Interested in thoughts on this

[Greenmama2]

Given a lot of parents of mathy kids worry about gaps & the advice here is often to not worry...

http://mathwithbaddrawings.com/2015/04/08/the-math-ceiling-wheres-your-cognitive-breaking-point/

 

(I neither agree nor disagree btw, I'm just curious what people think of the blog post)

[eternalsummer]

regarding the question of whether there is an intellectual ceiling (or more specifically a math ceiling), and whether that ceiling is different for different people - well, of course.  the exact nature of the ceiling is kind of difficult to define and changeable, though - einstein said, I think, that the nature of his intelligence was that he was able to stay with a problem longer.

 

whether a lack of conceptual understanding in early math can lead to a breakdown in understanding later - it can, but in my (very limited experience of just me), generally you work out these holes at some point fairly early during the process.  Coming at math from a variety of approaches/curricula/applications helps with that.

[eternalsummer]

this is a good comment from the post:

 

"One thing to consider–why do some students intuitively grasp the reality and other students just go through the motions to find the correct answer, even though they have the same teacher? Isn’t that evidence of the wall, or a cognitive limit? So yes, we can teach students properly and conceptually and delay the point at which they stall.

I went through calculus in high school, but my understanding stopped at algebra. Like another commenter, I came back to math when my son was struggling with geometry, where I first hit a wall. But I’d had 30 years of learning how to learn, and was better at challenging myself and understanding my stall points. And as a result, here I am a math teacher.

However, I’m well aware that my wall still exists. I’ve pushed it out, but the original hold up was the fact that my visual-spatial ability is nowhere near my verbal reasoning and logical abilities. I am compensating for this lack with logic–I redefine problems in ways I can understand. That won’t work out in the land of MVC and vectors and whatever else is out there."

I have had the same experience.

[kiana]

Hm.

 

I don't think that a gap because a skill has not yet been taught is a reason to worry. I *do* think that a gap in understanding when the child is just going through the motions but really doesn't understand what they are doing and why is a reason to not move on to more advanced skills until they understand the basic skills. For example, if someone really doesn't understand two-digit multiplication, it is stupid to try to move on to three-digit multiplication in a monkey-see-monkey-do fashion.

[Dmmetler]

I kind of resemble this-I ended up going into performing arts/music in part because I was "bad at math". (Bad at math meaning that it didn't come easily and intuitively for me like it did for most of the people in my math/science high school program). Loved science, but math just didn't always work. Sometimes it did. Sometimes, it seemed to, but then things got scrambled somewhere. And sometimes, I couldn't make head or tail of it. I knew it was supposed to make sense and always be the same, but my brain didn't always process it the same. Something that could click at 11:00 AM wouldn't click at 9:00 PM when I finally had a chance to do the homework. Stuff like that. One thing I learned in high school was to schedule math classes in the morning, because I could do math in the morning, but not in the afternoon. And when I went to college as an undergrad, I picked a major that only required two math classes, one of them statistics. I placed fairly high because I'd had a strong high school background, and I knew enough to get through the test, took my college math sequence, and closed the book on math. We just weren't going to get along.

 

I was in grad school-ironically, in a math education grad program (because despite being "bad at math" I had good GRE scores in it and had taken enough college level math classes to qualify-and math ed paid you, while other education programs that led to a degree and certification required that you pay them) when I was finally identified as having visual-spatial processing off the bottom of the chart-and what that meant was that the parts of math that I was having trouble with or just couldn't process were truly a case where my brain just couldn't process them as given.  I'd done so well overall because I have abstract reasoning skills off the top of the chart, and the two kind of balanced out to being able to do fairly well in high school math-but having no confidence in it. The whole purpose of that re-evaluation was to get a 504 plan for tests and a free bus pass (since I couldn't drive), but it gave me something that nothing previously had. It gave me answers. And that, coupled with the coursework I was doing at the time, gave me ways to work with the problem.

 

I had to fight crying when, in one of my first math education classes, we were handed manipulatives to use to demonstrate how to teach a concept. I about cried because suddenly, that concept clicked into place for me. I'd known it, I'd understood it, but seeing it drawn on paper, which is what math books at the time did just never straightened it out. I learned it algorithmically, and sort of understood it, but not quite. I had to be able to touch and feel it for it to click. And as a grad student in the heart of the constructivist era of math education, I had access to every manipulative in the ETA catalog, plus lots of books on how to use them.

 

I spent a good part of my graduate studies reteaching myself high school math now that I had access to all these tools. I ended up specializing in content mastery (which is Texas for "Remediation"), focusing on pre-algebra and algebra, because that is where my brain really just couldn't process it. Finally, it worked. Finally it made sense. I discovered that I was good at finding the kids who, like me, just couldn't speak the same language math books speak, and in translating for them.

 

 

And of course, DD speaks "math" like a native....

 

[kiwik]

I suspect for most people there is a point where effort far outweighs benefit. Maybe if they spent 8 hours a day for two years they would get it but what would be the point. I found maths easy but I was only taught algorithms by teachers who hated maths and didn't get New Maths anyway. If I had less natural number sense I wouldn't have recovered from the gaps and as it was I didn't until I had been out of school several years.

[regentrude]

Of course there is a natural ceiling. Some people lack the intellectual ability to grasp highly abstract concepts. My brother cannot count to ten. No genius teacher and immense effort on his part would change that.

 

But most people never reach their natural ceiling - their math ability stalls way before they hit their intellectual limit. Better teaching would go a long way to improve the math ability of the "non-mathy" people. Alas, good math teachers are in short supply; thus many people go through life thinking they can't possibly understand math... simply because they never had anybody who could have helped them reach their potential.

[Donna]

I would guess for some people, interest in the subject also plays a role. I was very good in math in school, taught the majority of my calculus class calculus during study hall in high school, but had no interest in pursuing it past what was required for my biology major. It had nothing to do with reaching my limit of understanding or ceiling. Maybe my interest could have been enhanced by a dynamic teacher or a mentor who had the ability to bring it alive rather than one who lacked the ability to teach well leaving me to figure it out on my own then show the rest of my class how to do it so it made sense to them.