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The Deepest Math Programs


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Long time sporadic reader here, but created an account with the intention of giving a review of our experience with Elements of Mathematics on that thread.  This is a spin-off of that thread.  Somewhere in that thread math programs that are "deeper" than AOPS are alluded to.  What are the programs or textbooks that would be considered deeper than AOPS?  (I tried to search for old threads on the subject but didn't find anything useful.)

 

 

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8 hours ago, ChickaDeeDeeDee said:

Long time sporadic reader here, but created an account with the intention of giving a review of our experience with Elements of Mathematics on that thread.  This is a spin-off of that thread.  Somewhere in that thread math programs that are "deeper" than AOPS are alluded to.  What are the programs or textbooks that would be considered deeper than AOPS?  (I tried to search for old threads on the subject but didn't find anything useful.)

Hello! I'm the resident mathematician and AoPS teacher 🙂.

I think it depends what you mean by deep. Care to clarify what that word means to you? 🙂 

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15 hours ago, Not_a_Number said:

 

I think it depends what you mean by deep. Care to clarify what that word means to you? 🙂 

No, I don't.  I am attempting to clarify what the Accelerated Learner Board thinks is deeper.  I'm genuinely curious.  

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8 hours ago, ChickaDeeDeeDee said:

No, I don't.  I am attempting to clarify what the Accelerated Learner Board thinks is deeper.  I'm genuinely curious.  

Well, I don’t know if there’s agreement on the board, but I think of things as deeper when they involve more abstract structure.

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9 hours ago, ChickaDeeDeeDee said:

I am attempting to clarify what the Accelerated Learner Board thinks is deeper.

Though there were a few people in that thread who referred to going deep, I thought it was just one person who alluded to something deeper than AoPS.  Perhaps you should try PMing them to see what they were thinking about.

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You got me curious, so I looked at that thread, and it appears what you're asking stems from this comment by 4KookieKids: `But I also recognize it's probably about on par (or even less) than many "deeper" options.'

https://forums.welltrainedmind.com/topic/706996-elements-of-mathematics-foundations-reviews-x-posted/?tab=comments#comment-8900268

I don't know what is being referred to here - just thought I'd pinpoint the probable source of your question. I'm curious too.

 

 

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21 minutes ago, HomeForNow said:

You got me curious, so I looked at that thread, and it appears what you're asking stems from this comment by 4KookieKids: `But I also recognize it's probably about on par (or even less) than many "deeper" options.'

https://forums.welltrainedmind.com/topic/706996-elements-of-mathematics-foundations-reviews-x-posted/?tab=comments#comment-8900268

I don't know what is being referred to here - just thought I'd pinpoint the probable source of your question. I'm curious too.

 

 

The second "it" in that comment (The one you linked to) is a bit unclear.  I actually took the the "it" in the deeper sentence to be referring to the other program being discussed in the thread (EMF).  I read it as saying essentially, "Among all the programs that could be described as 'deeper', EMF is on par or less deep." -  Rather than the opposite implication that AoPS is the "on par or less-deep" program.  Hopefully @4KookieKids can stop by this thread to clarify. 

Edited by kirstenhill
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That the poster was talking about their own kid using AoPS because it was cheaper and the use of the word "but" made me think that the comment was referring to AoPS, but I can see how it might have actually been about EMF.

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My ds went from the AoPS sequence directly into independently studying Analysis with baby Rudin. So I don't know that you actually need anything deeper than AoPS if you are using it to its full potential. To use AoPS well you need to develop deep problem solving by doing the hardest problems without getting hints. This takes quite some time. 

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45 minutes ago, lewelma said:

My ds went from the AoPS sequence directly into independently studying Analysis with baby Rudin. So I don't know that you actually need anything deeper than AoPS if you are using it to its full potential. To use AoPS well you need to develop deep problem solving by doing the hardest problems without getting hints. This takes quite some time. 

But with how many hours of Olympiad problems along the way?

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5 hours ago, Not_a_Number said:

But with how many hours of Olympiad problems along the way?

Honestly, I'm not sure. Not many. He didn't study for the the first 2 IMOs. He had no teacher, tutor, coach, or math circle throughout middle and highschool.  What he did do was Every. Single. Problem. in. All. the. Textbooks -- on his own without any hints.  That is what I mean by using the program to its fullest. 

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1 minute ago, lewelma said:

Honestly, I'm not sure. Not many. He didn't study for the the first 2 IMOs. He had no teacher, tutor, coach, or math circle throughout middle and highschool.  What he did do was Every. Single. Problem. in. All. the. Textbooks -- on his own without any hints.  That is what I mean by using the program to its fullest. 

He didn't do any practice IMO problems at all? Whoa. None at all?

I don't think the textbooks come with problems anywhere near as hard as the IMO problems. At least, none of the textbooks I've taught out of do. I was trying to pull some combinatorics problems for a Zoom class on IMO problems, and I can barely find anything that would be appropriate for a math olympiad in the Intermediate C&P textbook.  Maaaaaybe the hardest problems would be number 1s. 

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I suppose in that case, I'm claiming that math olympiad problems are deeper than AoPS textbooks 😉 . But I don't know if that's exactly right -- they are also more haphazard, and they often don't introduce you to interesting new structures as much. Rather, they require a lot more creativity and "ah-ha" moments than the AoPS books. 

College math for mathematicians is obviously deeper than AoPS books, but I don't know if that's too far down the rabbit hole for most people. 

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On 3/19/2021 at 5:43 PM, lewelma said:

Honestly, I'm not sure. Not many. He didn't study for the the first 2 IMOs. He had no teacher, tutor, coach, or math circle throughout middle and highschool.  What he did do was Every. Single. Problem. in. All. the. Textbooks -- on his own without any hints.  That is what I mean by using the program to its fullest. 

AOPS done like that is deep enough and builds strong conceptual understanding. After using AOPs, it is not difficult to fill in the gaps on topics missed by AOPs very quickly using other sources.

We use the full AOPS program in the same way as @lewelma describes - my son works on every single problem without looking up the solutions manual and he does the same thing with Alcumus as well without hitting the "give up" button (setting it to the "insanely hard" setting). The only help he gets is when he spends more than a full day on a single problem, I help him talk through his problem solving strategy with me and discuss what he has done to solve the problem, why his approach did not work, whether there are alternate approaches, whether he made use of all pieces of information in the problem statement, whether there any similar problems featured in the book etc. He is generally able to solve the problem at the end of the discussion. This is not an approach that I recommend for anyone on a schedule or anyone very goal oriented: e.g. finishing certain topics in middle school, doing post-calculus work before 10th grade and such. 

I supplement the AOPS curriculum with problems from other sources to cover the "applied math" part because I find AOPS theoretical. Same with Test Prep as well. 

Interestingly, my son is into extracurriculars and is not a Mathlete, but, he was encouraged to take the AMC's by the lady who runs his math circle and he went in with no preparation and took both the AMC 10 & 12 this year and managed to scrape through to AIME qualification scores even though his only exposure to competition problems has been from doing every problem in his AOPS books and turning Alcumus topics Blue.

 

Edited by mathnerd
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1 hour ago, mathnerd said:

Is this a class that you are running for your own students?

Nah, someone hired me to run this for a math club 🙂 . 

 

51 minutes ago, mathnerd said:

Interestingly, my son is into extracurriculars and is not a Mathlete, but, he was encouraged to take the AMC's by the lady who runs his math circle and he went in with no preparation and took both the AMC 10 & 12 this year and managed to scrape through to AIME qualification scores even though his only exposure to competition problems has been from doing every problem in his AOPS books and turning Alcumus topics Blue.

I would expect AoPS to be more than enough math to do AMC problems 🙂 . It's probably also enough for most AIME problems. It's just not enough for serious proof-based Olympiads. 

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46 minutes ago, Not_a_Number said:

I would expect AoPS to be more than enough math to do AMC problems 🙂 . It's probably also enough for most AIME problems. It's just not enough for serious proof-based Olympiads. 

This was the point I was trying to make while also eating my dinner that I did not express it well. If a student worked their way through the entire AOPS offerings, they will end up with enough depth and enough problem solving background that they need just a little test prep (for academic testing like AP) and very little preparation for AMC (and the AIME) because these skills are built into their curriculum.

Where I live, test prep for AMC and AIME is a thriving cottage industry and thousands of kids spend a couple of hours in those classes each week and attend summer camps, winter camps, spring camps etc to prepare for competitions and I was trying to convey that just working diligently through AOPS was enough in our case.

As for proofs, the Geometry text book has a good introduction to proof writing, though more proof writing practice is a good to have before attempting proof-based competitions.

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3 hours ago, mathnerd said:

AOPS done like that is deep enough and builds strong conceptual understanding. After using AOPs, it is not difficult to fill in the gaps on topics missed by AOPs very quickly using other sources.

We use the full AOPS program in the same way as @lewelma describes - my son works on every single problem without looking up the solutions manual and he does the same thing with Alcumus as well without hitting the "give up" button (setting it to the "insanely hard" setting). The only help he gets is when he spends more than a full day on a single problem, I help him talk through his problem solving strategy with me and discuss what he has done to solve the problem, why his approach did not work, whether there are alternate approaches, whether did he not make use of any piece of information in the problem statement, whether there any similar problems featured in the book etc. He is generally able to solve the problem at the end of the discussion. This is not an approach that I recommend for anyone on a schedule or anyone very goal oriented: e.g. finishing certain topics in middle school, doing post-calculus work before 10th grade and such. 

I supplement the AOPS curriculum with problems from other sources to cover the "applied math" part because I find AOPS theoretical. Same with Test Prep as well. 

Interestingly, my son is into extracurriculars and is not a Mathlete, but, he was encouraged to take the AMC's by the lady who runs his math circle and he went in with no preparation and took both the AMC 10 & 12 this year and managed to scrape through to AIME qualification scores even though his only exposure to competition problems has been from doing every problem in his AOPS books and turning Alcumus topics Blue.

 

What do you supplement with if you don’t mind me asking?  I need more practice for ds but I find even alcumus moves him along faster.

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7 hours ago, daijobu said:

There are some USAMO, IMO and Putnam problems in AoPS textbooks.  

Not a ton, and not the hardest ones. At least, not the books I’ve used. Which books do you mean?

 

7 hours ago, mathnerd said:

This was the point I was trying to make while also eating my dinner that I did not express it well. If a student worked their way through the entire AOPS offerings, they will end up with enough depth and enough problem solving background that they need just a little test prep (for academic testing like AP) and very little preparation for AMC (and the AIME) because these skills are built into their curriculum.

Where I live, test prep for AMC and AIME is a thriving cottage industry and thousands of kids spend a couple of hours in those classes each week and attend summer camps, winter camps, spring camps etc to prepare for competitions and I was trying to convey that just working diligently through AOPS was enough in our case.

Yeah, test prep is a thriving industry here, too. I’m not arguing there’s anything magic about the AMC and AIME! They are relatively straightforward contests if you know the concepts well (although the concepts aren’t the standard high school stuff.)

 

7 hours ago, mathnerd said:

As for proofs, the Geometry text book has a good introduction to proof writing, though more proof writing practice is a good to have before attempting proof-based competitions.

It’s not the proof-writing that’s the issue, lol. It’s the caliber of the problems. They don’t require the creativity that hard Olympiad problems do.

Look, I teach out of the AoPS books and I was also a serious IMO and Putnam person. I like the AoPS books, but they don’t prepare you sufficiently for that type of contest. They don’t provide enough of the “ah-ha!” moment for that. When they are hard, it’s mostly because you haven’t internalized an idea, not because a totally new angle is required. Olympiad problems are a different kind of beast.

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I'm thinking that deeper usually comes when there is a professional or very knowledgeable person who is able to custom curate materials and go really, really deep because they have the understanding, with a student who has the interest. I'm guessing that most kids wouldn't want to do biology to the degree L has, for example-and it really only has been achievable because of some absolutely amazing mentors. 

 

AoPS was plenty deep enough for L who wanted math for science's sake. And it is way too much for my bonus kids-the one who wants to be an engineer and is the most mathy is also dyslexic and would have real trouble with the amount of reading and writing needed to puzzle out the problems (I might try EMF, though---does anyone know if it's screen reader friendly?). Even Beast had too many words. 

 

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3 minutes ago, Dmmetler said:

AoPS was plenty deep enough for L who wanted math for science's sake.

I guess the thing I'll say about "mathy math" is that while you don't need proof-writing for sciences, I do think lining up one's reasoning like one does with proofs is a good way to train up linear thought. Having now spent a lot of time lining up DD8's reasoning, mostly in verbal format, I'll say that I can absolutely see the difference when we talk about other subjects. It's much easier for her to spot fallacies. It's much easier for her to figure out what she knows and doesn't. 

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13 minutes ago, Not_a_Number said:

I guess the thing I'll say about "mathy math" is that while you don't need proof-writing for sciences, I do think lining up one's reasoning like one does with proofs is a good way to train up linear thought. Having now spent a lot of time lining up DD8's reasoning, mostly in verbal format, I'll say that I can absolutely see the difference when we talk about other subjects. It's much easier for her to spot fallacies. It's much easier for her to figure out what she knows and doesn't. 

I think the proof writing was a big help, and there was a lot of benefit to AoPS for a science-focused kid. And for Singapore before that, since Beast didn't exist, because Singapore also had the deep focus on understanding of concepts and explanation. But any interest in competition math, or deeper math here faded by about age 10-11, and a big part of that is that by that point, the skills were present to dive deeply into the areas that were of greater interest. 

 

 

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2 minutes ago, Dmmetler said:

I think the proof writing was a big help, and there was a lot of benefit to AoPS for a science-focused kid. And for Singapore before that, since Beast didn't exist, because Singapore also had the deep focus on understanding of concepts and explanation. But any interest in competition math, or deeper math here faded by about age 10-11, and a big part of that is that by that point, the skills were present to dive deeply into the areas that were of greater interest. 

Yeah, that makes sense to me. Obviously, it's best to use the analytical skills for stuff you're actually interested in!! 

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15 hours ago, Ausmumof3 said:

What do you supplement with if you don’t mind me asking?  I need more practice for ds but I find even alcumus moves him along faster.

What level is he working on?

And does he have the "insanely hard" setting turned on in Alcumus? For us, Alcumus started moving slower at the Intermediate Algebra and precalculus levels.

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9 hours ago, Not_a_Number said:

Look, I teach out of the AoPS books and I was also a serious IMO and Putnam person. I like the AoPS books, but they don’t prepare you sufficiently for that type of contest. They don’t provide enough of the “ah-ha!” moment for that. When they are hard, it’s mostly because you haven’t internalized an idea, not because a totally new angle is required. Olympiad problems are a different kind of beast.

So, other than working on the old IMO problem sets and the Olympiad style books (Andreescu et al.) what resources are available for someone with a long term goal of attempting the IMO and preparing for a shot at a medal? I have been following all the threads by lewelma and Kathy in the past about the preparation their kids did. I have a few books on proof writing and my son attends an university math circle where mathematicians teach proof writing techniques a few times a year. It would be great to hear first hand from someone with that kind of background. OP, sorry to derail your thread.

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Just now, mathnerd said:

So, other than working on the old IMO problem sets and the Olympiad style books (Andreescu et al.) what resources are available for someone with a long term goal of attempting the IMO and preparing for a shot at a medal? I have been following all the threads by lewelma and Kathy in the past about the preparation their kids did. I have a few books on proof writing and my son attends an university math circle where mathematicians teach proof writing techniques a few times a year. It would be great to hear first hand from someone with that kind of background. OP, sorry to derail your thread.

Honestly, that's what I did -- I did old IMO sets. And old CMO sets. And old USAMO sets 😛. There are probably more efficient things to do, but I was almost entirely self-taught -- I did olympiad problems instead of listening to my high school classes 😛. But I was in Canada, where there are fewer layers between the initial contests and the IMO team -- from what I've heard, in the US, the competition is fierce. 

When you say they teach proof-writing techniques... what does that entail? Is this aimed at kids who kind of don't know what a proof is, or is it aimed at problem-solving techniques? Because if you want to have a shot at serious math olympiads, you really have to be able to tell whether you have a proof or not. Otherwise, you can have a wonderful idea, and you'll still get a 1/7 for not justifying it. 

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27 minutes ago, mathnerd said:

What level is he working on?

And does he have the "insanely hard" setting turned on in Alcumus? For us, Alcumus started moving slower at the Intermediate Algebra and precalculus levels.

Yeah he’s on counting/probability.  I don’t have it set to insanely hard because I assumed that would just give him harder problems - I specifically want more easier problems building toward the harder ones.  I keep finding that alcumus has moved him on past where we’re at in the text book if that makes sense.  

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2 minutes ago, Not_a_Number said:

Honestly, that's what I did -- I did old IMO sets. And old CMO sets. And old USAMO sets 😛. There are probably more efficient things to do, but I was almost entirely self-taught -- I did olympiad problems instead of listening to my high school classes 😛. But I was in Canada, where there are fewer layers between the initial contests and the IMO team -- from what I've heard, in the US, the competition is fierce. 

When you say they teach proof-writing techniques... what does that entail? Is this aimed at kids who kind of don't know what a proof is, or is it aimed at problem-solving techniques? Because if you want to have a shot at serious math olympiads, you really have to be able to tell whether you have a proof or not. Otherwise, you can have a wonderful idea, and you'll still get a 1/7 for not justifying it. 

So, it seems that looking at old IMO sets is a good place to start.

I know that the competition in the US is fierce and that many kids moved on to other olympiads (biology, physics, chem, comp sci etc) because of how competitive USAMO has become - also the reason why there are coaches charging top dollar to tutor prospective USJMO/USAMO students. I am trying to avoid that whole rat race and see how far along I can go down this path with a DIY approach.

The university math circle is a good one (in a place that you might already be familiar with 😉) and the instructors use Martin Aigner's proof book as a reference to teach proof techniques and the classes run in blocks of 3 sessions and they teach proofs for one block and then move on to another topic and another instructor might eventually come back to proofs at another time and so on. It is a good introduction to proofs but I think that it takes much more practice to get to the IMO level of proof writing.

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Just now, Ausmumof3 said:

Yeah he’s on counting/probability.  I don’t have it set to insanely hard because I assumed that would just give him harder problems - I specifically want more easier problems building toward the harder ones.  I keep finding that alcumus has moved him on past where we’re at in the text book if that makes sense.  

I just set Alcumus to "insanely hard" and saw a random Intermediate Algebra problem it gave me. Man, I really did not want to do this problem 😛 . I mean, I knew how to do it, but it was really tedious and boring and involved intersecting intervals and factoring cubics... no thanks 😛 . 

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1 minute ago, Ausmumof3 said:

Yeah he’s on counting/probability.  I don’t have it set to insanely hard because I assumed that would just give him harder problems - I specifically want more easier problems building toward the harder ones.  I keep finding that alcumus has moved him on past where we’re at in the text book if that makes sense.  

You could ask him to work on the topic that is already blue. He does not have to move on to the next topic because he turned it blue. Just have him pick that same old topic and work on it and they have enough problems there to last your son a long time. 

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3 minutes ago, mathnerd said:

So, it seems that looking at old IMO sets is a good place to start.

I know that the competition in the US is fierce and that many kids moved on to other olympiads (biology, physics, chem, comp sci etc) because of how competitive USAMO has become - also the reason why there are coaches charging top dollar to tutor prospective USJMO/USAMO students. I am trying to avoid that whole rat race and see how far along I can go down this path with a DIY approach.

The university math circle is a good one (in a place that you might already be familiar with 😉) and the instructors use Martin Aigner's proof book as a reference to teach proof techniques and the classes run in blocks of 3 sessions and they teach proofs for one block and then move on to another topic and another instructor might eventually come back to proofs at another time and so on. It is a good introduction to proofs but I think that it takes much more practice to get to the IMO level of proof writing.

The problem with IMO problems isn't the proof-writing -- it's that they require you to solve hard problems. I loved solving them, because I actually loved being stuck on problems and pondering them until the magic "ah-ha!" moment when I'd figure it out. It was incredibly exciting for me. I haven't really felt the same way about any other math problems except for my actual research problems -- problems in classes aren't hard for the same reason. 

And yes, you do need to have a sense about what it means to truly solve a hard problem -- you need to be able to tell whether your reasoning is fully logical or not. 

Let me know if you need some help with this one. I'm pretty happy to troubleshoot for boardies 🙂 . I probably can't do it for hours a week, but if you need anyone to check hard proofs, I can do it. 

Have you taken a look at WOOT at AoPS? They tackle actual Olympiad problems, and it's definitely a harder level than the books. 

Edited by Not_a_Number
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1 minute ago, Not_a_Number said:

I just set Alcumus to "insanely hard" and saw a random Intermediate Algebra problem it gave me. Man, I really did not want to do this problem 😛 . I mean, I knew how to do it, but it was really tedious and boring and involved intersecting intervals and factoring cubics... no thanks 😛 . 

lol yeah that’s my feeling with casework counting etc 😂. Mostly not exactly interesting but requiring a lot of care ... I love that people here say they did it for fun 🤩 and bits of it are but others  ... 

Im not sure if we’ll do AOPS for intermediate algebra or not yes, this year is a bit of a trial run.  I may end up outsourcing next year.  

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3 minutes ago, mathnerd said:

You could ask him to work on the topic that is already blue. He does not have to move on to the next topic because he turned it blue. Just have him pick that same old topic and work on it and they have enough problems there to last your son a long time. 

Ok i think I need to have a play with settings because ours keeps automatically changing without me realising 

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Just now, Ausmumof3 said:

lol yeah that’s my feeling with casework counting etc 😂. Mostly not exactly interesting but requiring a lot of care ... I love that people here say they did it for fun 🤩 and bits of it are but others  ... 

Yeah, I would be bored to tears, lol. Like, again, I can DO it, but I don't WANT to 😛 . I much prefer problems with some aesthetic appeal. 

 

Just now, Ausmumof3 said:

Im not sure if we’ll do AOPS for intermediate algebra or not yes, this year is a bit of a trial run.  I may end up outsourcing next year.  

I'm teaching that class this summer, apparently... I haven't taught it for a while. Maybe I'll report back whether the book looks reasonable 😄 . 

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Just now, Not_a_Number said:

Yeah, I would be bored to tears, lol. Like, again, I can DO it, but I don't WANT to 😛 . I much prefer problems with some aesthetic appeal. 

 

I'm teaching that class this summer, apparently... I haven't taught it for a while. Maybe I'll report back whether the book looks reasonable 😄 . 

Sounds good!

next year for me means next Feb so I’ll have plenty of time to work it out.

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Just now, Not_a_Number said:

Let me know if you need some help with this one. I'm pretty happy to troubleshoot for boardies 🙂 . I probably can't do it for hours a week, but if you need anyone to check hard proofs, I can do it. 

Have you taken a look at WOOT at AoPS? They tackle actual Olympiad problems, and it's definitely a harder level than the books. 

Thanks a lot for the offer. We are not at that point yet, but I will ask if I need help 🙂 

I am looking at WOOT, Awesome math, IDEA Math, Alphastar camps to see what they have to offer. I always thought that I would pick WOOT because it was the only online option available pre-covid and it is a tradition amongst IMO candidates, anyway. But now it seems everything is online. There is also a local class that teaches proof writing and I am considering it.

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5 minutes ago, Ausmumof3 said:

Ok i think I need to have a play with settings because ours keeps automatically changing without me realising 

If it automatically changes to the next topic after mastery, he has to go back and pick the old topic manually - for e.g. have him work on 5 problems on the topic that is already blue.

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1 hour ago, mathnerd said:

So, it seems that looking at old IMO sets is a good place to start.

I know that the competition in the US is fierce and that many kids moved on to other olympiads (biology, physics, chem, comp sci etc) because of how competitive USAMO has become - also the reason why there are coaches charging top dollar to tutor prospective USJMO/USAMO students. I am trying to avoid that whole rat race and see how far along I can go down this path with a DIY approach.

The university math circle is a good one (in a place that you might already be familiar with 😉) and the instructors use Martin Aigner's proof book as a reference to teach proof techniques and the classes run in blocks of 3 sessions and they teach proofs for one block and then move on to another topic and another instructor might eventually come back to proofs at another time and so on. It is a good introduction to proofs but I think that it takes much more practice to get to the IMO level of proof writing.

Is this the Aigner proof book? https://www.amazon.com/Proofs-BOOK-Martin-Aigner/dp/3642008550

Does it require Calculus and beyond level math skills or could a student who is finishing up Geometry and Alg 2 get something out of it? DS13 is highly disappointed in the lack of proofs in his geometry class and wants to explore proofs "for fun" - It actually hadn't occurred to me there might be whole books just about that topic. 

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1 minute ago, kirstenhill said:

Is this the Aigner proof book? https://www.amazon.com/Proofs-BOOK-Martin-Aigner/dp/3642008550

Does it require Calculus and beyond level math skills or could a student who is finishing up Geometry and Alg 2 get something out of it? DS13 is highly disappointed in the lack of proofs in his geometry class and wants to explore proofs "for fun" - It actually hadn't occurred to me there might be whole books just about that topic. 

I don't know about this specific book, but in general, no high school Olympiad require any calculus. If he wants to do proofs for fun, he can look up old USAMO/CMO/IMO/APMO problems and try to see if he can come up with solutions.

I also really liked this book as a teen: 

https://www.amazon.com/USSR-Olympiad-Problem-Book-Mathematics/dp/0486277097/ 

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23 hours ago, Not_a_Number said:

He didn't do any practice IMO problems at all? Whoa. None at all?

I don't think the textbooks come with problems anywhere near as hard as the IMO problems. At least, none of the textbooks I've taught out of do. I was trying to pull some combinatorics problems for a Zoom class on IMO problems, and I can barely find anything that would be appropriate for a math olympiad in the Intermediate C&P textbook.  Maaaaaybe the hardest problems would be number 1s. 

He was not really into competing. His math competitions were just an offshoot of his love for math. In addition, there were no other opportunities to be with mathy kids unless he did them. But it was *how* he used AoPS that was effective. Because he went so slowly and independently in the introductory series, and did every single problem without hints, he developed deep problem solving beyond just the content of the books. To give you a feel for how far AoPS got him, he solved this problem for his first camp selection problems, having only gotten through the 4 introductory AoPS books, and got full points for his proof. At this time, he was *completely* self taught. He had refused all help/teaching in math since he was 7.5 and had not started the AoPS classes at that point. So the only knowledge he had when he solved this problem was what was in those books, and this level problem was clearly not. Those books taught him deep problem solving because he used them to their fullest. 

In a sequence of positive integers, an inversion is a pair of positions such that the element in the position to the left is greater than the element in the position to the right. For instance the sequence 2,5,3,1,3 has five inversions, between the first and fourth positions, the second and all later positions, and between the third and fourth positions. What is the largest possible number of inversions in a  sequence of positive integers whose sum is 2014?

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1 hour ago, kirstenhill said:

Is this the Aigner proof book? https://www.amazon.com/Proofs-BOOK-Martin-Aigner/dp/3642008550

Does it require Calculus and beyond level math skills or could a student who is finishing up Geometry and Alg 2 get something out of it? DS13 is highly disappointed in the lack of proofs in his geometry class and wants to explore proofs "for fun" - It actually hadn't occurred to me there might be whole books just about that topic. 

my son is going through Peter Eccles book on proofs and is really enjoying it. It does not involve Calculus.

An Introduction to Mathematical Reasoning (Numbers, Sets and Functions) https://www.amazon.com/dp/0521597188/ref=cm_sw_r_cp_api_glt_fabc_186GE500676CN0AJJ80Y

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5 minutes ago, lewelma said:

He was not really into competing. His math competitions were just an offshoot of his love for math. In addition, there were no other opportunities to be with mathy kids unless he did them. But it was *how* he used AoPS that was effective. Because he went so slowly and independently in the introductory series, and did every single problem without hints, he developed deep problem solving beyond just the content of the books. To give you a feel for how far AoPS got him, he solved this problem for his first camp selection problems, having only gotten through the 4 introductory AoPS books, and got full points for his proof. At this time, he was *completely* self taught. He had refused all help/teaching in math since he was 7.5 and had not started the AoPS classes at that point. So the only knowledge he had when he solved this problem was what was in those books, and this level problem was clearly not. Those books taught him deep problem solving because he used them to their fullest. 

Well, I'm competitive, but I didn't do Olympiad problems because I was competitive -- I did them because they were fun. 

 

5 minutes ago, lewelma said:

In a sequence of positive integers, an inversion is a pair of positions such that the element in the position to the left is greater than the element in the position to the right. For instance the sequence 2,5,3,1,3 has five inversions, between the first and fourth positions, the second and all later positions, and between the third and fourth positions. What is the largest possible number of inversions in a  sequence of positive integers whose sum is 2014?

Cute. I think I know how to do that, but I'm going to check if I got this one right later -- gotta go for my "driving date," lol.

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23 minutes ago, Not_a_Number said:

Well, I'm competitive, but I didn't do Olympiad problems because I was competitive -- I did them because they were fun. 

I do think he did Olympiad problems and I know he enjoyed them, but I think he spent more time working on textbooks like baby Rudin. I know that he took at Grad class in combinatorics at MIT that had 5 prereqs that he didn't officially have, but yet somehow he had studied the content in high school. I just don't remember the resources he used as he often used online content.  So he improved his knowledge of combinatorics but I'm not sure he did that through Olympiad problems until the last year.

I'm glad you liked the problem! However, my point was that he solved that with only Introductory AoPS text knowledge. He could do it because of the deep problem solving he developed by how he used the books. I think that a LOT of kids that use AoPS don't use it to its full potential, especially kids taking the classes which go pretty fast and have way fewer problems. The kids also often get hints or work together, which undermines the development of their problem solving. So to answer the OP's question, AoPS is only as deep as how you use it.

Edited by lewelma
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1 hour ago, lewelma said:

I do think he did Olympiad problems and I know he enjoyed them, but I think he spent more time working on textbooks like baby Rudin. I know that he took at Grad class in combinatorics at MIT that had 5 prereqs that he didn't officially have, but yet somehow he had studied the content in high school. I just don't remember the resources he used as he often used online content.  So he improved his knowledge of combinatorics but I'm not sure he did that through Olympiad problems until the last year.

I'm glad you liked the problem! However, my point was that he solved that with only Introductory AoPS text knowledge. He could do it because of the deep problem solving he developed by how he used the books. I think that a LOT of kids that use AoPS don't use it to its full potential, especially kids taking the classes which go pretty fast and have way fewer problems. The kids also often get hints or work together, which undermines the development of their problem solving. So to answer the OP's question, AoPS is only as deep as how you use it.

I think I’m not communicating what I mean very well here. So, for instance, this problem is absolutely the kind of problem you should be able to do after AoPS. It requires discovering cool patterns, playing around, and otherwise being fearless with math. It’s true that you may not know the exact things this problem needs after AoPS, but I agree that AoPS would be enough, if used correctly.

What this problem and AoPS both don’t have is a moment of insight. The hard, really beautiful Olympiad problems have a serious “ah-ha” moment that this kind of problem doesn’t. So if you want that kind of experience, you’ll want to do harder Olympiad problems on top of AoPS. It’s a different kind of thing.

Edited by Not_a_Number
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Do you suppose it's true that the students who achieve USAMO and IMO level are not the ones who take classes and attend camps and work with tutors, but rather are the ones who study independently?  

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34 minutes ago, daijobu said:

Do you suppose it's true that the students who achieve USAMO and IMO level are not the ones who take classes and attend camps and work with tutors, but rather are the ones who study independently?  

The kids that qualify for USAMO that attend my son’s math circle have either gone through all the AoPS classes and are currently taking Woot classes or they have gone through all the Awesome math classes. 

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1 hour ago, daijobu said:

Do you suppose it's true that the students who achieve USAMO and IMO level are not the ones who take classes and attend camps and work with tutors, but rather are the ones who study independently?  

No, I think that kids with tutors and mentors that can direct their studies for IMO success would be far superior to what my son did to prepare. We couldn't find anyone to help, and I certainly couldn't, so he prepared as best as he could on his own. 

However, the self-teaching aspect of his approach was/is hugely important to his current success. He learned not only how to nut it out without help, he learned to struggle and not fear it. The day that an IMO perfect scorer knocked on my son's door to get help with his homework was a special day indeed. It showed him that his knowledge was different from someone who could well on the IMO, but it was not lesser. Basically, he couldn't do IMO problems in the time frame allotted, but if the questions were harder than IMO questions (questions that could take all day to solve), he was better than the other IMO kids. 

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