Day 2 CMP Digging Deeper

June 21, 2016

Orchestrating Mathematical Discussions

I have control issues. I like to be in control of making sure students learn. I would love to use my imagined superpower to open their brains and force them think and reason. I find this personality trait is common to many teachers. Several people have asked me how I am dealing with that using inquiry based learning in math. Honestly, orchestrating productive mathematical discussions gives me more control over learning than any traditional way I ever taught. You can't really force students learn (that doesn't stop us from trying though- or wishing for superpowers!) Students need to invest and connect in order to truly learn. During the summary of the problem students are primed and ready to make those connections and finally I can facilitate a semi-controlled learning opportunity by previously anticipating responses, monitoring discussion, selecting student work, sequencing the order, and finally connecting the ideas! Superpower!

Today we revisited our work from the orange juice problem and connected it directly to the excerpt we read on Orchestrating Mathematical Discussions We were given student work from the Connected Math site (from these we were given 7 different fairly randomly chosen pieces of student work (pieces 2,3,7,8, and 18). Then we were asked to look at the work from each of the 7 groups and choose which pieces of student work would be a part of our summary (selecting).  After selecting, we then we worked together to figure out how to best sequence them in a class discussion as well as questions we would use to connect. This was a very valuable experience, especially with the collaboration of our table groups. It was definitely not easy, but low risk (we weren't doing this on the spot) and high benefit (we saw our problem from new perspectives and got to wear our teacher hat without messing up our first period!)

We debriefed our choices with other groups; some chose a different order and also different groups' work to present. It was interesting to see how many different orders would still produce a math "story" and allow for students to recognize connections.

Linear relationships both proportional and non prorportional

As promised we continued to connect proportional reasoning beyond the Comparing and Scaling Unit from 7th Grade.  We explored problem 2.1 in Moving Straight Ahead where two boys race (one with a head start.) This problem is great opportunity for students to explore the difference between a proportional and a linear relationship that is not proportional.  

Inversely proportional relationships

In 8th Grade Thinking with Mathematical Models, students look at inverse variation (y=k/x) using several circumstances (holding the area constant while the side lengths both change- first explored in 6th grade, and holding the distance constant while the rate and time change problem 3.2.)  I have always understood inverse variation based on graphs and equations, but when I looked at the relationship between each variable in a table I really developed a deeper understanding. 

In a proportional relationship, when you multiply x by 2, you also multiply y by 2. However in an inversely proportional relationship you multiply x by 2 and divide y by 2! Of course this makes sense when you look at the equation  xy=24  (length x width = 24) because in order to remain equivalent, if you multiply x by 2, you must divide y by 1/2 to keep the equation balanced.  You are essentially multiplying by 2/2, just multiplying x by the numerator and dividing y by the denominator. 

We didn't teach this investigation this year due to time constraints, but I can't wait to see the connections my students will make. 

Proportional Relationship connection in 8th grade

Then we broke out into groups once again and explored other places that connected to revisiting proportional relationships.  Each group took on an upcoming lesson (jigsaw) to see how our work on proportional reasoning could be connected: Thinking with Mathematical Models 3.3, Looking for Pythagoras Problem 5.2, Growing Growing Growing Problem 1.3, Frogs, Fleas, and Painted Cubes Problem 4.3 and Say it with Symbols 2.1. Below are the results.

By the time I left today, I was even more convinced at how connected this math curriculum really is. This also seems to give me a little more of my desired "control" over student learning!