Why BP (Batting Practice or Benchpresses or ...)*?
We have all had the experience: you just finished the unit on area and volume, including a fantastic set of activities on the area of triangles. Two weeks later you revisit triangle areas, and half the class is asking you "How do we do that again? That's base times height, right?" Why does that happen?
In Moonwalking with Einstein: The Art and Science of Remembering Everything by Joshua Foer, I read about the different ways that chess masters and talented novices (pre-chess masters, I guess) use their brains when they play chess. Take a position 25 moves into a game, and place it in front of a talented novice, who also happens to be sitting in a brain scanner, and ask him what his next move would be. He will access mostly the analytical parts of his brain, trying to analyze the position and work out the best move. Now take the same position and place in front of a chess master. It turns out that she will, to a much larger degree, access her memory. She is recognizing this position; she has been here before, and it is familiar territory, just like you would feel in a city you know well. Not a lot of thinking required.
As a nascent jazz musician, I am also aware that no matter how much music theory I know, once a tune starts, there is no time to think and play. And yet I play the right notes (more and more often), from a seemingly magical place.
Below the thoughts in our awareness there is a great deal of unconscious thought going on. These are our emotional reactions and fears, habits good and bad, and also our automatic skills like juggling, riding a bike, driving home while we think about the school day - anything we can accomplish without conscious thought ... including deciding whether a fraction word problem requires addition or multiplication, or what computation will tell me how many gallons go into each of seven containers. I think these unconscious routines are the ones the brain calls on first, but they are relatively inflexible and therefore easily derailed. We see evidence of that when the students can tell you how to equally share 42 gallons between 7 containers, and 35 gallons between 7 containers, but come to a screeching halt or start rattling off nonsense when they need to share 32 gallons between 7 containers. Sometimes they can't even tell me that they need to divide; they may multiply instead, or just report "I don't know what to do". It seems that their brain, on autopilot, has run into what looks like a dead-end, and so it abandons the path, and heads off on the next most likely routine. And the reason they sometimes struggle to tell you what they did to get the other answers is because it may literally be missing from their memories, just like you can't remember exactly where all those chocolate covered almonds went.
Of course you can reason out where the almonds went, and likewise with math: reasoning is the critical resource that makes it possible to respond to new situations flexibly. And while we are excited about building the tool set of the conscious brain, increasing the repertoire of the unconscious brain is equally useful, expanding the range of the recognizable and automatic: I know how to find the area of a triangle, now how can I use that to find the area of this odd looking thing? Repetition and drills, which have gotten a bad name recently, seem to be the way to extend the expertise of that part of the brain. You feed it multiple examples (with feedback!), and it finds the patterns and builds the routines.
Chess masters play thousand of games, and musicians play thousands of scales and patterns, to give their conscious brains a head start on more and more sophisticated tasks. We should do the same for our students: pay equal attention to feeding both the conscious and unconscious math brains with the tasks each part learns from best. I'm not sure of the optimal way to do this, but I am experimenting, and so far benchpresses seem to be paying off. My students are less often stuck on the computation elements of problems, and better able to think about what's going on.
*P.S. "BP"? I used to call these sprints, but I was inspired by Jo Boaler's Week of Inspirational Math videos to emphasize the role of practice in strengthening the brain and remove the implied emphasis on speed. We called them benchpresses for a while, and then BPs, and that reminded me of batting practice... they all remind the students of repetition helping their brains get stronger.
Thanks to Robin Ramos and Bill Davidson for introducing me to the idea of sprints, and their excellent guidance. My approach is different than theirs...read about their approach at Robin's site.
In Moonwalking with Einstein: The Art and Science of Remembering Everything by Joshua Foer, I read about the different ways that chess masters and talented novices (pre-chess masters, I guess) use their brains when they play chess. Take a position 25 moves into a game, and place it in front of a talented novice, who also happens to be sitting in a brain scanner, and ask him what his next move would be. He will access mostly the analytical parts of his brain, trying to analyze the position and work out the best move. Now take the same position and place in front of a chess master. It turns out that she will, to a much larger degree, access her memory. She is recognizing this position; she has been here before, and it is familiar territory, just like you would feel in a city you know well. Not a lot of thinking required.
As a nascent jazz musician, I am also aware that no matter how much music theory I know, once a tune starts, there is no time to think and play. And yet I play the right notes (more and more often), from a seemingly magical place.
Below the thoughts in our awareness there is a great deal of unconscious thought going on. These are our emotional reactions and fears, habits good and bad, and also our automatic skills like juggling, riding a bike, driving home while we think about the school day - anything we can accomplish without conscious thought ... including deciding whether a fraction word problem requires addition or multiplication, or what computation will tell me how many gallons go into each of seven containers. I think these unconscious routines are the ones the brain calls on first, but they are relatively inflexible and therefore easily derailed. We see evidence of that when the students can tell you how to equally share 42 gallons between 7 containers, and 35 gallons between 7 containers, but come to a screeching halt or start rattling off nonsense when they need to share 32 gallons between 7 containers. Sometimes they can't even tell me that they need to divide; they may multiply instead, or just report "I don't know what to do". It seems that their brain, on autopilot, has run into what looks like a dead-end, and so it abandons the path, and heads off on the next most likely routine. And the reason they sometimes struggle to tell you what they did to get the other answers is because it may literally be missing from their memories, just like you can't remember exactly where all those chocolate covered almonds went.
Of course you can reason out where the almonds went, and likewise with math: reasoning is the critical resource that makes it possible to respond to new situations flexibly. And while we are excited about building the tool set of the conscious brain, increasing the repertoire of the unconscious brain is equally useful, expanding the range of the recognizable and automatic: I know how to find the area of a triangle, now how can I use that to find the area of this odd looking thing? Repetition and drills, which have gotten a bad name recently, seem to be the way to extend the expertise of that part of the brain. You feed it multiple examples (with feedback!), and it finds the patterns and builds the routines.
Chess masters play thousand of games, and musicians play thousands of scales and patterns, to give their conscious brains a head start on more and more sophisticated tasks. We should do the same for our students: pay equal attention to feeding both the conscious and unconscious math brains with the tasks each part learns from best. I'm not sure of the optimal way to do this, but I am experimenting, and so far benchpresses seem to be paying off. My students are less often stuck on the computation elements of problems, and better able to think about what's going on.
*P.S. "BP"? I used to call these sprints, but I was inspired by Jo Boaler's Week of Inspirational Math videos to emphasize the role of practice in strengthening the brain and remove the implied emphasis on speed. We called them benchpresses for a while, and then BPs, and that reminded me of batting practice... they all remind the students of repetition helping their brains get stronger.
Thanks to Robin Ramos and Bill Davidson for introducing me to the idea of sprints, and their excellent guidance. My approach is different than theirs...read about their approach at Robin's site.