Teach 180: Limits Aren't Just for Calculus (Day 22)

One of the things I love about being where I teach is that teachers are welcome to be a part of the student community.  A few years ago, I dusted off my violin and played with the string ensemble at school.  That doesn't work in my schedule this year, because I am teaching a class when strings is scheduled.  However, I am able to sing with chorale and singing as part of a larger group is something I really enjoy.  It helps me to see students as more than "just a math student".

The Processing Speed of the Fuzzy Brain

Near the end of chorale today students were starting to loose focus.  I know my mind was starting to get fuzzy, too.  Keep in mind that classes are 1 hour long this year due to a change in our bell schedule.  The director of chorale recognized we had reached our limit.  To switch things up he had us sing an acapella tune that he made up on the spot.  He started with the bases, giving them a line to repeat.  Next came the altos, then the tenors and finally the sopranos.  Once all the groups were singing together there was clapping and swaying and smiles on the faces of many of the students.  It was just the break we needed before singing through a piece one last time.

There are times when I see students are at their limits of focus, but too often I power though rather than taking a momentary break or detour.  What the teacher did in chorale today was clearly valuable and allowed the class to finish strong.  I'll need to think back to that moment when I encounter a similar loss of focus with my students in the future. 

Teach 180: A Distribution of Pennies (Day 21)

Today in Prob/Stat students worked in groups of 3-4 to create a distribution for the ages of pennies.  One group even organized their pennies in a dotplot of pennies.






Once we combined all the data we
had the following dotplot at the front
of the room. (The right tail

ended up at 59, but I erased part of it and then snapped this picture.) There are more younger pennies than older pennies, but it was clear to see that creating a dotplot like this by oneself would not have been very practical.  Our final dotplot had 180 observations!!  Clearly dotplots are better for smaller data sets.   What would be a better way to organize the data?  A histogram!

First we created the histogram by hand and then we used our TI-84's to make the histogram.  Students set the window manually to match the original histogram we made by hand.  Then, we used the Zoom Stat feature.  This allowed the calculator to determine the bar width. But what was the bar width?  6.875 years?  Since penny ages were recorded in years, on a quick glance the histogram at the left makes more sense than the histogram at the right.  Next up?  Measures of center and spread and boxplots.

 

Teach 180: Learning More About Card Sorts (Day 20)

Today we did a card sort activity in Calculus to review some basic parent functions and to remind students about creating "offspring" by using basic transformations of parent functions.  As a teacher, what did I learn from today's activity?

1) The teacher dashboard can be displayed to give feedback to students.  Students could see if their matches were correct and fix their errors.  This also allowed me to touch base quickly with the few pairs that were struggling.  Correct matches are in green and incorrect matches are in red.  It was interesting to note on the second slide that not all students used the same strategy, but there were two or three main strategies.

2) The end of activity "How did you do?" shows me that I need to circle back to one specific pair of students tomorrow.  But it also shows me that even though all students completed most of the activity correctly, they admitted to not easily remembering all of the parent functions.  This is something I need to keep in mind when I do anything with students that is supposedly review.  Review and recall does not come as easily for some students.

3) The "offsprings" of a vertical shift down 5 units or a horizontal shift right 5 units were reversed by some students. This will be something I need to pay extra close attention to in future units.

Teach 180: A Different Card Sort (Day 19)

Yesterday in Calculus we did a card sorting activity in Desmos that focused on linear and non-linear equations and linear and non-linear tables.  We began class today by talking about the importance of using precise language in math.  We critiqued the first answer seen here that was given by one pair of students. The summary question was "How can you determine if an equation is linear or non-linear?"

A few students hadn't been in class the previous day and wanted to know what a reciprocal function was.  Several students knew that a reciprocal function's graph wasn't linear, but couldn't recall what the graph looked like. Rather than pulling up desmos, we created a quick sketch by substituting a few values into 1/x.

Next, we critiqued the word "exponent". One student said that it wasn't clear that the exponent is to be on a variable.  He gave an example of y =3²x and the class agreed that this was linear, even with the exponent on the 3.  A second student pointed out that even linear equations have an exponent; it is a 1.  At this point, we discussed the form of the line that was used in the card sort activity, which was slope-intercept form.  Many students used this form to identify the linear equations.

After a discussion of domain and range, we did a different card sort activity.  There are 16 students in my class and each student was given an index card.  The white index cards had a domain and range listed and the blue index cards had a function listed.  So, there were 8 pairs of cards.  Students were told that they had to get up and move around and find their partner.  Once students found each other, they had to write the information found on their index cards on the board.  When we were done, we had 8 examples of functions and their domains and ranges displayed around the room to discuss.  We reviewed each of them and found one error.  The range of y=1/x and y=1/x² were reversed!  This gave us a 5 second penalty.

The table below shows the 8 functions I used and their domains and ranges.  Tomorrow Mr. Galitsky's Calculus class will be doing this activity.  Will they be able to beat our time of 2:06.91?

Teach 180: Desmos Card Sort (Day 18)

It's still early in the year and we are continuing to review concepts in Calculus. Our lesson today focused on functions in general and for part of the lesson students worked in pairs on a Desmos Card Sort activity.  In this activity, students worked with a partner to sort equations into piles of linear and non-linear equations and tables into piles of linear and non-linear tables.

Most students had no difficulty identifying the equations as linear or non-linear.  Where there was some discussion between students was when they were sorting the tables.  Within each pair, it seemed like one student knew that the slope had to be constant (or the same) when looking at how y changed in relation to how x changed.   

Where many students need a little more work is on using mathematically precise language.  The response below shows an example of an answer to the two summary questions.  The questions were: 1) How can you determine if an equation is linear or non-linear? and 2) How can you determine if a table of values represents a linear or non-linear relationship?

I think we will begin class tomorrow by critiquing this response to the two summary questions.  I'll blog about how that went along with a different sorting activity tomorrow.

Teach 180: One Simple Idea (Day 17)

Some days we do Desmos activity builder lessons and work with Fathom.  Other days we collect data as a class and ask "What do we notice?"  These classes often fly by for me and I am guessing my students find it that way, too.  Why?  Because they are engaged in what they are learning.

However, there is sometimes a need for direct instruction.  Within a class where the primary teaching method is direct instruction, I ask students questions and also have them work at their desks to solve problems.  Under our previous bell schedule, a class might have 10 - 25 minutes of direct instruction.  And within those 40 minute class periods, there was also time for a warm-up, reviewing homework questions and time for individual work on problems similar to the ones done together in class.  Students are also engaged in this class, but 25 minutes of direct instruction is the upper limit for my students with this more passive learning style.

Created by Freepik

With 60 minute periods, the 10 - 25 minutes of direct instruction becomes 15 - 40 minutes.  For me, the 40 minutes is too much.  And I am guessing for my students who start to zone out, it is too much for them.

Zoning out identification technique: Ask a question.  Call on student.  Student answers a totally different question or admits outright "I wasn't paying attention."

A few days ago I read a brief blog post on doing board work, called "Should Math Students Be at the Board Working?"  A simple idea really.  Give students problems to do in small groups at the board.  Since I normally have students do problems at their desks or have individual students put homework problems on the board, this was a quick and easy fix to my problem of keeping students engaged.

After reviewing one problem from the previous day and comparing its solution to a similar problem, we went over the homework by having students put the problems on the board.  One problem was assigned per table and I already had the problems put on the board.  I told the students that after about 20 or 30 seconds, I would say "switch" and someone else would have to write on the board.  This made sure everyone participated in getting the problem on the board and there was definitely more engagement by all students in the learning.  I could hear students in each group discussing the solution and correcting each other's errors. After the problems were up, we reviewed the strengths seen in each of the solutions.  The energy of putting the problems up on the board and having students discuss them carried us through into the direct instruction component of the lesson which lasted about 20 minutes. Not too bad for a Friday afternoon at 3 PM.

Teach 180: The Students Can Teach You, Too (Day 16)

One of the best things about where I teach is the inquisitive nature of the students.  They aren't afraid to ask questions.  Today I was giving a quiz in PreCalculus and some of the problems involved solving equations with rational exponents.  For these equations, we would remove the lowest power of the variable to get a quadratic and then solve the quadratic. Here is an example of one of these types of questions.

When you solve this equation, it appears that it has three solutions: 0, -3 and -5.  However, -3 and -5 are extraneous solutions.  We had shown solutions were extraneous by substituting the values into the original equation and finding it simplified to two different results for each side of the equation.  For this particular problem, we also know the negative solutions are extraneous, because the exponents have 2 in the denominator and taking the square root of a negative number leads to a non-real result.

Right before the quiz one of my students approached me and showed me the following two screens on his calculator.  On the left, we can see that he has entered both the right side and left side of his equations in the calculator.  On the right, we can see a table of values.  He asked if the error message is showing that these (meaning -3 and -5) are not solutions.  He also asked if the table shows that 0 is a solution, since both y1 and y2 showed the same function value of 0.  I told him that he was correct and then it dawned on me that I had not thought of checking for extraneous solutions this way.  Now I will need to think of how to incorporate this connection between tables of values and extraneous solutions into lessons in the future.

Of course you can do something similar in Desmos, and in my opinion Desmos wins.  It doesn't report an "ERROR" when evaluating the function.  Desmos calls it like it is "undefined".

I thought when I first became a teacher that I wouldn't be a learner anymore, that I would know everything about what I was to be teaching.  After 25 years of teaching, I know that is wrong and days like this remind me that we never stop learning or making new connections within our disciplines, even teachers.  Thanks to the student who stopped to make me think and learn today.

Teach 180: What Is Going On Here? (Day 15)

When you solve the following equation, you get 2 solutions.  At least that is what it looks like initially.  But then when you substitute the values into the equation, only one works!  Why does that happen?  What is going on here?  

In years past, I would have said something to the effect that squaring in this particular problem made a new equation with two solutions and yet one of the solutions was not a solution to the original equation.  Although this was true, I would often get blank stares from my students.

Today someone asked me this question and in an instant I was able to show the following two screenshots from Desmos.

By graphing both sides of the original equation, we can see that there is only one solution, x = 6.  Notice that squaring both of these expressions turns the red line into a parabola and turns the blue square root function into a line.  We can see that the line and the parabola have two points of intersection, when x = 3 AND when x = 6.  Transforming the equation by squaring both sides of the equation changed the graphs in such a way that an additional solution was created that was not a solution to the original equation.

Connecting ideas in mathematics was something that I don't recall from my days as a high school student or from the early days of my teaching career.  Now, it is so quick and easy to help students to form those connections with tools, like Desmos.

Teach 180: Plickers (Day 14)

Today in Calculus we did some review for our test on Probability by using Plickers.  I use it as a type

of formative assessment - a way for me to get a sense of what the class knows and doesn't know.  Plus it is a way for students themselves as individuals to see what they know and don't know.

Here is a quick video that describes how Plickers work.   

There are several benefits to Plickers.  First,

students aren't on a computer with other tabs or windows open to distract them.  Second, it is easy to encourage them to work together.  When we use Kahoot, it is about competition and getting your answer in first, even if that means guessing and not really understanding the problem.

When I scan the class using the app on my phone, I can immediately see how students did.  Green means the student answered correctly and red means the student answered incorrectly.  I can say something like, "It looks like many of you understand what you are doing", or "We may need to review this one." as I scan the room.  If a student changes his or her answer, I can quickly scan that student's plicker card again.  A second chance to re-enter an answer can't happen in Kahoot.

Finally, I can print off individual student results based on the plicker number I have assigned to him or her.  And here is a partial summary of my results from today.  Notice that some students did very well (their names are cut off the left side of the screenshot, but you can see the scores) and some questions were easier for the class as a whole than others.

Tomorrow I'll give students their individual one page report that shows how they they did on each of the seven questions.  They can use these results as they study individually for the upcoming test.

Teach 180: Games Day (Day 13)

Today there is just one class.  It is D period and I don't teach D period.  So, today I have no classes.  What am I doing today?  Today is Red and Gold Games Day.  It is a tradition that started in 2000 as the Millennial Games.  The entire student body (grades 1 - 12) are split into two teams - red and gold.  We play field day type games, like sack races, tug-of-war and 100 yard dash.  First grade students are paired with twelfth grade students and the twelfth grade students cheer on their first grader buddies as they race and help them with getting lunch.

What happens after lunch?  This year I have mandatory child abuse recognition training from 1 PM to 4 PM.  (There is required training for teachers once every 5 years.) Students will be participating in various activities including sports practices, scholastic scrimmage, working in the community garden, coda red music rehearsal, working in the art studio and my favorite - Who Wants to be a Mathematician qualifying contest!

Teach 180: Dealing with Student Absences (Day 12)

One of the most challenging parts of teaching is handling student absences.  Although it can be challenging for a student to get caught up on new content that was missed, it is even more challenging to get them to make-up a missed assessment.  Students can be absent for a variety of reasons – a dental appointment, a college visit, being sick or an early sports dismissal.  

On Tuesday, I gave a quiz in PreCalculus on solving quadratic equations.  All of the content onthe quiz was review from Algebra 2.  There were 3 students who were dismissed early for sports.  One student took the quiz early, one student took the quiz during a free period the next day and a third student did not have any free periods to take the quiz.  Rather than have him wait another day to take the quiz, I had him take it during class on Thursday.  (Note: With our new bell schedule this year, we did not have the PreCalculus class on Wednesday.)  This was less than ideal.  The student missed going over a review of the content he missed after the Tuesday quiz.  In addition, he did not do well on the quiz despite the fact that I told him that the most commonly missed question was the one on solving a quadratic equation by completing the square.  He earned 1 of 5 points on that question, because he got the right answer by factoring, but he showed no understanding of how to complete the square to solve the equation. 

After 25 years of teaching, you would think I would have developed a way to effectively deal with absences for tests and quizzes.  But the truth is that I usually have a list of three or four students that need to make up assessments at the end of a grading period.  Often these students have waited over a month to make-up the assessment.  At that point, they have forgotten much of the material that is on the quiz or test and usually don't do well.  Our school has a make-up policy that allows me to give them a grade of "0", but that doesn't sit well with me either.

If anyone has suggestions on dealing with student absences, please post them in the comments below.  Thanks in advance.

Teach 180: The Birthday Paradox (Day 11)

Due to our new schedule, I only taught two classes today – Probability/Statistics and Calculus.  In Calculus we are actually studying some elementary counting and probability ideas.  For about 25 minutes today, we explored the birthday paradox.  If you don’t know what that is, you can view it on this TED-Ed video.  I showed the first 30 seconds or so of this video and paused it when it got to the question.

The main question being “How many people do you need to have in a room for there to be more than a 50% chance that at least two people share the same birthday?”  This problem incorporates complementary events and independent events. 

I started by having each of the 16 students write two dates on a single slip of paper.  Their birthday and the birthday of someone they know.  This doesn’t quite match the scenario, because it is very unlikely for a student to have his or her two listed dates be the same.  However, in reality two people standing beside each other could actually share the same birthday.  We did this twice and both times we found a match, which students found surprising.

Next, we used technology (TI-84) to randomly generate a list of 32 integers from 1 to 365.  We stored these numbers in a list and then sorted the list to make it easier to identify if we had a match.  Unfortunately, only 10 people had calculators on them, but we had 80% have a match the first time we used the technology.  We had 80% with a match on our second trial and 100% match on our third trial.

To help students understand why this probability was so high, I played the remaining 4 minutes of the TED-Ed video.   So, how many people do you need to have in a room for there to be more than a 50% chance that at least two people share the same birthday?  The answer is about 23.

Teach 180: Why Randomization Matters (Day 10)

Today in Probability and Statistics we reviewed the article The Immediate Impact of Different Types of Television on Young Children's Executive Function from The American Academy of Pediatrics.  This article looked at four different tasks to measure executive function across the three treatment groups: fast-paced television, educational television and drawing.  Children were randomly assigned to one of the treatment groups and a table from the article (seen below) shows that the groups were fairly similar.  But does randomization always matter?  Will randomization tend to create groups that are similar?

To see if this was the case, we worked through an activity in the CollegeBoard's Curriculum Module on Random Sampling and Random Assignment.  There are 14 subjects listed on cards.  Each card has the subject's name, gender and IQ score.  Students shuffled the cards and split the cards into two piles of 7 cards.  Cards in the left pile represented the students that were randomly assigned to watch an episode of Sponge Bob Squarepants.  Cards in the right pile represented the students that were randomly assigned to the drawing group.

Next, students calculated the difference in the proportion of females in the two piles (Sponge Bob group - Drawing group) and the difference in the average IQ scores in the two piles (Sponge Bob group - Drawing group).  Each pair did this a total of five times and then we recorded our results on a dotplot.  Students noted that there was variability from sample to sample, but that the dots tended to center around 0 for both the difference in the proportions of females and the difference in the mean IQ.  Here you can see our class dotplot.

We finished the activity by watching a video done by Doug Tyson which uses Fathom to do the simulation.  If students weren't convinced by our dotplots that randomization matters, they were definitely convinced by the end of the video.

Teach 180: My First WODB (Day 9)

I have seen WODB posts and ideas on twitter and today I tried out my first one in class.  For those of you who are not familiar with the acronym, it stands for Which One Doesn't Belong.  We had just finished reviewing the discriminant on Friday and I used this WODB as my opener for class.

I asked students to think which one didn't belong and why.  Then, I called on Christian and he said the one in the top right.  I asked for a quick show of hands to see who else felt the top right was the one that didn't belong.  Almost all the hands went up.  When I asked why, Christian said, "It has no x-intercepts and that means it has a negative discriminant.  All the other ones have x-intercepts and discriminants that are not negative."  After that we had a brief discussion as to why the other graphs might be the ones that don't belong.  One student pointed out the the one on the top left was printed in red ink instead of blue ink.  (This was intentional on my part.)  This was definitely a fun opener, but next time I might assign each table one of the four graphs to create a justification for why it doesn't belong.  Note: Shout out to Desmos for the awesome graphs that made this WODB possible.

Teach 180: Did I Break Desmos? (Day 8)

We continued working with factorials and counting methods today in Calculus. Near the end of class a student asked what the graph of C(11,x) would look like?  Would it be like a parabola, going up and coming down? (Note: I love the students in my school for thinking like this and being willing to ask these questions.)  We quickly looked at a table of values and noted that the values went up and then back down as x changed from 0 to 11.  Then, I said "We have 2 minutes before the bell rings.  Let's look at the graph in Desmos."  Here is what we got.

Wait a minute!!!  What is going on here??? Why does this look like a step function???  Why is it plotting values for non-integer values of x???  Did I break Desmos???  At that point the bell rang and I told my students that I would ask Desmos what was going on here.  And I know that on Monday they won't forget to ask me what the folks at Desmos said.

Teach 180: Building Bracelets (Day 7)

In my Calculus class, we do a unit on probability.  Today, I decided to teach circular permutations by talking about bracelets. Each student was given 4 beads (black, blue, white and gold) and a piece of string with the question "How many different bracelets can be constructed with 4 different color beads?"  I should have had students spend more time struggling with this a bit themselves, but for the sake of time I asked one table to give me an ordering and I wrote it on the board.

One possibility is shown at the left.  Then I asked if anyone had something different.  We gathered 3 different bracelet orderings on the board, and then I did too much talking.  Rather than having a student point out that flipping the bracelet over didn't really produce a different bracelet, I showed that to the students.  They could see that the colors were still in the same position relative to each other, but to name the colors in the same order, we would be going in the reverse order around the bracelet. (clockwise vs counterclockwise)  I should have had students work together to come to this conclusion.

It was also at this point that I realized that the bracelet was actually a step beyond a simple circular permutation and the moment for deriving the formula of n!/n  or (n - 1)! had passed.  Although the students may have enjoyed this activity, I will likely try something different next time.  Rather than work with bracelets, I'll divide the students into groups of 4 and see how many ways they can arrange themselves in a circle.  This would get all students actively involved, talking and thinking.  Plus, I think the physical movement of realizing that rotating one spot doesn't change the order of the people (or beads) relative to each other would be more apparent.

Teach 180: Teaching in Longer Class Periods (Day 6)

We have moved to a dramatically different bell schedule this year.  Last year classes met for 40 minutes 4 days a week and 70 minutes one day a week.  Sounds pretty simple.

Now classes meet for 60 minutes 4 out of an 8 day rotation, 70 minutes one day out of an 8 day rotation and 50 minutes one day out of an 8 day rotation.  That means in an 8 day cycle a particular class that used to meet on a daily basis won't meet twice in the rotation.  Plus the periods rotate throughout the day.  So, sometimes I see students at 8 AM and sometimes I see those same students at 2:15 PM.  School started 6 days ago and at this point I think I have seen all my students 4 times in total.

Here is a graphic that may help you to understand the particulars of my teaching schedule.  You may want to click on it to enlarge it to see the details.

One of the adjustments I have had to make is knowing what can be reasonably accomplished in 60 minutes.  Reviewing a previous lesson, teaching a new lesson and having time for students to work on problems is not enough variety in 60 or 70 minutes.  Now instead of 2 or 3 changes in activities, there needs to be 3 or 4 changes in activities to keep students engaged for a longer period of time.  So, what do we do?

Here is what we did today in Probability and Statistics today:
1) Finish notes from a prior class - discussing sampling methods and problems that can happen with surveys.
2) Discuss all the homework problems.
3) Do a Plicker review activity to prepare for a quiz.
4) Video on Designing Experiments from Against All Odds Inside Statistics.

There was plenty of variety, classroom discourse and opportunities for participation to keep even the most attention-challenged students from zoning out.  But was it too much???  My one concern is that there may have been too much content for a one hour class.  Reviewing about a dozen ideas from previous lessons and introducing about five new ideas may not be optimal.  However, this course covers material that is on the AP Statistics exam.  In years past, I had just enough time to review (7 - 10 days of class time) prior to the AP exam.  With the new schedule, I am already concerned that I will be teaching new content up to the day of the exam.  By teaching old concepts and new concepts during the same class period, I hope I can at least salvage 4 days of review with my students in May.  Time will tell.

Teach 180: Reflecting on Reflecting (Day 5)

Reflecting on others reflections of my teaching sounds kind of meta, but that is what I am going to do in this blog post.  Reflecting on my teaching happens in several ways: notes to myself scribbled on the margins of papers, notes on my iphone, this blog, discussions with colleagues, and the more formal reflection that comes from National Board Certification application/renewal and applying for the Presidential Award for Excellence in Math and Science Teaching (PAEMST).  Today I received the following scores from three reviewers for my PAEMST application and was told that my application was not good enough to be a Presidential Awardee.

	Reviewer 1	Reviewer 2	Reviewer 3
Dimension 1	3	2	4
Dimension 2	3	1	4
Dimension 3	3	1	3
Dimension 4	4	2	3
Dimension 5	4	3	4

Note that 1 = Fair, 2 = Good, 3 = Very Good and 4 = Excellent.

I was told in the email that came with my scores (direct quote) that, "If you reapply these comments may be helpful."   Note that Reviewer 1 gave a total score of 17, Reviewer 3 gave a total score of 18 and Reviewer 2 gave a total score of 9.  That's quite a bit of variability. Some of the scores are not even "adjacent".  (For example, 1 and 2 are adjacent scores, but 1 and 3 are not adjacent scores.)

At first I was very confused, if three different teachers gave these scores to the same assessment done by the same student, teachers and parents would have a right to be concerned.  When I score AP statistics exams, there is a rubric that needs to be consistently applied to be sure that similar papers receive similar scores.  Looking at discrepancies in these scores, I would conclude one of two things.  Either Reviewer 1 and Reviewer 3 followed their scoring notes, rubric and benchmarked examples (their scores are very comparable) or Reviewer 2 followed the scoring notes, rubric and benchmarked examples. Based on the table, it appears that reviewer 2 should have plenty to say regarding my areas of weakness.  So, I went to Reviewer 2's comments.

Here is what reviewer 2 had to say about Dimension One: Mastery of Content Appropriate for Grade Level Taught. "Perhaps all misconceptions could have been addressed before the students brought them up.  You did not address them until the students did."  I typically anticipate student misconceptions as I plan my lessons.  However, to purposefully mention misconceptions before students have a misconception robs students of an opportunity to learn from their mistakes.  Essentially this comment is telling me to wrap my students in bubble wrap and steer them away from anything that might harm their understanding of mathematics.  That will not happen in my classroom.

Moving on to Dimension Two: Use of Instructional Methods and Strategies that are Appropriate for the Students in the Classroom and that Support Student Learning.  Reviewer 2 states, "You are not discussing instructional methods, strategies or tools.  One strategy was indeed wait time and building on prior knowledge, but you used many others." Score: 1 = Fair  So, I used many strategies that were evident in my video, but I guess I didn't talk about them enough.  Since I had already maxed out on the number of words for this section, I guess I need to also learn to write more succinctly.  Writing more succinctly will make me better at "use of instructional methods".  Really?  (Head scratch.)

Now for Dimension Three: Effective Use of Student Assessments to Evaluate, Monitor and Improve Student Learning.  Here is where it gets confusing to me. (I should say more confusing. ) Reviewer 1 states "Teacher described a variety of appropriate formative and summative assessments."  Reviewer 2 states "You didn't discuss any forms of formative assessment or q & a. Very good."  "Variety" and "didn't discuss any" seem like opposites to me.  Do I need to improve in this area or not?

The comments by Reviewer 2 are even less helpful for Dimension Four: Reflective Practice and Life-long Learning to Improve Teaching and Student Learning. "good.  This is very good. You did not discuss professional development, per se." However, Reviewers 1 and 3 mention my blog and actions I have done related to professional development.  They also mention me helping other teachers to improve on their own teaching.  This leads me to wonder what Reviewer 2 had in mind as the definition of "professional development".

Finally, Reviewer 2 saves his sparsest comment for Dimension Five: Leadership in Education Outside the Classroom. Comment: "This section is very well done." Score = 3.  Then, Reviewer 2 states "Please note that the overall quality is poor.  It is often very difficult to hear. Also, the lesson starts in the middle so we don't see the explanation."  Neither reviewer made any comment on video quality being poor or difficult to hear.  This makes me wonder if this reviewer needed a better computer or speakers.  It also leads me to wonder if his analyses and scores were impacted by his frustration with the overall "poor" quality.

Let's summarize.  Overall, reviewers 1 and 3 indicate that I am doing a Very Good or Excellent job in each of the five dimensions of teaching and there are no comments from Reviewer 2 that "may be helpful".  Although the letter sent by Charlie Wayne, Educational Assessment Specialist, says "We invite you to reapply for the PAEMST Program in the future and look forward to hearing from you again," I do not plan to apply again.  Applying for this award four times and not receiving it has been enough for me.  I don't need to apply for an award to reflect on my teaching.  I just need time, space to write my ideas, and colleagues to share them with.