Exhibit A

Then, I succumbed to a little-known corollary of one of Newton's Laws. The actual law is "A body at rest tends to stay at rest." The corollary is "An unwritten blog tends to stay unwritten." Today, I got some momentum going by spending the last hour working on a teaching and leading philosophy statement. Before my daughter and her four friends emerge from the basement (there was a sleepover last night), I thought I would use Newton's Law to my advantage.

Although not ideal for a #DITLife blog post, what I am about to write is being pieced together from memory. Friday, December 16th was the last day of classes before Christmas break.

Each pair of students gets a bag with slips of paper numbered consecutively from 1 to N. They are to mix the numbers in the bag and then pull out seven slips of paper. From the seven numbers on those slips of paper, they are to determine a way to estimate N. Popular statistics are based on doubling the mean, doubling the median, or finding the mean plus three standard deviations. This year two girls came up with a statistic that I had never seen before. (This is probably my eighth year of using this activity.) They calculated the average difference between consecutive pairs of numbers and then added it to the highest number. A sample calculation is shown here based on the numbers:

Note that many of the terms in the numerator sum to 0 and the resulting calculation is:

**Period A**- This forty-minute class started at 8 AM and was devoted to students working in pairs on the German Tank activity. We began the activity the previous day. The goal of the activity is to introduce students to sampling distributions and what makes one statistic better at making an estimate of a population parameter than another statistic. Ideally we want an unbiased statistic with small variability.Each pair of students gets a bag with slips of paper numbered consecutively from 1 to N. They are to mix the numbers in the bag and then pull out seven slips of paper. From the seven numbers on those slips of paper, they are to determine a way to estimate N. Popular statistics are based on doubling the mean, doubling the median, or finding the mean plus three standard deviations. This year two girls came up with a statistic that I had never seen before. (This is probably my eighth year of using this activity.) They calculated the average difference between consecutive pairs of numbers and then added it to the highest number. A sample calculation is shown here based on the numbers:

Here is the calculation:

It is interesting to note that this statistic really only uses the highest and lowest numbers and all of the other numbers are inconsequential.

I was curious as to how this statistic stacked up against the Partition method that was developed by statisticians during World War II. The Partition method assumes that the values are approximately equally spaced from low to high among the tank numbers. Drawing seven tank numbers essentially splits the number line from 1 to N into 8 partitions. The highest number is about 7/8th of the way when counting from 1 to N. Multiplying the highest number drawn by 8/7 would yield the number N. Based on the seven numbers used above, we would have 337(8/7) or about 385.1.

The actual number of tanks is 344. So, for this sample the "RedfernFerrari" method (named after the girls who created this statistic) produced a statistic that was closer to the population parameter than the Partition statistic. But how does it do in the long run? The screenshot below is from Fathom and shows 100 sample statistics based on samples of size 7. Which statistic would be better to use to estimate N - Partition or RedfernFerrari? This is the question that I will be posing to my students on January 3, our first day back from break.

**Period B**- During my forty-minute planning period, I started to plan for January 3rd by making copies and organizing papers. I can't recall the number of cookies I ate from the faculty room, but I know it was too many.

**Period D**- For the first fifteen minutes of class, I returned a test that students had taken the previous day. Many students had problems on the first page - matching polynomial equations to graphs with a focus on end behavior. So, I had the students work together in groups on a fresh copy of this page of the test before they saw their own test. As they did this, I learned that some students thought that to find the degree of the polynomial that you needed to add all of the exponents. This is true for f(x) = x(x - 1)

^{2}(x + 8)

^{3}, where the degree is 6. But it is not true for f(x) = x

^{4}+ x

^{3}, where the degree is 4. Had I not reviewed the first page in this way, I would have not discovered this misconception.

Next, I gave students time to work on their Birthday Polynomial project. (I had discovered this project during a twitter chat several years ago. I don't have the original link now, but there are many versions of this project online if you search "Birthday Polynomial". Here is Mr. Reed's version of the project.) The project is not due until Friday, January 6th and many of the students did not want to work on it. As it was the last day before break and there were Christmas movies showing in the dining room to celebrate meeting a school-wide service goal (students and faculty brought in over 275 items to donate to a home for a refugee family), I allowed students to go to view the movie for the last twenty minutes of class.

**Chapel**- Many students and faculty participate in our weekly chapel and today was no exception. The theme of the chapel was light and dark. I was surprised by the variety of photos students and faculty had contributed to make the opening slide show.

**Period C/Lunch**- This period was similar to my D period PreCalculus class, but I had about 1/3 of the class choose to come back after lunch to work on their birthday polynomial projects. We also sang two math-themed holiday songs - "Oh, Number Pi" (to the tune of "Oh, Christmas Tree") and "The Famous Four-Sided Shape" (to the tune of "Rudolph the Red-Nosed Reindeer").

**Period F**- In Honors Geometry, we had just finished a unit on similar figures and I decided to spend our last day before break doing a problem called "The Two Telephone Poles". Initially, I gave my students a blank diagram devoid of numbers or context and had them individually list what they noticed and what they wondered. In other words, what they knew was true and what they thought was true. They had about 2 minutes to do this.

Students came up with many ideas for "Notice" including vertical angles, BE + EC = BC, and there are five triangles in the diagram". The ideas for "Wonder" included statements about similar triangles, perpendicular lines, congruent angles and parallel lines. After I had students compare their lists at their tables, we created a class list from their individual lists.

Next, I added right angle symbols at B and D to show segment AB was perpendicular to segment BD and segment CD was perpendicular to segment BD. As a class, we decided which "wonder" statements could be moved to the "notice" column. When a student would give me a statement to move, they had to justify their answer.

Finally, I added some numbers to the diagram and gave them the context of the problem.

*Two telephone poles are erected perpendicular to the ground and 40 meters apart, as shown in the diagram below. The poles are 30 meters and 20 meters tall, respectively. Two of the supporting wires are shown, each running from the top of one post to the bottom of the other. How high is the crossing point of the wires off the ground?*
Because students had already spent time thinking about the diagram and what was true, they quickly solved the problem by using similar triangles and creating a system of two equations. After each group had the answer, I showed how coordinate geometry could be used. By setting AB on the y-axis and BD on the x-axis and having B be the origin, we could easily write equations for lines BC and AD and then solve the system of equations to find the point of intersection of the two lines.

At this point we only had about 8 minutes of class time left. I asked the students the following follow-up question.

*The public service commission has stipulated that the meeting point of the wires must be 15 meters off the ground. With the given conditions, would it be possible to position the poles either closer together or further apart so that this requirement could be met? Explain your answer.*

Ideally, we would have had about 15 more minutes of class time for exploration of this follow-up question. I had some students say to move the poles closer together, and others said farther apart. Very few said that it wasn't possible, which is the correct answer. Instead of having time for exploration, I showed the following video to dynamically demonstrate that it was not possible.

**Period E**- I was done teaching for the day at 1:40, but I still had work to do. My day was definitely not over. First, I talked to a colleague about plans for PreCalculus for after break. She shared with me information about the midterm exam, including review materials. Then, I had to finalize and print the line-ups for the home swim meet. As swim team statistician, I enter the swimmers into the twenty-two events for the meet, run the computer at the meet and generate reports for the coaches.

I leave school around 2:40 and arrive at the pool shortly before 3 o'clock. We are missing some students who normally record times at the scorers table and I recruit some swim alumni for the task. After making many changes in the line-up for the away team, I print off the meet line up and the meet starts. I realize that the computer is no longer communicating with the timing panel. I try some fixes with the computer and the cable, but to no avail. All times must be entered by hand into the computer. This also means there are no split times. The meet ends around 6 PM and I let the coach know that I will come in during break to work on the panel to try to figure out what went wrong. (Note: I went in today, December 27th and after about 30 minutes realized that a pin in the cable was bent. Using a paperclip to bend it back, it seems to be working for now.)

And now for the #DITLife reflection questions.

**1)**

**Teachers make a lot of decisions throughout the day. Sometimes we make so many it feels overwhelming. When you think about today, what is a decision/teacher move you made that you are proud of? What is one you are worried wasn’t ideal?**

I am most proud about recognizing the alternative statistic presented for the German Tank problem. I am very excited to show the class that the "RedfernFerrari" statistic is comparable to the one created by statisticians during World War II.

**The decision that was not ideal was rushing through the paradox of the two telephone poles problem in geometry. If I only had 40 minutes in the future for this lesson, I might not have students create individual lists of "Notice and Wonder". However, this would only save me about 3 minutes of class time. A better option would be to do this lesson during a 65 minute class on a lab day.**

**2) Every person’s life is full of highs and lows. Share with us some of what that is like for a teacher. What are you looking forward to? What has been a challenge for you lately?**

A low happened on December 9th when our Guatemalan exchange student returned to Guatemala. It was sad to see her go. We had so much fun - visiting New York City, going to ice hockey games, carving pumpkins and having a mini snowball fight. My whole family misses María Inés.

On December 16th, I was looking forward to break. But now I am feeling re-energized and I am looking forward to returning to school. We have two weeks of classes and then a week of midterm exams. With swimming thrown in there and some other professional growth opportunities (more in future posts), I know the next few months will fly by.

The biggest challenge lately is a lack of substitute teachers. This often leads to full-time teachers filling in during their planning periods without any compensation in terms of time or money. Teachers at my school fill in because they want to help their colleagues and do what is best for students. The absent teacher is always grateful to the teacher who filled in. However, there is an underlying sense that the administration is not grateful, or at the very least, does not express that gratitude outwardly. In an ideal world, we would increase the substitute teacher pool and be more aware of time as a valuable commodity.

**3) We are reminded constantly of how relational teaching is. As teachers we work to build relationships with our coworkers and students. Describe a relational moment you had with someone recently.**

On December 18th, I had the opportunity to see the Santaland Diaries with some of coworkers from school. The theater teacher was the sole performer in the show. The show was very funny with a few poignant moments. However, the best part of the day was having time to talk to the college counselor on the car ride to and from the show.

**4) Teachers are always working on improving, and often have specific goals for things to work on throughout a year. What is a goal you have for the year?**

My goal is to finally get around to reading the book that I am referencing for a talk in April. The book is called "Promoting Purposeful Discourse" and I know once I start reading it, I will be hooked. It the starting part that is getting to me.

**5) What else happened this month that you would like to share?**

It was announced in November that our current director (in public schools, that would be a principal) is not returning for the fall of 2017. We were in this same situation as a school three years earlier and at that time I briefly thought about applying for the position, but did not for a variety of reasons. After many hours of thinking and having discussions with trusted colleagues, I have decided to apply for the position.

At the opening chapel of school this year, I gave a birthday wish for the school. I stated, "

At the opening chapel of school this year, I gave a birthday wish for the school. I stated, "

*In our 275*^{th}*birthday year, I offer the following wish for our school, our community, our students and my colleagues. No, no, no. Don’t stick to the status quo. Be true to who you really are. Thoughtfully, not impulsively, let us stretch out of our comfort zones and reach beyond our status quo."*As I apply for this position, I will be stretching out of my comfort zone and even if I don't get the position, I will have learned more about myself as an educator and leader.
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