question about "discovery" method of AoPS

[Halcyon]

Yes, another AoPS post :tongue_smilie:

 

I think I need to more fully understand what is meant by "discovery" approach. I have the Pre-A book, and I have spent some time looking through it, but I am not 100% clear on a few things.

 

Here's how one is to go through the book, as I understand it from reading the intro to the book:

 

 

1. Your child goes through the beginning-of-chapter questions, without having the "tools" necessarily to solve the problems in the most efficient way, or even how to solve them at all. Your student may (or may not) get frustrated as he struggles to solve some of the problems. Other problems may come easily to him. Either way, there is no assistance or help to be offered by the teacher.

 

2. The child then reads the chapter to see if his approach "matches" the approach taken by the teacher, or if it's different. If different, is the book's way more efficient? Why? If the child couldn't answer the question(s) at all, then reading the chapter serves to teach the child the best approach.

 

3. He or she then applies the approaches explained in the chapter to the Exercises, Review Problems and Challenge Problems. The Review Problems should be fairly easy for the student, as they are simply a review of the concept. The student should re-read the chapter if he cannot do these.

 

4. The student then attempts the challenge problems. Some of these, as the author mentions, are "very, very hard". If your child cannot solve these, he should continue to work on them until he can. He can use the hints in the back of the book only after working on the problem without success for a reasonable amount of time.

 

4. If your student cannot solve the problem, even with the hints, eventually he should be allowed to look at the Solutions book. Then he should return to the question in a couple of weeks to solve it without looking at the Solutions book, to confirm he retains and understands the approach.

 

I think one of my concerns was that the challenging problems look.....challenging :tongue_smilie: But reading this intro, Richard seems to imply that some of the problems are very hard to solve, even for a child who is right for AoPS. I am used to my child being able to, eventually and without help, solve all the problems in a math book. I think I'd have to approach AoPS in a different way.

 

Also, (and here's my main question): _NOT_ understanding, being able to "sit" with that "not-understanding", and being okay with it...this seems to be a huge part of AoPS. Most math books assume that the explanation (which comes first) will be VERY CLEAR, allowing your child to then DO THE PROBLEMS correctly. AoPS, otoh, almost welcomes the confusion as a way to force the child to find his own way through the fog. And the child has to be comfortable in the fog, at least for a little ways. And bit by bit, the fog is lifted as more problems are worked, and solutions are presented. With standard math problems, there is, ideally, no fog before questions are attempted. The concepts are explained AS CLEARLY AS POSSIBLE before problems are attempted by the student. And it's the teacher's (or book's) role to explain things very carefully to the student, so that ideally the problems can be done without too much struggle.

 

AoPS seems to turn this on its head. The student needs to be comfortable with "not-knowing" and feeling his way around in the dark (or at least, low light :tongue_smilie:) and as more and more problems are worked on, the light begins to turn on.

 

So when evaluating whether AoPS is right for one's child, simply looking at the text isn't enough, right? One needs to determine if one's child is comfortable with figuring things out for him/herself, but more importantly, are both you AND your student confident enough to sit in a place of befuddlement some of the time? Or will that make you or your child feel "we're clearly just not getting this!"

 

I hope this makes sense.

Edited February 25, 2012 by Halcyon

[Iucounu]

What a thoughtful post. The challenge and discovery aspects are what draw me to AoPS.

So when evaluating whether AoPS is right for one's child, simply looking at the text isn't enough, right? One needs to determine if one's child is comfortable with figuring things out for him/herself, but more importantly, are both you AND your student confident enough to sit in a place of befuddlement some of the time? Or will that make you or your child feel "we're clearly just not getting this!"

I think you're right, but I wouldn't view this as an insurmountable obstacle. It's unfortunate but true that a lot of math instruction doesn't prepare children well for challenges; doing lots of drill and being led by the nose doesn't usually encourage intellectual courage.

 

However, I am a big believer in people's abilities to adapt and conquer. I think that even students trained to be comfortable in being mostly passive knowledge recipients could be retrained.

 

I guess what I'm stumbling towards is the idea that your assessment might be a good idea for anyone in doubt, but that maybe some sort of remediation could be done as necessary. Maybe by taking a break from instruction for a while, going back and doing a bunch of challenging problems from early conceptual levels on up. This wouldn't so often involve the student discovering new mathematical concepts on their own, but would get them in a more active thinking mode, and train them not to give up easily when the answer wasn't immediately apparent by following known rules.

[snowbeltmom]

 

I think one of my concerns was that the challenging problems look.....challenging :tongue_smilie: But reading this intro, Richard seems to imply that some of the problems are very hard to solve, even for a child who is right for AoPS. I am used to my child being able to, eventually and without help, solve all the problems in a math book. I think I'd have to approach AoPS in a different way.

 

 

When ds was taking the online geometry class with AoPS, I remember him commenting that the kids were told that if one could solve every problem that he was given, that the instructor was not doing a good job of challenging them. My ds took the geometry class a few years ago now, but at that time, no one ever had been able to solve every problem in the challenge set.

[onaclairadeluna]

So when evaluating whether AoPS is right for one's child, simply looking at the text isn't enough, right? One needs to determine if one's child is comfortable with figuring things out for him/herself, but more importantly, are both you AND your student confident enough to sit in a place of befuddlement some of the time? Or will that make you or your child feel "we're clearly just not getting this!"

 

I hope this makes sense.

 

I think the intention of AOPS is to teach the child to be comfortable to "sit in a place of befuddlement":)

 

Some kids have more of an aptitude for this than others. But I don't think the child necessarily needs to have experience with this or an ability for this before attempting the books. Just the potential.

 

That's why it's geared towards bright motivated math students. Those students are likely to be able to learn this skill. My son was not at all comfortable with challenge before AOPS (his "zone of proximal development" was paper thin) In the beginning I had to remind him once or twice that he didn't actually need to be able to answer ALL the questions. I walked him through the process of looking at the answer book and thinking about different solutions etc. But after a little handholding he took off like a rocket.

 

So I don't think your child necessarily needs to have this skill before attempting the books but they do need to have the ability to develop it early on.

[wapiti]

We did four chapters.

 

FWIW, chapter five is where my dd seemed to fall apart. After getting over that hump (that's where we switched to Dolciani briefly), the subsequent chapters have been much easier for her (that's not to say easy, but she's been pretty independent with the regular exercises; we also altered our approach to the lesson problems, FWIW, and dd spends more time with the videos, which she loves). We still need to go back to ch 5 and do the review problems; I'm keeping my fingers crossed that she'll be able to pull it together. Sometimes I think it would be best if she were working with a tutor instead of me, just for the (pre-teen) attitude adjustment :tongue_smilie:

[wapiti]

Looking ahead in it there seems to be easier chapters than the first four. I don't know if I was imagining that though.

 

I think you may be right! My dd is in for some review of those chapters. In particular, ch 2 and 3 come to mind. They seem to form the backbone of some of the thinking in later chapters, and so I think it's important to learn them well, but yes, they were not easy for her. The later chapters seem broken up into more managable chunks, or perhaps the topics are simply not as difficult, or at least they're somewhat more familiar.

 

You mentioned somewhere about more practice - if you have found the Alcumus problems to be too complex for that, options include Dolciani, which has loads of straightforward practice (eta, though not on exponents, which seems to be key here), and for even simpler straightforward practice you could look to Khan (though the few Khan videos I looked at were quite shallow in terms of the teaching of concepts so I would be reluctant to use it as a teaching resource). In retrospect, I wonder whether it may make sense for some kinds of students to have more practice after individual lessons, perhaps before even doing the bulk of the exercises, since the latter half of the exercises sometimes seems to kick it up another notch.

 

As you go along with MUS alg., It'll be interesting to hear your comparison between the two routes in terms of how they fit your student.

Edited February 25, 2012 by wapiti

[Halcyon]

The problem I found was at the end of each section I wondered if my son really got the concepts.

 

This is an important point that I forgot to mention in my OP. We have done 2 chapters together from Pre-A, and it went well enough. But at the end of the chapter, I didn't have a sense that he had a very, very firm handle on what he'd just learned. There weren't enough drill problems, and this is a child who does not need a lot of drill. But in this case, I felt a few more would have been useful. I am a proponent of over-mastery, and the two chapters we completed left me feeling unsure as to whether over-mastery had been achieved. In fact, I would say over-mastery had NOT been achieved.

 

I suppose adding in questions from another math text like Dolciani that are very similar and having him solve those using an "AoPS" approach might be an solution.

[stripe]

How do you think this compares with the style used in Japanese schools of throwing a problem on the board and letting the students wrestle with it?

[Halcyon]

How do you think this compares with the style used in Japanese schools of throwing a problem on the board and letting the students wrestle with it?

 

But they work in teams, no? I may have that wrong.

[Temperance]

Also, (and here's my main question): _NOT_ understanding, being able to "sit" with that "not-understanding", and being okay with it...this seems to be a huge part of AoPS. Most math books assume that the explanation (which comes first) will be VERY CLEAR, allowing your child to then DO THE PROBLEMS correctly. AoPS, otoh, almost welcomes the confusion as a way to force the child to find his own way through the fog. And the child has to be comfortable in the fog, at least for a little ways. And bit by bit, the fog is lifted as more problems are worked, and solutions are presented. With standard math problems, there is, ideally, no fog before questions are attempted. The concepts are explained AS CLEARLY AS POSSIBLE before problems are attempted by the student. And it's the teacher's (or book's) role to explain things very carefully to the student, so that ideally the problems can be done without too much struggle.

 

AoPS seems to turn this on its head. The student needs to be comfortable with "not-knowing" and feeling his way around in the dark (or at least, low light :tongue_smilie:) and as more and more problems are worked on, the light begins to turn on.

 

So when evaluating whether AoPS is right for one's child, simply looking at the text isn't enough, right? One needs to determine if one's child is comfortable with figuring things out for him/herself, but more importantly, are both you AND your student confident enough to sit in a place of befuddlement some of the time? Or will that make you or your child feel "we're clearly just not getting this!"

 

The skill that I most wish I had learned before college, and didn't, would be to be okay with not understanding. Higher-level physics and math textbooks routinely give you a small bit of new information and expect you to draw on the whole body of knowledge you have learned to solve challenging problems. Professors routinely give tests where the class average is below 40%. Nobody gets all the problems right. I'm not very familiar with AoPS yet, but the approach sounds great. It's much easier to learn to deal with being comfortable with not understanding when one is young and his financial aid, ability to get a job, etc. isn't adding extra pressure. So, IMHO, the question is, how far does he want to go with math or science? If he wants to go far, then when and how are we going to become comfortable with not knowing?

[stripe]

But they work in teams, no? I may have that wrong.

 

I am not sure, but I do have the gr 7-9 Japanese books from U of Chicago.

[Halcyon]

The skill that I most wish I had learned before college, and didn't, would be to be okay with not understanding. Higher-level physics and math textbooks routinely give you a small bit of new information and expect you to draw on the whole body of knowledge you have learned to solve challenging problems. Professors routinely give tests where the class average is below 40%. Nobody gets all the problems right. I'm not very familiar with AoPS yet, but the approach sounds great. It's much easier to learn to deal with being comfortable with not understanding when one is young and his financial aid, ability to get a job, etc. isn't adding extra pressure. So, IMHO, the question is, how far does he want to go with math or science? If he wants to go far, then when and how are we going to become comfortable with not knowing?

 

 

Older enjoys math, and finds it (for the most part), not difficult. Occasionally when faced with a CWP, he is stumped and needs guidance. I do not know what his career goals are, but he wants to go to MIT or Yale :) He's expressed interest in engineering, but he's only 9. I think he does need to learn to become more comfortable with not getting the answer right away. He is pretty good, for his age, with just "sitting" with a problem, but he has not yet learned how to back off and re-approach the problem from a new angle. He will keep hacking away with "his" approach until he gets too frustrated to continue. He really enjoyed Alcumus when we did it (I think in part because we worked through the problems together, which he enjoys very much).

[snowbeltmom]

How do you think this compares with the style used in Japanese schools of throwing a problem on the board and letting the students wrestle with it?

I am not familiar with Japanese schools, but this is my take on AoPS's approach: AoPS does not present a problem and then expect the student to then arrive at the final answer completely on his own. Rather, AoPS guides the student to the final answer by asking a series of intermediate questions.

 

The same approach is taken in the online classes. The instructor knows exactly how he wants the students to get from Pt. A to Pt. B, and he leads them along that path by asking a series of questions.

[catz]

FWIW, chapter five is where my dd seemed to fall apart. After getting over that hump (that's where we switched to Dolciani briefly), the subsequent chapters have been much easier for her (that's not to say easy, but she's been pretty independent with the regular exercises; we also altered our approach to the lesson problems, FWIW, and dd spends more time with the videos, which she loves). We still need to go back to ch 5 and do the review problems; I'm keeping my fingers crossed that she'll be able to pull it together. Sometimes I think it would be best if she were working with a tutor instead of me, just for the (pre-teen) attitude adjustment :tongue_smilie:

 

Interesting. We're talking about Intro to Algebra right? We are finishing Chapter 6. I found Chapter 5 the most challenging for us too. Chapter 6 has been no problem and Chapter 7 looks straight forward too. We actually were so sick of chapter 5, we just touched on some of the challenge problems. For us, I've considered good success on the review problem sets a good indicator of understanding. Usually, by the time we get to the review sets DS thinks they're pretty straightforward. I do occasionally print out free online algebra worksheets though if we need extra practice.

 

I think the kid that would enjoy AoPS is not the kid that feels like they're sitting in the fog waiting for it to lift. But the kid that thinks the problem is a game or puzzle to figure out. I have a math degree and I personally get a little giddy setting up some of their problems. Sometimes I don't know where they're going (especially the challenge problems). Sitting in a fog sounds really unpleasant!

 

I also think a kid should be very comfortable writing a full page of info and numbers (or more) without frustration before getting into AoPS. As someone with a tech degree, having a comfort level with not knowing where problems are going is a definite advantage in college (and out in the real world too. I worked in software design for 10 years.).

[wapiti]

Interesting. We're talking about Intro to Algebra right? We are finishing Chapter 6. I found Chapter 5 the most challenging for us too.

 

Ha, no! I was talking about the Prealgebra. However, your comments are helpful and duly noted for Intro to Algebra, if we ever get there! By the video list, I noticed that the first chapter of Intro to Algebra covers, in a much shorter timeframe, some of what is covered in the early chapters of the Prealgebra that Wendy and I were talking about.

[wapiti]

Halcyon, I'm now paying closer attention to your path. Today, in preparation for a teacher conference next week, I walked ds through the final exam for MM5, to see what he has accomplished in his classroom (where he is using MM5). He's all done except for the geometry chapter :001_huh:. I need to talk with his teacher, but I think I'll line up some of MM6. He's been very independent with MM in his classroom, which is surprising to me, but he's so different from his sister. I think he'll need to wait on AoPS Prealg. and I'm trying to tell myself that MM6 is the best approach for him next. Maybe we'll try a bit of AoPS when he's at home in the summer, as an experiment. I'm concerned about him having enough help in the classroom next year. Just needed to whine for a moment.... if he had any other teacher, I'd definitely have to hs him. (oops, he just passed the pretest. must. wait. longer.)

 

AoPS does not present a problem and then expect the student to then arrive at the final answer completely on his own. Rather, AoPS guides the student to the final answer by asking a series of intermediate questions.

 

:iagree: That is what I see in the Prealgebra. In fact, I find the series of intermediate questions to be pure genius :001_wub:

Edited February 25, 2012 by wapiti

[Halcyon]

Interesting. We're talking about Intro to Algebra right? We are finishing Chapter 6. I found Chapter 5 the most challenging for us too. Chapter 6 has been no problem and Chapter 7 looks straight forward too. We actually were so sick of chapter 5, we just touched on some of the challenge problems. For us, I've considered good success on the review problem sets a good indicator of understanding. Usually, by the time we get to the review sets DS thinks they're pretty straightforward. I do occasionally print out free online algebra worksheets though if we need extra practice.

 

I think the kid that would enjoy AoPS is not the kid that feels like they're sitting in the fog waiting for it to lift. But the kid that thinks the problem is a game or puzzle to figure out. I have a math degree and I personally get a little giddy setting up some of their problems. Sometimes I don't know where they're going (especially the challenge problems). Sitting in a fog sounds really unpleasant!

 

I also think a kid should be very comfortable writing a full page of info and numbers (or more) without frustration before getting into AoPS. As someone with a tech degree, having a comfort level with not knowing where problems are going is a definite advantage in college (and out in the real world too. I worked in software design for 10 years.).

 

 

Perhaps "fog" was the wrong analogy LOL. By "fog", I mean simply that not everything is easily discernible. The student can see the basic forms, the buildings, the road, but perhaps not sharply. As he works through the problems, the buildings and such all around him come into sharper and sharper focus.

 

I don't mean to imply that one's child should sit there, completely stumped, with no idea how to proceed. I meant more that there's a more....ginger-footed, less sure approach in the beginning. As opposed to other curricula, where the problems are meant to demonstrate mastery, or at the very least, to reveal something the child has not yet grasped so that further review can done.

 

My child does not find AoPS Pre-A unpleasant, far from it. He enjoys it a great deal. But I am concerned that there is less reinforcement than might be needed by some students.

[8filltheheart]

I am not familiar with Japanese schools, but this is my take on AoPS's approach: AoPS does not present a problem and then expect the student to then arrive at the final answer completely on his own. Rather, AoPS guides the student to the final answer by asking a series of intermediate questions.

 

The same approach is taken in the online classes. The instructor knows exactly how he wants the students to get from Pt. A to Pt. B, and he leads them along that path by asking a series of questions.

 

 

:iagree: I consider AoPS to be the epitome of Socratic dialogue. ;)

 

Halcyon, if ds gets a chance (I'm not sure that he will), I would be interested to see his response to your question. He has had math instruction both ways (direct and AoPS).

[Halcyon]

:iagree: I consider AoPS to be the epitome of Socratic dialogue. ;)

 

Halcyon, if ds gets a chance (I'm not sure that he will), I would be interested to see his response to your question. He has had math instruction both ways (direct and AoPS).

 

What question? LOL.

[Temperance]

Perhaps "fog" was the wrong analogy LOL. By "fog", I mean simply that not everything is easily discernible. The student can see the basic forms, the buildings, the road, but perhaps not sharply. As he works through the problems, the buildings and such all around him come into sharper and sharper focus.

 

I don't mean to imply that one's child should sit there, completely stumped, with no idea how to proceed. I meant more that there's a more....ginger-footed, less sure approach in the beginning. As opposed to other curricula, where the problems are meant to demonstrate mastery, or at the very least, to reveal something the child has not yet grasped so that further review can done.

 

My child does not find AoPS Pre-A unpleasant, far from it. He enjoys it a great deal. But I am concerned that there is less reinforcement than might be needed by some students.

 

What if we were to break it down into skills and content? Can he explain the problems he has done already? If not, is it a lack of understanding the content, or a lack in the skill of explaining the problem?

[Halcyon]

What if we were to break it down into skills and content? Can he explain the problems he has done already? If not, is it a lack of understanding the content, or a lack in the skill of explaining the problem?

 

Good question. I will pose the question to him on Monday and see what he says.

[8filltheheart]

What question? LOL.

 

:lol: I read him your post and that was actually his response. ;)

 

Interestingly, he said that he thought your fog analogy was off base. I had read the whole thread, but he hadn't, so I thought that was actually a very telling commentary.

 

FWIW, he says if had to draw an analogy it would be more along the lines of a jigsaw puzzle. He says that typical approaches are equated more with a puzzle you see already constructed, so you have the image of what you are going to build and you have seen how it is taken apart so that when you attempt to rebuild it, you have all those images w/you. AoPS is more like having the puzzle pieces before you and you know they fit together and can recognize pieces that should fit, but you don't have the image of the completed puzzle before you when you start constructing it. The image becomes clearer the more you assemble.

[heyyou]

My son is studying the Prealgebra book and we supplement with MathScore.com to learn and drill the basics he could get with a regular curriculum. It's very fast to learn and perfect exponents or integers, for instance, with this program. I runs from 2nd grade to 9th and is standards aligned.

[Scuff]

This thread is a good read. We're almost done with the first chapter. I'm also not sure how much ds is retaining vs a traditional approach. But he's enjoying i much more than mm. But yeah, some days my brain is tired and I don't want to do it!

[Halcyon]

My son is studying the Prealgebra book and we supplement with MathScore.com to learn and drill the basics he could get with a regular curriculum. It's very fast to learn and perfect exponents or integers, for instance, with this program. I runs from 2nd grade to 9th and is standards aligned.

 

Spam

[heyyou]

OK, perhaps you need some more explanation. My son hates, hates, HATES to write. So I told him that if he didn't want to really work hard at writing, he'd better be really good at math. I hired a private tutor to work with him, at (this isn't spam, either) a dollar a minute. After $60 worth of assessment, he told me that my son needed more drill since he wasn't able to skip steps in his calculations. And that his math books (standards-aligned) didn't contain enough of it. No surprise there, NONE of them do. The only one that does have enough drill is Rod and Staff, but it isn't a college-preparatory curriculum.

 

At the same time I'm dealing with that, I have a daughter who cannot seem to memorize any math facts, even after spending first grade in R&S. I sent her to public school for 2nd, since my husband believed her lack of success was due to my bad teaching. (Have you stopped laughing yet? Caught your breath?) Of course it didn't help at all, but Dad had to see it for himself. I began using a computer program called Timez Attack with her, and it seemed to work, but she kept forgetting the facts she had learned. Then, I found MathScore.com and using it did cement the facts. We were using BOTH of these programs with my daughter (back and forth between them) when the private tutor told my son he needed more drill.

 

So, I paid my son $20 to finish Timez Attack to make sure that he did know his facts. And then I assigned MathScore.com to him in addition to the specific drills his math tutor recommended, with the idea that what's good for the goose is good for the gander. Yeah, he hated it. He considered it infantile and beneath his dignity, mainly because he knew his sister sucked at math and he didn't.

 

I signed him up for the homeschool math club, and drove him 40 min. one way every Monday. I didn't know it at the time, but they were practicing for the MathCounts competition. All I knew was that the kids who are good at math back when I was in school all joined the math club, so he was going to join it, too. Turned out that he enjoyed that kind of thing. We took numerous MathCounts problems to the tutor, since I couldn't solve them, and neither could my husband. Nor could the Math Lab at the local community college! The tutor said that this is what today's kids needed to study BIG TIME. Problem solving was the key. I asked him what books I should buy, and he said that problem solving wasn't found in a book!

 

I took that as a personal challenge, and found AoPS, thanks to the Well-Trained Mind forums and Google. I brought it in to the tutor, and at a dollar a minute (not spam, just the truth) he looked it over and pronounced it rock solid. Only problem was, DS's standards-aligned 6th grade math book, Calvert Math, hadn't taught him calculations with integers yet (a prerequisite for AoPS Prealgebra). I checked out a book called Integers by Irving Adler from the library, which was excellent, very well done. But my son wouldn't study it since it required the use of manipulatives, which apparently are also infantile and beneath his dignity.

 

MathScore to the rescue! I told him to find every integer topic in MathScore and pass it, even if it's not listed in 6th grade (you can do that in MathScore). In two weeks he had passed them all, and we began AoPS. I suppose I could've bribed him with another $20 to get through the book from the library, but I didn't, and MathScore was probably faster anyway. He has been using MathScore for about 9 months at this point and has finished all of 6th and 7th grade topics, and some of 8th and 9th too. AoPS has been highly motivating for him with respect to his MathScore work.

 

My posts are not spam. I really am a homeschooling mother. I really have used these programs: Timez Attack, MathScore.com, and AoPS Prealgebra. The first two remediated a struggling math student and took her to above grade level (with respect to timed drills) in 9 months (not spam, just the facts). And the last two allowed me to push the math while simultaneously relaxing with my essay-hating son, and letting the IEW (Institute for Excellence in Writing) program take as long as it needs to (much slower than I had in school). Not spam, just an honest review of products that have worked well for me.

 

Speaking of which, this past month I have seen a fantastic change for the better in my son's writing. Two years of IEW seem to have broken through some blockage. Should I comment on that happy fact on these boards, or would that be perceived as spam as well?

[Rivka]

heyyou, I think the problem is that you've made five posts on WTM and four of them have been plugging Timez Attack and mathscore.com. If you don't like being taken for a spammer, try varying your subject matter.