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45 minutes ago, mathmarm said:

I am not qualified to speak to the experience of raising Neuro-Divergent children--on either side of the bell curve. It's not a part of my lived-experiences or my research.

May I ask why you’re convinced your kids are typical? You were posting about teaching your 7-year-old algebra, I recall. Now, there’s nothing wrong with that. I do that, too. But I’ve TAUGHT average kids, and they aren’t able to do that. They just aren’t.

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15 minutes ago, 8filltheheart said:

Some on this thread might find this article worth reading: Can Progressives Be Convinced That Genetics Matters? | The New Yorker

That was a great article. I need to hunt down that “The Cult of Smart” guy.

OP, my kids have an extremely good idea of their intellectual abilities, such as they are. We score off the charts in self awareness 🤣 Gifted in it, you could say 🤣

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2 minutes ago, Not_a_Number said:

Ooh, thanks for linking. DH was talking about this article earlier and I’ve been meaning to look it up.

I find the anti-genetics attitude on the left really unscientific. 

Yeah we acknowledge the impact of genes in the makeup of other physical organs, but some people want to pretend that brains are immune.

In my experience, people tend to think that things they do well are the result of their hard work, and by correlation those who don't do those things as well haven't put in the same level of effort. This applies to intellectual success, but I have also seen it play out in artistic and athletic endeavors. 

The truth is that there is no even playing field when it comes to base level ability and ultimate potential in any area. While no-one magically achieves high levels of competence in something complicated (whether that is mathematics or ballet) without putting in time and effort, the level of competence achieved for a given amount of time and effort, even with identical instruction, will always vary according to innate ability in that area.

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1 hour ago, Not_a_Number said:

May I ask why you’re convinced your kids are typical? You were posting about teaching your 7-year-old algebra, I recall. Now, there’s nothing wrong with that. I do that, too. But I’ve TAUGHT average kids, and they aren’t able to do that. They just aren’t.

I don't think average 7-year-olds can be brought up to speed with CA's old Algebra 2 standards. But I do think that algebra can be part of any average kid's math education starting from the very beginning. And that it would help them if it were.

At 5 my kid could solve 6-A = 4 and 5 x B = 35. I think this is something that can be taught to any average 7-year-old or even 4-year-old.

It was a while after that, maybe a full year, before she could also solve 140-C = 69, not because she was lacking any algebra knowledge.

It will be a while yet before she can solve 10-D^2 = 4+D, before I tell her what a "polynomial" is, and before I teach her how to find the greatest common factor of two them.

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34 minutes ago, UHP said:

I don't think average 7-year-olds can be brought up to speed with CA's old Algebra 2 standards. But I do think that algebra can be part of any average kid's math education starting from the very beginning. And that it would help them if it were.

At 5 my kid could solve 6-A = 4 and 5 x B = 35. I think this is something that can be taught to any average 7-year-old or even 4-year-old.

It was a while after that, maybe a full year, before she could also solve 140-C = 69, not because she was lacking any algebra knowledge.

It will be a while yet before she can solve 10-D^2 = 4+D, before I tell her what a "polynomial" is, and before I teach her how to find the greatest common factor of two them.

Your top examples are basic elementary math problems and how Horizons teaches math.  Your 140-c=69 example would be taught anytime double/triple digit addition/subtraction was taught.  

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11 minutes ago, 8filltheheart said:

Your top examples are basic elementary math problems and how Horizons teaches math.  Your 140-c=69 example would be taught anytime double/triple digit addition/subtraction was taught.

I'd like to see some of Horizons. My impression, just from a little bit of experience poking around, is that many teachers and textbooks are allergic to introducing letters (standing for variables or unknowns) to young children. Beast Academy leaves them until 3rd grade.

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1 hour ago, UHP said:

I don't think average 7-year-olds can be brought up to speed with CA's old Algebra 2 standards. But I do think that algebra can be part of any average kid's math education starting from the very beginning. And that it would help them if it were.

At 5 my kid could solve 6-A = 4 and 5 x B = 35. I think this is something that can be taught to any average 7-year-old or even 4-year-old.

It was a while after that, maybe a full year, before she could also solve 140-C = 69, not because she was lacking any algebra knowledge.

It will be a while yet before she can solve 10-D^2 = 4+D, before I tell her what a "polynomial" is, and before I teach her how to find the greatest common factor of two them.

Yes, I do this kind of stuff with my average students and find that they benefit. But my 7-year-old (and mathmarm’s) wasn’t doing that. She was doing genuine manipulations with good understanding (including with multiple variables) and factoring quadratics.

You know that I have strong opinions on teaching making a tremendous difference. But it’s definitely at most half the story.

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12 minutes ago, UHP said:

I'd like to see some of Horizons. My impression, just from a little bit of experience poking around, is that many teachers and textbooks are allergic to introducing letters (standing for variables or unknowns) to young children. Beast Academy leaves them until 3rd grade.

I do shapes and not letters. They are equivalent and conceptually easier. 

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26 minutes ago, UHP said:

I'd like to see some of Horizons. My impression, just from a little bit of experience poking around, is that many teachers and textbooks are allergic to introducing letters (standing for variables or unknowns) to young children. Beast Academy leaves them until 3rd grade.

I don't have my grandkids' books, so I can't take a picture.

But problems like n+3=5 start showing up as soon as they master basic addition/subtraction.  

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10 hours ago, UHP said:

At 5 my kid could solve 6-A = 4 and 5 x B = 35. I think this is something that can be taught to any average 7-year-old or even 4-year-old.

My son is turning 5 in a few months and I don't think he could solve 6-A=4. He would stare at the equation and wonder what he was suppose to do with it.

He could solve that problem if I framed it using concrete things for example "If I put 6 jelly beans on my plate and I now only have 4 left how many did my sister take." With the symbols and numbers in front of him he could take the word problem and put it in symbolic form. That however would be a separate activity for him than solving the problem.  

Cognitively I think there is too much abstraction in those problems for a 4 year old. So if your kids could do that then perhaps they are way above the normal curve. 

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This is an interesting topic to me because helping kids to be in a position to achieve their goals (through teaching, volunteering, etc) is important to me.  My experiences have left me with jumbled thoughts.  I do think that many kids are limited by poor teaching and lack of exposure.  But, it's also clear to me that there are differences between kids. My older was doing simple fractions, perfect squares, and simple mental algebra including negative numbers as a 4 year old.  Clearly kiddo wouldn't have been doing that had they not been exposed to it (I wasn't trying to create a math genius - we started with counting as we put away blocks and kid thought it was fun and it kind of got out of hand...).  But, it seems equally obvious to me that the kid that I volunteered with, who took a couple of years to understand 2 digit addition (they finally got it on their 2nd pass through 2nd grade), would not have grasped the advanced concepts even with excellent teaching, and may never grasp those concepts if their rate of learning addition is indicative of how long it will take them to learn other math ideas. 

Many people can be taught systematically, but some intuitively understand things.  If a student understands quickly, it's cumulative - over a year they can learn many more things than a student who has to spend several days, rather than several minutes, to figure something out.  And, as they acquire more knowledge, they have more hooks upon which to hang new ideas, which helps them accumulate information and skills even more quickly.  And, I once heard a talk that described intelligence in part as the ability to generalize. That may be simplistic, but differences in the ability to generalize may be why some kids can see one 2-digit addition problem with regrouping and then be able to add numbers of any size, while other students spend several days learning to add 2 digit numbers, and then later spend several days learning to add 3 digit numbers and only generalize it over a couple of years. 

And, for many people, the ability to generalize or learn quickly doesn't necessarily apply across disciplines - most aren't excellent at everything.  I don't know anybody who plays multiple instruments, speaks multiple languages, plays multiple sports, is an excellent artist, and is especially advanced/competitive in multiple academic disciplines.  There is only so much time in a day, and we gravitate to things that we find interesting or enjoyable, or that we are good at.  Some differences can be overcome with good teaching or practice, but there comes a point of diminishing returns.  To some extent, this is why many of us don't DIY every aspect of our lives.  We just paid a handyman to do a few building projects.  We could watch videos and figure out how to do it, but it's not worth the time that it would take.  The same can be true academically - the kid who took 3 years to learn to add would take an inordinate amount of time to learn the quadratic formula...to what end?  

I do think, though, that while there is a role for genetics there may also be a role for exposures during the toddler years, such that kids arrive at school-age with very different starting points.  It's even possible that some of the differences are not correctable, like the way that there is evidence for the idea that, if you don't learn some language during certain formative years, you won't ever be able to learn any language.  I could imagine that not being exposed to complex language, vocabulary, linear thinking, spatial relationships, music, or physical movement, whether through play or direct instruction, could lead to permanent or difficult-to-remediate deficits, and possibly that the reverse - early exposure could lead to big advantages later.  I'll admit that this idea is somewhat horrifying when we think about the implications.  But, it does make me think about the nature of acceleration and giftedness.  Acceleration can just be staring early and moving at a normal pace.  Are the intuitive people who were born with differently wired brains, or are they people who got some good early exposure and formed some great neural connections and then they are off and running?  And, where does receptivity fit in with this?  My kids have had tremendous differences in their interest in learning and their willingness to be coached.  The curious coachable will likely advance more quickly than the combative disinterested.  Some of that is likely to be environmental - I read an interesting proposal that disadvantaged kids needed to be 'taught harder' because their upbringing hadn't taught them to try to figure out how things worked, etc, but my own kids have grown up with books and activities and museums and they are still very different people with different levels of frustration tolerance and diligence and that affects their progress.  

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1 hour ago, Clarita said:

My son is turning 5 in a few months and I don't think he could solve 6-A=4. He would stare at the equation and wonder what he was suppose to do with it.

He could solve that problem if I framed it using concrete things for example "If I put 6 jelly beans on my plate and I now only have 4 left how many did my sister take." With the symbols and numbers in front of him he could take the word problem and put it in symbolic form. That however would be a separate activity for him than solving the problem.  

Cognitively I think there is too much abstraction in those problems for a 4 year old. So if your kids could do that then perhaps they are way above the normal curve. 

I think it helps to have some practice. I've definitely seen lots of kids be able to do this around 5 or 6 if they were taught to try some numbers and see what happens.  

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I always succeeded in academics more easily, with less work, than most people around me. I had the opposite experience with physical skills -- I do less well than the average person learning a new physical skill. In my experience, and looking at my kids experience, putting more work in only exacerbates natural differences, because the person innately better at it gets more out of each unit of work.

I enjoy backcountry skiing, but am a slow physical learner. I've put in quite a bit of work and practice, enough to be competent and enjoy myself at an intermediate level. My kids are, I'd guess, about average in their innate skiing ability. My son loves it, and, comes skiing with me about half the time. He gets more of a boost than I do from each practice. So he's already better than I am, and that difference will get larger and larger the more we go out. It's even more if we get any sort of teaching, because he can integrate the advice more easily. My daughter is so-so about skiing and doesn't come very often. I get enough more practice than her that I am still learning faster, despite her better innate ability. We can all improve, would improve more with perfect teaching, and are all better skiers than people who haven't practiced. But those innate differences are real. It seems like some people find that easier to accept for physical than academic skills, but it seems equivalent to me. 

My math/science kid has always been "ahead" in math, but didn't work hard OR have good teaching. In fact, when he was young, math was only occasional and incidental to other interests (balancing a chemical equation in his chemistry obsession phase, a math game hiking, etc...). I'm fairly child led, and when he was younger, I generally only directly stepped in and taught/required things if I was concerned I didn't see them naturally happening. So I taught him to read (which I never had to teach my non-dyslexic kid). But I basically didn't teach elementary or middle school math, and am not at all sure how he learned it, but he had access to books and videos and games and parents who could understand and talk about those things. Exposure was there in his world. Everyone needs exposure -- without it, innate ability can't help you. A brilliant natural skier who's never seen snow can't do much.

There are so many things you could potentially learn, and only a relatively few (math, reading), that we tend to elevate to enough importance and spend enough time teaching a majority of kids these things that it even makes sense to call kids "accelerated." But I wonder if we spent more time teaching these things, in a better and more individualized fashion, would we make kids generally more or less even in their skills? Everyone would improve, but not all at the same rate.

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2 minutes ago, mckittre said:

But I wonder if we spent more time teaching these things, in a better and more individualized fashion, would we make kids generally more or less even in their skills? Everyone would improve, but not all at the same rate.

I think you're right and we'd make them less even. It would benefit everyone, but the outcomes would be very disparate. 

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28 minutes ago, Not_a_Number said:

I think it helps to have some practice. I've definitely seen lots of kids be able to do this around 5 or 6 if they were taught to try some numbers and see what happens.  

I definitely focus more on the conceptual learning of math vs being able to do the symbolism part of it. 

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Just now, Clarita said:

I definitely focus more on the conceptual learning of math vs being able to do the symbolism part of it. 

Whereas I start out with the symbols and try to translate them into words and visuals. So something like this (with a square to fill instead of a letter) would be very familiar to kids I work with. 

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You might be interested to know that

Quote

Cognitively I think there is too much abstraction in those problems for a 4 year old. So if your kids could do that then perhaps they are way above the normal curve. 

In 1966 Dr. Engleman published a book for parents and care-takers, Give Your Child a Superior Mind and in this book he outlines exactly how to systematically teach your 4 and 5 year old child to solve these exact types of problems. Though, this book asserts that working the program will induce giftedness in your child.

...The program we have outlined in this book is a systematic attempt to increase the size of the child's collecting sufrace, and the child who is taught by this program will be gifted by the time he is 5 years old. (p.315, Give Your Child a Superior Mind)

If it's truly possible to induce "giftedness" into people, then it's highly likely that we've "induced giftedness" in our children by now. But I'm not so sure that you can induce giftedness as I understand it.

However, a picture is worth a thousand words, so a video should be worth a few million words.

13 hours ago, Clarita said:

My son is turning 5 in a few months and I don't think he could solve 6-A=4. He would stare at the equation and wonder what he was suppose to do with it.

If he's not taught how to interpret or understand what he's staring at then I can't imagine what other options he'd have.

You will see a group of Dr. Englemann's class of socially disadvantaged preschool students solving such problems if you watch this excerpt until 4m48s and watch this excerpt  until 7m12s. Dr. Englemann explains that the children in this video received 1-2 years of Direct Instruction, 20 minutes a day at his experimental preschool.

If you watch the first 19 minutes of the 28 minute video you will witness that group of poor, socially disadvantaged preschool students

  • calculate the answers to simple addition, subtraction, multiplication and division problems
  • solve 2-digit addition with regrouping,
  • solve word problems about money and fractions
  • add/subract fractions--after demonstrat
  • solve 1 step algebraic equations that involve fractions -- ie 2/3 *A = 6
  • demonstrate an understanding of algebraic notation such as you can write 4*C or 4C  or 4*8 for multiplication, but never 48 for multiplication
  • compare fractions to 1
  • calculate area of rectangles given either the sides or the total area

 

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30 minutes ago, mathmarm said:

If you watch the first 19 minutes of the 28 minute video you will witness that group of poor, socially disadvantaged preschool students

  • calculate the answers to simple addition, subtraction, multiplication and division problems
  • solve 2-digit addition with regrouping,
  • solve word problems about money and fractions
  • add/subract fractions--after demonstrat
  • solve 1 step algebraic equations that involve fractions -- ie 2/3 *A = 6
  • demonstrate an understanding of algebraic notation such as you can write 4*C or 4C  or 4*8 for multiplication, but never 48 for multiplication
  • compare fractions to 1
  • calculate area of rectangles given either the sides or the total area

I've watched his demos, and I always do wonder how much of what they are doing the kids understand. Now, mind you, I think what he's doing is vastly superior to what currently happens to those kids... but I do worry about whether teaching such a widely disparate set of topics leads to rote memorization as opposed to flexible skills. 

I'm not completely sold on DI for math and other 'reasoning' disciplines, to be honest. I liked TYCTR in 100 Easy Lessons a lot with DD9 and I absolutely see the benefit of DI for things where there's a small, limited set of things to learn... but it's not how I conceive of teaching math and science. For me, the whole point of teaching math is to teach the kids to reason even when they haven't seen the exact same problem before, and DI doesn't exactly seem to do that. 

 

33 minutes ago, mathmarm said:

In 1966 Dr. Engleman published a book for parents and care-takers, Give Your Child a Superior Mind and in this book he outlines exactly how to systematically teach your 4 and 5 year old child to solve these exact types of problems.

I do agree that these kinds of problems don't require particularly advanced skills. With most of the kids I've worked with (including the fairly average one), exposure is enough.

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That video was interesting.  To me, there is a tension.  It's entirely possible to be proficient at a skill but not really understand it.  I'm pretty sure that I did most arithmetic without thinking about what I was doing up until I started teaching it to my kids.  But, I also had the idea that I wanted them to see what was happening, so we did a lot with blocks and toy cars and drawing.  My older absolutely would not do anything that kid didn't undersstand, so I had to come up with good explanations for why each step in an algorithm worked.  There was a bit of that in the videos - 'so we put the tens here, and add 10 plus 30 plus 20...).  Advanced math will likely go better if kid has more understanding than 'do the algorithm'.  But, that being said...many of the kids that I've worked with have neither understanding nor algorithmic ability, and they can lead functional, happy lives if they can just do arithmetic whether they understand it or not.  I've worked with kids who didn't know that 5x6 was the same as 6x5. It's easy enough to show using an area rectangle or blocks, but even with no understanding of why they'd be better off knowing that they are the same thing.  I've had students tell me that the problem will be marked wrong if they draw 2 groups of 12 instead of 12 groups of 2, and that is worse than not knowing - it actually causes them to believe that the answers must be different.  It's like going from educational neglect to anti-education.  

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Memorization plays an important role in understanding. For many, memorization precedes understanding.

The word problems that Englemanns group solve offer an interesting preview into whether or not they understand and make connections back to the real world. The children immediately recognize the significance of the mistake when Englemann plants the wrong unit in a word problem. One boy is wowed at the idea that a pie would cost $20 as opposed to 20¢.

Direct Instruction isn't the only way to teach such skills. It's one tool that's known and highly effective for teaching. It's available to anyone who wants to use it because it's a set of principles, more than a packaged curriculum. I'm citing the video as a neutral example of average children exemplifying above average competence in something both concrete and highly objective.

Those are preschoolers and kindergarteners in that video. Obviously, if you keep them on that track extend the instruction, the work becomes more sophisticated

Direct Instruction is highly accessible because it doesn't require oodles of special manipulatives and difficult to acquire training for teachers like Montessori and to a lesser extent Gattegno/Cuisseniare which requires special blocks and books.

A well organized and executed Montessori Mathematics foundation established before 1st grade can put children at the stage of insightfully tackling algebraic concepts and skills--typically reserved for 9th and 11th grade--at the start of 1st or 2nd grade with understanding.

A well organized and executed C-Rod based Mathematics program can have children understanding and working sophisticated calculations. The class in this video are 6months into 1st grade.

There are a number of techniques that can be used to build up the understanding and abilities of averaged children.

 

I don't know if successfully applying these types of methods can truly make a child gifted as the term is understood today.

 

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2 hours ago, Not_a_Number said:

I've watched his demos, and I always do wonder how much of what they are doing the kids understand. Now, mind you, I think what he's doing is vastly superior to what currently happens to those kids... but I do worry about whether teaching such a widely disparate set of topics leads to rote memorization as opposed to flexible skills. 

What do you see in the video, that makes you worry about this?

I like this passage from "superior mind" (fourteen months ago that video was my impetus to search for the book), which sketches an argument against worrying:

Quote

Reading is not consistent; mathematics is. This means that in mathematics the number of rules is small compared to the number of applications. Not so with reading. The number of rules is much greater, the scope of application much smaller. There are so many rules and so many skills that it is not expedient to make all of them explicit. It's more practical to present the reading task is a semi-sloppy fashion, allowing the child to work out various rules on his own. No method of reading will ever achieve astounding results with all children because of the nature of the task. The situation is quite different with mathematics. Since mathematics is deductive, a teacher can take advantage of the consistency to teach operations in one year that children normally don't learn in eight. Astounding results are possible because mathematics is a neat and clean subject.

(My emphasis.)

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1 hour ago, UHP said:

What do you see in the video, that makes you worry about this?

Nothing. I'd need to talk to the kids to see how well they could do things outside this narrow set of problems, though. Proficiency with a narrow set of skills is very different from actually developing problem-solving ability and the ability to flexibly use the math. 

Most kids I teach math have some set of things drilled REALLY well. So, for example, almost every kid in the AoPS class I teach can rattle off the equation for a circle with center (c, d) and radius r. And the quadratic formula. And the fact that to get the inverse function, you reflect it across y = x. 

But can they explain why those things hold? No, not usually. And can they use the associated abilities outside this narrow set of questions? Not nearly as well, no. 

 

1 hour ago, UHP said:

The situation is quite different with mathematics. Since mathematics is deductive, a teacher can take advantage of the consistency to teach operations in one year that children normally don't learn in eight.

I'm very big on acceleration, but I don't believe this statement about squishing 8 years' worth of learning into 1 😛 . I just don't. Overstating things weakens one's case. 

We take math very seriously in this household. We don't take summer breaks and we move forward as quickly as we can. And I'm absolutely sure that despite all that, and despite the fact that my kids are very, very mathy, I am sure I can't stuff 8 years of learning into 1 😛 . 

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3 hours ago, Clemsondana said:

I've had students tell me that the problem will be marked wrong if they draw 2 groups of 12 instead of 12 groups of 2, and that is worse than not knowing - it actually causes them to believe that the answers must be different.  

No, it oughtn't make them believe that the answers are different. They can absolutely be taught as symbols for DIFFERENT visuals without thinking that the answer is different. I believe that the kids you're working with aren't taught this effectively, but that's the teachers' fault, not the fault of the idea here. 

Look, if you asked someone to illustrate 4+5 and they drew a 3 and a 6, they'd be graded wrong, right? Even though the answer is still 9? There's nothing wrong with that. It's the same thing with multiplication. 

So if you're taught in class that 3*4 means three fours, you NEED to draw a 

**** **** **** 

to illustrate the question. 

There's nothing wrong with that. That's just logical consistency. I teach this way as well. 

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3 hours ago, Clemsondana said:

It's easy enough to show using an area rectangle or blocks, but even with no understanding of why they'd be better off knowing that they are the same thing.

I mean, they are better off knowing all sorts of calculational tricks?? That's not the first property of multiplication I tend to show, frankly. I tend to start with distributive and associative properties and only once those are firm do I move to the commutative property, because it's the only one of those properties that's simpler in symbols than it is in visuals... and that means that kids can absolutely wind up memorizing it without having any understanding of it (or of multiplication!) whatsoever. 

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42 minutes ago, Not_a_Number said:

I mean, they are better off knowing all sorts of calculational tricks?? That's not the first property of multiplication I tend to show, frankly. I tend to start with distributive and associative properties and only once those are firm do I move to the commutative property, because it's the only one of those properties that's simpler in symbols than it is in visuals... and that means that kids can absolutely wind up memorizing it without having any understanding of it (or of multiplication!) whatsoever. 

I just finished chpt 3 of Foerster's alg 1 with my 6th grader.  She breezed through it  Why?  Bc all of the concepts/terminology were familiar to her bc she had learned how to use them correctly and  by name throughout elementary school. The chpt's focus is a review of all of the basic axioms and properties that students should have mastered up to that pt.  I imagine that chpt is a nightmare for kids who have no idea of what they are doing or why.

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1 minute ago, 8filltheheart said:

I just finished chpt 3 of Foerster's alg 1 with my 6th grader.  She breezed through it  Why?  Bc all of the concepts/terminology were familiar to her bc she had learned how to use them correctly and  by name throughout elementary school. The chpt's focus is a review of all of the basic axioms and properties that students should have mastered up to that pt.  I imagine that chpt is a nightmare for kids who have no idea of what they are doing or why.

Yeah, I see this all the time -- kids struggling with these things in algebra because they didn't really get the sense of it in arithmetic. Whereas, actually, the properties are much more obvious in arithmetic! It's serious overload to try to teach something like "the distributive property" in variables when a kid doesn't know why 

3*4 + 4*4 = 7*4

or the like. 

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38 minutes ago, Not_a_Number said:

I'm very big on acceleration, but I don't believe this statement about squishing 8 years' worth of learning into 1 😛 . I just don't. Overstating things weakens one's case. 

We take math very seriously in this household. We don't take summer breaks and we move forward as quickly as we can. And I'm absolutely sure that despite all that, and despite the fact that my kids are very, very mathy, I am sure I can't stuff 8 years of learning into 1 😛 . 

I followed his cookbook with my daughter, over the course of July-December last year. You are right that she wasn't ready for ninth grade by the end of it. I agree that it sounds outrageous. But:

1. There are only five months starting in July and ending in December

2. A 4-year-old would be finishing 6th grade at the end of 8 years, not eighth grade.

3. I'm not sure what K-6 math education was like in 1966. (It was the last few years of the "new math" era. A table here claims that in the 50s, 1/4 of high school students took algebra, 1/9 took geometry (down from 1/3 in the 1910s))

So I'm not quite ready to grant that he was overstating it, even if it sounds outrageous.

55 minutes ago, Not_a_Number said:

But can they explain why those things hold? No, not usually. And can they use the associated abilities outside this narrow set of questions? Not nearly as well, no. 

I don't believe that being unable to explain why something is true is an indication that someone doesn't understand. (After all, before I encountered Engelmann I wasn't able to explain to my daughter why 8+1 = 9.) Putting an idea into words is its own skill, I think usually more than 8 times as hard as learning the idea.

 

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5 minutes ago, UHP said:

I don't believe that being unable to explain why something is true is an indication that someone doesn't understand. (After all, before I encountered Engelmann I wasn't able to explain to my daughter why 8+1 = 9.) Putting an idea into words is its own skill, I think usually more than 8 times as hard as learning the idea.

I'm pretty good at drawing kids out. I don't mean that they can't explain clearly. I mean that they genuinely don't realize there IS a why. It's just taken for granted. 

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3 minutes ago, UHP said:

I agree that it sounds outrageous

I got distracted reading about the 1960s and I forgot my point. What I find interesting about this passage is not the factor of eight:

Quote

Since mathematics is deductive, a teacher can take advantage of the consistency to teach operations in one year that children normally don't learn in eight. Astounding results are possible because mathematics is a neat and clean subject.

It's "take advantage of the consistency." The rules of language, geography, even physics and biology, are arbitrary. The rules of math are not arbitrary, and this has some interesting implications for how you can teach it.

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For my DD, the understanding that she was gifted lifted a weight off her and made things make so much more sense.
She had felt a sense of alienation in every day school, did not fit in. She thought it was because we are immigrants and stopped speaking in our native language for a year, because she thought that was causing the differences. Can you imagine how painful that was?
When she was tested in 3rd grade and learned that her brain just works differently, that was eye opening. 

Kids aren't stupid. If they are profoundly gifted, they already know they are different. They deserve an honest explanation.

Edited by regentrude
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On 9/10/2021 at 8:11 AM, maize said:

In my experience, people tend to think that things they do well are the result of their hard work, and by correlation those who don't do those things as well haven't put in the same level of effort. This applies to intellectual success, but I have also seen it play out in artistic and athletic endeavors. 

Yeah, because people are in a very strange position: 
kids are taught not to take pride in their intellectual achievements. It's ok to be proud of being a good athlete, but heaven forbid you are proud you're good at math - that's "bragging". The gifted kid is not really allowed to acknowledge that their brain does this stuff better than that of most people. That means, they need to believe their accomplishment is not a big deal, and why can't other people get their act together and just DO what they are doing? That goes for math, for public speaking, for musical ability.
The realization that the things they do effortlessly are hard for other people cannot come until they are able to acknowledge that they have a special gift. Otherwise they cannot reconcile their perception with the observed reality.
If I have the choir piece memorized after two rehearsals and can NOT accept that this is a special gift most people don't possess, then I can only despair why some folks in the choir are so lazy that they are still making the same mistakes week after week even though the director corrected them. 
And it's especially hard for girls and women, because we are socialized to downplay our abilities for the sake of not hurting anybody else's feelings. We're just hurting ourselves by not acknowledging that we are awesome at things. It took me until I was in my forties to admit to myself that I am really, really good at what I do, and that tv was OK to feel this way.

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1 hour ago, UHP said:

It's "take advantage of the consistency." The rules of language, geography, even physics and biology, are arbitrary. The rules of math are not arbitrary, and this has some interesting implications for how you can teach it.

But the thing is that it's quite easy to get kids to FEEL like they are arbitrary. I'd say more kids I know think of math as arbitrary than think of physics as arbitrary. Physics they feel like they can kind of test out in the world (you throw something up, it does come down again... ta-da, law of gravity!) Whereas with math, a lot of kids don't wind up doing their own "real-life experiments," and that can make the rules feel very mysterious. 

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On 9/10/2021 at 4:52 AM, Not_a_Number said:

May I ask why you’re convinced your kids are typical? You were posting about teaching your 7-year-old algebra, I recall. Now, there’s nothing wrong with that. I do that, too. But I’ve TAUGHT average kids, and they aren’t able to do that. They just aren’t.

I’m thinking this may be a manifestation of the currently popular notion that everyone can be successful if they practice enough (a la Malcolm Galdwell’s 10,000 hours) and it allows parents to pay themselves on the back for their kids’ successes if they are the one requiring the practice. When I had just a couple kids, and they were young, I thought I had a lot more ability to lead them to success than I have realized is actually true. They are all their own individual people. I can help support them in that and provide them with the education and other foundations they need, but there is no program I could systematically take them through that would lead them directly to adult success, no matter how young they were when they could do algebra and write novels. 
 

I think there’s also an element, now that there is more understanding that giftedness does not always come without challenges, to want to think one’s smart kids are just “normal smart” not “weird smart.” Similar to the way I thought my kid’s quirkiness as a child was due to being very smart, rather than a sign of underlying neurodivergence. 

8 hours ago, mathmarm said:

You might be interested to know that

In 1966 Dr. Engleman published a book for parents and care-takers, Give Your Child a Superior Mind and in this book he outlines exactly how to systematically teach your 4 and 5 year old child to solve these exact types of problems. Though, this book asserts that working the program will induce giftedness in your child.

...The program we have outlined in this book is a systematic attempt to increase the size of the child's collecting sufrace, and the child who is taught by this program will be gifted by the time he is 5 years old. (p.315, Give Your Child a Superior Mind)

But why?  In all these kinds of things, I don’t understand the purpose in going through all these programs in order to teach very young children impressive academic tricks and skills. What is the purpose of that for a very young child? That’s an honest and not rhetorical question, because I don’t know what the benefit would be. 

1 hour ago, regentrude said:

Yeah, because people are in a very strange position: 
kids are taught not to take pride in their intellectual achievements. It's ok to be proud of being a good athlete, but heaven forbid you are proud you're good at math - that's "bragging". The gifted kid is not really allowed to acknowledge that their brain does this stuff better than that of most people. That means, they need to believe their accomplishment is not a big deal, and why can't other people get their act together and just DO what they are doing? That goes for math, for public speaking, for musical ability.
The realization that the things they do effortlessly are hard for other people cannot come until they are able to acknowledge that they have a special gift. Otherwise they cannot reconcile their perception with the observed reality.
If I have the choir piece memorized after two rehearsals and can NOT accept that this is a special gift most people don't possess, then I can only despair why some folks in the choir are so lazy that they are still making the same mistakes week after week even though the director corrected them. 
And it's especially hard for girls and women, because we are socialized to downplay our abilities for the sake of not hurting anybody else's feelings. We're just hurting ourselves by not acknowledging that we are awesome at things. It took me until I was in my forties to admit to myself that I am really, really good at what I do, and that tv was OK to feel this way.

I think this is all really true. I’ve heard it repeatedly from my (gifted to profoundly gifted according to definitions) kids—“Oh, I’m not any better at that than anyone else, I’m sure everyone can do the same if they want” with things where that definitely isn’t the case. I think it’s intended as a form of modesty, but I’m not sure. 

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1 hour ago, KSera said:

But why?  In all these kinds of things, I don’t understand the purpose in going through all these programs in order to teach very young children impressive academic tricks and skills. What is the purpose of that for a very young child? That’s an honest and not rhetorical question, because I don’t know what the benefit would be. 

I think it's fair to be concerned about a family that is racing through hard material, and putting a lot of pressure on their kids. Obvious pressure or subtle pressure. Our exalted hopes will return as disappointment. If that applies to me and my kid, I hope it is only in a limited way. I am grateful for "Give Your Child a Superior Mind," but the title still horrifies me.

But I can think of some kinder reasons to go fast. At least some kinder hypotheses. One of them is that, when you the tutor work hard (or use a program that has worked hard) to remove the awful parts of education — confusion, unnecessary repetition, boring lectures that are hard to pay attention to — then going fast will be a side-effect.

For instance, if your gentle expectations for a kid are that they will learn addition in first grade, subtraction in second grade, multiplication in third grade, and division in fourth grade (that's my memory of math class in childhood: one button on the calculator per year), you might not be doing them a favor. Even if you are right that the returns in adulthood to going fast are not very great. Instead you might be ensuring that math lessons are boring when they could have been fun.

Editing to put a big asterisk on unnecessary repetition: it's not a great example. Repetitio est mater studiorum. I didn't really know it before I started tutoring my kid, and I believe it with great intensity now.

Edited by UHP
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9 hours ago, mathmarm said:

If he's not taught how to interpret or understand what he's staring at then I can't imagine what other options he'd have.

You will see a group of Dr. Englemann's class of socially disadvantaged preschool students solving such problems if you watch this excerpt until 4m48s and watch this excerpt  until 7m12s. Dr. Englemann explains that the children in this video received 1-2 years of Direct Instruction, 20 minutes a day at his experimental preschool.

Interesting. (It's a lot to digest and I'm not trying to be rude just very different from other things I've read and trying to wrap my head around all of it.) We did briefly go through the mathematical symbols, I do think he'd be capable of telling me 2 + 4 = 6 symbolically and verbally.

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5 hours ago, Not_a_Number said:

No, it oughtn't make them believe that the answers are different. They can absolutely be taught as symbols for DIFFERENT visuals without thinking that the answer is different. I believe that the kids you're working with aren't taught this effectively, but that's the teachers' fault, not the fault of the idea here. 

Look, if you asked someone to illustrate 4+5 and they drew a 3 and a 6, they'd be graded wrong, right? Even though the answer is still 9? There's nothing wrong with that. It's the same thing with multiplication. 

So if you're taught in class that 3*4 means three fours, you NEED to draw a 

**** **** **** 

to illustrate the question. 

There's nothing wrong with that. That's just logical consistency. I teach this way as well. 

I understand that, but when you have upper elementary kids learning to multiply by 10, 100, or 1000 and they truly don't understand that you can write it either way, they can't count by 34 1000 times as easily as they can understand 1000 34 times.  It's not just a matter of nomenclature or stating groups of...they truly believe that they will get different answers.  I try to help them see 34 thousands, so how do we write 1 thousand?  5 thousand?  10 thousand?, etc.  But, many of us (tutors and public school teachers, for that matter) aren't in a position to go back and reteach this stuff from the beginning like it should be.  I get a 4th grader for an hour and I have to work with them where they are, doing the assignment that they are given.  They are afraid to rearrange the numbers because they don't understand that they will give the same answer.  I drew it all out  - dots in groups of 8 - and worked with a student who both didn't know that you could figure out 6x8 by doing 5x8 and then adding 8 more, but they also couldn't immediately see that 40+8 was 48.  I'm actually volunteering at a different place that where I used to be, in a different part of town, with a different population.  I am pretty confident that these misunderstandings are widespread.  The kids see the answers as almost 'magic tricks' and any rearrangement will cause something different to happen because I"m not sure that, even with drawing, they understand what is being represented.  

Would I love to see them properly taught?  Absolutely.  I mean, the reason that I've been doing this for years is the hope that some of them will have a few aha moments and actually understand what they are doing.  They are going to struggle if/when they get to more advanced math.  But, would they be better off learning arithmetic by memorization without understanding if they aren't going to move into higher math?  I'd say yes.  These kids are likely to have to take algebra to graduate, and even if they don't understand what they are doing at least the manipulations can lead to correct answers.  If they can't do arithmetic, they can't balance a checkbook or estimate costs or figure out much they need to finish a construction project.  I don't think that my grandmother ever learned algebra, but she kept books for a company for years and was great at mental arithmetic.  I've had similar experiences with other older people - they didn't learn to assemble and disassemble numbers, but they were incredibly accurate with mental or pencil-and-paper calculations.  Would this be limiting?  Yes.  It's certainly not the path that I've chosen for my own kids.  But, the utter bewilderment of kids who can't figure out what they are being asked to do, and the feeling of accomplishment that I've seen with some kids when they start to be able to solve problems, has caused me to think about it a little differently.  Obviously I don't think this is optimal, and if I could redesign some programs or change some teaching strategies or pacing decisions I would.  But, in my volunteer world the options aren't 'ideal understaning' and 'less good understanding', they're more like 'can do it' and 'has no idea and thinks it's magic'.    

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1 hour ago, KSera said:

I’m thinking this may be a manifestation of the currently popular notion that everyone can be successful if they practice enough (a la Malcolm Galdwell’s 10,000 hours) and it allows parents to pay themselves on the back for their kids’ successes if they are the one requiring the practice. When I had just a couple kids, and they were young, I thought I had a lot more ability to lead them to success than I have realized is actually true. They are all their own individual people. I can help support them in that and provide them with the education and other foundations they need, but there is no program I could systematically take them through that would lead them directly to adult success, no matter how young they were when they could do algebra and write novels. 

I think there’s also an element, now that there is more understanding that giftedness does not always come without challenges, to want to think one’s smart kids are just “normal smart” not “weird smart.” Similar to the way I thought my kid’s quirkiness as a child was due to being very smart, rather than a sign of underlying neurodivergence. 

Right. It's very humbling to realize that one is not in fact responsible for one's child's success to the extent that one wants to believe. I know that for me, teaching DD9's friends has been very useful perspective. I've always taught a range of kids, of course. But teaching a wide range of kids her age has been very interesting. 

To be fair, I did find that most of my teaching methods worked with kids who are close to the average. But it's still the case that it was all slower and felt different. 

 

1 hour ago, KSera said:

But why?  In all these kinds of things, I don’t understand the purpose in going through all these programs in order to teach very young children impressive academic tricks and skills. What is the purpose of that for a very young child? That’s an honest and not rhetorical question, because I don’t know what the benefit would be. 

I find it useful to give kids early exposure to concepts, because I find that you can't actually rush them, which means that if you start early, you're much less likely to "lose" a kid later so that they are no longer understanding. 

Now, my kids turned out to be really bright, so using this idea has resulted in really serious acceleration. But I'll say that I believe in early exposure to concepts for basically all kids. I think it sets the stage for success. 

That being said, I don't do anything like DI. 

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Just now, Clemsondana said:

understand that, but when you have upper elementary kids learning to multiply by 10, 100, or 1000 and they truly don't understand that you can write it either way, they can't count by 34 1000 times as easily as they can understand 1000 34 times. 

Actually, in some sense, 1000 34s is easier than 34 1000s 😉 .

 

1 minute ago, Clemsondana said:

They are afraid to rearrange the numbers because they don't understand that they will give the same answer.  I drew it all out  - dots in groups of 8 - and worked with a student who both didn't know that you could figure out 6x8 by doing 5x8 and then adding 8 more, but they also couldn't immediately see that 40+8 was 48.  I'm actually volunteering at a different place that where I used to be, in a different part of town, with a different population.  I am pretty confident that these misunderstandings are widespread.  The kids see the answers as almost 'magic tricks' and any rearrangement will cause something different to happen because I"m not sure that, even with drawing, they understand what is being represented.  

In my very strong opinion, they are afraid to "rearrange the numbers" because they have no visual whatsoever for what they are doing. I actually have not yet met a kid without a math learning disability who couldn't see that 

3*8 + 4*8 = 7*8

as long as they had the mental image that 3*8 is three 8s and as long as that mental image actually felt robust. (I'm sure they can rattle off what it means. It doesn't mean that they are genuinely comfortable with it.) 

Your drawing is only helpful if they fully understand what the multiplication sign stands for. If they don't understand why your picture shows either 6*8 or 8*6, then it doesn't help at all. 

 

5 minutes ago, Clemsondana said:

But, many of us (tutors and public school teachers, for that matter) aren't in a position to go back and reteach this stuff from the beginning like it should be.  I get a 4th grader for an hour and I have to work with them where they are, doing the assignment that they are given. 

Yes, I totally get that. My strong preference as a tutor is to spend at least some time building some stuff up from scratch while also moving forward. Would I teach a kid that NEED to be able to multiply for homework that you can rearrange the numbers? Absolutely. Would I also make time to backtrack and make sure the mental model was being developed, if belatedly? Yes to that, too, whenever I had the chance. 

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5 minutes ago, Not_a_Number said:

Yes, I totally get that. My strong preference as a tutor is to spend at least some time building some stuff up from scratch while also moving forward. Would I teach a kid that NEED to be able to multiply for homework that you can rearrange the numbers? Absolutely. Would I also make time to backtrack and make sure the mental model was being developed, if belatedly? Yes to that, too, whenever I had the chance. 

I would dearly love that. What I get, at different places, is coming for my hour or 2 and getting whoever needs help with homework.  I have asked for students who are just overall struggling with the idea of working with them alone on a consistent basis.  I offered to come over the summer and work with the same couple of kids all summer.  These places can't pick out who needs that help, and the parents don't ask for help as far as I can tell.  So, I may work with one student on place value or multiplication for a bit, leave a few problems, go call out spelling words to another student, help another student find paper to do some sort of drawing as part of their homework, then go back to the first student.  I may get a full 30 minutes with them.  They may or may not be back the next week.  If they come, I have no knowledge of what they need other than the homework that they bring in, which is almost invariably on a different topic.  I'm actually pretty good at figuring out the missing skills or knowledge, but at most I get to work with any given student for 2 hrs a month.  I'm playing around with the idea of doing more volunteering once my kids are out of the house, but I'm not sure that, even if I was able to go 3-4 days/week, there's any mechanism for me to actually work with the kids on what they need.  In other contexts, I've had teachers say 'trust the spiral' - if they don't get multiplication this year, they'll have another shot at it next year.  But, over time it moves from being taught to being review and the kids who didn't learn it early aren't having it explained any more.  Like I've said, I'd love to actually be able to properly remediate, but in the time that I have all I can do is hope that some of the visuals help.  Learning more than what is required from their teachers would be a very hard sell.  

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2 minutes ago, Clemsondana said:

I would dearly love that. What I get, at different places, is coming for my hour or 2 and getting whoever needs help with homework.  I have asked for students who are just overall struggling with the idea of working with them alone on a consistent basis.  I offered to come over the summer and work with the same couple of kids all summer.  These places can't pick out who needs that help, and the parents don't ask for help as far as I can tell.  So, I may work with one student on place value or multiplication for a bit, leave a few problems, go call out spelling words to another student, help another student find paper to do some sort of drawing as part of their homework, then go back to the first student.  I may get a full 30 minutes with them.  They may or may not be back the next week.  If they come, I have no knowledge of what they need other than the homework that they bring in, which is almost invariably on a different topic.  I'm actually pretty good at figuring out the missing skills or knowledge, but at most I get to work with any given student for 2 hrs a month.  I'm playing around with the idea of doing more volunteering once my kids are out of the house, but I'm not sure that, even if I was able to go 3-4 days/week, there's any mechanism for me to actually work with the kids on what they need.  In other contexts, I've had teachers say 'trust the spiral' - if they don't get multiplication this year, they'll have another shot at it next year.  But, over time it moves from being taught to being review and the kids who didn't learn it early aren't having it explained any more.  Like I've said, I'd love to actually be able to properly remediate, but in the time that I have all I can do is hope that some of the visuals help.  Learning more than what is required from their teachers would be a very hard sell.  

Yeah, this set-up would leave me deeply frustrated 😕 . 

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22 hours ago, Not_a_Number said:

I think you're right and we'd make them less even. It would benefit everyone, but the outcomes would be very disparate. 

This.  When you teach gifted kids to their level, they can go crazy far. 

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I have kids who are very good at math to incredibly gifted at math.  I have a granddaughter who struggles mightily with math (my dil has struggled her entire life mastering basic math; I'm sure there is some math disability there.)  From my experience with my sample size of 10 over multiple decades (my 8 plus teaching my 2 oldest grandkids), my perspective is that  bright kids can learn and master quickly.  Spending more time trying to accelerate a struggling student will do more harm then good.  Spending time more focused on simple steps in order to gain confidence in what they are doing goes farther than any other option.  (My granddaughter could not understand when she was in K that 1+2=2+1.  I spent days trying to figure out how to get her to understand something that I had never actually had to teach before bc my kids all looked at me like, yeah, obvious. It wasn't until I finally used dominoes with her and spun the domino tiles around and she counted and saw that the number of dots didn't change and they were exactly the same numbers just in switched positions that she finally had a lightbulb moment and was like, ohhhhh.)

But, my gifted kids are not like the others.  They can learn a simplistic concept and then apply it to unique situations that have never been introduced to them.  It is like when my ds "discovered" multiplication.  He had never seen multiplication.  He just told me that he had made a discovery of "rows of."  He had observed while playing with Legos the reality of repeated addition.  He mastered all of his multiplication tables without ever even having been introduced to the concept.  (We were baking cookies when he had the conversation with me.  It went something like, "Mommy, did you know if we put 5 rows of 4 cookies that there will be 20 cookies on the pan?"  I asked questions and he looked around and pointed to the window panes, and said 6 rows of 3 window panes is 18; then he looked at the door panels, tiles on the floor, etc.  He sees patterns.  He makes huge leaps in connections.  He was not accelerated bc he was taught via accelerated exposure/pace.  He was accelerated bc of the way he processed information.

No way my granddaugther will ever be the math student that my very good at math students are.  Nor will my very good at math students somehow ever think/see/process math the same way their brother does.  Some things are influenced by their "hardwiring processors."  

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1 hour ago, 8filltheheart said:

Spending more time trying to accelerate a struggling student will do more harm then good.

I would agree with the formulation: putting a lot of pressure on a struggling student will do more harm than good. Your formulation bothers me. Teaching a struggling student helps, it does not harm. (But: "if the student hasn't learned, the teacher hasn't taught.") If you find a way to use your student's finite reserves of attention and enthusiasm more effectively (for instance, by finding very effective explanations), you will have accelerated them. If you search for a way to do it, you are trying to accelerate them, and more power to you.

1 hour ago, 8filltheheart said:

(My granddaughter could not understand when she was in K that 1+2=2+1.  I spent days trying to figure out how to get her to understand something that I had never actually had to teach before bc my kids all looked at me like, yeah, obvious. It wasn't until I finally used dominoes with her and spun the domino tiles around and she counted and saw that the number of dots didn't change and they were exactly the same numbers just in switched positions that she finally had a lightbulb moment and was like, ohhhhh.)

Can't I retell your story this way? You located something that your granddaughter didn't understand. You spent a few days trying to remedy it, and found the solution. You taught your granddaughter why 1+2 = 2+1. To me, it is not a very much less auspicious story, than the one about a boy who invents multiplication.

Edited by UHP
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3 hours ago, KSera said:

But why?  In all these kinds of things, I don’t understand the purpose in going through all these programs in order to teach very young children impressive academic tricks and skills. What is the purpose of that for a very young child? That’s an honest and not rhetorical question, because I don’t know what the benefit would be.

I can not speak to every one who chooses to pursue academics for their youngster. For our own family, it was not a choice that we made lightly or without thoughtful research and reflection. There is absolutely nothing wrong with deciding to follow a traditional trajectory for your children, and there is nothing wrong with deciding to follow a less expected or less understood trajectory.

The most important thing is that children are loved and educated in nurturing environment.

However, to your question: you and I are not co-parenting children. I'm not going to enumerate the reasons that my homeschool is designed the way it is. The only one who needs to agree with my parenting vision or share in the educational values and goals that I have for my children is their father, and fortunately he does.

 

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1 hour ago, 8filltheheart said:

But, my gifted kids are not like the others.  They can learn a simplistic concept and then apply it to unique situations that have never been introduced to them.  It is like when my ds "discovered" multiplication.  He had never seen multiplication.  He just told me that he had made a discovery of "rows of."  He had observed while playing with Legos the reality of repeated addition.  He mastered all of his multiplication tables without ever even having been introduced to the concept.  (We were baking cookies when he had the conversation with me.  It went something like, "Mommy, did you know if we put 5 rows of 4 cookies that there will be 20 cookies on the pan?"  I asked questions and he looked around and pointed to the window panes, and said 6 rows of 3 window panes is 18; then he looked at the door panels, tiles on the floor, etc.  He sees patterns.  He makes huge leaps in connections.  He was not accelerated bc he was taught via accelerated exposure/pace.  He was accelerated bc of the way he processed information.

This is a really good explanation of the difference of being smart/working hard, or being gifted. 

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4 minutes ago, UHP said:

I would agree with the formulation: putting a lot of pressure on a struggling student will do more harm than good. Your formulation bothers me. Teaching a struggling students helps, it does not harm. (But: "if the student hasn't learned, the teacher hasn't taught.") If you find a way to use your student's finite reserves of attention and enthusiasm more effectively (for instance, by finding very effective explanations), you will have accelerated them. If you search for a way to do it, you are trying to accelerate them, and more power to you.

Can't I retell your story this way? You located something that your granddaughter didn't understand. You spent a few days trying to remedy it, and found the solution. You taught your granddaughter why 1+2 = 2+1. To me, it is not a very much less auspicious story, than the one about a boy who invents multiplication.

Sorry, but nothing you wrote makes sense to me in terms of real world functioning and the definitions of accelerated and gifted students. My granddaughter does not in any way meet the definition of "accelerated" in math in the context used in academics.  In terms of academics, accelerated has a defined meaning: 

Quote

Acceleration occurs when students move through traditional curriculum at rates faster than typical. Among the many forms of acceleration are grade-skipping, early entrance to kindergarten or college, dual-credit courses such as Advanced Placement and International Baccalaureate programs and subject-based acceleration (e.g., when a fifth-grade student takes a middle school math course).  Acceleration | National Association for Gifted Children (nagc.org)

She can be taught in a way that she can understand and master concepts, but her progress is slower than an avg student and it is only through my intense tutoring of her that she functions on grade level vs. behind.  Fluffing it up and putting a pretty flower on top does not change the reality of her situation in struggling to master very simple/basic math concepts.

In terms of "auspicious story," it isn't meant to be anything other than a factual retelling.  That ds went on to graduate with a 4.0 with physics and math degrees and then went to Berkeley for grad school in physics.  My granddaughter is now in 5th grade and based on my teaching her over the yrs, I foresee alg up as being incredibly difficult subjects for her.  To suggest that there isn't very real differences in fundamental ability and that those differences can somehow be be bridged through teaching is absurd.

 

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