Great question. My little motto for learning from worked examples (adapted from others) is analyze, explain, apply. Which is to say, first I ask students to analyze the solution, then to explain why it works in general, and then to apply it on their own.
When analysis is going well, I'll hear students say things like "oh wait that's actually smooth" or "I see what they did." I don't want to overstate this, but I see that as similar to the experience of breaking a code or cracking a foreign language. It's the fun of all the pieces sorting into place.
(I'm not really letting them struggle very much at this stage, is an important difference. Aiming instead to design a moment where as many kids as possible can be successful in "cracking" the solution.)
I'm not sure if explanation is itself the same sort of fun as the code-cracking, necessity-grasping things I'm talking about above. If math is partly about encountering necessity and also about using procedures, putting it all into words might be a third piece of the puzzle. That said, what we're trying to do at this stage is articulate the necessity that we might have only encountered implicitly before.
And with that third stage, application, we're sometimes simply using a procedure. Sometimes, especially when a problem appears different from the example to a student, there is more of that fun of puzzling out what needs to be the case -- "oh so it doesn't matter which side the variable is on," or "it'll still work even if..."
"When analysis is going well, I'll hear students say things like "oh wait that's actually smooth" or "I see what they did." I don't want to overstate this, but I see that as similar to the experience of breaking a code or cracking a foreign language. It's the fun of all the pieces sorting into place."
Okay yeah I'll buy that at full price. Thanks for using a couple of your remaining marbles to spell it out.
I agree that lots of games should be considered math activities and learning opportunities. I've been incorporating games from Ben Orlin's book Math Games with Bad Drawings (ISBN: 978-0-7624-9986-1) on a frequent basis in my classes with good results. Most of the games are easy to understand but the strategy is complex enough to be challenging for anyone.
How do you see the relationship between the necessity that people experience often in games / only occasionally in math and worked examples?
Great question. My little motto for learning from worked examples (adapted from others) is analyze, explain, apply. Which is to say, first I ask students to analyze the solution, then to explain why it works in general, and then to apply it on their own.
When analysis is going well, I'll hear students say things like "oh wait that's actually smooth" or "I see what they did." I don't want to overstate this, but I see that as similar to the experience of breaking a code or cracking a foreign language. It's the fun of all the pieces sorting into place.
(I'm not really letting them struggle very much at this stage, is an important difference. Aiming instead to design a moment where as many kids as possible can be successful in "cracking" the solution.)
I'm not sure if explanation is itself the same sort of fun as the code-cracking, necessity-grasping things I'm talking about above. If math is partly about encountering necessity and also about using procedures, putting it all into words might be a third piece of the puzzle. That said, what we're trying to do at this stage is articulate the necessity that we might have only encountered implicitly before.
And with that third stage, application, we're sometimes simply using a procedure. Sometimes, especially when a problem appears different from the example to a student, there is more of that fun of puzzling out what needs to be the case -- "oh so it doesn't matter which side the variable is on," or "it'll still work even if..."
"When analysis is going well, I'll hear students say things like "oh wait that's actually smooth" or "I see what they did." I don't want to overstate this, but I see that as similar to the experience of breaking a code or cracking a foreign language. It's the fun of all the pieces sorting into place."
Okay yeah I'll buy that at full price. Thanks for using a couple of your remaining marbles to spell it out.
I agree that lots of games should be considered math activities and learning opportunities. I've been incorporating games from Ben Orlin's book Math Games with Bad Drawings (ISBN: 978-0-7624-9986-1) on a frequent basis in my classes with good results. Most of the games are easy to understand but the strategy is complex enough to be challenging for anyone.
I love Ben's book!