Sunday, May 5, 2013

You didn't teach this...

Last class we introduced the binomial theorem. My approach was to have them expand \sum_{k=0}^{n} n_C_k x^n y^{n-k} without them knowing that. I explained it to them this way: your job is to identify n (the power to which the binomial is being raise). Example: (3a -2b)^4. Start by seeing that n=4 and start writing 4_C_0 (3a)^4(-2b)^0 + 4_C_1 (3a)^3(-b^1) + ... This was prefaced by a discussion about ascending and descending terms so they would know what it meant for the 3a terms to descend and for the -2b terms to descend. That was prefaced by expanding (a+b)^n for n=0,1,2,3 to see that ascending and descending terms naturally occur in binomial expansions.

The next day I assign two problems for a warm-up:
(1) (2a - 3b)^4
(2) (5x + y^2)^5

Most students make the error of expanding (1) without the correct parentheses around the 2a and -3b terms which I correct when they come to show me their work. They go on to successfully expand (2).

Several of my better students do (1) correctly but complain that I had never "taught" them how to do (2).

Wednesday, May 1, 2013

Pointcare's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles

I enjoyed the book very much because of its thorough account of the major and eccentric people who worked to prove or disprove the Poincare Conjecture. I was either working on my undergraduate or in grad school when the Perelman papers were published. I remember hearing about it from faculty and fellow graduate students but that is not why I am mentioning it here.

The descriptions of the amount of (sometimes "wasted") effort on solving the conjecture illustrate the difficulty of the problem and the tenacity needed to solve ANY problem. My students often seem to think they are either good or bad at math. Either opinion is reason to give up if the solution doesn't come quickly. I don't know how I will do this but I want to find a couple excerpts about how either people kept returning to the problem or were very occupied with it. I think a major point to discuss would be that a problem has an essential question to be answered (which will hopefully tell us something interesting and novel about the world) which is different from what is learned by working out the technical details. I would hope this would turn into a discussion about why we learn math which hopefully is because we have insatiable curiosities.

Today was a better day...

My pre-calculus class with 75% seniors was better today. It was on the binomial theorem and Pascal's triangle. I started with having them expand (a+b)^n for n=0,1,2,3 by hand. We then isolated the coefficients, found a pattern and soon had Pascal's triangle. We then segued into naming terms on the triangle by using n and r to stand for the line and term on the line where we start counting from 0 (my explanation was that the first line came from expanding from n=0 so we start from zero). We then moved to the choice function. Give your calculator n and r and it spits out the term you want! Go back to the expansion talking about how each of the terms ascends or descends and how the powers always add up to n (the power to which you rose the binomial).

Now we put it together. Want to expand a binomial? Let the power be n, set up the ascending and descending terms, find the coefficients with the nCr and simplify. Go over how subtracting in the binomial changes it (alternating signs). Go over how to deal with coefficients in the binomial other than 1 (don't forget to raise the whole monomial to a power).

Geometry had a test. Lots of kids still don't understand the idea of an exact answer. I don't blame them but I am tired of hearing, "what is the real answer? I have tan36." I understand to them math is numbers but I have not figured out how to convey that those are numbers...

My Daily Routine

If I take public transportation:

Wake at 5am
Catch the 5:30am train
Arrive on campus at about 5:50-6:00am, prepare by writing lectures, preparing notes, making quizzes and tests
7:45am - 2:30pm teach with one 1:20 free time
2:30-4:30pm coach
5pm exercise (hopefully for an hour)
6:30pm or after, get home help with family chores, feed kids, cook, etc.
9pm (hopefully) kids are laying down, light heartedly work on something
midnight usually go to sleep or sleeping

Monday, April 29, 2013

A hopeful start with a frustrating end

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.

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.