"Music is a pattern of sound waves that produces feelings of sensory beauty." Francis Su, Mathematics for Human Flourishing, p. 71
My connections and reflections land in two themes this week: representing mathematical ideas in more than one sensory mode and engaging in embodied tasks to understand. I also recalled an exploration I did in EDCP 550, Week 4 – Pattern, Beauty, and Truth.
Two Senses are Better Than One?
The Vi Hart videos and the article from the 2019 Bridges conference, Make Music Visible, Play Mathematics (Capozucca & Fermani, 2019) are exemplars of using more than one sensory mode connected to mathematics, all cases relating visuals, sound, and mathematics. Vi Hart’s videos (Mobius Music Box, Sound Braid, and Doodle Music) combine something to watch, such as the mobius strip moving through the music box or the animated doodles, with music. The music for the Mobius Music Box helped amplify the visual of the strip coming back to the beginning but how it keeps going with the music being played ‘backwards’ and then reverting to the original tune when the mobius strip has returned to the beginning. The videos with the doodling provide a sound to give another sensation of the visual pen strokes which are then combined into patterns (the braid or other doodles). The workshop activities designed by Capozucca and Fermani required participants to engage representing sounds visually and mathematically, mapping the notes on the chromatic circle and joining notes of chords in triangles. I would argue that all involved some sort of physical movement as well: the participants in the workshops physically produced notes with BoomWhackers and other instruments; Vi Hart turned the music box which in turn moved the Mobius strip; Vi Hart’s hands were moving to draw the doodles; and the animation of the doodles provided a sense of movement that the viewer could experience in their mind.
This made me think of the power of multiple representations and both Francis Su’s and Roger Antonsen’s ideas of really understanding by taking different perspectives (Su, 2021; Antonsen, ***) that I spoke about in the Week 1 post. Experiencing mathematical ideas with various senses gives the opportunity to take a different perspective with each one. Additionally, exploring musical intervals and chords mathematically offers a new perspective to deepen musical understanding. Capozucca and Fermani (2019), commenting on their workshop, note that they feel that both math and music benefit from being explored together. When I was working on replicating the art work with tessellating trapezoids in quilting, I felt like I was problem-solving not just about tessellations and trapezoids but also in quilting, deepening my understanding of both the mathematical and handicraft endeavours. This all makes me understand that not only does having our students explore math through art, music, crafts, etc. deepen mathematical understanding, it reciprocates to the other discipline, too.
Involvement for Learning
Ah yes, Benjamin Franklin’s famous line, “Tell me and I forget. Teach me and I remember. Involve me and I learn.” Cliché as it may seem, this was really evident to me this week in the art exploration and in how I approached the reading the Capozucca and Fermani (2019) article. In looking at Trapezia Pastiche by Margaret Kepner and reading her description of the piece, I felt I understood the mathematics. However, when I went to recreate it in fabric and consider quilting the small patterns of tiled trapezoids I realized I needed to spend more time to fully appreciate and comprehend the mathematics. I needed to play with sketching the tiling patterns to distinguish one from another and to figure out what she wrote about the patterns being of “primitive-cell size, fault lines, vertex configurations, and symmetry types” (Kepner, 2019). As I write this reflection, I am not quite done that exploration and am still coming to figure out each example. In order to quilt the patterns onto the 16 trapezoids that form the larger trapezoid, I really needed to dig deeply into how the patterns were created so I could replicate them; the need to quilt forced a deeper exploration.
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| Trapezia Pastishe, Margaret Kepner http://gallery.bridgesmathart.org/exhibitions/2013-bridges-conference/renpek1010 |
Reaching Back One Year
An exploration I did a year ago in Week 4 of Cynthia’s EDCP 550 course with Mathematics for Human Flourishing by Francis Su was to explore patterns through 12-tone music. 12-tone music is a 20th century genre of music that has a number of rules about the use of notes and what you can do with a string of 12 notes (and, of course, Vi Hart has a video about this kind of music!). I explored the prime (original string), the inversion of the prime (flipping it on the music staff, so to speak), the retrograde (the prime backwards), and the inversion of the retrograde to compose a short piece of music. I remember the struggle that came in figuring out inversions but that I felt that I really began to understand the idea of inversions and retrogrades of patterns by playing with them and sound. Here is the video of my composition in case you weren’t in my discussion group (I do not profess to any real musical integrity – it was just playing around for pattern exploration):
Sandra! I can’t wait to see what your finished trapezoid quilting project looks like! And I’m about to walk over to our piano and try out your chords you drew on the chromatic circle! *Plays the chords* To me, the only chord that *feels* equilateral is the C-major. Why? I don’t know. It just feels that way – that it forms a scalene triangle on the circle seems just wrong haha!
ReplyDeleteYour Questions to Ponder:
I like your first question – and I’m going to pause at the “as many ways of embodiment as we can” part of it. At first, I thought, “Yes! Totally! 100% we should!” but then I thought about it a bit. I think we totally should BUT not all at once, not at the risk of jumbling up our students understanding – I know I’ve definitely run into moments of, “wait, I’ve said too much, now you’re looking at me with the blank eyes of checked out children”. Hmm, is it possible to only use one sense or type of embodiment in an activity, or is it always a combination of a two or more? Probably most of us are always using sight and often hearing (with the exception of people with vision and/or hearing impairments), there’s generally some fine motor, even with art, and sometimes gross motor depending on what’s occurring, so I guess we’re most often dealing with multiple senses… but maybe the magic comes when everything is tied in together with an activity that’s very engaging/exciting, at least a bit challenging, and memorable, like you said. I found this week’s activity extremely engaging! I’m just waiting for some paint to dry so I can finish my piece. It was very engaging and challenging, but fun enough that I tried 5 different designs by choice, when there were definitely other things I could have been doing. I think THAT pull categorizes an activity as successful, especially for students.
Interesting question about separate disciplines and reporting on these disciplines – I wonder if my school may be a bit of an anomaly? Our English and Socials teacher routinely commented only on English or Socials as she blended her courses through the year. Her English comments often stated “Please see Social Studies comment”. Hmm! I’m interested to see what you and Natalie write in response to this question since I only teach Math and Science for Grades 8 and 9, and only Math 7. I definitely am in contact with the students’ other teachers every day, but we never touch base about reporting on concepts other than those falling in our subjects. I suppose:
A) I'm not working as cross-disciplinary as I should be, and
B) I didn't even think to report on anything other than the competencies and concepts that are required for my subjects.
After reading Cassie's comments about the "feel" of the chords, I opted to played the chords, too. Here are my 'felt findings': equilateral: C major and *C augmented
ReplyDeleteisosceles: C minor
scalene: C diminished
(*same as the findings in the article)
With the range of student learning styles, strengths, needs, experiences, and background knowledge, I'm an advocate of offering as many opportunities as possible for students to make connections and access curriculum- as long as the strategies are not too contrived or convoluted. I try to look for natural fits and extensions that bolster the intent of the lesson. Trying to force connections may result in cognitive overload or students focusing on the logistics of performing all of the parts rather than the intended learning goal. Time needs to be provided for processing, practice, and reflection. Based on my professional observation, embodied learning is particularly helpful for students who are ELL, need more time to process information, have attention issues, or struggle with written output.
ReplyDeleteThis year, I was learned that it is helpful for both students and parents to offer explicit connections between big ideas, content and competency objectives and cross-curricular tasks. I explain the task and select the specific objectives from a range of subject areas to display on my FreshGrade posts. When I need to report on discrete subjects, I focus on the aspects of the thematic learning or cross-curricular task that is relevant to each discipline. For example, we recently completed a school challenge wherein each student created a self-portrait out of materials from home that they recycled or repurposed. My students brainstormed ideas and completed a planning sheet (Language Arts) prior to the ADST activity and then shared a verbal reflection and wrote a procedural text after the activity (Language Arts).
Wonderful post and commentary -- thank you everyone! I'm really interested in the multisensory and crosscurricular discussion. My thoughts on this: I encourage everyone to try to bring in sensory modalities that have typically been ignored in mathematical representations; and that includes aspects of vision (like colour and pattern) and hearing (like music and poetry), as well as small and large scale aspects of tactility and movement, and maybe even smell and taste. But you wouldn't want to pile on a hundred sensory representations all at once, for no real reason...we need to design lessons and learning units to direct students' attention, engage them so that they are really involved in exploration, and move between sensory experiences and recording, notating, sometimes calculating, definitely reasoning and for sure inquiring into aspects of mathematical ideas and patterns!
ReplyDeleteIn early October, my students created Gratitude Math Poems inspired by Becky Shillington's Appreciation Equations included in Thanku: Poems of Gratitude by Miranda Paul.
ReplyDelete