Collaboration in the Mathematical Sciences Community on Mathematical Modeling Across the Curriculum
As representatives of the Society for Industrial and Applied Mathematics (SIAM) who are engaged in education outreach, we are working to bring together the mathematical sciences communities to help facilitate mathematical modeling in the K–12 arena. This effort embraces all aspects of modeling: mathematical, statistical, computational, and data-based, as well as science-/physics-based. Indeed, for most interesting applications, these aspects interweave and no one sphere of expertise will have all the answers. We will use the term “modeling” as an umbrella for all of these.
The emergence of data science—and data-enabled science—as a major area of research and study makes this blending of skills and understanding from the entire mathematical sciences spectrum critically important. When we say “mathematical sciences,” we include statistics, operations research, data science, and algorithmic approaches—pure, industrial, and applied mathematics. Certainly, modeling will be a key element of educational programs that prepare students for work in data-rich environments. Arguably, modeling is key to moving from data to information and making good decisions.
Another major motivation for this article is the fact that applied mathematics and statistics share much of the raison d’être for K–12 education in the mathematical sciences in the first place. (See also the Usiskin article, “The Relationships Between Statistics and Other Subjects in the K–12 Curriculum,” in CHANCE 28.3.) We share the responsibility for answering typical questions, such as “Why do we have to learn this?” or “When will I ever use this?”
In recent years, SIAM has been the driving force behind an initiative called Modeling across the Curriculum (MaC). Here, “across the curriculum” deliberately includes multiple disciplines, such as applications fields, and the full K–12 and college experience.
The introduction of modeling in early grades has also been facilitated by the Common Core State Standards for Mathematics (CCSSM), released in 2010, which include eight Standards for Mathematical Practice that describe “processes and proficiencies” mathematics educators at all levels should seek to develop in their students. We note that these CCSS in Mathematics include statistics, so reference to mathematics here should be seen as inclusive of statistics and early exposure to data-based reasoning.
Modeling has a privileged place in the CCSSM. It is the only topic that is both a practice and a content standard. Mathematical modeling is also the only mathematics standard that is also a science standard. Thus, mathematical modeling has been elevated to a new level in CCSSM.
We would like to encourage collaborative action with an explicit invitation for the statistical community to get involved in several initiatives: MaC, the new SIAM activity group on Education (SIAG/ED), the Moody’s Mega Math Challenge, and more. We hope others will join our modeling-based projects to advance the collective actions of SIAM and ASA in K–12 education.
One area where ASA and its members can bring discipline-specific expertise is in assessment of these efforts. We need to use evidence-based approaches to revise and improve our programs and practices. The ASA’s K–12 GAISE report has been, and will continue to be, a guide to pedagogy in statistics teaching. This report has inspired a work in progress: Guidelines for Assessment and Instruction in Mathematical Modeling Education (GAIMME), with a report funded jointly by SIAM and COMAP.
The Modeling Across the Curriculum Workshops
SIAM received two National Science Foundation (NSF) grants to work on increasing mathematical modeling and computational applied mathematics in K–12 and college curricula. These efforts grew out of discussions between SIAM and NSF representatives early in 2011 of the topics of undergraduate and K–12 courses and programs, college readiness, and career preparation. Representatives from educators spanning K–12 and undergraduate education, as well as mathematics education specialists in mathematics and statistics, participated.
The overarching goal for this project is to “Engage and Keep Young People in STEM Disciplines, from K–12 through Undergraduate (and Graduate) Studies, and into the Workforce.” This objective is simply stated, but less easily achieved.
Two modeling competitions for high-school students—the Moody’s Mega Math Challenge, organized by SIAM, and the hi-MCM, organized by CoMAP—help demonstrate the importance of increased modeling activity. These competitions have helped increase awareness of modeling and its importance for real-world problems. Its impact on student motivation is clear from the following email that was sent to the Moody’s team:
Dear Mr. Kunkle:
Our entire school, Sacred Heart Academy, is thrilled to learn that our two teams of 7 students have advanced to Round Two. Our girls have been so elated since they have completed the challenge a few weekends ago because they loved the problem they were working on and were thrilled that they had some sense of accomplishment in solving it to the best of their ability. They were elated the following Monday morning, acting as if they won the lottery! The exhilaration of embracing the challenge Moody presented and tackling it for 14 hours was transforming for them. One of the team members told me she did not realize how much she loved math and its power to solve problems until after the Challenge weekend. For that one comment, I am eternally grateful to all of you who sponsor the Moody’s Mega Math Challenge. It produces within our participating students Mega Math benefits!
Kathryn Gniadek, MS
Mathematics Department Chair
UCONN ECE Site Representative and Instructor
Co-moderator of Mu Alpha Theta
In addition to providing mathematics students with fun, team-based modeling activities, the challenges have built a bank of accessible problems and methodology for evaluating mathematical modeling work. The team approach is important, because it can help build the vital communication and career skills required in business, industry, and government (BIG) jobs that are not always emphasized in the traditional high school classroom where students may sometimes work in teams but are generally tested individually.
The main themes of the second workshop (MaC II) focused on investigating ways to increase modeling across undergraduate curricula and developing modeling content in the K–12 educational arena. As stated above, the term “modeling” should be interpreted very broadly in this context. So-called “physics-based” (non-stochastic) models are still important, but data-based models are increasingly critical to work in many fields. The latter, especially, demand both statistical and computational expertise and, therefore, education and training in these fields require collaborative efforts from across the mathematical sciences spectrum.
The workshops addressed several key issues raised in both the President’s Council of Advisors on Science and Technology (PCAST) report, “Engage to Excel,” and the influential National Academies report, “The Mathematical Sciences in 2025.” These include increasing student preparedness for science, technology, engineering, and mathematics (STEM) majors and overall enhancement of STEM education in the first two years of college. The results of the discussions may provide helpful responses to criticisms of the implementation of the Common Core State Standards and provide pathways to increasing modeling and application-based learning in school curricula.
A common criticism of mathematical education has highlighted its lack of real-life relevance, at least in the eyes of the students receiving that education (and perhaps among the teachers, too). Such a lack of relevance damages our collective ability to contribute fully to the development of a broadly STEM-literate workforce.
Modeling has the potential to increase interactions and interconnections among various STEM areas. Through modeling, students in K–12 can prepare for STEM college majors and careers, thus increasing the pipeline of scientific and technical talent in America. This career-readiness imperative was a central focus for the INGenIOuS project—another joint collaboration to bring together educators at all levels with mathematicians, statisticians, and industrial users of mathematical science skills.
Infusion Model vs. Standalone Debate
The first Modeling across the Curriculum report stated that all K–12 students should have significant modeling experiences by graduation, possibly in part through a well-designed course supported or endorsed by SIAM, ASA or others. Content accessibility, whether through digitally delivered modeling content, explicit modeling courses, or modeling laboratories, is critical for teachers (pre- and in-service), students, and school leaders.
Developing and then gaining acceptance for a new AP-like course in modeling may be possible, but the idea of infusing modeling content throughout the K–12 (and undergraduate) curricula was attractive to many. MaC I included this explicit call:
We should explore the possibilities of the ‘Trojan mice’ approach: infusing modeling into the full K–12 curriculum through many entry points, in many small ways, that strengthen what is already being taught. Certainly a high-school course or capstone experience in modeling should not be the only place in K–12 where students experience and know they’re experiencing modeling.
The Trojan mice approach addresses this issue and embodies the spirit of Modeling across the Curriculum.
Mathematical Modeling in the Early Grades
The second Modeling across the Curriculum Workshop paid real attention to the early grades, K–6. This new focus recognized that improving the output from the mathematical pipeline requires attention to the entire educational process. We want to form partnerships with administrators, teacher-educators, mathematics education researchers, teachers, and modeling contest coaches, to discover how best to teach mathematical modeling in the early grades and determine what will work in the classroom.
There are several motivations for incorporating modeling in the curriculum from the earliest stages. For example, a study by the Cognitive and Technology Group at Vanderbilt University on the Jasper Woodbury series found that “[s]tudents who worked on real-world problems demonstrated less anxiety toward mathematics, [and were] more likely to see math as relevant to real life, more likely to see it as useful, more likely to appreciate complex challenges.” The study also found a positive effect for both previously high- and low-achieving students.
While applying mathematical thinking to real-world applications can have advantages for learners of all ages, there may be advantages specific to introducing modeling early.
Thinking creatively may come more easily to children when they are first learning and exploring concepts.
Young students have a high potential to become fluent in the mathematical sciences. Teachers can lay the groundwork for modeling through pre-modeling activities, such as making simplifying assumptions about a situation.
Students can be coached to use trial and error to approach problems for which they have not been shown the solution approach.
Because early-grades teachers are generalists, they can address several subjects simultaneously through modeling activities.
Teachers may have the flexibility to seize on a moment when modeling can happen throughout the curriculum. Students can also learn to recognize these times. For example, a class could graph the “happiness” over time of a particular character in a story.<img src="http://chance.amstat.org/files/2015/11/mathmodelingfig1small.jpg" alt=" Mathematical modeling motivates students in a wide variety of learning domains. From K. M. Bliss, K. R. Fowler, and B. J. Galluzzo, Math Modeling: Getting Started & Getting Solutions, SIAM, Philadelphia, 2014. ©2014 Society for Industrial and Applied Mathematics. Reprinted with permission. All rights reserved.” width=”450″ height=”291″ class=”size-full wp-image-10079″ />
Mathematical modeling motivates students in a wide variety of learning domains.
From K. M. Bliss, K. R. Fowler, and B. J. Galluzzo, Math Modeling: Getting Started & Getting Solutions, SIAM, Philadelphia, 2014. ©2014 Society for Industrial and Applied Mathematics. Reprinted with permission. All rights reserved.
Potential barriers to the introduction of mathematical modeling at the elementary-school level are commonly held misconceptions about what is meant by modeling in the mathematical sciences. A 2013 PhD thesis by Heather Gould at Columbia University aimed to determine the conceptions and misconceptions held by teachers about mathematical models and modeling to aid in the development of teacher education and professional development programs through four research questions: (1) How do teachers describe a mathematical model? (2) How do teachers describe the mathematical modeling process? (3) What do teachers believe to be the purpose of mathematical modeling? (4) What are the misconceptions evident in the teachers’ descriptions of mathematical models and the mathematical modeling process?
Most teachers correctly understood that:
(a) Mathematical models can be equations or formulas; for example, a quadratic equation or d = rt, the distance-rate formula.
(b) Mathematical models can be used to explain the underlying causes in a given situation.
(c) The mathematical modeling process involves determining whether a solution makes sense in terms of the original situation.
(d) Repeating steps and making revisions may or may not be part of the mathematical modeling process.
However, a majority of the teachers held these misconceptions:
(a) Mathematical models can be physical manipulatives; for example, fraction tiles, pattern blocks, or three-dimensional solids (such as cubes, octahedra, and other polyhedra).
(b) Mathematical modeling situations come from “whimsical” or unrealistic scenarios.
(c) The mathematical modeling process always results in an exact answer or exact answers.
The teachers did not realize that the mathematical modeling process necessarily involves making choices and assumptions. While this report focuses on the response of the majority, if even a 10th or a quarter of all teachers carry a misconception, many students will likely be affected.
Thus, when we communicate with teachers, we must acknowledge that they may commonly use the term modeling differently from the way it is used in the mathematical sciences. We might suggest that, in some of these non-modeling cases (such as demonstrating a technique, employing visualizations such as tape diagrams, and using manipulatives such as pattern blocks), teachers use the word “represent” rather than model. This way, we can distinguish between the practice of mathematical modeling versus representations of mathematical concepts.
To focus on solutions to these issues, the Integrating Mathematical Modeling, Experiential Learning, and Research, through a Sustainable Infrastructure and an Online Network (IMMERSION) program, will address many of the action items from the working group, including curriculum development and repository, professional development, and connections to testing organizations.
As an effort to support teachers in the implementation of modeling in the classroom, SIAM developed a handbook, Mathematical Modeling: Getting Started and Getting Solutions (2014). The purpose of the book is to provide guidance to students and teachers who are not confident about or experienced with modeling. There is a strong emphasis on communicating results and technical writing to produce a final “product.” In addition, the book provides examples on different levels, from basic mathematics to introductory calculus and differential equations, so it is accessible to a broad high-school audience.
An important message in the introduction is the distinction between modeling and solving word problems. The key feature of math modeling is the opportunity to make assumptions and create a unique approach to solving an open-ended problem. Word problems are a powerful tool for putting context to mathematics, but ultimately all the assumptions have been made for the reader. Modeling is presented as an iterative process with components that are revisited, often in no particular order, while building and testing the model. These are key ideas for both K–12 students and teachers.
Developing an optimal recycling program for a generic town is one of the three modeling examples explored in detail throughout the book:
The population of Yourtown is 20,000 people and 35% of them recycle their plastic water bottles. If each person uses nine water bottles per week, how many bottles are recycled each week in Yourtown?
How much plastic is recycled in Yourtown?
Question 1 is a straightforward word problem in which all the information is provided so the student can calculate the answer. Although it takes place in a real-world context, it is not a rich mathematical modeling question, because the students are not making decisions about how to refine and approach the problem. In contrast, Question 2 is entirely open-ended and requires a variety of possible assumptions, and the collection and analysis of data, before it can be approached. Students with varying skills sets would be likely to approach it differently and arrive at equally valid but widely different solutions.
An even broader modeling question could ask, “How much plastic will wind up in landfills over the next 10 years?” This was the first part of the 2013 SIAM Moody’s Mega Math (M3) Challenge, which is also included as an example for learning the modeling process in the guidebook. To answer that question, students foraged through resources and found a data set provided by the Environmental Protection Agency (EPA) reporting on the tons of discarded plastic waste annually since the year 2000.
One approach to the problem would be to fit a curve to the data and build a predictive model. In that approach, students fit a logistic model and a linear model, analyzing R2 values and the standard deviation of the residuals to try to draw conclusions. They found the difference in predicted plastic waste was roughly 7 million tons when comparing the two models, but concluded that, since the population of the United States is increasing linearly, the linear model was more plausible as a predictor.
This example shows that a simple question can be rich in mathematical and statistical analysis, but also be accessible on many levels. Some solutions used calculus, while others found data sets generated when scientists had sampled densities of landfills over time. Others incorporated probabilistic approaches assuming that recycling efforts over the next 10 years would increase the likelihood of less plastic entering the landfill.
We emphasize the fact that high-school students can come up with innovative approaches to challenging real-world problems. In fact, the guidebook provides a complete student solution and the M3 website provides an archive of 10 years’ worth of problems and the top six solutions for each challenge.
As modeling assumes a greater role in the educational process, it will be essential to strengthen the assessment and evaluation of the changes and their impact. To this end, SIAM is taking a leaf out of ASA’s book in creating guidelines similar to the GAISE report for statistics education.
As mentioned above, one recommendation from the MAC II workshop was the creation of a GAIMME Report. The inception of this report was influenced by the success of the Guidelines for Assessment and Instruction in Statistics Education (GAISE) K–12 Report, which is widely viewed as instrumental to the greater footprint of, and focus on, statistics in the United States K–12 curriculum.
The motivations for GAIMME, as implied throughout this article, are to fill gaps in understanding about how modeling plays a role in STEM education. It will provide K–12 teachers with guidance on the teaching and learning of both building and analyzing mathematical and statistical models. The report will also discuss the need for consistent assessments of student modeling work, which has special challenges since it is a process with multiple possible outcomes.
We have highlighted some of the initiatives currently underway at SIAM in collaboration with the ASA and others to improve, and increase, the modeling content in the mathematical sciences curriculum. Our focus has been firmly on the K–12 arena, although the Modeling across the Curriculum project certainly includes undergraduate study in the mathematical sciences and across the broader STEM spectrum. We must consider many mathematical arenas—statistics, data (-enabled) science, and computational and traditional applied mathematics.
The full reports on both MaC I and MaC II are available on SIAM’s website for free download, but the following three summary recommendations from MaC II are indicative.
First, SIAM should create an activity group on applied math education. This recommendation has already been implemented.
Second, develop strong professional development and teacher training programs, materials, and support networks to provide experience, understanding, and skills in mathematical modeling at levels appropriate for use in early-grades classrooms. This is a major and multifaceted undertaking—but, again, one on which action has already begun. The early-grades group initiated a successful NSF proposal to create the IMMERSION project, to develop professional development in mathematical modeling education for teachers from Virginia, Montana, and California in collaboration with local school districts.
Third, develop strong professional development programs, curricular and assessment materials, and working groups to investigate different strategies for introducing modeling into the high school. One suggestion is to produce GAIMME along the lines of the ASA’s GAISE Report, which is in progress.
In pre-college education (and at college, too), modeling provides a natural focus for a coordinated approach to the study of STEM disciplines. It also reflects the nature of modern scientific and technical work, which is necessarily interdisciplinary, whether we are thinking of economic or weather modeling and forecasting; engineering design and optimization; environmental protection and use of resources; biomedical research; materials design and development; business and data analytics; or animation, special effects, and entertainment.
PCAST’s Engage to Excel called for 1 million additional STEM graduates for the workforce and a real-world–relevant approach to mathematics teaching at all levels. Workforce preparation was also the focus of the national INGenIOuS workshop, which called for greater relevance and applicability in our curricula. The Modeling across the Curriculum project, although its inception predates those reports, represents a concerted effort to address precisely these issues.
We have established the need for collaborations among the mathematical sciences community with interests in applications. SIAM and the ASA typify that focus. Therefore, we reiterate the welcome and invitation to all who want to be involved in Modeling across the Curriculum. The partnership is, and needs to be, broader than just the ASA and SIAM. The National Council of Teachers of Mathematics (NCTM) and Mathematical Association of America (MAA) have also been increasingly involved. Other complementary initiatives are broadening this endeavor. These initiatives rely critically on participation across the full mathematical sciences spectrum.
Bliss, K.M., K.R. Fowler, and B.J. Galluzzo. 2014. Math modeling: Getting started and getting solutions. Philadelphia, PA: SIAM.
Cognitive and Technology Group at Vanderbilt University. 1992. The Jasper series as an example of anchored instruction: Theory, program description, and assessment data. Educational Psychologist 27:291–315.
Franklin, C., G. Kader, D. Mewborn, J. Moreno, R. Peck, M. Perry, and R. Scheaffer. 2005. Guidelines for assessment and instruction in statistics education (GAISE) report: A Pre-K–12 curriculum framework. Alexandria, VA: American Statistical Association.
National Research Council. 2013. The mathematical sciences in 2025. Washington, DC: National Academies Press.
Society for Industrial and Applied Mathematics. 2012. Modeling across the curriculum: Report on a SIAM-NSF workshop (PDF download). Philadelphia, PA: Society for Industrial and Applied Mathematics.
Society for Industrial and Applied Mathematics. 2015. Modeling across the curriculum II: Report on the 2nd SIAM-NSF workshop. Philadelphia, PA: Society for Industrial and Applied Mathematics.
Zorn, P., J. Bailer, L. Braddy, J. Carpenter, W. Jaco, and P. Turner. 2014. The INGenIOuS project: Mathematics, statistics, and preparing the 21st century workforce. Washington, DC: Mathematical Association of America.
About the Authors
Peter R. Turner is dean of the School of Arts and Sciences at Clarkson University, where he is a professor of mathematics and computer science. For six years (2009–2014), he was the Society for Industrial and Applied Mathematics (SIAM) vice president for education and instigator of the Modeling across the Curriculum initiative. He now chairs the SIAM Activity Group in Applied Mathematics Education and is a SIAM Fellow. Although his recent research activities have been in educational topics, his fundamental research was in computation and computer arithmetic.
Rachel Levy is an associate professor of mathematics at Harvey Mudd College and the associate dean for faculty development. She serves as vice president of education for SIAM and is on the editorial board of Math Horizons magazine. At Harvey Mudd College, she is a faculty mentor for the mathematics clinic program, in which teams of students solve problems for business, industry, and government sponsors. She co-directs an NSF-funded research project on flipped classroom instruction and the IMMERSION program on mathematical modeling for elementary grades. She also is the creator of the blog “Grandma got STEM,” which celebrates the stories of senior women in STEM fields from all over the world.
Kathleen Fowler is a professor of mathematics at Clarkson University. She is the director of the New York State Education Department (NYSED)-funded Integrated Math and Physics for Entry to Undergraduate STEM (IMPETUS) Program, a year-round outreach program serving local grade 7–12 students in rural upstate New York. She is a judge and a member of the Problem Development Committee for the SIAM Moody’s Mega Math Challenge, and a coach for Clarkson University’s COMAP Mathematics Contest in Modeling. She is committed to undergraduate research and has mentored 28 students in the last eight years, resulting in 13 peer-reviewed scientific publications with undergraduate students.