The last two years, I started my second semester (of this NON-AP course) with probability. Now, most of the probability in the class is destined to be empirical because of my overall approach—for example, you get a P-value by simulating a sampling distribution and counting the cases in the tail—but I had a notion that this was the place to learn some of the nuts and bolts of solving probability problems “by hand”—using area models and tree diagrams and learning when you add and when you multiply, and why. I even flipped the classroom a bit, making a series of videos describing the tools and techniques I wanted students to use.
Now I’m not sure that was such a good idea. It took a lot of time, it confused some of the students, and frankly we didn’t use it much. The only place we used the concepts was in addressing some issues in conditional probability—and a lot of that might be approached elsewhere (e.g., with the Markov data game). And in a traditional class, the main place you use probability is probably to get a value from the cumulative Normal distribution, and that certainly doesn’t require understanding of tree diagrams and vocabulary like “mutually exclusive.”
Instead, if I were to do this again (though that looks less and less likely, and I find myself regretting that more and more) I’m thinking I might start out that second semester with random walks.
Why? Because they’re all about variation, and from them, you can use modeling to get the root-N dependence, and it’s a good setup for learning the effects of sample size. Which have gotten short shrift this year, and I think they’re really important. (See this video, by the way, for an example why.)
Variations on the traditional random walk can also help you see conceptually what bias is all about.
Here we have something interesting that’s outside the narrative thread. On the AP Stat list serve, Chris Talone asked this question:
Is there a way to set up a Fathom simulation to illustrate how the slope of a line of best fit will vary when choosing ordered pairs from a population of ordered pairs? My students are having a hard time understanding the purpose of the linear regression t-interval and the linreg t-test. I would like for them to see how the slope can vary depending on the sample of points chosen. Ideally, I’d like to set up a population of ordered pairs, graph a scatterplot and find the line of best fit for the population, then have Fathom randomly select 2, 5, or 7 of those ordered pairs, graph a scatterplot of the sample chosen, find the line of best fit for the sample chosen, and also plot the sample slope on a dot plot, and then repeat many many times….
I posted a response there, but we can’t give illustrations. We can here! This is where we’re heading:
I know the language changes. We’re losing the subjunctive. “Hopefully” is hopeless. And words that have specific technical meanings take on new meanings in less-technical contexts. But I’m wondering: any other math teachers out there flinch when people use exponentially to mean dramatically?
Here are some examples from a quick search on the New York Times website. Some are clearly figurative. Others we can’t be sure. But a scary thing is that I got over 4700 hits for the last 30 days:
With HBO filming the Jets for its annual training camp series, with Ryan’s pronouncements growing bolder in his second season, with expectations that have grown exponentially, Jenkins and the Jets feel a greater sense of urgency.
—Greg Bishop, July 19
Regulation is more necessary than ever before because the severity of harm that can be caused by human technology is exponentially increasing.
—Dennis Montana, comment, July 26
This one has a curious juxtaposition of drama and the exponential function:
As night falls across the country, the number of calls increases dramatically, and their drama increases exponentially.
—Ashley Gilbertson, Lens, July 30, about a suicide hot-line.
This one may actually use the word correctly, even if we disagree:
…if we do not plan an energy descent plan to deal with the approaching decline in hydrocarbon production, the economy WILL collapse rather than gradually decline. We cannot have an exponentially growing economy without exponentially growing hydrocarbon production. Those days are history.
—Joshua M, July 26, 2010 (in a comment)
Nothing is quite as fun as getting stuff. Especially stuff in quantity. I’m sure exactly how I will use these—the obvious is to do a hands-on simulation of exponential decay, but I bet I can come up with something cooler.
A first draft: “We’ve taked about craps, a game that uses two dice. Most games with dice use two dice. Suppose you wanted to design a game that used two hundred dice. What could it be?”
Anyway, I’ve had it on my list to buy dice because I just think I ought to have a lot of them for this class. You veterans probably already have your dice in vast quantities, but for the record, in Summer 2010, the best deal I coud find was at, of all places, Amazon. $11.70 per hundred. These are the translucent, full-sized red ones in the photo. A little more expensive, but with more variety, is a product from Chessex (also cheapest at Amazon in my searching) called “pound-o-D6,” which incudes about 100 dice (seconds, but who cares?) in various colors and sizes, mostly a little smaller than the red ones.
I am not quite done with jet lag. I think I’m trying to hang onto being away, it was so delicious.
I love travel. In my other life as a freelance science-and-math educator, I attended conferences and traveled a lot. Now the schedule does not permit it much, so I guard my opportunities carefully. Summer is the big chance.
So: ICOTS. The International Conference On Teaching Statistics. Held every four years. 2010 was in Ljubljana. (Huh? The capital of Slovenia. Oh. You mean Slovakia? No, Slovenia. Just to the right of the top of Italy. Under Austria. Part of the former Yugoslavia. Forests. Mountains. Caves. Castles. Gelato. Tied USA 2-2 in the World Cup even though we maybe shoulda won.)
People will often make the case, “We can’t be that stupid, or we would have been evolutionarily wiped out as a species a long time ago.” I don’t agree. I find myself saying, “Well, no. Gee, all you need to do is be far enough along to be able to get three square meals or to solve the calorie problem long enough so that you can reproduce. And then, that’s it. You don’t need a lot of smarts. You don’t have to do tensor calculus. You don’t have to do quantum physics to be able to survive to the point where you can reproduce.” One could argue that evolution suggests we’re not idiots, but I would say, “Well, no. Evolution just makes sure we’re not blithering idiots. But, we could be idiots in a lot of different ways and still make it through the day.”
I’m inspired to write this by a small group of what seem to be like-minded but more experienced math teachers around the blogosphere (whom I will reference as soon as I learn how). They are scary smart, young, and energetic, and can probably actually type. They sure as hell can write. And you seriously want your kid in their math class.
If they’re so great, why should I blog? Although I can dream that my unique contributions may help others, the real reason is to help me record what I’m thinking, what I try, and what happens in the classroom. I’m in such a good situation, it seems to demand some sort of record.
Put another way: What happens when you have no excuses? I’ll try to live up to the trust implied by all the resources I get from the school community, and I’ll try to make something good out of a situation most teachers would give various limbs for.
I’m also hoping that by putting some of my plans out there—such as using standards-based grading, OMG, I can’t believe I actually put that in print—I’ll feel some obligation to follow through.
I’m an older, part-time high-school teacher. In August 2010, I start my third year. So I’m relatively new. I’ll be teaching one section of statistics. And the real kicker: I get small classes, lots of administrative support, helpful, often brilliant colleagues, motivated students, and all the resources I need.