Wednesday, April 21, 2010

Why Teaching Probability Messes with My Mind

It's hard to measure things that haven't happened yet. It's quite a challenge to keep front and center the question: Just how sure do we need to be right now?

Remember this?



Hurricane Ike made U.S. landfall in Galveston, Texas.

WCYDWT: Fiestaware

I know darn well this isn't going to knock any adolescent socks off...and honestly it's barely a WCYDWT. But, it's the best I have so far, and maybe if I put it out there, someone has something better.
Traditional Lesson: 


(Complete the example. Practice practice practice. Quiz. Quiz. Done until end-of-year review time.)
That formula, with the combination and the n's and p's and r's and q's, is like a bouillon cube. Very useful, but not intended to be consumed raw. (To rip off an Ellen Kaplan analogy.) As awful as that slide is, this is what is going on in classrooms across America, because the sobering volume and ill-conceived spiraling of content means we are rushed beyond reason. I know Algebra 2 teachers whose ulcers live in fear of a snow day for the loss of instruction time.

So here's my idea and how I intend to deploy it in class. I was indeed fortuitously inspired. I didn't think of this just because Bernoulli trials are coming up. 

Please bear in mind this anticipated dialog is boiled down, like bouillon, and there will be much waiting, hemming, hawing, frowning, clarifying questions, etc etc. And, it will go differently in all three classes.

"When I go visit my mom, it's my job to set the table for dinner."



"I was visiting for five days last week, and on Thursday, this happened:"



"My mom said, 'Hey, did you do that on purpose?'

'No' I said. 'That's weird.' 

Why do you think we were surprised?"

(The bowls and plates match.)

"Now, my mom thought this was a miracle. A once in a lifetime Fiestaware occurrence. So, what do you think is my question?"

(What are the chances you choose matching plates and bowls.)

Here, someone will assert that I subconsciously tried to make them match, skewing the chances of a matched set. I will acknowledge this but ask if we can, for the sake of learning, pretend I was choosing plates and bowls randomly.

"So, listen. I wasn't so sure this was so improbable, because remember, I set the table for five days."

(It would be more likely the more times you did it.)

"Right. So in order to convince my mom this wasn't so unlikely, what would I need to calculate?"

(The probability of choosing two matched sets one time in five days.)

Write this on the board.

"I'd like you to discuss possible approaches with your partner. You may not be able to solve the problem yet. That's ok. I'm going to give you four minutes to discuss. I want you to make progress. What would we need to know to make progress? Write it down. And if you can't, what questions do you need answered that would help you make progress? Write them down. Is there an easier problem, that you can solve? Write it down. Go."

I don't think I can script any more after this. It depends on what they come up with. They are likely to be able to, in order of increasing difficulty/decreasing likelihood:
  • calculate the probability of choosing one matched set
  • calculate the probability of choosing two matched sets, performing the experiment once
  • calculate the probability of choosing two matched sets on a particular day during the five days
After the four minutes, I would ask each pair what was their finding, or what is their question, and write all these on the board. And then I would go from there. With the intention of deriving the formula for Bernoulli trials that day or the next day, with copies of some practice problems at the ready to hand out, once we got this one.

Suggestions? Better ideas? I readily admit this question kind of sucks, because there's no intrinsic buy-in. Nobody really cares about the dishes at my mom's house. Or maybe it is the best I can do in my classroom, because I thought of it, so I can do it justice. What do you think?

Friday, April 16, 2010

Regression Project Debrief

These are intended to be notes for me for next time. Though if they help you to deliver this experience better than I did, awesome.

Here is the project description. The students' work is published to the class blog.

Percent of students who were able to:
  • Transcribe the data into an accurate, useful table of values: 100%
  • Model the scatterplot with an appropriate regression equation: 77%
  • Correctly use the equation to make predictions about future values: 60%
  • Interpret the meaning of the variables in the regression equation (or would-be equation): 94%
  • Interpret the meaning of the constants in the regression equation in the context of effect on the graph: 31%
  • Interpret the meaning of the constants in the regression equation in the context of the data: 20%
Those last two kind of bug me. I understand that it was a difficult thing to ask them to do, but I can't decide how important it is. If they know what x and y represent, they can use the equation to make predictions and solve useful problems. Is it really important that they know that in y = (0.44)(1.1)x, that the 1.1 means 10% growth? I don't have a good answer for that.

 

Changes for next time to the project description:
  • Write more explicit instructions for uploading images and making and editing blog posts.
  • Add a template for each problem that they can copy and paste.
  • Add more implicit instructions to try several types of regressions and compare r-values to find the best one.
  • Delete 'A Piece of Cake' or consider curmudgeon's comment below to just delete the 'cake' part and add a known data point that allows them to check their model.
  • Delete interpreting the equation constants from the quadratics problems.
  • Started out requiring four problems - one from each group - amended that to two problems. Four class periods were only enough time for them to reasonably complete two.
  • Make the easier tasks (with high percentage of success based on this year) worth most of the points, so you won't have to fudge the scoring such that everyone wouldn't fail.


Examples of Excellent Work:

Jarrod and Stephen: Sinusoidal Regression

Harry and Matt: Exponential Regression


And...I'm on goin' on break.

Tuesday, April 6, 2010

Regression Project Day n

Day 3

Oy. So the file-save errors from yesterday were caused by running out of room on our network servers. Luckily I had a couple flash drives lying around we could save to. I learned that kids don't as a rule carry around flash drives.

A student said "This is too hard. You should teach honors, not regents." Apparently using actual data and not made-up crap = "honors."

There are a number of changes I need to make if I ever undertake this insanity again:

"Shiver in My Bones" needs to all be graphed on the same axes.

"Piece of Cake" just needs to be cut from this project because it is the one thing that is not like the other.

"Walking after Midnight" needs to be made at least as attractive as "Harder Faster Stronger" because it was not nearly as popular as it should have been. Also specify that it must be recorded in miles vs hours and not the other way round and not in seconds or...non-miles.

"Will the Ball Hit the Can" needs to de-emphasize the meaning of the coefficients and cut it down to 1 to 2 throws, and emphasize "figure out whether or not will the ball hit the can?"

I need to throughout the year ask more often what numbers in an equation mean. Like what the a and the b mean in y=a*b^x. They were universally, alarmingly, totally stumped by these questions.

(more to come soon. this post is a hydra of a post.)

Day 2

Goods:
  • Had a student teacher helping today who was really smart on tech so, wonderful. 
  • More victories from refusing to tell them what to do.
  • Asking pointed, leading questions is just as bad as telling them what to do. I wasn't playing that game, either.
Others:
  • Google docs wouldn't let us insert an image. It was giving a weird, stupid error. I think it was a google problem and not an us problem.
  • As a result, we gave them the alternative of writing up their analysis as a Blogger post. Dear Google: people don't realize that Blogger is Google and they can use the same login. Just call it GBlogs already.
  • Blogger is not as good a solution for this as Gdocs was. No tables, no equations. And no I'm not teaching tenth graders html and latex in 2 days so don't bother.
  • The student accounts are not able to use TI-Connect to upload a screenshot with the USB cable, because they don't have administrator privileges. Super annoying because I was expecting this to work.
  • Yesterday we had a problem saving the TI-Smartview emulator state because of a "cannot save to that location" error. (I hate that. Why the hell not?) Super frustrating when they did all that work, and it wouldn't let them save. I was going to have kids who were able to save, just email their files to the kids who couldn't.
  • But, some of the kids who did save their emulator state weren't able to open them. Unreadable file something something.

Summary: The tech is very much getting in the way of the learning. Is this still worth doing? I think so. Real data imposes a priceless logic and inevitability that I will continue to walk over broken glass to get to.


Day 1
  • Kids get a laptop and instructions to head to class blog and start reading instructions. Many dire warnings about reading all instructions before starting, reading carefully, blah blah blah. Still get kids going "Can I be in Group A?" Not until you learn how to read, babe.
  • Only a few need to create or recover a Google account, and it is once again surprising to me how hard this is for them.
  • Some don't know how to make a new Google document (add instructions for that somewhere)
  • 17% successfully invite me as an editor as instructed.
  • Pretty sure everyone has at least created the document to edit and started tabulating data.
  • Lots of complaints along the lines "You mean we have to use all this data?!" Only if you want a good model. And have fun in SUPA Stats next year because you thought it was going to be easier than Precalc.
  • Some IEP-ers have trouble with written instructions and breaking large tasks down into smaller ones. Sent lunchtime email to Resource teacher, pleading for help. There are a lot of instructions. I can't give them all verbally.
Group A: In Facebook World Domination, I am patting myself on the back for not telling them how to tabulate months. They kept asking for validation. I kept saying "Does that make sense on a number line? Are data that are 6 months apart going to be 6 apart?" etc etc. They all figured this out and got a little shot of whatever brain chemical you get when you do something intellectually satisfying and I'm glad I didn't rob them of that.

In Blink, lots of kids graphed Transistors vs Years instead of Years vs Transistors. Let it get to the scatter plot, when they realize they don't know a function that will fit that shape? Or nip it in the bud first thing tomorrow? They also used years like 79, 84, 93 but then when they got to 2002 they called it 02. Headsmack. I'm letting that one go until they get confused by their scatter plot.

Group B: Everybody, I think, who got that far, picked A Piece of Cake because it looked easier than measuring and timing the pendulum. They have been miserable at generating data for a Piece of Cake, though. Three cuts only gets you 6 pieces! What do you mean, maximum pieces?! Nobody cuts a cake like that!

Miscellaneous Notes:
  • Plug in all the computers during lunch or they will start dying during 7th period.
  • Scrounge as many TI link cables as you can before tomorrow.
  • This is suffering as much from the "Get it done while learning as little as possible" mentality as anything that ever happened in a school, even though the whole point was for it to be interesting. How in the world do I tackle that 900 pound banana, I have no idea. (Why don't I let them analyze whatever data is interesting to them? Because it was hard enough for me to find manageable, relatively easily-tabulated data that would fit a function appropriate to the course content. And they're supposed to think of something interesting and go find bivariate data about it? Puh-lease.)

Day 0 and less than 0
Collected ideas for fun data to regress. Wrote project description. Got some helpful Twitter feedback and made a few last minute changes.

Thursday, April 1, 2010

Formulas? What Formulas?

I banished the textbook slope, distance, and midpoint formulas from Geometry class this year. I feel like such a rebel.
First of all, kids have trouble with subscripts. They write them like exponents and then madness ensues. Second, that big long radical sign freaks them out. Third, if they learn and remember it conceptually, they won't try to memorize a formula they don't understand and dink it up later.
So here's what they did learn, and even though I'd like a more organic way for them to get there, the outcome was not bad at all.
Distance
Really Just the Pythagorean Theorem. Distance2 = (subtract the x's)2 + (subtract the y's)2. (To be followed next week by Standard Form of a Circle: Really Just the Pythagorean Theorem.) This is awesome because they heart them some pythagorean theorem.
First I asked how far away are these places as the crow flies. (We had to discuss what "as the crow flies" means.) (Thanks Visalia, CA, for laying out your streets on a square grid.)
Then we practiced a little treating the segment whose length we wanted as the hypotenuse of a right triangle, drawing the right triangle, using PT:
Then...dun dun duuuuuun:

We go back to the ones we can graph, and figure out how we can get the legs of the right triangle from the coordinates:


Then we are basically home free:

Midpoint
(average the x's, average the y's)
Same basic plan. Practice a few times, throw one in that doesn't fit on the graph, backtrack.
Slope 
Subtract the y's, subtract the x's, divide. Yep we played the song. (We sang the song. We claimed we wanted to write our own song about the distance formula and film it and put it on youtube. We are probably not that ambitious.)

Which part of the roof would you rather stand on in a flood?


How do we compare their steepnesses numerically? They've seen slope before.

Do the y's or the x's go on top? Well...which of these slopes do you think should come out to be the bigger number?

Put it together:

I was a little uneasy about not making them copy down and memorize the formulas as they appear in books...am I setting low expectations? Will they get to college and fail because they are unfamiliar with subscripts? But then we were settling in for a quiz this afternoon, and someone said all panic-stricken "What's the midpoint formula?!" And someone else said, "Calm down, you just average the x's and average the y's." And I think that's all right.