Intuition Math - Please Help

[Mukmuk]

Ds is nearing the end of his AoPS Algebra 2 class, and it's been wonderful for him. He talks math all the time, comparing it to beauty, and thinks of it as the key to humanity. He's doing well in the class.

I hired a tutor to help with the homework because Ds had difficulty initially when he started the class. It's worked out well - Ds thinks best when he can bounce ideas and talk through them with someone, and the chemistry with his tutor has been great. After each session, Ds continues solving the other problems (always lots of homework!) over the next few days by himself, often looking up the transcript or the textbook. The tutor is not familiar with AoPS, so it's a great partnership if anything- I don't want Ds to be spoonfed.

Anyway, the issue arises when Ds is unable to crack the problem. The tutor attempts to teach Ds, but he doesn't seem interested to listen. Instead, he intuits and often cracks the problem. "How did you intuit that!?" is a phrase I hear a lot from the tutor. It's happened so often, and to be honest, It made me rather proud about Ds' abilities. But it just dawned on me- some intuition is great, sure. But will this impact his learning at higher levels of math? Is intuition important? Ds has always had difficulty taking anything at face value, which I believe is a strength (okay. It can be a pain. But it's mostly a strength). His tutor thinks the creativity is wonderful, but is concerned that he's not using the tools that are available to him. I gather that the basis for his intuition can be rather shakey sometimes too. He thinks Ds should learn more math theory. I've let Ds handle the demands of the class and he's doing a superb job. But he may need some gentle steering?

Any advice here is appreciated!

[Mike in SA]

Intuition is a common trait among mathematicians.  Depending on your son's age, using intuition may be nothing to be concerned about.  It is just as appropriate to work backwards from an answer as it is to start from the beginning, provided that you eventually can connect the dots between the two.

 

If your son isn't ever solving the problems, but is just coming up with an answer without a reason, then there is some work to do.  If he is solving complex problems this way, then he already has the tools needed, but may not yet be able to verbalize them.  He will need to learn to show how the result can be achieved.  He can do so by working forwards or backwards, but he should be able to show a process.

 

If the concern is about his lack of desire to systematically derive a result without using intuition, well, that may not matter.  As a mathematician, eventually building a large bag of tricks will become important, but can you say that he will definitely be a mathematician?  Is he even within two years of college?  Mathematical maturity takes time...

[Woodland Mist Academy]

Ds is nearing the end of his AoPS Algebra 2 class, and it's been wonderful for him. He talks math all the time, comparing it to beauty, and thinks of it as the key to humanity. He's doing well in the class.

 

I hired a tutor to help with the homework because Ds had difficulty initially when he started the class. It's worked out well - Ds thinks best when he can bounce ideas and talk through them with someone, and the chemistry with his tutor has been great. After each session, Ds continues solving the other problems (always lots of homework!) over the next few days by himself, often looking up the transcript or the textbook. The tutor is not familiar with AoPS, so it's a great partnership if anything- I don't want Ds to be spoonfed.

 

Anyway, the issue arises when Ds is unable to crack the problem. The tutor attempts to teach Ds, but he doesn't seem interested to listen. Instead, he intuits and often cracks the problem. "How did you intuit that!?" is a phrase I hear a lot from the tutor. It's happened so often, and to be honest, It made me rather proud about Ds' abilities. But it just dawned on me- some intuition is great, sure. But will this impact his learning at higher levels of math? Is intuition important? Ds has always had difficulty taking anything at face value, which I believe is a strength (okay. It can be a pain. But it's mostly a strength). His tutor thinks the creativity is wonderful, but is concerned that he's not using the tools that are available to him. I gather that the basis for his intuition can be rather shakey sometimes too. He thinks Ds should learn more math theory. I've let Ds handle the demands of the class and he's doing a superb job. But he may need some gentle steering?

 

Any advice here is appreciated!

 

Thank you so much for posting! I am not a math mom, but I can relate to your post. You have put into words what I am experiencing with my daughter. Only instead of not being interesting in listening to some gentle teaching about a tough problem, my dd becomes furious.

 

She has become upset at times because she says that I can't see the math the way she does. Sometimes that's true. She sees things I don't - sometimes things that are hard for me to wrap my mind around. (Not that it takes much for me to feel like that... ;) ) There have even been times when two or three solutions were offered in the AoPS solutions manual, but she approached it a different way. Once she explained it, her way made the most sense. (Of course, maybe there was some reason we don't know about that made it less ideal...)  There have also been times when she can't explain at all how she is getting from one point to another but she is getting the correct answer in the end. She just sort of skips 3 steps without understanding how to express what she is doing. This is incredibly frustrating to her.

 

Similar to what your son's tutor mentioned, Dh and I have concerns about the wonderful creativity, but lack of tool use. There are other concerns, but that's a big one.

 

So, no advice.... ;)

Thanks for your post, though. I have been rather concerned about my daughter, so I'll be watching the replies.

 

[Luckymama]

The situation may improve with maturity, both age-wise and math-wise.

 

Dd does use a lot of intuition in her solutions. However, since she is forced by AoPS to write proofs for so many problems, she has gotten much better at explaining individual steps in non-proof solutions. She also has improved her ability to attack a tricky problem from different angles when she gets stuck on a first attempt.

 

She still does many problems in her head----but then again, so do I :)

[Mike in SA]

Thank you so much for posting! I am not a math mom, but I can relate to your post. You have put into words what I am experiencing with my daughter. Only instead of not being interesting in listening to some gentle teaching about a tough problem, my dd becomes furious.

 

She has become upset at times because she says that I can't see the math the way she does. Sometimes that's true. She sees things I don't - sometimes things that are hard for me to wrap my mind around. (Not that it takes much for me to feel like that... ;) ) There have even been times when two or three solutions were offered in the AoPS solutions manual, but she approached it a different way. Once she explained it, her way made the most sense. (Of course, maybe there was some reason we don't know about that made it less ideal...)  There have also been times when she can't explain at all how she is getting from one point to another but she is getting the correct answer in the end. She just sort of skips 3 steps without understanding how to express what she is doing. This is incredibly frustrating to her.

 

Similar to what your son's tutor mentioned, Dh and I have concerns about the wonderful creativity, but lack of tool use. There are other concerns, but that's a big one.

 

So, no advice.... ;)

Thanks for your post, though. I have been rather concerned about my daughter, so I'll be watching the replies.

 

 

Your experience with AoPS is familiar.  Sometimes their solutions are NOT the most elegant or accessible.  As long as your solution is well-connected, it's fine.

 

Learning to make a well-constructed solution is what is important.  Skipping three steps is dangerous because it can mask a deficiency in reasoning.  We're working on the same issue with our younger son, but he has lots of time to grow into that ability.  At 8, he doesn't have the maturity to assess whether his arguments are well-constructed, so we tolerate a few leaps here and there.  It will all come together in due time.

[wapiti]

I can't comment on mathematicians at all, but intuition I know.  It's my bread-and-butter, how I did well in school, on standardized tests, analyzing the big picture (I used to write briefs backwards all the time and then rearrange them, LOL), how I make difficult decisions about issues with my kids, etc.  I think it's very much a VSL thing.

 

I agree with the importance of being able to connect the dots, backwards, forwards, whichever way works.  If he can "prove" it by talking through the reasoning, even if he had to find the answer first and work backwards to make the connections, I'd be happy with that.

 

FWIW, dd13 does this all the time, and it's a downside of short answer problems.  In both Alcumus and the on-line short-answer challenge problems, sometimes she will guess intuitively and be correct even though she hasn't explicitly connected the dots.  (I know because she will tell me.)  If she can't explicitly connect the dots, that means to me that she hasn't learned what she is supposed to learn.  Usually, it's not so much that she can't, but hasn't taken sufficient time and effort to do it due to not having fully absorbed something, didn't do the section in the book first, was slacking, etc.  Taking the time and effort at such juncture will save her in the long run.

 

I think math intuition may be a huge asset on the SAT.

[Mukmuk]

Thanks so much for your replies! These are good ideas. Admittedly, I have a hands off approach, because like WMA, my son used to be so furious when I corrected him. He's so much happier with professionals, not a fellow learner. My pace and lack of understanding frustrated him (Ack, WMA, I *know*. :grouphug:).

 

Mike, my son is 11. He does have reasons for coming up with the answer, but not what the chapter is explicitly teaching. I just took a look at his weekly AOPS scores. The last problem of the challenge set involves proof writing and is an area that the tutor never seems to get to work on with Ds (an hour a week is very short!). Ds often hunkers down with a grim look to do it, but has been happy with himself lately. He's now hitting 7s for technical writing (it was 0 to 1 in the initial weeks). His writing style though, isn't so hot at 0.3-0.8 (I don't know what the total score here is, but I'm assuming it's also 7). Does anyone know what "style" is? I will have the tutor help him in this area regardless.

 

AOPS has a two week break. I will ask the tutor to turn on his math charms with Ds if we can find time to stick in a slot. There is nothing Ds likes better than a lively math discussion, so it could be a slight shift in the discussion dynamic that the tutor is not picking up on. Ds is highly sensitive this way.

[Woodland Mist Academy]

Thanks so much for your replies! These are good ideas. Admittedly, I have a hands off approach, because like WMA, my son used to be so furious when I corrected him. He's so much happier with professionals, not a fellow learner. My pace and lack of understanding frustrated him (Ack, WMA, I *know*. :grouphug:).

 

 

 

The bolded is where we are. The plan is to try to see her safely through Algebra 1, then turn math over to the professionals. Thanks for the :grouphug: . It's been a rough few weeks....

[Mukmuk]

... The plan is to try to see her safely through Algebra 1, then turn math over to the professionals. ....

That's what happened here. Quark persuaded me to l.e.t.g.o. We've never looked back since! It will be fine, WMA. Hugs.

[Luckymama]

Mukmuk, I believe the writing style score goes from 0.1 to 1.0, so the 0.8 would be pretty darn good.

 

Like others, I will be leaving dd's math to professionals. She'll be starting the calculus sequence at the university in the fall. (Biking distance, baby!) She has decided that she wants an in-person teacher and classmates after this year.

[Mukmuk]

That's good to know, Luckymama!

 

And great to hear about your daughter's in-person class. From my very limited experience with math circles, many math people just love other people to share ideas with.

[quark]

Mukmuk, :grouphug: ! Your DS is an amazing young man.

The situation may improve with maturity, both age-wise and math-wise.

Dd does use a lot of intuition in her solutions. However, since she is forced by AoPS to write proofs for so many problems, she has gotten much better at explaining individual steps in non-proof solutions. She also has improved her ability to attack a tricky problem from different angles when she gets stuck on a first attempt.

 

This. We noticed so much more improvement with math maturity as he tackled more and more problems. Not all were fully solved and some were just abandoned but he worked consistently enough daily to have exposure to many, many problems. And he also verbalizes his steps apart from only writing them down and that seems to be helping him see them clearer...maybe something to do with seeing, doing, hearing it spoken aloud, many senses coming together?

 

I remember how terribly frustrating some of his AoPS problems were (and those fast-paced classes though immensely interesting were just so intense). It's like he knew something needed to be done (intuitively) but didn't understand how to flesh it out. Or knew the answer but struggled on how to break it down in steps. His solution would be correct and the key would say something else and I didn't have enough expertise myself to assure him he was right or alert him that he was wrong. He was working on math like this in addition to what the tutor was giving him and the tutor just didn't have time to help him with all the extra things DS was doing. Sometimes we just put it aside and other times, I made note of what was hard/ conflicting/ intutuitve-but-unsure and wrote it in the book for him to come back to later. But something told me to trust him and my gut (intuition again!) to keep moving ahead and that eventually something will spark some thought somewhere (or more maturity will kick in) and it will all right.

 

Knowing that we had plenty of time helped so much! He and I could make many mistakes over and over again and there was no rush to learn from them just yet...mistakes are wonderful. Struggle is wonderful because that's usually the lowest point you are at.

 

I can't comment on mathematicians at all, but intuition I know.  It's my bread-and-butter, how I did well in school, on standardized tests, analyzing the big picture (I used to write briefs backwards all the time and then rearrange them, LOL), how I make difficult decisions about issues with my kids, etc.  I think it's very much a VSL thing.

 

I agree with the importance of being able to connect the dots, backwards, forwards, whichever way works.  If he can "prove" it by talking through the reasoning, even if he had to find the answer first and work backwards to make the connections, I'd be happy with that.

 

[...]

 

I think math intuition may be a huge asset on the SAT.

 

This is me. I actually used to solve some of my higher math homework problems backwards by looking at the key first. Those back of the text keys didn't show you the steps...they just told you the answer and I used to intuit it backwards because it made more sense to me to know the answer first and I started to understand math better this way. My teachers had to handle 35-40 students in each class...they didn't have time to explain everything. I didn't go on to major in math so I can't tell how beneficial this method is in the long term but it did a ton of good for my confidence in math...enough to approach the teacher periodically during office hours for help (although I used to be so fearful of coming off as being stupid). I recently took a college placement exam (multiple choice) and same thing...I didn't have the key but intuition told me how to solve at least a couple of the questions that stumped me.

 

Yes, I believe it's an asset in the SAT too.

 

Thanks so much for your replies! These are good ideas. Admittedly, I have a hands off approach, because like WMA, my son used to be so furious when I corrected him. He's so much happier with professionals, not a fellow learner. My pace and lack of understanding frustrated him (Ack, WMA, I *know*. :grouphug:).

Mike, my son is 11. He does have reasons for coming up with the answer, but not what the book is explicitly teaching. I just took a look at his weekly AOPS scores. The last problem of the challenge set involves proof writing and is an area that the tutor never seems to get to work on with Ds (an hour a week is very short!). Ds often hunkers down with a grim look to do it, but has been happy with himself lately. He's now hitting 7s for technical writing (it was 0 to 1 in the initial weeks). His writing style though, isn't so hot at 0.3-0.8 (I don't know what the total score here is, but I'm assuming it's also 7). Does anyone know what "style" is? I will have the tutor help him in this area regardless.

AOPS has a two week break. I will ask the tutor to turn on his math charms with Ds if we can find time to stick in a slot. There is nothing Ds likes better than a lively math discussion, so it could be a slight shift in the discussion dynamic that the tutor is not picking up on. Ds is highly sensitive this way.

 

It's sad but I've learned to trust that my son needs other teachers and that I slow him down when I try to learn along. It's lovely to learn along, but some kids just need that other person...but him knowing you are learning along? That's fantastic! You are a role model for him. He will remember that his mom was there...and similarly, there are many things you learn faster than he does...and a good humbling experience I think for them to see this too. :D

 

If you are using the same tutor we were, I suggest talking to him. I used to give him a call every 4-6 months or so to report what I was observing at home and what I felt needed some extra attention.

[lewelma]

I'm not a math mom, just have a math son, and he did not do decent proofs until age 13.  He did the Intro Algebra by him self, so I don't even know if he wrote proofs. Like others, ds refused help.  But when he hit geometry, I decided that I *would* get involved and make sure he understood the idea of a proof.  And he. did. not!  We started with the paragraph form that AoPS teaches, and I soon realised that he was not clearly linking each assertion with a reason -- it was just too muddled.  So we switched to the column proof for a few months until he really understood that every assertion needed a reason.  That was age 11-12.  It was not until this past year (8th grade 13-14) that ds got the all important feedback on his proof writing.  One year later it is 7s and .9s all the way.  He now sees when he has a hole that needs to be filled, and he also understands what needs to be included and what does not.  He has taken 4 AoPS classes this year (inter number theory, algebra 3, inter counting, and Oly geo (on now)), so around 120 graded proofs.  He trusts and respects the graders, and takes their comments to heart. This feedback has been critical.

 

To me it sounds like you are on the right track, and you just need to be patient as the maturity is developed and more problems are experienced/solved.  Your ds sounds so dedicated!

 

Ruth in NZ

[Mukmuk]

Thanks for the btdt, Quark and Ruth! The best lessons are the real life ones. I really appreciate it.

 

Ruth, it's good to know about proofs. I never did any and don't understand it. I'm glad that Ds seems to have gotten the hang of it for the recent chapters, but this changes every week. Definitely, he needs to learn it in a more systematic fashion.

 

This week, the intuition-meter seems to be jumping very high for Arithmetic Sequences, which is why I hear more talk than usual. The tutor wants him to learn the algorithm, while Ds instantly sees this as a puzzle - fun, fun, fun, with ideas all over the place. Some assumptions are rather shakey, but it seems to work. I hope I'm accurate in describing the situation as I'm peering from afar. I'm not sure which is the best way to learn this. On my own, my instinct would have been to let Ds run on his program - he can back track later when he runs into problems at a higher level. But I'm advising Ds to talk it through with the tutor at the next session. Without the immediacy of the problems to solve, he's not reluctant.

[Arcadia]

 

This week, the intuition-meter seems to be jumping very high for Arithmetic Sequences, which is why I hear more talk than usual. The tutor wants him to learn the algorithm, while Ds instantly sees this as a puzzle - fun, fun, fun, with ideas all over the place. .

 

Arithmetic sequence does seem like a puzzle. He will probably get into a more systematic approach when he is doing arithmetic sequence with proof by induction. e.g. from MEP http://www.cimt.plymouth.ac.uk/projects/mepres/alevel/fpure_ch6.pdf

 

If your son sees math as puzzles, can the tutor let him derive the algorithms rather than learn them?

[Mukmuk]

Arcadia, knowing AOPS, I bet Ds has derived the algorithm already. He always loves the derivation. But each new question brings out some different angle. I can safely say that Ds doesn't feel the need to practice what he just learnt. He keeps reinventing the wheel. :huh:

[Arcadia]

 I can safely say that Ds doesn't feel the need to practice what he just learnt. He keeps reinventing the wheel. :huh:

 

Just treat it as a mental exercise as long as he knows that reinventing the wheel would cost him time in a timed test.  How my just turned 10 behave in daily work is different from when he is doing a timed test. Sometimes I tell my older that he can either just get it done or reinvent the wheel umpteen time while I read a book.

 

Your avatar reminds me of mochi (its dinnertime for me)

 

ETA;

I think its the age of independence for my older.  The 'I want to do it my way' stage while they assess their own strengths and weakness..

[lewelma]

. He keeps reinventing the wheel. :huh:

 

Mukmuk, my ds took almost 3 years to get through the Intro Algebra book.  I'm sure he reinvented the wheel over and over again. I know for a fact having looking at his notebooks, that he did waaaaaay more steps with each problem than he needed to.  In fact, I finally forced him to drop some steps, because it was becoming ridiculous.  But in the end, he needed to do it his way, and we had the time, as do you.  So don't worry if his approach is somewhat inefficient.

 

I will also add that AoPS classes encourage both intuitive thinking and proof writing.  Those short answers don't require a proof, if you can get it right with no work or using some weird way, you still get the points.  So kind of a mixture of speed-based intuition mixed with really pickily-graded proofs to force to you think it all out.  Kids would rebel if you made them write every single problem out at a proof level.  I think that AoPS knows what they are doing.

 

However, I will also say that my ds did not switch to the classes until he had done 4.5 years of AoPS - 3 for Intro algebra and 1.5 for geometry.  In those years he could really delve into the material, with no sense of assignments being due or sense that you only have to do what is assigned.  AoPS classes move FAST, and I'm not sure that doing a steady diet of them at the expense of self-directed study is a good idea.  My ds knows some kids on the board who have done the classes from the start, and raced through the AoPS material because of it.  But what has happened is that they don't have the math-maturity to deal with the harder material -  intermediate counting comes to mind.  So you may want to consider, taking a few months off here and there and having him work through the challengers in the book he has just finished. 

 

Ruth in NZ

[Mukmuk]

... My ds knows some kids on the board who have done the classes from the start, and raced through the AoPS material because of it. But what has happened is that they don't have the math-maturity to deal with the harder material - intermediate counting comes to mind. So you may want to consider, taking a few months off here and there and having him work through the challengers in the book he has just finished.

 

Ruth in NZ

Ruth, Thank you so much for this priceless insight. My son loves speed, and AOPS classes cater precisely to this inclination. Sometimes, I wonder how to slow him down because he misses the details this way. Getting him to do the challenge problems makes sense. I was worried about thoroughness - how can 6 Alcumus questions lead to a green?!? He's already planned his classes for next year (3!), but I will stretch out the intervening periods.

 

Arcadia and Ruth, I hammed it up a bit on Ds reinventing the wheel. Truth is, he does it a lot, and because we homeschool, we have no time pressure. He can't help himself- it's part of the dyslexic profile. I always think of it as him looking at things from yet another perspective. :D

[lewelma]

Ruth, Thank you so much for this priceless insight. My son loves speed, and AOPS classes cater precisely to this inclination. Sometimes, I wonder how to slow him down because he misses the details this way. Getting him to do the challenge problems makes sense. I was worried about thoroughness - how can 6 Alcumus questions lead to a green?!? He's already planned his classes for next year (3!), but I will stretch out the intervening periods.

 

 

Oh, ds sees quite a few of these kids in his classes.  What happens is that they can understand the material, but they just don't have the insight to solve the problems independently.  But the moment they get a hint, that little hint *is* the insight that they need to develop.  The material just needs time to sink in, and the best way to do this is to do *more* problems.  And work on the problems without help.  Doing 1 problem in 2 hours on your own is often more useful in developing math maturity than doing 6 problems in 2 hours with little hints and help.  If you don't take time to consolidate your knowledge and work hard problems with persistence and ingenuity then you are likely to move into the harder material without the problem solving skills that you really need.  Seems better to me to develop the problem solving skill while the material is still pretty straight forward.

 

There has been more than one kid in his classes who have decided to *retake* all the classes because the first time through they just didn't get as much out of them as they would have if they were older or had more math maturity.  Now these kids were young (like age 11) but they realised that they were too reliant on others helping them through the material; they basically could not do any of the proofs without some serious hints.  The problem was that they raced through the classes and finished them in just a couple of years.

 

Have him do *all* the challengers from every book he completes a class for, and do these challengers without peaking at the solutions. :001_smile:  This should take him months for each book (at least it did for my ds).

 

ds just spent 22 hours this week on TWO geometry problems for the Olympiad Geometry class.  So tell your ds that it really is ok to struggle and not see the answer right away!

[Woodland Mist Academy]

Ruth,

Could you elaborate on this? I was tossing around the idea of the classes, but this concerns me. Are there no checks and balances with the classes? How is it possible to pass them not truly knowing the math?

 

I was thinking since dd seems to understand what she's learned better than I gave her credit for (she is going quickly though Foerster) that maybe the feedback from the classes might be enough to help order her thinking. Your last post gives me pause. Did the instructors not alert the parents the student was not ready to proceed so quickly? So I wouldn't be able to trust the classes totally to the instructor's judgment? I would still need to test, etc?

 

Thanks.

 

Eta: Yes, I know I had this all sorted out, but that was yesterday.... Then I reread this thread.... ;)

[Woodland Mist Academy]

Actually, I would even need to assign and check extra work beyond the class, correct? Does this not seem a poorly designed class? Or is it set up to just be a partial class with the parent or student knowing what else needs to be done...hmmm....

[Mike in SA]

Oh, ds sees quite a few of these kids in his classes.  What happens is that they can understand the material, but they just don't have the insight to solve the problems independently.  But the moment they get a hint, that little hint *is* the insight that they need to develop.  The material just needs time to sink in, and the best way to do this is to do *more* problems.  And work on the problems without help.  Doing 1 problem in 2 hours on your own is often more useful in developing math maturity than doing 6 problems in 2 hours with little hints and help.  If you don't take time to consolidate your knowledge and work hard problems with persistence and ingenuity then you are likely to move into the harder material without the problem solving skills that you really need.  Seems better to me to develop the problem solving skill while the material is still pretty straight forward.

 

There has been more than one kid in his classes who have decided to *retake* all the classes because the first time through they just didn't get as much out of them as they would have if they were older or had more math maturity.  Now these kids were young (like age 11) but they realised that they were too reliant on others helping them through the material; they basically could not do any of the proofs without some serious hints.  The problem was that they raced through the classes and finished them in just a couple of years.

 

I find this very important, as well.  DS12 is one of those who feels a need to race through courses.  So, we construct overlapping curriculum to support his need while still giving him the time to develop the maturity.  It's very similar to the kids retaking classes.  AoPS is one of our cornerstones because of how well it teaches one how to think -- advanced mathematics (we're talking post-calculus here) doesn't involve the luxury of rote execution.  One *must* be able to think independently, and sometimes for long hours.

 

In our case, we can teach with or without AoPS, but it is a beautifully constructed curriculum.  To get the most from it, it's best to have the "rote" portion of the knowledge down BEFORE you start the course.  So, if you were to ask me "Foerster or AoPS," I'd respond "Foerster THEN AoPS."  They can even be concurrent, but I'd let AoPS lag half a course, because it is significantly more advanced in its approach.  This is precisely how we do it at home (we used Martin-Gay & Saxon before AoPS).

 

In fact, because of the intuitive nature of his learning style, we've been careful to keep a heavy dose of more formal construction in his diet.  He actually did geometry before algebra, AoPS counting & probability and number theory before intermediate algebra, and will do symbolic logic and set theory before precalculus, and introductory modern algebra before calculus.  He loves the theoretical diet, and it allows him to look at the more "routine" stuff in a very different light.  Even though he is mathematically gifted and overly inclined to race, he is still only going to do calculus in 10th, multivariate in 11th, and statistics in 12th, similar to other kids.  It's just that he'll have lots of unique depth to go alongside it.

 

If you have a kid who likes the competitions (ours does NOT), AoPS has a series of great books and courses, like the aforementioned olympiad geometry, that probably do just as good of a job, if not better, than we are prepared to do.

 

[wapiti]

Ruth,

Could you elaborate on this? I was tossing around the idea of the classes, but this concerns me. Are there no checks and balances with the classes? How is it possible to pass them not truly knowing the math?

 

FWIW, I totally agree with Ruth here:

 

Oh, ds sees quite a few of these kids in his classes.  What happens is that they can understand the material, but they just don't have the insight to solve the problems independently.  But the moment they get a hint, that little hint *is* the insight that they need to develop.  The material just needs time to sink in, and the best way to do this is to do *more* problems.  And work on the problems without help.  Doing 1 problem in 2 hours on your own is often more useful in developing math maturity than doing 6 problems in 2 hours with little hints and help.  If you don't take time to consolidate your knowledge and work hard problems with persistence and ingenuity then you are likely to move into the harder material without the problem solving skills that you really need.  Seems better to me to develop the problem solving skill while the material is still pretty straight forward.

 

...because I see this with dd sometimes.  

 

Editing out a long post to say, IMO bottom line, the key is working through the book *prior* to class.  However, the book problems do not need to be turned in.  Not using the book can lead to trouble, because there wouldn't be enough regular practice, just challenge challenge challenge.  

 

For the geometry on-line course, if a student works through the book even without the book challenge problems, between the regular book problems, Alcumus and the weekly on-line class challenge set, at least for my dd13, there's more than enough practice and challenge IMO (for example, I pulled up a vintage Regents exam for comparison to a "regular" geometry course and for the topics she's covered so far, it would be a walk in the park for her).  The times dd has failed to work through the book are the times that lead to trouble - getting lost in class followed by getting stuck on the challenge sets and procrastinating on the proof too late on the night it is due.  I don't worry too much about her getting all the challenge problems because a few of the proofs have been so hard that I can't even help her, but I'd like her to get many of them with as little to no help as possible.

 

If there's one thing missing from the on-line class, I would say it would be a few easier proofs, maybe one every other week or so, or maybe toward the beginning of the course, purely for mechanical purposes, wording, things like that - and the grader's comments thereon.  When a proof is so hard that she can't even get it, then she doesn't get a chance to work on those things.  She still doesn't feel sure about them and could use a smitch more explicit instruction, something very basic that is taken for granted (probably because I helped her too much on the first couple but can't seem to explain to her).

[Luckymama]

On my phone so I can't easily multiquote.

 

Agreeing w above. If taking an AoPS class that has a corresponding book, the student needs to work through the relevant book sections before class.

 

Once time dd didn't. She never did that again :lol:

[go_go_gadget]

In our case, we can teach with or without AoPS, but it is a beautifully constructed curriculum.  To get the most from it, it's best to have the "rote" portion of the knowledge down BEFORE you start the course.  So, if you were to ask me "Foerster or AoPS," I'd respond "Foerster THEN AoPS."  They can even be concurrent, but I'd let AoPS lag half a course, because it is significantly more advanced in its approach.  This is precisely how we do it at home (we used Martin-Gay & Saxon before AoPS).

 

My son prefers to speed through the AoPS, and then use Alcumus and Khan Academy exercises intermittently to keep fresh, rather than doing the rote first and AoPS second. This is similar to my approach in my graduate math courses; I've discovered that I can.not memorize algorithms long-term until I've fully understood the proof. I can do it in the short term, but then have to re-memorize the next week. I do better with fully internalizing proofs first. Certainly some may learn better in the way you've described, but I think describing it as ''the best way'' to use AoPS is perhaps a bit overly prescriptive.

[daijobu]

Ruth,

Could you elaborate on this? I was tossing around the idea of the classes, but this concerns me. Are there no checks and balances with the classes? How is it possible to pass them not truly knowing the math?

 

 

 

FWIW, I've only used the online classes for fun, not as core curriculum.  We do the AoPS books at home, because I want her to at least attempt all review and challenge problems.  AoPS online classes are a fun supplement for whatever electives she wants exposure to, and also to meet up with her friends online. 

 

I do this mainly because the online classes are too fast paced for us,  and it takes up too much of our time.  But as a fun social outlet, that's okay.  

[Mukmuk]

So interesting. Thank you all for contributing!

 

For the record, Ds does the assigned reading and the learning problems before the class. He likes to talk to me about the ideas although I can't add anything more than "uh-huh" and "cool". He only has blue bars for class participation (he loves getting featured), which means to me that he's following. Sometimes, he does the homework ahead of time, but he can't complete without the class or more reading.

 

What definitely can be beefed up is the Challenge Section after each session. When Ds started the class, the transition from Dolciani was brutal. Ds spent every weekend, every day, working on the homework. His Alcumus score fell dramatically and he was wailing so much, it was awful! But he was so determined to "get with the program" that he wouldn't stop. (Outwardly, I commended DS's struggle to the tutor, who was all for it. But I confess that I wrote in to AOPS to ask if there was a setting to get past Alcumus without actually passing it, or at least stopping it at a finite mark of say 10 or 15hours. They told me to beat it, politely. :blush: ) He's now worked it down to about 4-6 hours a week. I was rubbing my hands in glee because, with the time freed up, he can get to other interests. I'll try to talk him round to doing the Challenges. It won't be easy, but I'll give it a shot.

[lewelma]

Ruth,

Could you elaborate on this? I was tossing around the idea of the classes, but this concerns me. Are there no checks and balances with the classes? How is it possible to pass them not truly knowing the math?

 

I was thinking since dd seems to understand what she's learned better than I gave her credit for (she is going quickly though Foerster) that maybe the feedback from the classes might be enough to help order her thinking. Your last post gives me pause. Did the instructors not alert the parents the student was not ready to proceed so quickly? So I wouldn't be able to trust the classes totally to the instructor's judgment? I would still need to test, etc?

 

WMA, please know that these are only my observations given the 4 classes my son has taken this year and the different kids he has worked with.  Clearly, there are many paths to math excellence, and I certainly don't want to scare anyone off.

 

I will state upfront that the AoPS classes are the best thing my son has ever done in his homeschooling career.  He LOVES them!  He organises his schedule so that he can talk to the other kids in the chat room, which for us at this time of the year is from 1 to 3 in the afternoon.  They talk a lot, and he gets to know the different kids and their strengths and weaknesses.  Many of them are in multiple classes together.  Some of the kids are amazing, others need serious help.  But they always discuss how to develop that all important insight -- they know it is critical so they talk about it.

 

In my mind, the way AoPS works is that you do fewer hard problems rather than more easier problems in order to master the material.  12 hours of week of homework is pretty typical in AoPS which is usually for 8 short answers and 2 proofs.  Imagine how many problems you would complete in that time if you were doing Saxon.  A LOT!  But here's the problem, the moment you get a hint for a hard problem, it becomes an easy problem.  So if you get hints for everything (which some kids do) then you are basically only doing 10 easy problems a week, which is not enough to master the material.  The trick to mastery using the AoPS approach, is to struggle.  To struggle long and hard over every problem and only go for a hint if you are *desperate*.

 

The reason why ds spent 22 hours last week on 2 geometry problems is because he did NOT want a hint.  He really wanted to get them on his own.  Olympiad geometry has 2 proofs a week for 12 weeks.  There is no book and there are no short answers.  The class is known for being so hard that most kids don't get more than 7 or 8 of the proofs during the class. So my ds is taking this as a challenge, and although he will solve all the problems in the end with hints when required, he wants to solve more than 8 *without* hints.  So this week, he finally broke down and got a hint from a kid he trusts who had already solved it.  After FOUR hours, ds proved that the approach was wrong!!  He told the kid, who freaked, and went back to the drawing board.  This is the struggle I am talking about.  ds only went for a hint after 5 hours on that problem, and then followed a wrong hint for 4 hours.  In this class, every time he gets a hint from a moderator, he can solve the problem in about 30 minutes to an hour.  And this is just a tiny hint, but it *is* the insight.  So 11 hours of struggle to get it on his own or 1 hour with a hint.  You can just guess which approach will teach him more.

 

As for the additional work ds does beyond the class problems: when there is a book, my son reads the chapter before class but has never done any of the exercises.  He has never done the review sets or challenger sets in the back of the book during or after the class, and he has never done Alcumus.  The difference between my ds and the way some others use AoPS, is that ds spent 4.5 *years* on 2 books (Intro Alg & Geo) developing the insight *before* he started the classes.  Now he has the insight, he can solve most problems without help, and he does not need additional work to master the material beyond what is provided in the class.  However, if you *start* with the classes, then perhaps you need to spread out those 4.5 years of extra effort that my ds did and do the exercises, challengers, and alcumus. Seems to me that you will know how much extra needs to be done by finding out how many hints they are using/asking for.  To move forward well in the sequence, they need to be earning a blue with few hints.  The kids I was describing in my previous post were earning a blue, with lots and lots of help.  I know, because my ds was helping them. They simply did not have the problem solving skills even though they understood the material.

 

Hope that makes sense, and happy to answer more questions based on what my ds has seen in the classes.

 

Ruth in NZ

[mathwonk]

Although there are obviously many fixed formal math techniques, and these are what is taught in school and books, math is also an open ended world of new and related problems and these continually call for new ideas and insights, which spring from intuition.  So the further one goes in math the more important intuition becomes.  Math research is solving a problem no one else has solved before, and the reason is often that existing methods were not quite adequate, i.e. the solution calls for some new insight.

 

The way to prepare for the day when a problem is encountered that does not yield to the known methods is to pursue ones own way, via ones own intuition, in solving more elementary problems.  One of our profs at college, the great Raoul Bott, said we should try to do the problems he assigned by our own methods "before your heads get so full of other people's ideas, you are no longer able to generate your own".

 

So learning to do math is more than just learning the existing methods that past mathematicians have discovered for doing those parts of it they have mastered.  But when we do succeed in solving a problem by our own methods, we need to have the discipline to clarify our intuition, to reveal why it works, both so others can understand us, and so we can apply the same principle again.

 

So intuition gives rise to methods, but only if it is made precise and clear.  And since we may not always have a bright insight, we need to possess a toolbox of tried and true methods as well.

[Mike in SA]

My son prefers to speed through the AoPS, and then use Alcumus and Khan Academy exercises intermittently to keep fresh, rather than doing the rote first and AoPS second. This is similar to my approach in my graduate math courses; I've discovered that I can.not memorize algorithms long-term until I've fully understood the proof. I can do it in the short term, but then have to re-memorize the next week. I do better with fully internalizing proofs first. Certainly some may learn better in the way you've described, but I think describing it as ''the best way'' to use AoPS is perhaps a bit overly prescriptive.

 

Fair enough.  Same concept, though.  ;)

[Woodland Mist Academy]

Thanks, everyone! Dd currently has research projects she's assisting with, so she really doesn't have the time, inclination, or need for comradery or struggle with the classes. She has enough pondering and trial and error elsewhere. The responses were sooo helpful! I feel even more at peace with the Foerster decision for this year. Thanks, again!

[SeaConquest]

This is such a fascinating conversation. I truly appreciate the insight the mathematicians and more experienced parents bring to this discussion (and this forum generally). As the parent of a wee one, I'm wondering now if I'm doing him a disservice by our more socratic style of math. I was thinking that I was leading him to learn how mathematical insight happens when the answer isn't immediately apparent, but perhaps this isn't the best strategy. How much struggle do you think is appropriate in the early stages of problem solving? I've tried to give him the most challenging problems, but do them socratically. But, if the hint is the insight, perhaps I should change my approach.

[kiana]

I think the more important thing is that a child learn to put in an age-appropriate amount of struggle.

[wapiti]

This is such a fascinating conversation. I truly appreciate the insight the mathematicians and more experienced parents bring to this discussion (and this forum generally). As the parent of a wee one, I'm wondering now if I'm doing him a disservice by our more socratic style of math. I was thinking that I was leading him to learn how mathematical insight happens when the answer isn't immediately apparent, but perhaps this isn't the best strategy. How much struggle do you think is appropriate in the early stages of problem solving? I've tried to give him the most challenging problems, but do them socratically. But, if the hint is the insight, perhaps I should change my approach.

 

Thinking out loud, a socratic style is not the same thing as giving hints.  Leading questions contain some amount of hint, but pure socratic questioning isn't leading, is it? (e.g., recall your most socratic first-year law classes)  I think there is a continuum from socratic questioning at one end, to leading questions toward the middle, and at the other end, direct instruction.  (Eta, maybe the continuum is pure "discovery" at one end, socratic questioning or guided discovery a little closer to the middle, and then the rest...)

 

What I'm not sure about is development of intuition.  It seems to me there is some amount of natural ability and some amount of experience/knowledge/understanding that combine together, or is it all experience tucked away in a hidden location that forms the basis for intuition?

 

Eta, I don't have advice on the correct amount of struggle except as Kiana mentions, age can be a factor, and there may be other factors as well (personality, etc.).  I think it works out best when the motivation to struggle comes from within, motivated by the pure enjoyment of puzzle-solving.  (Just don't ask me how to make the jump from family fun strategy games to independent motivation in the context of learning math.  For my ds8, we have an ongoing conversation at random times, a lot of thinking out loud about math.)

[Woodland Mist Academy]

Regarding leading... technically AoPS leads the student through the discovery. At least that's how it seems to me.

 

Some ideas and terms in the thread are murky, with myriad factors and unknowns...

[quark]

This is such a fascinating conversation. I truly appreciate the insight the mathematicians and more experienced parents bring to this discussion (and this forum generally). As the parent of a wee one, I'm wondering now if I'm doing him a disservice by our more socratic style of math. I was thinking that I was leading him to learn how mathematical insight happens when the answer isn't immediately apparent, but perhaps this isn't the best strategy. How much struggle do you think is appropriate in the early stages of problem solving? I've tried to give him the most challenging problems, but do them socratically. But, if the hint is the insight, perhaps I should change my approach.

 

Please don't stop doing what you are doing if it is working and you feel your child is thriving!

 

I agree with wapiti's thinking aloud.

 

For us, we had several math strands running when kiddo was younger, and at least one was always of the Socratic style with a lapboard or whiteboard to help doodle out the math (for me and/ or him) when necessary. I still hold on to those memories...once they are so independent, you miss these things you know. Doing it together builds a great "math" bond between you too apart from being a good amount of age-appropriate guidance for him. :001_smile:

 

[1053]

.

[Mukmuk]

I think the more important thing is that a child learn to put in an age-appropriate amount of struggle.

I mulled over this thread last night. How do you *love* these posts from Ruth, Quark, Luckymama, Mathwonk, Mike, Go-Go, and so many others who have contributed their experiences? "Like" just doesn''t cut it!

 

In my mind, Kiana's message rings loudest, within the context of the other posts. And not just age-appropriate, but what the child *wants* to do and *learns to want* to do it. I have a feeling Ruth's son is so successful because he learnt, by himself, that being thorough is what works. If I imposed a prescription of challenge problems without Ds' buy-in, it will not have the same effect. Actually, quite the opposite!

 

Dear Ruth and everyone here, thank you for being so generous in putting your personal experiences for the benefit of the hive and me. Please don't stop. I've learned so much! My mileage will vary however, and I need to take stock.

 

My plan is to talk to Ds and get his buy-in. I'll see what happens after!

[Mukmuk]

 

 

The way to prepare for the day when a problem is encountered that does not yield to the known methods is to pursue ones own way, via ones own intuition, in solving more elementary problems. One of our profs at college, the great Raoul Bott, said we should try to do the problems he assigned by our own methods "before your heads get so full of other people's ideas, you are no longer able to generate your own".

 

 

Priceless.

 

The trick is to balance between intuition and learned ideas. I guess this will always be a life-long shifting, from one hand to the other, but both parts are necessary. Thank you.

[lewelma]

Oh Dear!  I hope I did not put my foot in it.  It is hard to talk to so many different people all at once because I can't tailor my advice to the circumstance.  If I could I would segregate it out like this:

 

To Rose and WMA: We've had this conversation in the other thread, but just for others listening in here: Your kids do not need AoPS to be successful in STEM careers.  There is more than one way to learn the material, and the discovery method does not cut it for all kids (my younger included).

 

To Seaconquest: My older boy did not do an formal math until he was almost SEVEN!  We played did number puzzles, played games, estimated how far it was that tree, made up oral word problems for each other, played shop, etc. We did ALL math together.  Don't give this up because you are reading about my boy at age 14, prepping for Squad selection for the IMO.

 

To Mukmuk: I only give a cautionary tale of kids racing through the AoPS classes, just so you go in with your eyes open.  Clearly it depends on your child.  And between you and your tutor, you guys can make sure that your ds continues to love maths and has success in it.

[Mukmuk]

Oh Dear! I hope I did not put my foot in it.

 

Not at all, Ruth. :001_wub: Your son is amazing, and so are you. I'm learning so much your experience. Thank you for your generosity!

 

 

 

To Mukmuk: I only give a cautionary tale of kids racing through the AoPS classes, just so you go in with your eyes open. Clearly it depends on your child. And between you and your tutor, you guys can make sure that your ds continues to love maths and has success in it.

Ack, this does strike fear in my heart, so my eyes are peeled. I'm treading softly.

[Mike in SA]

This is such a fascinating conversation. I truly appreciate the insight the mathematicians and more experienced parents bring to this discussion (and this forum generally). As the parent of a wee one, I'm wondering now if I'm doing him a disservice by our more socratic style of math. I was thinking that I was leading him to learn how mathematical insight happens when the answer isn't immediately apparent, but perhaps this isn't the best strategy. How much struggle do you think is appropriate in the early stages of problem solving? I've tried to give him the most challenging problems, but do them socratically. But, if the hint is the insight, perhaps I should change my approach.

 

Socratic is WONDERFUL imho.  That is the basis for modern mathematical reasoning, if you think about it.  It isn't easy, though - I'm impressed that you take this approach!

[lewelma]

I want to talk about age-appropriate struggle, what it is, and how you identify it.  I'm hoping others can give some ideas because I struggle with this with my younger.

 

First, the moment there are tears, something is wrong.  The material is too hard for that child at that particular moment.  In my house, when there are tears, there is a mandatory break.

 

But how do you teach kids to struggle without it getting too hard and leading to tears?  You start slowly!  You expect a little child to struggle with a problem for 10 minutes, a bigger child 30 minutes, a teen for an hour.  Where these cut offs are depends on *your* child, which topic it is (my younger will struggle for longer with geometry than algebra), the time of day (my older struggles the best in the morning), and what else is going on in their life (my older just spent 22 hours because we had finished all other classes for the year, so he had nothing else to do). 

 

So how do you judge the struggle?  I can tell by body-language.  And if I am noticing trouble brewing, I sit in the room (this is true even for my older.)  I find that camaraderie helps kids develop more interest in struggling.  So when my older had set this goal to get these 2 problems without help, the first day was fine. The second day he started to complain and want to kick ideas around with me. The third day, he started to get anxious and frustrated and this is when I started doing my work in the room with him.  I did my computer chores and then my physics in the room with him.  Physics is my weak spot, and I struggle, so occasionally I would interrupt him to talk about *my* struggle.  Sharing struggle is a good thing. :001_smile: By the fourth day, when I was starting to see the beginnings of tears, I began to push (hard) that he contact the professor which he does not like to do.  After a few more hours, I *required* it, and I sat an typed the PM that my son dictated.  When at last he got the problems on the 5th day, I was there to celebrate.  My point is that he did not struggle by himself in a room for 22 hours and emerge with a solution.  He *needed* me, and I was there.  I didn't have any hints (because I can't do this stuff) but I was there emotionally and physically.

 

With my younger it is a different story, he has a very low tolerance for struggle, and I have to judge every. single. day what he can handle.  I have actually had to reduce my expectations, make the work basically too easy to build up his desire to do it. He doesn't want the work too easy because "its boring" and he doesn't want it too hard because "I can't do it".  Some days it feels like :willy_nilly: .  So I sit with him for maths, every day for the entire time.  I have learned that I need to sit across the room rather than right next to him or he will constantly ask for help.  I have found that he struggles better with more practical problems than with theoretical ones.  I work *hard* everyday on developing the desire to struggle, and I do it very very carefully in all subjects.  I see it as one of my major long-term goals for him.  I figure I still have 6 more years.  How do you eat an elephant?  one bite at a time.

 

For me, one of my main ways to help my kids learn to struggle, is to see *me* struggle.  I try to learn something hard and then I talk about my struggles and how I overcome them.  Right now I am learning physics and mandarin.  So I talk about how I overcome something -- what strategies I use (go for a walk, find another book, ask a friend for help, reread, redo all the problems, etc) .  I talk about this everyday.  And with my younger I make sure to explicitly discuss issues like "When I can't do Optics, I don't feel stupid, I just know I need to redo the chapter.  I don't mind because I want to learn it."  My dh just finished a PhD and so my kids saw his struggle to.  He would talk about stuff at the dinner table.  Basically, we model the behaviour we want to see.

 

I think that being willing to struggle is one of the most important things I can teach my kids, so I work at it, everyday. 

[Chrysalis Academy]

I want to talk about age-appropriate struggle, what it is, and how you identify it.  I'm hoping others can give some ideas because I struggle with this with my younger.

 

First, the moment there are tears, something is wrong.  The material is too hard for that child at that particular moment.  In my house, when there are tears, there is a mandatory break.

 

But how do you teach kids to struggle without it getting too hard and leading to tears?  You start slowly!  You expect a little child to struggle with a problem for 10 minutes, a bigger child 30 minutes, a teen for an hour.  Where these cut offs are depends on *your* child, which topic it is (my younger will struggle for longer with geometry than algebra), the time of day (my older struggles the best in the morning), and what else is going on in their life (my older just spent 22 hours because we had finished all other classes for the year, so he had nothing else to do).  and hormones!! Yours and theirs. 

 

So how do you judge the struggle?  I can tell by body-language.  And if I am noticing trouble brewing, I sit in the room (this is true even for my older.)  I find that camaraderie helps kids develop more interest in struggling.  So when my older had set this goal to get these 2 problems without help, the first day was fine. The second day he started to complain and want to kick ideas around with me. The third day, he started to get anxious and frustrated and this is when I started doing my work in the room with him.  I did my computer chores and then my physics in the room with him.  Physics is my weak spot, and I struggle, so occasionally I would interrupt him to talk about *my* struggle.  Sharing struggle is a good thing. :001_smile: By the fourth day, when I was starting to see the beginnings of tears, I began to push (hard) that he contact the professor which he does not like to do.  After a few more hours, I *required* it, and I sat an typed the PM that my son dictated.  When at last he got the problems on the 5th day, I was there to celebrate.  My point is that he did not struggle by himself in a room for 22 hours and emerge with a solution.  He *needed* me, and I was there.  I didn't have any hints (because I can't do this stuff) but I was there emotionally and physically.

 

With my younger it is a different story, he has a very low tolerance for struggle, and I have to judge every. single. day what he can handle.  I have actually had to reduce my expectations, make the work basically too easy to build up his desire to do it. He doesn't want the work too easy because "its boring" and he doesn't want it too hard because "I can't do it".  Some days it feels like :willy_nilly: .  So I sit with him for maths, every day for the entire time.  I have learned that I need to sit across the room rather than right next to him or he will constantly ask for help.  I have found that he struggles better with more practical problems than with theoretical ones.  I work *hard* everyday on developing the desire to struggle, and I do it very very carefully in all subjects.  I see it as one of my major long-term goals for him.  I figure I still have 6 more years.  How do you eat an elephant?  one bite at a time. The bold describes my younger dd to a T.

 

For me, one of my main ways to help my kids learn to struggle, is to see *me* struggle.  I try to learn something hard and then I talk about my struggles and how I overcome them.  Right now I am learning physics and mandarin.  So I talk about how I overcome something -- what strategies I use (go for a walk, find another book, ask a friend for help, reread, redo all the problems, etc) .  I talk about this everyday.  And with my younger I make sure to explicitly discuss issues like "When I can't do Optics, I don't feel stupid, I just know I need to redo the chapter.  I don't mind because I want to learn it."  My dh just finished a PhD and so my kids saw his struggle to.  He would talk about stuff at the dinner table.

 

I think that being willing to struggle is one of the most important things I can teach my kids, so I work at it, everyday. 

 

This whole post is lovely, but the bolded especially resonate with me, and describe my teaching and learning philosophy, and basically how school goes at our hourse almost every day.  The kids really need to see that I am on their team.  I need to *be* on their team!  This means different things to each of them, at different times.  But being a learner with them helps keep it fresh.

[lewelma]

Since I am procrastinating on my packing (we leave today to go tramping!), I'd like to open a conversation about hints and help.  I'm not a mathematician, so I don't really understand the different types of hints that a person can give.  But I kind of see it like this:

 

1) Hint on the type of tool that might be useful to answer a question

2) Hint as to which thing you should consider first

3) Hint as to which area of the figure you should focus your attention on.

4) Hint that you are going down the wrong path

 

These are the types of hints I have seen ds get from AoPS. 

 

Since I am physically with my son, I give different types of hints/help than these.  They are more help on problem solving skills, rather than help on the specific problem:

 

1) Let's go through your book and see what you have learned recently.  Perhaps one of those theorems will help.  And then we sit together and evaluate each one, one at a time.

2) Can you work backwards?  Lets think about what the last step would look like?

3) Can you draw an image?  Can you color code it?  Should you draw it again but in a different way so you might see things differently?

4) Can you solve a simpler problem?  How can we make it easier? What have we learned by solving the easier one?

 

These types of help model approaches to problem solving.  They teach kids how to interact with the material in original ways for a longer period of time. 

 

For my younger, he struggles right now with organising data and drawing graphs for more complex, messy statistical problems.  So I'm looking to help him sort through the material in his mind. 

1) What do you need to write down? How can you organise it into a table?  Is there more than one way?  Why would that way be better than this way?

2) What kind of graph do you need?  Let's look in your book at the options?  Why would you choose that one over this one?

3) How can you organise the material on the page? Where should you put it?  Does it need to be bigger or smaller? 

 

These are very open ended thinking type hints/help.  I try to make sure that I don't have an expected answer in my head.  I want to just lead him towards judging his different ideas and evaluating their effectiveness. 

 

The other thing I always do is make sure my kid generalizes the hint/help.  After the problem has been solved with a direct hint, I ask 'what did you need to see that you did not see'.  'How can you make sure that you never need this type of hint again.'  'Name the different approaches you can use when you get stuck.'  I think that this consolidation is really important.  Otherwise, kids never take the time to reflect.

 

So I think that there are different types of hints/help you can give.  Clearly it depends on the child and the situation.  But once again, I model the type of thinking that I want them to have.

 

 

[lewelma]

Ack, this does strike fear in my heart, so my eyes are peeled. I'm treading softly.

 

Eeek!  I did it again.  No no no.  I think you are doing great!  Just by asking the questions, means that you are on the right track!  Your ds sounds so delightful and enthusiastic!

 

[wapiti]

Awesome, Ruth!  I think we need to pin those two posts!!!  I think I need to print the second one out and laminate it, so I can leave it on my kitchen counter.  "Ruth's How to Encourage Problem Solving"

[wapiti]

We have a lot of "I don't want to be alone" in our house... when dd13 is working on geometry, she wants me to sit with her.  Then every time I open my mouth to speak, for any of the aforementioned hint reasons, she commands me to not say anything at all, and I have to sit there and bite my tongue, probably a good idea in any case (never mind that it's late and I want to go to bed!).  Every time I try to get up, she commands me to sit back down.  Oh, and I'm not supposed to browse the forums either.  I'm supposed to pay attention and "help" but say nothing, LOL.

 

Eta, I think for younger kids, working together on a white board is one of the better ways, especially if it can be light and fun and puzzle-ish, which may mean few problems in a sitting, or just one.  Buddy math.  "What should we do next?"  The thrill of figuring out the puzzle can be shared.

 

For dd, I have in fact said, "what is the topic of the chapter?" if I'm allowed to squeak out some words.  On the proofs, I get a lot of "how should I start? NO don't tell me what to say but what should I say?"  Um.

[daijobu]

As the parent of a wee one, I'm wondering now if I'm doing him a disservice by our more socratic style of math. I was thinking that I was leading him to learn how mathematical insight happens when the answer isn't immediately apparent, but perhaps this isn't the best strategy. 

 

When I was first exposed to competition-style AoPS math in high school (not AoPS, but stuff from the AHSME), I froze and couldn't solve any problems.  It was when I saw my math teacher solving problems at the board that I finally realized "Hey, I could have done that."

 

Only then did I develop the courage to try approaches that feel right but may or may not lead to the solution.  So not only was I getting hints, I was watching problems actually being solved, with no struggle on my part, before I myself could do real problem solving.  

 

So what I'm trying to say is that there are many paths and many teaching approaches that lead to problem-solving competency.