1. James Tanton has a video and series of videos and lessons called “exploding dots”. They start with a very intuitive look at number systems and trace through arithmetic, polynomials, decimals, sequences & series, irrational numbers, negative bases, and more. As another teacher at PCMI remarked, “Every time I watch, I am amazed at how far these concepts reach. If our kids could see this from when they were little, think of what they would understand.” This one was found by Matt. Watch Tanton here:

http://gdaymath.com/courses/exploding-dots/
2. While you’re working with Tanton’s exploding dots and number systems, you might want to get a better sense of scale for what 10^9 or 10^-5 really mean. Check out The Scale of the Universe, and interactive applet that lets you zoom in and out and see objects at different scales. Thanks to Chris for this one:

http://scaleofuniverse.com/
3. Ilana Horn is a professor at Vanderbilt University who wrote the book

*Strength in Numbers: Collaborative Learning in Secondary Mathematics*. In it, she outlines how to effectively use group work to create a learning environment in the secondary mathematics classroom. She goes on to explain how students experience learning mathematics in collaborative settings, and how teachers can develop tasks, concepts, strategies, and tools that create successful group work. She has some interesting views on group work, some I agree with, and some I would modify. Here is a link to her blog:

https://teachingmathculture.wordpress.com/about/

4. We had a discussion about group quizzes today. In her book

*Strength in Numbers*, Ilana Horn suggests that group quizzes may be used as a review for an upcoming test. Groups of students are asked to complete two to four big problems without the teachers help or any help from outside the group. Upon completion, the teacher collects and grades just one paper from the group (pg 59). While this practice has its merits, some teachers at PCMI felt that students would tend to let the highest achieving student do most of the work, and then copy off of his or her paper before they were collected. Some solutions to this issue include having students work alone first, grading every students paper, having a higher level of supervision during the work, and asking students to grade each other participation. However, the best solution is to give the group 15 minutes to work together right from the start. They can use this time to ask questions, formulate ideas, or attempt the problem. After 15 minutes, they receive a small grade for their collaboration, and then must finish the task individually. I like this solution because it promotes collaboration and questioning, yet still places the final burden on the individual student.

5. Joe introduced the group to an intuitive way of measuring irrational numbers. I believe it was presented at an NCTM conference, and has been reposted in a few different places. Yet, it’s a good activity that deserves to be passed around. Here is one incarnation:

https://thescamdog.wordpress.com/2011/05/31/radical-ruler/
6. (Extra Credit) The International Congress on Mathematical Education (ICME) is being held in Germany next year, and they’re looking for representatives from PCMI to go. Hmm…

7. (Extra Credit) I presented both a 10-minute share and a 5-minute short today at PCMI. The 10 minute share focused on mathematics and music. Although there were some technical issues, the presentation went pretty well. The 5 minute short was with Matt, and we explained how to have students create a survey, and then use those results to create game-show style questions. We gave a survey to the PCMI teachers and created some game show questions – the results (without the statistics) are below.

*What is the distribution of number of years teachers have been in education? Normal with a positive skew.*

*Who likes the food at lunch more: tea drinkers, coffee drinkers, or both the same? Coffee drinkers.*

*What percentage of PCMI teachers are from an east coast state? 56% ± 15% teachers.*

*What do teachers enjoy more: Darryl and Bowen's comments or the comics before RoP? Darryl and Bowen's comments.*

*Which had a lower standard deviation: how much teachers enjoy afternoon cookies, or how much teachers enjoy goats? The cookies (although both have high standard deviations, and are very divisive topics).*

*Who likes goats more, teachers with dietary restrictions (vegetarian, vegan, gluten intolerance, etc) or those without? Everyone loves goats the same (difference was not statically significant).*

*Is there a correlation between teacher’s enjoyment of breakfast food and lunch food? Not as much as you’d think, r = 0.45.*