By Cory Saunders, 2016 Spring student at BSM
On Wednesday afternoons, my classmate Erin and I walked ten minutes to Derkovits School in Budapest, Hungary to host a math club for seventh graders. With the guidance of workshops led by Réka Szász and other collaborative resources, we created and led the activities for 4-8 students each week. In this reflection, I have compiled my observations into “lessons” I learned about my teaching.
Lesson One: Strategies for working through a language barrier.
The seventh graders in our group had a particularly strong grasp of English. I was impressed when I was explaining the rules of SET which involves several different adjectives and a concept of “same” and “different.” I was speaking at a normal pace and many of the students showed their understanding by playing correctly.
It was common for some students not to catch all of our explanations. They would explain to their classmates in Hungarian. I appreciated that this extra explaining was not seen as shameful or embarrassing, but rather as a natural and helpful gesture. Erin and I would always be open to explaining again. I noticed that we would rely on the students with the most confident grasp in English to guide the others. I think that this limited the interaction between the quieter students but I hope we provided ways for them to participate verbally or nonverbally.
I was also impressed with how well the students spoke English to us. I often forgot that they were English language learners. The students learn English at an early age and so they are fairly fluent by seventh grade. In that respect, it did not seem like there was much of a communication problem but I still outline some strategies that helped us.
Take an opportunity to practice Hungarian
Erin was much more comfortable with speaking Hungarian than I was. During name games, she would offer to speak in Hungarian so as to model her own language learning to the group. I was much more timid doing this, but I think my fear for looking silly or stupid may be what some students experience trying to speak English to native English speakers. It reminds students that we are much worse at Hungarian than they are at English and perhaps gives them confidence. With the mix of English and Hungarian during quick number games, we often spiraled into collective laughter at the confusion.
Not everything needs to be in English
Oftentimes, a student would show most of their understanding through hand gestures or partial English and we’d ask them to explain to their classmates. For some students, it is a lot of effort to think of a full, fleshed-out explanation in English. Therefore, Erin and I usually offered the option, “You can say it in Hungarian,” to help speed up the process. Sometimes it was less important for us to completely understand their explanation than for them to share their mathematics with each other.
Lesson Two: Mathematical games are great for differentiation.
Réka hosted a wonderful workshop on using mathematical games in the classroom. Often these games were in pairs. The game would be explained and often there would be a condition where one player would win at the end. Then we asked what the “winning strategy” was and who should go first to guarantee this condition. I love these games because as the student figures out the best way to win, they are using math to solve it, often without knowing it!
We played the two-player chip game where students are given 18 chips and students take turns taking 1 to 5 chips. The person who takes the last chip wins. Our students struggled with this and even I had a difficult time keeping track of who had gone first and who would end up being the winner. We had four pairs that day that were going at different paces. I noticed that for students who struggled, it helped to suggestively group the chips in groups of six and then let them discover why that format was useful. For students who had finished early, more questions could be asked, such as changing the initial number of chips, the number of chips that were allowed to be taken, or the winning condition.
In terms of a collaborative game, Erin and I also introduced the students to the Tower of Hanoi problem by using forints as the various-sized disks. For some students, figuring out the minimal number of moves is a wonderful challenge. Others who were struggling were given the minimal number of moves and then tried to see if they could achieve it. Students who finish quickly and want to learn more can easily experiment with four coins instead of three. Erin showed at the end how through recursion, we could actually generalize how many moves we would need in general. We’re not sure how much of that was understood by the students but giving students exposure to the interesting mathematics we’re learning can spark an interest for some students.
Competitive and collaborative
It’s also an interesting dynamic because although we are using the words “game” and “win,” the competitive environment leads to a collaborative environment where both players are trying to figure out the system. Often students of mathematics can be pitted against each other in terms of speed, skill, or creativity — and so the “loser” is actually a “loser.” The type of mathematical games we learned in the workshop are an opportunity for students to realize that “winning” is not really “winning,” but learning about the system is the true end goal.
Lesson Three: Manipulatives and role-playing guide understanding.
In connection to the first and second lessons, having tangible manipulatives is extremely helpful. We found this useful with the mathematical games (Lesson Two) but also later on when we started bringing logic puzzles to the group. For the various riddles involving weights, we would have chips represent the weights. At one point, I even acted as the scale and the students would test out their hypothesis by weighing the weights on the “scale” (my two hands). I just had to designate secretly which weight was the special one and then act accordingly. It also helped that Erin had made a decision tree for them on the chalkboard to organize their actions!
Sometimes preparing the manipulatives required extra time. Erin and I brought in a riddle related to data encryption which involved two people trying to communicate by sending a message through boxes with locks. I made the “box,” “messages,” “locks,” and “keys” for the students and they sent the box across the “ocean” (table). It was helpful with keeping track of which pieces were where. After they had figured out a solution, it was just a matter of refining, replaying, and recording their solution.
At other times, the role-playing was impromptu. Erin and I brought the flashlight bridge problem to the students. It turned out that the number of people in the puzzle matched the number of people we had, so we asked people to stand up and play out their proposed solution. We used a piece of chalk as the “flashlight” and I kept track of the time by holding up fingers on my hand. The students really enjoyed it and I like the idea of dispelling the idea that math only exists on pencil and paper while sitting still in a chair.
Lesson Four: Great planning leads to a great session.
Erin and I got together on Tuesday mornings for half an hour to plan our sessions. We greatly appreciated the suggestions from Réka through the activity log. We planned a warm-up which often was not math-intensive such as Zip, Zop, Zap or Buzz, Buzz. This broke the ice and let students get up and move around to get their energy going. Then we’d plan one or two major activities, such as a game or a set of riddles. We estimated about how much time it took, often overestimating to make sure we had enough time. Lastly, we had a filler activity or a few back-up extension activities. For us, this was the game SET, which students loved playing at the end. Often SET would be a reward for finishing an activity or an incentive. Since SET can be stopped at almost any point, requires little setup, and can be cleaned up easily, it was the perfect “filler” activity.
Along with planning the activities, Erin and I assigned each other to tasks such as “Pick up the colored pencils” or “Choose and print out riddles.” Great communication and fair division of labor made our sessions run smoothly.
The Derkovits internship was a low-stress environment to interact with Hungarian students and share math. We loved to ask the students questions about their lives and classes. There was plenty of autonomy for us to share the activities we wanted but also plenty of support from Réka and our peers who were hosting clubs on other days. Most importantly, Erin and I looked forward to our time at Derkovits School every week. I’ll never forget the excitement of the students speaking rapidly in Hungarian about a puzzle we had presented, bouncing ideas off each other, and feeding off each other’s energy! I remember thinking, “This is exactly what I want my future classroom to look like” as the students were shouting a phrase to each other. I later found out that it meant “I’ve got it! I’ve got it!”