Learning More about Conway’s Rational Tangles

I just led the Philly Area Math Teachers’ Circle and did a version of John Conway’s Rational Tangles.  I don’t claim to know everything about them by a long shot, but I put this together in case some of the participants want to explore more beyond what we covered in math circle.


My interest in this started when my husband traveled with me to the Northwest Math Conference in Whistler in 2018. He’s a retired high school math teacher, so I told him to go to Fawn’s session because she would make him do math.  They did this activity in her session, so I decided I’d do it when I was leading our local math circle this February.

Fawn Nguyen: Students Embroiled in Conway’s Rational Tangles [Blog post]. Fawn learned about this activity in a math circle.  Here she reports out on trying it with her students.  Be sure to watch the charming video at the end!

Tom Davis: Conway’s Rational Tangles [PDF]. Everyone seems to point to this or at least mention it, so I figured I better too.

Tom Davis: Conway’s Rational Tangles [Video]. Don’t want to read the PDF above? Head on over to YouTube and watch all 21 videos in the playlist of Tom leading this activity in a math circle.

James Tanton: Understanding Rational Tangles [PDF].  Want to dig into the math of this?  So did James, and he wrote about it.

Matthew Salomone: Math 596: Topics in Algebra and Knots [Video]. Ever wondered what an non-Abelian group is? Check out these support materials for Matthew’s course. (Hint: non-Abelian comes up in the third video and you will totally understand it.)  Oh, and it also explains, with visual aids, a bunch of the math about rational tangles.

Sense-Making, Math, and Literacy at CMC South

My thanks to everyone who participated in my session.  I’m sorry I didn’t take any pictures during our mingle, because there was a lot of activity and sharing going on!  (Wonder what I mean by a “mingle”?  I guess you should have come.  It was great.)

I added pictures of all your “roles” posters to the slides, in case you want to check out the others more closely than we did during the session.

Download a PDF of the slides

Emily and Margaret’s blog post about sense-making with kinders

Brian Bushart’s blog post about numberless word problems (and graphs!)

My 5-minute Ignite talk that summarizes our session today, Sense-Making: It Isn’t Just for Literacy Anymore

I look forward to hearing more from some of you about things that you try when you get back to your classrooms, or how ideas coming from my session start productive, or at least interesting, conversations with your colleagues.  Leave me a comment here, or tweet me at @MFAnnie.

Fun Times at the Northwest Math Conference! #NWMath

Thanks to everyone who got up and did math and thinking about sense-making with me so early this morning!  I hope you had some good conversations with your partners in those moments where I asked you to chat. Hopefully some of you even made some new friends.

As promised, below is a PDF of my slides, as well as some links to related resources.

At the end of the session, it sounded like many folks talked to their partner about what they might try next week.  My thanks to those of you who made public declarations.  Thanks for being brave.  I’d love to hear from all of you.  So leave a comment here, or tweet at me (@MFAnnie) and tell me how it went, or what you’re wondering about, or whatever.  Thanks!

My Sessions at #NCTMRegionals Hartford

My thanks to those of you who came to my sessions. I look forward to hearing from some of you going forward, so drop me a comment here or find me on Twitter.

Session 23: The Power of Ideas

Grab a PDF of the slides

Here’s the blog post I mentioned: Emily, Kindergarteners, and Sense-Making

You might be interested in these 5-minute Ignite videos:

The Power of Ideas, NCSM Annual Meeting, 2017

Sense-Making: It Isn’t Just for Literacy Anymore, NCTM Annual Conference, 2015

Session 52: Using Technology to Increase Conceptual Understanding in Algebra and Geometry

Grab a PDF of the slides

Links to the technology I highlighted

Types of Triangles, Desmos

Runners, Web Sketchpad (The NCTM Illuminations version used to be linked from here)

Galactic Exchange vending machine, Java

Algebra Tiles, the National Library of Virtual Manipulatives

You might be interested in this Ignite video, which is basically a five minute version of this hour-long session:  Technology. Why Bother?

PCTM 2018

Thanks to everyone who came to my PCTM session.  Next time, bring space heaters!  Here’s a copy of the slides.  I’ll add a link to the video of my Ignite when it’s available.

PCTM 2018 Slides [PDF]

CAMT 2018

My thanks to all who came to my sessions.  Keep me posted on ways in which you end up using some of these ideas either in your planning or in your actual classrooms in the fall.  Good luck!

The Power of Ideas, Grades 3-5, Monday, 10-12

Slides [PDF]

Historically, the teaching of mathematics is particularly guilty of giving students the message that their ideas aren’t important and that they only need to memorize and regurgitate the ideas of others.  If students are to become critical thinkers, they need to realize that the ideas they bring to the classroom are worthwhile.  They should also have opportunities to express and refine those ideas, as well as to learn to listen critically to the ideas of others.  We’ll talk about some examples of ways in which we as mathematics teachers ignore students’ ideas, then explore different routines and strategies for making students’ ideas play a more central role in the classroom.

Sense Making: Is It at the Core of Your Classroom?, Grades 6-8, Monday, 3-5

Slides [PDF]

The National Research Council points to a “productive disposition” as one of the key strands of mathematical proficiency.  A major part of this strand is viewing mathematics as something that makes sense.  Are your students making sense of the mathematics they explore? Do they feel that mathematics is an inherently sensible endeavor?  We’ll look at ways in which students don’t make sense of mathematics, consider why, and discuss strategies for making it a larger part of the expectations in your classroom.

My NCTM 2018 Session

Here are some links to resources mentioned in or related to my Thursday afternoon session.  A PDF of the slides will be available after the session.

Here’s the link to the slides [PDF].

This session is the long version of an Ignite I did at NCTM Boston in 2015. You might use this 5-minute version to introduce colleagues to some of these ideas and start some conversation.

  • Joe Schwartz’s Blog – Joe is a recently-retired elementary math specialist in central New Jersey. He blogged about using Noticing and Wondering at his school, among other things, and I’ve pointed you to a N&W Sampler that he wrote in January 2015.
  • Numberless Word Problems – Brian Bushart has done a great job blogging about and preparing numberless word problems, including the mouse problem we looked at during the talk.
  • Tina Cardone’s blog post about how using noticing and wondering saved her time.
  • The NY Times Learning Network’s What’s Going On in This Graph?
  • Beth Brandenburg is a lead teacher in Washington County, Maryland, and makes a lot of use of Noticing and Wondering in her school (and her district – she used to be a district-level lead teacher). I’ve pointed you to a post she wrote in August 2015.
  • You can find more of my Ignite talks by rooting around on the Math Forum YouTube Channel.
  • Check out tweets posted about Noticing and Wondering, using the #NoticeWonder hashtag (and add your own!).

My #NCSM18 Session with Joe Schwartz

It was a stroke of genius on my part to invite Joe Schwartz (@JSchwartz10a) to bring his perspective as a school-based math coach to my sense-making session at NCSM. He offered some ways in which he encouraged the greater use of sense-making tweaks in math instruction in his school and his district. You can get a copy of the slides and related links at his blog.

If you’re going to Twitter Math Camp 2018, look for a chance to think about this with Joe and me.

Emily, Kindergarteners, and Sense-Making

Time for a guest blog post! Emily Payán is a beginning teacher at a high needs elementary school in a large suburban school district north of Minneapolis. Her mother, Margaret Williams, is the district’s Teaching and Learning Specialist for K-2 Math. Margaret is also an adjunct instructor for a local university and happened to be Emily’s Math Methods instructor. I know Margaret because she is part of the Minnesota “math family” that I’ve gotten to know while serving on the leadership team for Math On-a-Stick at the Minnesota State Fair. As a prelude to inviting me to lead a day of professional development for all the K-5 teachers in her district this August, she buttered me up by telling me this great story about the huge effect my advice from the previous August had had on her daughter’s experience teaching kindergarten. I replied, “Sure, I’ll do the PD, but you and Emily have to write up your story and I’ll post it on my blog.” All parties kept up their end of the bargain, and here’s what they wrote.

Margaret: I was looking forward to hanging out with Annie while volunteering at Math on a Stick during the 2016 MN State Fair. My daughter, Emily, had just finished her first year of teaching. Her second graders made wonderful progress in math but, as is often the way for beginning teachers, she was being moved to kindergarten. Being able to talk with Annie and get suggestions regarding the K math experience was going to be awesome!

Emily: To say I was nervous would be an understatement. I had completed my student teaching in a kindergarten classroom in a neighboring district. In this classroom, worksheets were the modus operandi and manipulatives were frowned upon. Needless to say, I had no idea where to start or what skills to engender in my new students to set them up for success in their schooling.

Margaret: Right! It was frustrating for me, as her mentor, to see and hear how the student teaching experience had been. Once she had her own classroom, we worked together a lot in planning math lessons for second grade using a Cognitively Guided Instruction framework for implementing Everyday Math. What would that look like in kindergarten? I was excited to talk to Annie. I was pretty sure I knew what she would say: The children should have lots and lots of opportunities to count collections. I was wrong. When Annie and I had a chance to talk, she told me that the most important thing Emily could do, as a K teacher, was to hold her students accountable for making sense.

Emily: While this made sense for a mathematician of any age, I had to wonder How on earth do I start when only three of these students have had a preschool experience? Less than half of my class exhibited predictors that would indicate foundational skills and concepts for number sense and future success in mathematics. My mom’s Math Methods course included lots of conversations around Boaler and Dweck’s work regarding mindsets and mathematics. I knew that if I could just get these kids to engage in real world math without the fear of mistakes, they would be successful and confident – regardless of what their initial assessments indicated.

My mom helped me collect items such as buttons, pine cones and blocks as well as pictures of real world things that would be interesting to five year olds: puppies, turkeys, plants (after all, math is EVERYWHERE!). I also used resources from Christopher Danielson. His book, Which One Doesn’t Belong and his open-ended approach to counting in picture books inspired my students to value multiple perspectives and to think of math as a creative domain. Trying to pull all this together felt overwhelming. Looking back, I think this was good. It wasn’t long before these “struggling” kindergarteners were mathematizing the world around them and, just as importantly, talking about their thinking. They were all in.

Conversations started flowing as they worked through their reasoning. I challenged them with simple questions like, How do you know?, Do you agree with what Ylva said? Giving an answer wasn’t enough, they had to be able to talk the class through their thinking. Sometimes the justification didn’t make sense to someone and they began to (respectfully) challenge one another and ask questions of their classmates. It was fantastic! I had eighteen 5 and 6-year-olds engaging in deep, meaningful questions about all types of mathematical tasks: addition and subtraction, multiplication and division.

This means that mistakes were inevitable. I intentionally modeled making mistakes and talked about Boaler and Dweck’s research on the brain growth that happens when a mistake is made and thought through. We watched videos on synapses and even had class meetings in which we all shared something from the day that was challenging. We congratulated each other on making our brains stronger by ‘sticking with it’ and not giving up. By the middle of the year, excitement beamed across their faces when a mistake was made. They knew that this meant they were on to something, and they accepted the challenge.

For example, my mom (known in my classroom as “Nana Math”) had taken a picture of a group of turkeys and sent it to my class to think about what was mathematical. Autumn decided she would count the legs of the birds – three turkeys were standing and two were sitting. Counting the legs she saw, she decided there were five legs. Dakota studied the photo a little more and thoughtfully explained, “I think there are more legs, Autumn. I think there are legs we can’t see. Each turkey should have 2 legs, right?”

Margaret: These little problem solvers were on fire. It was amazing to listen to them explain their thinking. I would go into Emily’s classroom to help with assessment interviews and wonder when was I going to get to the low achieving students. These kids were confident and supportive of each other even though they often challenged each other’s ideas. It was a math teacher’s paradise! Emily will be teaching Grade 2 this coming year. We look forward to another year of focusing on reasoning and making sense!

I can’t wait to hear about Emily’s second grade experience this year when I see Margaret again next summer in Minnesota. Maybe she’ll send me some updates before then. And by the way, y’all should come join us at Math On-a-Stick!