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Chikae's avatar

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.

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Benjamin Riley's avatar

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.

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