Campbell and von Renesse's article Learning to love math through the exploration of Maypole patterns they demonstrate how seeing Maypole dancing through a mathematical lens can excite Liberal Arts students to explore mathematics more deeply. Starting with sharing their foundational perspectives on their math class. Later delving into the mathematics discovered by the Fall 2016 class and Julianna Campbells independent study. Exploring how many non-equivalent ribbon patterns are there, given the number of dancers and numbers of colours?
Julianna Campbell was a student in Prof. Von Renesse's 'mathematical explorations' class, at a small public university, this course was a general core class for non-math majors, the course description is based around meta-goals and doesn't require specific content goals,
The meta -goals for the course include that students will:
- appreciate mathematics as a human endeavour which is one of our most fundamental intellectual pursuits
- strengthen their reasoning skills and become better problem solvers.
- strengthen their skills in reading, writing, argumentation and speaking.
- will become more self-monitoring, reflective learners and take greater personal responsibility for their learning.
- approach mathematics more positively and gain a balanced perspective of mathematics.
- improve their mathematical confidence.
- develop awareness of the negative impact of broadly held societal views.
- be capable of and interested in considering mathematics outside of the confines of the classroom, understanding the value of lifelong learning in mathematics
(Campbell and von Renesse, 2019, p.132)
The classroom pedagogy centered around inquiry, von Renesse shares a definition of inquire from a study by Laursen et al., defining inquiry-based learning courses to be "characterized by:
- Learning goals focused on problem-solving and communication
- A curriculum driven by a carefully constructed sequence of problems or proofs
- Driving toward a small number of big ideas
- Course pace set by students’ progress through this sequence
- Class time used for a mix of active and collaborative problem-solving tasks
- Instructors who guide student work instead of delivering information."
(Campbell and von Renesse, 2019, pp. 132-133)
Campbell reflects on the course as the first time she was encouraged to ask questions and search for answers without the fear of seeming less intelligent, motivated by the desire to understand and the motivation to question. Sharing that every time they thought they had reached the end of questioning Prof. von Renesse would ask ""But how do we know for sure? Will this always happen?'"
Prof. Von Renesse alongside colleagues profs. Fleron, Hotchkiss, and Ecke developed content and pedagogy materials for using inquiry in teaching mathematics for liberal arts courses, which can be found at www.artofmathematics.org.
This mathematics class was stable as a fully integrated learning community with an English composition honors class. Working alongside colleagues Jennifer DiGrazia, they were able to model how to be students in each others subject area, to take risks, be curious, and make mistakes. The topic chosen for this course was from the arts, this gave students who felt negatively about mathematics a chance to start new and experience mathematics differently.
Julianna Campbell who co-authors this article as previously mentioned was a student in this course. Campbell had experienced math as a source of struggle and insecurity, until experiencing Westfield State university's math and English learning community. Feeling failed by the route memorization and formality of her K-12 math education, it wasn't until she was encouraged by Prof. Von Renesse to follow her natural instincts of curiosity to question and struggle with mathematics that she found joy in mathematics. Campbell shares the insecurity she felt first walking into that class, feeling confused by conversations about factors, questioning her idea of factors, to the encouragement she was met with and the progress she has made to have completed an inquiry-based mathematics independent study, presented at a math conference and co-authored this paper with her professor.
The class was spent in small groups initially searching for conjectures and taking steps to prove them, with students volunteering to share work without fear as there was and understanding that each mistake led them closer to an answer, being encouraged to develop an argument around their answers and thought process.
Campbell describes herself as the other breed of mathematician, saying others can quickly solve a problem and logically find a solution, but she is strong in areas of inquiry, curiosity, and perseverance with mathematics. She says mathematicians like herself have her to be encouraged to pursue mathematics in K-13 education, and it wasn't until collage that she found her place in mathematics, stating students are experiencing a misinformed perception of math, it can be used to enrich our lives, to answer questions your students truly care about, creatively and as a means of bringing students together as a team.
The article goes on to describe a mathematics classroom that is active, engaged, with arguments, discussions and developments of proofs, where math assignments are pages long of handwritten theorems, filled with inquires, points of confusion, and proof of hours of hard work; a math class of non math majors. Working in small groups on particular problems, before gathering as a whole class to discuss ideas, Prof. von Renesse would listen to ideas, ask questions, but let students lead with their findings. Students collaborated to understand and figure out problems that initially seemed impossible, doing things that now felt new and exciting.
Campbell recalls the day that Prof. von Renesse brought a tall PVC pipe and a bag full of colour stripes of ribbon, and the class danced the Maypole dance, launching a 6-week undertaking for the class and a semester long independent study for Campbell. Creating space in the class they assembled the pole and practiced dancing, creating an over-under pattern. Once the pattern had been danced and the ribbon was wrapping the pole, the students observed and were asked "what do you notice? what are you wondering about?" Without fear students could share their observations and conjectures, no idea wrong, just a step closer to a bigger lead. They explored what happened if leaders held all one colour and followers all another, if some leaders held some of each colour and followers some of each as well. considering how these alterations and manipulations changed the picture. As they continued questioning they determined they needed to represent a Maypole dance without actually doing the dance. This was considered to be their biggest challenge, there was no known answer, it was new, students worked together to explore different ways to represent this 3D representation into a 2D image, they considered different viewpoints of above and to the sides but it didn't look right. A student named Austin figured out a way, he 'cut' the pole vertically and unwrapped the picture so that it could be viewed as a rectangle, this was later termed by Prof. von Renesse and Campbell as the tree representation. Austin showed the class how to draw the dance and later another student, Olivia helped refine the technique by using graph paper to help make the picture neater and more systematic, supporting students quest to search for patterns.
Originating from a traditional folk dance, the maypole dance involves a tall wooden pole with various coloured ribbons attached at the top, from the bottom of the pole people hold the ribbon and dance in patterns. "Figures 4 and 5 show an example of a ribbon pattern with 3 couples. Assume that there are 3 black ribbons (called 1, 2, and 3) and 3 red ribbons (called A, B, and C). Each black ribbon is paired with a red ribbon. All of the people holding black ribbons move to the right (i.e. mathematically positive when looking at the pole from above) and go first over the red ribbons. At the same time, the people holding red ribbons move left (i.e. mathematically negative from above) and go first under/inside the black ribbons. After this first step, the black ribbons would go under while the red ribbons go over. This pattern keeps alternating. While the dancers move around the pole, the ribbons will cross over and under to generate different patterns depending on the colour and order of the ribbons." (Campbell and von Renesse, 2019, p. 136)
With the tree diagram determined by the students, von Renesse introduced an efficient way to label the different dances with letter representation. With the expanding scope of patterns hand-drawing these representations became more challenging, leading to independent student and professor exploration of excel sheets to attempt to automize drawing the ribbon patterns.
The following definitions were determined throughout their process.
"Definition 5.1 (Tree Representation): We call the representation of a ribbon pattern using diagonal ribbons as in Figure 4 the tree representation. The ribbon pattern on the maypole is ‘cut vertically’ so that the cylinder can be unwrapped to see the full pattern. Leaders are represented by numbers and followers are represented by letters. Leaders are defined as ribbons which first ‘go over’ in the dance. Accordingly, followers go under first in the dance. In the image, 1, 2, 3 are leaders and A,B,C are followers; we can also see that the 1-ribbon first passes over the A-ribbon, the 2-ribbon passes over the B-ribbon, etc.
It was a big step for the class to notice that the tree representation doesn’t need to be ‘coloured in’. Although it is tempting to colour these diagrams, and often very helpful when looking at specific examples, this diagram is more useful for proving results in its general form.
Definition 5.2 (Letter Representation): We can also represent a ribbon pattern by the shorter letter representation 1A2B3C. We call this in short the letter representation. Here the letters and numbers are used as described in Definition 5.1. 1A2B3C is the basic unit of all dances for 3 couples. If we want to talk about a specific pattern, we can, for instance, write BWBWBW for the patterns resulting from 3 black ribbons (leaders) and 3 white ribbons (followers).
To avoid having to draw too many ribbon patterns by hand, we explored a Microsoft Excel sheet to find the coloured patterns more quickly. You can find the excel sheets at https://www.artofmathematics.org/node/705. This led to a new definition.
Definition 5.3 (Screen Representation): We used Excel to automize the pattern drawing and received a pattern in which the ribbons were now horizontal and vertical instead of diagonal. We call this the screen representation of a ribbon pattern. Figure 6 shows how the tree representation relates to the screen representation. The leader ribbons (numbers) are now horizontal while the follower ribbons (letters) are vertical.
In the screen representation on Excel, we can easily extend the pattern horizontally to see the larger pattern.
Definition 5.4 (Fundamental Domain): A fundamental domain of the ribbon pattern is defined as a smallest region that repeats itself inside the whole pattern. See Figure 7 to see a fundamental domain of 1A2B3C in screen representation. Notice that a fundamental domain has to exist since the dance repeats itself, but that it is not unique." (Campbell and von Renesse, 2019, pp. 137-139)
Initially the classes guiding questions sought to understand how the maypole dance worked. Hoping to predict the ribbon pattern for any given dance without having to dance, to determine this they narrowed the dance to four and then six dancers. With a growing understanding of the dance, students wondered how many different ribbon patterns there are given the number of ribbons. The class manipulated the letter representation to determine if new geometric patterns would emerge and if there would be commonalities with the original pattern. Identifying four ways, the letter representation could be manipulated with the ribbon patterns being essentially the same.
The article goes on to define what equivalent ribbon patterns are and discuss proofs of four corresponding theorems. Later seeking to determine how many different ribbon patterns are produced by specific numbers of ribbons and colours, they use powers with a as the base representing the number of colours and b is the exponent representing the number of ribbons. They use this alongside their previously established definitions and proofs to identify the different ribbon patterns, concluding the following.
Number of ribbons- Number of colours: Number of different ribbon pattern
6-1:1 ribbon pattern
6-2:5 ribbon pattern
6-3:7 ribbon pattern
6-4:5 ribbon pattern
6-5: open problem, pictures do not have much visible structure, not as aesthetically pleasing
6-6:1 ribbon pattern since all ribbon patterns, are colour changes of each other the pattern is a checkerboard.
(Campbell and von Renesse, 2019, p. 150)
Stop 1:
"Students are experiencing a misinformed perception of math." (Campbell and von Renesse, 2019, p. 134)
This was more of a sad stop, I agree that many students experience a misinformed perception of math. I sometimes as a secondary teacher feel I am fighting against the math ideas and identities students have already started establishing for themselves. Throughout this program I realize just how misinformed my perception of math was, I am still wrapping my head around the various connections that are and are still to be made with mathematics. With the videos this week, specifically thinking of the TedX talk with Karl Schaffer and Mr. Stern I felt myself trying to see the math before they fully revealed it, I thought of common multiples but hadn't fully made the connection to fractions and fraction subtraction before Schaffer revealed it. I find myself feeling the furthest distance from embodied mathematics with music and performance and am hoping to continue to bridge more of that distance and start to find ways of utilizing these components in my teaching practice.
Stop 2:
"In our class we sat in small groups, working together on a particular problem, and then gathered as a whole class to discuss our ideas." (Campbell and von Renesse, 2019, p. 135)
This sentence was a stop for me as the description reminded me of a very specific time in my math education. I had left an engineering program, spent a year taking a variety of classes before switching to a different university and going into a math degree and my first course was a proofs course. In that course I experienced a style similar to this, where we were regularly working in small groups all contributing to working through a proof together on one of the chalkboards around the room, before coming back to the class as a whole and having each small group present their proof to the rest of the class. I remember in that class not really knowing my classmates as I was joining in a small university into second and third year courses, but all of my courses with this prof felt like a little community, her style of teaching did just as this article said, it welcomed mistakes as a part of the journey of learning and through that built connection among students. While we didn't experience embodiment in the way that this article described, many other key features discussed aligned strongly with my experience in courses taught by this Professor.
I apologize for the length of this summary/post and the amount of directly quoted information, I struggled to summarize their definitions and proofs without feeling that the details needed to follow the work were getting lost, and found it very interesting to wrap my head around these proofs and wanted to include those details for folks who were also curious. If interested in the more detailed information of the proofs and accompanying visuals check out page 139 onward.
Question: How do you think we can work against students experiencing a misinformed perception of math?
Campbell, J., & von Renesse, C. (2019). Learning to love math through the exploration of Maypole Patterns. Journal of Mathematics and the Arts, 13(1–2), 131–151. https://doi.org/10.1080/17513472.2018.1513231
For a long time now, I've felt that our work here in MAE3 has been to answer exactly what your question asked: How can we change student perception around Mathematics? We are fighting against the abstraction created by decades of modern curricula that turn students into human capital for the sake of “the market”. The skills that are deemed “necessary” are disjoint from reality and the body because that’s what post-secondary schools are looking for. Many students will still find a love and understanding of mathematics, despite the procedural nature of many western curricula.
ReplyDeleteWith respect to your first stop–I think collaborative mathematics is another way of changing our perception of mathematics. I think it was Skovsmose who noted that reflective mathematical practice was missing from modern mathematics education, and what better way to reflect about mathematics than by having people to converse with? Reflection, despite being a seemingly singular act, can be made quite impactful when multiple people are there to question and affirm one’s beliefs. It’s what I’ve been feeling as part of this cohort, and for that I am grateful.
Your first stop felt sad to me as well and resonated deeply. The idea that students are operating from a “misinformed perception of math” is both accurate and disheartening. As teachers, it can sometimes feel like we are working not only with content, but against years of already-formed math identities. I encounter this to a lesser degree in elementary, students have had fewer years to internalize negative narratives, and primary math is often engaging and playful, but even then, those identities can begin forming surprisingly early. I appreciated your honesty in recognizing how your own perception of mathematics has shifted throughout this program. That kind of reflection models exactly the growth mindset we hope to cultivate in students.
ReplyDeleteYour reflections also helped me realize something important: embodiment does not have to look like dance or performance to be transformative. It can live in collaboration, shared struggle, discussion, and collective reasoning. When students are actively constructing understanding together, mathematics becomes something experienced rather than simply completed.
In response to your wondering, I think part of the answer lies in intentionally designing embodied, socially responsive, and community-connected tasks that meet the needs of diverse learners. If we thoughtfully apply the tools and perspectives we’ve developed throughout this program, we can create classrooms where mathematics feels human, relevant, and accessible. I would like to believe that even if not every approach reaches every student, at least one experience might interrupt a misinformed perception and spark a small moment of mathematical wonder.