Micro-teaching: how to tie a figure eight knot

 Lesson Topic: How to tie a figure eight knot

Materials: small strand of rope for each of the four students, laptop to show video on

Intro: 2 mins

- students sitting around table, instructor also sitting at table

- assessment of pre-knowledge: Has anyone gone rock climbing before? Did you use a rope? Did you tie a figure eight knot? Have you heard of one / tied one for a different reason before?

- key concepts: how to tie a figure eight knot, and why we use it in climbing: by the end of this lesson you will be able to tie a figure eight knot and explain why we use it in rock climbing

Video: 1 min

Demon and follow along: 3 min

- students sitting around table, instructor standing up - holding rope up so everyone can see the demo, rope is now handed out so students have it as well

- i give demo of figure eight knot being tied, go step by step, so learners can follow along at each step

- do demo twice

Practice time: 3 min

- students sitting around table, instructor circulating around table

- give students time to practice themselves

Competition: 1 min (can skip if we dont have time)

- who can tie the figure eight knot the fastest?

Closing question:

- why do we use this knot for climbing

Extension: (can do this if we have time left)

- let's try a double figure eight knot

- show video 

Math Art Project Reflections

 (1) Group reflection:

Our group recreated Seven Linked Surfaces by Eve Torrence. The piece is made up of seven crocheted knot surfaces that all have the same boundary of two unknots linked through each other twice. The one that we chose to recreate is homeomorphic to a torus with two punctures, as seen later.

For one of our recreations, we contacted the artist to get some help. She sent over the pattern that she used, and directed us to some videos showing the technique. Even with this help it was still difficult to recreate the original piece and we didn’t fully succeed. The crochet technique is more involved than we thought, but it was really fun to try! There’s definitely still a lot to learn about topological crochet. Re-creating the piece gave good insight about the shape of it, and the math behind it. Following the method that Eve used allowed us to understand more about the knot and how to go from a knot to a surface. 

We used different colours in our recreations, and one of the recreations followed a totally different technique from Eve’s. We thought it might be interesting to crochet a knot link similar to the boundary of the seven linked surfaces following a technique published by Shiying Dong. We made two chains of length 100 stitches—or 10m and 10n respectively where m=n=10 stitches—and marked each one with stitch markers at the crossing points (8m/2n, 3m/7n, 9.5m/0.5n, 4.5m/5.5n, 2.5m/7.5m, and 2.5n/7.5n) so that the pattern would be symmetrical. Then we did single crochet stitches along the chains, going either over or under the stitch marker points depending on the type of crossing. At the saddle points where the surface twists, we did double crochet stitches to get the desired twist effect. 

Another of the recreations that we made was done without any instructions, working only from the photograph. A picture of this is shown below this paragraph. We started by creating three ‘kissing’ chain loops (three chain loops joined at two points in a row). Then we worked double crochets into the chain loops around until we reached an intersection. At each intersection we then had to very carefully determine which of the three paths to continue onto based on the picture. In the end the result was surprisingly similar to the original piece despite us using a very different method to design and create it. This goes to show that there are often many ways to approach and solve a given problem and that different people’s intuitions will lead them down different paths.

We also made a similar piece that is much simpler in order to better understand the techniques at play. This piece is simply a torus with two punctures and no tangling on the edges. Making this piece was very helpful in understanding what can be done with the technique that we used and in understanding how the more complex structure can be formed. Making this we realized that we could make the more complicated structure by taking this twice punctured torus and introducing twists into four of the linear sections. This was the basis of our class activity.

For our class activity we had students create the same surface as in Eve Torrence’s piece except out of paper. This involved creating three loops of paper each with a 360 degree twist and then taping them together to form the final product. The specific instructions are below.

Making a Linked Unknot Surface Activity

Duration: 5-10 minutes (longer than expected).

Materials:

    • 3 paper strips (approx. 1 cm by 10 cm) per student

    • clear tape

Instructions:

Make a clockwise twisted strip:

1. Pinch one short end of your strip in your left hand and pinch the other short end in your right hand.

2. Flip the end of the strip in your right hand clockwise twice.

3. Tape the surface to itself.

Make another clockwise twisted strip.

Making a counter-clockwise twisted strip:

1. Pinch one short end of your strip in your left hand and pinch the other short end in your right hand.

2. Flip the end of the strip in your right hand counter-clockwise twice.

3. Tape the surface to itself.

Tape the first two of the loops together at a 90 degree angle.

Opposite where you joined the two loops, tape the third loop to the second loop.

(2) Personal Reflection

I found this project very challenging! Looking at the artwork we wanted to re-create, I never imagined just how complex and involved the process would be. I still have a lot of learning to do around topological crochet. My favourite part of this project has been my correspondence with Eve Torrence. I sometimes forget just how amazing the math community is, and how friendly people are when you take an interest in their research. It's so cool that you can look up math art online, send an email, and then start a collaboration with someone. This is something that I've been missing since completing my masters degree - collaborating, and just the community at large! 

I've certainly learned some math from this project - prior to this I had very little knowledge of knot theory, and limited experience doing topology. I've also refined some of my crochet skills, so that's been nice. As a teacher, I learned most from presenting the hands on activity. I always forget just how long it takes students to do certain tasks. I thought that it would take a lot less time to cut out the paper and twist them into the correct shape. It was a good reminder to always give more time than you think you will need. 

The most frustrating part of this project was absolutely the recreation. I still haven't managed to re-create the piece successfully, but I feel I've invested so much time, I don't want to walk away until I have. Doing this art project was very reminiscent of trying a hard math problem. You try, try, and try again and it's still not right, but each time you've uncovered a new dimension to the problem, and come away with a new understanding. 

In my own classes, mathematical art will always be a part of what I teach. Having visualizations and engaging with math in a hands on way is so powerful. It also works to deconstruct the narrative that math isn't creative and thus can only be done by 'left brained individuals'. Most mathematicians I know are highly creative, artistic individuals. I think the limitations of doing mathematical art as a class is that the materials have to be readily available, and the art can't take too long to create. I don't think it would be reasonable to assign a topological crochet project, but making the structure out of paper and tape is a great way to bring this idea to the classroom. 

The Locker Problem

 

Favourite and least favourite math teacher: reflections

Unfortunately I have had many bad math teachers and a a few rare good ones. I thought about several of my less favourable teachers and tried to think of characteristics they shared that made me not enjoy their classes. Something that stood out in my mind was their inability to explain concepts in multiple ways across multiple modes. I often felt that the problem with my university professors was that the material they taught came so readily to them, that they couldn’t comprehend students inability to understand, or where confusions might arise. An example of this was when I was taking an introductory algebra class. I was so confused about modular arithmetic. I just could not wrap my head around it. I went to the professors office hours and the only explanation he could provide me was the one he had already provided in lecture- this formal definition of the modulas and the remainder, etc. I didn’t understand it just as much as I didn’t understand it the first time around. I repeated that I still didn’t understand and he repeated this explanation to me, making no attempt to bridge this concept to things I was familiar with - the click for example. After the third time I around I knew that it was useless, and I also knew that any other concept I didn’t understand, I had no hope of him explaining it to me. Ultimately I failed this course - literally got 10%, and I felt so bad about myself. As a teacher I think it’s important to not give up on students. Sometimes I’ll explain a concept in four different ways and the student doesn’t get it, but that’s okay, I’ll try a fifth! I also felt very judged when asking questions to this professor. This is something that’s important to me in my practice - that there are no bad questions, and to please ask if you have them. 

Similarly I reflected on the good teachers I had and what qualities they shared. These teachers were engaging when they taught (used tones of voice to convey excitement about the material, asked questions and allowed us to actively participate), kind, open to adjusting their lesson based off of where students are with tho the material, fostered discussions where everyone felt safe to make mistakes, conveyed material in a variety of modes (drawings, writing, gestures, videos), and a sense that they genuinely cared about our learning. These are all qualities that I strive to embody in my own teaching. I’m always learning from my instructors. When they do something I don’t like, I think about how I can be different as a teacher, when they do something I like, I think about how I can incorporate it into my own teaching. 

I also think that if I open up this concept of who my math teachers were, to include all people who have taught me math, then my favourite math teachers were my friends. I think it’s because they were the people who I felt safest around and could ask the most questions too. I suffer from the fear of asking a “stupid” question, so it can be difficult for me to put up my hand in a room full of people. Among my friends however, I always felt that I could ask them anything. 

The Educational Imagination: Reflection

 This article was both familiar and unfamiliar to me. There were many ideas that were new to me and a few that I had encountered before. The first thing that stopped me in the article was this idea that because classrooms are crowded, children need to learn to delay their gratification, as the teacher cannot give every student their attention all the time. This is something that I've subconsciously been aware of, and something that I would have categorized under learning 'social skills', and regarded as a positive. Reading the article however, they don't necessarily frame this as a positive thing. It really made me think about what other things I view as positive in the classroom, without critically examining them. For instance, is it good that we're teaching students to delay their gratification in this way? What is it preparing them for? Reflecting on my own learning, the best experiences for me were often in very small groups, where the class was ten students or less. Something that contributed to my learning was this instant gratification from my instructor - I could participate more freely, ask a question - get an answer right away, they could instantly check my understanding by asking me a question. Are there many places in the 'real world' where we encounter such large groups that we must work within? Or is this an artifact of schools that we are socializing kids to just because of resource constraints?

Another thing that stopped me in the article was 'The cultivation of imagination is not a utopian aspiration'. As a person, I deeply value art, imagination, and creativity. Something that I try to communicate about mathematics is how deeply creative and artistic it is. I think that math is often taught in a very 'left brained' way - meaning that it follows a predicable pattern, and everything is very algorithmic. To me, this way of thinking about math could not be farther from the truth. Good mathematics is out of the box, it's creative, often times it's brand new! It requires the ability to relate seemingly unrelated things, to build connections cross-disciplinarily, to think around problems, to visualize strange concepts (how do you really think in n dimensions? what does that even look like?!), the list goes on and on. Good mathematics require your whole brain, the left and right hemispheres, and this should be cultivated in schools today, not in some far off utopian future where art can be pursued because we've 'solved' all the worlds problems and can now take a break. 

I think that the BC curriculum does connect with Eisners ideas. I wasn't too familiar with the BC curriculum so I looked it up for math 10. Some things that stood out to me were 'Think creatively and with curiosity and wonder when exploring problems', 'Develop, demonstrate, and apply mathematical understanding through play, story, inquiry, and problem solving', 'Take risks when offering ideas in classroom discourse', and 'Use mistakes as opportunities to advance learning'. I think these things speak the hidden curriculum that Eisner discusses. The last two points highlight social learning - taking risks in the classrooms, learning from mistakes. The first two speak to valuing art and creativity as much as we value science and math. I actually really like the BC curriculum and I'm excited to work with it! 

Skemp on two approaches to teaching and learning mathematics: A Reflection

This article deeply resonated with me. I have always felt that I had two different experiences learning math, and that the math I studied in university varied drastically from the math I had learned in high school. I knew that it didn’t necessarily have to do with the content, but rather how I was learning and my engagement with the material. Reading this article, it became clear to me that I had experienced an instrumental learning of mathematics in high school and a relational understanding of mathematics in university. Something that prior to this I would have called under the names “learning math” and “doing math”. This brings me to the first thing that made me stop when reading the article. Skemp says “I now believe that there are two effectively different subjects being taught under the same name, ‘mathematics’”. This gave me pause because I had felt the difference between these two subjects, and I wholeheartedly agreed with him. I stopped again when he wrote “This difference is so marked that even relational mathematicians often use instrumental thinking.”. I stopped because I felt ‘called out’ by this, I do think that I am a relational mathematician who uses instrumental thinking. The most obvious example of this to me was when I was writing my thesis. I had to go through every line of mathematics with a fine-tooth comb and re-teach myself why everything worked the way that it did. Of course, when I had initially written it, I didn’t fully understand every step that I was taking, instead I was relying on my instrumental understanding of certain concepts. So again, in this idea I agree with Skemp.

Another thing that stopped me in the article was when Skemp wrote “… since an imaginary opponent who thinks differently from oneself is a good device for making clearer to oneself why one does think this way.” It made me stop because it allowed me to see the necessity of discourse amongst differently thinking people. Being in an echo chamber is nice – it’s validating, it’s easy, and it can be exciting to be surrounded by like-minded individuals. I think that I fall into echo chambers easily because they are comfortable. It is uncomfortable however, to have your ideas challenged, pushed, and prodded. This idea that it is good for one’s own growth to push back against your own ideas is not new to me, but something that I need to be reminded of, to keep pushing into the uncomfortable dialogues and re-evaluating my own ideas and values.

Hello World!

 Hi, my name is Danielle and I love arts, crafts, and math! :) Here's a picture of me in the Louvre this past spring.