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AoPS Paradigm Shift - Pros, Cons, Struggles


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If a child could not solve the vast majority of exercises and review problems completely independently (without any parent input), then I would think that AoPS would not be a good fit. At least with my two younger kids, they would be completely overwhelmed if I simply gave them the book and told them to read and work the example problems themselves.

 

 

I feel like you're saying contradictory things here--could you clarify? It sounds like on the one hand, if a child can't do AoPS completely independently, it's not a fit. But I gather your youngers are using AoPS, with your help? :confused:

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So Ester Maria, would you say AoPS is ONLY for the math-obsessed, utterly passionate, can't stop talking-reading-breathingmath student, and ONLY that type?

I do not know enough about math to answer that. My impression, and based on my kids' reactions (and my kids are a limited sample :tongue_smilie:), led me to that direction in thinking, yes. But I may be wrong?

 

I think you could use it with any very bright and relatively interested child, but it will come at a price - in terms of time and effort allotted for this subject. For many people, my guess is simply that the price may get unreasonable if it is not an area of a particular interest for a child.

 

It is like emphasizing Latin by dedicating a dozen hours weekly to it. Sure, it is a legitimate academic goal and a legitimate approach, but probably not something you would opt for with every child. With most children, even if you DO emphasize Latin for its cultural importance, and even if you DO require Latin through graduation, you still allot more "reasonable" time to it, you do not go as wide or as deep, your choice of texts studied reflects that and you do not deal with some obscure ones, etc. In my view, AoPS is a bit like that (although I have exaggerated as regards time, LOL) - a challenging and legitimate task in and of itself, but something you would think only kids who passionately love math will be willing to be so invested into... and it does require far more investment than your typical math program.

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I feel like you're saying contradictory things here--could you clarify? It sounds like on the one hand, if a child can't do AoPS completely independently, it's not a fit. But I gather your youngers are using AoPS, with your help? :confused:

 

Sorry I was unclear. Each lesson in the book contains two sections: example problems and exercises. Each chapter also has a review section.

 

I think the program would not be a good fit if the child could not work the exercises and review problems independently.

 

With my two younger kids, I go through the text example problems with them. Usually, that means that my dd (or ds) would be at the whiteboard and I would be leading them through the text example problems. After they had worked through the example problems and we discussed the text comments, they would then complete the exercises completely independently (along with Alcumus and the end of chapter review problems.)

 

With my older son, I have no involvement at all. He works the example problems and reads through the text comments on his own. This would be my definition of "independent" since I am not involved at all - I give him the book and say "have at it."

 

I am not exactly sure what other pp mean by the term "independent." My guess would be that other pp would not consider my two younger kids to be working independently in the program.

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So Ester Maria, would you say AoPS is ONLY for the math-obsessed, utterly passionate, can't stop talking-reading-breathingmath student, and ONLY that type?

 

I am not EsterMaria, but none of my kids would fit the above description, yet AoPS works extremely well for them and is the perfect fit.

They like math, are interested in math, good at it, enjoy discovering relationships, find math fun - but they also like plenty of other things, and neither has the slightest desire to participate in math competitions. They are not math obsessed.

 

Btw, it is not necessary to devote an extraordinary amount of time to math when you have decided to use AoPS. You can start early and go as slow as you like (my son works only 45 to 60 minutes daily on math, not more). What you need is the willingness to work through the hard problems, plus a natural aptitude. It is not the curriculum for a struggling math student.

Looking at times: the Intro to Algebra text is a lot more than a traditional algebra 1 course (that would be only the first half of the book), DD took about 220 hours to complete the book. The Intro to Geometry text took her a total of 130 hours (we omitted one chapter); so pretty much in line with what a normal traditional math credit would involve - only for a strong student with a challenging text. (Had we used an easy text, she'd be done in half that time. But it is not as if AoPS eats up your school day or completely shifts the focus of your homeschool.)

Edited by regentrude
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Although my ds has done AoPS independently using the discovery approach from the beginning, I definitely think it can be used without the discovery approach and for kids who are not passionate about math. So, for example, you work the discovery problems together and then the kids do the practice exercises and the review problems independently, and skip the challengers.

 

The benefits to this approach is that a child who is not passionate about math can still get the complexity and lack of repetition that AoPS is known for. The costs, as I see them, include 1) using a very wordy text book which is really only needed if you are doing the discovery approach, otherwize there are much more direct ways to teach. So teaching section will necessarily take more time. 2) The book covers a lot of material that a traditional algebra book does not cover. This means that algebra will take more time (2.5 years for my ds:001_huh:). Does your child need this additional material if he is not passionate about math? Depends on if you are trying to slow them down because they have flown through elementary math. But if the child is not passionate about math, then there are just lots of extra material that could be left out to leave time for their passions (like foreign languages or science or whatever), and I personally would not know which chapters to skip. This extra material will not be found on the SAT or the subject tests and will not help with the math required for the Sciences. So, if the child's passion is not math, I would spend those extra hours on his/her passion.

 

Ruth in NZ

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Does your child need this additional material if he is not passionate about math? Depends on if you are trying to slow them down because they have flown through elementary math. But if the child is not passionate about math, then there are just lots of extra material that could be left out to leave time for their passions (like foreign languages or science or whatever), and I personally would not know which chapters to skip. This extra material will not be found on the SAT or the subject tests

 

I agree up to this point.

 

and will not help with the math required for the Sciences.

 

but with this I strongly disagree.

Everything in the Intro to Algebra book is extremely useful for sciences. It's just that some material is typically taught as part of an algebra 2 or precalculus course (sequences and series, for example).

The only material I have seen so far that I, as a scientist, would feel comfortable to omit is Power of Point in the Geometry text, and chapters 17-21 in Intermediate Algebra, which is basically competition math.

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Looking at times: the Intro to Algebra text is a lot more than a traditional algebra 1 course (that would be only the first half of the book), DD took about 220 hours to complete the book. The Intro to Geometry text took her a total of 130 hours (we omitted one chapter); so pretty much in line with what a normal traditional math credit would involve - only for a strong student with a challenging text. (Had we used an easy text, she'd be done in half that time. But it is not as if AoPS eats up your school day or completely shifts the focus of your homeschool.)

I have no idea how much time it takes for DD because she does not log hours.

 

However, I think that 220 hours is an AWFUL lot to ask of a typical student for whom math is not a particular subject of interest. For us this would equal to 6-7 hours weekly, which in my view is a LOT. For comparison's sake, when I was in high school I had 2-3 math lessons weekly and with homework, if I bothered to do it, we could round it to an average of 4 hours weekly (for a very bright student, but one who could care less about math). I am not sure I even dedicated 6-7 hours to Latin every week, and I was in a classical school.

 

I can see your point if one skips chapters, though. DD does not skip; maybe AoPS could be adapted for my eldest with picking and choosing what she gets to do from it and avoiding the challengers. I was speaking with the assumption one is actually going to do the whole book.

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Just chipping in with my 2 cents here. My DD lives and breathes numbers, and we've been doing AoPS Pre-Algebra since August. She's been doing extremely well and has had no major problems with the regular exercises and gotten about half of the challenge problems right on the first try.

 

This week, I switched *my* math program from AoPS Algebra (so I can keep up with her for next year) to Dolciani Algebra 1. She asked me if she could do it with me instead of AoPS this week for a break. Now after a week, I think she wants to make a permanent switch - she's hasn't brought it up directly but she's said several times how much more SENSE Dolciani makes than AoPS, and how much she's ENJOYING her 'break'. She loves the AoPS videos, but I think she's not liking the discovery method and the roundabout ways of AoPS as a daily diet.

 

So, there's an example of a very mathy kid who's neither thrilled with AoPS or hating it. She's in sixth grade, and she has always wanted to be an engineer. She likes the step by step, and while she likes to work the puzzles and creative stuff in math, I think the steady diet of it in AoPS is wearing her down.

 

So I think at this point what we are going to do is do mostly Dolciani, and supplement with some of the relevant AoPS videos or parts of chapters for fun. We may not ever go back to AoPS full time, or maybe we will when she gets a little older, I really couldn't say - I can't really tell if this is something that she will change as she becomes more mature, or if it's just her 'engineering-brain' personality getting irritated with too much round and round :001_smile: Time will tell. (I just wish she'd quit confusing ME! :lol: )

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DS11 has only completed Pre-algebra through chapter 2 (took about 18 school days). Here are my observations so far...

 

1) Reading this thread, I realized I am the discovery type of learner! I never knew what it was called...I just thought it was the slow type:) I am an engineer and extremely successful student but I had to rework everything from the ground up for every test. I had to totally understand why! And it helped me to have the big picture first and then start from the beginning. Of course there rarely was a big picture until right before test time and so then I had to start at the beginning again. I LOVE AOPS. I get excited about the problems...perhaps too excited that I blurt out too many clues!

 

2) DS is not working independently (but he's only 11 and I think that's a big issue with this curriculum...it's directed at a mature student) and he is not a math lover. He does enjoy math olympiad and he is good at math but has developed a dislike for math the last couple of years....probably my fault. However, given the choice between going back to Saxon pre-algebra and sticking with AOPS, at this point he is definitely on the AOPS side!!! So far so good. I think he might need a little repetitive reinforcement with a worksheet here and there, but he is happier and that is better for him (and me!) overall.

 

Brownie

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This is all very interesting to me. I would say, at this moment in time, my son's passions are history and Latin. And sharks :tongue_smilie:. He takes pride in doing math well, and I think his ability to spend time on problems is growing. But, as Deniseibase mentions, he likes step-by-step, straightforward instruction....but then again, when he finds a shortcut in a math problem, or finds a different way to solve it then the book shows, he's thrilled. Sigh. I am glad I can get Dolciani and Lial's so cheaply!! :001_smile:

 

Right now, DS works about 5 hours a week on math, without complaint. He would happily work more. But then again, he's doing MM not AoPS. Looking through AoPS, I don't think he could sustain the mental concentration of more than an hour a day on those problems. Plus, he's working hard on Henle Latin (probably more than 5 hours a week) which takes a lot of his brain cells (and mine, and I don't have many left! :lol:)

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She likes the step by step, and while she likes to work the puzzles and creative stuff in math, I think the steady diet of it in AoPS is wearing her down.

 

 

I think this might happen more frequently to a younger student. Your idea of using AoPS as an "extra" might be spot on.

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and will not help with the math required for the Sciences.
but with this I strongly disagree. Everything in the Intro to Algebra book is extremely useful for sciences.

 

 

Perhaps it has to do with the sciences one has studied. I know you are a physicists, and I am an ecologist. In University, I took 1 year of Physics, 1.5 years of Chemistry, and 10 years of Biology, (plus 1.5 years calculus and 4 years of Statistics). My dissertation was interdisciplinary between Ecology and Statistics (publishing in both ecology and statistics journals). The math that I did in high school was NOT of the complexity of AoPS (I think I used Jacobs), and I was not hindered in any way studying the sciences. I got good grades from a top tier university and published in top tier journals.

 

My point is that if your student is not passionate about math but is passionate about science, you do not NEED the rigor of AoPS to succeed in science.

 

Ruth in NZ

Edited by lewelma
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Wow, this thread is so rich with interesting views, differing perspectives, nuances and personal experiences from the Front Lines of math education that I am loving it!!! Thank you all who have contributed. I am glad intellectual discussions can occur with differing views and outcomes. After all every child is uniquely gifted in the way they learn and every parent different in the way they teach and guide their children. We sometimes guide toward similar goals (e.g. STEM careers) in very different ways. I find your individual journeys facinating.

 

Thanks again,

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Just chipping in with my 2 cents here. My DD lives and breathes numbers, and we've been doing AoPS Pre-Algebra since August. She's been doing extremely well and has had no major problems with the regular exercises and gotten about half of the challenge problems right on the first try.

 

This week, I switched *my* math program from AoPS Algebra (so I can keep up with her for next year) to Dolciani Algebra 1. She asked me if she could do it with me instead of AoPS this week for a break. Now after a week, I think she wants to make a permanent switch - she's hasn't brought it up directly but she's said several times how much more SENSE Dolciani makes than AoPS, and how much she's ENJOYING her 'break'. She loves the AoPS videos, but I think she's not liking the discovery method and the roundabout ways of AoPS as a daily diet.

 

So, there's an example of a very mathy kid who's neither thrilled with AoPS or hating it. She's in sixth grade, and she has always wanted to be an engineer. She likes the step by step, and while she likes to work the puzzles and creative stuff in math, I think the steady diet of it in AoPS is wearing her down.

 

So I think at this point what we are going to do is do mostly Dolciani, and supplement with some of the relevant AoPS videos or parts of chapters for fun. We may not ever go back to AoPS full time, or maybe we will when she gets a little older, I really couldn't say - I can't really tell if this is something that she will change as she becomes more mature, or if it's just her 'engineering-brain' personality getting irritated with too much round and round :001_smile: Time will tell. (I just wish she'd quit confusing ME! :lol: )

 

This is an incredible story. Thank you for sharing your experience. Just to clarify are you saying you jumped ahead to Dolciani Algebra 1 to practice for next year. Then your DD decided to jump ahead with you from AoPS Pre-Algebra to Dolciani Algebra 1 (not Pre-A) and liked it better? If thats the case that is quite a jump IMO. :tongue_smilie:

 

Does she mind so far not having all the extra goodies that come with AoPS such as the instructional videos, alcumus, etc..?

 

Any reason why you chose Dolciani over Foerster for Algebra 1? I wonder if there is a way to combine some type of video instruction with Dolciani like AoPS, Khan Academy, Math without Borders, etc?

Edited by dereksurfs
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how can a self teaching book not be adaptable to be taught? I CAN fathom getting up in front of a child and writing out the intro problems on a board and asking the student to attempt them, talk about them, and then go over the answers and explanations with them.

 

Thanks for the food for thought :tongue_smilie:

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I don't understand the comments about this not being appropriate for a child who cannot work mostly on their own.

 

I would not use it with a student that needs to be taught the material. My personal POV is that AoPS is designed specifically for the student that needs and wants to struggle and work through concepts on their own. Giving occasional hints that do not give away the process that they are supposed to be discovering is not the same as working with them on the problems. I think a lot is lost from the program with direction/teaching/assisting.

 

IMHO, AoPs has 2 major aspects of the program that create an environment that fosters true mathmatical excellence for the very top students: 1) The very challenging problems and 2) the discovery method. Obviously, many strong students would benefit from #1 even if they do not do #2. However, the discovery aspect of the program is part of the mathematical training. It is kind of like math research for the young; like doing a masters thesis in math but scaled down to the young. Researching new mathematical approaches is similar to science research, but just like "real" science is not done very often most high schools (most experiments are really demos), real math research is not done either. AoPS gives the mathematically gifted a hint of what could be if they would pursue a career in mathematics. So, as 8filltheheart said "a lot is lost from the program with direction/teaching/assisting". This does not mean that your student still cannot benefit from the very challenging problems (#1 from above).

 

The fact I'm using this book with a younger student does require I make some adaptations. In my mind that is no different than me writing my son's narratives for him before he was able to type them out himself or turning the pages of a book he wanted to read because it was difficult for him. This is why I homeschool.

 

This I can understand. I am considering AoPS without the discovery method for my younger son. But just go in with your eyes open for what is being lost by this approach.

 

Ruth in NZ

Edited by lewelma
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This is an incredible story. Thank you for sharing your experience. Just to clarify are you saying you jumped ahead to Dolciani Algebra 1 to practice for next year. Then your DD decided to jump ahead with you from AoPS Pre-Algebra to Dolciani Algebra 1 (not Pre-A) and liked it better? If thats the case that is quite a jump IMO. :tongue_smilie:

 

Yup, that's exactly what is happening, but it's really not that big of a jump, at least so far. We had already completed the first half of the AoPS book, which includes the chapters on properties and equations that were the main things we had not covered in previous years. The rest of the AoPS book covers things like percents and angles, which we have studied in previous years, or things like counting problems and problem solving techniques, which are not generally covered in a typical pre-algebra course anyway. So, either last year or this year, she's already covered most of the material that would be in a typical pre-algebra. AoPS goes WAY beyond a typical pre-algebra text!

 

And, keep in mind, we've only done this for a week and have just covered the first two chapters of the Dolciani, the chapters on sets, which Dolciani presents as if the student has not encountered them before, and properties, which we studied earlier in the year. If we run into something later in the year that she's not prepared for, there's no reason we can't pause and go over background material at that time, but so far so good.

 

 

Does she mind so far not having all the extra goodies that come with AoPS such as the instructional videos, alcumus, etc..?

 

She hasn't complained so far. *I* miss the videos, Richard is pretty funny sometimes! :001_smile: We didn't use the Alcumus all that much as I kept forgetting about it :blushing:

 

 

Any reason why you chose Dolciani over Foerster for Algebra 1? I wonder if there is a way to combine some type of video instruction with Dolciani like AoPS, Khan Academy, Math without Borders, etc?

 

I chose Dolciani over Foersters because of advice I received in this thread - http://www.welltrainedmind.com/forums/showthread.php?t=338961 Basically, it seems Dolciani is 'closer' to AoPS than Foersters.

 

As far as the videos, I see no reason why you couldn't. The AoPS videos and Khan Academy videos are laid out in a fairly organized fashion, it would be fairly easy to just check to see if there are videos on the topic you are currently studying and watch them as a supplement. I'm not familiar with Math without Borders, but this is the third mention I've heard of them this week so maybe I should take that as a sign to go check them out :001_smile: Thanks!!!

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One other thing I realized...you really can't use the discovery method AFTER you've already learned the material in another course. It takes away much of the benefit. I had intended to use AOPS pre-alg next year after using saxon pre-alg this year to buy us another year. I didn't think we were ready for AOPS yet. However, DS was bored and frustrated so we gave it a try. When we hit concepts he already knows, I can see the difference in his absorption of the material. It's hard for a kid to back up and re-learn with the discovery method what he already knows.

 

Brownie

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What is lost and how is it lost? Unless I completely skip the beginning problems and go straight to the explanations I don't think anything is lost. The only difference is that instead of me handing the book to my son and telling him to go at it, I'm reading some of the beginning sections to him and watching him as he attempts the problems.

 

Thinking back to those RR articles on problem-solving, the more the student does on his/her own, the greater the problem-solving skills they might be developing. However, I do see a benefit in adapting the approach for the student - I do think they still might get a lot out of this book even with some modification, the kind you describe where they still have to think about it, where the alternative is to not use the book at all. (However, no one wants a student having a cloudy understanding - if cloudy understanding was the result, then Dolciani might be the better alternative.) Also, I've noticed in at least a few instances, some of the exercises, particularly toward the end of the exercise section, do explore concepts further in a way similar to the lesson problems - sometimes foreshadowing the next lesson.

 

I don't even think that is true. I've learned all of the concepts in the past that we have encountered so far in AoPS and I have learned a great deal more from AoPS.

 

This has been my experience as well. In some cases, AoPS takes concepts deeper, and in other cases uses mathematical tools that I am familiar with but in ways that I was never taught and had never occurred to me. My perspective is quite changed by it, and I wish I had had the opportunity to be exposed to this when I was younger. This is why I'm still trying to make AoPS work for my dd, even though I also love the Dolciani (which is really math as I learned it). Even if she has to do it later, I want her to do some of it.

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I have no experience with AoPS at the younger age/pre-alg book, but I am assuming that it is the same as all the higher level books.

 

It goes to the core of how AoPS is written. The process of figuring out on your own is an actual objective. Consider it a mathematical Socratic method where the answers are not given, but only more questions asked. It is the way that AoPS teachers teach as well. They do not explain problems, they ask more questions via hints to guide their thinking.

 

My ds could have spent hrs on a problem and email his teacher with a question and the answer will be "Have you considered this? I can't help you any more w/o giving away the entire process." Discovering the process w/o being taught it is the essence of AoPS. Conversely, in a traditional math class, if a student didn't know how to work a problem, the teacher would explain the process and teach via example problems how to do the process and want the student to replicate what they have been taught.

 

You can obviously teach students directly. It is the way the most math books approach material. Teachers normally teach the material and then ask students to do the work. It is simply just not the way AoPS is designed.

 

FWIW, ds says he does not think he would have liked AoPS at a younger age. He doesn't think he would have had the maturity to want to spend the time doing it even though he capable of doing that level of math. He was 13 when he started he started his first AoPS course and the methodology was a good fit by that pt.

Edited by 8FillTheHeart
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I have no experience with AoPS at the younger age/pre-alg book, but I am assuming that it is the same as all the higher level books.

The AoPS pre-algebra book format is the same as the other AoPS books.

 

It goes to the core of how AoPS is written. The process of figuring out on your own is an actual objective. Consider it a mathematical Socratic method where the answers are not given, but only more questions asked. It is the way that AoPS teachers teach as well. They do not explain problems, they ask more questions via hints to guide their thinking.
:iagree:

 

I think that WendyK and I are using the same Socratic process with our younger kids even though our kids are not working completely independently in the book. The difference is that we (the parent) are guiding the student by reading the step by step textbook questions/problems to them, rather than the student reading the textbook himself. I think that the end result is very similar in that both methods result in the child discovering the concept.

 

My ds could have spent hrs on a problem and email his teacher with a question and the answer will be "Have you considered this? I can't help you any more w/o giving away the entire process." Discovering the process w/o being taught it is the essence of AoPS. Conversely, in a traditional math class, if a student didn't know how to work a problem, the teacher would explain the process and teach via example problems how to do the process and want the student to replicate what they have been taught.
:iagree:

This has been my oldest son's experience as well. It is amazing how many times he has been able to complete a problem with one simple hint. By contrast, based on my own personal experience in school, being able to replicate the problem after being shown an example, did not result in understanding even though I would get the future problems correct.

 

You can obviously teach students directly. It is the way the most math books approach material. Teachers normally teach the material and then ask students to do the work. It is simply just not the way AoPS is designed.[/Quote]:iagree: I wonder if when other pp say that the child needs to work independently to get the full benefit of AoPS, maybe a different way of interpreting that statement would be that to get the most out of AoPS approach, the child should discover the material on his own. Whether that discovery is done completely independently or parent guided, the important thing is that the discovery is taking place.
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Another thought I had was that AoPS offers online courses using these books. So the material is being taught (otherwise why would anyone pay for such a course).

 

Well, the student is supposed to have read the text and worked the problems at the beginning of the chapter before they attend class. The prealgebra class as a whole is continuing to work the concepts from the book and then expanding on that concept. It's hard to explain but to me it feels more like reinforcement of what the child should already have learned and moving out from there versus traditional teaching. More of the Discovery approach in a group setting than what I see as traditional teaching. For my DD it's a time to show what she learned and becomes more of a race to see if she can answer correctly and quickly. I imagine that may change as you move to the higher classes.

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I think that WendyK and I are using the same Socratic process with our younger kids even though our kids are not working completely independently in the book. The difference is that we (the parent) are guiding the student by reading the step by step textbook questions/problems to them, rather than the student reading the textbook himself. I think that the end result is very similar in that both methods result in the child discovering the concept.

 

That's a helpful description :). We started out sort of that way, and it worked well, but then I guess I got busy :001_huh: and left dd to her own devices with the book, which worked for a while (she loves not being taught by me), but then led to a lot of frustration and wasting of time. She lost some confidence and then didn't want to think through anything (oddly, nor was she willing to look at the lesson problem solutions when she was quite stuck - on the one hand, she really wanted to figure it out herself and I was certainly not permitted to tell her the solution lest a tantrum ensue, but on the other hand, she was experiencing intermittent episodes of lazy brain). Perhaps it's worth trying again, with more, selective involvement from me.

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I feel like you're saying contradictory things here--could you clarify? It sounds like on the one hand, if a child can't do AoPS completely independently, it's not a fit. But I gather your youngers are using AoPS, with your help? :confused:

 

I thought of an example to articulate a bit better what I mean by this statement.

 

Last year when my middle child had completed AoPS Alg I (which is the 1st half of the Intro to Algebra book), I moved him into AoPS Intro to Counting & Probability book. We followed the same process in the C&P as we had for Alg I: I would guide him through the example problems (with him working on the whiteboard) and then he would complete the exercises independently.

 

However, unlike in Algebra I, where he was able to correctly complete the vast majority of the exercise/review/challenge problems completely on his own, this was not the case with the C&P book. Once we got into the 3rd chapter of C&P, he struggled with many of the exercise problems, arrived at incorrect answers, and needed my help to solve them. He clearly did not have a solid grasp of the concepts and he was not ready for this AoPS book. He was not learning the material since he could not apply the concepts learned in the discovery example problems and apply those concepts "independent of my help" to solve the varied exercise problems.

 

At that point, I put the book away. He is working through the C&P book along with finishing the Intro to Algebra book this year. This time around, he is getting the concepts and is able to work independently and arrive at the correct answers the vast majority of the time.

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Well, the student is supposed to have read the text and worked the problems at the beginning of the chapter before they attend class. The prealgebra class as a whole is continuing to work the concepts from the book and then expanding on that concept. It's hard to explain but to me it feels more like reinforcement of what the child should already have learned and moving out from there versus traditional teaching.

I think that is a great explanation on how the classes operate. The kids are supposed to be familiar with the material before they attend the online lecture session. By having those "hooks" already in place, they are able to understand the concepts in much more depth in the online session.

 

More of the Discovery approach in a group setting than what I see as traditional teaching. For my DD it's a time to show what she learned and becomes more of a race to see if she can answer correctly and quickly. I imagine that may change as you move to the higher classes.
In my experience, what you have described does not change as you move to the higher levels; you have described very well how the classes at all levels operate. (Although I don't have first hand experience yet with the Calc class, but I wouldn't think it would operate any differently.)
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Another thought I had was that AoPS offers online courses using these books. So the material is being taught (otherwise why would anyone pay for such a course).

The online AoPS courses that DS has taken were run very much like the books... No one says "here's how you do the problem, everyone watch" -- it's more like the teacher posts a problem and asks where to start, and the students start throwing out ideas, and the teacher pounces on the good ones and asks "why that", and they nibble away at it applying what they've used before or trying things that might not work, with the guidance of a teacher who can immediately correct something trivially wrong (arithmetic, for instance) but let them chew on the strategies for a bit.

 

I run my own math team this way... I give them hard problems and then I sit back while they hack at them. Sometimes there's a strategy they already know from previous problems that we can build on, but sometimes it's hidden in a very different way than they've seen before. I can jump in and say "wait, check that calculation" before they get too far with an error, but they sit around discussing whether their approach will work and explaining their reasoning to each other. If I think they're way off base I'll make them explain it to me while I pick apart their logic. If they come up with two reasonable strategies, I'll have them do both and then compare the results and figure out why they both work or why one doesn't.

 

The key, IMO, isn't independence vs. teacher guidance. It's who's doing the work. The work, in this case, isn't just the problems themselves but the reasoning. And the payoff, for me, isn't in doing a lot of problems right, but in having an extensive toolbox of potential strategies that could work in different situations... and the skill and experience to apply those strategies appropriately. Where that develops (in my experience) is from being faced with very challenging problems that draw on a variety of skills in an unpredictable way, with only the most minimal guidance of a breadcrumb here and there. Where I get concerned about people using AoPS isn't in whether the kid can be plunked in a corner with the book or not, but in whether the parent is figuring things out for the kid and then showing them exactly what to do. I like the balance that DS and I have, where he works primarily on his own but comes to me when he's stumped. If I have him explain what he's tried and where it fell apart he can usually see his own mistakes, and when he can't I can ask a question that nudges him in the right direction without saying "do this, then this, then this."

 

That's the independence I think you need. Not dumping a kid in a bare room with a book, but making absolutely certain that they do all their own thinking, that they're faced with challenges in figuring out the strategy (not just in performing it), and that they get a chance to work through situations where things could go twenty different ways and they have to dig through their toolbox of math skills to find one that works.

 

DS enjoys math but doesn't "breathe numbers" -- he prefers science and engineering. I know that AoPS is more work than he really needs to do for a science or engineering career, but it's worthwhile work on its own merits. (And IMO he's still too young to rule out other career possibilities, especially in fields where he has quite a significant aptitude.) Really though, whether he uses the specific math strategies he learns through AoPS or not, what I want him to take away from all this is an ability to tackle hard problems efficiently, without panicking and without floundering... and to get through to the end and distill the process to a straightforward explanation and answer. That ability really will serve him well in any field.

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The key, IMO, isn't independence vs. teacher guidance. It's who's doing the work. The work, in this case, isn't just the problems themselves but the reasoning. And the payoff, for me, isn't in doing a lot of problems right, but in having an extensive toolbox of potential strategies that could work in different situations... and the skill and experience to apply those strategies appropriately. Where that develops (in my experience) is from being faced with very challenging problems that draw on a variety of skills in an unpredictable way, with only the most minimal guidance of a breadcrumb here and there. Where I get concerned about people using AoPS isn't in whether the kid can be plunked in a corner with the book or not, but in whether the parent is figuring things out for the kid and then showing them exactly what to do. I like the balance that DS and I have, where he works primarily on his own but comes to me when he's stumped. If I have him explain what he's tried and where it fell apart he can usually see his own mistakes, and when he can't I can ask a question that nudges him in the right direction without saying "do this, then this, then this."

 

That's the independence I think you need. Not dumping a kid in a bare room with a book, but making absolutely certain that they do all their own thinking, that they're faced with challenges in figuring out the strategy (not just in performing it), and that they get a chance to work through situations where things could go twenty different ways and they have to dig through their toolbox of math skills to find one that works.

 

...

Really though, whether he uses the specific math strategies he learns through AoPS or not, what I want him to take away from all this is an ability to tackle hard problems efficiently, without panicking and without floundering... and to get through to the end and distill the process to a straightforward explanation and answer. That ability really will serve him well in any field.

 

Thank you, Erica, for fleshing this out. This is very helpful. I think I'll be reading this over a few times. This reminds me of the RR articles on problem-solving, and it's one of the reasons I continue to be drawn to AoPS.

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The online AoPS courses that DS has taken were run very much like the books... No one says "here's how you do the problem, everyone watch" -- it's more like the teacher posts a problem and asks where to start, and the students start throwing out ideas, and the teacher pounces on the good ones and asks "why that", and they nibble away at it applying what they've used before or trying things that might not work, with the guidance of a teacher who can immediately correct something trivially wrong (arithmetic, for instance) but let them chew on the strategies for a bit.

 

I run my own math team this way... I give them hard problems and then I sit back while they hack at them. Sometimes there's a strategy they already know from previous problems that we can build on, but sometimes it's hidden in a very different way than they've seen before. I can jump in and say "wait, check that calculation" before they get too far with an error, but they sit around discussing whether their approach will work and explaining their reasoning to each other. If I think they're way off base I'll make them explain it to me while I pick apart their logic. If they come up with two reasonable strategies, I'll have them do both and then compare the results and figure out why they both work or why one doesn't.

 

The key, IMO, isn't independence vs. teacher guidance. It's who's doing the work. The work, in this case, isn't just the problems themselves but the reasoning. And the payoff, for me, isn't in doing a lot of problems right, but in having an extensive toolbox of potential strategies that could work in different situations... and the skill and experience to apply those strategies appropriately. Where that develops (in my experience) is from being faced with very challenging problems that draw on a variety of skills in an unpredictable way, with only the most minimal guidance of a breadcrumb here and there. Where I get concerned about people using AoPS isn't in whether the kid can be plunked in a corner with the book or not, but in whether the parent is figuring things out for the kid and then showing them exactly what to do. I like the balance that DS and I have, where he works primarily on his own but comes to me when he's stumped. If I have him explain what he's tried and where it fell apart he can usually see his own mistakes, and when he can't I can ask a question that nudges him in the right direction without saying "do this, then this, then this."

 

That's the independence I think you need. Not dumping a kid in a bare room with a book, but making absolutely certain that they do all their own thinking, that they're faced with challenges in figuring out the strategy (not just in performing it), and that they get a chance to work through situations where things could go twenty different ways and they have to dig through their toolbox of math skills to find one that works.

 

DS enjoys math but doesn't "breathe numbers" -- he prefers science and engineering. I know that AoPS is more work than he really needs to do for a science or engineering career, but it's worthwhile work on its own merits. (And IMO he's still too young to rule out other career possibilities, especially in fields where he has quite a significant aptitude.) Really though, whether he uses the specific math strategies he learns through AoPS or not, what I want him to take away from all this is an ability to tackle hard problems efficiently, without panicking and without floundering... and to get through to the end and distill the process to a straightforward explanation and answer. That ability really will serve him well in any field.

 

:iagree: This is an excellent description. I don't think that reading the book to them would be an issue at all. I think the main difficulty with that approach would be having to prevent myself from putting myself between the material and my child and wanting to add information. ;)

 

All the upper level classes my ds has taken are taught in the same method. Questions asked in a similar "Socratic" approach w/o directly teaching the material. The only difference between the cal book and the others is that there are a higher percentage of application problems.

 

I did want to make a comment on the bolded. I agree that it is more than necessary on one level, but the mental development via its approach is significant.

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The online AoPS courses that DS has taken were run very much like the books... No one says "here's how you do the problem, everyone watch" -- it's more like the teacher posts a problem and asks where to start, and the students start throwing out ideas, and the teacher pounces on the good ones and asks "why that", and they nibble away at it applying what they've used before or trying things that might not work, with the guidance of a teacher who can immediately correct something trivially wrong (arithmetic, for instance) but let them chew on the strategies for a bit.

 

I run my own math team this way... I give them hard problems and then I sit back while they hack at them. Sometimes there's a strategy they already know from previous problems that we can build on, but sometimes it's hidden in a very different way than they've seen before. I can jump in and say "wait, check that calculation" before they get too far with an error, but they sit around discussing whether their approach will work and explaining their reasoning to each other. If I think they're way off base I'll make them explain it to me while I pick apart their logic. If they come up with two reasonable strategies, I'll have them do both and then compare the results and figure out why they both work or why one doesn't.

 

The key, IMO, isn't independence vs. teacher guidance. It's who's doing the work. The work, in this case, isn't just the problems themselves but the reasoning. And the payoff, for me, isn't in doing a lot of problems right, but in having an extensive toolbox of potential strategies that could work in different situations... and the skill and experience to apply those strategies appropriately. Where that develops (in my experience) is from being faced with very challenging problems that draw on a variety of skills in an unpredictable way, with only the most minimal guidance of a breadcrumb here and there. Where I get concerned about people using AoPS isn't in whether the kid can be plunked in a corner with the book or not, but in whether the parent is figuring things out for the kid and then showing them exactly what to do. I like the balance that DS and I have, where he works primarily on his own but comes to me when he's stumped. If I have him explain what he's tried and where it fell apart he can usually see his own mistakes, and when he can't I can ask a question that nudges him in the right direction without saying "do this, then this, then this."

 

That's the independence I think you need. Not dumping a kid in a bare room with a book, but making absolutely certain that they do all their own thinking, that they're faced with challenges in figuring out the strategy (not just in performing it), and that they get a chance to work through situations where things could go twenty different ways and they have to dig through their toolbox of math skills to find one that works.

 

DS enjoys math but doesn't "breathe numbers" -- he prefers science and engineering. I know that AoPS is more work than he really needs to do for a science or engineering career, but it's worthwhile work on its own merits. (And IMO he's still too young to rule out other career possibilities, especially in fields where he has quite a significant aptitude.) Really though, whether he uses the specific math strategies he learns through AoPS or not, what I want him to take away from all this is an ability to tackle hard problems efficiently, without panicking and without floundering... and to get through to the end and distill the process to a straightforward explanation and answer. That ability really will serve him well in any field.

 

Excellent post, on so many counts!

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Okay, so I'd like to give a "Real Life Example" to see whether this would be considered independent working on AoPS or overly guided. I really am having trouble getting my head around this, I'm not sure why. With math up until now, I have rarely had to explain anything to DS. So today, I was working through some of the problems in AoPS Pre-Algebra just on my own for fun, and DS comes over and I ask if he'd like to do a few. "Sure mom". So he gets his pencil and paper and begins to work on the following question from the beginning of the book (again, discovery approach right? So no instruction at all)

 

 

 

Compute the sum of (2+12+22+32)+(8+18+28+38).

 

 

 

He looks at it for a moment, and then says (as I expected) "Well, 2+12 is 14....so I could take that and add 22....and get 36....." I interrupt him (Wrong?) "Is there an easier way to do this problem?" He thinks for a bit. "You could make 10 by adding." Okay, so looking at it another way. He begins to add 2+8 and then pauses. "Wait! 12+18 is 30!" and he does the rest of the problem that way. When we look at the answer, he sees how the book shows that making 40 (2+38 and so on) is even easier, probably, than the way he does it, and he's pleased to see that, while he got the right answer, there's an even easier way to tackle it.

 

 

 

He starts to work on one of the exercises at the end of the chapter. 1999+2001+1999+2001+1999+2001+1999+2001. He "sees" how to do it and solves it correctly. He probably gets the approach because he saw it done in the explanation of the previous problem (ie. rearranging the numbers to find an easier solution).

So, too much help? (granted, it's Sunday night, and he's tired, and this was "for fun")...I am more asking whether my "assistance" would be considered too much were this to be our main curriculum. I'm trying to get a handle, I guess, on what it means to "teach" AoPS. Does one "teach" AoPS? Or is the student supposed to be on their own, perhaps with the teacher sitting nearby for moral support? :tongue_smilie: Also, I fully recognize that the difficulty ramps up quite a bit.

Edited by Halcyon
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How much guidance is too much guidance?

 

 

The one thing I think you don't want to do, is guide them (with little hints) to make sure that they go down the right track the first time. AoPS is about trying one thing, finding it does not work, and trying something else (and maybe trying many more approaches depending on how hard the problem is.) It does take much longer to try many ways before finding the right answer, and this is where the frustration and persistence come in. I have seen my son do an entire PAGE of equations (2 columns worth) only to realize that the method would not work. IMHO, if I had guided him to the right approach the first time -- that would be too much guidance. Real science and engineering problems require you to try, try, and try again. AoPS sets you up to expect this kind of difficulty.

 

Ruth in NZ

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I don't think an adult can't use the discovery method with material they already know. The issue is a child...most children won't do the hard thinking if they already know the answer. It is difficult to backpedal when the material is already fully grasped, at least as far as the child is concerned. I'm sure there are children who will appreciate it nonetheless...I think these are the same students begging for AOPS to begin with.

 

But I quickly realized with my kids it won't work to relearn with AOPS. Partly I would have no idea of knowing whether they grasped the material in the "new" way...they'd still get the problems right either way.

 

Brownie

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I love AoPS for my oldest. We sit together and I usually read the text... he loves doing it together, thought he is capable of working on his own. He's great at math, but has severe dysgraphia so I let him answer orally or I scribe for him. It's not unusual for me to help him along the way, but by the end of the chapter he can do all the review problems and challeging problems himself. I know he's learning the material and he doesn't forget it once learned.

 

One "drawback" of AoPS is that it does not go back and review material. It covers a subject in a chapter until it is mastered and then expects the student to know it later even if it hasn't been reviewed. This is perfect for my oldest as review is completely annoying and unnecessary. It is not the way that all students learn.

 

I've seen some say that AoPS is to teach toward competition math. While a few of their online classes are specifically to teach toward a specific contest, I don't think the books are that way at all. Many of the problems in the chapter are from math contests, but this is just because they are great math problems that are on the topic being taught. Other chapters have very few problems from math contests as these topics don't show up much in competition math. I think AoPS is pure math for the mathematically inclined student.

 

According to Richard Rusczyk, they estimate only about 5% of their online students are homeschoolers. By far, most of their students are supplementing their public/private school courses.

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The key, IMO, isn't independence vs. teacher guidance. It's who's doing the work. The work, in this case, isn't just the problems themselves but the reasoning. And the payoff, for me, isn't in doing a lot of problems right, but in having an extensive toolbox of potential strategies that could work in different situations... and the skill and experience to apply those strategies appropriately. Where that develops (in my experience) is from being faced with very challenging problems that draw on a variety of skills in an unpredictable way, with only the most minimal guidance of a breadcrumb here and there.

 

:iagree:

 

This is exactly how I approach mathematics in my home. We work hard to fill up the "toolbox" so that those tools will be handy when the time arises for their use. I really can't see working through a course of mathematics for elementary any other way. The tools need to be in place first, in order to face the challenges later.

 

What we are doing to achieve that end is working through our math program, extending the lessons at times to twist things up a bit, changing the problems to ramp up the difficulty level. That has helped tremendously, and I'm hoping it is paving the road for further travels along our path in mathematics. (for example--looking for patterns, using properties of mathematics, looking for connections, etc.)

 

I can't see why an older Dolciani text, or any algebra text for that matter, wouldn't work for this purpose. There are many vintage texts free on google books, for access to "muscle math" problems. Why not write a problem or so a day on the dry-erase at the beginning of the school day and see if it can be solved independently, based upon the skill set which the child owns at present? It's what we're doing with a 60's Dolciani, and it's working so far. (A sling-it-out-there-and-see-what-happens kind of thing. Let them work as far along as possible.)

Edited by Poke Salad Annie
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:iagree:

Why not write a problem or so a day on the dry-erase at the beginning of the school day and see if it can be solved independently, based upon the skill set which the child owns at present?

 

That's what we're doing with AoPS right now. It's been eye-opening and enjoyable.

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Personally, I let my son do the problem his way. THEN I say, let me show you how I would solve it. Sometimes it's not even any simpler (because it's my way...not the book's way) and I don't realize that until I'm done...it's just different. He is learning from watching me that there are many ways to approach a problem. Usually my way is more elegant and usually it matches the approach in the book. We're content with this. If the kid doesn't see the elegant way and is getting the right answer, how on earth is he going to "discover" the elegant way? How does an 11 year old even grasp "elegant" until he's seen the different a few times?

 

Saying "Is there an easier way?" is a GREAT approach, but then if he says no...well I just don't think it's an issue to show the child. You are modelling thinking about the possibility of a more elegant solution and hopefully eventually they will take more time to seek out these solutions on their own but until then I'm demonstrating the approach.

Brownie

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Based on its unconventional nature I am wondering from those who it worked for what you did to prepare for it?

 

I'm not using aops prealg as intended by the authors. I skip around. We do it together on the white board. Sometimes dd is the teacher. Some days I am. Regardless of how we use it, it is still beneficial.

 

For instance, today we did Exercise 5.1.1 on the white board. We used different colors of markers. Dd solved the problems a few different ways. She can answer them all 'in her head' of course but I had her go back and show me how she got the answers.

 

The aops police didn't come confiscate my text. There really is no hard & fast rule as to how to use aops prealg. Otherwise I'm in big trouble.

 

We are working in chapter 5 of the text. We may be here for a few months. It's all good. I love the approach of aops but would NEVER use it exclusively for dd at this age. Dd is young and not extremely mathy so we are not your typical aops customers. I like too many other programs which teach dd directly. Aops helps to keep life interesting. :)

 

HTH you, Derek. For $57 it is worth it to purchase aops to use it as reference and for variety if you don't choose to use it as your main program. I know you just purchased HoE so it may seem you are spending your dc's college fund on math products. I know how it feels.

 

I am not an aops expert like the other moms here. My older dc didn't use this method and do extremely well in high school math. Dd8 is graphing equations and learning function, domain, range, etc in her other math programs. Aops prealg doesn't cover that yet so we are really turning the typical prealg time line on its head here. Oh, well. We're having fun in the journey.

Edited by Beth in SW WA
grammar fail
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Personally, I let my son do the problem his way. THEN I say, let me show you how I would solve it. Sometimes it's not even any simpler (because it's my way...not the book's way) and I don't realize that until I'm done...it's just different. He is learning from watching me that there are many ways to approach a problem. Usually my way is more elegant and usually it matches the approach in the book. We're content with this. If the kid doesn't see the elegant way and is getting the right answer, how on earth is he going to "discover" the elegant way? How does an 11 year old even grasp "elegant" until he's seen the different a few times?

 

Saying "Is there an easier way?" is a GREAT approach, but then if he says no...well I just don't think it's an issue to show the child. You are modelling thinking about the possibility of a more elegant solution and hopefully eventually they will take more time to seek out these solutions on their own but until then I'm demonstrating the approach.

Brownie

 

I'm guessing you were gearing this response towards me, but either way, it's helpful :tongue_smilie: I appreciate it!

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The aops police didn't come confiscate my text. There really is no hard & fast rule as to how to use aops prealg. Otherwise I'm in big trouble.

 

 

:lol:

 

 

Aops helps to keep life interesting. :)

 

I agree. I have yet to figure out how we're going to incorporate it this coming year, but I'm glad I own it. We'll use it in some way; if not as our main curriculum, then as a fun add-on.

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I'm guessing you were gearing this response towards me, but either way, it's helpful :tongue_smilie: I appreciate it!

Yes - it was sort of in response to you. Your post was a reminder to me to ask my son first if he sees a simpler way! I just don't see the issue with using AOPS the way you want to, though it can be helpful to realize if you're not using it as intended. Maybe the "ideal" way would work for some people but they just don't recognize that it is "ideal" so of course it is good to know how others are using it.

 

I have an 11 year old who was finding pre-alg boring and making a bunch of careless mistakes anyhow in Saxon. Is it really sacreligious to use AOPS not 100% as a discovery method? Even if I teach it straight to him, it's better than Saxon which just says "here's the distributive property...memorize what you can do".

 

With AOPS I keep telling my son (who by the way has an incredible memory!) "Buddy you don't need to memorize the rules for exponents...just think it through. You can figure out the rule each and every time." I want him to understand math like that, not just memorize the rules and AOPS helps us on our way. Plus with all those crazy problems with negatives and parenthesis everywhere, it still mandates that he learn to be more organized with his math in order not to make a silly mistake! That was a bonus I didn't expect from AOPS.

 

Brownie

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I'm not using aops prealg as intended by the authors. I skip around. We do it together on the white board. Sometimes dd is the teacher. Some days I am. Regardless of how we use it, it is still beneficial.

This is good to know that there is more than one way to use AoPS.

 

The aops police didn't come confiscate my text. There really is no hard & fast rule as to how to use aops prealg. Otherwise I'm in big trouble.

 

Haahaa, glad you didn't get arrested or anything. That is definately the funniest comment of the thread! :lol: Humor is a good thing when things get so serious during such discussions.

 

HTH you, Derek. For $57 it is worth it to purchase aops to use it as reference and for variety if you don't choose to use it as your main program. I know you just purchased HoE so it may seem you are spending your dc's college fund on math products. I know how it feels.

 

I am not an aops expert like the other moms here. My older dc didn't use this method and do extremely well in high school math. Dd8 is graphing equations and learning function, domain, range, etc in her other math programs. Aops prealg doesn't cover that yet so we are really turning the typical prealg time line on it's head here. Oh, well. We're having fun in the journey.

 

I'm glad to see you are having fun with it, even if tailored to use in a different way that works for your family. I'm thinking there may be value in using AoPS even when not according to the official approach. Many have stated that if its not fun/enjoyable/interesting for the child then its not really good to use anyway. So I don't think it has to be all or nothing. I am surprised to hear some using it only as a supplement to keep things interesting with their regular curriculum. I think I may ask our library to order Algebra or Pre-A just to check it out. They are pretty good about that when given a valid reason to purchase it.

Edited by dereksurfs
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Sorry for the sidetrack here: Can a child use AoPS pre-algebra after SM 5? Do we need to finish SM 6 before we start? What about finishing SM 6 and going to AoPS Algebra without the pre-algebra?

Thanks!

 

We're going to go right into Pre-A (Aops/Lial's/Dolciani or some combo thereof) after MM5 (he had been using SM until this this year, but I decided to switch to MM for a number of reasons). So yes, I think you can do it assumign your child is having no problem with math.

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Sorry for the sidetrack here: Can a child use AoPS pre-algebra after SM 5? Do we need to finish SM 6 before we start? What about finishing SM 6 and going to AoPS Algebra without the pre-algebra?

Thanks!

 

We are working through SM 5B and supplements (Life of Fred etc) and plan to go to AoPS pre-algebra next semester.

 

At least that's the plan right now. Threads like this one sometimes make me question everything. ;)

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Sorry for the sidetrack here: Can a child use AoPS pre-algebra after SM 5? Do we need to finish SM 6 before we start? What about finishing SM 6 and going to AoPS Algebra without the pre-algebra?

Thanks!

Fwiw, my dd finished SM 6 before she started the AoPS pre-algebra, however, had the pre-algebra book been available last year, I think she would have been fine skipping SM6 and going straight to the pre-algebra.

 

Imo, there is a big leap between SM6 and AoPS Introduction to Algebra. Whether or not you decide to complete SM 6, I would definitely complete AoPS pre-algebra before you begin AoPS Introduction to Algebra (especially if your child is elementary age)

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