By Rachel Levy
Mathematical modeling can be taught at every stage of a student’s mathematical education, from kindergarten to undergraduate school and beyond, as a basis for developing problem-solving skills and mathematical habits of mind.
That is the premise of the national Guidelines for Assessment and Instruction in Mathematical Modeling Education (GAIMME), a new report co-published by SIAM and the Consortium for Mathematics and its Applications (COMAP) and in cooperation with the National Council of Teachers of Mathematics. The recommendation to create the report originated at two NSF-SIAM workshops on Modeling Across the Curriculum, held in August 2012 and January 2014, and was motivated in part by the American Statistical Association’s highly successful GAISE report, which promoted statistics education in K-16.
The Guidelines for Assessment and Instruction in Mathematical Modeling Education (GAIMME) offer suggestions for incorporating mathematical modeling in classrooms.
Written by faculty members as well as K-12 teachers engaged in teaching and applied/industrial mathematics research, the report was reviewed by K-12 teachers, mathematics teacher educators, mathematics education researchers, and mathematics faculty. The first release of the report was showcased in a panel and three workshops at this year’s National Council of Teachers of Mathematics Annual Meeting and Exposition in San Francisco (April 2016). Future showcase locations include the SIAM Annual Meeting in Boston (July 2016), the SIAM Conference on Applied Mathematics Education in Philadelphia (September 2016), and the Joint Mathematics Meeting in Atlanta (January 2017).
In mathematics education, the word ‘modeling’ is used for many things, such as demonstrating a mathematical process (for example, solving an equation), using manipulatives (such as utilizing blocks to represent addition), and describing mathematical techniques (for example, repeated addition as a model for multiplication). The GAIMME report begins with a general chapter that describes mathematical modeling. The chapter emphasizes that mathematical modeling, both in school and in the workplace, employs mathematics to answer big, messy, reality-based questions by quantifying phenomena and analyzing relationships. The report shares experiences from the classroom that allow students to engage in genuine modeling activities. With appropriate facilitation from teachers, students can then use mathematics to answer meaningful questions that could help enhance their futures.
The report’s opening chapter explains that merely adding a context to a mathematics problem, as with many traditional word problems, does not constitute mathematical modeling. However, transforming 2+3=5 to the elementary-level problem 2 apples + 3 apples = 5 apples moves on the continuum towards mathematical modeling. The question “How many slices of apple should be in your lunch?” moves much further. The most genuine modeling questions are generally open at the beginning (as not all necessary information is provided), open in the middle (so students have the opportunity to choose their mathematical tools), and open at the end (where many answers are possible and an answer’s usefulness should be discussed).
The chapter ends with the following five guiding principles:
1. Modeling (like real life) is open-ended and messy. It may seem like a good idea to help students by distilling a problem so they can immediately see which are the important factors to be considered. However, doing so prevents them from doing this on their own and robs them of the feelings of investment and accomplishment in their work.
2. When students are modeling, they must be making genuine choices. The best problems involve making decisions about things that matter to the students, and help them see how using mathematics can help them make good decisions.
3. Start big, start small, just start. After reading this report, you may feel ready to jump in and make big changes, and if so, that is great! However, even small changes to things you already do in your classroom can encourage students to engage in mathematical modeling.
4. Assessment should focus on the process, not the product. Mathematical models (and the results they produce) are intimately tied to the assumptions made in creating the models. Assessment should be in service of helping students improve their ability to model, which will, in time, translate to a better product.
5. Modeling does not happen in isolation. Whether students are working in teams, sharing ideas with the whole class, or going online to do research or collect data, modeling is not about working in a vacuum. The problems are challenging, and it helps to know you have support as you seek answers.
The next three chapters of the report focus on more grade-specific content. One section focuses on early and middle grades, another on high school, and a third on college. The goal is to help people “look over the shoulders” of experienced modeling teachers and their students to see how modeling is enacted in the classroom. The final sections of the report provide tools for assessment and a set of further modeling resources.
The GAIMME report will exist as a living, growing document with an associated online repository of resources. Its writers hope that with the increasing awareness of math in everyday life—its continued need in academic/research environments, the growing prevalence of mathematics and data science in business, industry, and government (BIG) jobs, and the push for students to develop 21st century workforce skills like communication, collaboration, creativity, critical thinking, and citizenship/stewardship—teachers and faculty will view mathematical modeling as an effective way to prepare their students for the future. The GAIMME report and its associated materials will help educators facilitate and develop modeling abilities in their students, as well as pave the way for the inclusion of modeling as part of the established curriculum in a balanced mathematics education.
View and download the full report here.
Rachel Levy is SIAM VP for Education, as well as an associate professor in the Department of Mathematics and Associate Dean for Faculty Development at Harvey Mudd College.