I love this idea! Here's my my mind went: how does this apply to a few generalizations I teach? What I'm thinking is that there is a relationship between the complexity of the specifics and the complexity of the generalizations.
Example one, unit rates. A really helpful generalization in 7th grade math is "find the unit rate and use that to solve the problem." Students often have a pretty good intuition for these. "You go 10 miles in 2 hours, how many miles do you go in an hour?" is a problem lots of kids just know how to solve or can solve with some pretty minimal nudging. Same with "you are biking at a speed of 15 miles per hour, how far will you go in 3 hours?" The generalization is more complex -- unit rates are hard to explain, though with enough examples they start to make sense. So you can do lots of small specific examples, and then try to climb that ladder to generalization. Specifics are simple, generalization is complex.
Example two, two-step equations. The generalization here is something about inverse operations or undoing things or doing the same thing to both sides. But the specifics are hard! It is hard to understand how to solve 3x+2=14. I love tape diagrams as does the IM curriculum so we spend a lot of time on those. Here the generalization is still complex but so are the specifics. This generalization is always tough for me, we spend so much mental energy on the specifics it feels tough to really generalize. I still think your framework is helpful here! Just different, because of the complexity of those specific cases. Specifics are complex, generalization is complex.
Example three, complementary angles. Here I think both the specifics and the generalization are pretty simple, and not that different from each other. This topic relies on some prior knowledge and really benefits from students who have good intuition for angle size in general. But we do some specific examples and the generalization is mostly just recognizing right angles in places where they're not obvious and applying complementary angles in problems that include other stuff as well. Specifics are simple, generalization is simple.
All of those are my subjective judgments based on my experience. Maybe I'm trying to shoehorn this stuff into a box it doesn't fit into. (Also, are there topics where the specifics are complex and the generalization is simple?) But when I read this post I was thinking "whoa this applies so well to some things I teach but not to other things, why is that?" and this is what I came up with as a first stab at trying to understand.
I think that what I'm talking about here has almost varying time and size scales. Like, I think even if you're trying to get a kid to learn how to solve 3x + 2 = 14, there is a jump to generalization from something else that they already know (which may be a tape diagram of 3x + 2 = 14). And it may be that there are smaller generalizations in some cases that we want to aim for on the road to larger generalizations.
So I think you're right to put pressure on this idea, but I want to clarify that I'm not just talking about "big-ticket", curriculum-size generalization.
Makes sense. You're really talking about Pershan's Koch Snowflake of Greatness, though that's probably a little less catchy of a tagline. But good point about those two-step equations, there's a lot of generalizations happening at different levels.
How do you handle the tension between the implications of this framework and the desire to validate students’ own approaches, when those approaches may be less “principled”?
When I ask an open-ended question, a few kids solve the problem in the simplest “principled” way, but most will hack their way through using common sense or something. It feels disrespectful to highlight the “principled group’s work and then try to demonstrate that the other groups were using a disguised (less-efficient) version of principled method. Doing so creates a hierarchy of methods and communicates, “you got the answer, BUT…” to the “common-sense groups.
How do you recommend handling this?
Maybe it’s okay to just have straight-up conversations about efficiency, and maybe I avoid those too much.
I think I choose different moments for my open-ended questions and student-thought validation. For example, like so many of us I'll often open class with a short warm up problem. That problem is going to be something kids can answer on their own that gets them ready for the day's lesson. That's often where I'll ask my open-ended question, because it's lower stakes and it's not trying to make new generalizations. It's review and prep for the lesson, essentially, so I'm quite eager to hear what students say and think.
BUT when it comes time to generalize, I'll call on students to share ideas in a more directed way. Because now I've got a very specific and important goal, which is to help everyone make generalizations.
You're right though -- what I'm saying is in tension with something like the 5 Practices or 3 Act approach.
I think this makes a lot of sense. I’m not a 3-Act kind of teacher (usually), but I do like the 5 Practices a lot. Yet this scenario comes up and makes me think I need to plan better for the generalization discussions to make sure the focus is on the target generalization.
I think what’s wrong with it is that it can cause students to stop volunteering to share their ideas in class. If the teacher discusses their contribution merely to demonstrate how it’s less efficient than someone else’s contribution — and even if that happens just once! — it can end up leading to a “Bueller? Anyone? Bueller?” type of classroom.
But there really are better and worse ways of solving a problem. If a student uses an inefficient or incorrect method, isn't it reasonable to say, "you got the answer, BUT ..."? If not, that student will persist, and it will hurt them in the long run when later on they need to use the proper approach.
This seems to me to be the worst sort of pandering to people's sensibility. Part of growing up is learning how to receive feedback and take criticism. I agree that the teacher has to deliver the criticism constructively, but I'm alarmed by the suggestion that they shouldn't receive that criticism at all.
How would you handle that situation? If a student proposes an approach that's either incorrect or simply inefficient, do you correct it? If so, how?
I love this idea! Here's my my mind went: how does this apply to a few generalizations I teach? What I'm thinking is that there is a relationship between the complexity of the specifics and the complexity of the generalizations.
Example one, unit rates. A really helpful generalization in 7th grade math is "find the unit rate and use that to solve the problem." Students often have a pretty good intuition for these. "You go 10 miles in 2 hours, how many miles do you go in an hour?" is a problem lots of kids just know how to solve or can solve with some pretty minimal nudging. Same with "you are biking at a speed of 15 miles per hour, how far will you go in 3 hours?" The generalization is more complex -- unit rates are hard to explain, though with enough examples they start to make sense. So you can do lots of small specific examples, and then try to climb that ladder to generalization. Specifics are simple, generalization is complex.
Example two, two-step equations. The generalization here is something about inverse operations or undoing things or doing the same thing to both sides. But the specifics are hard! It is hard to understand how to solve 3x+2=14. I love tape diagrams as does the IM curriculum so we spend a lot of time on those. Here the generalization is still complex but so are the specifics. This generalization is always tough for me, we spend so much mental energy on the specifics it feels tough to really generalize. I still think your framework is helpful here! Just different, because of the complexity of those specific cases. Specifics are complex, generalization is complex.
Example three, complementary angles. Here I think both the specifics and the generalization are pretty simple, and not that different from each other. This topic relies on some prior knowledge and really benefits from students who have good intuition for angle size in general. But we do some specific examples and the generalization is mostly just recognizing right angles in places where they're not obvious and applying complementary angles in problems that include other stuff as well. Specifics are simple, generalization is simple.
All of those are my subjective judgments based on my experience. Maybe I'm trying to shoehorn this stuff into a box it doesn't fit into. (Also, are there topics where the specifics are complex and the generalization is simple?) But when I read this post I was thinking "whoa this applies so well to some things I teach but not to other things, why is that?" and this is what I came up with as a first stab at trying to understand.
I think that what I'm talking about here has almost varying time and size scales. Like, I think even if you're trying to get a kid to learn how to solve 3x + 2 = 14, there is a jump to generalization from something else that they already know (which may be a tape diagram of 3x + 2 = 14). And it may be that there are smaller generalizations in some cases that we want to aim for on the road to larger generalizations.
So I think you're right to put pressure on this idea, but I want to clarify that I'm not just talking about "big-ticket", curriculum-size generalization.
Makes sense. You're really talking about Pershan's Koch Snowflake of Greatness, though that's probably a little less catchy of a tagline. But good point about those two-step equations, there's a lot of generalizations happening at different levels.
How do you handle the tension between the implications of this framework and the desire to validate students’ own approaches, when those approaches may be less “principled”?
When I ask an open-ended question, a few kids solve the problem in the simplest “principled” way, but most will hack their way through using common sense or something. It feels disrespectful to highlight the “principled group’s work and then try to demonstrate that the other groups were using a disguised (less-efficient) version of principled method. Doing so creates a hierarchy of methods and communicates, “you got the answer, BUT…” to the “common-sense groups.
How do you recommend handling this?
Maybe it’s okay to just have straight-up conversations about efficiency, and maybe I avoid those too much.
Great questions!
I think I choose different moments for my open-ended questions and student-thought validation. For example, like so many of us I'll often open class with a short warm up problem. That problem is going to be something kids can answer on their own that gets them ready for the day's lesson. That's often where I'll ask my open-ended question, because it's lower stakes and it's not trying to make new generalizations. It's review and prep for the lesson, essentially, so I'm quite eager to hear what students say and think.
BUT when it comes time to generalize, I'll call on students to share ideas in a more directed way. Because now I've got a very specific and important goal, which is to help everyone make generalizations.
You're right though -- what I'm saying is in tension with something like the 5 Practices or 3 Act approach.
I think this makes a lot of sense. I’m not a 3-Act kind of teacher (usually), but I do like the 5 Practices a lot. Yet this scenario comes up and makes me think I need to plan better for the generalization discussions to make sure the focus is on the target generalization.
What's wrong with creating a hierarchy of methods?
I like saying "you got the answer, BUT...". That seems to me to be a good thing.
I think what’s wrong with it is that it can cause students to stop volunteering to share their ideas in class. If the teacher discusses their contribution merely to demonstrate how it’s less efficient than someone else’s contribution — and even if that happens just once! — it can end up leading to a “Bueller? Anyone? Bueller?” type of classroom.
But there really are better and worse ways of solving a problem. If a student uses an inefficient or incorrect method, isn't it reasonable to say, "you got the answer, BUT ..."? If not, that student will persist, and it will hurt them in the long run when later on they need to use the proper approach.
This seems to me to be the worst sort of pandering to people's sensibility. Part of growing up is learning how to receive feedback and take criticism. I agree that the teacher has to deliver the criticism constructively, but I'm alarmed by the suggestion that they shouldn't receive that criticism at all.
How would you handle that situation? If a student proposes an approach that's either incorrect or simply inefficient, do you correct it? If so, how?