Wk 9: Making and Doing

 First stop: Could we make durable dog leashes out of plastic bags? In my "free time" I train service dogs, as well as have my own pet lab. I love animals in general, and my dogs are very close to my heart! The idea of creating a leash (or other supplies) in an eco-friendly way is really interesting to me. 

Second stop: That bridge is equally fascinating and terrifying!! I love that the man is stepping on the rope with his bare feet...really getting in there! The fact that they can just let the river wash away the old bridge is very neat. 

Common Threads: Weaving Community Through Collaborative Eco-Art caught my attention when it unpacked "back-sourcing". The idea focuses on the notion of taking back what we've become dependent upon out-sourcing. The multiple videos from this week's outline gets at the same idea: How can we keep things local? Is it necessary to use supplies from other parts of the world, or could we find materials in our everyday lives? 

Going back to my first "stop", I decided to try braiding plastic grocery bags into a strand, ultimately to see if I could turn it into a dog leash. Below are pictures of my progress. 

My thoughts were:

• "Smaller strips of bags would lead to more flexibility and proper sizing"
• "This is going to take a long time!"
• "It would be really handy to have a leash that could just be run under the tap to clean off!"
• "This would be so much work..."
• "I think I might try it again!"

Working with my hands is something I find very calming. When I was younger I would spend hours making friendship bracelets, just for fun. The patterns are beautiful, and I like the physical act of folding and tying. It would be fairly easy to tie these types of activities into pattern and sequencing units in math. What are your thoughts on more complicated connections? 

Wk 8: "Touching" Math

 Watching the "Easy Miura-Ori Origami Tutorial" was very similar to a lesson I did with my Grade 8's a few weeks ago. We have a new group of international students from Japan and they were really excited about teaching their new friends how to fold origami. We watched this TED-Ed video first: https://www.ted.com/talks/evan_zodl_the_unexpected_math_of_origami?language=en

When I was looking at the video about orbifolds and shibori dyeing, I was overcome by a sense of relief. I've avoided tie-dying for years, because it stresses me out to have such random patterns being created at my fingertips...I feel like I've finally found the answer to this in the patterns created by Carolyn Yackel. 

Exploring Ratios and Sequences with Mathematically Layered Beverages was a new take on "math with food". It's very common that we use baking as a food example of mathematics; having students measure out ingredients and calculate conversions is a hands-on way to work with the concepts being covered in class. 
I really appreciate the desire to use food as a "vehicle for conveying mathematical ideas to the general public" (519). The main purpose of the workshop was to have students use beverages to represent different mathematical fractions, ratios, and sequences. Discussions surrounding the amounts of sugar to the amount of beverage opened up the path to deeper understanding of concepts. Different layers and colours led to exploration around ways to represent certain sequences, including the Fibonacci sequence!

The nice thing about the activity in the article is the very visual and kinesthetic way to grasp math concepts that are often misunderstood by students. The fear around fractions is universal, but this way of representing shows the different pieces that go into equivalent fractions very clearly! 

Overall I can't say that this week really captured me. I like the movement and rhythm from past weeks, and this week feels like it has the potential to be more confusing than helpful. But maybe that's just me? 

I'd love to hear your thoughts! How has this week challenged you? Are there ways you plan to use this week's activities in your classes? 

*Pictures of activity to come. Phone died, so will update tomorrow!

Wk 7: Mathematics, Poetry, and Novels - Oh My!

Two years ago I took an unexpected hiatus from teaching High School Math in order to teach Grade 8 English Language Arts, Math, and electives. To this day, it is my least favourite year of teaching, and almost resulted in me leaving my current school. Not a fan...BUT during that year I discovered a love of poetry that I didn't know was in me. I've really enjoyed this week!

For the five Bridges poets, I chose to listen to Marian Christie, David Greenslade, Lisa Lajeunesse, Amy Uyematsu, and Ursula Whitcher. It happened to work out that they all had different takes on the way that the melded mathematics and poetry together, with some using mathematics, some writing about mathematics, and others writing to mathematics and mathematics students. Out of the five, Marian Christie and Lisa Lajeunesse caught my attention most. 

Marian Christie read four poems: Pathways, Elevenses, The Primary Hourglass, and Clothoid. All of her poetry utilized the mathematics they were about to communicate information about them. Pathways was based on the Fibonacci sequence, Elevenses on the number 11 (side note: Much to my dismay, it does not reference The Hobbit at all 😞), The Primary Hourglass used the largest two sided prime number (739397), and Clothoid took the shape of carvings Christie had seen on a holiday in Malta. All of the poems took the shape of their topics and paid attention to the number of letters, words, or syllables that was connected to their inspiration. Below is The Primary Hourglass, which fascinated me, as not only do the number of words in the line mirror the number of numbers in each line, but each word has the corresponding number of letters! 

Lisa Lajeunesse shared two poems, Dear Linear Algebra Student and Life and Love. I really appreciated the way she approached her poetry, reading as if the audience she was speaking to was right in from of her, familiar and relatable. She read in the first to her new students in a Linear Algebra course. I was amused by the way she compared Linear Algebra and Calculus and quickly connected with the things the said. The second poem was inspired by the Fibonacci Sequence in a different way that Christie. Beginning with the words "Life" and "Love", these words were repeated over and over again with the small addition of joining words for each phrase. 

"Surfing the Möbius Band: An Example of the Union of Art and Mathematics" 

This short article described how Marvel creators used the Möbius Band in their comics, allowing characters to become the vehicle for some mathematical content. I have no idea this was a thing! 

Marvel superhero, Silver Surfer, has a new-ish series, and Issue 11 is titled "The Moebius Madness of Silver Surfer". Below are two examples of Möbius Bands in comics. 

My take-aways?

I am a visual learner. I gravitate to that which I can see and study and observe. I think that's why I'm so drawn to the poems that take the shape of their topic and the ideas that can be sketched out and made sense of on paper. I am fascinated by the artists who work abstract concepts into tangible pieces. 

I think it is so cool that comic book creators have found ways to use the Möbius Band and other academic content in their creations, as this exposes children (and adults!) to bigger ideas without them even realizing it. 

Listening to Mike Naylor, I also realized that I am drawn to rhythm, which he really leaned upon for his poetry. 

All in all, this has caused me to stop and think about the way that I present my material in math class. Am I only presenting it in ways that I understand? Are there more creative ways I can engage students? How can I reach the students that really grasp humanities and arts? 

I had a good time creating these two Fib poems: 

COVID. Year 3.

Now

is

the time.

We look forward; 

We long for change...

A time where "normal" and "new normal" meet.

The Same Question

What

are

we learning?

My students enter,

their faces sharing their hearts.

Math is an experience; no two the same.

Wk 5: In the Classroom

This entire week has truly captured my attention and my mind has been racing with many ideas I'd like to try! Every video I watched and lesson plan I read, I found myself thinking, "Oh! And then I could do..." 

This is exciting! I love feeling like my lessons are interesting and engaging and "real". 

But then I came up against the worry that my Middle School and High School students would think this was "dumb" and that I would have little to no buy-in. At which point, what am I to do? 

The "Math of the Move" lesson plan I looked at was the "Math in Your Feet Starter Kit". It was the video that I thought I would use right away and it sparked an idea for a lesson I can try in the coming weeks. 

My Grade 8 students and I are currently looking at patterns and linear relations -- this fits perfectly! It is a little advantageous to have them up and dancing, as they are in the throws of being too cool for everything, but I think I can get away with handshakes. 

Similar to the symmetry and movement that is worked with in the movement variables, I plan to have my students create handshakes that represent patterns they've constructed with numbers. Here's what I'm thinking: 

1. Students use the numbers 1 to 8 (or whatever number you want to cap it at) to create, in partners, a pattern. This pattern can be as simple or complex as they want. 
2. Once all students have created a number pattern, as a class, determine what each number represents. For example, 1 = left-hand high five; 2 = clap; 3 = ... and so on. 
3. Have students use this information to try out their newly created pattern handshake. 

Looking around the room, you should notice that all students have a unique, mathematically-based handshake. Discussions could lead to what needs to be tweaked in order for the handshake to flow, which in turn should be reflected in the numerical pattern as well. 

My hope is that this activity could then lead to something like the Math in Your Feet lesson plan in which students are up and moving. This also connects to the Karl Schaffer & Mr. Stern TedX  talk (2012) video we watched this week. 

I LOVE when students get to engage with math with their bodies. I danced for years while I was growing up, and I still find myself coming back to the basics that I learned there when I am stressed or nervous. Counts of 8 calm me in nerdy dancer ways that others will never understand. How cool would it be if, in 15 years, my students find themselves coming back to mathematical movements when they need grounding? How much deeper will their understanding of mathematics be if they are able to turn to a physical movement that immediately returns them to the learning they engaged in in my class? The ways in which doors will be opened by the movement of bodies excites me in many ways! I'm looking forward to try this out!

Wk 5: When the 2016 Rio Olympics gave way to Mathematical conversation

"Reenacting mathematical concepts found in large-scale dance performance can provide both material and method for ensemble learning"
Wow...that is a long, packed title! Based, interestingly enough, on the Rio 2016 Olympic Opening Ceremony, this study talks about "foraging" through the internet for "found objects" in public media to use as the inspiration/basis for mathematical discussions. The authors refer to this as "re-use" of the material. 

If you watch the youtube clip, there is a chunk from 12:10 to 15:15 in which an ensemble is moving through a choreography with large foil squares. This becomes the basis for the study. The individuals participating in the performance work in a variety of combinations: "quartet" (4 people), "group" (groups of quartets), and "whole" (all performers together). The mathematics arises in the movements, which are similar to rotations, reflections, translations...symmetry is heavily depended upon. 

Using this clip, participants in the study watched the clip and were led into discussion and challenges around how to recreate and build upon the ideas. 

The first group was a group of 8th Grade girls. The girls watched and quickly became excited, claiming they knew "exactly what to do" in order to recreate the movements! They quickly realized that it would not be as simple as they had originally thought, and ended up needing to move into conversations that had new terms ('overs' and 'unders') and specific instructions about how to move and when. 
When asked, it was difficult for the girls to comment on the mathematics connected to these ideas. 

After this, STEM educators attempted the task. They began with a common goal: to create a "dancy", fluid performance. They were attentive to the patterns in movement and actions, and worked together to state and reflect upon new terms that allowed them to differentiate between the people, the positions of the people, and the parts of the 'prop' they were connected to. That being said, even with the focused attention to detail, the educators were surprised by their eventual success!

To close the study, a group of learning scientists were brought in to try to understand what the STEM group had done. Rather than jumping in to attempt the movements, they began by discussing the ideas behind, studying the recording of the educators, and freezing the frame at points when the educators made certain movements. Then, in their own reenactment, they took time at each of these points, determined to find a formal description that would lead to deep mathematical understanding. 

As I read through this study and watched the clip of the Olympics, I was captured by the coordination that was required. The choreographer had a complicated task! The symmetry and fluid movements caught my attention and I immediately found myself noticing patterns and shapes, and I was asking myself questions about how they knew where to move, and how things could have been different, had they changed the direction they were travelling in, or the timing of the movements they made. 

Connecting to this week's topic, this is a fascinating example of using movement to draw people into mathematics. While the initial purpose of the performance was not to look at mathematical movements, I love that this piece was "foraged" for a deeper purpose. It makes me wonder about how one can find more pieces to use. Could this be a project for students? Is there a way to connect the movements to tangible mathematics in the classroom, or is this an idea that is best left to curiosity and wonder? 

Very cool, nonetheless! 

Wk 4 - Art-ifying Mathematics

 To begin my experiment, I started simple. Based on the concentric circle representation of Binary put together by Ali and Colin, I tried my hand at representing base 4. It wasn't until I was partway through that I compared it to their base 4 representation and found many differences! 

Here is mine: 

It made sense to my brain to connect the location/size of the circle, the colour, and the number. So, in the first circle, the number 1 is represented by both the size of the circle (smallest) and the colour (red). I used the existing colour(s) until they could no longer combine in addition to create the next numbers. At this point, a new colour and layer of circle was introduced. Because there were only 4 circles, I ran into a problem at circle 15. 

While creating this, I found I had to pay close attention! I struggled to consider which circles would work together to create the number I needed. It is clear that my brain is not used to mixing art and math. When I finished, I found myself pondering many questions: 

• How could I make it so that I could use numbers multiple times? (For example, use 2 2's to make 4, rather than 3 and 1?)
• Is it necessary for the circles and colours to both represent a number? What would change if only one represented a number?

My thought, which I might try and update later, is that the circles and colours can both represent colours, so when a colour is in a specific circle, it results in those two numbers being multiplied! Things might get crazy, but I'm excited! 

Wk 4 - A New Vision for Mathematics Education: Mathematics as Narrative

 "I am talking about a culture of schooling in which more importance is placed on exploration than on discovery, more value is assigned to surprised than to control, more attention is devoted to what is distinctive than to what is standard, more interest is related to what is metaphorical than to what is literal."

Elliot Eisner, 2002

"What Mathematics Education Can Learn from Art: The Assumptions, Values, and Vision of Mathematics Education" presents readers with a seemingly radical idea about a new method of mathematics teaching. Dietiker suggests that mathematics classrooms are often "too tidy" and society assumes that math should be precise and carefully deduced. Instead, she presents, we should be thinking about mathematics as story. 

"Interpreting mathematics as a story repositions mathematics curriculum from an instruction manual or a collection of facts to a form of art, intentionally crafted to offer aesthetic experiences for a set of students, whether positive or negative." (2)

In her years of teaching and working with educators, Dietiker has experimented with incorporating narrative and the information we know about story into the mathematics classroom in what she calls "Mathematical Adventures". Textbooks are traditionally the source of information and practice questions, resulting in students who can solve very specific types of problems and become bored when faced with the same types of questions over and over again. 

"I wondered why many of the teachers I met through professional development, who had earned by respect, rarely questioned their textbooks." (4)

Stories teach lessons about life, so why do we not use stories in math? Mathematical stories must also have characters and plot points and setting that move the story along. Questions can be asked to guide the process, such as "How might this mathematical story compel a student/reader in parts of the lesson to be interested in the final outcome?"

While I was reading the article, I stopped at many points mentioned above, and I kept thinking about a method I use in my own classroom that I call "Smudge Math". Working in groups, I present students with the information we are learning that day, but rather than saying "The quadratic equation can be factored to look like this.", instead I say, "I know that this quadratic formula can be factored, and the factor looks like this, but I can't quite remember what goes in those blanks that have been smudged out...you need to figure it out!" We start with a lot of assistance, so most of the factored form is filled out, but as we progress through types of problems, I give less and less information, and students draw on the vast experience they have to come up with an answer. I love this because it makes them think! I hate the idea of math class being experiences where students copy the work on the board and then replicate it in various forms -- tests, quizzes, assignments....we need students who are capable of critical thinking and problem solving! Engaging them in story and challenge changes the task from "completing the question" to "going on an adventure", which I think is extremely beneficial!