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Carl’s NCTM Experience: Hyman Bass And Theory Building

2014 NCTM Annual Meeting and Exposition

This year’s annual meeting took place one short drunken stumble away from Bourbon Street at the New Orleans conference center.  The entire space was used by the NCTM conference, and they still had sessions hidden away in deep recesses of the center as well as in the Hilton which was a onerous 5 minute walk down the street. Selecting talks at this event was a constant struggle between dozens of excellent speakers.  The best algorithm that I found to choose which talks to go to was to attend the talks that people smarter than you recommended. Using this method, and a little ‘phonebook’-style searching, I ended up attending and learning from the following talks. After three action packed days I came away with a lot of ideas, and a lot of notes, but nothing really concrete.  So in order to squeeze every little bit of learning out of the experience I am going to write up as many talks as I can remember here on the blog.

The Mathematical Practices of Finding Structure and Making Connections

Hyman Bass – Slides

This talk had a number of situations that push students to think deeply about solving problems and making connections between the ways you can solve problems.  For example, slide 2 shows a problem that students can understand as a visual problem or as a problem that can be dealt with by adding the pattern of odd numbers.  He went on to show how a generalization of this problem could be expanded to larger squares, (but the slides shared on the NCTM website don’t seem to have all of the things that he talked about).  The connection between a geometric representation of something that can be expressed with an equation was the big concept and slides 3 and 4 show two geometry problems that can have similar math connections and produced genuine “Aha’s” from the crowd of math educators.

Following the 3 problems, Bass went into a talk about how to make students into “Problem solvers and Theory Builders” which wasn’t really reflected in the slides.  My notes show that when people look to create more rigorous or deeper learning situations that they can to place students in, there is a huge thrust to push students into applied situations, where mathematics is used to solve specific problems in a specific way.  This focus on calculation only lets students participate in a small part of what it means to be a mathematician.  Mathematicians were thought of as thought leaders through history, as they also used their brains to examine evidence, analyze patterns, and create theories about numbers.  While all of the math that children study in school has been examined and studied to an exhaustive degree, students need to come out of math class with this theory building skill set.  


Math teachers have to create new problems that put situations to build their Problem Solving/Theory Building skills.  Bass boiled it down to a few steps that were important in helping them build theories off of patterns by doing the following with a given pattern in order of which shows the most understanding:


  1. Repeat the given patterns or duplicate the pattern (Ex. ‘star star circle’)

  2. Extend or continue a sequence of the pattern (Ex. ‘star star circle star star circle’)

  3. Reproduce the pattern with new materials (Ex.  given new shapes: ‘diamond diamond triangle’)

  4. Explain why the reproduced pattern is has the same pattern as the original (Ex. “Same, Same, Different, Same, Same Different”


To expand this to more advanced math he showed a problems set that had students look through 4 different example of combinatorics problems which are shown on the 5th slide.  These problems are all interesting in their own right, but when put in a set, it gives students the opportunity to think about how these problems should be solved and perhaps talk about how the structure of these problems.  Once students see the similarities in the problems, teachers could look to ask students to apply the thinking to new problems, create their own situation when given a different context, and explain all the similarities between the mathematical thinking required to finish all of the tasks.


One quote that stood out was while he was talking about making problem sets to get students to think differently.  Bass said, “Give kids problems that force them to fail enough to where they see the benefits of the kind of thinking that would be useful.” This means having a bunch of the same kinds of problems that have simple procedures is going to be harmful for their development since they won’t get a chance to learn from their mistakes.  


Next, Bass started talking about the Tea and Wine problems, which I saw detailed in this article Theory Building in the Mathematics Curriculum?.  This article also includes a proof of the situations in slides 6 and 7.  These problems are part of Common Structure Problem Sets (CSPS).  My notes at this part of the talk were not very useful as I was also texting my school to see how everything was going while I was gone (it all was fine).  


The talk ended before I was able to get more an understanding for these kinds of problem sets, and google isn’t too much help either, but I think the pattern of repeated problem solving strategies is definitely something I am going to employ the next time I want to explicitly teach mathematical thinking.


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