Student Questioning

I annoy students terribly. I love teaching geometry and teach it the way I love it. To me, geometry is the foundation of questioning, exploring, extending thinking, inquiry. I think Algebra should be, but for our students it seems, at least at this point, geometry is where that begins. Until now, Algebra has been taught procedurally, like their previous math classes. Then comes geometry, with Mrs. Ryan. Oh no!

The first thing my students learn is that I never run out of “why?” I also rarely answer questions, but respond with a question. When they say they don’t understand a problem, I ask them what they do understand about it. I guide them to find an entry point, I push them to figure out what they know and what they need to know. I walk away when they shrug their shoulders at me and tell them to call me back when they figure out where they are. I make them explore, inquire, dig, re-read, look at examples and re-answer the same questions until they go, “OOOOOOHHHH.” I get calls from parents telling me their son or daughter has told them I don’t teach them anything. They accuse me of refusing to show students how to solve problems. I try to explain what I do, but they don’t get it. It doesn’t look like math has looked for their previous 10 years in school.

Some years my students begin to see and understand in the first semester. Usually this occurs about November. I’ve had a couple of years when this has occurred near the end of October. Some years it takes to the second semester, January or February. This year, it still is not occurring. I had a parent-teacher conference yesterday where I heard once again from the parent that her daughter says I refuse to answer her questions. When I responded that I always answer, I just tend to answer with a question, the daughter starting laughing hysterically. We all looked at her and she said, “that’s exactly what she does.”

After the conference, I had a geometry class. We were working on solving some trigonometric problems and students were asking questions about HW problems. I always ask them to tell me what they tried and where they got stuck. They have to have tried something, in fact that’s one of my chants, “try something.” We worked together on creating a visual model on the board, labeling values and making sure that we had a right triangle in the model to work with. I solved the problem the way they told me to solve it, then asked, “are there any problems with what we’ve done here?” They all looked at me like I had lost my mind. The problem started with a plane 10,000 feet off the ground, we were looking for the angle of elevation for the plane to get to 20,000 feet. They used 20,000 feet as the opposite leg of the triangle. When we finally figured that out, and changed the values, students began to solve again. One student asked, “can’t we just divide the angle measure by 2?” I said, “I don’t know, can we?” He responded, “I guess not.” I said, “why do you guess not?” He said, “because you just asked me if we could like I had asked a crazy question.” You’d think they’d be used to me by now! I said, “I asked you because I thought it was a worthwhile question and thought we should figure it out.” After a few more questions, a few more changes in values to test conjectures, determining what types of values we needed to test to determine if the conjecture worked, we came to a conclusion. And they understood the relationships between the ratios and the sides and angles of the triangles better. We didn’t get to what I had planned for the day, but I think we got a whole lot more mileage out of what we did do.

When will they learn?

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Right Triangle Trigonometry

Monday was our first day back from spring break. We had started looking at right triangles, pythagorean theorem and special right triangles before the break. After a week off, I was sure they had forgotten some of it. I wanted to move on to trigonometry, so I began searching out some ideas for how to present this. I have learned in the past that just trying to teach them the ratios and show them how to use them rarely resulted in good understanding of the ratios or how to determine which one to use when. I wanted to do something that would really drive home that these were related to the acute angles in the triangle and the ratio of specific sides in relation to the angles.

I found a couple of activities that others had done, but was not quite satisfied with the way they were written or convinced that they would really drive home what I was hoping to help students understand. I decided to write something myself expanding on some of the ideas I had seen, and as I worked on it I realized that we needed two days to do this. Monday was a pre-discovery activity, mainly to refresh what they knew already about right triangles and to begin to encourage them to start relating the angles and sides. They worked on this in groups, discussing and writing out their conclusions together. This went well, better than I had expected, and I gave them a problem to work on after finishing it, which I thought they would need to take home and finish. A large portion of the students finished in class and were able to clearly explain what they were thinking and how they found an entry into the problem. I was pleasantly surprised at the work.

Tuesday I had a trigonometry discovery activity for the groups to work on together. Unfortunately, my planning was not quite as organized as I would have hoped. Tuesday’s are our short days, our PLC teams meet for an hour in the morning, so we start school one hour later. This shortens our periods approximately 10 min, which doesn’t seem like much, but this can be crucial for an activity. It definitely was for this one. I tried to get the students started right away, but kids will be kids. Several groups did not make it to the 3rd problem, and I even told them to skip the second, explaining that it was to verify that the results of the first problem would be consistent no matter what type of right triangle they used. I gave them some homework problems to work on identifying trigonometric ratios in several right triangles in both fraction and decimal form, but I felt uneasy about what might happen.

Today I decided that we would go back to the 3rd problem in the discovery and discuss it as a class. It turned out more students that I had anticipated actually did understand what they were seeing, and we went over a couple of the homework problems and the class really did have a vision of how they should be setting up the ratios. We looked at a couple of problems where they had to find the angle measure using the inverse functions and made sure everyone’s calculators were set properly to use the trig functions.

All in all, a good activity and the students are really verbalizing understanding of the functions better than I have ever seen in the past. We will work on application problems tomorrow and friday, which will give me a better idea of how well they are truly understanding this, but I feel better about it right now that usual.