Nevan is measuring the length of an aquarium when he exclaims:
"Dad, this thing is 3 feet long!"
"No it's not, its 36 inches."
"Hmm." *looks at the tape measure* "Oh, 36 inches and 3 feet are the same."
"No they're not. One is 3 and the other is 36. 36 is waaay bigger than 3."
"Dad. Look. Inches are smaller than feet. There's an amount of inches in a foot."
"Oh. How many?"
"12."
"Ok."
- Posted using BlogPress from my iPad
Friday, August 12, 2011
Wednesday, July 27, 2011
Popping Popcorn in a Popcorn Popper
Popcorn Question from David Cox on Vimeo.
Most of my previous attempts with these story problems have resulted in me slapping on a timecode and cutting the video. This one had me thinking a bit. I'm not sure I got it.
Your assignment:
1. What question does this provoke?
2. If you have a tough time answering #1, what question do you think I was after? And what can I do to help that question along?
Act 2's a Killer
I need a little help. I think I've nailed the question:
Barbecue Q2 from David Cox on Vimeo.
But I can't figure out what to give students to help them through Act 2.
Here's the raw footage and the current conversation and Greg's run at the data. (Thanks to @maxmathforum for archiving)
Any help would be appreciated.
Barbecue Q2 from David Cox on Vimeo.
But I can't figure out what to give students to help them through Act 2.
Here's the raw footage and the current conversation and Greg's run at the data. (Thanks to @maxmathforum for archiving)
Any help would be appreciated.
Wednesday, July 6, 2011
Virtual Conference on Core Values: Treat 'Em Like They're My Own
Conference is here.
What's at the center of my classroom?
It's the same thing that drives my parenting: I'm raising them to leave.
My home is differentiated. My wife and I have boys ages 12, 9, 6, 4 and 1.5. We don't treat them according to their age; we treat them according to how ready they are to be independent. It's based on this unwritten authority/influence continuum that seems to be as dynamic as anything I've ever encountered. At one end, we have authority which is determined by the decisions we make for our children. At the other, we have influence which eventually becomes the decisions they makes for themselves. And in between? We have all kinds of decisions we make together.
We start out by feeding, burping, bathing and changing. It's a no-brainer, because an infant can't do these things for himself. Trust me; I've tried.
As the child grows, he begins to do more for himself. The part we have to embrace is the messiness that ensues as the spoon leaves our hand and moves into the hand of the child. Let go of the spoon too soon and you'll be cleaning the ceiling for months; let go too late and the child may never learn to feed himself. But as you begin to let go, make no mistake, it's going to be messy. That part is hard because sometimes it's just easier to feed the kid yourself. Parents get this all kinds of messed up and the child eventually pays for it. We've all seen it: helicopter parents who make decisions for their kids, resolve their conflicts and clean up all their messes.
There may be times when the child may look like he's taking responsibility, but all he is doing is following the lead of his parents.
The way this works into my classroom is simple: Never do for them what they can do for themselves.
Our initial placement on the continuum is critical. The only way to assess that is to provide activities that can be differentiated in terms of our involvement: How much do we show them? How much do we explicitly tell them? What questions do we ask and how helpful should they be? We can't turn them loose too soon, but the goal has to be to let go.
We are preparing them to leave.
I think once we wrap our mind around that concept, the continuum begins to look more like this:
See, most of the teaching I've been around assumes the teacher to be the source of all knowledge, like a breathing encyclopedia. I realize we are all dealing with mandated curriculum and most students probably aren't quite ready to choose their own adventure anyway. I'm not trying to discuss what we teach (that's a topic for another post); I'm talking about how. But sometimes, the spoon never leaves the hand of the teacher because it's assumed the student can't feed himself. The complexity of the material may increase, but the cognitive demand of the delivery never leaves modelling and explicit directions in the form of statements--or maybe, if we're lucky, closed questions. The student may look like he's functioning at a high level, but all he's doing is following the lead of his teacher.
I have to constantly take a look at how much of what I say ends with a period and how much ends with a question mark. I also have to be aware that not all questions are created equal--are my questions pointed and closed or are they open, allowing for multiple entry and exit points? If the best I can do is offer closed questions to my students, then the best they'll do is depend on me to be the one asking the questions. The really interesting part is how a student and I can move all over the continuum during a single conversation. Sometimes we may start with the open questions (which is always best, in my opinion) and based on the student's response, I may have to just get out of the way and let him go--or we keep moving towards modelling until we find a spot where the student is comfortable. It's really up to him. The key is to let the student lead. This may be problematic at first because, just like learning the tendencies of a new dance partner (sorry for changing metaphors there, but it had to happen. Besides, I do dance with my kids, so it kinda fits.), many students are conditioned to follow. They'll sit and stare for a while until you let them know, "the music's playin', kid, time to bust a move."
So, whether we are talking about feeding a child or dancing, the point remains the same: the student determines my level of involvement and it's important to never underestimate how independent he can truly be. It's a tough call sometimes because the little boogers'll sandbag, for sure.
So now what? How do you move the spoon from your hand to the hand of your student?
Sorry, it's your spoon.
What's at the center of my classroom?
It's the same thing that drives my parenting: I'm raising them to leave.
My home is differentiated. My wife and I have boys ages 12, 9, 6, 4 and 1.5. We don't treat them according to their age; we treat them according to how ready they are to be independent. It's based on this unwritten authority/influence continuum that seems to be as dynamic as anything I've ever encountered. At one end, we have authority which is determined by the decisions we make for our children. At the other, we have influence which eventually becomes the decisions they makes for themselves. And in between? We have all kinds of decisions we make together.
We start out by feeding, burping, bathing and changing. It's a no-brainer, because an infant can't do these things for himself. Trust me; I've tried.
As the child grows, he begins to do more for himself. The part we have to embrace is the messiness that ensues as the spoon leaves our hand and moves into the hand of the child. Let go of the spoon too soon and you'll be cleaning the ceiling for months; let go too late and the child may never learn to feed himself. But as you begin to let go, make no mistake, it's going to be messy. That part is hard because sometimes it's just easier to feed the kid yourself. Parents get this all kinds of messed up and the child eventually pays for it. We've all seen it: helicopter parents who make decisions for their kids, resolve their conflicts and clean up all their messes.
There may be times when the child may look like he's taking responsibility, but all he is doing is following the lead of his parents.
The way this works into my classroom is simple: Never do for them what they can do for themselves.
Our initial placement on the continuum is critical. The only way to assess that is to provide activities that can be differentiated in terms of our involvement: How much do we show them? How much do we explicitly tell them? What questions do we ask and how helpful should they be? We can't turn them loose too soon, but the goal has to be to let go.
We are preparing them to leave.
I think once we wrap our mind around that concept, the continuum begins to look more like this:
See, most of the teaching I've been around assumes the teacher to be the source of all knowledge, like a breathing encyclopedia. I realize we are all dealing with mandated curriculum and most students probably aren't quite ready to choose their own adventure anyway. I'm not trying to discuss what we teach (that's a topic for another post); I'm talking about how. But sometimes, the spoon never leaves the hand of the teacher because it's assumed the student can't feed himself. The complexity of the material may increase, but the cognitive demand of the delivery never leaves modelling and explicit directions in the form of statements--or maybe, if we're lucky, closed questions. The student may look like he's functioning at a high level, but all he's doing is following the lead of his teacher.
I have to constantly take a look at how much of what I say ends with a period and how much ends with a question mark. I also have to be aware that not all questions are created equal--are my questions pointed and closed or are they open, allowing for multiple entry and exit points? If the best I can do is offer closed questions to my students, then the best they'll do is depend on me to be the one asking the questions. The really interesting part is how a student and I can move all over the continuum during a single conversation. Sometimes we may start with the open questions (which is always best, in my opinion) and based on the student's response, I may have to just get out of the way and let him go--or we keep moving towards modelling until we find a spot where the student is comfortable. It's really up to him. The key is to let the student lead. This may be problematic at first because, just like learning the tendencies of a new dance partner (sorry for changing metaphors there, but it had to happen. Besides, I do dance with my kids, so it kinda fits.), many students are conditioned to follow. They'll sit and stare for a while until you let them know, "the music's playin', kid, time to bust a move."
So, whether we are talking about feeding a child or dancing, the point remains the same: the student determines my level of involvement and it's important to never underestimate how independent he can truly be. It's a tough call sometimes because the little boogers'll sandbag, for sure.
So now what? How do you move the spoon from your hand to the hand of your student?
Sorry, it's your spoon.
Wednesday, June 8, 2011
Adventures in Pedagogy: No Solution
Dawson (12) is ready to begin 7th grade and I'm taking over the curriculum duties. We are starting off with some of James Tanton's Math Without Words
One of the early puzzles looks like this:
Dawson jumps in and devours these things. Right up until he encounters a puzzle like this:
"Dad, I don't get it."
"Get what?"
"This puzzle. I don't understand it."
"What's the rule?"
"I have to connect the two dots by going through each box."
"Is that the whole rule?"
"Yeah. No. I can only go through each box once and I can't go diagonally."
"Does that rule work for all the others you've done?"
"Yeah."
"Hmm."
I go back to doing the dishes as Dawson and Nevan (9) discuss what's "wrong" with this particular puzzle. Once I'm finished, I chime back in.
"So, have you figured it out?"
"No. I just don't understand?"
"Have you considered that maybe this particular puzzle doesn't have a solution?"
*perplexed*
"You mean, that's allowed?"
"Yeah. Sometimes problems don't have answers."
*points to a different puzzle on the page*
"Oh, then this one doesn't have an answer either."
- Posted using BlogPress from my iPad
One of the early puzzles looks like this:
Dawson jumps in and devours these things. Right up until he encounters a puzzle like this:
"Dad, I don't get it."
"Get what?"
"This puzzle. I don't understand it."
"What's the rule?"
"I have to connect the two dots by going through each box."
"Is that the whole rule?"
"Yeah. No. I can only go through each box once and I can't go diagonally."
"Does that rule work for all the others you've done?"
"Yeah."
"Hmm."
I go back to doing the dishes as Dawson and Nevan (9) discuss what's "wrong" with this particular puzzle. Once I'm finished, I chime back in.
"So, have you figured it out?"
"No. I just don't understand?"
"Have you considered that maybe this particular puzzle doesn't have a solution?"
*perplexed*
"You mean, that's allowed?"
"Yeah. Sometimes problems don't have answers."
*points to a different puzzle on the page*
"Oh, then this one doesn't have an answer either."
- Posted using BlogPress from my iPad
Friday, May 27, 2011
And Now a Word From Elisabeth
Elisabeth decided to define her own project. She became interested in the Golden Ratio. She came to me with a boatload of questions she wanted to tackle ranging from: 1) How can I create the Golden Spiral? to 2) Why is the Golden Rectangle aesthetically pleasing?
I said, "yeah, that's some good stuff there. I trust you. Do it."
Two days later, I sit down with her as she's working and watch her think. She has this perplexed look on her face as she is shuffling through her notes. She has about a half-dozen different rectangles sketched with different dimensions along with more questions than what she started with.
She looks up and says, "The more I look through all this, I'm wondering, 'what's my question?'"
That's right, Elisabeth. Don't ever forget that.
I said, "yeah, that's some good stuff there. I trust you. Do it."
Two days later, I sit down with her as she's working and watch her think. She has this perplexed look on her face as she is shuffling through her notes. She has about a half-dozen different rectangles sketched with different dimensions along with more questions than what she started with.
She looks up and says, "The more I look through all this, I'm wondering, 'what's my question?'"
That's right, Elisabeth. Don't ever forget that.
Thursday, May 26, 2011
Something Different
This year, I decided to take a much more hands-off approach when it came to student projects. There were some homeruns, but there were too many swings-and-misses. Some students opted not to even step to the plate. I suppose that's what happens when students are offered more autonomy. But, I didn't do enough to prepare them to make decisions in such an open ended environment. I think I was too hands-off.
For the final project, I gave my 8th graders seven choices; one of which was to determine the angle that would maximize the distance traveled by a projectile.
What they knew:
Last year I had students do an investigation on trig ratios prior to working with projectile motion. But due to a shortened school year and the fact that all of my students will be taking geometry next year, I had to cut something.
It took a few short conversations for the group to get the fact that horizontal and vertical work together to determine the path and that they needed to use the vertical motion model to determine how long the ball would be in the air. From there, they could figure out how far it would go.
But there was one problem: they didn't know how fast the ball was travelling which made it impossible to determine the vertical and horizontal components.
The Process
Q: How fast is the ball travelling when it is hit?
A: I didn't specify, did I?
This led to a nice conversation on how we need to eliminate as many variables as we can.
Solution: Pick a velocity and work with it. They chose 100 ft/sec.
Q: So how fast is the ball travelling vertically and how fast is it travelling horizontally?
A: That depends.
Q: On what?
So we took turns pushing Joey around the room from behind and the side simultaneously. Each time one person pushed harder than the other.
Conclusion: If the person from the back pushes harder, Joey goes forward more. If the person from the side pushes harder, Joey moves to his left more.
Then we talked about how the velocities can be modeled using vectors and we can use what we know about triangles. Since the forces are perpendicular, we have a right triangle.
Q: If all we know is the hypotenuse of the right triangle, how do we find the other lengths?
A: Is that really all you know?
Solution: They settled on using a 45-45-90 since that is the only way they could figure out the other two sides.
Q: But what do we do for other angles?
A: Yeah, that's kinda tough, huh? Why don't you use a protractor to draw the angle you want, build the triangle you want and measure.
Q: Can we use GeoGebra?
A: Or that.
They used an applet with a fixed hypotenuse of 100 and gathered data on the other two sides.
Q: Is there an easier way?
A: Yeah. It's called sine and cosine. See how these ratios don't change as long as the angle remains constant? (it took a little longer than that, but you get the point)
They were off and running.
Conclusions
I think you physics folks would say something like this:
For the final project, I gave my 8th graders seven choices; one of which was to determine the angle that would maximize the distance traveled by a projectile.
What they knew:
- Linear motion model.
- Vertical motion model.
- Vertical and horizontal motion do not affect one another.
- How vertical and horizontal motion work together to determine the path of a projectile.
- Trig ratios
Last year I had students do an investigation on trig ratios prior to working with projectile motion. But due to a shortened school year and the fact that all of my students will be taking geometry next year, I had to cut something.
It took a few short conversations for the group to get the fact that horizontal and vertical work together to determine the path and that they needed to use the vertical motion model to determine how long the ball would be in the air. From there, they could figure out how far it would go.
But there was one problem: they didn't know how fast the ball was travelling which made it impossible to determine the vertical and horizontal components.
The Process
Q: How fast is the ball travelling when it is hit?
A: I didn't specify, did I?
This led to a nice conversation on how we need to eliminate as many variables as we can.
Solution: Pick a velocity and work with it. They chose 100 ft/sec.
Q: So how fast is the ball travelling vertically and how fast is it travelling horizontally?
A: That depends.
Q: On what?
So we took turns pushing Joey around the room from behind and the side simultaneously. Each time one person pushed harder than the other.
Conclusion: If the person from the back pushes harder, Joey goes forward more. If the person from the side pushes harder, Joey moves to his left more.
Then we talked about how the velocities can be modeled using vectors and we can use what we know about triangles. Since the forces are perpendicular, we have a right triangle.
Q: If all we know is the hypotenuse of the right triangle, how do we find the other lengths?
A: Is that really all you know?
Solution: They settled on using a 45-45-90 since that is the only way they could figure out the other two sides.
Q: But what do we do for other angles?
A: Yeah, that's kinda tough, huh? Why don't you use a protractor to draw the angle you want, build the triangle you want and measure.
Q: Can we use GeoGebra?
A: Or that.
They used an applet with a fixed hypotenuse of 100 and gathered data on the other two sides.
Q: Is there an easier way?
A: Yeah. It's called sine and cosine. See how these ratios don't change as long as the angle remains constant? (it took a little longer than that, but you get the point)
They were off and running.
Conclusions
- 45 degrees maximizes distance.
- Complementary angles yield the same distance.
- Oh, and this:
I think you physics folks would say something like this:
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