Week Nine: Mathematics & traditional and contemporary practices of making and doing

I chose to do the rope making activity and scavenged some dead grass from areas where the snow has melted off the hill behind my house. Lacking helpers, I chose to do the simple version where the grass was twisted into strands then wrapped back on itself in the opposite direction.

My z-twisted rope resembled a double-helix and was somewhat cylindrical. It was interesting to see the differences in process and product for multi-stranded braiding and rope making when watching the videos. The ropes were all cylindrical helices and each strand could be followed down the rope without going over or under another strand. In contrast, braids were flat and the alternating pattern of strands going over and under each other gave a more woven appearance.

 

 

 

I noticed that the grass still held the smell of summer despite having been dried and frozen for months. When making the grass rope I noticed a pattern of making three twists then wrapping the strand. The combination of twisting and the smell reminded me of when my sisters were young and used to make braided daisy chains and purple clover crowns in the summer. I also used to make hemp bracelets and the process was similar to the net making.

 

 

 

 

 

It was interesting to see such a similar perspective on the value of traditional knowledge and knowledge keepers from people in such different cultures. The videos showed Scandinavians (Closed by Hand), North Americans (Wildfibres), and South Americans (Weaving the Bridge at Q’eswachaka) all expressing the need to learn traditional methods from those who came before us. 

This quote from Undrum grabbed my attention, “There was no other way to learn. I had to learn from someone who knew how” (Ensby, 2016, 6:39). My husband's parents immigrated from Italy in the 1960s and they have a similar respect for traditional knowledge, including braiding garlic and onions for storage through the winter. 

 

 

 

 

 

 

The Astrom and Astrom (2021) article described the historical context and properties of rope. The mathematical description of rope has a lot of potential for a classroom, especially a physics class (which is just logic and applied mathematics in many ways). There could be discussions of the mathematical modelling used to describe the appearance of rope, or relate the diameter of the rope to the number of strands and their diameter. The authors also give equations used to approximate these values which could lead to a discussion of the assumptions made to create the models and equations. Their description of physical properties including linear density could also be used in a calculus course. When watching the video, "Weaving the bridge" I wondered how heavy the ropes would be and how much the final bridge weighed. The equations provided by Astrom and Astrom could be used to estimate the weight which could be compared to the weight of a more traditional steel construction. The woven bridge would also be a great exercise in all three areas of the Core Competencies. Communication, problem-solving, collaboration, positive cultural identity, and more.

Week Eight: Mathematics & fibre arts, fashion arts and culinary arts

My capstone project is going to be an exploration of factors that help keep girls engaged in mathematics through high school, so this week’s theme of challenging binaries and gender expectations resonated with me. I appreciated the acknowledgement in the introduction that men may be excluded from traditionally feminine endeavours, just as women can be excluded from traditionally masculine endeavours. My preliminary research and literature review indicates that women are underrepresented in STEM fields, especially computer science and engineering. Gendered stereotypes and expectations around STEM fields seem to play a significant role, but I wonder about the gendered stereotypes and expectations around “nurturing” fields like nursing and teaching. Men are very much underrepresented in those fields, and I wonder how many men rule out certain careers simply because they are considered feminine?

The introduction also mentioned social class stereotypes. My husband is a journeyman mechanic so the idea that trades people are lower class and/or unintelligent is familiar and annoying. While the specific math concepts encountered may not be extremely complex, there is an incredible amount of mathematical thinking and problem-solving involved in the trades. And yet, as the introduction suggests, the stigma persists and even pervades the school system. Students often view PreCalculus as “smart kid math” and Workplace math as “dumb kid math” despite its potential to be a really interesting and valuable course. But, I fear I digress…

On to the activity! I chose to do the origami because it is something that I am familiar with and enjoy doing. When working weekends in university I used to make little origami animals from post-it notes and hide them around my boss's office. I even made him a little zoo in an empty fruit tray container one time. 

The Miura-Ori pattern here is a relatively simple series of folds resulting in tessellations of parallelograms. It reminded me a little bit of a “magic ball” or “dragon’s egg” which is a slightly more complex pattern involving tessellations. 

 

 

 

 

 

 

 

 

 

In Quilts as Mathematical Objects, Gerda de Vries spent a significant portion of her presentation discussing tessellations, their properties, and their appearance in the art of quilting. The discussion of aperiodic tiling was quite interesting. In addition to tessellations, she also played with permutations and algorithmic design in her quilting. It was fascinating how many different patterns she could make with right isosceles triangles, three coloured fabrics, a bit of asymmetry, and an arbitrary set of rules for combining her triangles. 

 

 

It reminded me of the creativity that  arises from the tension between freedom and constraints which we explored through poetry last week. She talked about blindly following her rules then being surprised and delighted by the final product, which seems similar to the way PH4 poems unfold. Mathematical transformations, such as translations and reflections, and classifications of patterns were also analyzed using the context of quilting.

 

 

 

In the discussion of quilt design I thought there was a missed opportunity to talk about ratios and proportions as many patterns involved similar geometric shapes in different sizes. Ratios and proportions are also an important part of food preparation which is utilized in Hawksley’s article. She used coloured, layered beverages as a means of experiencing mathematics through the senses of sight and taste. Further claiming that “unlike most examples of food-based mathematical art, where the math is purely visual with no effect on the actual flavor and experience consuming the food, the math in our beverages is conveyed entirely by the ingredients and flavor” (p. 519). Ratios, proportions, and fractions can be explored by calculating the relative amounts of sugar or flavouring in each layer then comparing the differences in flavour when tasted. Hawksley also discusses the potential to use layered beverages to investigate arithmetic sequences, but this honestly feels like a stretch to me.

 

 

 

I teach the senior grades and this activity seems like it would be most appropriate for a math 8 class, but I could possibly see using it as a review activity in upper grades. Up to this point, I have only taught PreCalculus or Calculus courses and have been finding it difficult to apply much of what we have been learning to those courses. With the fairly specific yet assorted content to be covered, this has been a fairly consistent challenge throughout the program and my own investigations into more authentic learning activities. However, I am trying to remain open minded and make strong attempts to think of ways I could adapt the material. I need to keep reading Maria's blog for more "wonky math" ideas!

 

I wonder if this beverage activity could be adapted to look at rates of change or gradients if the layers were similar enough to slightly mix at the interface and/or have several layers? Another possibility is having students design quilt blocks, or copy an existing pattern, using mathematical functions in Desmos. Then they could use appropriate transformations to complete a full quilt design.

 

Week Seven: Mathematics & poetry and novels

Activity
The 5 poets that I chose were: Cindy Lawrence,Tom Petsinis, Lisa Lajeunesse, Robin Chapman, and Susana Sulic.

There was an interesting variety and two of the three types of mathematical writing that were described in the introduction were represented. None of the poems I read or listened to seemed to be about mathematical concepts or mathematicians. My parents always give us "kids" books for Christmas and many of them would fall under this first category. The idea of a visual poem was thought provoking. I'm not entirely sure if I "got" Sulic's Spatial Contamination, but I thought Naylor's Entirely Nothing was quite fun. I wonder if he entirely reverse engineered it or played with the words in the middle "frames".

The poems of Lajeunesse, Chapman, and Petsinis alluded to mathematial concepts, but in a way that was more imaginative than informative. I laughed while reading The Travelling Salesman Problem is NP Difficult by Chapman. My friend is a computer programmer and we once had a conversation about the complexities of timetabling and the potential for AI to help. I also particularly enjoyed Dear Linear Algebra Student by Lajeunesse and could clearly recall my first experience with Linear Algebra after three years of Calculus courses! The instructor's teaching methods certainly didn't help. I will forever remember him reading a textbook page that had been photographed, uploaded to his computer, opened in windows picture and fax viewer, then projected on to the classroom wall, all the while pointing at each word he read with an extended TV antenna..  It was all a bit of a muddle until my physics classes when we actually used vectors, dot products, cross products, projections, eigenvalues for wavefunctions... Context and application shed a little light on the murky situation.  

Returning to the matter at hand, I was intrigued by the interplay between freedom and constraints in fib and PH4 poems. Lawrence used the number of syllables for her Fibonacci sequence while others had used the number of words, and PH4 poems can use words or short phrases. The use of alternate spelling and punctuation to change or enhance meaning in the PH4 poems is also an example of artistic liberty.

The choice of punctuation reminded of a poster in a colleague's classroom and inspired my first PH4 poem.

The Oxford Comma

eats shoots and leaves
shoots, eats leaves and?
shoots leaves, eats, and?
leaves shoots and eats?
leaves and shoots eats
and leaves eats, shoots
and eats leaves’ shoots
eats and shoots leaves
eats, shoots, and leaves

 

My second poem is inspired by the feeling of an egg-head after scrambling.

The masterminds?

masters students’ lost minds
student’s masters’ minds lost!
students’ minds, masters lost
minds students, lost masters
minds lost students’ masters
lost minds masters students?
lost masters minds students
masters lost students’ minds!
masters students lost minds

Gerofsky wrote, "words that can be interpreted in multiple ways or as multiple parts of speech (for example, words that can be treated as either a noun or a verb) often yield the most interesting results" (p. 275). In my first poem "shoots" and "leaves" can be nouns or verbs, and "eats" could also be interpreted as a colloquialism for food. In the second poem "minds" and "masters" can be nouns or verbs. Both of these, and also "students", can be interpreted as plural, possessive, or plural possessive which adds some extra punctuation fun.

She also writes "there are 4! or 24 possible permutations" and "the PH4 selects a third of the all the possible permutations ... dependent upon initial row order" (p. 274). Thinking about factorials, I wonder about using this type of poem to look at permutations and combinations in a probability unit. Or possibly as a problem-solving activity where students choose four words then figure out the ideal starting combination to have the highest number of sensical phrases. Or even as an exercise in mathematical argumentation. For any 8 randomly selected possibilities, is there guaranteed to be a starting combination that would include all 8? Why or why not?

Reflection
de Adana, F. S. (2018). Surfing the mobius band: An example of the union of art and mathematics [Paper]. Bridges 2018 Conference Proceedings, Stockholm, Sweden.https://archive.bridgesmathart.org/2018/bridges2018-423.pdf

This article considers the intersection of mathematics and art in popular culture using the example of mobius bands in comics such as the Silver Surfer. As with many of the poems we read this week, the mobius band tends to be treated metaphorically in popular culture, often representing the idea of an endless and inescapable cycle. The authors acknowledge that the exact nature of the mobius band is not, and cannot, be fully represented in graphic narratives. Instead, the combination of mathematics and popular culture inspires "the collective imagination" (p. 426) and makes a mathematical idea more interesting and accessible.

The authors use the mobius band as "an example of how mathematics can be part of an artistic narrative" (p. 423) combining story with geometry. It reminded me of previous conversations and course work on the use of story in mathematics teaching. Not my area of strength! I wonder if the added visual elements make it easier to combine math and story, or not. In this case, the mobius band itself captures interest by defying expectations and the geometry is more easily appreciated when it can be seen. It is like the geometry version of a discrepant event in science. Or one of those "tricks" involving a bunch of algebra that essentially "undoes" itself or sneakily divides by zero. But what other concepts might work well with graphic narratives? Could similar cyclical metaphors work with fractals or sinusoidal functions? What other concepts or objects defy expectations or capture attention? Maybe geometric optical illusions?

Week Six: Mathematics & dance, movement, drama and film

I chose to try the rope polygon activity to try, but I had limited assistants available to me. I improvised with a small scale version using string and thumbtacks to represent where the rope would be held. 

As I played around with making regular and irregular polygons I was thinking about ways to adapt and extend this activity for the students I teach in grades 10-12. 

 

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.

A potential line of inquiry could be: If you are limited to using the knotted points and whole numbers for sides is there a specific number of regular polygons that can be formed? How can you know? What if the knots are simply to mark distances and the rope can be held anywhere? Does it make a difference? Now think about irregular polygons…

  

When I was playing with different triangles I wondered about an exploration of non-right angle triangles which are covered in Pre-Calculus 11. I was thinking specifically SSA triangles and the ambiguous case which students find quite challenging in my experience. The activity is based on a Geogebra I have used. 

 

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.

Final Project Outline

 Here is a link to the draft outline of my final project inspired by the Kamloops Art Walk.

Week Five - Developing Mathematics Pedagogies

I chose to work with the binary circles and ended up rather far afield by the end of my exploration. I had curricular integration with an introductory unit on exponents in mind when contemplating the following extensions. In the introduction video, Susan spoke of inquiry and variations as integral components to the activities so I envision the first extensions as a "What do you notice, what do you wonder?" assuming students have done the original activity of addition with binary or base three numbers. 

Extension 1: Using colours to represent digits as Ali and Colin did, the first extension compares the patterns made by two different bases. I used binary and base four as an example, but students could choose any two bases and their own colour schemes.

I envision guiding questions such as:
What do you notice about the patterns made?
How would you arrange the circles to make patterns more obvious?
Do circles that are coloured the same from the sets with different bases represent the same number (in decimal)?
Can you find a relationship between them beyond looking the same?

 

Extension 2: Using repeated addition with the same bases as before, can we figure out how to multiply?

Guiding questions could include:
What do you notice about the patterns made when you add the same number?
What "rules" can you find for colour changes?
How are the rules for different bases the same or different?
What about adding or multiplying numbers with different bases?

Is the first number in the last addition base 2 or base 4?


Extension 3: At this point I went off trail and asked myself, "What if we used shapes to indicate the base? Can fractals be used to illustrate numbers in exponential form?"

Possible guiding questions:
How can we represent different bases?
How do components of the fractal images correspond to parts of the notation for exponential form?
How can a number like 32 be represented in different artistic ways?

Brainstorming Sketch - Math 9 Exponents and Exponent Laws

1. What do we know about binary, bits, & bytes?
   
2. Kandinsky circles inspired binary numbers and addition activity.
3. Binary expressed in exponential form with base of 2. Notation, coefficients, bases, exponents.
   
4. Converting larger binary numbers to base 10 using exponential form. How many bits in a megabyte? Ex: 00110101 = 0*2^7 + 0*2^6 + 1*2^5 + 1*2^4+ 0*2^3 + 1*2^2 + 0*2^1 + 1*2^0 = 53
   
   
5. What about other bases? Hexadecimal, duodecimal, e?
   
6. Extension 1: Comparing different bases
7. Can you use the principles from binary to decimal conversion to find a formula for converting base ___ to decimal? What about bases that are multiples of one another?
   
   
8. What about multiplication?
9. Extension 2: Multiplication by repeated addition
10. What are the rules? How would you express the rules in mathematical notation?
    
    
11. Extension 3: Exponential form and fractal art
12. Move on to Exponent Laws

Reading and Reflection
It probably would have made more sense to choose the embodied activity rather than the art activity since my reading from Riley et al. reported results from a study integrating physical activity with mathematics through a program called EASY minds in grade 5/6 classes in New South Wales, Australia. Yet, I feel a little more comfortable finding ways to integrate outdoors and activity so I chose to challenge myself with Kandinsky in Binary.

The authors reported positive results in both mathematical attitudes of students and teachers as well as increased mathematics learning after a six week intervention to embed physical activity within the math curriculum. The authors specifically noted that math became a subject students looked forward to. The activities were either practice for procedural fluency, such as jumping rope while reciting multiplication tables, or finding math in their surroundings, such as estimating distances.

Some interesting thoughts from the teachers were "that the preparation took more time and effort because it involved teaching concepts in a new and practical way" (p. 1666). Time and effort seem to always come up as barriers to being more innovative in our classrooms. I know it is something we have discussed previously and I don't see any way around it. It is a matter of being content to take baby steps and making slow but steady progress towards the goal.

"The teachers also talked of EASY Minds having forced them to be more creative and forward thinking as well as having to structure their planning to get EASY Minds aligned with the scope and sequence of the existing curriculum" (p. 1666). This is something I have found as well. Since there are limited ready-made resources at the high school level it does require creativity and thinking outside the box to develop activities. This seems especially true if the goal is curricular integration and continuity rather than the orphaned or frivolous activities Susan cautioned against in her introduction. I am not sure how long it took me to create three somewhat cohesive extensions aligned with curriculum, but it was significant and included several thinking walks. I know the shift is towards competencies, but the content is also important.

The authors also suggested that a teacher's attitude towards mathematics and their pedagogical repertoire strongly influenced student's attitudes. I can certainly see how this could be true. If someone is passionate about a topic they engage with it in a very different way. Though, I do feel there are limits to the second-hand enthusiasm that can be generated. Just look at the way parents and children engage with Minecraft!

Several students claimed the multi-tasking required to combine math with physical activity was beneficial. I wonder about this. Did the multi-tasking help with learning process or did it facilitate higher engagement? Is this related to shortened attention spans (apparently now legitimately shorter than a goldfish)? Do the activities and math concepts use the same neural pathways? The researchers noted the physical activity before the learning was most beneficial. Is it partly burning off some of the extra energy of 11-year-olds? Related to this, are there gender differences in attitudes and learning combined with physical activity? Lots of questions, lots to think about...

Week 4: Mathematics and the arts

My apologies for the post and responses being late. We went north to dig my mother-in-law out after a significant snowfall and cell service and internet were both down. It would have been enjoyable to unplug if it weren't for things to do!

I looked at the artworks and chose several to examine more closely. Despite a relatively strong background in math, many of the principles behind the pieces were outside of my linear algebra and calculus area of experience. I noticed that most of the pieces I gravitated towards had a lot of symmetry. The one I chose to recreate is called "The Stijlish Seed of Life" by Emanuela Ughi. I was curious why six circles perfectly overlapped the central circle so I used a compass to start drawing it. I drew one circle, then added a second centered at a random point on the first circle's circumference. When I added a third circle centered on the intersection between the first two, I found that I could draw an equilateral triangle with side lengths equal to the radius of the circles. As a result, the arc length is pi*r/3 so six circles would be 2*pi*r, equalling the circumference of the circle.

I was also fascinated by the fact that a single circle has continuous rotational symmetry and infinite axes of reflection.  If you ignore the different colours, the entire piece has six fold rotational symmetry and six axes of reflection symmetry. Once the colours are considered the piece no longer has any rotational or reflection symmetry because the four pieces of each colour are different shapes and sizes. Yet, at the same time the puzzle can be put together many, many different ways because the curvature is the same in all pieces. Any convex edge can be matched with any concave edge. Below are just a few examples!

I appreciate the goal of bridging mathematics with art and culture. I have found that people assume I am not interested in things like art or  literature because I am a "math/science person". I do have interests beyond math and science, but will admit that I do view the world through a lens that is a bit more technical than artistic. When reading Spinning Arms in Motion: Exploring Mathematics within the Art of Figure Skating I was honestly more interested in how biomechanics and physics engaged with the geometry. The authors explored the geometry of a figure skater's arms in relation to their body while performing an upright spin. They presented an elementary and secondary version, including appropriate mathematical questions. The article referenced modeling, but when I followed the links the privacy settings did not allow me to view the videos. I felt like the link between artistry and geometry was a little forced to be honest and I am not sure that I would use this particular idea.

I think it would be interesting to look more closely at things like sacred geometry, how math can be used to create depth and perspective, or looking for the Golden Ratio in art and architecture. But perhaps that is my overly analytical brain must watching to know how things work? My mother does quilting and she told me that if the entire quilt is perfectly symmetrical the brain decides it is boring so you have to add one subtle thing to break the pattern, like a colour that doesn't quite match. Then your brain finds it interesting and subconsciously tries to figure out why it is not quite right.