As I extend my knowledge of mathematical modeling I realize there are many facets of modeling I still don’t quite understand. I’ve gotten pass the hurdle of believing modeling is just something the teacher does for the students at the beginning of the lesson. Reading Modeling with Mathematics Authentic Problem Solving in Middle School by Nancy Butler Wolf has not only stretched my thinking but has provided next steps for deepening my own understanding.
In her new book, Butler doesn’t just stop with telling readers what is mathematical modeling, she goes much deeper. My first aha moment came when reading about the types of problems students encounter. Due to my recent engagement with CGI problems, my expectation was most modeling problems were set up like CGI problems. Butler discussed four types of problems and gave a comparison to the four depth of knowledge levels. Computation problems, your most basic problem type are equated to DOK 1. Word problems and problem solving aren’t as in depth as modeling problems according to Butler. You may, as I did, question what’s the significant difference between word problems and modeling problems? Butler’s examples throughout her book display problems which cover the concept but also require students to apply reasoning, use prior knowledge and problem solving in order to work through the problem or task. Tasks such as those found on the MARS site, 3-Act tasks developed by those such as Dan Meyer, Graham Fletcher, Andrew Stadel and Mike Wiernicki are resources mentioned within the book. I would also add Robert Kaplinsky’s problem based learning tasks to the list as well.
I never realized until now the numerous factors of mathematical modeling. Exploration, effective questioning and a lot of reasoning are key components to mathematical modeling. Butler paints a clear picture of each of these components within the book. Her task examples and explanation of how they played out within classrooms provided me with more insight on creating opportunities for mathematical modeling.
In essence, mathematical modeling is students making sense of mathematics and representing the mathematics to provide a solution to a real world problem. It is also allowing for multiple representations. This means, when I would show students what table to construct or how to organize their data, I really was not emphasizing SMP 4 like I thought I was. Now it makes sense to why some students struggled to interpret the data or solutions, I was doing it for them. There really wasn’t much thinking they did, just simply inputting information into my organizer.
If my last couple of sentences hit home for you, Modeling with Mathematics discusses students interpretation of solutions within the process of mathematical modeling. With that, the idea of unrealistic answers is also mentioned. This book provides examples of tasks, student work samples as well as means to assess student modeling. It has proven to be a great resource for me as I move forward with personal and professional learning with mathematical modeling. If you find yourself on this same journey at any place along the way, you should pick up a copy of Modeling with Mathematics Authentic Problem Solving in Middle School it’s available now!