Recently I was asked to design a mathematics education course for K-12 STEM teachers enrolled in a master’s degree teacher education program. The course is for teachers from all grade levels with specializations in mathematics, biology, physical sciences, engineering and technology education. Here I describe my process of conceptualizing the course, including how I determined the mathematical focus and the design principles that guided the development and selection of course materials.

### Role of Mathematics in STEM Education

To identify the mathematical focus for the course, I started with considering the role of mathematics in STEM education. As one who appreciates mathematics for its logic and coherence, even apart from its applications, I wrestled with how to conceptualize a course that would be true to the discipline of mathematics while simultaneously supporting broader efforts in STEM education. Historically in STEM education, mathematics has been viewed in service of science, technology, and engineering as an algorithmic solution to problems (Fitzallen, 2015). This perspective has led to some STEM education programs that focus on science, technology, or engineering content and contexts and only draw on mathematics when needed, thus not emphasizing the “big ideas*”* of mathematics (Stohlmann, 2018, p. 311). However, this view of mathematics as the machinery for problem solving, rather than a language of logic, patterning, and even generality (Mason, 1999) does not accurately reflect the historic relationship between mathematics and the sciences. Mathematics has had a role independent of, yet instrumental to, the advancement of the sciences. For instance, the development of knot theory and non-Euclidean geometry resulted from mathematicians playing with the rules of logic long before the usefulness of these constructs to describe real-world phenomena was recognized by biologists and astronomers (Orlin, 2018). If K-12 STEM education relegates mathematics to a service discipline, key opportunities to develop the next generation of mathematical thinkers are being missed. Viewing mathematics as only a tool for solving problems and leaving all conceptual depth of the discipline to pure mathematicians limits students’ understanding of mathematical content (Fitzallen, 2015). Indeed, “… STEM education must involve significant mathematics for students. Otherwise, the M in STEM is silent” (Schaughnessy, 2013, p. 324).

Stohlmann (2018) advocates for a model of integrated “steM” education, where mathematics content is explicitly integrated with other STEM disciplines. In this model, knowledge in two or more disciplines is required to develop a solution to a problem, and mathematics is always one of them. Stohlmann posits that the design of such an integrated steM model is best done through engineering design challenges, mathematical modeling, and open-ended mathematics integrated with technology. Stohlmann’s proposed steM model is a welcome shift away from the *incidental nature* of mathematics in STEM learning activities to a stronger focus on the *instrumental nature* of mathematics (Fitzallen, 2015). Moving beyond algorithmic applications of procedures, an instrumental approach focuses on designing tasks and lessons that deepen students’ conceptual understanding of mathematics while connecting to STEM contexts.

### Mathematical Focus: Modeling in STEM Education

After these initial considerations about the role of mathematics in STEM education and a review of the extant literature, I selected modeling with mathematics as the focus of the graduate level mathematics education course for K-12 STEM teachers. Mathematical models are useful in STEM contexts when the underlying structure of the mathematics aligns with the structure of a real-world situation. However, if mathematical models are used only procedurally rather than conceptually, students miss opportunities to see the analogous structure in real-world situations that apply the same type of mathematical model. For instance, students can understand why both conversions between Celsius and Fahrenheit and distance-time graphs for constant speeds are modeled by linear functions if they recognize the mathematical structure of linear patterns. Indeed, the power of the generalizations that emerge from linear, quadratic, and exponential function models are only utilized when students understand the mathematical nature of the generalized rule (Lannin et al., 2006). Another benefit of focusing on mathematical modeling is that the modeling cycle directly aligns with the Next Generation Science Standards’ (NGSS Lead States, 2013) Science and Engineering Practice Standards, providing an explicit connection between the work of mathematicians and the work of scientists. Mathematical models are used to provide insights into real-world phenomena, and there are multiple methods and approaches to answering questions for which models are useful (Cirillo et al., 2016; GAIMME, 2019). Choosing to design a mathematical modeling course aligned with my goal of helping teachers deepen their mathematical content understanding while concurrently showing how the STEM disciplines are intertwined, advancing Stohlmann’s (2018) model of steM to K-12 STEM educators.

After selecting the main focus of the course, I developed course materials aimed at making modeling with mathematics relevant and accessible to K-12 teachers in all STEM fields. Even with this wide variety of teachers, all courses in the graduate teacher education program are intended to be relevant to their everyday work in their classroom. Knowing that STEM teachers have varying levels of experience with mathematical modeling, I structured the course to support the teachers as both learners of the modeling process and as professionals who will apply their work as mathematical modelers in their work as teachers. To this end, I developed four guiding principles to inform the selection of content, contexts for modeling, tasks, and assignments aligned with the course goals. These principles were developed from the literature on mathematical modeling and a review of modeling resources for elementary and secondary grades.

### Guiding Principles for Course Design

As the first guiding principle, I centered the content of the course around the major conceptual themes that are present in each grade band of the K-12 curriculum to highlight key ideas in mathematics: discrete mathematics, functions and patterns, geometry, and statistical reasoning. After selecting the mathematical concepts that would be the focus of each unit, I chose modeling contexts and classroom activities appropriate to the middle school grades. By keeping the core mathematics of each class session at the middle school level, the teachers in each grade band studied prior and horizon knowledge relevant to the content they teach. This approach also supported science and engineering teachers to consider core mathematical ideas and how those ideas support the models used in their respective fields. After learning the middle grades content as a class, the teachers aligned the models with their grade-specific content standards, which provided for rich discussions across the grade bands. Figure 1 illustrates how this principle provided an overall structure for each modeling unit.

Figure 1. Modeling Unit Structure for K-12 Teachers

While mathematical models generally start in the real world and move between real and mathematical worlds (Cirillo et al., 2016; GAIMME, 2019), there may be times when starting in the mathematical world lays the conceptual foundation to connect the mathematical structure to the models that scientists and engineers already use. This observation became a second core guiding principle for the course design. When initially highlighting a mathematical concept helped the teachers link the mathematical structure and the real-world structure, the tasks started in the mathematical world. One example is the approach to functions as patterns described above, which places an emphasis on identifiying the mathematical pattern structure in real-world situations. However, when starting in the real world provided opportunities to develop models that could be adapted as situations became more complex, the tasks started in the real world. One example where starting in the real world was advantageous was the study of combinatorial counting problems using Kahn Academy and Pixar’s collaboration titled Pixar in a Box (Kahn Academy Partner Content, 2020). The teachers carefully studied various contexts for counting that increased in complexity, causing them to alter their counting techniques. Because the teachers developed the counting procedures, rather than being given counting formulas, they began to recognize real-world situations with similar structures and could apply a counting method appropriately aligned. For instance, they could recognize that the counting procedure for the number of animated robots a designer can draw for a scene in a Pixar movie is analogous to determining the number of committees that can be created to serve in Congress. Capitalizing on the movement between the real and mathematical worlds makes mathematical formulas and generalizations become more than just algorithms for producing an answer. They become symbols and operations that align mathematical structure with real-world contexts.

As a third guiding principle, I recognized the need for scaffolding in the modeling cycle, both for the teachers as learners of the modeling process and their application of the modeling process in their classroom. Considering their role as learners, I structured lessons and activities in which the teachers could focus on different aspects of the modeling cycle, such as making assumptions or testing a model to help develop their skills in a focused way. For instance, the teachers used Three Act Math Tasks (Meyer, 2015) to focus on making assumptions, asking questions, and gathering appropriate data to create a model. When considering their role as K-12 teachers, I utilized the Illustrative Mathematics (Illustrative Mathematics, n.d.) modeling cycle rubric to help them evaluate potential tasks for their own classrooms by classifying the level of “lift” or difficulty for a specific step of the modeling cycle. The teachers could implement tasks that support their current curricula, but still help their students develop mathematical modeling skills.

As additional support for the teachers to bring modeling into their own classroom, the course used many different representations of mathematical models. While equations and statistical models allow for predictions and are often the models relied on by scientists, I took a broad approach to the ways the teachers represented real-world situations mathematically, including simulations and various graphical models. This led to the fourth design principle that moved beyond just the mathematical goals of the course. Mathematical models provide insight into real-world phenomena that cannot be ascertained through other means, and these models can be presented in ways that help people engage in community life and productive citizenship. While mathematical models can certainly be developed in isolation from community, leaving out the humanity in mathematical modeling would present an incomplete perspective of the models that we as mathematics teacher educators want our K-12 students to know and be able to use. In this course, the teachers examined ways in which the STEM fields could address important topics for equity and access, including issues surrounding urban planning, food insecurity, and nutrition.

In sum, described here is a vision for K-12 STEM teacher education that aims to position mathematics as instrumental in STEM education. As mathematics teacher educators, we have the opportunity to help pre-service and in-service teachers in STEM fields envision the role of mathematics in STEM. By advocating for keeping the conceptual depth of mathematics in STEM education programs, we can support the education of the next generation of mathematical thinkers. We owe it to our K-12 students, and dare I say society, to ensure that when mathematical models are needed in science, technology, and engineering in the future, they will be there.

#### References

Cirillo, M., Pelesko, J. A., Felton-Koestler, M. D., & Rubel, L. (2016). Perspectives on modeling in school mathematics. In *Annual Perspectives on in Mathematics Education: Mathematical Modeling and Modeling Mathematics*, pp. 3-16. Reston VA: NCTM.

*GAIMME: Guidelines for Assessment and Instruction in Mathematical Modeling Education, 2 ^{nd} Ed*. (2019). S. Garfunkel and M. Montgomery (Eds.). Philadelphia, PA: Consortium of Mathematics and Its Applications (COMAP) and Society for Industrial and Applied Mathematica (SIAM). Retrieved at: https://www.siam.org/publications/reports/detail/guidelines-for-assessme....

Fitzallen, N. (2015). STEM education: What does mathematics have to offer? In M. Marshman, V. Geiger, & A. Bennison (Eds.). *Mathematics education in the margins (Proceedings of the 38th annual conference of the Mathematics Education Research Group of Australasia)*, pp. 237-244. Sunshine Coast: MERGA.

Illustrative Mathematics. (n.d.). Mathematical Modeling Prompts. Retrieved on Feb 19, 2021: https://curriculum.illustrativemathematics.org/HS/teachers/mathematical_modeling_prompts.html.

Lannin, J. K., Barker, D. D., & Townsend, B. E. (2006). Recursive and explicit rules: How can we build student algebraic understanding? *Journal of Mathematical Behavior, 25*(4), 299-317.

Mason, J. (1996). Expressing generality and roots of algebra. In N. Badnarz, C. Kieran, & L. Lee (Eds.), *Approaches to algebra: Perspectives for research and teaching* (pp. 65-86). Dordrecht, The Netherlands: Kluwer Academic Publishers.

Meyer, D. (2015). Missing the promise of mathematical modeling. *The Mathematics Teacher, 108*(8), 578-583.

Kahn Academy Partner Content. (2020). *Pixar in a Box*. Retrieved May 25, 2020, from https://www.khanacademy.org/partner-content/pixar.

NGSS Lead States. (2013). *Next Generation Science Standards: For States, By State*s. Washington, DC: The National Academies Press.

Orlin, B. (2018). *Math with bad drawings: Illuminating the ideas that shape our reality*. New York, NY: Black Dog & Leventhal Publishers.

Schaughnessy, J. M. (2013). Mathematics in a STEM context. *Mathematics Teaching in the Middle School, 18*(6), 324.

Stohlmann, M. (2018). A vision for future work to focus on the “M” in integrated STEM. *School Sciences and Mathematics, 118*(7), 310-319.