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.