or Context Clues (part 2)
Nevan's helping me build planter boxes by holding the 2x6s in place while I screw them together when I ask:
"Nevan, are those two pieces flush?"
"Yeah, Dad."
"Do you know what 'flush' means?"
"Mmm. No."
"What do you think it means?"
"I assumed it meant 'on track' or something."
So when I'm asked, "how do you know that they know?"
That's how.
The challenge is to care enough to ask.
- Posted using BlogPress from my iPad
Thursday, March 17, 2011
Adventures in Pedagogy: Percentages
Sitting here in the car with the boys as my wife runs into Target to pick something up "really quick." (I believed her. She can be quick—in Target, right?)
Anyway, I find a spot that gives us a clear view of the entrance and Dawson says:
"Hey Dad, see that white car in front of us? Why don't you park in front of it? It's closer."
"Can't. See that other car? It's like halfway in the parking space."
"Halfway? That's a bit of an exaggeration."
"Alright, then what percentage of the space is it taking up?"
"About 1/11."
"I'll give you that, but what percentage?"
"About 9 point something percent."
"Not 10%?"
"No, a little less."
Right about then, my wife returned. She was pretty quick.
And she brought me a Drumstick.
- Posted (on location) from my mobile
Anyway, I find a spot that gives us a clear view of the entrance and Dawson says:
"Hey Dad, see that white car in front of us? Why don't you park in front of it? It's closer."
"Can't. See that other car? It's like halfway in the parking space."
"Halfway? That's a bit of an exaggeration."
"Alright, then what percentage of the space is it taking up?"
"About 1/11."
"I'll give you that, but what percentage?"
"About 9 point something percent."
"Not 10%?"
"No, a little less."
Right about then, my wife returned. She was pretty quick.
And she brought me a Drumstick.
- Posted (on location) from my mobile
Monday, March 14, 2011
Adventures in Pedagogy: Problem Solving
Dawson is playing a game on his DS and I plop down next to him and ask:
"Whatcha doin'?"
"Oh man, Dad, I'm on this level that took me forever to figure out."
"Really? So what do you do when you get to a level and you don't know what to do?"
"I experiment."
"What do you mean?"
"I try stuff and keep trying until something works."
"You mean you don't just sit there, throw your hands up and say, 'I don't know what to do, so I quit?'"
"Nope."
- Posted from my iPhone
"Whatcha doin'?"
"Oh man, Dad, I'm on this level that took me forever to figure out."
"Really? So what do you do when you get to a level and you don't know what to do?"
"I experiment."
"What do you mean?"
"I try stuff and keep trying until something works."
"You mean you don't just sit there, throw your hands up and say, 'I don't know what to do, so I quit?'"
"Nope."
- Posted from my iPhone
Monday, March 7, 2011
Similar Triangles and Algebra
Dr. Hung-Hsi Wu pg. 58:
So, why do you think that is?
- Posted using BlogPress from my iPad
The reason for the critical need of a definition of similarity is that a working knowledge of similar triangles is absolutely essential for students to achieve algebra. Without this knowledge, they would have no hope of understanding the interplay between a linear equation of two variables and its graph, which is a major topic in beginning algebra.
So, why do you think that is?
- Posted using BlogPress from my iPad
Thursday, March 3, 2011
Fraction Multiplication
I've been spending some time lately thinking about how to help elementary school teachers teach math conceptually even if they don't consider themselves strong in math. I keep coming back to interactive applets (read: GeoGebra) because it's kind of a one-stop shop. Yesterday, I tried a few out with a 4th grade class and they responded very favorably. So, here are the latest two I've done on fraction multiplication.
Click on "File" and "Save" if you'd like a copy of either applet.
Please forgive the crowding, I had to minimize everything in order to fit them into the post.
Click on "File" and "Save" if you'd like a copy of either applet.
Please forgive the crowding, I had to minimize everything in order to fit them into the post.
Wednesday, March 2, 2011
Another First
Usually kids raise their hands to show what they know.
Today, M.C. politely asked me to get out of the way, walked up to the document camera, put her work under it and said, "I need your help, guys. Really. I want your feedback. I don't know how to do this."
The class responded. No one ridiculed her. She asked them questions. They asked her questions.
All I could do was sit down and watch.
Today, I caught a glimpse of what school should be.
Tuesday, March 1, 2011
Let 'Em Drive
The Problem:
Me: How many different ways could we do this?
Class: Quadratic Formula, completing the square and factoring.
Student 1: Mr. Cox, why does y = 0? Isn't this a function where y can equal a bunch of things?
Student 2: Yeah, but we want y = 0 because that's when it crosses the x-axis.
Student 1: Ok, so we are just focusing on one possible value of y.
Student 3: So then, if we let y = 0, then we are finding the x values that make y = 0, right? But what if we want to know when y = 1? Can we do that too?
Student 4: I guess we should just let y = 1, but we'd have to subtract it from both sides so we get:
0 = x2 + 10x + 15. So now we are finding the x-intercept again.
Me: Is this an x-intercept? What does y equal?
Student 4: Oh, no, y = 1.
Student 5: Mr. Cox, can we see what this looks like in GeoGebra or something?
We graphed it and then moved a point around on the parabola to verify our results and looked at how the points where y =1 maintained the same symmetry with the x-intercepts. Then, a kid pipes up:
Me: Hmm. Maybe. I'll tell you what...you guys give me a function that has x intercepts at -4, 0 and 3. You have 15 minutes. Go!
About 5 minutes later, J.V. walks up with this:
And I ask him, "How did you come up with this?"
J.V.: Remember the other day when we had to come up with parabolas with x-intercepts? I figured I could just work backwards like we did then.
I'm amazed at how often these kids will take the lesson into places I wouldn't have thought to go. It's just a matter of letting go a little. And the more I look for open ended opportunities, the return is exponential.
Find the x-intercepts of y = x2 + 10x + 16.
Me: How many different ways could we do this?
Class: Quadratic Formula, completing the square and factoring.
Student 1: Mr. Cox, why does y = 0? Isn't this a function where y can equal a bunch of things?
Student 2: Yeah, but we want y = 0 because that's when it crosses the x-axis.
Student 1: Ok, so we are just focusing on one possible value of y.
Student 3: So then, if we let y = 0, then we are finding the x values that make y = 0, right? But what if we want to know when y = 1? Can we do that too?
Student 4: I guess we should just let y = 1, but we'd have to subtract it from both sides so we get:
0 = x2 + 10x + 15. So now we are finding the x-intercept again.
Me: Is this an x-intercept? What does y equal?
Student 4: Oh, no, y = 1.
Student 5: Mr. Cox, can we see what this looks like in GeoGebra or something?
We graphed it and then moved a point around on the parabola to verify our results and looked at how the points where y =1 maintained the same symmetry with the x-intercepts. Then, a kid pipes up:
Are there graphs that have more than two x-intercepts?
Me: Hmm. Maybe. I'll tell you what...you guys give me a function that has x intercepts at -4, 0 and 3. You have 15 minutes. Go!
About 5 minutes later, J.V. walks up with this:
y = (x + 4)(x - 0)(x - 3)
y = x3 + x2 - 12x
J.V.: Remember the other day when we had to come up with parabolas with x-intercepts? I figured I could just work backwards like we did then.
I'm amazed at how often these kids will take the lesson into places I wouldn't have thought to go. It's just a matter of letting go a little. And the more I look for open ended opportunities, the return is exponential.
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