September 4, 2012 by ray_emily
Last week, while reading Mark Driscoll’s Fostering Algebraic Thinking, I nearly shrieked “eureka!” upon turning to page 11 and discovering “Crossing the River.”
I was elated about this problem for a multitude of reasons. I knew that it would: (1) serve as the topic of my third New Blogger Initiation Post, (2) extend and up the rigor of my Week #1 whiteboarding activity, and (3) welcome kids back to school with a bang, after their unexpected week-long hurrication. (During this electronics/internet-free break from school, I had the pleasure of reading Driscoll’s book, in its entirety. I highly recommend it as a source of challenging, worthwhile problems; it also has gotten me thinking about how I might help smooth my students’ transition from arithmetic to algebra.)
Anyhow, take a look at the problem, first, which I slightly revised from its original form to increase the clarity for my little people. See if you’re as excited as I was, okay?
Eight adults and two children need to cross a river. A small boat is available that can hold one adult or two children. (In other words, there are three possibilities: 1 adult in the boat; 1 child in the boat; or 2 children in the boat.) Everyone can row the boat. How many one-way trips does it take for all of them to cross the river?
Bonus: Can you describe how to work it out for 2 children and any number of adults? How does your rule work if you have 100 adults?
Finished early? Think about how to visually represent your solution.
In case you’re not in the mood to play around with this puzzle, here is one version of the solution. (Or, way more fun option: Scroll down to check out my kids’ solutions! The level of clarity varies from whiteboard to whiteboard, but all boards contain accurate information.)
And now that you’re familiar with this lovely little problem and its answer, I’m sure you’re dying to hear about how my kids attacked it.
First: I gave everyone a solid five minutes to make sense of the problem, by reading, re-reading, and note-taking. The room was silent, and most kids (judging by facial expressions) were pretty perplexed.
Next: I let students join heads for a few minutes. What I noticed, here, was that only a very small handful of kids had a clear idea about how to begin breaking down and solving the problem. Those kids – the ones who “got it” – helped their peers wrangle their way through the facts and make sense of the information provided.
Now, the fun part! After frustration had just begun to swell and some kids were feeling a little stumped, I gave one envelope to each table – each of which contained the pieces that you see, here.
I didn’t need to explain what the pieces were. (In fact, two students were working on their own concrete, movable models before I distributed my envelopes: so great!) As soon as envelopes were opened, students knew exactly what to do. They were no longer stumped, but eager to use the simple, multi-colored, cut-out shapes to tell the story of the river-crossing.
After ten minutes of playing with these low-tech, low-glam models, all of my students noticed that for each adult, the boat made four trips across the river. (“Why are you making these poor kids do all the work!?” one observant student inquired. “This is child abuse! The adults are hardly doing anything!”) Alternatively: Students independently discovered that they needed to seek out structure within this problem, and then they did just that – without my saying a word.
Side note: This problem is so beautiful because it is nearly impossible for kids to find the solution without noticing that certain events are repeated over and over and over within their narrative. That there are eight adults (rather than, say, three or four) forces even the most ridiculously stubborn of kids to pause and think, “Huh, haven’t we repeated those steps a whole bunch of times?” (I’m talking about those students intent on plodding through, step by step by step, and steering clear of anything remotely resembling pattern or structure or a shortcut. You know the type, eh?)
Next, I gave students time to plan their visual/pictorial models. After plenty of lively discussion, here’s what they came up with. (Click on any of the collages, below to take a closer look – or zoom in much, much more by visiting the collection of photos, here.)
Many students used some variation of the models that were the clearest and most compelling way back when we solved the first river-crossing puzzle. In other words, they opted to recycle previous work and ideas, rather than considering the unique variables facing them in the present moment. Alas: These groups’ models (though lovely and correct) did not quite capture how the solution involved cycling through the same four steps eight times in a row.
The other groups—the ones that developed novel, often circular models—ended up with pictorial representations that better (IMHO) captured the essence of this problem and its solution. Regardless, I’m pleased by the variety of approaches and the abundance of creativity – and also at the notion that my kids are developing more nuanced ideas about clear mathematical communication. (Students did, of course, have a chance to look at and critique one another’s boards, at the end of the activity.)
The simplest ‘model’ of the solution, which we explored at the end of class, was of course an algebraic equation: T = 4a + 1, where T = total trips and a = the number of adults. That my kids got to develop and make use of an algebraic equation was just a little added bonus – icing on the cake! – although I can imagine using this problem as a jumping-off point for all sorts of fun algebra-related analysis. (Driscoll also suggests altering the number of children to really make things interesting.)