I chose to work with Sarah Chase’s “3 against 2” phasing activity this week; I found her video beautiful, simple and complex all at once. I understood how 2x3 takes 6 movements before repeating, but I wanted to take a closer look at even numbers or multiples to better see the patterns there. I did 2 against 2, and then tried 4 against 8. The pattern was too big for me to master, even trying the word trick that Sarah mentioned, so I had the idea of making it into a team activity. One person (my daughter Kaitlyn) moves 4, a second person (me) moves 8, a third person (my daughter Ella) provides the beat to move to, and a fourth (my husband Ryan) could record the number of movements it takes until realigning/repeating. (This could extend to more people joining the phase as well, for example looking at 2 against 3 against 5.)
Right away, my brain went to looking for patterns with lowest common multiples and factors. When I mapped the process out on a big white board, I could see that the reset was happening at the lowest common multiple.
I think phasing like this could also work well when learning how to read an analog clock. One student could walk 60 steps in a circle (second hand) and when they get back to the top a second student can take one step forward (minute hand). In this way the phasing becomes, like Sarah says in the video, an abacus that marks and counts tracking relationships. Similarly, one could also use it to look at regrouping and place value with base 10 and other bases.
Drawing on my reading from Deitiker (2015) this week, using this embodied practice could invite students to explore the process of mathematical relationships first and formalize structured definitions/equations later. I have tried to pose a lesson sketch that would invite students into the story of this experience.
- When can the friends 3 and 2 meet up? Explain, and have students demonstrate, the phasing activity of 3 against 2 to see how long until the friends can get together.
- Group students (2 actors, a beat keeper and a recorder) to introduce new friendships. Students can take on the role of acting the number character and test friendships like 3 and 4, 4 and 5, 4 and 8, and 2 and 6. Have groups predict when the friends will meet and design a notation to record their experiences - as Antonsen (2016) said in his video during week one, it is OK for students to make their own notations and definitions while making meaning.
- Have groups analyze their notations to look for trends and make a conjecture to try out. Can they find a counterexample for their conjecture? For another group’s conjecture?
- Share conjectures in a whole class discussion and when it is realized that get togethers happen at the first shared number, introduce the formalized term and a mini demo on lowest common multiple.
- Choose your own adventure ending - groups can choose to try using new language and notation for LCM to predict outcomes for when three friends can get together (ex: 2 and 3 and 5) or identify which mystery friends might be meeting based on a set of clues like: they meet on beat 24, and one other time before 30, then justify their choices using the new language.
Hopefully, by grounding in a shared physical experience this way, learners will have a place for the abstractions to land and view mathematics as a way of describing patterns in the world rather than as a set of disconnected procedures. It flips the lens from starting with “this is a least common multiple and here is how we find it” to “wait…why did that take 6 beats? Will the same pattern happen in this context?” I know the second option would engage me more in the story.
References:
Antonsen, R. (2016, December 13). Math is the hidden secret to understanding the world [Video]. TED. YouTube. https://www.youtube.com/watch?v=ZQElzjCsl9o
Chase, S. (2016, November 10). Dancing combinatorics, phases and tides [Video]. Vimeo. https://vimeo.com/251883173
Dietiker, L. (2015). What mathematics education can learn from art: The importance of considering form and experience. Educational Studies in Mathematics, 89(1), 27–44. https://doi.org/10.1007/s10649-015-9592-3
Nichola, I love the notation you used on the whiteboard. I was very clear to see the set of patterns prior to the repeat. The lesson sketch of discovering the idea of LCM is also so valuable especially when used to describe a natural occurrence of the real world: in general, I need to do this AS DEFAULT FRAMEWORK in math class (self-reminder). Start with a discoverable concept, explore, conjecture, then connect to standardized terminology, rather than the reverse in which there’s no intrigue. Thank you for the reminder of this framework.
ReplyDeleteTwo wonders:
1. How can we encourage our high school students to create a more visceral movement for this activity involving the spine?
2. How can we bring authentic ways of movement to introduce this instead of “make a 3 movement sequence” in which they would be wondering “why?” and get suspicious? Here’s an idea, but isn’t as full-body focused as I like… nor does it promote a student to keep moving: The librarian scenario - Get one student to (1) grab a textbook from a pile (2) open to cover (3) push away, while the other student (1) inks a stamp, (2) stamps the book. Something like that.
I was thinking a factory worker scenario that involved a procedural construction / packaging of something that needed to go on certain beats regularly but maintaining different roles and speed. Anyhow, I need to keep thinking on this one!
Thanks for the thoughts! 😊
I like your idea of telling a story within the movement pattern. It makes it more authentic, and easier to remember more complex patterns I think :)
ReplyDeleteNichola, I really liked how you took this concept and spread it across multiple people. Because I too, struggled to understand how someone can remember all these patters with such high numbers... EXTREMELY impressive.
ReplyDeleteOlly, I love you extension here with adding "real life" movements to the sequences!
Fascinating post and conversation, everyone! Nichola, I love your notation and the way you've integrated it with the activity -- a great model of how to connect our students' learning through embodied activities AND notation! And I love the idea of working with four or five people as a team, because it really does take several people to do the movement, keep time, and keep track as the numbers get larger, as Taylor says. The clock idea is a very cool one -- and so is Oliver's idea about the old fashioned librarian (or other) assembly line process! I also really like Oliver's suggestion about finding activities that move the spine as well as locomote around the space. Beautiful! I'm interested to see where this all takes you in your teaching too.
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