Math and its sequential nature

[Not_a_Number]

I love “Math with Bad Drawings” in general, and this post is great:

https://www.google.com/amp/s/mathwithbaddrawings.com/2015/04/08/the-math-ceiling-wheres-your-cognitive-breaking-point/amp/

[Little Green Leaves]

2 hours ago, Not_a_Number said:

I love “Math with Bad Drawings” in general, and this post is great:

https://www.google.com/amp/s/mathwithbaddrawings.com/2015/04/08/the-math-ceiling-wheres-your-cognitive-breaking-point/amp/

This is very cute but as a not especially mathy person I resent being compared to a broken futon...

I mean, maybe I am not a futon at all. Maybe I was always a chaise longue.

 

[Momto6inIN]

I know from listening to my oldest DS talk about his something or other analysis and algebra class (where they do algebra but on, like, squares and things that aren't numbers  😲🤤😝 whatever that means) that I will never get some upper level math concepts. I don't know if that's because of how I was taught math (do this algorithm) or a limitation of my actual intelligence or a consequence of my interests and likes/dislikes or a combination of all 3 interacting and influencing each other,  but I feel it in my bones to be true. It literally makes my brain hurt to hear him talk about some of it lol

Edited November 5, 2020 by Momto6inIN

[Not_a_Number]

45 minutes ago, Momto6inIN said:

I know from listening to my oldest DS talk about his something or other analysis and algebra class (where they do algebra but on, like, squares and things that aren't numbers  😲🤤😝 whatever that means) that I will never get some upper level math concepts. I don't know if that's because of how I was taught math (do this algorithm) or a limitation of my actual intelligence or a consequence of my interests and likes/dislikes or a combination of all 3 interacting and influencing each other,  but I feel it in my bones to be true. It literally makes my brain hurt to hear him talk about some of it lol

I would actually guess I could teach most of those concepts to you. They aren't as scary as they sound. Would you be the most brilliant student ever? Probably not, lol. But they aren't actually hard. 

[Momto6inIN]

46 minutes ago, Not_a_Number said:

I would actually guess I could teach most of those concepts to you. They aren't as scary as they sound. Would you be the most brilliant student ever? Probably not, lol. But they aren't actually hard. 

I appreciate the vote of confidence lol, but I'm not sure you're right 🤣

[Not_a_Number]

1 minute ago, Momto6inIN said:

I appreciate the vote of confidence lol, but I'm not sure you're right 🤣

Eh, I'm pretty sure I am! 😄 Some things are really hard, but these things just SOUND hard. 🙂 

And yes, you can do "algebra" with objects that aren't numbers! It's pretty cool 😄.  

[EKS]

I think it's a combination of having all of the requisite skills in place and a hard ceiling--meaning that people who don't have the skills in place find themselves hitting a lower--frequently much lower--ceiling than they would have otherwise.  

But that doesn't mean that there isn't a ceiling, just as there is a ceiling on other things people learn.  Ability counts.  I think that high-ability people tend to seriously underestimate how much ability facilities their own learning, so they tend to buy into the idea that ability is overrated and that hard work made palatable by interest is what really matters.

[Not_a_Number]

5 minutes ago, EKS said:

But that doesn't mean that there isn't a ceiling, just as there is a ceiling on other things people learn.  Ability counts.  I think that high-ability people tend to seriously underestimate how much ability facilities their own learning, so they tend to buy into the idea that ability is overrated and that hard work made palatable by interest is what really matters.

It's possible. I've known so few people who've hit what I think of as their theoretical ceiling that it's hard for me to know. 

I tend to think the ceiling will come out more in the ability to do creative problem solving and speed than in an ability to understand concepts and to make logical deductions. At least, I've seen MANY kids whose ability to use logic in math was about 2% of their ability to use logic in real life... I'd say it's most kids. 

[EKS]

7 minutes ago, Not_a_Number said:

At least, I've seen MANY kids whose ability to use logic in math was about 2% of their ability to use logic in real life... I'd say it's most kids.

This is probably because their knowledge of real life is far better than their knowledge of math.  When you know a lot about something, it's easier to think about it flexibly.

As an example, I had an adult math student who needed help learning how to solve eighth grade level word problems.  Every time I gave him a problem involving money, he would breeze through it, even if it had the exact same structure as problems he was having trouble with.

[Not_a_Number]

1 minute ago, EKS said:

This is probably because their knowledge of real life is far better than their knowledge of math.  When you know a lot about something, it's easier to think about it flexibly.

As an example, I had an adult math student who needed help learning how to solve eighth grade level word problems.  Every time I gave him a problem involving money, he would breeze through it, even if it had the exact same structure as problems he was having trouble with.

Yep. Exactly. So then my question is what people's ceiling is if we spend more time letting kids honestly getting to know math, and not doing whatever the heck it is they do in school. There are really only SO MANY things to know in math. 

Anyway, it's an experimental question 😉 . 

[Clemsondana]

I remember reading years ago that people (researchers? educators?  I don't remember) and worked with low IQ people to see if they could teach them complex, challenging tasks.  What they found was that it was possible to teach people to do almost anything, but IQ seemed to determine the students' ability to generalize the task and use the same skills to do different-but-similar tasks.  It wouldn't seem crazy to me that the same thing applied to our 'IQ' for different types of tasks, in the same way that athletes seem to have skills that transfer across sports so that they can learn a new one much faster than a non-athlete.  The man mentioned above who could only do the problems if they related to $ might not be good at math, so he can do the specific work as applied to money, but can't generalize it to other types of problems.  Or, his brain might not process that sort of problem using 'math' circuits because he's done that type of problem so often that it encodes in the 'money' or 'business' neuron pathways.  Brains are weird, particularly with processes that have become automatic for us so that we don't consciously think about how we do them any more.

That doesn't mean that somebody couldn't practice the methods until they could do them in many areas, or learn to think in 'math similes' so that they can substitute.  I also think that some people really do struggle with abstract concepts.  So, they might struggle to solve algebra problems but have no problem doing the same problem in a construction or money situation because then it isn't really abstract to them.  I've read some of Barbara Oakley's books about learning how to learn and how she went from being terrible at math to being an engineer, so I do think it's possible to rewire a brain to learn new tasks even if they don't come naturally.  But, I wouldn't be shocked to find that we have different abilities to do that, similarly to how I think we can pretty much all learn to make a lay-up in basketball, and probably a free throw, but we probably wouldn't all be able to play competitively. 

I also think that people can make the completely rational decision that it isn't worth it to invest as much time and energy as it would take to learn something because it would take more work for them than is 'reasonable', whatever their standard is for that.  I struggle with where that line is, though.  On one hand, I don't want to track students out of opportunities at an early age.  On the other, having some students learn algebra or geometry in an abstract way as taught in school might be a frustration for both them and their teacher, but teaching some of the same skills practically (by business math and carpentry, for instance) might be a great fit. I had a friend who struggled to get through high school math (I'm not sure he got through algebra 2, so I'm not talking about calculus - he may have just gotten through algebra I and then done business math).  He got a good paying job as a machinist and was great with his hands.  A few years out of high school, he was talking about maybe taking a class to learn trig because he was seeing how it was used at work.  He said that it would have been a waste in high school because he didn't see the use, and probably didn't 'see it' at all, but after years of using it he wanted to learn how it worked so he could use it better.

 

[Not_a_Number]

@Clemsondana -- but see, I think the "abstract way" we teach it at school isn't abstract at all. It's just the wrong way to teach it. 

[Xahm]

My understanding based on observation and the stuff I've read is that some people definitely get better returns on their investments in studying different areas, but pretty much anyone can learn more than they know now. Some people have an aptitude for math or music or whatever, and they can advance more easily than many others. That doesn't mean that others can't learn or shouldn't be encouraged to keep trying, but I think it's something that each person considers as they make choices. I am terrible at drawing. I have improved some with a bit of work, and I know I could improve more with more work. I've known this for years, but it hasn't been important for me to be able to draw, so I have chosen not to use my time that way. I think I have a lower than average aptitude for drawing, but if I had a goal that needed decent sketching ability to meet, I'd put in that time, and if I had a greater aptitude and so could get a quicker/greater return on my investment, I likely would even without a specific need. As it is, I don't feel it's worth my time right now and I'm ok with that. I also suspect that if I threw myself into it, I'd find that the law of diminishing returns applied. I'd get to be decent with a reasonable amount of work, but becoming really good would take most of my time and concentration, and there's just not enough time in the world to make me Van Gogh. I think it's probably like that in math. Most people can get to basic calculations with decent teaching and a bit of effort, but this will be extremely easy for some and extremely difficult for a few. As the level and complexity increases, the percentage of people for whom it comes easily decreases. To get to the highest levels requires a fair investment for anyone, but it requires a lot more investment for some than others. For most people, that investment just wouldn't be worth it. For a few or perhaps for many, there probably isn't time in the world to get there. A highly motivated student with a goal should never be discouraged by any talk of aptitude, but it's reasonable for a person who doesn't need higher math to choose not to put in the work needed to get there. There may well be a number of students for whom the work needed to get to an average standard is just too much to make sense unless they are motivated to do it. I think we should give every student excellent instruction starting in the early years (a bit of a dream), but we shouldn't scoff at those who make reasonable decisions to stop their studies before Calculus, for example, or even Algebra. Time and attention are limited resources, and applying them wisely should be encouraged. (This may be very rambling; I'm sorry. I'm suffering from a bit of a deficit of attention at the moment)

[Not_a_Number]

8 hours ago, Xahm said:

I think we should give every student excellent instruction starting in the early years (a bit of a dream), but we shouldn't scoff at those who make reasonable decisions to stop their studies before Calculus, for example, or even Algebra.

Oh, sure, I don't disagree with that. It's just that I think MOST people stop because they missed a key step, not because they've hit their limit. If everyone gets excellent instruction that honestly immerses them in mathematical thinking, then I can think that people are simply making good decisions. Right now, I don't feel that way at all. 

(By the way, I'm pretty decent at drawing and enjoy it, and I do think that the ability to draw is a lot less crucial to one's functioning than mathematical REASONING. Note I said reasoning and not "higher mathematics." I would much rather explain probability to someone who stopped at fractions but understood them than someone who got to calculus and never really got fractions.) 

Edited November 6, 2020 by Not_a_Number

[kbutton]

In terms of remediation, I think that both IQ and a good teacher are crucial. By remediation, I mean anything from someone has an LD to missing class on a day of instruction (even with a good teacher) for a concept was apparently crucial to that specific student.

For instance, my son has a pretty profound, but narrow, problem with expressive language. He's also profoundly gifted. Remediation for his language issues has been amazing, but I know that another student whose issues were more entrenched and/or who has a lower IQ and/or whose logic is less developed would not necessarily make that level of progress.

But in the sense that my son missed something fairly sequential about language development, his remediation went back to preschool/Kindergarten levels, and nothing short of doing so helped him out. Once that was in place though, he's been able to accelerate right on through the other levels at an increasingly rapid pace, but someone else would not necessarily be able to do make that progress without a lot more effort and more in-depth re-teaching. The initial problem my son had was deep and difficult, but once he had it, the rest of the remediation has been light because after a point, he could globally incorporate that information going forward. Someone else might need every one of those connections forged anew.

[Not_a_Number]

8 minutes ago, kbutton said:

But in the sense that my son missed something fairly sequential about language development, his remediation went back to preschool/Kindergarten levels, and nothing short of doing so helped him out. Once that was in place though, he's been able to accelerate right on through the other levels at an increasingly rapid pace, but someone else would not necessarily be able to do make that progress without a lot more effort and more in-depth re-teaching. The initial problem my son had was deep and difficult, but once he had it, the rest of the remediation has been light because after a point, he could globally incorporate that information going forward. Someone else might need every one of those connections forged anew.

I've seen this with math. I have lots of kids in my AoPS classes where I KNOW exactly where I'd have to go back, but I can't possibly do it in the context of the class 😞 . And yes, it requires figuring out the missing piece of the futon before you can remediate 😉 . I have enough experience to be good at that, though. 

Interestingly, DD8 is obviously gifted at math, but one thing I've noticed teaching her from this perspective is that she unexpectedly talks like a teeny mathematician and not like a kid. Like, she honest-to-goodness doesn't have gaps. She knows what she understands, and she understands it all in a good way. It's actually kind of uncanny. I see very few kids like that at twice her age, and I include very smart kids. 

(I actually don't think she'll turn out to be a mathematician, but it's a very unusual presentation in a little kid. She writes better proofs than almost all the kids in my college classes, even the math majors, because they really did NOT know what they knew and what they didn't. Their confidence was pretty eroded by years of teaching they had only half understood.) 

[drjuliadc]

I love anyone who references neuroplasticity in regard to education. There is no way the entire brain is plastic, but intelligence isn’t. IQ and any other form of intelligence is just as changeable and improvable as any other aspect of neurology. You just have to exercise it. People have too fixed of a mindset about math.

Yes, there is a ceiling. There is a ceiling on everything because there are only 24 hours in a day. The ceiling is way, way higher than where almost anyone stops.

[Not_a_Number]

2 hours ago, drjuliadc said:

People have too fixed of a mindset about math.

They really do. And I think our culture encourages it. 

[drjuliadc]

It is definitely cultural, the fixed mindset about math. Asians seem to have a less fixed mindset about it. Maybe Russians too, although I only see Asians and Indians at the math enrichment things around here. We are the lone caucasians.

I know Indians are technically Asian, but they are culturally different enough that it is hard to put them in the same category.

It is only “Asians” at my kids piano stuff, no Indians.

I’m in the company of “tiger moms” a lot. I’m too much of a slacker about it to actually fit the category and too athletic. I mostly use hacks, games and technology to accomplish things too. I’m sure there are limitations to that.

 

[Not_a_Number]

20 minutes ago, drjuliadc said:

It is definitely cultural, the fixed mindset about math. Asians seem to have a less fixed mindset about it. Maybe Russians too, although I only see Asians and Indians at the math enrichment things around here. We are the lone caucasians.

I definitely see some Russians at math enrichment things. When I give AoPS grades, it unmasks the names for me, so I see who the people are -- and yeah, it's largely Asians, Indians, and Russians. 

 

20 minutes ago, drjuliadc said:

I know Indians are technically Asian, but they are culturally different enough that it is hard to put them in the same category.

It is only “Asians” at my kids piano stuff, no Indians.

I’m in the company of “tiger moms” a lot. I’m too much of a slacker about it to actually fit the category and too athletic. I mostly use hacks, games and technology to accomplish things too. I’m sure there are limitations to that.

Where do you find tiger moms to hang out with, lol? I'm not a Tiger Mom, though, I don't think... or, at least I care a bit too much about what the kids like and don't like for that title, I think. Although I haven't actually read the book, so I could be wrong! 

[drjuliadc]

I don’t make time in my life for socializing, (probably not good because I love people) but I have incidentally found tiger moms at Abacus class, math circle, piano recitals, and I have a friend of a friend who is a tiger dad. He moved to our area just for the centuries old $21,000 per year private school that I would like except for all the homework.
 

I am really more of a Doman mom, but a slacker at that too.  I find them at gymnastics, agility training ( called Ninja class at a karate studio, but another Doman mom pointed out to me that it was actually agility training), and by going to Doman seminars In Philadelphia.
 

I used to find them on Brillkids forum, which isn’t active now. That was my favorite place. That is where I found out about this forum. Although, if people are Doman oriented, they don’t mention it here.  You could get bloodied and bruised by the hatred for that type of thing on this board. I just don’t care and I love everything about Glenn Doman and I still mention him anyway. It is not really hatred, but a misunderstanding and lumping ANY early learning into “developmentally inappropriate” category. 
 

If you start out with kids in the academically gifted category to begin with, it can be torture to them and invite a lot of misbehavior to not be Intellectually stimulating them at a pretty high level. It is easier to just teach them algebra, or let them think they invented it. My one son substituted letters for numbers in math equations on his own and I told him that was algebra.  He liked the idea that he “invented” algebra and I just went with it. We are working on humility with him.
 

Btw, I even forgot arithmetic from disuse, so I haven’t taught anyone algebra, I am referencing you teaching your daughter algebra.

 

 

Edited November 13, 2020 by drjuliadc
Too many actuallys

[Not_a_Number]

2 hours ago, drjuliadc said:

Btw, I even forgot arithmetic from disuse, so I haven’t taught anyone algebra, I am referencing you teaching your daughter algebra.

Ah, yeah, she loves algebra. She’s been using it relatively fluently for a year now. She would have probably seriously disliked math due to boredom at school. 

What’s a Doman mom? I know Tiger Moms from our Hebrew School, but it’s not quite my thing — I like my kids to have a fair amount of free time, and I care about their social time.

[drjuliadc]

On 11/13/2020 at 7:47 AM, Not_a_Number said:

What’s a Doman mom? I know Tiger Moms from our Hebrew School, but it’s not quite my thing — I like my kids to have a fair amount of free time, and I care about their social time.

Glen Doman was a physical therapist who treated brain injured children. His children carry on his work at “The Institutes” in Philly, also known as IAHP.

They often start with children in a coma and regularly get them out of a coma. Their success rate in teaching brain injured children to read is either 85 or 88%. I can’t remember exactly. That includes those that started out in a coma, blind, deaf or all three. They have treated Downs Syndrome children who go on to graduate from college.

He described neuroplasticity several decades before anyone else, and utilized it in treatment. He had a deep respect for mothers and what they could accomplish with their children. He advocated homeschooling, maybe even zealously.

He just accidentally found that if you applied his methods to a neurotypical child, you get a result that is a little hard to believe. I have applied these methods to my own children (in my slacker fashion) and the results are a little hard to believe.

[Not_a_Number]

48 minutes ago, drjuliadc said:

Glen Doman was a physical therapist who treated brain injured children. His children carry on his work at “The Institutes” in Philly, also known as IAHP.

They often start with children in a coma and regularly get them out of a coma. Their success rate in teaching brain injured children to read is either 85 or 88%. I can’t remember exactly. That includes those that started out in a coma, blind, deaf or all three. They have treated Downs Syndrome children who go on to graduate from college.

He described neuroplasticity several decades before anyone else, and utilized it in treatment. He had a deep respect for mothers and what they could accomplish with their children. He advocated homeschooling, maybe even zealously.

He just accidentally found that if you applied his methods to a neurotypical child, you get a result that is a little hard to believe. I have applied these methods to my own children (in my slacker fashion) and the results are a little hard to believe.

Oh, fascinating. Any suggested things to read? 

[drjuliadc]

I have only read What to Do About Your Brain Injured Child and How Smart is Your Baby. I read the first one first because that is an intense area of interest of mine.  I didn’t have children then. I don’t think I would have liked the second as much if I hadn’t been so impressed with the first.

[drjuliadc]

I need to add that I so wholeheartedly believe in everything the Doman people say, do and believe and that is so unusual for me.  My political affiliation is a tree hugging libertarian who doesn’t believe in legalizing drugs, just one example how I can never stay entirely in one camp.

The only exception is they love discovery methods and I don’t. Haha.  See, they would agree with you and you are probably right.

Actually, the way they teach neurotypical kids is very much a form of classical education. They don’t call it that, but it really is, another reason it is a little boggling that WTMers are so dead set against their methods.

They advocate a classical method and militantly recommend homeschooling. What not to like? They start it earlier than deemed acceptable to WTMers.

They don’t explicitly say this either, but I feel they advocate homeschooling because if you use their methods, you are going to end up with a gifted child even if they wouldn’t have been gifted otherwise, so gifted that either regular or private school is inappropriate.

I have one of those children, adopted from foster care. His biological father worked in a factory and his mother worked at MacDonalds. She had drugs in her system when he was born.  He was believed to be deaf in one ear and one eye dropped. He had many retained primitive reflexes when we got him at 22 months old. 

Last year, his gifted program 3rd grade teacher, who has been one for 30 years, said he was one of the most intelligent in his class. He is good in all subjects other than spelling, and he is acceptable at that. He was the only one who got straight As one grading period and he was the only one who had perfect attendance. He truly is a walking encyclopedia in certain subjects.

Edited November 17, 2020 by drjuliadc

[drjuliadc]

The Wikipedia on IAHP reads a lot like the Wikipedia on chiropractic, coloring both as ineffective and unsafe. Hence, why I personally wouldn’t put any stock in either.

1/2 of our practice pays completely out of pocket for our services. (They have coverage but their copay is as high or higher than our fee). If we weren’t wildly effective, they wouldn’t.

Our practice was completely fueled by referrals from other patients for decades, until Google reviews became a thing. Now we get a significant amount through internet search because our reviews are so good.

My malpractice insurance is between $40-50 per month for 3 million of coverage, unheard of in any specialty. If we were hurting people right and left, it wouldn’t be that cheap.

[kiwik]

On 11/17/2020 at 8:16 PM, kand said:

It’s not just that WTM’ers are dead set against t their methods, it’s that the medical and scientific communities around the world have spoken out about their methods and the damage done to children and families. Here is just one statement: https://www.saphysio.co.za/media-your-health/press-releases/position-statement-the-doman-delacato-patterning-therapy-ddpt/

Also, as a side note on your statement about your son’s progress being remarkable because his birth parents worked at McDonalds and a factory, that doesn’t say anything about their intelligence level. There are a lot of brilliant people working in such jobs, but not everyone has the same life opportunities (or interests for that matter). I’m sure your son has benefited a lot from having you as devoted parents who are invested in his education, that statement just made me feel defensive on behalf of those who work in unskilled labor jobs and for their children, who can be just as smart as anyone else’s children (not that being smart is the end all either—I can get defensive on that front as well 😬). 

In fact a lot of gifted people end up working such jobs after they has succumbed to depression etc brought on by a poor educational fit.  And a lot of 2e people.

[jboo]

On 11/11/2020 at 7:06 PM, Not_a_Number said:

Where do you find tiger moms to hang out with, lol? I'm not a Tiger Mom, though, I don't think... or, at least I care a bit too much about what the kids like and don't like for that title, I think. Although I haven't actually read the book, so I could be wrong! 

The book's actually (and intentionally) hilarious, but there are quite a few people who read it who don't seem to get the humor. I found it well worth a read.

I have a bunch of friends that think I'm a Tiger Mom, but there are a lot of high-SES Koreans, Chinese, and Indians in the area that make me look like a kitty cat. Some I know from work, others from my older son's (very, very Asian) public school gifted program.

[Not_a_Number]

26 minutes ago, jboo said:

The book's actually (and intentionally) hilarious, but there are quite a few people who read it who don't seem to get the humor. I found it well worth a read.

I have a bunch of friends that think I'm a Tiger Mom, but there are a lot of high-SES Koreans, Chinese, and Indians in the area that make me look like a kitty cat. Some I know from work, others from my older son's (very, very Asian) public school gifted program.

Yeah, I really don't inspire to the height of Tiger Mom-ness exhibited by some Asian immigrants, lol. I do actually care about what my kids' preferences are 😉 . 

[drjuliadc]

The tiger mom book WAS hilarious. She definitely meant it that way.  No one who wrote anything about the book seemed to get it though.  I thought that was funny too. They probably didn’t even read it. I am just glad people “not getting it” led to a lot of publicity for her.

[Not_a_Number]

2 minutes ago, drjuliadc said:

The tiger mom book WAS hilarious. She definitely meant it that way.  No one who wrote anything about the book seemed to get it though.  I thought that was funny too. They probably didn’t even read it. I am just glad people “not getting it” led to a lot of publicity for her.

Ok, I'm definitely adding it to my list of books to read, lol. 

[ieta_cassiopeia]

Some people definitely hit a cognitive ceiling (for a few, it's bevfore algebra, for a rather more substantial group of people, it's some way into algebra and statistics). For most people, I think it is less a question of there being a cognitive ceiling as a cognitive scaffolding assembly.

 

Some mathematical concepts (easy ones) are well within one's grasp, and can be done with a small amount of scaffolding (or even with the ladder you keep in the shed). Other concepts (difficult ones) require every piece you have, a boxful of attachment pieces and a crew of assistants to ensure it's safe and doing its job correctly. In short, the more complex the concept, the longer and more difficult it is to support the concept's construction successfully.

If your relatives have borrowed your scaffolding kit (i.e. you have lots of demands on your time), no capacity for difficult maths until they've finished and returned it, and you've documented that everything is back where it belongs.

As people age (up to a certain point), they gain more resources they can use to help them mathematically. So a practical restriction now is by no means obliged to still exist a year or three later.

If you know you're building somewhere, you may leave the scaffolding in place so you can build on it. But not forever because if it's not doing anything useful, it just looks ugly and perhaps it's more needed elsewhere.

Any of these can lead to a situation where it's reasonable, given someone's personal situation, to stop building maths. They could go further, but it's not efficient for them, or their resources are needed elsewhere.