Below are some of the highlights from our UNL Math Day trip last week.
Our group boards the bus in Scottsbluff. At the opening ceremony, Scottsbluff High School was recognized for driving the longest distance to the contest (400 miles!) and for being the best represented school in Class II with 50 students.
Our certificate for being the best represented school.
Another group of 24 students sat in on a math capstone course for undergraduates that intend to become secondary math educators. UNL Professor Yvonne Lai invited SHS students to speak about the special mathematics curriculum students enjoy in the SHS Math Theory and Problem Solving class. A group of teaching methods students (center semicircle) talk to SHS students about college opportunities and the value of mathematics.
Andrew and Kyle address the capstone class. Pictured is the UNL Math Day promotional poster that Andrew (left) designed for the University.
Our students (left) compete in the Math Bowl competition. We ended up losing this match to Lincoln Pius X - had a chance to tie it on the last question! - 7 to 6. In the second round, we lost on a tiebreaker. Despite the setbacks, the students really had fun preparing for and participating in the competition.
This is an exciting time to be a math teacher. New technologies give us the chance to refine the art and science of teaching through video study. I spent a semester last fall looking into ways to create and to use classroom video of my own teaching to improve my instruction. I'll share a few things I learned in my travels below.
I think teachers can learn a lot from NFL players. NFL players have much easier access to massive collections of film. Sportscasters and fans alike circulate tales of how the best NFL players are students of the game and spend massive amounts of time in film study. In an article entitled A Former Player's Perspective on Film Study and Preparing for an NFL Game, there's a line that jumps off the page:
The position an NFL player plays also determines how much film that player needs to watch.
Suppose this idea generalizes to the population of teachers. Does the physical education teacher need to study more film or less film than the social studies teacher? While the metaphor may break down, it stands to reason a mathematics teacher - sometimes faced with daunting differentiation loads and a compressed time schedule which sometimes leans more towards content coverage than understanding - would probably need to watch a lot of film... especially when it comes to asking good questions, maximizing class time, and identifying and redirecting quickly students that are off-task.
The cognitive demand on math teachers to adjust instruction based on events in the classroom is high, particularly for novice teachers with little experience from which to draw. The book Mathematics Teacher Noticing: Seeing Through Teachers' Eyes provides wonderful insight on how teachers can use video to improve their ability to pay attention to classroom events, specifically those events that occur during instruction. This book is a collection of some of the best field research currently available. From my perspective as a classroom teacher, you will be pleasantly surprised at the style of writing in the book... this isn't dry academic canon. The authors of these articles breathe life into the classrooms and the dialogues ongoing between teachers and students. If you haven't heard of it, and you teach mathematics, I'd highly recommend picking up a copy. There's a reason why I only see 2 available on Amazon this morning.
Suppose a teacher is thinking of filming his or her own classroom for the purposes of reflection. There are several things to consider before a teacher can just plop a camera in the corner. First, how do we address student privacy? Often, school policy lags behind technology. A parental permission form will be needed (of some kind) for every student. Here is a simple example from the New York City Department of Education. The teacher will also need a plan if a student opts out of being filmed. It probably isn't acceptable to force a student to sit in the corner if he or she is the only one not granting permission to film.
Second, what technology do we use for filming the classroom? In every case, it is critical we obtain the highest quality image and audio possible. Many teachers use Flip cameras. These are quick and easy to use. Simply charge the camera and move files by plugging the camera into a USB port. I personally use my 64 GB iPad2. Regardless which type of camera a teacher selects, a wide angle lens is absolutely essential for capturing classroom action, along with a tripod for stabilizing the image. There is a pronounced difference between using a wide angle lens and not using a wide angle lens. If a teacher uses an iPad, one solution I have come across that I have been very satisfied with is the Makayama Movie Mount.
Third, how will the teacher address sound issues? It depends on the purpose of the video. If the teacher is using the video strictly for personal use, adding subtitles for student speech is a possibility. Through trial and error, I have found with the iPad it is best the person filming circulates the room to follow action. The built-in microphone on the iPad 2 captures sound pretty well within an 10-12 foot (3.05m - 3.66m) radius. Entities like the National Board for Professional Teaching Standards and the Presidential Awards for Excellence in Mathematics & Science Teaching do not allow subtitles to be added to video. This is a common practice among entities reviewing classroom videos for teaching awards. If anyone out there finds a strong, functional sound solution for the iPad, please let me know. If the teacher uses a Flip camera or a traditional video camera, the place to start would be to look for an omnidirectional boundary microphone.
Fourth, what are the risks to the teacher? If the video is shown to others, the teacher may want to consider the implications. Teaching is an unbelievably personal act. The teacher would need to be comfortable with the other parties viewing the video. The working environment would need to focus on teacher growth and reflection - and not run the risk of being punitive either explicitly or implicitly to the filmed teacher. I have not done a lot of study in this particular area since my studies in filming the classroom have been limited to myself.
Fifth, what if a teacher isn't ready to film his or her own classroom? Not a problem. Others are doing it now and sharing their videos. Check out the Teaching Channel website, where phenomenal teachers are posting videos of teaching from their classrooms. Teachers can reflect on the practices they see in these classrooms. A teacher could start a video club, perhaps during a common planning period or PLC time, to discuss the effective practices demonstrated in these videos and how to incorporate those practices into instruction.
In the twenty-first century, film study is essential in teacher preparation programs. Here is a statement below from a replication study using classroom film study with preservice teachers, based on Sherin's work, at Harvard:
We found that preservice teachers began the methods course with relatively poor observational skills, and after a course focused on improving their ability to notice a full range of classroom events, preservice teachers were better observers of both mundane and important events.
You can find a summary of the study here. When I think of my own daughter and the future of her math education, my hope is that her future teachers, likely early in their careers and in methods classes now and for the next few years, are studying video and using video to become more effective teachers.
This was great to wake up to on a Sunday morning... I am officially a National Board Certified Teacher!
The application process was time consuming but rewarding. I hear that NBPTS is making some changes to streamline the process. I would encourage any teacher interested in developing their craft to pursue National Board Certification.
Sorry for the mid-week hiatus. We got back from UNL Math Day last night. I will post some pictures and stories about it later.
Zombies. Kids love them. I, personally, am a rabid fan of the entire genre, maybe even more so than the kids. My favorite TV show is The Walking Dead. Sunday nights are always a protected time in the Aaberg household.
I am wondering how to use a part of this video as a hook for potential probability questions. The clip is taken from the movie World War Z. In the clip, Brad Pitt's character (Gerry Lane) speaks with Ludi Boeken's character (Jurgen Warmbrunn) about how Israel knew to finish its exterior walls prior to the zombie outbreak. Jurgen Warmbrunn refers to the principle of the "tenth man" [01:43]. "If nine of us with the same information arrive at the exact same conclusion, it is the duty of the tenth man to disagree."
I am thinking about how this problem relates to the Binomial distribution. An example of a question I would ask students would be, "Suppose the chance that one person agrees with another person is 50%, and agreement between any two people is equally and identically distributed. What is the probability that exactly nine of ten people agree?"
I would like to branch out and ask some more sophisticated questions with the video as a hook. My purpose in posting this is to seek ideas for ways to include this situation as a hook for writing a lesson in a statistics class related to the Binomial distribution.
Paradoxes are a fun way to introduce logic to students. For example, Chuck Norris' roundhouse kick is so powerful, it is a paradox he does not roundhouse kick himself out of existence. After all, the laws of physics clearly state every force has an equal and opposing force. So, technically speaking, when Chuck kicks an object, it hits back with the same force.
On Friday afternoon, a teacher announces to her class, "There will be a test one day next week, but the day will be a complete surprise."
After giving the matter a little thought, Anne reasons that the test cannot be left until Friday because if it has not taken place on Thursday, everybody will know for sure that it has to take place on Friday. Thus, it will not be a complete surprise.
But then, says Tashi, the test cannot take place on Thursday: if it has not taken place by Wednesday afternoon, everybody will know it has to take place on Thursday because Anne has shown it will not take place on Friday. Hence, Thursday is out, too, because otherwise how could the test be a complete surprise?
Using the same reasoning, Caroline deduces that the test cannot take place on Wednesday, and Deon immediately notices that it cannot take place on Tuesday. Hui then clinches the paradox with the observation that the test cannot take place on Monday.
Triumphantly the students present their reasoning to their teacher.
1. Can you explain this paradox?
2. Can the teacher get around the paradox somehow and still surprise her students with a test?
Many of my students use older TI-83 and TI-84 calculators that are hand-me-downs from an older sibling. I still use the old operating system on my TI-84 because I am familiar with it. As an instructor, I must also stay current on the new operating system and new features while still meeting the needs of those in the classroom using the old operating systems.
Linear regression is a tough process to teach on the old TI-83/84 because it involves so many button presses. Below is a PDF of a series of step-by-step instructions I made a few years ago using TI-SmartView and the screen capture tool Jing.
Probability and counting questions associated with rolling two dice are frequent at math contests. For example, one might ask, "What is the most common sum when two dice are rolled?" The grid below helps the student see the most common sum.