# Broken Pipe in a Classroom: Student Work (Sequel)

Here is a link to the original post on the broken pipe in a classroom problem.

This is the email I received from a teacher at a different school:

I don't have all the details and may not be able to get them... a pipe in my room burst on Monday and caused water damage in the hallway end and a total of 6 classrooms in full or in part... I have been told varying amounts as to how high the water was in my room and mere estimation by our custodial staff on how much water they themselves eliminated... the plumber said that the little copper pipe was spewing 12 gal per min...

Some of the outcomes of this activity:

• Students gained a stronger understanding of Excel. Many students chose to write formulas and use the fill handle to brute force the time it would take to fill the various bathtubs.
• Students deduced the connection between casework, writing out values, finding a trend, and generalizing through trying to write formulas for the volume of each bathtub as time passes.
• Students discovered finding the "center" of a semicircle is more challenging than they anticipated. (see the half-cylinder tub on the original worksheet) Several used Geogebra to find the center through geometric construction.
• One of the students made some progress on understanding 3-dimensional graphing with Geogebra 5.0 Beta.

Here are some examples from student work on Excel:

This student (below) chose to write the functions for fill and drain as linear functions in Geogebra:

Here is some work by a student trying to find the center of a semicircle:

A student modeled a theoretical room using Geogebra 5.0. (Apologies in advance for the sound quality.)

The students submitted written reports summarizing their thinking in this activity. I wanted to see a progression in their thinking from specific cases - like the percentage of a tub filled after 10 minutes - to generalized cases.

# Principal Pac-Man 1.0 (Chasing Behavior)

Disclaimer: The following is an idea I have been thinking about this week. I have absolutely no idea whether it would work. I don't know if anyone anywhere is approaching principal work in this way. Some of this has been daydreaming or thinking during a long drive to a conference. These ideas may seem disconnected, but I will try my best to explain the relationships I see between these ideas.

We had some professional development days to start this week. I enjoyed two presentations Monday by David Webb from the University of Colorado Boulder and the Freudenthal Institute. His morning and afternoon sessions focused on formative assessment in mathematics. When he saw many morning participants planned to stay for the afternoon session, he quickly talked about something he and his colleagues use to teach early computer programming concepts to middle school students.

Dr. Webb posed this question to the audience: how do we design intelligent ghosts that will actually chase Pac-Man? The mathematical process, known as collaborative diffusion, describes a possible method for programming ghosts to effectively chase Pac-Man. Here's a link to an academic paper on collaborative diffusion by Alexander Repenning. A screenshot from the paper appears below.

Think of the spaces around Pac-Man as the yellow mountain above. The ghosts want to climb the mountain - and effectively destroy Pac-Man - by climbing to the top of the mountain as quickly as possible. I was thinking about this idea of how the ghosts are chasing down Pac-Man. Then, I thought about how we often in school try to chase down behavior. For example, when a teacher is in the hallway greeting students, sometimes amorous couples try to hide from the teacher's line of sight. If the teacher has to help a student in the class, and cannot man the hallway post, then the threat of punishment is gone. Speeding tickets then came to mind. I thought about how punishment rarely works well as a behavioral deterrent. Drivers may choose not to speed when a police officer is nearby, but once the police officer leaves, look out.

To tie this stream of consciousness back to teaching, think about how often teachers must identify, on the fly and while making mental decisions regarding content delivery, students misbehaving in the classroom. Proximity works well as a deterrent - walking near the student, pointing at the open book on the student's desk as the teacher walks by - but this technique also has its limitations. As soon as the teacher walks away, the student may misbehave again.

Then I thought about how tough it can be to be a principal. Here's a great post on how to navigate the frequent interruptions a principal faces. The principal position can sometimes be very similar to the function of a police officer - a deterrent. But, as a principal leaves, so does the threat of getting into trouble, and the idea is the same as the teacher that walks away from the student's desk. How do we address this behavioral piece while teaching? How do we keep students on task?

The possibility of being called on randomly.

While thinking about police officers, I thought back to another article I read in a grad class about The Santa Cruz Experiment. The article, which appeared in Popular Science, described predictive police work. <think Minority Report> A mathematician designs an algorithm based on data for allocating patrols. Though random phenomena may be wildly unpredictable in the short term, long terms trends and patterns emerge.

Tying this back to the principal idea... if the ghosts chase Pac-Man... doesn't the principal chase the behavior? Suppose we try to incorporate a random mechanism into the principal's behavior in an effort to make chasing this behavior - just like the patrols in Santa Cruz - more efficient. Let's set up an imaginary simulation. We will declare the following events as things the principal could do.

Let
0 = observe 1st floor hallways
1 = observe 2nd floor hallways
2 =  observe 3rd floor hallways
3 = observe 1st floor classrooms
4 = observe 2nd floor classrooms
5 = observe 3rd floor classrooms
6 = observe school entrance / parking lot exit
7 = monitor stairwells
8 = monitor cafeteria
9 = monitor library

Then, we could use some sort of random process to generate a random behavior for the principal.

Looks like today's focus will be first floor classrooms. Because all outcomes are equally likely, we now have a mechanism like the Popsicle sticks in the classroom. This will be a more efficient approach to deterring negative behaviors among students as well as teachers. This would also give the impression to students that the principal could be anywhere. Thinking back to collaborative diffusion, and Pac-Man emitting a scent that can be chased down by ghosts... the metaphor places data in the role of the scent. We have plenty of sources of data on student misbehavior. Also consider certain events more likely given certain days of the week and months of the year. Isn't a student more likely to get a discipline referral close to a vacation, after a long block of no days off from school, because teachers' behavioral tolerance is lower? Isn't a staff member more likely to violate dress code on a Friday? Aren't students more likely to be off-task close to passing periods? We could use data (and a different random digits assignment scheme) to make an attempt at 'predictive principalship' much like the predictive policing in the Santa Cruz Experiment article.

It would be interesting to see whether this is a viable strategy for administrators to use.

P. S. If you've made it this far in the article, please be sure to read the disclaimer at the top of the article a second time.

P.S.S. I know the title doesn't quite jive with what was discussed here... since the metaphorical principal is the ghost and the metaphorical Pac-Man is the behavior... but the title is way more catchy this way.

# Interpreting Study ‘Results’ Can Be… Tricky

When I first introduce experimental design to students in AP Stats, we consider the diagram below.

A fun exercise in stats class is to locate stories in the media and comment on how the story's author interprets findings. Here's one such example from my morning Internet surfing.

Why Being Tired Can Make You Thinner and Healthier
Source: Yahoo News, Shine section

The following passage is taken from the article:

The urge to take care of our bodies when we're sleepy could be biological. “We proposed that people are more motivated to engage in healthful behavior when they are depleted and perceive their safety to be at stake," wrote study authors Monika Lisjak, assistant professor of marketing at Erasmus University in the Netherlands and Angela Y. Lee, professor of marketing at Northwestern University.

In the study, researchers asked subjects to read about the dangers of kidney disease and early detection, those with a family history included. Afterward, those who were feeling exhausted expressed a higher likelihood of being tested than their energized counterparts. In another study, subjects were asked to complete a survey on health and fitness, either before or after hitting the gym. After the survey, everyone was told to choose a gift of either sunblock or moisturizer. Those who had worked out were more likely to select the skin-saving sunblock.

Questions for students:

• Does the treatment suggest participants will get 'thinner' when tired?
• What is the author's motivation behind the wording in the article's title?
• What is the author's definition of 'healthier'?
• Could we design further research to confirm or disconfirm the author's claims?
• What implications do the results have on our daily lives?
• Is the source trustworthy? How would we make this decision?
• What would we need to know about the sampling method to make an informed decision about the results of this study?
• What would we need to know about the population of interest to make an informed decision about the results of this study?

# I Don’t Stretch Point Totals on Tests to 100. I Do This Instead.

This post is a quick hitter about the way I have approached grading tests for several years. I purposefully try not to stretch the point total to 100 or some other tidy multiple of 10 or 20. Instead, I assign points to each problem based on sub-steps. I use my trusty TI-84 to compute values quickly for grade book entry.

Suppose a test turns out to be worth 68 points. Here's what I do.

This is the linear function with slope 1/68 and y-intercept 0. One approach is to construct a table of values like below.

But the trouble occurs when grades vary wildly. Say some students are getting poor scores, like a 41 out of 62. Then we have to either reset the initial table value or, even slower, scroll up.

Instead, I set the viewing window and make a graph.

Just ignore the delta-x and TraceStep at the bottom of the screen. I set the window to have an Xmin of 0, an Xmax of 68 (because these x-values span all possible raw scores), and then set the Xscl to 1. The Xscl assigns integer values to tick marks on the graph, but for our purposes, this is really a personal preference, not a requirement.

Setting Ymin = 0 and Ymax = 1 with Yscl = .1 computes the percentage for the student's score. I could obviously do a little extra work by graphing Y = (X/68)*100 and make window adjustments accordingly, but for the sake of speed, I will just mentally convert the decimal to a percent.

By pressing the TRACE key, I can now manually enter any score and see the decimal corresponding to the student's percentage instantly. For example, suppose a student answered 44 items of 68 correctly.

I know there are other ways to do this... but, this way works well for me. This is another of the many things that students in math teacher ed programs may not see or talk about before student teaching experience, or in some cases, at all before entering the classroom.

# Who’s the ‘Better’ Scholarship Candidate? (z-scores)

The following problem is what I use each year I introduce the notion of z-scores and converting position values on a Normal distribution with mean μ and standard deviation σ [ N(μ, σ) ] to scores on the standard Normal distribution with mean 0 and standard deviation 1 [ N(0, 1) ].

We use YMS The Practice of Statistics, 3rd edition, in the AP Stats class I teach. Standardized scores makes its first appearance in Chapter 2. We cover this content in early September, a time where many college bound students are busy filling out college applications, preparing resumes, and requesting letters of reference. Since our school is in the Midwest, virtually all students are familiar with the ACT. Few know about the SAT; in particular, few know the maximum possible score on sections of the SAT. This activity leads to a nice thought experiment also, where the students must put themselves in the shoes of scholarship committee members making decisions that affect students' lives.

Here's what the example above looks like worked out on the Promethean board:

This decision is pretty simple to make. The student with an ACT math score of 33 has a relative performance far more impressive than the student with the SAT math score of 705. I start with this example because the values are fairly clean and the decision is easy to make.

But what happens when the computations reveal values that do not yield an 'easy' decision? Here's the example I use to immediately follow the ACT vs SAT issue.

I like this example because it requires the students to reflect on the choice they will make about units of length. Should we convert the feet & inches measurements to decimal feet? Or to inches? Many students choose to use inches. I show students the problem and put three minutes on a countdown timer.

Then I circulate the room as students work through the problem. I listen carefully for the discussion, for the argument of which student would be the better scholarship candidate. I randomly select a student to go to the front of the room to show the work they did and to explain their thinking. An example of some student work is below.

Listening to the students argue about this decision is fascinating. Some will insist that because the female has a z-score that is a whole unit higher (5.11 versus 4.109), the female deserves the scholarship.

Others will argue that because the normalcdf command on the TI-84 yields the same value to four decimal places, it does not matter which candidate we choose; both are equally good. (This four decimal claim is because the rounding convention we use as a default is to the nearest ten-thousandth when not specified).

Another school of thought amongst the students is that because the procedure does not yield a clear result, further analysis is needed, such as academic performance, financial need, or an assessment of each athlete's moral character. These factors of consideration are student-centric.

I challenge students to think from the perspective of the athletic team or the institution. Perhaps the conference is loaded with strong female athletes, so we need a strong female athlete to be competitive. Perhaps we need to choose the athlete whose family could provide more financial support in the event we have to split the scholarship value later. Many of these institution-centric considerations do not occur to the students naturally.

Using high quality problems like this one provides another hidden instructional benefit. I always have a conceptual hook on which to hang the process of standardizing scores. If I ask "Do you remember the process for standardizing scores on the Normal curve?" and get little to no positive responses, I can always quickly follow with, "Think back to the scholarship problem, where you had to compare two different candidates to see which candidate was better." This cuts down on the time I have to spend reteaching and allows us to be more efficient during class time.

As I look to summer when I have additional time to better my practice, this is one of the first problems I will look to film when I test the waters of 'flipping' the classroom.

# Staff Meetings: Can We Do Better?

I had a conversation yesterday with a strong second year teacher. Afterwards, I thought about what types of things would have been helpful for me to know or think about when I was a second year teacher. What follows below is my view on staff meetings. Suppose we ask some theoretical high school math teacher in the U.S. the question, "Why did you become a math teacher?" Here are some 'said no one ever' types of responses:

• "I would like to spend the better part of my twenties, thirties, forties, and fifties grading papers."
• "I would like to teach a class where kids bring negative attitudes toward the subject matter."
• "I want to teach a class where students will incessantly ask me, 'When will I ever use THIS in my life?' "
• "I really enjoy staff meetings."

Let's examine the last one for a moment. Staff meetings. Despair.com has some hilarious de-motivational posters. Here's one of my favorites.

In my teaching experience in two very different buildings, I have been at staff meetings that were useful and productive. Unfortunately, these tend to be the exception, not the rule. A contributing factor is that the person or persons running the meeting sometimes do not follow good teaching practice. What would students do if the teacher talked non-stop, without asking for feedback, for 45 minutes? For 90 minutes? The human form is not designed to sit for extended periods of time. Staff meetings serve their function of being a way for a person to communicate information to large groups in one shot, but the problem with passive lecturing is just like a broadcast from a radio tower: if no one tunes in, what's the point?

In my building, I teach on the 3rd floor of the classroom wing. It is rare that a person walks up a set of stairs just to say hello. This means there are teachers in the building, on a staff of roughly 60, I often go without seeing for long periods of time. I ran into a teacher in the hallway last night I hadn't seen for a while, and we had a great conversation. I found myself wondering why staff meetings often do not produce high quality conversations that have impact on teaching, learning, and school culture. What prevents us (teachers) from delving into difficult issues or having productive conversations?

• Ideas have inertia. It takes force to overcome inertia. That force is someone else piping up and saying, 'yes, I agree with that,' before others jump on board. Sometimes it's too early in the day... or too late in the day... or the person that would have agreed wasn't listening.
• The vocal minority often do not speak for the entire staff but act as if they do. The staff becomes frustrated because the opinion being expressed does not reflect what is typical of the staff.
• Meetings decompress to the time allotted. Saying 'we have 15 minutes to make a decision' is very different than saying 'we have 45 minutes to make a decision.'
• Democracy tends to be a very inefficient way to govern. Power dynamics often play  a key role when we open up large group discussion in a staff meeting. The same good idea sounds very different (and garners a very different reaction) when it comes from the mouth of the venerable veteran as opposed to the first-year rookie.
• And the list... like some meetings... goes on and on and on and on...

So... can we do better? What does better look like? Here are some ideas I have about adapting staff meetings to better serve the intended audience:

1. Seek ideas from sources online. I'm not the only person that has considered this staff meetings issue. Here's a great post on finding purpose in staff meetings. And here's another one. And another one. The World Wide Web is chock full of great ideas. Some are not so great and require a bit of sifting, but mistakes can be made in a good direction.

2. Change our perspective. There's plenty of drudgery in mathematics, for example. Many times, I have worked through exorbitant amounts of algebra to discover I made an early error and wasted an awful lot of time. Sometimes we have to endure in order to be rewarded. Staff meetings can provide valuable insight into teaching and the culture of our school building. The conversation can go places that will benefit students but we must first put ourselves in the right frame of mind.

3. Take information sessions and make them webinars. It is pretty frustrating to teachers when nothing happens to the teacher that failed to show up to the 7:00 am staff meeting. The principal 'confronts' said teacher: "Oh, you missed the meeting? It's ok, I just went over some information, catch up with me later for 10 minutes <when the meeting was 45 minutes long!>" Make a video and send the link through email. Tell teachers to watch it before the meeting. Make the meeting a productive discussion about the information in the video... which leads into my next suggestion...

4. Provide opportunities for the conversation to go where it needs to go. We've all seen the ambitious agenda, the bulleted list the administrator or presenter intends to 'cover,' but our subconscious quietly chuckles in the background knowing full well the presenter cannot possibly span the agenda in the allotted time.

5. 45 minute meeting? Do it in 25. Take the allotted time for a meeting, cut it in half, then add 5 minutes. When I share ideas on Twitter, for example, the 140 character limit pigeonholes me into being succinct. The 140 character limit forces me to focus only on the important stuff. Cutting staff meeting time down is analogous to the 140 character limit on a Tweet. Just the important stuff, please.

6. Invite staff members to speak as often as possible. Think about the typical classroom. Kids often tune out the instructor, but as soon as a kid is called to the front of the room, that kid often commands the attention of the room simply because he or she is a peer, not a viewed authority.

7. Don't waste time. Time is the most precious currency of the school and of the teacher. If a meeting is called, it better be worth it to all parties involved.

8. Have a plan. A bulleted list or an agenda is not sufficient. We should be clear about the purpose and the function of the meeting. What if an administrator walked into a classroom and the teacher was entirely shooting from the hip? Or that the teacher had not anticipated student misunderstandings or misconceptions? We should have the same high expectations for staff meetings, too.

9. Consider the man-hours that go into a meeting. On a staff of 60 teachers, for example, a meeting lasting one hour, at a rate of pay of, say, \$30, means the taxpayer just shelled out \$900. What was the outcome of said expense? Was it worth the expense? Did it improve learning outcomes for students? Did it improve the culture of the school? Was progress made? Ultimately, we are accountable to our customer base. In public education, it is the taxpayer.

10. Teach participants how to interact. This goes beyond 'establishing norms' to ensure respect and professional decorum. What I'm talking about is that teachers often do not receive training on group dynamics or on the psychology of group work. Understanding group dynamics can produce better outcomes from group work and alleviate some of the social friction we sometimes see after meetings conclude.