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/