Wow, this really put words to my own wonderings about how different students perform in schools and how they perceive themselves as capable learners. As I was reading this, I found myself reflecting on the two extremes of students I often saw struggle in high school math:
1) Students who had a high fluid intelligence and had got to about the middle of middle school by banking on the fact that they could reason out the answer without really understanding the conceptual underpinnings. I firmly believe that most of math are "open middle" type of experiences where there are multiple routes to the answer... but many of them had middle school teachers who told them they were "doing it wrong" if they didn't follow the specific algorithm. So first those students started mentally checking out of class (because nobody likes being told they're doing it wrong if they're still getting the right answer!!) and then they missed all of the building blocks they needed to build up their crystallized intelligence and then they were in a state where they felt like they "should get it" but they no longer knew what was going on. Those students would often be called "lazy" or "not that smart" by their teachers (or even their parents), so by the time they got to me in high school, they would just kind of shrug and say they "weren't good at math."
2) On the other extreme, I had students who were force fed "stuff" in middle school (i.e. taught overly advanced curriculum in class or in tutoring), so they had crystallized intelligence, but the crystalline structure (so to speak) was not well-formed. These students would do well in 9th grade, but by the time they hit advanced concepts in 10th or 11th grade, where they had to synthesize their conceptual understanding (not procedural knowledge), they didn't know how the pieces really fit together. Those kids often had teachers and parents around them saying they were "good at math" and "smart"... and had a really tough time with the abrupt cliff drop in ability and understanding. It's almost like if you buy knock-off Lego sets... the bricks may possibly fit with all of the others in a particular set, but the pieces might not universally fit with each other if you mix them all up, because the manufacturing process is not as rigorous.
Interestingly, 1) isn't far from what my experience in middle/high school was like. I agree with you about 2) - I want to write a post about what you describe as the "crystalline structure," that's a great way to think about learning. Crystallized intelligence isn't something that just adds up endlessly, the structure underneath is really important!
I saw the title and read the opening paragraph and not gonna lie Dylan, I was worried...but I should have had more faith. You've brilliantly articulated the core ideas around intelligence that I think every teacher should know, particularly around "crystallized intelligence" aka knowing stuff in long-term memory. What a great post.
Thanks Ben. I was thinking of you while writing it -- this is a followup on that post about learning speed from the spring that we talked about. It's a tricky topic so it took me a long time. There's a side I didn't get into here that connects to your symphonies not salad dressing post. Empirically we see persistent inequities in education and it's easy to ascribe those to innate factors, but it's really this complex interplay between a bunch of different factors. Schools are one of those factors. I think you could look at this post as a description of potential confounders of some of the relationships we've talked about re: deBoer, Harden, etc.
"No teacher wants to give up on a student, but we do."
Schools are primary care and they're the hospital too. Decent at the first but bad at the second.
Some primary care docs are better than others at solving things themselves. But all of them reach a point where they - not give up per se - but refer to the second tier of expert providers.
Schools lack that second tier capacity. Special ed providers of very mixed competence, diluted by red tape. For non SPED kids, smidge of tutoring running on vapors, maybe a counselor providing talk therapy (empirically not so hot). Student is discussed at the grade team meeting, perhaps, but absent heroic intervention by one of the teachers....
I think there's capacity, and there's also losing the forest for the trees. We get distracted by what's right in front of us. Kid is failing Algebra I so we put resources toward helping them with Algebra I, rather than saying, "maybe they're struggling in Algebra I because they missed a bunch of fraction skills that are holding them back." Or if we do that it's putting them on IXL, i-Ready, whatever, and hoping the computer will magically solve it for us. Imo there's a lot of wasted capacity that goes into helping kids with what's in front of them, rather than figuring out root causes. Metaphor would be treating the symptoms rather than the disease. Keep going back to the doctor for the same thing. The fever gets better in the short term, kid passes the quiz, etc, but it comes back with a vengeance until it's just too much.
Wow, this really put words to my own wonderings about how different students perform in schools and how they perceive themselves as capable learners. As I was reading this, I found myself reflecting on the two extremes of students I often saw struggle in high school math:
1) Students who had a high fluid intelligence and had got to about the middle of middle school by banking on the fact that they could reason out the answer without really understanding the conceptual underpinnings. I firmly believe that most of math are "open middle" type of experiences where there are multiple routes to the answer... but many of them had middle school teachers who told them they were "doing it wrong" if they didn't follow the specific algorithm. So first those students started mentally checking out of class (because nobody likes being told they're doing it wrong if they're still getting the right answer!!) and then they missed all of the building blocks they needed to build up their crystallized intelligence and then they were in a state where they felt like they "should get it" but they no longer knew what was going on. Those students would often be called "lazy" or "not that smart" by their teachers (or even their parents), so by the time they got to me in high school, they would just kind of shrug and say they "weren't good at math."
2) On the other extreme, I had students who were force fed "stuff" in middle school (i.e. taught overly advanced curriculum in class or in tutoring), so they had crystallized intelligence, but the crystalline structure (so to speak) was not well-formed. These students would do well in 9th grade, but by the time they hit advanced concepts in 10th or 11th grade, where they had to synthesize their conceptual understanding (not procedural knowledge), they didn't know how the pieces really fit together. Those kids often had teachers and parents around them saying they were "good at math" and "smart"... and had a really tough time with the abrupt cliff drop in ability and understanding. It's almost like if you buy knock-off Lego sets... the bricks may possibly fit with all of the others in a particular set, but the pieces might not universally fit with each other if you mix them all up, because the manufacturing process is not as rigorous.
Interestingly, 1) isn't far from what my experience in middle/high school was like. I agree with you about 2) - I want to write a post about what you describe as the "crystalline structure," that's a great way to think about learning. Crystallized intelligence isn't something that just adds up endlessly, the structure underneath is really important!
I saw the title and read the opening paragraph and not gonna lie Dylan, I was worried...but I should have had more faith. You've brilliantly articulated the core ideas around intelligence that I think every teacher should know, particularly around "crystallized intelligence" aka knowing stuff in long-term memory. What a great post.
Thanks Ben. I was thinking of you while writing it -- this is a followup on that post about learning speed from the spring that we talked about. It's a tricky topic so it took me a long time. There's a side I didn't get into here that connects to your symphonies not salad dressing post. Empirically we see persistent inequities in education and it's easy to ascribe those to innate factors, but it's really this complex interplay between a bunch of different factors. Schools are one of those factors. I think you could look at this post as a description of potential confounders of some of the relationships we've talked about re: deBoer, Harden, etc.
"No teacher wants to give up on a student, but we do."
Schools are primary care and they're the hospital too. Decent at the first but bad at the second.
Some primary care docs are better than others at solving things themselves. But all of them reach a point where they - not give up per se - but refer to the second tier of expert providers.
Schools lack that second tier capacity. Special ed providers of very mixed competence, diluted by red tape. For non SPED kids, smidge of tutoring running on vapors, maybe a counselor providing talk therapy (empirically not so hot). Student is discussed at the grade team meeting, perhaps, but absent heroic intervention by one of the teachers....
I love your off-the-wall metaphors for schools.
I think there's capacity, and there's also losing the forest for the trees. We get distracted by what's right in front of us. Kid is failing Algebra I so we put resources toward helping them with Algebra I, rather than saying, "maybe they're struggling in Algebra I because they missed a bunch of fraction skills that are holding them back." Or if we do that it's putting them on IXL, i-Ready, whatever, and hoping the computer will magically solve it for us. Imo there's a lot of wasted capacity that goes into helping kids with what's in front of them, rather than figuring out root causes. Metaphor would be treating the symptoms rather than the disease. Keep going back to the doctor for the same thing. The fever gets better in the short term, kid passes the quiz, etc, but it comes back with a vengeance until it's just too much.