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Thinking about highschool math...


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I've enjoyed reading along the math thoughts threads and have been trying to get my head around some math-related stuff here! Would love some insight from you wise and clever people!

I'm pretty comfortable with using 'curriculum' as a teacher support for elementary/lower-middle  school maths, I'm comfortable with the topics and how I prefer to teach and introduce them.

I'm not yet comfortable to the same level with upper-middle/highschool maths. I have aops algebra (intro), dolciani, saxon, assorted other textbooks. I like different things about them so am finding it hard to settle. I realised that I don't have a good overall vision/direction/scope for highschool maths so I'm getting distracted. I'm still relying on the book to tell me/my kid what to do because I feel on unsteady ground.

So I'm wondering, what topics and subtopics would you like to see in a highschool maths course? Do you prefer them separate (like Geometry one year, algebra, calculus) or intertwined (grade 10 maths that has a little bit of all those subjects)? Do you prefer - as an example - to cover Pythagoras/similar triangles etc before quadratic equations (as in one textbook I own), or after as in the AOPS algebra before geometry sequence? I feel like I'm missing what the maths-worldviews are that explain those different trajectories? I'm sure that ultimately the order doesn't make a huge difference, I'm just trying to understand the overall big picture of highschool maths. I'm very much a big picture person, I need direction and an end goal otherwise I get distracted and unmotivated by only the details.

I guess simply, what is your favourite highschool maths course, scope and sequence, book series? And why? Any other recommended reading or sources?

For background, my oldest is in 9th, quite maths adept (like me) but very independent. I have 3 more coming up behind her, of varied natural aptitudes. My highschool maths experience was the grade level mixed topics kind, it's the norm here (Aus). I took higher level maths in years 11 & 12 but struggled with terrible teachers and zero encouragement, which I still kind of resent, so I never went higher than pre-calc stuff.

Looking forward to your thoughts. Thanks in advance!

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I like AoPS because it's very similar to how I studied math in high school.  Single subject at a time with plenty of contest math practice thrown in.  And we all took the AMC every year.  

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I think you'reasking the right questions!  If you arent someone who knows what is supposed to be taught, you have to find something that you can trust will teach it and teach it well.

That is actually how I found the WTM forums yrs ago. I had purchased MUS's alg and geo program at a homeschool conference when my oldest was finishing Horizons 6. I knew he wasn't ready to make the leap from workbook to high school alg textbook, but he was ready for alg..  The worktext of MUS seemed great.....until we started completing it. Even without a math background I knew it was not covering concepts in a way that challenged ds enough.

I started looking to see what math books were in the online bookstores at top prep schoolsl or looking to see if I could find their math syllabi online. Foerster's was the book most frequently used. I asked on a Catholic yahoo loop if anyone knew anything about it and someone said that they'd bet someone on the WTM forums would. (and of course they did. 🙂 )

Anyway, once I found Foerster's was considered  a solid textbook that taught well, I focused on it.  That gave me the framework I used from there. (For example, our local engineering school used either Sullivan or Larson for precal/cal. It just depended on the section. When I saw Chalkdust and DO used those texts, I was ok with them for their teaching videos, etc.)

I didnt hear about AoPS until our youngest ds was in 8th grade.  AoPS is great for some students, ds loved their courses. Dd strongly disliked the approach. I knew by then that our regular textbook approach was solid enough bc my oldest had graduated from college with honors with his chemE degree. 

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Thank you 8fill, I really appreciate your post. 

Yes, that's where I am. I spent some time this afternoon comparing the contents of algebra texts from Dolciani, Aops, Cambridge Essential Maths 9 and Jacobs. I struggled to find a table of contents for Foersters. I may also look at a few others tomorrow. I was trying to get an overall sense of the general consensus, common topics and common sequence.

I want to do the same for 10th/11th/12th as well, that's when I realised that this question was probably too big for a simple forum post 😄

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I really started relying on the book in the way you're saying when we got to geometry. After that, I decided to bow out and put them in a class beginning with Algebra II. I feel zero sense of regret about it. Teaching high school is a lot. I do parts, but I don't feel like I have it in me to do everything and I'm okay with that. I'm really, really glad we went through algebra I with me though, because I know their foundation was pretty good.

Which is not to say that you can't push on through. For me to feel successful, I think I have to be several steps ahead. Because my background is in teaching humanities and that's what I studied, I'm maybe hyperaware of the things I see homeschoolers doing a bit off when they get to high school when they don't have a college humanities or liberal arts gen ed level education. And that's not to say it's some terrible failure. Maybe they're more able to break free of some conventions that I'm hold on to? But also, I always have the sense that because I don't have an education beyond pre-calc and stats that I am missing some fundamental pieces even when I'm helping with Algebra II homework. 

But, again, that's a personal choice.

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Thanks Farrar! Thinking about this is more for me, I always liked maths and felt robbed of being able to pursue it further. Whatever my individual kids end up doing for each class, I want to know for myself 🙂

As it happens, my oldest much much prefers to work her way through AOPS at her own pace, using the videos on their website if she gets stuck. Kind of like a flipped classroom, we go through it together afterwards and if I think she needs more practice/understanding, we'll pause and cover the topic from another angle (either my own or another book). AOPS is very different to how I learned maths so not fully knowing 'where' it's going was a bit disorienting, and that's kind of what led me here. 😄

I appreciate you sharing your experience Farrar!

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Well that's very reassuring Square! Thanks for your input! I feel like those are also my priorities, we spent a lot of time on the why of manipulating fractions - eg. showing with pictures and manipulatives why multiplying the reciprocal works. I like that discovery aspect of aops actually, that they have to discover the why rather than just practice a given algorithm.

Have you come across a favourite sequence or book series for highschool maths?

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4 hours ago, square_25 said:

I think if you like the AoPS style, then it's really a pretty good sequence! I do think it'll require filling in, though. Like, I really like the discovery aspect, but I don't find that they spend quite enough time on any given concept. On the other hand, if you use the book and not the classes, you have a lot more latitude about how much time to spend with each idea. 

I agree, and there seems to be a lot covered in those books. I love Mr Ruczyk's videos though! We took about 18 months to get through their pre alg, I'm gearing up to do it again with my 7th grader soon (he's working through a more basic 7th textbook plus life of Fred's pre alg books first.)

Maybe I'll make a comparison chart, with all my free time 😄

Is the general US sequence Alg 1, Geometry, Alg 2, Pre-Calc/something else? Here we have Year 9 maths, year 10 maths, then different choices for 11 & 12, basic maths and two harder classes, maths methods and specialist maths. I'm going to dig out those scope and sequences...

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I really like AoPS, but my kids were not getting enough exposure to  build certain skills.   We are in the second half of AoPS Algebra and I have added Saxon into the rotation.  We are using Saxon because I had it on-hand and it is solid.  It is similar with how I was taught in math class.  I wish I had had Beast/AoPS for competition math.

We are currently on our Saxon phase and I expect AoPS Algebra will go smoother when we return to it.

We do math year-round which gives us the time to go take the Saxon side-track. 

Surprisingly my kids like the predictability and certainty of Saxon.

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3 minutes ago, square_25 said:

Having taught AoPS, I'd be careful with mixing it with Saxon, at least from what I've seen... at least, I'd definitely do some sort of rotation where they had to work on concepts from a conceptual standpoint for a WHILE before drilling it. Because my experience is that even with AoPS, things become rote unless you get reminded of the motivations. 

I've just been doing square roots with my prealgebra class... we do it for two days and then the kids are supposed to remember all the square root simplifications 😕 . Like, that just doesn't work. I'm making them constantly square things to check which identities work. Like, is sqrt(2)*sqrt(3) the same thing as sqrt(6)? Well... square sqrt(2)*sqrt(3) and see if you get 6, lol. But if I didn't make them check, I'm sure they'd wind up just memorizing it and the conceptual stuff would escape them. 


Thanks for the feedback.   I often torture my kids by making them explain why.

My kids were understanding the conceptual components, but they were sloppy in execution such as in dealing with negative numbers in the distributive property.  I felt they needed more practice and actually wanted more consistent and quicker processing for it.     We have been doing Saxon over the summer and so far have not covered much that wasn't covered by AoPS, but they have gotten more consistent in giving written solutions and working through a set of questions every day.

It helps that in Saxon my kids are on the same lesson which makes it easier for me to teach/correct.   A kid can get stuck on an AoPS section for quite a while; I find monitoring progress on a self-discovery approach a lot more challenging.  

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1 hour ago, Eilonwy said:

How did you get to this point, @LMD? I’m thinking about some of these questions too, so I’d be interested to hear what your approach has been so far, and what has been important along the way.

Hoo boy that's a question 😄

I guess it started when my oldest was about 7 and working through singapore math grade 2. She started crying and resisting, which was unusual for her. I had to realise that the book was a tool for us, not a master. I let her take the chapter review as a test (she aced it) and we moved on. She was literally bored to tears with subtraction and regrouping. She still tends to prefer the challenging maths, even though she gets moody and frustrated (teen).

We have meandered a lot on maths, I don't know that we have ever finished a book in a year. We jump around, use 2,3,4 different books at once and move on when we're ready. What this means is that I've looked at a lot of books and scope and sequences. I'm on my 4th run through elementary math now and I have a pretty solid idea of what is important to me, what the overall goal and next destination is, and how I prefer to get there. 

I think it's just a lot of exposure, I have my favourite books, but the actual topics covered in elementary math aren't that different between books. I was helping a (very non mathy) friend who was worried that her dd was behind, we went through the actual topics needed to be understood before moving on to algebra and she was very reassured. I figured there must be a similar 'actual topics are pretty much standard' idea for highschool maths - so here I am!

Over the years I read some teacher development stuff too, Liping Ma, AIMS stuff (especially math circles), I took Prof Boaler's class through Stanford U (beta testing) 'how to learn math' etc. Just generally being immersed in the education space - not just math - and forming my own priorities. I've been brushing up on my own math skills - just using Khan and YouTube mostly, since I haven't done much above algebra since highschool, which helps too.

So, what has ended up being important to me? Conceptual understanding is a big one. I had one kid (my second oldest) who could do the algorithm but get totally lost with the concept. We spent a long, LONG time doing long division with base-10 blocks, then a long time drawing base-10 blocks for every question. Once he had the concept ingrained, when he got stuck then simply asking him to explain what each number represented was enough to get him on track again. Ditto fractions & recently with simultaneous equations with my oldest. Real world maths, puzzles & wonder/fun are important to me too, reducing math to dry tedium is horrible - this isn't the same as silly/fad type stuff, maths for its own sake, done well, is satisfying. Number sense and place value are important too. I don't spend a lot of time on memorizing (we do some, but not to a high level of quick recall or anything), my kids tend to just remember with enough in-context practice.

What I tend to use now is: Miquon/MEP/Singapore for K/1st. Beast Academy/Singapore/whatever I have on hand for particular topic reviews for 2-6ish. My kids really like life of fred too so I use it as a treat alongside the others. 6th/7th I like to explore, shore up and review, my oldest did (some of) jousting armadillos and Russian math 6 and Key to Alg amongst other things. She started aops pre alg in 7th and alg in 9th. I know that many of the clever maths kids around here do algebra earlier but I was more interested in mastery than speed. 

My kids are all so different, a standardised path just wouldn't work as well. My 3rd kid is very naturally mathy, and he absolutely balks when I try to show him an algorithm. He much prefers to think it through and find his own way to an answer. He will passive aggressively just not learn something if he doesn't get the why, but if he gets the why then he makes amazing connections in his head. I can't just give this kid a Singapore workbook to practice a concept because he figures out patterns and tricks and ignores the lesson 😄 He's my most creative, most challenging, very clever kid, and he has challenged a lot of how I teach and parent...

Sorry that's probably rambly and I'm not sure I've answered the question, I've been dipping in and out of writing this during the day! 

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56 minutes ago, square_25 said:

I'm definitely not against drilling once the concepts are down! On the other hand, I've found that I vastly underestimate how long concepts I'm already fluent with take kids. I've been doing the unit circle in precalc with my kids, and I think a few months in, most of them have got it down, lol. Well, maybe half of them have gotten it down, which is already pretty good... the other half are either checked out or are using weird mnemonics from who knows where

 

I'm curious, are the students using these mnemonics in class or on the homework?  And how do you know they are "checked out"?  Not participating in class?  I'm not challenging you, but sort of curious because it's an online class, it must be difficult to glean insight to your students.  Thanks!  

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Thanks, @LMD, for describing your path so well! It gives me lots to think about and some ideas to try, as well as a better understanding of what led to your question.  I’m in the elementary to middle transition with my oldest, and observing how different math people interact with math styles, so I can see there will lots going on in the next few years. I am with you in wanting to see the big picture to move forward.

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4 hours ago, square_25 said:

The checked out ones just don’t participate or do the homework :-). AoPS classes have relatively high attrition rates.

 

Thanks!  I imagine AoPS has collected tons of data correlating homework performance, class participation, and discussion board participation.  I'm wondering if you are aware of any insights AoPS has gleaned over the years?  

Also, it's interesting to me how observant you are of the students in your class.  As someone who has taken a couple of AoPS courses and a mom of kids who have taken several, my impression has been that teachers drop in to run the lecture and maybe occasionally answer discussion board posts, but that's where there involvement and interest in student life ends.  Kind of like a college professor.  Do you find yourself to be unusual in the interest you take in your students?  Do you ever feel the urge to nudge a student  with a quick communication to encourage them to continue?  Has AoPS every studied something like this to reduce their attrition?

Finally, is there greater attrition lately during the pandemic compared with earlier years, perhaps owing to greater numbers of unqualified students?  

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I am not sure this answers the original question, but once I decide on what to use for Pre-A, I looked at what others (providers, WTM forum people, etc.) were using in the sequence after that so that I could have an idea of what sequence other providers felt worked. I guess there are multiple levels to that such as the scope and sequence, how one book feeds into another, and also how the texts are written/structured.  For example, between Dolciani Alg and Jurgensen Geo, there are not a huge amount of differences in how the text "looks." So even though there is a different author, it doesn't feel like a whole new experience, if that makes sense. 

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21 hours ago, LMD said:

What I tend to use now is: Miquon/MEP/Singapore for K/1st. Beast Academy/Singapore/whatever I have on hand for particular topic reviews for 2-6ish

Would you say that you use a similar mix of resources for each, according to what works best for you to teach and what you have the most enthusiasm for, or much more individual according to what seems to best suit each child? 
The biggest picture is probably helping your kids develop adult math skills and interest, which can go in a lot of different directions. How to get there - just thinking it over, and since you have so many resources, and are comfortable using a variety, would choosing a table of contents that makes sense to you from one of your resources work as an outline?  AOPS, say, or MEP for an intertwined one as used in Aus/Can? Then you could choose material to cover that objective from wherever you felt it was best covered.  This is just a random idea I got from your description.

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9 hours ago, square_25 said:

They do have tons of data, but I get the sense they are much more focused on growing the company than on fixing specific things in the classes. So I'm not sure the data gets used to an optimal extent. It's a real treasure trove... I've made really vast use of the precalc transcripts to see what kids stumble over when I've done my rewrites. I also did some following of cheating on the short-answer problems (I made an accidental "natural experiment") and came away with really depressing answers, lol. 

 

I did notice one new feature.  It used to be you could click the Get Help link for certain problems, even before you attempt your first answer.  I notice in my intermediate c&p class now, I notice that they seem to keep track of the most frequent wrong answers, because I'm now receiving additional hints based on the wrong answer I submitted.  It's really neat, kind of like a very basic AI.  

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36 minutes ago, Eilonwy said:

Would you say that you use a similar mix of resources for each, according to what works best for you to teach and what you have the most enthusiasm for, or much more individual according to what seems to best suit each child? 
The biggest picture is probably helping your kids develop adult math skills and interest, which can go in a lot of different directions. How to get there - just thinking it over, and since you have so many resources, and are comfortable using a variety, would choosing a table of contents that makes sense to you from one of your resources work as an outline?  AOPS, say, or MEP for an intertwined one as used in Aus/Can? Then you could choose material to cover that objective from wherever you felt it was best covered.  This is just a random idea I got from your description.

Thank you, yes using the table of contents as a kind of outline and mixing/matching is exactly where I'm going with this. I actually haven't looked closely at MEP's 10/11/12 years... I should.

Yes I tend to use a similar mix of resources, because I have my preferences about language/explanation, sequence and layout. I'm familiar enough with what I have now that I know the contents and know how to teach it. E.g. I really really like beast academy, but I would have no problem stretching it out for the whole of end of 1st/2nd to 6th grade, even into the 1st half of 7th. I feel like it is rich enough and solid enough, and I can see that it is building and preparing for higher maths, that I wouldn't rush through it, and I would pause where they get frustrated with other material.

How that has actually worked in practice is much more responsive, I won't plan ahead that I'll teach x concept using chapter 2 from this book and some exercises from chapter 4 of another book. What tends to happen is we use a spine book and as I see how they grasp and interact with the material, I might decide to cover it from another angle, grab more drill-like practice exercises or more puzzle/word problems. A very simple recent example would be with my 1st grader, he grasped the addition/subtraction number bonds concept through Miquon, but needed more practice before moving on so we are currently going through some lessons with MEP, which he loves, which focus on different ways to build one number at a time within 10. All the while we move through a Singapore textbook as a spine. 

My method is more like, read through material and plan a quite detailed path, throw away the plan and follow the gist of the path as my intuition leads. 😄 I wouldn't have been so confident to do this when I started 10 years ago, but I am now. I guess I just haven't got the familiarity with high school level stuff yet and I'm trying to rush it!

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9 hours ago, square_25 said:

That sounds like DD8!! She totally refused to just keep practicing addition with regrouping by itself. The concept was easy to her and the practice was tedious. We had to work around her desire to stay challenged. 

 

We spend SO LONG drawing base 10 manipulatives. We'd draw boxes and dots -- the boxes would contain 10 dots, although we couldn't see them 😉. And more boxes and dots. And more boxes and dots. 

Interestingly, we started talking about fractions when DD8 was like 5 or 6, and then when she was 7, I wanted to move to doing formal fractions via pictures... and she balked. Apparently, over the 2 years before that, she had internalized verbal ways of working with fractions, and that's all she was willing to do. She'd do pictures to oblige me, but they didn't match her mental model. I actually yelled at her about it one time, then felt horrible... but yeah, the pictures didn't help her at all. I laid off of her after that. 

 

It sounds like you took an excellent path 🙂 . You taught your kids exactly where they were and were mindful of what they were interested in. I strive to do the same... it's easy for me in math, and not so easy in other things! 

Thanks for your response! That is what I strive to do and is one of the reasons we homeschool, I'm not sure how successful I've been, time will tell I guess 😄 so far none of them hate maths and they're all pretty on or above grade level, capable and fairly confident.

One thing I emphasize a lot, I forgot to add in my super long post, is mistakes/wrong answers. We experiment a lot, trying one thing, estimating, just trying something. I always say that wrong answers are wonderful pieces of information, they signpost you in the direction to the correct answer. One thing I learned in Prof Boaler's class was that a wrong answer fires more synapses than just getting everything correct the first time (even if you don't realise you made an error). Getting 100% in math isn't a cause for celebration here, it means that you weren't challenged, learned nothing, didn't grow your brain. I like that about AOPS too, they encourage just trying something to see what happens, building with what you have to try and find what you're missing.

Thanks everyone for letting me just waffle on in this thread, it's helped my thought processes 😄😄😄

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21 hours ago, LMD said:

E.g. I really really like beast academy, but I would have no problem stretching it out for the whole of end of 1st/2nd to 6th grade, even into the 1st half of 7th. I feel like it is rich enough and solid enough, and I can see that it is building and preparing for higher maths, that I wouldn't rush through it, and I would pause where they get frustrated with other material.

I feel better reading this, because we are about on track for the first half of 7th grade with finishing  BA here. I don’t want to rush and push when I can see she is steadily learning and has a good attitude to math.

It’s been helpful reading your “waffle” and the responses to it, thank you for sharing it!  I am inspired to play more games and to step back more readily when something isn’t clicking to look at it from a different perspective. 😀

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On 8/22/2020 at 7:50 PM, LMD said:

For background, my oldest is in 9th, quite maths adept (like me) but very independent. I have 3 more coming up behind her, of varied natural aptitudes. My highschool maths experience was the grade level mixed topics kind, it's the norm here (Aus). 

My high school maths experience is similar to MEP maths since mine was the Cambridge exams.  My kids did geometry with algebra, calculus with statistics. They reinforced their applied maths skills with physics. 

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2 hours ago, Arcadia said:

My high school maths experience is similar to MEP maths since mine was the Cambridge exams.  My kids did geometry with algebra, calculus with statistics. They reinforced their applied maths skills with physics. 

Thanks Arcadia, did they use a particular book series?

Eta, I would love to integrate physics but my own physics knowledge is pitiful. Did you 'match' the physics and maths somehow? My own maths learning in highschool would have been much helped with some physics, would have helped all those graphs make sense! My dh and I each have a piece of the puzzle in that sense, he gets the physics and not the maths, if we work together we can usually puzzle things out 😄 

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19 hours ago, Eilonwy said:

I feel better reading this, because we are about on track for the first half of 7th grade with finishing  BA here. I don’t want to rush and push when I can see she is steadily learning and has a good attitude to math.

It’s been helpful reading your “waffle” and the responses to it, thank you for sharing it!  I am inspired to play more games and to step back more readily when something isn’t clicking to look at it from a different perspective. 😀

Thank you, I'm glad you've enjoyed it 😄

I should clarify re beast academy that I'm talking about my state/country (Aus) standards for maths. I did go through our state standards and compared it to beast, and I could cover to the end of 6th grade maths without anything more than (I *think*) 5a... and that to a much, much higher standard. 

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1 minute ago, Arcadia said:

Algebra and Geometry was using AoPS.

Calculus was AoPS, dual enrollment for DS15,  Larson textbook for DS14

Statistics was Yates (https://www.amazon.com/Practice-Statistics-Daren-S-Starnes/dp/142924559X) but MEP maths would have worked too

So they did aops alg and geo concurrently? Did they alternate chapters? How long did that take? 😄

Sorry for the rapid fire questions!

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17 minutes ago, LMD said:

So they did aops alg and geo concurrently? Did they alternate chapters? How long did that take? 😄

 

DS15 was intermediate algebra with intro to geometry 

DS14 was intro to algebra with intro to geometry

I didn’t keep track but it was about 18 months continuously since we school year round. They started with the books then did the online classes then finish the books. 
Math was basically the only subject they spent much time on since they spent less time on the other subjects.  That was in middle school so I didn’t need to care about accountability/grades.

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3 hours ago, LMD said:

I should clarify re beast academy that I'm talking about my state/country (Aus) standards for maths. I did go through our state standards and compared it to beast, and I could cover to the end of 6th grade maths without anything more than (I *think*) 5a... and that to a much, much higher standard. 

I’ve done the same comparison with our provincial standards, and year 5 covers almost everything in that grade range here too. Graphing is the exception, I think. It’s easy to get twitchy about saying you’re doing “Gr. 5“ into early Gr. 7, though, even though I know the material is reasonably  equivalent.  

What has been your experience with doing AOPS/BA or other challenging-type math programs with kids for whom math is not their passion and their primary interest? Do you have this situation in your family? Would you emphasize it less or choose more procedural approaches based on interest, or is it worth digging into for all your kids?

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12 minutes ago, Eilonwy said:

What has been your experience with doing AOPS/BA or other challenging-type math programs with kids for whom math is not their passion and their primary interest? Do you have this situation in your family? Would you emphasize it less or choose more procedural approaches based on interest, or is it worth digging into for all your kids?

I can share our family's experience with gifted math students who are older.  I have a son who started alg at age 10.  We didn't know about AoPS until he was in 8th grade.  At that pt he had already completed alg 1, geo, and alg 2.  He took AoPS intermediate in 8th.  He loves math and theory.  AoPS fit his personal needs.  (He graduated from high school with a large stack of notebooks full of thought experiments.) He continued with AoPS through cal and is now a 3rd yr physics grad student.

I have a dd who is currently a college sr who is equally good at math (or was until she stopped pursuing math at his level).  He convinced her to take AoPS alg.  She did not enjoy its approach at all and didn't want to continue with it.  I also have a 5th grader who is probably the most gifted of all of my kids.  I tried BA with her and she also didn't like it (her dislike was stronger than her older sister's dislike of their alg.)

FWIW, I do disagree with the bolded comment.  Simply bc other texts provide more direct instruction of concepts does not mean they are "procedural."  There is a wide range of possibilities between AoPS and procedural math programs.

 AoPS tends to dwell more in theory.  Other textbooks tend to focus more on  applied math.  As a matter of fact, I recently had a conversation with him about his littlest sister (age 10) and whether or not he thought I should consider AoPS for her (she is currently on the same math trajectory as he.)  I expected his answer to be an emphatic yes bc he had been so adamant with his other sister.   (Those 2 were incredibly close growing up and obviously much closer in age.)  Anyway, he said no.  He said to have her go through Foersters for alg bc in hindsight that he realizes that Foerster was just as vital in forming his math foundation as anything he learned through AoPS.  (Not what I was expecting him to say.)  (FWIW, I also have an older ds who is a chemical engineer who graduated from our homeschool long before I ever heard about AoPS.  Our non-AoPS approach equipped him well for engineering. )

That's one family's experience.  AoPS is a good program for those who enjoy it.  But, it is not the only solid approach.

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30 minutes ago, Eilonwy said:

I’ve done the same comparison with our provincial standards, and year 5 covers almost everything in that grade range here too. Graphing is the exception, I think. It’s easy to get twitchy about saying you’re doing “Gr. 5“ into early Gr. 7, though, even though I know the material is reasonably  equivalent.  

What has been your experience with doing AOPS/BA or other challenging-type math programs with kids for whom math is not their passion and their primary interest? Do you have this situation in your family? Would you emphasize it less or choose more procedural approaches based on interest, or is it worth digging into for all your kids?

Oh Yes, I'm definitely susceptible to the twitch 😄

Hmm, I wouldn't say that maths is a passion or special interest for any of my kids. Some are naturally more capable in some ways, I would describe my more naturally capable ones as being generally more intuitive people, the less obviously capable ones are the over thinkers. I see it in their music too, one kind tends to pick it up easily and 'feel' their way through - they spend 20% of the energy learning the first 95%, then the last 5% takes most of the work, the other kind tends to need explicit instruction and repetition to master the tricky parts first then it all comes together. They are both capable of similar outcomes, with similar effort, the effort just comes in different spots and needs different support - does that make sense?

So I still use largely the same materials but with different emphases as needed. For example, my current 7th grader did most of beast academy through the book 4s, alongside that he did just the reviews in a grade 6 textbook (which he found easy -  actually I recall he was motivated to work through that before sitting a maths competition paper). He is currently doing a run through a year 7 textbook, the life of fred preA series and dips in to BA5 occasionally. He finds the yr7 book very easy. I would like him to try aops pre alg afterwards but I honestly have no idea how he'll go with it, he may like it or hate it 😄 stay tuned I guess !

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2 minutes ago, square_25 said:

I don't agree with that, sorry. I haven't seen much truly applied math in other textbooks. Ultimately, an elementary textbook is hampered by the fact that it has to communicate very basic stuff. So, the "word problems" tend to be fairly boring stuff -- not truly applied. 

For me, where AoPS fails is expecting all kids to like being frustrated. But just because the AoPS way isn't the only way to communicate the concepts doesn't mean the concepts don't matter. I just think you've figured out ways to communicate them that work better than AoPS for your kids. I have, too 🤷‍♀️. I know my kids better than a generic program. But it's not like our program is less "conceptual" because we don't do it the AoPS way. 

Disagree all you want, but it does not mean that others have to agree with you.  My ds may not have his PhD in math, but he does have his BS in math (and BS and Master's (so far) in physics).  He isn't making an off the cuff comment without any validity to his perspective.  He does have insight to his own math experience in what he has experienced as a student in an applied math heavy field. 

FWIW, I don't consider AoPS an elementary program, but high school. Foerster's is full of applied word problems and his texts have equipped my kids to succeed in applied math fields.  If my ds says to stick with it for his sister, I value his real world hindsight perspective.

 

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2 minutes ago, square_25 said:

Yep, I have zero expectation of people I disagree with agreeing with me. Pretty much by definition. 

Did he specifically say that Foerster's provided value because it was more applied? Or because the style was more helpful for him? I've heard good things about Foerster's and am in no way disputing that it had been helpful for him, but I also tend to think that questions in math textbooks tend to be quite contrived compared to the messiness of the real world. It was even the same for calculus, and calculus is in fact a VERY applied field. But it's hard to do hard applied problems when you're just learning the material. 

I also would argue with the idea that AoPS is less applied because it's more theoretical. I personally wind up using probability estimations more often in my daily life than any other kind of math. And probability is essential for statistics. So for me, something like Intro to Probability is actually very much an applicable class, because it comes up. Yes, they are approaching it from a theoretical perspective, but that's not the only way you can use it. 

Considering I stated that he loved his AoPS courses bc he loved the approach and thrives in theory, I think that is a clear answer that he valued Foerster's for its applied approach, not bc the style was "more helpful for him."   Considering that any discussion  about students HAS to take place in terms of students learning the material vs. an adult using the material in their daily life, applied problems may be limited to the material they are learning, but Foerster's does an excellent job of teaching students how to think about scenarios and how to set up equations for solving.  That is a useful skill in and of itself. 

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Just had to share that I found a copy of Foerster's for only $50! Including shipping! Now to wait the eons for it to arrive in my part of Aus 😄

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12 minutes ago, square_25 said:

Post when you get it! I'm curious how you'll like it 🙂 . 

I wish one could see more samples of textbooks online... it's so annoying having to decide without having enough information. And I have very strong opinions on these things, lol. 

Haha I will. 

Yes it is frustrating. I snapped this one up because I've never found Foerster for less than nearly $200! 

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10 hours ago, 8FillTheHeart said:

FWIW, I do disagree with the bolded comment.  Simply bc other texts provide more direct instruction of concepts does not mean they are "procedural."  There is a wide range of possibilities between AoPS and procedural math programs.

Thanks for responding with such a detailed description of your family’s experience with various math curricula. I agree fully that there is a wide range of worthwhile possibilities besides AOPS. I was actually just using it as an example of a program that I was familiar with the approach (at least at BA level), and I’m sorry that it came across as a comparison. 

My question was more about teaching philosophy, I think. Do you strive for having kids who learn the concepts to a challenging degree right through high school maths, and even when maths isn’t their passion?

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57 minutes ago, Eilonwy said:

Thanks for responding with such a detailed description of your family’s experience with various math curricula. I agree fully that there is a wide range of worthwhile possibilities besides AOPS. I was actually just using it as an example of a program that I was familiar with the approach (at least at BA level), and I’m sorry that it came across as a comparison. 

My question was more about teaching philosophy, I think. Do you strive for having kids who learn the concepts to a challenging degree right through high school maths, and even when maths isn’t their passion?

Yes.  I understand your question and agree.  There are definitely varying degrees of depth and challenge in different math programs.  AoPS is definitely a challenging program.  The main difference in the approach to teaching concepts.  There are math programs that equip students just as well as AoPS in understanding concepts and how to use math in new/unique scenarios (though AoPS often exceeds the scope of high school topics. I don't remember for sure which class, but I think it was real analysis where ds said that something he had learned in AoPS was covered in that class).  Foerster's alg, for example, is a great math text.  His explanations are thorough and clear.  He incorporates multistep problems that they can't solve without understanding what they are doing.  He has them prove formulas like the quadratic formula.  

 

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1 hour ago, square_25 said:

I’m not sure what it means to learn the concepts to a challenging degree. I find that you can learn concepts very well without that much challenge. 


Hmm, I think I mean in a way that continues to stretch the student, to keep looking for ways to understand what isn’t simple for them.  But the question you’ve asked suggests that there may be a more useful way to look at things. Do you find that most  of your students learn more/better with less challenge? 

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On 8/27/2020 at 10:09 PM, LMD said:

They are both capable of similar outcomes, with similar effort, the effort just comes in different spots and needs different support - does that make sense?


Yes, that makes sense, and is a helpful way to think about it.  Thanks!

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7 hours ago, square_25 said:

I just find them orthogonal. Learning the concepts well usually involves working on them as long as a kid needs, finding useful mental models, seeing them from many angles. But I don’t find that it’s super related to how challenging the work is. Some kids thrive with challenge and some don’t.

Thanks for this mental image, that makes sense to me as a useful way to think about teaching. There aren’t any particular math concepts that are causing me difficulties right now. I’m trying some new ideas for showing the distributive property to my 9 year old following this discussion and I think it’s going well. 😀 

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On 8/28/2020 at 8:05 PM, square_25 said:

So, this is just my take, but I did everything topsy-turvy and had my kid USE the distributive property to multiply all the time, so that by the time we got to it in symbols, it was really obvious. 

I think we’ll play around with this for a bit, he knows it to a degree but not to the point where it’s really obvious.  
Thanks for the examples!

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On 8/28/2020 at 1:40 PM, square_25 said:

I don't agree with that, sorry. I haven't seen much truly applied math in other textbooks. Ultimately, an elementary textbook is hampered by the fact that it has to communicate very basic stuff. So, the "word problems" tend to be fairly boring stuff -- not truly applied. 

For me, where AoPS fails is expecting all kids to like being frustrated. But just because the AoPS way isn't the only way to communicate the concepts doesn't mean the concepts don't matter. I just think you've figured out ways to communicate them that work better than AoPS for your kids. I have, too 🤷‍♀️. I know my kids better than a generic program. But it's not like our program is less "conceptual" because we don't do it the AoPS way. 

I don't think AOPS expects all students to use their books though.  

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20 hours ago, square_25 said:

I know, but even the kids who sign up for the classes have lower frustration tolerances than we tend to expect!! 

One of mine is OK with frustration the one with ASD would probably hide the textbook.

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