Sorry that all I’ve been posting lately is things I’ve been writing for my teaching class. My next big software project is in the works and I’ll write about it when I have something non-trivial. For now, though, I offer up my experiences in observing a teacher in a Boston public high school for about 20 hours this semester.
Speaking of teaching, I’m also doing a bunch of Splash classes this weekend, and a sample lesson for 11.124 tomorrow! I’ll keep a record of how each of these classes went, and what I noticed about my teaching (plus links to any lecture notes or slides I use).
Cyber Security – October 8th
The first day was a whirlwind. I met Mr. Teacher Man early in the morning before the morning announcements. It was very strange to be back in a high school, sitting in a desk and listening to the teacher while taking notes. Besides the strange familiarity and confusion as I navigated the new school, I was able to see how to address computer science concepts without using any programming or even a great deal of technical terminology.
Class began with a slow, slow start. The school itself has no bells that ring when classes end, so students trickled in and out of their classes. To compete with this problem, though, classes begin with something called a “Do Now,” where the stduents work on some type of activity that allows them to explore a concept or practice solving particular problems. Here, they were considering the question “what is computer intelligence”? After thinking and writing for a few minutes, the class came together for a guided discussion about what it meant to solve problems and think. Mr. Teacher did a great job at introducing complex ideas like the Turing Test, by explaining a simpler and condensed main idea, and not giving it a name (which I believe makes students worry about memorization). Overall, this seemed to get the students thinking about what problems are important to address without being distracted by technical details.
The class then moved onto discuss topics in cyber security, and what it meant to have security. This involved two useful components. The first component was a set of definitions the students needed to learn. The definitions we done first by students, and each student defined a word (like “hacker”) using only 3 words. They would do this for each word. After some set amount of time, students were asked to offer up their definitions to be used as the class definition. After this, Mr. Teacher would write the definition he used. This was an interesting method, as it gave students the chance to think about the words as opposed to blindly copying what the teacher had written. This allowed students to work through ideas, and then discuss as a group, and thus allowing more connection to the material. Unfortunately, having student discussion meant the class was slowed, which means a teacher needs to be careful evaluating and moderating commentary.
Once definitions were taking, students watched a video made by PBS about the challenges that occur with having everyone connected via the internet (and how vulnerabilities in computers can effect a greater network). I noticed that students seemed very disengaged and many did not even appear to be paying attention to the video. While it is important to show clips and have different perspectives, I wonder if there is a way to have video and also have students interact and engage with it. The students then got laptops in order to work with a web-based programming challenge (similar to Scratch) that allowed them to see how they can write instructions in a systematic way. This was loosely based on security concepts, but this seemed lost on students. I believe this could have been remedied with a bit clearer explanation of the task (ie, think about a program like this, and the interface works in this way). Additionally, many of the students failed to even start the assignment, and instead turn their computer screens away from Mr. Teacher.
It’s unclear how to make students accountable for these actions. Both of these materials are excellent tools for learning, but students lack the interest to follow through. One option that my high school teachers used was to give worksheets in which we answered questions about the video or activity as we went along. This can’t always work, however, as stuents should also be able to learn skills of responsibility and autonomy in learning. I think another possible solution is to have a post-assessment quiz, but that can lead to a stresfful activity as opposed to exploratory.
Regardless, I think that Mr. Teacher did his best to vary the activities in the classroom and have interesting activities that guide students through the material. Monitoring a classroom is certainly a challenge I do not know how to address.
Inverses of Functions – October 15th
One of Mr. Teacher’s greatest strengths in the classroom is creating worksheets that lead the students through the material while still allowing them to make guesses about how something will work. Class began with another Do No, where students were given about 5 minutes to work through a couple problems. Mr. Teacher reviewed the homework with the students, and afterward they called out their score (for accuracy and completion), and I immediately noticed bias in student answers, and a student near me had a blank paper, but called out a perfect score. There certainly should be a better way to collect and score these homework assignments.
After this, students used their understanding of composition of functions to examine compositions of functions, like and where and . Students wrote down what they noticed about these functions, and saw that when they were told that two functions were inverses of each other, the compositions just equaled . They then moved into an activity where they sketched graphs, and made tables of inputs and outputs. Because of this, they were able to observe patterns before being told how to rotely find the inverse function. (I can’t recall my own education, maybe I should ask Mrs. O about this)
Outcry came from the class when Mr. Teacher tried to formalize the idea of a function, and introduced the symbols and . This raises the question of how to introduce mathematical notation, and the idea that we can generalize statements to be true for all numbers. It’s also interesting to watch Mr. Teacher explain these concepts — he falls into mathematical language and sentence structures rather easily (as he was a math major in college) which I think confuses the students. Fortunately, they are quick to call out and complain about the issue.
Mr. Teacher is very big on the idea of guiding students through concepts, and explaining after they have intuition. This is also how I tend to teach, so I note the bias in my recollection of the class. It also possible that it is not entirely his decision — the school appears to encourage (or maybe require) teachers to follow the form of Do Now/work/Exit Ticket. Though the class went rather slowly, due to student tardiness and talking during class, the material was presented clearly and concisely.
Student Interactions – October 22nd
This day I spent a great deal of time working individually with a student, and then listening in and participating in a discussion during “academic advisory.” Instead of observing the computer science class in the morning, I working with a struggling student who was in Mr. Teacher’s Pre-calc class. This student was preparing a cheat sheet for an upcoming exam. This was to be a double sided sheet to review material from the first quarter of the year. (Ironically, I was also working on making my own cheat sheet for my algorithms class). In this, I tried to both explain the concept of making a condensed sheet of notes that have things you don’t remember and also teach this student concepts they didn’t quite understand.
This resulted in me being very confused about how to explain limits. It’s difficult to come up with an easy example while not glossing over important details (such as the fact that the limit of a function as it approaches a point is not necessarily the value of the function at that point). Additionally, it’s hard to keep coming up with different phrases for the same concept in order to ensure that I’m not repeating myself over and over again with the student becoming more frustrated. I think the maor problem was that limits are so engrained in my academic life, and I work wih the concepts so often that I can no longer remember how to intuitively explain them. I had the same problem while trying to algebraically find the minimum or maximum of a function (I could only remember how to use calculus).
Something I noticed from this hour-long review was that it is incredibly difficult to assess when a student actually understands a topic, or if they are trying to hide the fact that they are confused. As a result, I started making examples on the fly (which occassionally failed). If there are any suggestions about how to better engage and assess student understanding, I would greatly appreciate pointers.
Academic advisory also allowed me to gain a glimpse into how teachers can interact with students. In this class, seniors were discussing their future plans and anxieties about college. This started by addressing study skills, and recognizing that going to college would be harder and that they were unsure if they would be prepared for the challenge. Mr. Teacher discussed how he had a difficult time adjusting, and when one of the students said “But you went to (prestigious university) of course you had a hard time,” he snapped back about how the student didn’t understand. Something I am never sure about is, in an educational situation, how much do you allow the students to see and how much do you hide. For example, students might benefit from knowing that I have failed in college over and over. But that might come at the cost of respect. The balance is unclear.
This also brings the point of discipline, where it’s unclear how to draw a line between allowing students to engage with you and have a closer relationship, and where student behavior becomes distracting or takes away from the educational environment. I’m sure that people have thought hard about this problem, and there are certainly papers about the topic. I think that, while discussing how to educate is important, teachers should also be taught about how to be a mentor.
Subtle Graph Theory – November 5th
The day of my 6.042 exam, of course the computer science class is covering graph theory. Of course, it was not explicitly called graph theory. This is what I love most about Mr. Teacher’s style of instruction: students are solving problems and learning strategies without being scared by the fact that they are doing a somewhat complicated problem.
Here the class was started by considering the concept of efficiency, and looking at how students define good problem solving. This also included the equation “efficient = smart + lazy” which I found amusing. Anyway, after the students were given a problem about handshakes: if everyone in a group of 10 people shakes hands with one another exactly once, how many hand shakes occur? I personally solved this problem by looking at the degree of the edges in a graph (where edges represented handshakes and nodes represented people). Mr. Teacher and I walked around the room and talked with students about their solutions.
Unfortunately, almost all of the students thought about the problem that they had previously done, where everyone goes down a line shaking hands, meaning their solution was 90 instead of 45. As I walked around, I noticed that most students did not care about this problem, did not listen when asked to work on the problem, and instead talked about random events and gossip with their groups. This class seemed to be the least efficient of all the classes I have observed. Students wee distracted and didn’t solve the problems with extra time, and had difficulty explaining how to generalize their solutions.
I’m curious if the class would have gone over better if the idea of graphs were discussed or if they were given a smaller number (say, 3, 4, or 5) to think about and work through completely. I think that this is one instance in which the method of introducing the problem before the tools was ineffective in the classroom. This makes me wonder if there is something fundamentally different about this, perhaps related to the fact that students were given a very open-ended problem with little instruction or direction.
Overall, I have enjoyed working with the students and Mr. Teacher, and I look forward to continuing my math/science education licensure.