This past fall, we collaborated across the disciplines of mathematics and education to create a mathematical modeling task for secondary pre-service teachers (PSTs), aligned with grades 7-12 mathematics content standards in both Nebraska (NSBE, 2015) and Kansas (KSDE, 2019). The purpose of this paper is to emphasize the flexibility of implementing modeling tasks across disciplinary boundaries to connect mathematics content and pedagogy. In this paper, we first describe the rationale and development of our modeling task, the Sprinkler Task. Then, we share reflections for our two implementation contexts.

In addition to being a high school content domain, mathematical modeling is important in the K-12 curriculum as a mathematical practice/process (CCSSI, 2010; KSDE, 2019; NSBE, 2015). A modeling diagram, often in the form of a cycle, illustrates the nonlinear, iterative nature of modeling (e.g., Figure 1). Due to the cyclic structure of modeling, the process of revising mathematical work (possibly multiple times) is emphasized as a justification tool for formulating a realistic solution to an authentic problem.

Mathematical modeling tasks benefit students in many ways, such as connecting multiple content areas and developing problem-solving skills and creativity (e.g., Tidwell et al., 2021; Mousoulides et al., 2008). They are also productive activities for PSTs, as modeling requires knowledge of relevant mathematical content and demonstration of mathematical processes and practices as indicated in the AMTE Standards for Preparing Teachers of Mathematics (i.e., standards C.1.1 & C.1.2; AMTE, 2017). As modeling can be challenging for teachers to attend to, it is crucial to include experiences with modeling in teacher preparation courses (Anhalt & Cortez, 2016; Anhalt et al., 2018). 

Figure 1. A Cyclic Diagram of the Modeling Process (Anhalt et al., 2018)

Designing the Sprinkler Task 

This task originated from a collective interest to generate a regionally relevant modeling task for our PSTs in the Great Plains. We were inspired by our past work with colleagues to create a community-based (M2C3, 2018) and authentic (Aguirre et al., 2019; Anhalt et al., 2018) task as a foundation to draw on PSTs’ funds of knowledge (Moll et al., 1992) and multiple mathematical knowledge bases (Turner et al., 2012). We decided on the topic of watering a backyard garden based on our own lived experiences with water conservation and gardening in the Great Plains. 

The task prompted PSTs to develop a watering plan for an imagined neighbor, based on a schematic of the neighbor’s backyard and available resources from a local gardening store (see Appendices A and B). The schematic was drawn on square grid paper, where each square represented a 3ft-by-3ft area of the yard, and indicated plant locations, including two established trees and multiple irregularly shaped flower beds. We also compiled a document with names and pictures of sprinklers, hoses, splitters, and timers available from a local gardening store.

There is a second part to the task, intended for use after PSTs develop their watering plan, where they are provided new information from their neighbor (see Appendix A). The purpose of the new information was primarily pedagogical, since it would likely cause PSTs to revisit their assumptions and possibly revise their model (e.g., to include stationary sprinklers with timers), emphasizing the iterative nature of the modeling process (see Figure 1). This new information in the task was authentic, in that it was inspired by our experiences balancing garden preservation and water conservation during the hot and unpredictable Midwest summers, particularly while away on summer trips. 

 We anticipated two probable approaches to solving the task: 1) conservation focused, where one places sprinklers to water only specific plants and not the entire lawn, or 2) coverage focused, where one places sprinklers to water the entire area of the backyard. Within these two approaches, we expected PSTs to consider the shapes created by the chosen sprinkler(s), the length of hose required to support sprinkler placement, and how movement of sprinklers might affect the area watered. As such, the task connected to concepts such as the Pythagorean Theorem, equations of circles and other geometric shapes, rigid transformations, areas of irregular and overlapping shapes, and a coordinate system suggested by the scale grid on the backyard schematic. Aligning with the open nature of modeling tasks, we intentionally did not cite any particular mathematical concept or tool in the prompt.

Implementation

We implemented this task in two distinct courses, each located in different teacher preparation programs at separate universities. One course was a mathematics content course, and the other was a mathematics methods course; both were required for senior-level, secondary PSTs. In the content course, modeling was an emphasis and consistent practice of the course; in the methods course, modeling had only recently been introduced and this was the first formalization. Below, we share our noticings and suggestions for other mathematics teacher educators implementing modeling in either context.

Launching the task

In the methods course, we introduced the modeling process (see Figure 1) and discussed routines for making assumptions (M2C3, 2018) to highlight the importance of including relevant information and factors into a model. We spent considerable time in the launch phase of the Sprinkler Task lesson, which was necessary, but led to time constraints when we reached the final reporting out stage and peer analysis of models.

PSTs in the content course were familiar with the mathematical modeling process at this point, so the launch of the task was brief and began with the simple question, “What do you consider when taking care of your yard or garden?” We appreciated how these initial conversations during the launch allowed PSTs to share their experiences with gardening and set up the open nature of the task.

Looking across models

Mathematical models can take various forms; thus, we asked PSTs to reflect on, “What is the model?” in this task. We encouraged PSTs to consider not only their annotated diagram of sprinkler placement as the model, but also their general plan and approach for how to place sprinklers based on their objectives for watering.

While the final stage of the lesson in both contexts included PSTs reporting out their models, we felt it was insufficient to support deeper comparisons and connections across the models produced in the class. Afterward, we considered other structures for future implementations, such as a gallery walk where PSTs use a graphic organizer to give other groups feedback on the assumptions made and mathematics used.

Unpacking pedagogical choices

As future teachers, our students were simultaneously considering how they might teach this type of task as they solved it as learners. Thus, to make our pedagogical decisions visible, we shared design choices we made when creating the task, additional options or features we considered, and strategies for facilitating or adapting the task.

For instance, we explained our decision to provide a curated list of images of sprinkler options from a local gardening store. The purpose of the list was twofold: 1) the images of available options kept the task authentic and locally situated, and 2) the narrowed-down options reduced the complexity of the decision-making in the task and allowed more time to focus on the mathematical models. We reflected afterward that this time devoted to having transparent pedagogical conversations was worthwhile, particularly in a mathematics content course where such considerations are not typically discussed.

Conclusion

          The purpose of sharing the implementation of the Sprinkler Task in two contexts for senior-level, secondary PSTs is to highlight the versatility of modeling and discuss the implementation across disciplines. Despite the two different course contexts, we found this task to be largely successful in both classes, which we attribute to alignment to the modeling cycle and regional significance. While we would alter the implementation of this task for future iterations (i.e., include time to support introductory knowledge of modeling; restructure the comparison of models created), we believe that the work of developing and implementing authentic mathematical modeling tasks for PSTs is important. Our challenge to fellow mathematics teacher educators is to explore issues in your local geographical areas and consider how PSTs can model these situations with mathematics.

References

Aguirre, J. M., Anhalt, C. O., Cortez, R., Turner, E. E., & Simic-Muller, K. (2019). Engaging teachers in the powerful combination of mathematical modeling and social justice: The Flint water task. Mathematics Teacher Educator, 7(2), 7-26.

Association of Mathematics Teacher Educators. (2017). Standards for Preparing Teachers of Mathematics. Available online at amte.net/standards. 

Anhalt, C. O., & Cortez, R. (2016). Developing understanding of mathematical modeling in secondary teacher preparation. Journal of Mathematics Teacher Education, 19(6), 523-545.

Anhalt, C. O., Cortez, R., & Bennett, A. B. (2018). The emergence of mathematical modeling competencies: An investigation of prospective secondary mathematics teachers. Mathematical Thinking and Learning, 20(3), 202-221.

Common Core State Standards Initiative. (2010). National Governors Association Center for Best Practices and Council Of Chief State School Officers. http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

Kansas State Department of Education (KSDE). (2019). 2017 Kansas Mathematics Standards Grades K-12. https://community.ksde.org/LinkClick.aspx?fileticket=Fk5h7Uw24Kw%3d&tabid=5276&mid=15449

Mathematical Modeling with Cultural and Community Contexts (M2C3 Project). (2018). https://m2c3.qc.cuny.edu/teacher-resources/selecting-sports-team-tasks/soccer 

Moll, L., Amanti, C., Neff, D., & Gonzalez, N. (1992). Funds of knowledge for teaching: Using a qualitative approach to connect homes and classrooms. Theory Into Practice, 31, 132-141.

Mousoulides, N. G., Christou, C., & Sriraman, B. (2008). A modeling perspective on the teaching and learning of mathematical problem solving. Mathematical Thinking and Learning, 10(3), 293-304.

Nebraska State Board of Education (NSBE). (2015). Nebraska Mathematics Standards.  https://www.education.ne.gov/wp-content/uploads/2017/07/2015_Nebraska_College_and_Career_Standards_for_Mathematics_Vertical.pdf

Tidwell, W., Anhalt, C. O., Cortez, R., & Kohler, B. R. (2021). Development of prospective elementary teachers' mathematical modelling competencies and conceptions. International Journal of Mathematical Education in Science and Technology, 1-21.

Turner, E. E., Drake, C., McDuffie, A. R., Aguirre, J., Bartell, T. G., & Foote, M. Q. (2012). Promoting equity in mathematics teacher preparation: A framework for advancing teacher learning of children’s multiple mathematics knowledge bases. Journal of Mathematics Teacher Education, 15(1), 67-82.

Appendix A –The Sprinkler Task

The Sprinkler Task Prompt

New information in the task to encourage revision

Appendix B – The Garden Schematic