a fun, game-based academy for younger students. The site is divided into different sections, each focusing on a different area of math and different grade levels. The main divisions are AoPS Online, Beast Academy, and AoPS Academy.

AoPS Online provides older students with lessons on everything from pre-algebra foundational skills up to Math Olympiad training. There's also a substantial forum where students can interact with one another to ask -- and answer -- various math problems. The Beast Academy is for younger students and has a fun interface with comic characters, videos, games, and more for grades 2 to 5. Finally, the AoPS Academy is an in-person academy that students can attend. 

Overall, the Art of Problem Solving provides a variety of resources to help struggling students succeed and to encourage and build enrichment for students to challenge themselves. The fact that it's all presented together in one place may also have the bonus effect of encouraging once struggling students to take on greater and greater challenges -- and perhaps be the next Math Olympians. 

The choice of videos, books, forums, online interactive classes, and games (for the younger students) means that students have a number of different ways to learn the material. The Beast Academy, in particular, is very good at providing students with different learning options. The bright colors and fun characters also lend an air of friendless to the site without losing the focus on teaching. 

The forum promotes collaboration among students, but it's hard to navigate, and some of the posts are less about math and more about "kids being kids" in an online environment. But for those who take the time to explore, there's a rich source of information to be found.

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A variety of content formats gives students of different abilities multiple options for accessing the materials. The forum, though hard to navigate, encourages kids to engage with one another in a safe, friendly way. 

Math content covers a large number of concepts and is structured differently by grade level: webinar lectures for older students and games and comic book characters for young kids.

The Beast Academy supplies kids with lots of hints, and online, interactive classes give students direct access to the teacher. Digging through the forum may provide additional help.

Strong program with emphasis on deep understanding of concepts and love of math. Would highly recommend above most curriculums I have tried.

The K-5 program is strong and could easily be used in any k-6 program. The content is challenging but so strongly done that any level of student could use the program with support.

The art of problem solving program for 6-12 is very challenging. It could be used for all students, but again, would need support. The online classes are excellent, books alone would need strong classroom support. Clearly aimed at high functioning students. This is a math program that goes beyond basic understanding and rote memorization to emphasize true, deep understanding and complex thinking. There are a plethora of problems available and a rich online community.

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I am an Art of Problem Solving Drop-out.

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By Jenn in Mo , September 21, 2010 in High School and Self-Education Board

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We're a mathy family here, so I'm feeling kind of like a failure here, but I closed the book today and said, "Enough." My math-loving son was diligently struggling through hours of math a day and getting nowhere. It took nearly two months to get through two chapters and I know he couldn't pass any review tests. He understands algebra, but this is....different. My husband and I are struggling to understand half of the problems ourselves...and this is our *thing*, ya know?

I haven't heard a negative word about AoPS yet, so it's kind of embarrassing to admit, but I just do not "get" some of these questions. Anyone that gets it feel like helping to clear the mud on this one?

2.38 What number must be in the blank in the expression 3(x+7)-_(2x+9) if the expression is the same for all values of x?

The solution manual says that "if the expression is the same for all values of x, then the x's in 3(x+7) must cancel out with those in _(2x+9).

My question is...WHY must they cancel out? It's a random, unfinished number sentence. What is in the original problem that tells me they must cancel? What am I missing here? Knowing they must cancel, we can work the problem, but I'm not grasping why I should know they cancel each other out. :svengo:

My other question is, what do I do with this book now? Do I take him through NEM and then come back to this or avoid it forever? I've never given up on a program before. I chose it because the other algebra books were too easy. We've ran into hard things in other books before, but stopped and worked through and moved on. Every day I see him patiently ramming his head into the AoPS wall and it's not something I can help him just "work through" this time. Even though I know that, I still hear Frankie Avalon in my head singing AoPS Drop-Out. :blush:

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2.38 What number must be in the blank in the expression 3(x+7)-_(2x+9) if the expression is the same for all values of x?   The solution manual says that "if the expression is the same for all values of x, then the x's in 3(x+7) must cancel out with those in _(2x+9).   My question is...WHY must they cancel out? It's a random, unfinished number sentence. What is in the original problem that tells me they must cancel? What am I missing here? Knowing they must cancel, we can work the problem, but I'm not grasping why I should know they cancel each other out. :svengo:   My other question is, what do I do with this book now? Do I take him through NEM and then come back to this or avoid it forever? I've never given up on a program before. I chose it because the other algebra books were too easy. We've ran into hard things in other books before, but stopped and worked through and moved on. Every day I see him patiently ramming his head into the AoPS wall and it's not something I can help him just "work through" this time. Even though I know that, I still hear Frankie Avalon in my head singing AoPS Drop-Out. :blush:

(Okay LOL on the Frankie Avalon - I love that song!)

But if the value of the expression is the same for all values of x, that means there won't be any x's left when you simplify it. Otherwise the value of the expression would change for each different possible x.

Have you considered the classes? I know it's a lot of money for a maybe (although I think you have three weeks to bail and get a refund) but I find that there are things I should know but can't always blurt out right when DS needs them... and I like the idea of having someone else who is prepared to do it and other kids who can add to the discussion. Even one math jam left me going "wow - this guy rocks"... :) DS is starting two AoPS classes this year, so I'm not really speaking from experience yet, but I know I'd be stuck going "uhhhhhh...." even on things I really should be able to get. It is hard stuff, and not the "I can teach this in my sleep" of other algebra and geometry programs. I've tutored those for years, but AoPS really is quite a bit more.

8filltheheart

Well, I was in a serious minority (a group of 1, me ;) ) on the k8 board when a mom asked about AoPS for a slow, methodical avg student and I said I would not recommend it. Everyone else was saying that they were accessible to all students.

I have a lot of good math students (I had 1 take alg in 5th, 1 in 6th grade, and 3 others in 7th), but I only have 1 child that I would use AoPS with. He is not just good at math; he is passionate about math. He lives and breathes math. He has taken several AoPS classes and they are tough. He spent hours this summer on their alg 3 challenge sets. Thing is........he wants the challenge. That sort assignment literally makes him smile and jump right in.

My other kids do not desire that type of math depth. They used/are using programs like Foerster. I do not think they are being deprived at all. My ds that loves AoPS thinks mathematically and sees the world that way. The others are just good at math. :)

FWIW......I don't think you have anything to be ashamed of. The books are designed for the top 2-3% of all math students (not all students). They just aren't for everyone. (I couldn't teach them if I tried. He either takes their online classes or has a math coach for them. They are wayyyyyyyyyy beyond my abilities and my dh's as well (and he is an engineer! ;) )

ETA: Thought I should share that if I had known about AoPS umpteen yrs ago, I would have wanted my oldest to try them. I do think they stretch mathematical comprehension beyond the norm......basing that on the fact that my 14 yos now talks some strange language that I don't have a clue about what he is saying. All that said, I **know** that the rest of my older kids would say no way. (as a matter of fact my 11th and 6th graders did!! )

Guest Cheryl in SoCal

Well, I was in a serious minority (a group of 1, me ;) ) on the k8 board when a mom asked about AoPS for a slow, methodical avg student and I said I would not recommend it. Everyone else was saying that they were accessible to all students.   I have a lot of good math students (I had 1 take alg in 5th, 1 in 6th grade, and 3 others in 7th), but I only have 1 child that I would use AoPS with. He is not just good at math; he is passionate about math. He lives and breathes math. He has taken several AoPS classes and they are tough. He spent hours this summer on their alg 3 challenge sets. Thing is........he wants the challenge. That sort assignment literally makes him smile and jump right in.   My other kids do not desire that type of math depth. They used/are using programs like Foerster. I do not think they are being deprived at all. My ds that loves AoPS thinks mathematically and sees the world that way. The others are just good at math. :)   FWIW......I don't think you have anything to be ashamed of. The books are designed for the top 2-3% of all math students (not all students). They just aren't for everyone. (I couldn't teach them if I tried. He either takes their online classes or has a math coach for them. They are wayyyyyyyyyy beyond my abilities and my dh's as well (and he is an engineer! ;) )

:iagree:I haven't used it yet but have been looking through the Intro to Counting and Probability and a bit of the Intro to Algebra and from what I have seen I wholeheartedly agree. No math curricula is the answer for every student but IMHO AoPS is going to be a good fit for the minority of students. One of my ds's would DIE if I had him use AoPS and another I think will like it because he thinks that way, though we'll only know after we try it.

I had a look at the algebra and geometry and found the layout visually confusing, which is kind of weird since in some ways it is not as busy or distracting as textbooks in general. But it is packed solid, and hard to read as a text, if that makes any sense. And algebra is my own strength. My dd went for Discovering Algebra instead.

The geometry book struck me similarly as user-unfriendly, but that may be again just my visual reaction to such a dense text.

However, that similar packing did not bother dd with the data and probability books, which she greatly enjoys. So maybe it depends on the kid (and grown-up) as well as on the area of math involved.

(Okay LOL on the Frankie Avalon - I love that song!)   But if the value of the expression is the same for all values of x, that means there won't be any x's left when you simplify it. Otherwise the value of the expression would change for each different possible x.

Ha! Lightbulb! Thank you! Regardless of what x is, the expression will equal the same thing. You're speaking my language. The book is not. :tongue_smilie:

I am relieved to hear these responses. He loves math and enjoys a challenge, but I couldn't at all say that he is passionate about math. I had forgotten that Foerster was on my list of maybes before I heard that AoPS was the crème de la crème of algebra texts. So....NEM or Foerster....thoughts?

I know absolutely nothing about NEM, so I can't offer any insight there. I have used Foerster with all of my older kids, including my ds that now uses AoPS. He used Foerster for both alg 1 and alg 2. His first AoPS courses were Intro to Counting and Prob and their alg 3 courses online and he owns several of their other "off normal track" math books that he works through on his own.

My oldest used Foerster and it more than prepared him for university level cal and an engineering major.

I'm not sure I can explain the differences very easily. Foerster is a solid math program and they absolutely learn the math skills they need to perform high level math problems, etc. No question.

AoPS helps them process math in a completely different fashion. Ds goes around "discovering" concepts and "proving" things like why x equals some bizarre thing or how an indefinite number of fractions add.......I don't know......I just nod my head!!! :lol: He thinks he wants to double major in astrophysics and math. For him it is a match. I have only ever met 2 other people that think that way.......one is now majoring in math and the other has a phD in math. For the rest of mere mortals, I think we are fine w/o it!

In The Great White North

Dc used/are using Foerster. They arde very solid math books. I have the AoPS Problem Solving, Counting & Probability and Number Theory books. Dc don't like them. They are not straight forward. Dc like to have an example, an explanation and get the problems done. Foerster does this for them. AoPS does not.

AoPS is for people who like to PLAY with their math. If your idea of a great time is to go round and round with a math problem til you figure it out, AoPS is for you. Yes, the top 2-3% of math students in the country are more likely to enjoy that, but even they don't always like that.

I've seen but not used Foerster, so I can't compare... but NEM is good challenging work. Most of it is more straightforward than AoPS (examples first, exercises second) but there are Challenge sections and Mathematical Investigations that could be frustrating in the same ways AoPS can be. They do (in those sections) throw you in and expect you to dig yourself out. And to some extent they leave some gaps in the regular explanations too -- nothing huge, but enough that sometimes the student (or the teacher) has to connect the dots.

C_l_e_0..Q_c

I'm in the opposite situation. We're a NEM drop-out. I'm so much happier as a mom with AoPS *and* the online class. Someone else is teaching math to my math kid!

(and I"m good at math too, I did math all the way to a Master's level, although it was not my major - I'm an engineer, we need math )

My son is truly having a blast with the online class. He did Algebra 1 over the summer. The kids at the summer pool who learned he was doing a math class assumed it was remedial, until they saw a problem! LOL. Sweet nerd revenge. :)

The math classes are totally worth the cost. Plus they force the kids to move forward, and not spend 2 months on 2 chapters. In 15 weeks, they did 14 chapters (possibly 13, I'm not sure) so the interest is kept high.

at the beach

I agree that AoPS is for people who enjoy playing around with math. I think I am one of those who said something along the lines that more people can be successful with it than the top few percent. But I think I also said that sometimes the solutions/explanations gloss over things or skip a step, assuming readers will see that step in the problem when they might not. I felt it could be used by more than just the top few percent because I think that kids often underestimate their abilities. I also agree with the pp that even kids in the top few percent who can do it may not want to do it because it does take a lot of time and puzzling.

I was terrified of math from 8th grade on. I never took math in college but took logic to avoid it. When I signed my daughter up for AoPS this summer, I decided to give the book a try. I did the first three chapters. I wanted to help my daughter if she needed it, but she was the one helping me :001_huh: and the class moved so fast that I couldn't keep up.

The problem that you mention, to me, looks like it is asking you to make the 2x the same as the 3x so the only thing that works is 3/2. How do you know it will cancel? I think you know that because it's a subtraction problem. Maybe I am oversimplifying this.

Maura in NY

NEM is an integrated program - Algebra/Geometry/Trig and rolled together. It's challenging, in a good way, but it doesn't fit the typical Algebra/Geometry/Algebra II/Trig-PreCalc/Calc track.

FWIW, my ds used NEM 1 in 7th, thinking he would stick with it for 4 years. Instead, we switched. He didn't use Foerster's Algebra I, but he did use Foerster for Algebra II and PreCalc. Great real world application problems.

We used AoPS for several months and also found some of the problems strangely worded. But, our biggest problem with AoPS was not enough repetition.

We are using Foerester's Algebra now. It has more than enough problems for us, but it's much easier to eliminate problems than try to add more in.

Miss Marple

Not a drop-out yet, but we are doing AoPS very slowly and I've added NEM2 back as the primary math. Ds is still working through the geometry book (45 minutes per day), but we've put the algebra book away because he already did LOF Beginning Algebra and the NEM should cover the bases thoroughly.

I'm finding that hormones cause my son to be less inquisitive and less dedicated right now. He does great with math, so I chose the other program I love - Singapore NEM, and it is going very well.

If we never finish the AoPS books that will be OK because NEM is the primary curriculum (which we will finish). I know that any exposure ds has had with AoPS has helped fine tune his mathematical abilities.

Well, I was in a serious minority (a group of 1, me ) on the k8 board when a mom asked about AoPS for a slow, methodical avg student and I said I would not recommend it

:iagree: I cannot imagine how frustrated a student and a parent would be trying to work through this program and thinking that it is accessible for all students!

The problems in NEM can be plenty challenging at times. My son did NEM 1 before switching to AoPS.

Regarding AoPS Algebra 1, they introduce Algebra 2 concepts, too. That doesn't make it easier! Ds thought the middle of the text was difficult and often trudged through the problems, but then he flew through the last few chapters. He thought they were easier, even many of the challengers. I don't know if they really were, or if he just started to get the hang of the text and problems.

When it comes to math, it's so important to pick what works and then bite off and chew well at a pace that's suitable for the student. I wouldn't hesitate to step back and reconsider if something wasn't working.

  AoPS is for people who like to PLAY with their math. If your idea of a great time is to go round and round with a math problem til you figure it out, AoPS is for you. Yes, the top 2-3% of math students in the country are more likely to enjoy that, but even they don't always like that.

I agree entirely with the bolded.

If anybody is deciding whether to go with the AoPS classes, there's a Math Jam on the subject this afternoon/evening (7:30 Eastern, I believe). It will include sample questions from the class, so you can try them out.

Math Jams are free informational sessions, usually hosted by Richard Rusczyk, who is always responsive to questions. If he can't answer your question within the confines of the session, he'll likely invite you to e-mail him and he'll help you decide whether a class is right for your family.

We're mathy. . . and we're AoPS drop outs as well.

Dd & I tried Number Theory for a couple/few months. It actually went OK so long as *I* worked through *every section* and *every problem* on my own before dd tried it (or simultaneously sitting together).

The exploration and problem solving was interesting. I rather enjoyed it. But, I just *don't have time* to do it right now.

We dropped it after a few chapters -- which were successful but slow b/c I just couldn't make myself find the time to do it often enough to keep up w/ dd's needed pace.

So sad. . .

Now y'all are scaring me. My dd loved the AoPS Number Theory book so much this summer, I bought Alg I for her (but we don't start for another year...) I had been planning on Foerster's, so I've got both ready to go (well, I still need that pesky Foerster's Solutions manual...)

We're mathy. . . and we're AoPS drop outs as well.   Dd & I tried Number Theory for a couple/few months. It actually went OK so long as *I* worked through *every section* and *every problem* on my own before dd tried it (or simultaneously sitting together).

Dd and I worked through all the examples together, but then she did the problem sets pretty much on her own (except I let her skip the challenge problems at the end of the chapters). Was your dd just needing you to sit with her for the "lesson" part, or also the problem sets? I didn't mind working through the lesson with her... I actually learned a lot myself and had fun... :tongue_smilie: For some reason I really like the layout and presentation of these books... OTOH the Lial's layout really drove me crazy when I first saw it (I'm getting used to it now that my other dd's working through BCM).

But now you guys are intimidating me a bit... :001_huh: I do think the Algebra text looks more challenging than the Number Theory, but I'm hoping after she's gotten through Singapore DM1, she'll be preparerd (or I guess we'll do Foerster's after all...)

I found the solutions manual for Foerster this afternoon! I finally called the publisher and it turns out that they renamed the book and it is now called Classic Edition instead of Foerster's. The ISBN # is 0201861003. I can't find it used anywhere, but it is available new at pearsonschool.com.

I'm still torn though. I loved Singapore so well and would like to try going on with NEM. But I don't want to commit to it being his math for all of high school. I hear that Foerster does an excellent job with proofs, which I think are important. I may end of tossing a coin before this is over.

This is exactly what I was trying to express in the other thread! It's just as much about the approach as the difficulty, imo. Yes, you need to be a good math student, but I dont' think you need to be a TOP math student. Not in the sense of the top 2% to 3%, at any rate. I can't imagine my dd would score anywhere near that high, but she loves AoPS.

And you can be a top math student, but if you hate this approach, it will be very hard to persevere and succeed.

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The Art of Problem Posing

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The new edition of this classic book describes and provides a myriad of examples of the relationships between problem posing and problem solving, and explores the educational potential of integrating these two activities in classrooms at all levels. The Art of Problem Posing, Third Edition encourages readers to shift their thinking about problem posing (such as where problems come from, what to do with them, and the like) from the "other" to themselves and offers a broader conception of what can be done with problems. Special features include: an exploration of the logical relationship between problem posing and problem solving; sketches, drawings, and diagrams that illustrate the schemes proposed; and a special section on writing in mathematics. In the updated third edition, the authors specifically: *address the role of problem posing in the NCTM Standards; *elaborate on the concept of student as author and critic; *include discussion of computer applications to illustrate the potential of  technology to enhance problem posing in the classroom; *expand the section on diversity/multiculturalism; and *broaden discussion of writing as a classroom enterprise. This book offers present and future teachers at the middle school, secondary school, and higher education levels ideas to enrich their teaching and suggestions for how to incorporate problem posing into a standard mathematics curriculum.

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Chapter 1 | 11  pages, introduction, chapter 2 | 7  pages, two problem-posing perspectives: accepting and challenging, chapter 3 | 12  pages, the first phase of problem posing: accepting, chapter 4 | 29  pages, the second phase of problem posing: “what-if-not”, chapter 5 | 41  pages, the “what-if-not” strategy in action, chapter 6 | 15  pages, some natural links between problem posing and problem solving, chapter 7 | 34  pages, writing for journals of editorial boards: student as author and critic, chapter 8 | 6  pages.

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