If you’d be willing to extend your invitation to others, I’m looking for people to help me pilot XYFlyer:

http://puzzleschool.com/puzzles/xyflyer

Here’s a quick demo video:

http://www.youtube.com/watch?v=EUkgXfgbJwE&feature=youtu.be

Let me know if it would be a good fit for your students. I’m in San Francisco, so happy to come down there and work with you on it.

1) Students found the screen with circles/stars confusing. The students thought they signified spots in the sand instead of time estimates. Can the graph be placed ABOVE the picture? Also, would be cool to be able to toggle incorrect answers on and off.

2) When determining best path in the final screen, would love to see y-value (seconds) while sliding across the boardwalk in addition to the x-values (which is less interesting to the problem).

3) The text box seems to be missing for the question “What does this graph mean and how can it help us find the fastest path?” I’m not sure if the teacher-side of this is built, but I presume the idea is that the teacher can see the different responses – very cool feature!

Teaching at a school that is committed to individualized instruction via technology as much as possible while preserving good pedagogy and math practice standards, I find this adaptation of this problem exactly the tool I need. If you ever want to pilot stuff like this at Summit San Jose just say the word.

The next step conceptually (from a technical rather than pedagogical standpoint at least) is to make it work inside of say, iBooks Author. The most logical implementation would be to have it pull live results if it has network connectivity and otherwise fallback to predefined ‘example’ classmates.

That feels counter-intuitive on the surface and seems like it should be a good source of discussion in the classroom on two levels:

Explain in general terms why this might be true.

Show it mathematically – which involves making a lot of simplifications, which I think is an important skill in mathematics. Too many students seem to feel that math has been invented to make easy problems more complicated instead of the other way around.

To fool this I sometimes resort to “guessing badly” — in Dan’s work this is the “give me a number you know is too high” stuff. I might say here “Make a second pick you think is worse than your first pick, and find out if it is.” That takes away the implicit goal of finding the best possible answer, and can help students focus on guess-and-check toward a generalization.

I agree with Bowen, that having different representations that use different numbers is probably too much of a swamp to wade through.

Related to Bowen’s comments, I might give them a few minutes to decide on a second pick after seeing the scatter plot of all the guesses and corresponding times. I think running the calculations a few times is essesntial for kids that struggle with variables and generalizations.

It is a tough balance. The hook is the kid gets to pick a number and see if they “win”. The more you ask a kid to do things in a particular way (show work for a method that would only involve hitting enter on the calculator one time), I suspect the less commited they become to the problem.

It’s definitely a balance. If each student were asked to do three numeric examples, then generalize, I’d have them do the whole thing completely on their own and see what generalizations come. Multiple generalizations are awesome, and it’s really fun to compare them.

In this case you were having each student do one numeric example then generalize through sharing. In my opinion, when students do one complex numeric example and report back, you get a sea of responses. For the Taco Cart, some kids will resolve the addition and subtraction in the numerators. Others will make a common denominator. A lot of kids will just report back a decimal, the result of hidden calculations. All of these answers are correct.

Then comes a tough phase, showing that these answers are the “same”. But: the numeric answers are different. I’d rather do this comparison phase with multiple generalizations instead of multiple numeric answers. With multiple generalizations, you can actually show they are the same! Students usually want to chase this question, since they’ll want to know who’s right. (Everybody!). With multiple numeric answers, students may just accept that the answers are different — of course they’re different, we used different numbers!

So definitely yes: different representations and the discussion about equivalence are terrific and I love them. I’d just rather have that discussion after each student makes their own generalization. If there is a specifically targeted “form” of generalization (like what appears in Step 5), working out one calculation as a group makes it more likely that students will compute their own the same way, allowing the generalization to come from the different students’ work.

One example that comes to mind for me is the equation of a line, generalized from the slope formula — if students don’t put the points in the same order each time, the generalization won’t come.

Either way, it’s still a lot better than the “here’s the formula now evaluate it” manner in which such problems are often presented to students, and I feel these overlooked phases are where the real mathematical thinking is done.

In class, my tendency is to work out one example as a full class in an attempt to push everyone to the form I’m after.

Not to be that progressive educator, but aren’t those different representations and the interesting discussion about their equivalence nine tenths of the fun of teaching math?