# Cultivating Teamwork in Math Class

Establishing community in the classroom can be a challenge. Here's an activity my students participated in on the first day of school. I learned about this activity while participating in the Advanced Educator International Space Camp in Huntsville, Alabama. The objective is for two crews of astronauts to exchange positions in cramped quarters when a new crew shows up to relieve the old crew at the International Space Station.

## How the Game is Played

1. Only one person can move at a time.
2. Only movement forward (in the direction a person faces) is allowed. In the above diagram, orange players can only move right, while blue players can only move left.
3. A person can move to an empty space in front of them.
4. A person can jump an opposing team member in front of them.

## What Does It Mean to Win the Game?

The teams win the challenge when they have exchanged their original positions. See the ending position diagram for an example of what this looks like.

## What If...?

A teacher could use an agility (speed) ladder for this activity. Or if a ladder isn't handy, use tape. This is a shot of my classroom the first day. It's pretty unlikely a teacher would have only eight students. I put down three tape ladders in my classroom on the floor. If the number of students is not a multiple of eight (like my class was), the teacher could place the extra students on the side as coaches. To up the responsibility of the coach, add the rule that no one inside the ladder can talk to anyone else.

## Teacher Moves

While this activity was going on, I floated between the groups and listened very carefully. I wanted to learn about which of my students would step up and take initiative; which would be a leader; which would be concerned about the frustration of others and take action to minimize other students' discomfort/anxiety. This activity helped me better understand how to assign groups for course work in a meaningful way.

## Examples of Student Moves

Below is an example of what some students might do.

If the third blue player from the right jumps the lone orange player, the blue team has a problem. With two blue players in adjacent cells, the game is gridlocked and ends.

## Computational Thinking

Once the students came up with the solution, I gave them the sequence "1-2-3-4-4-3-2-1" and asked them how it relates to this situation. Think of this sequence as the answer key.

1: Orange moves first
2: Blue moves next - twice
3: Orange moves three times
4: Blue moves four times
4: Orange moves four times
3: Blue moves three times
2: Orange moves two times
1: Blue moves one time

## Low Entry, High Ceiling (Extending the Task)

• Ask the students to come up with some pseudo-code to describe how they would build this game on a computer using programming applications.
• Ask the students whether the strategy remains the same if there are teams of five? Or if there are two empty middle squares? Three empty middle squares?
• Ask the students to write a program that allows the user to watch the game. Then ask the students to write a program that allows the user to play the game.
• Example from my classroom I had two students come up with different lines of thinking for coding this game on a computer. One student thought of a number line to label each cell, using the values -4, -3, -2, -1, 0, 1, 2, 3, 4. Another student thought of simply have numbers represent each student. The starting configuration would be
1 2 3 4 _ 5 6 7 8. Then, each move would be a shuffling of the sequence. The second row would be 1 2 3 _ 4 5 6 7 8. The third row would be 1 2 3 5 4 _ 6 7 8. The fourth row would be 1 2 3 5 _ 4 6 7 8. We had a really spirited discussion of the issues that could arise from each organizational coding strategy.

# Differentiation in Math Class: Make Students Write

My elective math class has all grade levels represented within it. While not ideal, this is a feature of our scheduling system. So I have freshmen in Algebra I all the way up to seniors that have already completed Calculus AB. This poses a huge classroom differentiation challenge each Monday, Wednesday and Friday we hold class. Here is an instructional strategy I use that gets students writing about the mathematics they do in class.

Students worked on one of four things in the computer lab last Friday.

1. Construct 2013 Probe I Problem 7 diagram (2D)

2. Construct 2013 Probe I Problem 23 diagram (2D)

3. Construct 2013 Probe I Problem 11 diagram (3D)

4. Work on Alcumus problems independently

After we spent approximately 55 minutes in the lab, we returned to my classroom for a writing activity.

Here is the writing prompt I put on the board:
(Writing exercise on a separate sheet of paper to turn in to me)
Think about what you learned about the diagram or diagram(s) you built in Geogebra. Write a letter to the you of October 12. How did building diagrams in Geogebra help you understand the problem better?

I put eight minutes on the clock and informed students they would need to continuously write for the eight minute timeframe. Here are some samples of student writing from the activity. For convenience, I have inserted another copy of the problem. Immediately below each problem appears student writing associated with the problem.

The Students revealed their thinking about these math problems throughout their writing. While some students chose to concentrate on the construction process in Geogebra, others also revealed some of the mathematical structure they encountered while making the diagrams.

Writing samples from students that worked on Alcumus:

These writing samples revealed to me the depth of student thinking going on in the classroom. If I could have a superpower, I would be a mind reader. Then I wouldn't have to guess at what my students are thinking. Having the students write for an extended period of time gives me insight into how they are seeing the mathematics and gives me ideas on how I can help further their understanding and guide them as they struggle.

I collected these writings immediately after students completed them. I ran the pages through my ScanSnap scanner and converted them to a PDF for me to review later. I told students we would get these writings back out in a month's time, emphasizing the need for specificity on what they were working on and what they learned that day.

Going forward, these writings help me be more efficient with respect to differentiating classroom instruction. We don't need to be working on the exact same thing at the exact same time at the exact same pace for the students to engage in meaningful problem solving.

# 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.

# What Goes on the Wall Matters • 2 Comments

Here's my annotated Friday philosophy: we have a culture of laziness on Fridays in the U.S. Students think Friday is a day to take it easy. I view Fridays as an opportunity. The standard work week is Monday - Friday (five days). If we work hard on Friday, we will outwork people with which we are competing 20% of the time.

What you post on the walls of your classroom matters. The stuff on the wall indicates to your students the things you think are valuable. What you post should suggest to your students that success in mathematics is possible for everyone through a malleable view of intelligence, a passion for hard work, and the notion that good mathematics is about knowing how to behave when you don't know how to start.

Below are some examples of things posted on the walls of my classroom.

Exhibit A: The "I CAN'T" poster. I don't even remember where or how I stole this acronym. My wife made the poster file in Microsoft Publisher. Then, we took it to Staples to print it on poster paper. This poster is one of my favorite things to reference, particularly when I call on a kid that I know doesn't have an immediate answer. Philosophically speaking, many of the motivation issues we see in high school math students are symptoms of a lack of effort or a lack of efficacy (feeling good about one's performance in mathematics).

Exhibit B: The only Quantum Learning key I like - THIS IS IT - and my movie theatre style, "Please Turn Off Cell Phones" sign. The key suggests to students every day matters. Every day is a chance to add another brick to a student's pyramid of understanding. Every day is another chance to instill in students skills that will make them confident waiting in a waiting room for a job interview while up against other strongly qualified candidates. The cell phone sign demonstrates we live in modern times. Kids have cell phones. Let's acknowledge the fact kids have them. I have all my students turn off their cell phone and place it in plain sight at the front of their table. We acknowledge cell phones are distracting to adults, especially in meetings. We don't want cell phones to distract us from the important learning we need to do.

Exhibit C: My Math Club display. Something I need to work on as a teacher is to update my Math Club display with more photos of students doing club activities. While the Escher drawings are nice, this is something I can do to demonstrate to students I value their hard work and dedication to our program.

Exhibit D: The "Working Hard" clock. Our school has analog clocks hard wired into the walls. The last thing I want at the front of the room, above my Promethean board, is a constant reminder of what time it is. Instead, I cover the clock with a "Working Hard" sign and put up a clock at the top of the back wall in my classroom for my instructional reference. This way, it becomes really obvious when kids are checking what time it is. They have to turn all the way around in their chair. That's a signal to me to step up my game - make an adjustment to an activity, get the students moving, to do something to make the class more engaging.

Exhibit E: Color photocopies of my college degrees. (The real copies are at home, of course). If we want more students to prepare for college and take that preparation seriously, we need to be very vocal about the pride we have in our collegiate accomplishments. Education changes life trajectories and can transform life outcomes for our students. It certainly did for me and my family.

Have a great Friday at school. Don't get outworked!!

# Cheating & Calculators • 1 Comment

Many standardized math tests (the ACT, the SAT, and AP exams) allow students to use graphing calculators. The most common model here in the U.S. is the TI-83/84 family. I suspect many test proctors do not know how to check these devices for programs that may be used to cheat on these exams.

No, I have not had to deal with this particular problem... this year.

I have a classroom set of calculators on which I periodically check to make sure there are not any programs or apps that might help students cheat. Every time I administer an assessment, if a student uses their own graphing calculator, I walk around the room and physically check for programs and apps that might help them cheat.

Here is a PDF I made on how to check for cheating on the TI-83/84. This covers programs only. A student could potentially download an app with content disallowed on a test. To look at the apps, a teacher would need to press the APPS key and look for suspicious apps that are not standard installs on the calculator.

Checking a TI Calculator for Cheating

I have also posted images from the PDF below.

# Classroom Management: Behavior and Cascading Errors • 1 Comment

In 2010, I had the privilege of attending a USATF Level 2 Coaching School. One of the interesting presentations was by a field event coach using Dartfish. The video featured a world class rotational shot putter. He lost a competition despite tying with another competitor for his best throw. The tiebreaker goes to the next best throw. This shot putter fouled the other five of his six throws at the competition. Using Dartfish, the coach superimposed the video of the athlete's best throw with his worst throw - the throw in which the athlete fell out of the ring.

The video reminded me of playing Mario Kart on the Super Nintendo. After setting a personal record on a particular track, a player would race against his or her own 'ghost,' a recording of his or her best race. This throws video looked the same, where the best throw was the ghost. The software synchronized the two throws perfectly for diagnostic comparison.

A person could easily get lost in the details, wondering how it was possible for this elite athlete to fall out of the ring. The instructor encouraged us to pay close attention as he played the clip in slow motion reverse. The catastrophic error, the fall out of the ring, could be traced back to the thrower's first step with the left foot. The first step on the worst throw was roughly three inches different than the first step on the best throw. The instructor called this a cascading error. He stated many coaches spend too much time coaching later errors that are the result of the initial error. The trick, he said, is to diagnose and correct the initial cascading error.

This idea stuck with me. In my experience coaching football, basketball, and track & field, I have found good teaching and good coaching have a lot in common. Conversations with inexperienced and pre-service teachers often include classroom management strategies. Teacher preparation programs often point to classroom management as a struggle for new teachers.

These ideas bring me back to the notion of a cascading error. If your spider sense is tingling, and you sense a child is about to misbehave, try to diagnose the cascading error before the behavior blows up. Diagnosing a cascading behavior means identifying the root cause of the misbehavior and working to correct the child before misbehavior really gets started. Below is a wonderful exercise for both new and experienced teachers alike. Edward Sabornie at NC State has a video aimed at identifying good and bad behaviors by both students and teacher. Think about cascading errors while diagnosing the behaviors by both teacher and students. What would you do to prevent student misbehavior in this room? Click on the image below to access the video.

What does an effective classroom manager do differently than an ineffective classroom manager? The teacher skilled at classroom management:

• Leverages student relationships to correct behavior and optimize conditions for learning
• Communicates classroom rules and procedures effectively and efficiently
• Creates and executes engaging lessons that reach all students
• Maximizes and conserves time in class for learning
• Lets administrators handle discipline but not before conference one-on-one with the student and placing a phone call home to the parent or guardian

My approach to classroom management is an amalgamation of ideas from many different sources. These are my views on effective classroom management.

1. If your lessons are interesting and engaging, and the kids are thinking about the content, the probability of having a behavioral episode in your classroom approaches zero. Administrators conducting observations for evaluation purposes often record passive compliant behaviors as engagement. I have seen many students in my classroom and while observing other teachers feigning attention in a lesson. Passive compliance does not imply engagement. Even active compliance does not necessarily imply engagement in the lesson. If students are actively wrestling with the content, many issues resolve themselves. If you know a lesson may have a long lecture period (by long I mean more than 5 minutes), embed opportunities to socialize within the lesson (e.g. "Turn to your table partner and explain which of the three sampling techniques we should use in this experimental design to minimize bias; I will call on two people after the minute timer is up.")
2. Establishing consistent procedures and high expectations translates to student success and teacher efficacy. I attended a presentation a few years ago by Harry and Rosemary Wong. They have a wonderful book titled The First Days of School. Harry and Rosemary Wong assert students coming to class on the first day have the following questions that a teacher must address to start the teacher-student relationship strong.
• Am I in the right room?
• Where do I sit?
• What are the rules in this room?
• What will I be doing this semester?
• How will I be graded?
• Who is my teacher as a person?
• Will my teacher treat me fairly?

Another big idea I took from this book and presentation is the difference between classroom discipline and classroom procedures. Classroom discipline concerns how students behave. Discipline has penalties and rewards. I have yet to meet a teacher that felt truly triumphant after a power struggle with a student. Classroom procedures, in contrast, concern how things are done and have no penalties nor rewards. Procedures put the focus on what the students should do.

3. Assume everyone is on the same basic level. If students are to learn how to behave properly inside and outside school, we must maximize on the time students spend in our classrooms. It is gambling at best to assume students 'already know how to behave' or 'students should know how to behave.' Assume nothing and carefully communicate your expectations of your students. Students are bound to forget your procedures. Keep calm and gently remind students, particularly in the first three weeks of class, how you would like them to request to leave the room. Students tend to have several classes, particularly at the high school level. The student is not trying to offend you; he or she simply has a lot to remember. Model the behavior you want to see, and be prepared to teach students how to behave. As the old adage goes, "an ounce of prevention is worth a pound of cure."

4. A phone call home is the single most powerful tool at a teacher's disposal. Use it early and whenever possible. I can remember feeling anxious and nervous calling home, particularly in my first three years of teaching. My view on the matter is definitely more complete now that I have a daughter of my own. If my child were misbehaving or struggling in school, I would want to know about it. I would want the teacher to call me. Calling home in week one, no matter how awkward the interaction, makes a tough conversation about academic performance in week six go more smoothly. Telling a parent or guardian you would like help connecting with his or her child gives you an ally at home, increasing the likelihood the student will correct his or her behavior in class.

5. Use positive reinforcement when students perform a procedure correctly. As one of my assistant principals told me this summer, "If we wanted to catch students doing something wrong, we could probably do that every single minute of the day." Here is an example of positive reinforcement In my classroom, I post the materials students need to have ready at the bell. As I walk in the room after greeting students in the hallway, I will quickly survey the room to catch students ready for class. "I see Ben and Alexis are ready for class. I know that because both of them have their graphing calculator and pencil on their table. Oh look! They even have their textbooks open to page 98. Man, that helps me out a bunch. Thanks for being ready." Kids won't see it as cheesy if you don't sell it as being cheesy. Make the comment and move the focus to the objectives for the day.

6. Any person visiting your classroom will know within 30 seconds whether or not you are effectively managing your students. Lew Romagnano, in his book Wrestling with Change: The Dilemmas of Teaching Real Mathematics, describes his experience as a researcher working with a practicing teacher. In the gap between theory and practice, Romagnano asserts while teachers do have problems to solve, there will also be unsolvable dilemmas in the classroom teachers must manage. Managing these dilemmas effectively will maximize the probability students will learn in your classroom.

7. Cold calling is highly effective in keeping students on task. Seasoned law enforcement agents and war veterans will say time almost seemed to slow down during firefights. As a teacher builds more teaching experience, the cognitive load during teaching does decrease slightly over time. I can validate this claim empirically; having taught how to solve a system of equations by substitution for several years, the cognitive load has gone down considerably from the first time I taught the lesson. Instead of being comfortable while teaching and going through the motions, I choose to focus on how many times I have called on each student. I am reading the body language of every single student in my room and looking for off-task facial expressions and body language. I will call on students I know do not have the answer. I will call on students I see change posture from sitting upright to slouching. I will call on students whose line of sight has drifted as soon as I identify it. I never let a student get away with saying, "I don't know." If a student knows they can be called on at any time, the student will likely choose to stay on task to avoid embarrassment in front of his or her peers.

8. Proximity is a powerful tool. More than management by wandering around, move with purpose about your room. Teachers that sit at the computer for the entire period lose out on valuable opportunities to assess students as they are working. Roaming the rows purposefully lets you spot both potential cascading behaviors and content struggles. Actively collect evidence as you roam the room on which students are getting it and which students are not.