First, if you have not tried the problem yourself (and read through Dan's post on this), you should do it. It's a great problem, I think. At first glance it seems fairly obvious, but upon further examination, it isn't. I'm still not sure if I took the "easiest" approach or not, and I'd be curious if you solved it an entirely different way.
But, solving the problem and creating opportunities for students to solve the problem are two entirely different beasts. I think if I were to use this problem, it could be "taught" in steps by breaking it down as shown below, but it's not clear 1. whether all students (or most students) would understand this problem, and 2. what I hope the students would gain or retain from this problem. I guess the latter bothers me more.
(In the picture below, "All Students" means all students would be encouraged to independently explore/figure out the important relationships. "Teacher guided" is more like places where I would intervene and provide hints.)
How would you present this problem? Would you scaffold more than what I propose above? The idea is that, in my class, I would give kids a few hints, one at a time, and see how many of them can run with it. If they can't, I'd guide them more heavily and hope that I can stop a bit further down the line for them to re-gain independence on the problem. But that's a fine line, because at that point, are they doing the thinking or have I stripped them of that opportunity?
Do you find the process of maintaining a cognitively demanding task challenging? How do you manage it to balance the need to make the task accessible to students (not just upon the entry point, but throughout the task) and still not over-simplifying the cognitively demanding parts for them?
I solved it while in bed last night. I had gotten stuck before, and when your post showed up in my Goggle Reader yesterday, I saved it to read after solving. When I had trouble sleeping, I started in on it. I used x and y coordinates, but was following a similar path to yours. I wondered how it could be done without the coordinates, and you've answered that for me.
ReplyDeleteI loved this problem, but Sue, do you think it's accessible to a majority of students? My approach was fairly particular, and I wonder how I could guide them along without giving it all to them.
ReplyDeleteI have no idea! I almost always overestimate what my (college) students can do. Can you give it without scaffolding at first, and charge (points or something like that) for hints, giving this version as your hints?
ReplyDeleteSo here's my solution
ReplyDeleteThe hardest part for me with the scaffolding is first getting kids engaged in the problem (what exactly is it asking? what exactly are we trying to find?), and what to do about the kids that just have no idea where to even start, without taking all the fun out of it by giving too much away. (I'm realizing that that's basically what you're saying in your last paragraph, Mimi, except you said it much better.)
I like how you changed the problem to "the sides are 10, find the areas" instead of "find the ratio" because asking them to find a ratio always seems to throw them for a loop.
I would resist having a pathway in mind. I find myself frequently saying "ok we don't see how to do this, so let's solve easier problems. What /can/ you find? I don't expect you to solve the whole thing right away - but you can all make progress - be ready to report on your progress in five minutes." With this prompt they could start finding some things we know they are capable of: angle measures or similar triangles for example. After they reported back, I'd probably be like "What would we need to know to find areas? What do we not know yet that would be really nice to know?" Let them chew on that for a few minutes - it would be really nice to know the location of P. OK so now we're all looking at a new problem: where the heck is P? I wouldn't do it for them, but I'd probably draw a highly suggesting diagram like your red triangles, or I'd overlay some axes to suggest coordinate geometry, or both? Or maybe put these on hint cards that would be available for those that wanted it?
It is a really nice problem but really tough to think about how you'd keep a whole class engaged in solving it! Mimi if you use it in class, I hope you'll write up a reflection so we know how it went.
Hi Mimi,
ReplyDeleteThanks for sharing this! I like the way you've broken down a problem to solve (with such nice pictures) and explained what you think you as the teacher would scaffold and what the students would do. It's a neat format for sharing about problem solving.
Reading the follow up comments made me think a lot too! One thing I noticed was that in your post you focused a lot on *what* the students could see or do to be successful. In Kate's comment she focused more on *how* the students need to think and explore to be successful. Did you notice that too?
I wondered, re-reading your post, what problem-solving strategies or habits of mind led you to the *what* that you saw? I noticed some "Solve a Simpler Problem" type steps, like Kate mentioned ("gee, wouldn't it be easier if...") and some "Change the Representation" ("how could we change the picture to show more relationships?" "what other problems does this problem remind us of? how could we make it look more like that?") as well as "Understand the Problem" ("what do you notice?" "what calculations can you do?" "what can we figure out about the scenario without paying attention to the question?")
I wonder if thinking about some of those mathematical strategies and practices, and getting good at them, might also help your students with the *what*?
Max
@Kate Nice solution!
ReplyDelete@Kate I also appreciate the detailed scaffolding plan. It sounds great! Unfortunately I don't have plans to teach this soon... I think it'd be a really nice toward-end-of-the-year thing if we're doing great on pacing everything else.
ReplyDelete@Max Yeah, I'm still thinking about the Habits point. Unfortunately, mathematical habits aren't built in a day, so I feel kind of like if the kids don't already have those habits before this lesson, they're not likely going to be drawing many connections on their own without heavy hints.
I'm late to this party, but I recommend letting the side length be 6 or 12, not 10. Since you know there will be some 1/3 and 1/12 kicking around in the work, side lengths 6 or 12 avoid fractional side lengths and areas. It makes the work a little cleaner and faster.
ReplyDeleteI say Less Scaffolding! Have those helps available in your back pocket and use 'em when you need 'em.
- Bowen
Can you explain where the 1/3 and 1/12 come from? It's not obvious to me. :(
ReplyDelete