Christian R. Hirsch - Western Michigan University

Hirsch might claim that modeling is at the core of the Common Core, but at a glance it looks like a standard without standards. Yes, the fourth Standard for Mathematical Practice is "Model with mathematics," but the high school content standards chooses to mark standards in other domains as related to modeling instead of grouping the modeling standards together. This makes it more difficult to see the modeling connections across the high school standards, but that shouldn't reduce their importance.

Christian Hirsch has been at Western Michigan for 40 years and he is probably best recognized as the principal investigator for the Core-Plus Mathematics Project. Along with IMP, Core-Plus is one of the most recognized secondary, integrated, NSF-funded curricula to come out of the post-

*Standards*curriculum development period in the 1990s.

Christian Hirsch |

Hirsch opened his talk by detailing how all of the mathematical practices can be addressed with a modeling-focused framework of curriculum and instruction. "Real world problems, if even solvable, take a lot of time and perseverance." To Hirsch, Standards for Mathematical Practice 1 and 4 are the focal points of the entire process, at least in classrooms with good instruction. "I'm talking about classrooms where classes begin with problems. I'm not talking about classrooms where the problems are saved until the end."

The key to modeling and making mathematics problematic, says Hirsch, is to identify problems in context, study those problems through active engagement, and reach conclusions as the problems are at least partially solved. The learning lies not only in the solutions to the problems, but the new mathematical relationships that are discovered along the way.

Hirsch used several examples of problems involving modeling in this presentation. The first dealt with the business prospects of a climbing gym. Assuming a survey had been conducted that found the number of expected climbers is related to price \(x\) by the equation \(n(x) = 100 - 4x\), how many daily climbing wall customers should the gym expect? I didn't catch all the details of this problem, but the next question involved finding the optimal and break-even revenue points for the gym, which is nicely modeled by a quadratic. Hirsch advocated using a computer algebra system to assist with the calculations, and advised to help students realize that rounding to the nearest cent, if necessary, also slightly moves answers away from their true zeroes or maximums.

Hirsch's next problem dealt with finding the optimal location for an oil refinery with wells 5 km and 9 km from shore. I sense that this and the previous problem are in Core-Plus, but unfortunately that wasn't made clear and no handouts or downloads for this talk have been provided. While I don't like leaving presentations early, at this point I had a pretty good sense for this one and left to catch an overlapping presentation starting at 2:45. The problems Hirsch chose and the approaches to solve them were pretty solid 30 or more years ago and are still pretty solid today, and I wasn't feeling like the presentation was suddenly going to break new ground. (For me, at least. I totally understand that problems and approaches like this might be new ground in many classrooms.)