Preparing for our Teacher's Circle!

The portions of the workshop that we can adapt are the actual tangle dance, the transition to algebra and writing the tangle number. The dance was a fun visual and really connected to what we were learning, then using the functions to write the tangle number is very concrete.

I think we should start the teacher’s circle with what knot is and how we can make a knot into a tangle using the ropes. After we introduce that we can start by doing the tangle dance to create the tangle. This should be done a couple of times so teachers can really understand what is happening when we twist or rotate. After we complete the dance, we should start talking about the algebra behind the tangle what functions make each. Then we can start with the tangle number working to untangle then write out how to create the tangle and draw it. Once the tangle is drawn, we can talk about how to color the tangle. I believe that this would take the whole 2-hour time period, giving teachers an inside glimpse to knots and tangles.

I would like to take responsibility for untangling the tangle then working to draw it!  

My Favorite Theorem--> Laura Taalman

Listening to the podcast My Favorite Theorem is fun! It is refreshing to hear people talking about math and making jokes, that is what I liked particularly about this most recent episode. I think that the theory Laura Taalman talked about was very interesting however I was very confused at first. The way this theorem was talked about really shows how interesting and exciting knot theory is. It is difficult, intricate and in some ways still developing. This theorem was only proved years ago, very different from many mathematical theories that were published many, many years ago. I like that Laura compared her favorite theorem to champagne. She said that this is something that should be celebrated and I agree. I want my students to understand that math is something that should be celebrated. Mathematicians work so hard to develop theorems with proofs and it is exciting!

I think my understanding of knots and algebra has evolved very much. This course has forced to me to think outside of the box, like WAY outside the box. It has forced me to become very visual and artistic. I think that it is making me a better teacher in a way, its giving me an outside perspective as a teacher for what my students think about and how they learn. They things that have worked best for me so far are the videos of our previous classes. I like to go back and review them while I am completing homework or prepping for our next class. Another thing that has worked for me is all the activities we have done. It really brings home what have learned and I use them as a visual if I am struggling to figure something out.

I have had many challenges this semester because I am used to understanding everything. When I am sitting in class, I understand what we are doing but when I get home I draw a blank. I am not sure what I could do differently to fix that. I also struggle with the fact that there really isn’t much information on knot theory, that I can find online, that would be helpful. In the past if I was struggling with something I would just look for resources and how-to’s online but there isn’t anything.

Math Teachers Blog o Sphere

After searching through the MTBoS I came across many fun activities. Something cool that I found relates to what we did last class with being colorable.

This math teacher gave his students a page of knots. Using this we could have them explore the concept of being "3-colorable or 4-colorable" by giving a few more instructions like alternating or not letting any colors touch. Students could make their own thoughts about this.

Something else I say and liked related to Algebraic thinking.

This teacher used pictures to represent variables and had the students create a equations to help them solve for each variable. This definitely gets the students to think about algebra and how to create equations. Pictures really make this task easier on students. I think that when they see so many letters and numbers together it can be scary and overwhelming.

I think that teachers would be looking for fun, engaging activities that make SENSE to students. You don't want the concept to go over their head where they can't even grasp the concept. It needs to be on level with where your class is so they can really have fun with the activity. The activity also needs to be interactive... There have been times when I have been in a PD and I am sitting there bored out of my mind. I have gained the most from Professional Developments that are interactive and hands on.

I think our content area is so unique. When I first thought about taking this class I was confused about how knots can be a part of math and even after the first class my mindset changed. It is a fun, different way to think about math. With doing this PD for teachers it will allow us to show them how math and knots come together.

Professional Development

Although we have already done these in class my favorite activities that relate to knots and algebra is the tangle dance. It gets you moving and really makes you think about how to create a tangle and what the proper over/unders are like. I think this would be really fun and easy for high school students to do. Right now we are talking about inverse functions and I think it would be really cool to connect it somehow. I tried to be original and come up with something on my own, I think this would be the most fun!  

My Favorite Theorem

After listening to the podcast of my favorite theorem with Dr. Candice Price, I am happy to say that her theorem has been one of my favorite concepts of this course. I love how we are able to find a rational number given a tangle and how we are able to sketch a tangle given a number. In their discussion, Dr. Price explained a tangle as webbing inside of a ball. This reminded me of what we did with the ropes inside the envelope. Even though we have discussed this in class, I think introducing the rational tangle dance to the younger students will give them a different perspective on math, add some excitement behind how “math” can be applied. Especially since fractions are so “scary” to all students K-12, showing them that they have applications and are not that scary could make them excited to use fractions in the rest of their math career. 

At the end of the podcast, they talked about how to pair the theorem with something "real". I think that this is a great idea and brings life to these concepts. Dr. Price said that she would pair her favorite theorem with a Neopolitan Shake: three different flavors--> chocolate, vanilla and strawberry all equally interesting just like Rational Numbers, DNA and topology and when they are all mixed together they create a wonderful combination. 

Some new aspects of rational tangles that Dr. Price brought up was how rational tangles show up in DNA topology. DNA are long thin strands that wrap around their selves creating rational tangles. This allows scientists to use what they know about rational tangles and apply this to DNA.

Some questions that surfaced for me was the concept of non-rational tangles and how they are classified into prime and locally knotted. I think we have talked about this before briefly in class, but I couldn’t find the video for more information. From what I listened to the prime tangles were similar to what we discussed with Dowker. I would love to have more information or even visuals of what non-rational tangles looked like and is it ok to say “irrational”?

I know this is something that we are going to talk about in class and work on but I feel very passionate about introducing the concept of rational tangles to younger students. I would like learn how to accurately and confidently explain the concept of rational tangles to my high school students. Also I would like to make sure I find a way to bring rational tangles to a level where my students can understand and not feel intimidated. Something else that would be beneficial in explaining this to our students would be to come up with a resource that they can go back to reference when needed. 

Ka

I found that the paper by Tanton and the paper by Kauffman and Lambropoulou have the same Theorem 1 about isotopy. Kauffman and Lambropoulou's discussion on theorem 1 is lengthy and without any diagrams. Tanton's explanation is direct and to the point.

I found it very difficult to read Kauffman and Lambropoulou's paper. It was very wordy and detailed. As much as I love math and find it extremely interesting but I am not a fan of reading papers like this. It takes me a while to read and if I want to absorb the information, I need to read and re-read each part. So with their paper it took a very long time for me to get through it. And to be honest, I am not sure if I understand it fully yet. On the bright side, I really like their diagrams and drawings. I found those easy to understand and useful.

I think Tanton's paper is easier to understand and more reader friendly. I found it helpful to break up definitions and explanations with diagrams. It allowed for us readers to digest what we just read.

I believe that the audiences for both of these papers are fairly similar. I think that audience for Tanton's paper is grad school students who might not have much experience with  reading though papers or dealing with difficult concepts. I think Kauffman and Lambropoulou's paper is more for teachers and professors looking to dive deeper into the concept. Maybe readers with a background in the content area.

Professional Development

   Professional Development

       I would first like to say that the activity was so much fun! It really connected what we were learning during class yesterday and this concept will stick. Teacher professional development can oftentimes be boring. Depending on the timing of the professional development, teachers may think that it is a pain and they could be doing so many other things with their time. I have experienced this a few times over the last 2 years. When PD came around, I could see the look of annoyance on teacher’s faces and comments of “I could be planning or grading right now”. The purpose of attending these professional development sessions is to have the opportunity to make ourselves better. I embrace that opportunity to the fullest. I am always looking of ways to expand what I know about teaching and learning, professional development sessions help in that aspect.

       Over my 5 years of teaching, I have attended many different professional development sessions and the ones that stick out in my mind, were the ones that were fun and was something I could use in my classroom. I really enjoyed how “hands on” our activity was and this reminds me of a very fun PD I went to. The professional development I was able to attend was on Common Core Math and this PD came at a time when I was struggling to develop my own curriculum for my school. I was the only math teacher at the school, I was given the Common Core Standards and a good luck. I thought to myself “ I don't really know what these standards are so how can I make this fun?”. When I went into the PD I was pleasantly surprised by the hours we spent with our instructor. We did so many fun, hands on activities like learning to juggles and using manipulatives. It was wonderful and I was able to bring back so many things to my classroom.

       Another PD that reminded me of what we did yesterday was a technology one I went to in October. We learned the different ways to do a quick assessment in the classroom. We created our own games and played them as a group. Again, there were something that I could easily use in my own classroom. The instructor gave us easy to implement ideas and strategies rather than a lecture and a pamphlet.

       I wish professional development workshops were move vibrant and for the choices to be topics that are currently going on in the schools. It would be beneficial if more instructors did hands on, interactive activities with the teacher so we could use it as a tool in our classroom. Another idea for professional development is to give us teacher the opportunity to brainstorm with each other about ways we can bring what we just learned back into the classrooms. Isn’t that what these professional development days are meant for? For us to learn and implement things into our classrooms. I believe that if teachers had some time at the end of each PD to discuss ways to use what they learned in their OWN classroom, there would be less huffing and puffing about PD days and more excitement generated!

Algebraic Thinking

    Algebraic Thinking

           When I was young, I was (and still am) a lover of Algebra, so I want algebraic thinking to only involve letters, equations and solving for one answer. However this is not the case, as I have moved on in my math career and into teaching my thoughts on algebraic thinking have grown, matured and have changed. Algebraic thinking typically means for me looking for patterns and relationships. It allows for generalizations to be made about something, putting an equation or rule in place. I also think that algebraic thinking means taking some abstract concept and putting a face on it, or applying some type of rule. It is similar to other forms of mathematical reasoning because it challenges the student to think critically and provides a certain level of understand (some more than others). Algebraic thinking differs from other forms of mathematical reasoning like deductive reasoning. Deductive reasoning takes known facts and makes a conclusions.

         My students most use algebraic thinking when they see a concept for the first time. I usually start each unit out with some type of investigation. For example I gave my students many graphs of polynomial functions. I asked them to find some type of relationship or pattern between the number of zeros and the degree of the polynomial. This allowed for many conversations to take place, they had to develop some sort of rule to explain why a 4^th degree polynomial can’t have 5 roots, but a 5^th degree polynomial can have 4 roots. It took a little while for each group to sort through the graphs and come to some type of generalization. I think that this type of thinking paid off in the long run because I did not just give them the rule and memorize it, but figuring it out by themselves. 

        I imagine that algebraic reasoning can mean problem solving, justifying and proving. I am currently struggling with my students and their constant question of “why do I need this in real life?” In real life most of my students are not going to have to solve a quadratic equation by factoring or multiply rational expressions, but they will need to problem solve. The problem solving skills and thought process to answer the questions will help with real life problems. Being able to figure out what the “problem” states, what resources you have and what you need, setting up a way to solve the problem, actually doing it and then checking to see if you are right can translate into most careers.

          Math and the skills used in the math classroom (algebraic thinking and all methods of mathematical reasoning) all us to “sharpen” our critical thinking skills for us to use later on in life when we are given a problem that we do not know how to solve right away. We can use our algebraic thinking skills that were learned in a math classroom to overcome the problem.  Algebraic thinking also does not always have to be for the math classroom or on a homework problem. Anything that calls for problem solving or making generalizations will use algebraic thinking.