Math is a technology for counting and measuring. Procedures were invented and improved over time. The procedures illustrate the concepts much more than the concepts reinvent the procedures. Even though the latter is possible it's not going to happen like that. And besides we already know that students who already know procedures are the ones who gain from discovery lessons, which are supposed to magically (in that moment) construct the knowledge, a mistaken approach to foster "constructed knowledge"
Thanks for sharing your perspective on this. I do really resonate with part of what you said that people spend their entire lives studying and teaching us new ideas about math! And sometimes when we try to have children “invent/derive” this knowledge with some prior knowledge and hints it baffles me that this approach seems like a good usage of time. There are other, more meaningful, ways for children to engage in deep, meaningful application of math concepts.
Thank you. Nice starting point, Corey. I appreciate the way you lay out the question. You are so correct that words can be sticky things especially when we don't all use the same ones. One idea I might toss out from my perceptual perch is as follows: At foundational levels of math (employing some dual coding here), when I think of conceptual understanding of basic math, I want to "see" what the math looks like. That means- little to no language. I want to see objects, manipulatives, "things" illustrating that addition is putting together, that multiplication is making many, that division is separating apart, that fractions constitute part of a thing or group, that slope intercept form comes from a constant rate of change after a starting value. To me, those are the pieces that make the math make sense. Then, I can map those on to words and procedures. Often that is accomplished simulatneously. I like to think that math vocabulary should be experienced. Concepts are visualized or experienced. To me seeing the foundation concepts as constructions, annimations without symbols involved helps...I find it helps my students by association, to build and construct meaning as they build models. They see some of the meaning behind the math. The goal is always the application. That is the leap or link. Procedural knowledge answers the "how" but that emantates from understanding what is going on and perhaps the "what or why."
Thanks for sharing your thoughts here! They do give me something to think on.
Something I have been wrestling with recently it just the fact all of math learning gets “sorted” in this debate as either conceptual or procedural knowledge. My issue becomes lots of things kids do end up pulling from both conceptual and procedural knowledge. And two, the procedural knowledge domain ends up looking pretty small - i.e., easier to operationally define what we are aiming to see kids do.
ALL ELSE in math is defined as conceptual - which ends up looking like a BIG A$$ sandbox (pardon my language :)). So how useful is it to say, “this child is struggling with conceptual knowledge” OR “this teacher is not providing conceptual knowledge instruction”? I should have maybe included more of this in that post but that was what I was intending by saying how we get lost in abstractness of these terms
Math is a technology for counting and measuring. Procedures were invented and improved over time. The procedures illustrate the concepts much more than the concepts reinvent the procedures. Even though the latter is possible it's not going to happen like that. And besides we already know that students who already know procedures are the ones who gain from discovery lessons, which are supposed to magically (in that moment) construct the knowledge, a mistaken approach to foster "constructed knowledge"
Thanks for sharing your perspective on this. I do really resonate with part of what you said that people spend their entire lives studying and teaching us new ideas about math! And sometimes when we try to have children “invent/derive” this knowledge with some prior knowledge and hints it baffles me that this approach seems like a good usage of time. There are other, more meaningful, ways for children to engage in deep, meaningful application of math concepts.
Thank you. Nice starting point, Corey. I appreciate the way you lay out the question. You are so correct that words can be sticky things especially when we don't all use the same ones. One idea I might toss out from my perceptual perch is as follows: At foundational levels of math (employing some dual coding here), when I think of conceptual understanding of basic math, I want to "see" what the math looks like. That means- little to no language. I want to see objects, manipulatives, "things" illustrating that addition is putting together, that multiplication is making many, that division is separating apart, that fractions constitute part of a thing or group, that slope intercept form comes from a constant rate of change after a starting value. To me, those are the pieces that make the math make sense. Then, I can map those on to words and procedures. Often that is accomplished simulatneously. I like to think that math vocabulary should be experienced. Concepts are visualized or experienced. To me seeing the foundation concepts as constructions, annimations without symbols involved helps...I find it helps my students by association, to build and construct meaning as they build models. They see some of the meaning behind the math. The goal is always the application. That is the leap or link. Procedural knowledge answers the "how" but that emantates from understanding what is going on and perhaps the "what or why."
Thanks for sharing your thoughts here! They do give me something to think on.
Something I have been wrestling with recently it just the fact all of math learning gets “sorted” in this debate as either conceptual or procedural knowledge. My issue becomes lots of things kids do end up pulling from both conceptual and procedural knowledge. And two, the procedural knowledge domain ends up looking pretty small - i.e., easier to operationally define what we are aiming to see kids do.
ALL ELSE in math is defined as conceptual - which ends up looking like a BIG A$$ sandbox (pardon my language :)). So how useful is it to say, “this child is struggling with conceptual knowledge” OR “this teacher is not providing conceptual knowledge instruction”? I should have maybe included more of this in that post but that was what I was intending by saying how we get lost in abstractness of these terms