Chapter 8: What's My Line? Continued (Days 63-64)

Here in California, this week is Veteran's Day, which is observed on November 11th no matter what day of the week it is. This year it falls on a Thursday. Years in which Vets Day falls on a Thursday (or Tuesday) are problematic, because it leaves a lone workday on Friday (or Monday). Some districts, including mine, have a simple solution -- there's no school on Friday either. Thus Veteran's Day forms a four-day weekend in my district.

Unlike the past few weeks, when a minimum day shifts the block schedule up to Monday, this week there is no change to the block schedule -- losing both Thursday and Friday keeps the schedule balanced. And so yesterday was a regular all-classes Monday, today is odd periods, and tomorrow will be even periods.

Moreover, this week marks the end of the first "quarter" -- a misnomer, since it really indicates that we are two-thirds of the way through the semester. (The flagship high school had progress reports at the end of the first term, but our school waited until the end of the second term.) Even then, Day 64 is a bit late even for that -- the one-third mark of the full 180-day year was last week, and two-thirds of the way through the 84-day semester was Day 56, two weeks ago. This week doesn't really mark any mathematical milestone in the year, yet that's when our progress reports are coming out.

(Ironically, schools that start after Labor Day really are through a quarter of the year at Veteran's Day. In fact, for several years, the district I attended in grades K-8 considered Vets Day to divide the quarters.)

Originally, both Veteran's Day and the end of the "quarter" were both considerations when I was planning the day of the Chapter 3 Test in Calculus. The last quiz was two weeks ago, and so I wanted to give a test this week. Moreover, by doing the test today, it would be before the four-day weekend (that is, four days for the students to forget what they learned) as well as before grades are due.

But that went by the wayside due to the observation by the department chair, and the recent problems I've been having in Calculus class. By taking that extra day last Friday for test corrections, it pushed the test back to Monday. Thus I can no longer include the test on the progress report. Moreover, any test on a Monday is asking for trouble, especially after a long weekend. But I can't push the test back any further to avoid having it on a Monday, since I want to finish Chapter 4 before the end of the semester.

Today in Calculus, I finish Section 3.9 on linearization -- the last lesson before the test. Unfortunately, there are still a few problems with the way I present this lesson, and so I can't really be sure that my students fully understand it.

After doing Examples 1-2 during the Monday observation, I begin with Example 3. The text asks students to determine how accurate the linear function L(x) = 7/4 + x/4 approximates the original function f(x) = sqrt(x + 3) near x = 1 -- in particular, where is error 0.5 or less, and where is it 0.1 or less. The text directs them to graph sqrt(x + 3), sqrt(x + 3) + 0.5, sqrt(x + 3) - 0.5, and 7/4 + x/4 in order for students to see where the line crosses the square root graphs.

The problem is that this is a trial-and-error process -- specifically, figuring out what viewing window on the calculator is the best to discern the graphs and tell see they intersect. I try a few windows myself, then simply ask the students just to use the window given in the text.

But then I give my own example -- inspired by my figuring out sqrt(901) in my head decades ago, I ask them to find where the error between the graph of sqrt(x) and the tangent line near x = 900 is less than a certain value -- I believe I chose 0.01.

Unfortunately, I made two mistakes here. The first is a typo on the actual linearization function itself -- it should have been L(x) = 15 + x/60, but I write something else on my presentation. It wouldn't have been that bad, except that today I finally hand out guided notes to go with the lesson -- and now the students have copies of notes with an incorrect function.

The second mistake is that even if I had written the function correctly, it's not an ideal example to do on the graphing calculator. Recall the well-known example -- consider a polynomial function whose zeroes are 0, 0.001, -0.001, 1000, and -1000. Any viewing window that's small enough to discern the three zeroes at 0, 0.001, and -0.001 will be too small to see the zeroes at 1000 and -1000. Likewise, the functions sqrt(x) + 0.01 and 15 + x/60 intersect at two points (that is, the endpoints of the interval where the error is at most 0.01). Any window small enough to distinguish between the square root and linear functions near one of their intersections will be too small to see the other intersection.

This is one reason why I should stick to the problems in the text (including even-numbered problems when the assigned homework is odds), especially when the problems involve graphing. After all, the problems in the text are hand-chosen to have convenient graphs. While giving my example a personal touch by connecting it to sqrt(901), this isn't good for graphing. Indeed, the square root function beyond one-digit numbers will appear indistinguishable from its tangent lines on the screen, and so any problem based on finding the error between these by looking at the screen is doomed to failure.

Of course, any graphing problem where the optimal viewing window is unknown will be tricky. In fact, there is an upcoming lesson in Chapter 4 on using Calculus (critical points, points of inflection, etc.) to determine the best graphing window for a function. But we haven't reached that lesson yet -- in this current section, the text glosses over finding optimal windows, since the task is to find the interval where the error is small, not learn how to graph functions.

Then again, there's a way to find this interval without relying on a graph at all. Instead, we use the TABLE key on the TI calculator. We place the original function in Y1, the linearization in Y2, and then let Y3 be Y2 - Y1. Then we use TBLSET to choose a suitable table. Since the linearization for Example 3 in the text is centered at 1, and since the answers to that example are rounded to the nearest tenth, this suggests using TblStart = 1 and delta-Tbl = 0.1. Then we can just scroll up and down the table to see how long it takes for the error in Y3 to exceed 0.5 or 0.1.

This is not how the text does it -- but then again, a good teacher knows better than to rely solely on the text, and chooses other ways to teach a lesson if it will lead to more student understanding. But if I do this, I need to make sure that the students know what buttons to press -- in this case by indicating the key sequence in these same guided notes. I could make it easier for the students by having the students subtract the square root function from the linear function all in Y1 (rather than require them to know how to use Y-VARS to enter Y3 = Y2 - Y1), but then again, the purpose of the lesson is to show them how well the linear function in Y2 approximates Y1. So it's worth showing them the graphs in Y1 and Y2 before using the table to find Y3.

Notice that if I use a table, then it becomes OK to use the sqrt(901) example, since I won't have to worry about graphs. Notice that the difference between 15 + x/60 and sqrt(x) near x = 900 is so small that it appears as E-notation on the table (for example, at x = 901, Y3 is 4.6E-6, or 4.6 * 10^-6). So if I use the example, then I'd have to explain E-notation, or tell them to scroll until E goes away -- which it does once the table value reaches 0.001 (at x = 915). For this example, I can tell them to find the value where the error reaches 0.002 (which is at x = 921). With values like these, I should tell the students to make delta-Tbl be 1 rather than 0.1.

After wasting time on sqrt(901), I finally get to Example 4, on differential notation (dx and dy). Once again, I find myself rushing through this, and I can see that the students are still confused by what purpose dx and dy even serve. Since I couldn't ask the students a question about Example 3, I finally get to ask them a question on Example 4 -- but I know that these differentials are so confusing.

The reason I rush through Example 4 was to get them to start reviewing for the test. I follow my partner teacher's suggestion and assign them an Edpuzzle video followed by a DeltaMath lesson to review, as well as the next Michael Starbird video -- Lecture 7: "Derivatives the Easy Way." (I'd always planned on doing Starbird lectures on regular Mondays -- there should have been one yesterday, but that we the day of the observation, and there's no point having my partner teacher observe a video. Also, I always wanted to make sure that Lecture 7 occurred before the Chapter 3 Test -- and I already knew coming in that Parent Conferences would mess up the block schedule in October.)

And so I'm still worried that I'm not teaching my students effectively -- and indeed, I'm worried about what this lesson and its mistakes look like to the concerned girl from last week. During the period, almost every student is summoned to the counselor's office for an unknown reason. Most of the students are summoned early in the period and return after a few minutes. But the concerned student, who is called out later in the period (around the time I start the Edpuzzle) takes all her belongings and stays out the rest of the period. I'm wondering what she's thinking -- something like with all these mistakes in the lesson, this teacher isn't teaching me anything, so I might as well sit in the counselor's office the rest of the period. Moreover, with all of these Edpuzzle and DeltaMath (and possibly Starbird, though she was gone before I began it) videos, the thought could be of course the teacher's relying on all these videos, because he doesn't know how to teach it himself.

And today was my last chance to demonstrate the concerned girl (and any other possibly concerned students in the class who haven't spoken up) that yes, I can teach a lesson properly. Instead, I just kept on making one mistake after another.

Why am I having so much trouble setting up strong, effective lessons, anyway? One problem I have is how much time it takes me to set up a lesson. To prepare a good, polished lesson -- one where, from Warm-Up to Exit Pass and every moment in between, I move from one part of the lesson to another, with no pregnant pauses, searching for lost papers on my desk or files on my desktop, bad calculator graphing examples like today's, etc. -- takes me about the same length of time as the class itself (that is, about one hour to prepare for a one-hour class, 100 minutes for a block period, and so on). This is a lot of time, and it's partly because I'm inexperienced. But any efforts to cut out some planning time resulting in some part of the lesson being missing. (Perhaps I posted the wrong Warm-Up, or I forget to post the homework to Google Classroom, or I'm still struggling to determine what a suitable HW assignment is, and so on.)

Indeed, once I was interviewing for a teaching position at another school. Part of the application process was to prepare a good one-hour lesson. It took me about an hour to do so -- and the interviewer asked me how long it took me. Upon hearing my response, he asked, "Is that really sustainable?" The correct answer, of course, is no -- it's not sustainable to spend as much time planning as teaching.

With first period now a conference, I should be able to use first period to prepare for third. Today, I end up spending all of that time making sure that I posted strong Edpuzzle and DeltaMath assignments -- especially making sure that the DeltaMath review assignment covers the entire chapter and is truly representative of what the test will be (which led to problems on earlier assessments). This gave me less time to make the lesson itself strong by double-checking the graphing examples.

And of course, all this time planning for Calculus meant less time to prepare for Stats. Today's Stats class is also preparing for the upcoming test. I assign even problems from the book again -- but this time, I also peek at the test from the online test bank and ask some questions that are similar to the ones on the test.

I need to find ways to improve my lessons -- and improve my teaching and planning habits. I need to find a way so that I'm spending less time preparing lessons -- yet automatically coming up with effective ways to teach them.

Well, I'll have plenty of time to think about it over the four-day Veteran's Day weekend. This is my last blogpost of the week. As usual, I will be tweeting something about Ethnostats tomorrow.