Colorful Tangle Dance

When thinking about building a lesson plan for a 2-hour teachers' circle around our theme, I think it is important to keep it both engaging yet concise. I think it is also important to understand the people who will be attending the workshop. Yes they are math teachers, but they may not have done upper level mathematical thinking in years, so it is important to keep things at a semi-basic level, so that they understand the concepts. I don't really have a preference as to which piece of the workshop I plan, but I think I would enjoy planning the opening best. Planning an activity to have everyone start with and give them a basic introduction as to what the objectives are and what we plan to cover during the workshop. I am assuming we are each going to be in charge of a 30 minute portion, so the first half hour would work perfectly. 

I really like the idea Tanton uses with his tangle dance After the dancers have danced for a while, an envelope is placed over the tangle and another person yells out another series of instructions, then stop. The envelope is then removed resulting in their tangle untangled. This may spark the question, how did this work? Depending on how many people we have at the workshop, we could split them into groups if needed, and have 2 different tangle dances going. I think it is important to incorporate all three parts of Tanton's tangle dance activity, before diving into coloration. 

Knot Theroy

The podcast discussed much of the following:
Knot theory is so abstract resulting in obnoxious formulas & numbers stumping calculators, making knot theory interesting, challenging, and valuable. Conclusions are "funny" from research & proofs . The more mathematicians look into knot theory, the "better" the conclusions/formulas become. I thought it was interesting that she connected the theorem to Champagne, but the reasoning made sense. Champagne is an unpractical drink that could cause a headache, much like knot theory. 

Mathematics is constantly being analyzed. Mathematicians expand upon each other's work and manipulate theories in different ways. There are many ways to explain and visualize various things. I think this idea is extremely important for both students and teachers. Just because one person or resource says to do something one way, doesn't mean that's the only way to do it. Same goes for learning styles. Not all kids learn the same way, so utilizing various strategies and methods helps students learn more efficiently. 

Going into this course, I knew absolutely nothing about knot theory and how it connected to the algebra world. This course has been truly fascinating, and I am so happy I took the leap and tried it! Everything we have covered so far has been new and enlightening; from tangles, to knots, to tri-coloration & tangle numbers. 

This course definitely has been challenging in many ways. It took a good couple weeks to finally nail down how to draw the tangles and knots we were working with, but I am finally getting the hang of it! Also, being able to take the concepts we covered in class and directly try to manipulate them at home on my own has been challenging. There have been some weeks where I go home lost and I think to myself, "What the heck did we just do?", but there have also been weeks where I go home feeling confident. I think my biggest challenge is being able to apply the theorems we uncover to specific exercises. I usually work very well with seeing examples first, so that when practicing, I have something to mirror off of. 

I have loved how hands on this class has been. I definitely am a learner that needs hands on instruction when developing abstract concepts. It helps me visualize what is happening, so that I can get the full picture. 

Teachers & Math

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   1) The human knot activity was the activity that peaked my interest most. I have done this activity before and it is loads of fun, but is also a great team builder/strategy building activity. This is an activity we could do to start the event off, to help everyone get comfortable with each other, and loosen everyone up, before diving in to knot theory. I still believe that hands on activities will be best to keep teachers engaged and will help keep things related to them and their students. 
   2)On the MTBoS blog, I found a post written by an elementary school teacher, who was writing about a post she saw regarding complex fractions. The post she was referring to, was a post about complex fractions in higher level math. She was trying to connect complex fractions at the higher level to division in 3rd grade and 5th grade. She had lots of great questions throughout her post like, "How does the way we represent fraction division relate to one or both of these ways to think about fractions?". I really enjoyed reading this post, because I would never expect an elementary teacher to even think about complex fractions. When I think about the mathematics taught at the elementary level, I think about foundational concepts and basic problem solving skills, when really the ideas children are exposed to are in direct correlation to what they will see in high level math, they just don't know it yet. 
   
   When thinking about in-service for teachers, I think it is important for teachers have the ability to ask questions and reflect. Having a question area or a "parking lot" for teachers to write down their thoughts or questions throughout a workshop, I think will not only benefit the educators, but also help drive the workshop in various directions. Teachers are always thinking about how to introduce their students to topics in various ways in order to meet the needs of all students, so being able to work together to answer each other's questions I think is very important. Reflection is also a huge piece of educating as well. The author of the blog post mentioned that when she first looked at the complex fraction post by another educator, she thought it would be way too complicated to even look at, "I have to admit, when I saw her Twitter post with the words pre-calculus and simplifying complex fractions, my inclination was to skim right by because I would not understand the post anyway." This is something that we as educators have to help encourage ourselves to do just as we do with our students. Reach out of our comfort zone. Talk to educators of all age levels. Reflect. Reflect. Reflect. 
   
   https://kgmathminds.com/2017/11/26/fraction-division-and-complex-fractions/

Project Introduction Ideas

Most activities we have done in class have hit every factor on the list of what makes a good math circle problem. For example, engaging, hands-on, minimal lecture, low-level entry, , abstract, etc. I particularly think that the most important thing is that the activities are hands-on and include something that teachers can take back to their classrooms. Two activities that we have done in class come to mind. 

The first activity is the activity we did the first day of class where knots were displayed on the front table, we then had to draw them and describe them in various ways on the board, which sparked a great discussion. This activity I think would be a great starter activity to get teachers moving and talking about knots. 

The second activity I think we can use in a variety of ways is the tangle dance. Every time we break out the ropes and actually take the time to manipulate things, it really brings everything together and allows us to have great discussions. 

I have not yet thought of any specific "problems" per say, but I think it is important to introduce knots on an entry-level basis, with small hands-on activities that will keep teachers engaged throughout the PD event. 

My Favorite Theorem Podcast

Notes from Podcast:

Candice Price: John H Conway's Basic Theorem on Rational Tangles
1 to 1 correspondence between rational numbers and rational tangles
Rational Tangle Dance
Non-rational tangles-prime & locally knotted
DNA topology: DNA is coiled around itself (twisting)
Pair with Neapolitan shake: take 3 flavors mix together -DNA, topology, tangles

Thoughts:

This was the first time I have ever listened to a math teacher podcast and I am very intrigued! I felt my nerd side come out a bit :)

It was great listening to professors discuss tangles like we have been learning about in class. Everything Candice Price discussed regarding the Basic Theorem on Rational Tangles entailed aspects of concepts that we have already discussed in class. Mainly the idea that tangles can be represented by rational numbers and that it is possible to have non-rational tangles. One part of the podcast that really sparked my interest was the discussion of DNA and its' comparison to a tangle. I have read about this a little bit before now, and we have discussed it a bit in class, but it again peaks my interest a bit more. I would love to dive more in to tangles outside what we have seen in class, specifically its relationship to biology or other sciences. I also thought that Candice's pairing of a Neapolitan shake with tangles was very interesting. It made me think of what I may pair tangles with.... the first thing that comes to mind is hair braids. I am constantly braiding my hair, and growing up, I was always looking up new ways to braid and twist my hair. Tangles continue to intrigue me more and more with each lesson!!

Compare & Contrast

Both papers discuss similar aspects of tangles, but in very different ways. Tanton writes to an audience that may not have much background in math and/or knots. Tanton breaks down each idea step by step with lots of diagrams, making it very easy to follow. Kauffman and Lambropoulou write much more academically focused. In order to be able to fully follow and understand Kauffman and Lambropoulou's article, you must have some sort of background in mathematical knowledge (especially proofs). Both papers discuss what it means for tangles to be isotopic as well as operations on tangles, but as I have previously mentioned, Tanton's paper, is much easier to understand and follow. 

The advantage to Tanton's paper is that anyone could read it and follow it. The disadvantage is if the reader was someone well versed in mathematics and knot theory, than they may be bored with the basic explanations and humor that is embedded throughout the paper. 

The advantage to Kauffman and Lambropoulou's paper is that it goes much more in-depth with each concept, thoroughly proving and explaining every idea. The language used is also much more academic compared to Tanton's. This could be a disadvantage depending on who the reader is. Also, Kauffman and Lambropoulou do not use as many visuals in their paper, which may put them at a disadvantage to a reader who is more of a visual learner. 

As a reader with a mathematics degree, I was able to follow both papers, but I connected most with Tanton's paper. As my knowledge of knot theory is developing, Tanton's paper was easy to follow, and I was able to make connections and develop take-aways rather quickly. I also enjoyed the whit and humor that was embedded throughout Tanton's paper. I felt that Kauffman and Lambropoulou's paper was extremely dry, and it was difficult to get through. Yes I could understand the majority of what was being discussed and proved, but it was just SO in-depth, it was very easy for me to get distracted. 

If I were to give math teachers one of these readings to introduce them to tangles, I definitely would have them read Tanton's paper. I think that teachers would be thoroughly engaged in the paper, and they would not feel like they are swimming in a pile of academic jargon. 

Professional Development

          The activities we have done in class so far, has made learning about knots a blast. Working hands on to develop an understanding for knots, makes it much easier to understand, than if we were just given definitions, formulas, etc. When it comes to professional development, this rarely is the case, especially at the secondary level.
          I have been teaching for three years and I have not experienced any hands on professional development. I would say about 90% of the professional development I have had over the past three years has been curriculum work. Due to NEASC, schools have to have all curriculum and common assessments documented formally on unit plan templates, so that is what most of our PD time is spent doing. When we are not doing curriculum work, we are watching speakers talk all day long. For example, this past fall, I attended a two day PD on trauma. The speaker was extremely knowledgeable and I was interested in learning about the content, but it was EXTREMELY difficult to sit and listen to this man talk for 2 days, 8 hours straight, each day. The presentation was not engaging at all, causing me to lose interest and focus very quickly. This is what we see in our classrooms too when lecturing to much/often. Yes, we are adults and have a longer attention span, BUT we get very little out of being talked out for hours on end.
          I understand that curriculum is important and I do believe it should be documented, however I do think there should be a mix between curriculum work PD and engaging presentation type PD. Providing teachers the opportunity to experience PD that entails hands-on activities like we are doing in class, I believe would give them the tools and motivation to want to be more hands-on in the classroom.