Now the students are all at work; they are in groups of 2-4 (they tell me they get some degree of choice about with whom they can work), each at a whiteboard, and they are diligently working through a sheet titled “Investigation 7.”   Students tell me there is no textbook, and that this is the class-time norm; students are given problem sets called investigations, and they work their way through them, one or two a week maybe.

This one begins with a heading: “in previous investigations you studied how sine, cosine, and tangent can be used to help you model the behavior of a right triangle.   In this investigation, you will extend those ideas to include triangles which do not contain a right triangle.”
“Two park rangers are in fire watch towers. They both spot a fire in the distance; one of the ranger recognizes the location and knows that it is 4.9 miles from that tower.  The rangers also use surveying equipment to measure the angles shown in the diagram at right.”
Following are five or six questions following the logic of the scenario.   Looking around the room, I am struck by the fact that though this 80 minute math period is immediately following lunch, there are no droopy eyes.  Students can sit, stand, move around, or sit on a table– there is flex for their physical posture learning preference.  They are using graphing calculators a little.  They are explaining their reasoning to each other, and have the burden (good) of articulating their understanding.   The teacher is circulating, working on a tablet laptop and making her own drawings on the white board alongside the students’ work.   “Wouldn’t the same pattern hold if you changed the numbers here?”   A student offers a remark, unsolicited: “I like this Katie, this is fun!”
I go group to group, asking students to tell me how this way of learning is different from at their previous school, and by which method they better learn the material.  Students agree widely that the method is very different here: at every previous school, they tell me, they had a textbook and the teacher just assigned homework problems and they came in and the teacher taught them the material and the problems.   Here, it is so different, they work in groups and have to learn it themselves.  Some say his way is hard, or harder, than the other way; some say they enjoy working in groups and prefer that.  When I ask them if the learn more, they aren’t sure at first; they hesitate as they think about it, trying to go back and compare (the question is a stretch, especially if they haven’t thought much about it).  But after some cogitating, most tell me they think they do; because they have to figure it out themselves, because they write it twice, once on the whiteboard and once on their notes;  because they have to explain it to each other; and because by having to the work themselves in class, they have to pay attention rather than tuning out.

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