Doing it the hard way

As the first full week of classes came to an end yesterday, I was working with my Honors Geometry class on Friday afternoon.  We had all worked this problem for homework:

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As I redid the problem myself prior to class, I found myself redrawing the diagram on a larger scale (which I find my students don’t often do), and I also knew it would be a question that not everyone could solve.  As the class went over homework together in small groups, I asked four different sets of people to put their solution on the board.   One group imposed a scale of 0 to 100 on the problem, placing point R at 0 and point S at 100.  They then found the numerical placement of the other midpoints in order to find the result that WZ=3x.  When they presented their ideas, they said they were thinking of 0-100%, which is what led to that scale.

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A second group did something similar, but chose to place point R at 0 and point S at 8.  They also reached the same solution, that WZ=3x, but all of the points they placed on their “ruler” were at integer values.  pic6

A third group did an identical scale, although one boy shared as the group presented that he had actually used 0 to 24 on his paper, not 0 to 8.  pic3

The fourth solution on the board was drawn by one person [let’s call him Steve] who had gone to the board early in the class so he could explain his work to his two partners, who hadn’t figure out the problem.  Steve opted for a more algebraic path to a solution, but reached the same end result.  He had included some statements which included defining some lengths as being 5/8 of other segments in the problem.pic5

As Steve explained his solution to the class, I could see the confusion in other’s eyes as to how he had approached the problem.  We worked through this solution together to make sure everyone understood Steve’s idea, and then paused.

I asked the class to consider the three different solutions we had up on the board (since two were the same).  Was any path a better one than another?  Are all of the solutions reasonable?  What should we take away from having done this problem? One girl raised her hand right away and said, “I think we should all be using Steve’s method because it’s the hardest.”

As soon as the words were out of her mouth, I realized how much of the typical student experience with high school math could be summarized by that one line.  To many students (and perhaps to many teachers too), the hardest path is the one we should be taking.  I felt pained as I heard this girl say this out loud, but at the same time, I felt grateful that she was willing to put that thought out there.  I immediately shared my feelings with the class, and our conversation continued, but I find myself still thinking about her comment now.  I shared the thought right away with a few of my colleagues, almost in a joking way.  However, it feels like there is so much more to draw out of the remark.

If most students think that they have a path to a solution, but they doubt themselves because it isn’t “hard enough”, then I have work to do as their teacher.  If we all share solutions in class as described above, debrief, and students still leave the room thinking that the hardest path is the one they “should have” taken, then I have work to do too.  Is there any way to combat this feeling in students of math?  Why doesn’t an easier path seem plausible?  What has happened in students’ prior math experiences to make them think that harder is better?

As I experiment with a variety of different classroom structures this year, I want to be sure to remember to find the moments to point out the multitude of solutions, and try to emphasize that the path that makes sense to each person has validity and does not have to be hard.  I also want to try to work to change my own mindset.  When I wrote my solution to this particular problem, I wrote this:

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It didn’t even occur to me to place a ruler-type scale on top of this image to simplify the work.  I think that is the math teacher in me, feeling like I should be able to express the situation algebraically, even there is a simpler path.

Overall, I’m grateful to my students for showing me an easier path to a solution, and eager for the year ahead to try and help all of my students understand that math does not need to involve choosing the hardest path.