Figure 1 |

Figure 2 |

By solving the escalator problem (see Dan’s blog) he and his collaborators were doing a far more interesting albeit more challenging problem. The contrived version (figure 2) remains in the textbook still to be conquered or ignored as many of you might consider doing until you realize that not only might this problem be on one of those dreaded standardized tests but it's also a part of the way that Algebra works its magic and is very interesting to folks who like math. And if I want kids to appreciate math don’t we want them to be able to understand how the powerful ideas in Algebra makes solving problems so elegant?

*It turns out the escalator problem may not be the best way to motivate the algebraic approach. (See Frank, Rob, Dan and Colin’s responses.)*

**Sidebar:**I’ve started to think about a way to rescue our disparaged kayak problem and I think I made some progress. What I like to do is give my students a challenge (containing some perplexity** and fun) that’s in their zone of interest and skill level to pursue using manipulatives (virtual or physical) to help them concretize the formal language that will be used later when discussion about algebraic solutions comes up. (I like to think of formulas or rules as a shortcut way of solving the problem. But the caveat is they should only use formulas after they understand how and why they work. Part of mastery is being able to derive a formula when memory fails you.)

So I started looking for a manipulative to use with the Kayak problem. After a quick Google search I found a virtual manipulative*** written in Geogebra. I used the data I collected from this simulation and a spreadsheet to help me solve this in a way that I think would make sense to kids.

This is my initial collection of data. From my chart I see that that sum of the speed of the boat and the current is the same as the distance traveled divided by the time it took.

In the kayak problem since we know that distance = rate x time and distance = 12, and time upstream = 3 hours then the average speed of the kayak is 4 mph. Going down stream the kayak takes only 2 hours, so it's traveling 6 mph.

The only current stream speeds that are possible for the times to travel both up and down stream in the same time is either 5 or 1. Why? If stream speed is 5, then the kayaker could cover the distance downstream in 2 hours with a boat speed of 1 mph and vice versa – switch the 5 with the 1. The same works for upstream. This time subtracting the current speed from the kayak speed has to equal 4. And 5 and 1 do that as well.

So in review, is this a contrived problem? Yes, indeed. Do we need better ways to teach it. Very definitely. The way to do it is for teachers to put on their creative caps and work with colleagues who can bring something to contribute to the WCYDWT curriculum party! And that’s what Dan is doing at his blog site right now.

**Next time (Part 2):**a powerful idea for connecting the abstract with the concrete. Inspired by Dan’s discussion of motivating students to move past counting rise and run from a graph with the more abstract short-cut: y

_{2}-y

_{1}/ x

_{2}-x

_{1}

**My “most students” theory comes from my experience of teaching algebra word problems to freshman high schoolers. I didn’t say all because there were individual and classes of students that loved them and treated them like puzzles. As a student I personally wasn’t very fond of them because they weren’t the right kind of hard for me or in other words out of my zone, but I did them conscientiously so I could do well on the test. Sigh…*

***Thanks, Dan, for the cool word.*

****Online applet works best using Firefox (Mac users)*

I was thinking about why the kayak in the river should be a "hard" problem. Shouldn't most math problems be "challenging" so that the students will need to think about solving them? We all like to be able to solve problems easily, but doesn't learning happen when we pushed to go beyond what we currently know?

ReplyDeleteReply to David: Challenging works if the problem is in the student's zone of understanding and interest. If it's not, they either ignore it or try to do it but only because they have to. For me the word interest is the key to get a problem i.e. challenge into a student's zone. Most contrived problems are only interesting to those who like doing these kinds of problems. I usually found those student in my honors classes, but only occasionally in others. By adding the manipulative the contrived problem becomes more interesting though it still may not be enough to get real student engagement.

ReplyDeleteIt may be that many aspects of education are contrived (choosing what to teach) and students are often made to do things. Within those constraints, it often helps to make topics as interesting as possible. But one aspect of teaching is to be a cheerleader, an encourager, a motivator, to find the hook for an uninterested student to engage with a "contrived" problem?

ReplyDeleteIt certainly helps to try to motivate a contrived problem. But isn't it better and easier to motivate a problem that the student wants to solve? In a Math 2.0 world students are engaged in a "wannado" curriculum rather than an extrinsic "havetodo" one.

ReplyDelete