I taught two classes today. Honors Geometry and Pre-Calculus. The first class went very well. We had a review session. Several students either started or completed work on extra credit assignments. It was good.
The last period of the day arrived. It started out well enough. Students worked on review problems from the previous class. I introduced new material then everyone just kept talking and talking and talking. Half the class is seniors. I'll just stop here. This has been cathartic enough.
Monday, April 29, 2013
Saturday, April 27, 2013
Nearing the end
As I reach the end of my first year teaching high school, I am thinking about what was good and what wasn't.
Calculators and decimals. In geometry, many of the students have a hard time finding an exact solution: tan36 becomes a decimal, sqrt 2 becomes a decimal, pi becomes 3.141. At my school, it is considered "okay" to do this until we get to volume, surface and circles then the students are overwhelmed. Questions I have heard, "is r=(1/5)^(1/3) an answer?", "can I take the square root of pi?". Through the year I have told some students they weren't allowed to give decimal answers. These students had no problem with the switch. I told the other students to give the exact solution time permitting. Side note: my class had the highest average on the entrance exam into algebra 2 so I wonder if forcing them to give exact solutions from the beginning helped.
I have taken to writing lecture notes in latex. I include exercises after each topic. As a class we go over a new concept, do an example problem or two and I set the class loose on in class exercises. Some students enjoy this and do well. Some are able to read the examples, do the exercises, check with me and then start their homework. I have several students in my regular pre calculus class that need to see a "similar" problem worked out first (the problems can only differ by constants). Generally this has meant my notes are organized by solution method. How do you find an explicit formula for an arithmetic progression given a_1 and d. How do you find an explicit formula if you are given a_n and a_m. I have avoided this somewhat in my honors class by introducing an idea and then including exercises which lead the student to discover what they need to know. Example: an arithmetic progression has a common difference between terms. Have them write out 5 terms using the recursive definition. Suggest that a_2 can be written as a_1 plus something. a_3 then can be written as a_2 plus something and then a_2 can be written in terms of a_1 and so on. They figure out the nth term formula. Some students find it frustrating. Other don't bother and just go to the book and look up the formula. Some enjoy figuring it out. I have found that it's difficult to engage all of my 32 students in my class.
Review days. These have been pretty good. I post a set of in class problems the class before the test. Everyone works on them. I go around asking probing questions and directing people towards the answer. Generally it has shown me what I have taught well and what I haven't. It also gives me an opportunity to introduce new methods of solving problems. Example: we had been computing partial sums of arithmetic progressions where the first term is not a_1. I had originally demonstrated how that is equal to S_n - S_m where m is the index of the first term in the partial series. In the review I had the students compute the same partial sum but each problem had a different explicit formula for the terms and a different starting and ending index. This lead to a discussion about how the same number of terms were being added whether we were adding terms 2-5, 9-12 or 25-28 and that a_2=a_9=a_25 so the a_1 in the partial sum formula could stand for "the first term you were adding" as opposed to "the first term in the sequence." This has been good because the class cycles back to topics we had discussed and the students view them in a new light but it also means that some students don't figure those things out until right before a test. This seems to mostly be a pacing issue. The school syllabus usually calls for one day per section in the book. I find it difficult to discuss the many ways a problem can be approached in just one day and add in an extra day. The most "actual learning" seems to occur on the review days.
Calculators and decimals. In geometry, many of the students have a hard time finding an exact solution: tan36 becomes a decimal, sqrt 2 becomes a decimal, pi becomes 3.141. At my school, it is considered "okay" to do this until we get to volume, surface and circles then the students are overwhelmed. Questions I have heard, "is r=(1/5)^(1/3) an answer?", "can I take the square root of pi?". Through the year I have told some students they weren't allowed to give decimal answers. These students had no problem with the switch. I told the other students to give the exact solution time permitting. Side note: my class had the highest average on the entrance exam into algebra 2 so I wonder if forcing them to give exact solutions from the beginning helped.
I have taken to writing lecture notes in latex. I include exercises after each topic. As a class we go over a new concept, do an example problem or two and I set the class loose on in class exercises. Some students enjoy this and do well. Some are able to read the examples, do the exercises, check with me and then start their homework. I have several students in my regular pre calculus class that need to see a "similar" problem worked out first (the problems can only differ by constants). Generally this has meant my notes are organized by solution method. How do you find an explicit formula for an arithmetic progression given a_1 and d. How do you find an explicit formula if you are given a_n and a_m. I have avoided this somewhat in my honors class by introducing an idea and then including exercises which lead the student to discover what they need to know. Example: an arithmetic progression has a common difference between terms. Have them write out 5 terms using the recursive definition. Suggest that a_2 can be written as a_1 plus something. a_3 then can be written as a_2 plus something and then a_2 can be written in terms of a_1 and so on. They figure out the nth term formula. Some students find it frustrating. Other don't bother and just go to the book and look up the formula. Some enjoy figuring it out. I have found that it's difficult to engage all of my 32 students in my class.
Review days. These have been pretty good. I post a set of in class problems the class before the test. Everyone works on them. I go around asking probing questions and directing people towards the answer. Generally it has shown me what I have taught well and what I haven't. It also gives me an opportunity to introduce new methods of solving problems. Example: we had been computing partial sums of arithmetic progressions where the first term is not a_1. I had originally demonstrated how that is equal to S_n - S_m where m is the index of the first term in the partial series. In the review I had the students compute the same partial sum but each problem had a different explicit formula for the terms and a different starting and ending index. This lead to a discussion about how the same number of terms were being added whether we were adding terms 2-5, 9-12 or 25-28 and that a_2=a_9=a_25 so the a_1 in the partial sum formula could stand for "the first term you were adding" as opposed to "the first term in the sequence." This has been good because the class cycles back to topics we had discussed and the students view them in a new light but it also means that some students don't figure those things out until right before a test. This seems to mostly be a pacing issue. The school syllabus usually calls for one day per section in the book. I find it difficult to discuss the many ways a problem can be approached in just one day and add in an extra day. The most "actual learning" seems to occur on the review days.
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