Week 5 - Critical Reflection

     I feel like I am rushing to gather my thoughts because I went down so many exploratory paths this week – pursuing more of the tessellating trapezoids from last week (see my Scratch coding addition), thinking I would pursue more of Sarah Chase’s movements than the New Teachers’ Riff on Kandinsky in Binary but playing with that as well, and getting caught up in the possibilities of visualizing with technology.  I hope my ideas come together here.

Scratch cat tessellating trapezoids

    First off, in Susan’s introductory video this week she mentioned that mathematics is part of world heritage and that we owe it to students to bring this heritage into our classrooms. I see this as meaning that we do need to offer ways to explore mathematics that are authentic to the way math is used and engaged with across human endeavours. The arts are a heritage, too, and in our tasks so far in this course, we have seen that math and the arts are linked. We can/should bring heritages together in our classroom because they work together outside our classrooms. Part of the interconnectedness is the sense of play that exploring mathematics through arts brings and I think that is why I keep jumping down twisty-turny rabbit holes – because it is just so much FUN! I get myself into “flow” states and suddenly 2 hours or more have gone by or it is 1:30 in the morning!

Q: How would you manage students that might go down rabbit holes and not pursue the intended outcomes? Does this need managing? Why or why not?

    As part of the data analysis in the EASY Minds study, the authors found that the bit of criticism a few students had toward the program was that sometimes they got frustrated with the physical activity and they commented that this frustration interfered with them learning the mathematics (Riley et al, 2017). Upon reflection, I experienced this, too, in my attempts at the activities, both with the physical coordination and with technology. This makes me think about implementation with students. The EASY Minds study wasn’t looking to refine physical literacy but just to get students active more throughout the day, specifically in math lessons. So, I think it might be important to 1) be sure the focus is on just moving, not refining physical skills and 2) to introduce some of the physical skills needed for an embodied math task separately from this task so students develop competence and confidence in that skill. This involves some planning ahead and predicting what could get in the way of success.  The same would go for using technology to create visual representations; the students might be more successful connecting with and deepening understanding of the mathematics if they already know how to use the app or program or coding platform that will be used for exploration.

Q: How can we mitigate frustration with the embodied, outdoors, or art aspects of our integrated mathematical activities so students can engage deeply with the mathematical concepts?

I don't think 6 and 8 are wrong

When Sarah Chase talked about doing the dance patterns with other people and that they might choose the “wrong” numbers, meaning pairs or groups of numbers that might have a lowest common multiple smaller than their product, I will admit my back went up a bit. In thinking about why, I came around to there being interesting discoveries with what she calls “wrong” numbers and just because her intended outcome is to have a long sequence of movements before they repeat, it does not mean that other people cannot have other outcomes, intended or not. “Happy accidents” can lead to interesting discoveries. I saw the “wrong” numbers as an opportunity to explore multiples and lowest common multiples (which Cassie mentioned in a text chat she is immersed in with her students working with fractions) and that is
what my movements and notations explored. The “wrong” numbers could also lead to an interesting discussion about number choice and how it effects the length of a dance sequence that is choreographed in this way that Chase uses.

Q: How can we as teachers ensure that we are open to ‘happy accidents’, to explore these with students?

iPad piano is providing fun!

If you look back to Week 4 posts and the first image in this post, you’ll notice played with coding tessellating trapezoids on Scratch. I also played with PowerPoint to work with circles and base 5 in the Kandinsky binary activity and to try some illustrating/animating combinatorics of 5, 3, and 2 with clothing. My iPad piano came out this week, too. Working with technology allowed me to make more observations about the mathematics and how it worked. Are coding and working with technology to create visuals embodied activities? I feel like they are – I am doing something to create a visual representation so my sense of sight is used to understand the mathematics. Also, when I am working to figure out how to represent ideas in code or to make them animated on sequential slides, I am doing a heck of a lot of visualization and manipulation in my head. This reminds me of the 2013 Fernandes and Healy paper I read about students who are blind and their experiences working with symmetry; I understood from this paper that manipulating physical representations in the mind is an act of embodiment where the physical and abstract are working together.

Q: What do you think of the idea that work on a screen with coding or other applications being representative of embodied learning?

Week 5 - Activity - Piano, Movement, Sketching, Curriculum Idea

    I couldn’t resist playing around with both activities but settled on Sarah Chase’s movements to do the required activities.

    But first – let me quickly tell you how I explored the New Teachers’ Riff on Kandinsky in Binary…

    I jumped right into extending the activities in the video. Ali commented that he wasn’t much of a musician but still came up with a way using 3 notes to add sound to the animation. I wondered if there was a way to play around with some notes and a different base. I chose Base 5 because I could work with a pentatonic scale – one without semitones that is supposed to eliminate discordant sounds but the one I chose has a major 2^nd in it (C D E G A) which I find discordant. Anyway, I did play around with circles and colours on PowerPoint slides but I didn’t save my work and it disappeared…not even a “do you want to recover this file?” notification next time I went into PPT. Sigh. What I did on my iPad keyboard was zoom out so I could get 2 full octaves and another 6^th. I set C as 0, D as 1, E as 2, G as 3, and A as 4. I used the upper most octave as the ones place, the middle octave (starting at middle C) as the 5s place, and the lower octave as the 25s place (is my elementary teacher terminology making people cringe?). I assumed a 0 in the 5s and 25s from the beginning – I wrestled with that and went with that...having a sound for 0. I played up to (255)₅ .  Here is the result with a few slips (wish I had a real piano to play on):

    I had to pause that to go do life things and came back to find my PowerPoint gone so didn’t pursue this further.  I wanted to put the music to the PowerPoint slides of coloured concentric circles that were like Ali and Colin’s template.

    So I moved onto Sarah Chases’ movement.

    I have a physical memory of her 2 and 3 movement – my jazz dance teacher when I was a teenager used to make us do that as part of our warm up. It was interesting to see it used it for combinatorics.

    I did 3 and 4 and 4 and 5.  My introvert didn’t want to be filmed or photographed today so after doing some choreography in my living room (many tries to get through the 4 and 5 but practicing each separately helped), I notated them on paper.

 

Notation of movements for 3 and 4 - common multiples circled

Notation of movements for 4 and 5

I thought this is another way of extending the activity – 3D movement into a 2D representation.

    I took exception to Chase saying that numbers like 6 and 8 together would be wrong but she might let people try it and see that they were wrong. It seemed they were “wrong” because they have a common multiple that is less than their product. I’ll expand on this more in my reflection.

I wondered if 6 and 8 could be notated in a circle some how to find common multiples, lowest or not.  This is what I came up with:

 

The blue lines divide the circle in 8ths and the orange lines 6ths. Then I used 2 colours (green and pink) to ‘count’ around the circle, noting when I wrote the same number on the last 6^th and 8^th number which would be a common multiple – they would start again together on the vertical radius.

I also tried some combinatorics on PowerPoint – 5 shirts, 3 pants, 2 hats. I’m having trouble recording it as an animation.  I might come back to it…I remembered to save!

 

Curriculum Idea Sketch

Issues – common multiples, combinatorics

Guiding Questions – How can you use body movements representing two (or more numbers) to understand how their cycles coincide? How can you notate this? How does this relate to common multiples? How is using strictly prime numbers different than including composite numbers? What kind of problems can we solve with these movements?

The Story – watch part of Sarah Chase’s video (opening and 2 and 3 movements); students try 2 and 3 and then explore other combinations of two or more numbers; explore ways of notating; how can pare down (e.g., write down only the numbers that end the cycle for each number – leads to listing multiples and finding least common); apply to combinatorics – e.g., clothing combinations, lunch combinations, etc;

Integrating embodied learning and other learning – move from 3D to 2D with notation, problem solving

Possible extensions - find another way of notating (e.g, on circle); coding, shuffling on PPT slides, stop motion videos

Week 5 - Reading Summary

 Riley, N. et al. (2017). Movement-based mathematics: Enjoyment and engagement without compromising learning through the EASY Minds Program. EURASIA Journal of Mathematics Science and Technology Education. https://doi.org/10.12973/eurasia.2017.00690a

     This article is part of a larger study on the effectiveness of the Encouraging Activity to Stimulate Young Minds (EASY Minds) program in New South Wales, Australia. The research presented here focuses on the teacher and students perceptions of the program after a 6-week experience. Interviews were conducted with 66 upper primary (elementary) students and 4 teachers.

    The EASY Minds program was designed with two purposes: to increase physical activity through the school day with a specific focus on mathematics lessons and to increase student engagement with mathematics tasks in primary school. Prior to the study, the program underwent a successful efficacy trial. The study upon which this article is based was done in a randomized controlled trial. Teachers that were to implement the program received one day of professional development prior to implementing lessons. The activities in the program are designed to use physical activity to promote fluency of number operations and to use places in and around the school to look for mathematics. The program is based on a pedagogical model from the New South Wales Quality Teaching model in which lessons entail intellectual quality (IQ), teachers provide a quality learning environment (QLE), and have meaningful lessons for students or “generate significance (SIG)” (p. 1656).

    The larger study included 8 public schools in New South Wales, Australia using grade 5/6 students in intervention (6) and control (4) groups. In addition to the professional development mentioned above, teachers received a packet of resources and equipment to support lessons. Data was collected using a focus group methodology for the 66 students and interviews for the four teachers and both types of events were digitally recorded and transcribed. An inductive qualitative analysis approach was used by a researcher independent of the program and data collection.

    The analysis identified themes to student and teacher perceptions of the EASY Minds program. Both groups indicated that the enjoyment of mathematics and engagement had increased. Students also noted that the activities felt freeing and that the expenditure of energy helped them learn. Higher performing students noted they were able to take mathematics to new levels and students who tend to struggle found the hands-on format allowed them to understand concepts. Additionally, students noted that they felt their teacher enjoyed teaching mathematics more than before the program was implemented. Teachers observed that students were not only more engaged during mathematics lessons but also after the lessons. The focus group and teacher interview data revealed that the program met the required components of the NSW Quality Teaching Model for IQ, QLE, and SIG. Students’ statements reflected aspects of higher order thinking and deep understanding and any criticism was related to struggles with the physical activities (like hitting a target with a bean bag) distracting from mathematics content. In regard to QLE, data showed that students identified components of self-direction and social support. The lessons also supported SIG through a feeling of connectedness between the activities and the real world.

    Some specific teacher observations were noted about the movement-based lessons. Teachers recognized that the lessons took longer to prepare than other formats but they felt they might become more fluent at preparing these kinds of activities with practice. Additionally, teachers noted that they had to be “more creative and forward thinking” (p. 1666) and felt they would continue with the program after the study. The idea of extending the program to other subjects and to the whole school was communicated.

    In discussion and conclusion, the authors note that this study confirms and explains findings of the larger study of increased engagement in mathematics lessons. They connect previous studies’ findings that teacher attitude in regard to mathematics is a major predictor of students’ attitudes to how the teachers and students in this study found increased enjoyment of mathematics. The authors also note that the day of professional development contributed in part to the success of the EASY Minds program. Limitations such as cost and availability of equipment are discussed. They conclude by stating that quality teaching and learning is supported by movement-based activities being embedded across the school day.

Week 4 Follow Up - Obsessed with Tessellating Trapezoids

I continue to be fascinated with tessellating trapezoids to the point that I see them inside my eyelids when I go to bed and close my eyes!

Then, this week I did a 2 part Scratch coding workshop through the Edith Lando Virtual Learning Centre (Carol was there in part 1, too!).  The presenter showed quite a few examples of coding geometry concepts and other math concepts.

I had caffeine on "today" (really yesterday, it's 1:30am now) (which I have become so sensitive to for some reason that even decaf can give me a slight buzz) and after going for a late night walk to get steps in for the Meandering Math Teachers Walk for Joy team (Week 1 - 43rd out of 129 teams) and finishing doing my reading, I didn't go to bed but played on Scratch.

So - next rabbit hole - could I get the sprite to draw tessellating trapezoids?

Quilting the trapezoids last week helped with this a bit because I had some insight into how the thread or pen could move continuously to draw a couple trapezoids and some sketching got me to how to code 6 sides and the remaining open side would be closed with a repeat.

So here is my late night ala-Mozart creative time product:

Week 4 - Exploring Mathematical Art through Quilting

 Well, this week was a take-me-down-a-fun-and-twisty-rabbit-hole week for sure! When my creative juices are sparked, I can be like a dog with a bone, playing with the ideas and not letting go, even staying up way past my bedtime. I thought the art activity this week was going to be my one and only exploration and then I started making videos for the reading summary! Ha. But this post is about the art activity.

I was assigned Bridges 2013 (along with Joy). The piece I selected was one of two by Margaret Kepner, an independent artist based in Washington, D.C. (Here is her website: http://mekvisysuals.yolasite.com/

 

Here is the piece from the Bridges 2013 section of Mathematical Art Galleries, specifically on this page:

Trapezia Pastiche
Margaret Kepner, 2013

(click on the image for a larger one you might be able to zoom in and out of on your device to see the tessellation patterns)

This archival inkjet print is an exploration of tessellation of trapezoids/trapezium. The larger trapezoid is made of 4 coloured trapezoids which in turn are made of 4 trapezoids. Then, each trapezoid is tiled with a different pattern of trapezoids. Kepner, in the description of the art, notes that there are infinite possibilities for tiling trapezoids.

I was attracted to this piece for a few reasons. I liked that as simple as it seems on first glance, it is more complicated when you look carefully. It seems so beautiful in its pairing of the simple and the complex. The tessellation was interesting to me, too, because I really enjoy doing projects with tessellation with students. The artwork also makes me think of quilting in how it is pieced together and seemingly stitched with patterns; this is where I decided to go in exploring this piece deeper - to quilt it.

And so began the leap down the rabbit hole!

My mom is a quilter, a hobby she has taken up in retirement and even more so since she moved to a smaller town (Armstrong, BC) that has a quilting store that holds classes.  She gives me her scraps for my classroom projects so I was able to scour through my bin at work for some nice pieces to re-create this artwork in fabric.

I also called my mom for advice on how to piece together trapezoids with sewing. Over the phone it was tricky and she didn't really understand what I was getting at, so I sent her a copy of Kepner's art to see what she would come up with. I thought she might actually try sewing a few trapezoids together, but she just got as far as cutting the printout of the art into pieces.  We were definitely on different wavelengths for what I was asking for because my sister sent me this picture with "Mom says the green and blue have to be trimmed to fit":

Mom's take on re-creating Kepner's art

Whereas my fabric pieces looked like this:

Fabric pieces cut and ready for sewing

When Mom and I talked again (I had sent a picture of mine back), she said, "But you told me I could cut it up into pieces!" Yes, yes, I did. I didn't get any helpful advice for piecing the trapezoids together to form what Kepner's art looks like so I tackled it on my own.

Piecing the trapezoids together was a bit tricky because of the need to sew more than one edge together on 2 pieces at times.  It made me appreciate how many sides are being shared between trapezoids in the tiling pattern for the 16 coloured trapezoids. I was also trying to figure out how to most efficiently sew the pieces together so most of the time I was joining the trapezoids on one edge, rather than two.  I forgot to document that process in pictures - I was in the "zone" or in a state of "flow".

I layered the top piece of trapezoids with a layer of batting and another piece of fabric for the batting and outlined the 16 trapezoids in stitching.

Next came the tiling of each of the 16 trapezoids and I knew this step was not going to get finished for this post - so I will edit another day with an update (or maybe a few along the way).

I had to play with sketching the tiling patterns to understand them and to find the base pattern/root.  I have only done a few so far.

Sketching the tiling patterns

While I was sketching, I asked, "How does Kepner know that there are infinite patterns to tile trapezoids?" and as I was sketching for trapezoid B, I found an answer (see image above).

I spent some time hand-stitching the tiling for parts of trapezoids A, I and F (I didn't sketch F but after stitching A, I understood F).

Stitching trapezoid A

Closer look at the stitching on trapezoid A

Trapezoid F

Trapezoid I - triangles that need fixing for scale

There is a lot more stitching to go!

There is a lot more to do to cover the whole piece in the tiling by stitching. However, the process of just getting started really helped me explore the patterns in the art. For trapezoid A (blue), when I was sketching, I didn't at first see that adjacent "rows" were slightly different and by checking that I could repeat the row, I realized I needed to rotate two trapezoids that form a hexagon on alternate rows. By stitching the A pattern, and then exploring other patterns, I realized I could do the tiling for trapezoid F quite easily based on some of the patterns in A. When I went to stitch trapezoid I, I discovered that the tiling patterns on adjacent trapezoids on Kepner's art feed into each other and though I didn't reflect that in my piece (at least not yet - I may decide to correct it), I wouldn't have noticed that the tiling patterns join together if I hadn't been examining them so closely to figure out where to set up the stitching. I also made an assumption that for pattern I, I could fit it in the same width of "row" that I had done for A and F - and only after stitching it did I realize that my stitched trapezoids were too small and that the "row" for triangles has to be wider than the ones for a hexagon made of 2 trapezoids.

At first I worried that I had chosen a piece that was too simple and wouldn't stretch my thinking but now I continue to uncover new understandings of the tiling patterns each time I work with it. As I continue to work with this piece (I am quite determined to finish it sometime!), I want to explore more explicitly some questions Kepner asks about her patterns:

"Which ones have a primitive-cell size of 4? How many are fault-free? Hexagon-free? Are there tilings with only 1 vertex type? Any with 6? Which patterns have rotational symmetry? Mirror lines? No symmetry?"

Check back for updates!

References

Kepner, M. (2013). Margaret Kepner. Mathematical Art Galleries. http://gallery.bridgesmathart.org/exhibitions/2013-bridges-conference/renpek1010

Week 4 - Mathematics and the arts introduction - Critical Reflection

"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.

Trapezia Pastishe, Margaret Kepner
http://gallery.bridgesmathart.org/exhibitions/2013-bridges-conference/renpek1010

     

When I read the article Make Music Visible, Play Mathematics, I had an instinctive need to draw out what the authors were explaining, to go beyond their one example of fitting chords into the chromatic circle as triangles. The same instinct made me want to hear the chords and I downloaded a keyboard app onto my iPad and worked with sound. The experience of doing some of the work the participants would have done helped me understand the math and the music theory as well as the message the author’s wanted communicated, that interdisciplinary approaches benefit understanding in both disciplines. I’m glad I explored this reading in an embodied way; it was time consuming but I will not forget it!

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):

Questions to Ponder

If at all possible, should we be connecting as many ways of embodiment as we can in activities? Is there evidence that this is more powerful to learning than one mode of experiencing embodiment? Is it possible to only use one sense or type of embodiment in an activity or are there always multiple senses/embodiments, explicit or not?

How can we help students reflect on the learning they acquire in both (or more) disciplines when exploring through multiple disciplines? When we still have to report on discrete subjects, how might we make the connections between disciplines explicit for students and families?

References

Capozucca, A. & Fermani, M. (2019). Make Music Visible, Play Mathematics. Bridges 2019 Conference Proceedings. https://archive.bridgesmathart.org/2019/bridges2019-647.pdf

Kepner, M. (2013). Margaret Kepner. Mathematical Art Galleries. http://gallery.bridgesmathart.org/exhibitions/2013-bridges-conference/renpek1010

Su, Francis. (2020). Mathematics for Human Flourishing. Yale University Press.

Su, Francis (2021, October 22). Keynote presentation. Northwest Mathematics Conference, virtual.

TED. (2015). Roger Antonsen: Math is the hidden secret to understanding the world. [Video], YouTube. https://www.youtube.com/watch?v=ZQElzjCsl9o

Week 4 - Reading Summary (embodied)

 Capozucca, A. & Fermani, M. (2019). Make music visible, play mathematics. Bridges Conference Proceedings, p. 647-650.

Presenting my posts for this week in a little different order and starting with my reading summary as I am not finished playing with my art just yet!

I found that I had to do some embodied learning myself to fully understand this article, so I am offering it to you as well by embedding some videos for your aural enjoyment!

This article, stemming from Bridges 2019 conference proceedings, describes a 5 part workshop done with music and math teachers with the aim of "making maths audible as well as visible, and...use this geometrical method to foster a new approach towards music theory where to show how music could be effectively displayed" (p. 647).

I did attempt a bit of a trig review to get through the "Things We Need to Know Before We Get Started" and did some scribbling (that may or may not be correct) before deciding to trust the authors that the trig would help determine the triangles later discussed in the paper.

Here is a quick video of the musical knowledge from the "Things We Need to Know" section:

The paper describes the 5 sections of the workshop, each that had a "minds-on activity, a hands-on activity, and a consolidation" (p. 648). Not each of the 3 parts of each section was explicitly described.

I'll describe the first 3 sections with videos.

Section 1 - Do they play nice?

In the first section of the workshop, participants were each given a BoomWhacker (BW) (percussive tube played by striking it on the hand or other place on the body) corresponding to one of the notes on the chromatic scale. They moved around the room to meet with other participants and played their BoomWhackers together. In the consolidation, participants discussed which notes were pleasant when played together. These notes were described in how far apart they are on the chromatic circle.

Here are some videos that demonstrate how these intervals sound:

Plus and minus 3 - minor 3rd, major 6th

Plus and minus 4 - major 3rd, minor 6th

Plus and minus 5 - perfect 4th and perfect 5th

Plus and minus 6 - augmented 4th/diminished 5th

Section 2 - Build up the triangles

During the second section of the workshop, participants worked with the chromatic circle and lengths of straws to map out the 'pleasant' intervals and build chords (3 or more notes played together) from these. They looked at triangles formed by the notes of the chords and specifically the four triangles that were fully inscribed inside the chromatic circle. Chords and triangles were classified (triangles with the use of the Law of Sines).

Major and minor chords

Diminished and augmented chords

Section 3 - What happens if?

In this third section of the workshop, participants were asked to move or otherwise manipulate the triangles of the 4 chords to see the effects on the chords.

Rotating a triangle around the chromatic circle (major chord)

Section 4 - It's time to play

In the fourth section, participants chose songs from a list and examined them for the geometry in the chords. They also played and sang the songs together.

Section 5 - It's time to compose

In the final section of the workshop, participants used the discovered traingles/chords to make their own compositions, what the authors "a tasty dish for the ears and eyes by using suitable ingredients (notes0, proper tools BW or other instruments) and some geometric recipes" (p. 650).

Conclusion

The authors found that the format of workshops of this collaborative type allow for conversations about the connections between math and music and propose that this can be extended to other interdisciplinary endeavours and across age levels. They suggest that both disciplines of math and music (or whatever combination of subjects) "can gain mutual benefit from this approach" (p. 650).