Starting with the rope itself and regular polygons, you could have a conversation about discrete and continuous data then get into curricular competencies like mathematical argumentation.
Students could use a wall or other long, straight line of indeterminate length as one side of the triangle. Another side would be created by the rope at a fixed angle with the wall (I used 30 degrees in my pictures, but it could be something else or even random). The remainder of the rope forms the final side. The line of questioning is similar to the regular polygons. How many triangles can be made using whole numbers for the rope sides? How do you know when you have found them all? Once they have discovered there are two possible triangles for some of the lengths, Can you find the rules for when there are one, two, or no possible triangles?
Reflection: Vogelstein, Brady and Hall (2019) Reenacting mathematical concepts in large-scale dance performance
I was struck by the imagination and creativity that went into many of the mathematical performances we viewed this week. From the slightly slapstick, combinatorics inspired dance of Shaffer and Stern to the intricate and mesmerizing longsword dances. The spatial reasoning required to plan moves that result in the correct overlaps and geometric shapes is really impressive. I had some difficulty with the Kieth Terry rhythm videos as I experience some sensory overload from repetitive tapping noises and cluttered environments. Reason 192,765 that I could not be an elementary teacher and have the utmost respect for them.
I honestly believe that everyone is creative and imaginative, but not always in the same way. Dance is not an area where I feel confident or talented so my mathematical creativity does not come out in this way. I would struggle to make connections the way Shaffer, Stern, and Terry do, but I can certainly see and appreciate them.
That is why I appreciated the approach of Vogelstein, Brady, and Hall. They found a meeting place between culture, embodiment, and math through detailed examinations of large, choreographed performances. This type of activity would be less of a stretch for those of us who would not feel comfortable trying to create our own dances connected to mathematics. The study focused on ensemble learning defined as “fundamentally collective and performative, where learners recognize the need to act together”(p. 332) and utilized performances from the opening ceremonies of the 2016 Olympic games in Rio. Quartets were given the task of watching and reenacting a video clip from the performance, exploring the mathematics involved (geometry and transformations), then creating their own performance. The activity required very high levels of problem-solving, communication, and true collaboration since the participants had to dissect then physically recreate the synchronized moves involving a large prop. An activity like this would be an excellent way to address some of the Core Competencies from the BC curriculum!
Patterns, combinatorics, and geometry have emerged as common concepts that lend themselves to artistic and embodied learning. This has me wondering if they can be integrated with all mathematical concepts. Should we be finding ways to incorporate these “new to us” ways of learning with as many concepts as possible? Is it fine, or even desirable, to simply make the connections that feel most natural or obvious and leave other teaching strategies for the places they fit? For example, matching embodiment and arts-based activities with geometry and social justice with financial literacy.
Hi Danielle,
ReplyDeleteI appreciated your ingenuity in adapting the rope polygon activity. As I read through your post I was thinking about the challenges that come with group work. How much learning potential is lost or gained by highly collaborative work?
I find there is a lot of work I need to do to help develop an environment where students will work well with each other. I can only imagine if the task is not something students feel comfortable in trying. I would hope that selecting approaches that feel natural would make it easier for students to see the purpose in the activity.
I also was thinking about the challenges some students face with overwhelming sensory input. During the activity I tried with my students this week, one student was not willing to try it until the rest of the class went inside. He was quite overwhelmed by all the moving around. It made me wonder if the range of student engagement may not only be a willingness to participate, but it may an indication that there are some barriers for some students. This opens up the question of what accommodations/differentiation should be considered for these activities?
You make good points about the challenges of group work and need for accommodations. I think collaborative activities might be a little easier in the older grades? I have a clear expectation that students will work with anyone else in the class. My "Beaker of Destiny" often chooses groups when I do collaborative work, but I do let them choose their own groups sometimes.
DeleteYou are right, sometimes the core competencies being learned through group work can overshadow the curricular content. There are students who have difficulty navigating the social dynamics that come with group activities and work better on their own (I always did). I also find excessive group work can contribute to some students relying on peers and not having confidence in their own abilities. I guess what I am trying to say is that I think group activities and independent work are both important, and students comfortable with one or the other need the opportunity to stretch and grow by participating in activities that are not preferred, with support if needed. I am not above making sure the kid with anxiety is "randomly" selected to be in a group with someone they trust.
Hey Lana and Danielle
DeleteYou have touched on some important pieces here. There is so much that goes into making collaborative work beneficial - it is certainly not plug-and-play, especially in a subject like math where collaboration is not normalized. It's a tricky situation because some students might need a partner or group to get started (task initiation is difficult for my designated students), but as you mention Danielle, students could end up relying on each other. What works best for one class or student might not work for the next class or student (or even for the same class the next day), and that's okay. Accessible entry points are key here - we want to push our students to try new things, while simultaneously creating that comfortable environment Lana described.
As I'm going through these courses in these programs, I think of all the strategies as tools in a toolkit. If I'm building a house, I'm not going to use every tool at the same time - I'm going to pick out the tool(s) suited for that particular day/project, and use the rest for a different project. At certain stages, some tools are more helpful than others.
Hi Danielle
ReplyDeleteI appreciated your question at the end about natural connections. I'm often thinking about authenticity. I think our students are smart enough to notice when a connection feels forced instead of natural, and it might bring up those what's-the-point why-are-we-doing this questions, particularly when it takes less time and energy to do a worksheet. Some of our students even prefer pencil and paper algorithmic tasks, so getting them to buy into something new means we have to be genuinely on board ourselves. Your example is great. I don't think dance is necessarily the best fit for financial literacy in grades 10-12. Surely there is a balance between searching for new connections that we may not have known existed, while simultaneously ensuring the connections we do choose are authentic to ourselves. How do we know when to push ourselves and when to 'sit back'? This goes back to not throwing the baby out with the bath water. Ultimately, we have teacher autonomy for a reason. We can choose what works best for us when it works best for us and leave the rest for another time.
That is my conclusion as well. To continue with your toolbox visual, you do not use the whole toolbox for every single concept. You use the ones that make most sense in a given situation, but we should also continue accumulating new tools and finding appropriate places for them to be used.
DeleteHi Danielle
ReplyDeleteI agree that everyone is creative and imaginative -and in different ways! My mom always quotes her dad throughout my life, "it takes all kinds to make the world go 'round!" I feel that quote when it comes to our educational learning, some of us learn abstractly and emotionally through examples and 'for instance's, others of us learn literally and concretely with proofs and examples. You stated, "...found a meeting place between culture, embodiment, and math through detailed examinations." Isn't this the truth?! We have had many different types of courses through this Program and each one has made us focus on ɑ different perspective to help encourage our students to see and learn mathematics in every aspect of their lives, not just through symbols and equations written on ɑ page.
It seems that video games and tiktok is the standard practice for students right now, maybe this idea of dancing mathematics is ɑ great segue-way for them to experience mathematics on ɑ different level; "what can you see in this guys great tiktok that would explain the angles etc in his dance?" "how could you improve your shot in this game, what angle could you utilize to improve the trajectory of it hitting the target?" etc.
I found myself wondering along with you while reading your last paragraph, "Patterns, combinatorics, and geometry have emerged as common concepts that lend themselves to artistic and embodied learning...wondering if they can be integrated with all mathematical concepts." To which I say "why not?" Cynthia continues to tell us mathematics is all around us!
My grandfather used to say the same thing!
DeleteI appreciated your connection to the different course themes we have encountered during this program. That is what I had on my mind when asking some of the questions at the end of my post. We have all of these useful and interesting lenses to help us see the math surrounding us, it is just a matter of discerning when to use each of them!