“How much energy do we use at school every day? Do you think we can make enough to sell energy back to the grid? Maybe then we could have an extra field trip!”
“If we put solar panels over the courtyard, we could make energy and have shade. It’s so hot out there in the summer.”
These aren’t typical fourth-grade conversations, but these comments are examples of the big thinking generated when students are given the opportunity to pursue curiosities stimulated by classroom explorations that integrate mathematics and science. A modeling approach to mathematics instruction unlocks these possibilities, as guidelines suggest, “Teachers can launch mathematical modeling problems from questions that children naturally ask about science,” (Garfunkel & Montgomery, 2019, p. 30), and accomplish both mathematics and science standards while tapping into new levels of student motivation.
Following an introduction to the topics of energy, fossil fuels, and their impact on the environment during science instruction, fourth-grade students in Ms. Curria’s class wondered how they could reduce the environmental footprint of their school. Ms. Curria leveraged the students’ natural passion for sustainability to fuel an inquiry-based learning sequence exploring energy and conservation that authentically intertwined mathematics, science, and engineering.
Connections among these disciplines have been recognized and affirmed in national standards. For example, the application of mathematical ideas and skills within science has been highlighted in national science standards for over 20 years. More recently, the Next Generation Science Standards (NGSS; NGSS Lead States, 2013) incorporated computational thinking (Wing, 2006), engineering design, and a K-12 strand of performance expectations in engineering, technology, and applications of science. Careful planning is essential to make good use of these synergies, as Shaughnessy (2013) warns against integrated projects that keep mathematics silent; the National Council of Supervisors of Mathematics and National Council of Teachers of Mathematics (2018) add that mathematics should be addressed “with integrity to the grade level’s mathematics content and mathematical practices” (p. 1).
To ensure opportunities for mathematical development within a science context, a focus on mathematical modeling can be productive (Stohlman, 2018): a process of using mathematics to represent, analyze, and make predictions regarding complex, reality-based phenomena (Garfunkel & Montgomery, 2019). The importance of developing students’ capacities for mathematical modeling at every grade level is underrealized, yet it offers noteworthy advantages for deep mathematical learning while also charting a way forward in mathematics and science integration. Mathematical modeling fosters intrinsic motivation by triggering a need-to-know mathematical content and showcases the power of mathematics to answer important cross-disciplinary reality-based questions, such as those asked by Ms. Curria’s students (Garfunkel & Montgomery, 2019).
The process of mathematical modeling (see Figure 1) involves six components: (1) specify the problem, (2) make assumptions and identify variables, (3) represent the idea in mathematical terms to reach a solution, (4) analyze and assess the solution, (5) iterate, and (6) implement the model (Garfunkel & Montgomery, 2019). Progress through these six phases is not linear, however, as modelers often revisit the assumptions made at the beginning or bounce back and forth between phases as they refine their approach. Traditional mathematics textbooks typically emphasize the two phases of modeling on the right: representing with mathematical terms to get a solution and (sometimes) assessing the solution (Meyer, 2015). Despite the prevalence of “word problems,” textbooks do not typically provide opportunities for students to hone the decision-making and analysis skills necessary to produce and refine a model—skills necessary for recognizing the power of math to model reality-based problems in scientific fields and beyond. For more insights into the modeling process, please refer to Fry and English (2023).
The Mathematical Modeling Process is Not Linear
Citation: Mathematics Teacher: Learning and Teaching PK-12 117, 5; 10.5951/MTLT.2023.0200
Note. Source: Garfunkel & Montgomery, 2019, p. 13.THE PROJECT
This lesson sequence fits well near the end of a fourth-grade science unit—on energy, conservation, clean energy production, and sustainability—which provides students with the background knowledge needed to contextualize the issue. After students had learned about fossil fuel extraction and combustion to generate electricity, Ms. Curria initiated the first phase of the investigation, specify the problem to be solved (Day 1), by prompting students to consider how they might contribute to energy conservation, clean energy production, and sustainability. After forming randomized groups of four to five students, she presented the following challenge: come up with an informed plan of action to decrease their school’s environmental footprint. Table 1 provides an overview of the project’s stages, timeline, and content area blocks used.
Phases of Integrated Mathematics and Science Sustainability Project
| Phase | Day/Time | Learning Activities |
|---|---|---|
| 1: Specify the problem to be solved | Day 1 (science block) 30–45 min |
|
| 2: Make assumptions & define essential variables | Day 2 (math or science block) 30–45 min |
|
| Day 2 (science or literacy) 45–60 min |
|
|
| 3: Develop & refine the math model | Days 3–7 (math block) 45–60 min/day |
|
| 4: Create physical representation | Days 3–7 (science block) 30–45 min/day |
|
| 5: Communicate solutions | Days 7–9 (science or literacy) 45 min/day |
|
| 6: Present final project | Day 9–10 (science block) 60 min/day |
|
SPECIFYING THE PROBLEM AND DEFINING VARIABLES
As students discussed plan options on Day 1, Ms. Curria circulated while anticipating what corresponding information and mathematical skills students would need. She had to ensure that students’ inquiries allowed them to encounter and deepen their understanding of critical fourth-grade mathematics standards such as area and perimeter, multi-digit multiplication, division, and geometry. If a group was producing a plan too complex for quantifying or solving with relevant fourth-grade standards, Ms. Curria asked them to think of a second option or guided them toward the most feasible investigation to ensure integrity to grade-level mathematics standards. Final proposals included the following: installing a wind turbine, solar panels, sink motion sensors, and the addition of skylights in all classrooms.
“You will need to prove to our principal that, without a doubt, your environmental sustainability plan is the best option for our school. To do that, what information do you need?” This was the question posed to transition groups to the second phase on Day 2, making assumptions and defining essential variables. Students began talking excitedly in groups, jotting down lists of questions: “How big is our roof, and is there anything already up there?,” “How big is a solar panel and how many could we fit?,” “How much room would we need for a wind turbine? If we need to cut down trees that wouldn’t be very green,” and “What size skylights are there? If it’s too small, we won’t get enough light!” Ms. Curria circulated to guide through probing questions, ensuring students accessed all the opportunities for mathematical questions. In particular, the skylight group needed guidance to form questions such as, “What is the best size for a skylight?” and “How big is the area of the classroom ceiling?” Table 2 organizes the teacher planning and scaffolding necessary for each group project, including anticipated student questions and teacher scaffolding questions. In addition, the digital asset (link online) includes teacher-sourced texts and focal mathematical concepts.
Group-Selected Projects and Associated Teacher Scaffolding
| Project | Anticipated Student Questions | Teacher Scaffolding Questions |
|---|---|---|
| Wind Turbines | Where can we place our turbine? How tall will it need to be? Do we have enough wind here? Do we need to chop any trees? How much power will our wind turbine produce per day/week/year? |
What do you already know about wind turbines? How does proximity to a building affect the wind received on a windy day? How do we decide if a wind turbine is a good option for our school? |
| Solar Panels | How much space do we have? How many panels will fit? How much energy will they produce? How much sunlight do we have? |
What do you already know about solar panels? How would you determine how many solar panels to install? |
| Sink Motion Sensors | How long does the water run when we wash our hands? How long should we be washing our hands? How much water do we use? How many people are in the school? How many times a day do they wash their hands? |
How long should you wash your hands to ensure you are killing all the germs? How frequently do you wash your hands? How long should a sensor allow water to run? How can we investigate the potential water savings? |
| Skylight Install | How large should a skylight be for optimal efficiency? Can all the rooms at our school have skylights? |
I just hired you to install skylights for the whole building. What do you need to know? Could we put a skylight in this room? How do you know? |
After scaffolding the groups’ selection of foci, this phase also required responsive planning: teacher-sourcing multimodal texts providing the necessary data to support grade-level appropriate mathematical modeling aligned with groups’ defined variables. Providing annotated texts (i.e., articles that have been marked up by the teacher) is one way to ensure students can make sense of the information provided (Phillips & Norris, 2009). In addition to using teacher-sourced data, some groups collected their own data: the water sensor group surveyed peers to determine reasonable assumptions for daily in-school handwashing. This opportunity to build students’ reasoning and decision-making capacity is key to authentic mathematical modeling, engineering design, and scientific problem solving.
MODEL WITH MATHEMATICS, ASSESS, AND ITERATE
With their variables defined and data extracted, the groups set off toward vertical non-permanent collaboration spaces (Liljedahl, 2021) to the phase of developing and refining the math model (Days 3–7). For about 45–60 minutes per day during the math block, Ms. Curria monitored and scaffolded groups’ progress with developing and refining models. She allowed them to go down their own paths, while scaffolding student thinking toward efficient strategies. With probing questions, she pushed them to locate and correct their own mistakes and arrive at deeper conceptual understanding. Student-led mathematics was everywhere, and it looked impressive, but it was also often cluttered and disorganized. At times students ran into numbers or situations that did not make sense. Instead of relying on the teacher to reroute them as they often would, they became their own detectives, assessing their steps and revising their model as needed. The pride and ownership students took over their group inquiry was gratifying to witness. Students came to school excited each day to continue their work. Positioning students as thinkers, decision makers, and modelers transformed fourth graders into engineers: motivated and empowered. Through daily gallery walks and share-outs, students began to see the value of labeling, making charts, and sketching diagrams to help their thinking make sense to themselves and others. Figure 2 shows the solar panel group’s in-progress work on Day 4 as they developed their mathematical model.
The Solar Panel Group’s Work on Model Development
Citation: Mathematics Teacher: Learning and Teaching PK-12 117, 5; 10.5951/MTLT.2023.0200
REPRESENTING AND COMMUNICATING RESULTS
Simultaneously during the science block, students worked on creating physical representations (Days 3–7), making decisions about how to convey their mathematical models through posters and 3D representations. Figure 3 shows the solar panel group’s poster. Another layer of model refinement occurred when groups switched posters and checked each other’s work. Groups diligently checked each other’s calculations, eager to try to find another group’s mistake. The group checking the solar panel poster found an error in the final answer: while the group had correctly calculated that their proposed 48 solar panels would create a total of 14,400 watts per hour, when multiplying that by their total hours of sun (13 hours of sunlight in a summer day), they incorrectly reported 182,000 watts of total energy. It is unclear whether it was a computational error or a copying error.
The Solar Panel Group’s Poster Presentation
Citation: Mathematics Teacher: Learning and Teaching PK-12 117, 5; 10.5951/MTLT.2023.0200
On Days 7–9, students used time in their literacy block for the phase of communicating solutions. They practiced authentic communication of their model through persuasive writing, studied earlier that year, with letters to their principal arguing the validity of their plan. Flexibility in allocating instructional time across content areas maximized the student experience of interconnection and synergy. Figure 4 displays the solar panel groups’ persuasive letter.
Persuasive Student Letter Summarizing Findings for the Principal
Citation: Mathematics Teacher: Learning and Teaching PK-12 117, 5; 10.5951/MTLT.2023.0200
The culminating phase, presenting final projects, occurred on Days 9–10 through a showcase where students displayed their final models and recommendations to an audience consisting of their principal, administrators, and fellow students. Third graders attending the showcase became excited about the possibilities they hoped to create next year. When a third-grade teacher later asked her students what they were interested in exploring during a time set aside for third-grade lab explorations, one student responded, “Something to make the world a better place like fourth grade did!” Using mathematical modeling to explore an authentic and critical issue was not only engaging to the fourth graders, but it was inspirational to others as well.
INTERDISCIPLINARY STANDARDS ALIGNMENT
Authentic, project-based learning that integrates mathematics and science naturally leads to a variety of high-priority learning outcomes. When freed from their silos, there is much overlap among mathematics and science practices and standards. The mathematical modeling process has much in common with the scientific inquiry and engineering design processes outlined through the NGSS Science and Engineering Practices (SEP). Throughout the learning sequence, these processes and practice standards are closely intertwined, creating simultaneous overlap. With content-focused standards, the analogy of a relay race describes their integration: one set of standards leads off the race (e.g., starting with science standards for fuel and energy) and then hands off the baton to the next set of standards (e.g., operations, geometry, and measurement) to begin their leg of the race. Table 3 presents the mathematics and science standards in which students actively developed competencies, with opportunities for feedback and guidance. Alignment with additional content areas’ standards is outlined in the supplementary material (link online).
Overview of Project Phases with Corresponding Interdisciplinary Standards
| Prominent Standards Addressed | Phases of Project | ||||||
|---|---|---|---|---|---|---|---|
| 1a | 2b | 3c | 4d | 5e | 6f | ||
| Science & Engineering | 4-ESS3-1 Obtain and combine information to describe that energy and fuels are derived from natural resources and their uses affect the environment. | X | X | X | X | X | |
| ETS 2.A Knowledge of relevant scientific concepts and research findings is important in engineering. | X | X | X | X | X | ||
| SEP Asking Questions and Defining Problems | X | X | |||||
| SEP Obtaining, Evaluating, and Communicating Information | X | X | X | X | X | X | |
| SEP Using Mathematics & Computational Thinking | X | X | X | X | |||
| SEP Constructing Explanations and Designing Solutions | X | X | X | ||||
| Mathematics | 4.OA A. Use the four operations with whole numbers to solve problems. | X | |||||
| 4.MD A. Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. | X | ||||||
| 4.MD.A.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems. | X | X | |||||
| 4.MD.B Represent and interpret data. | X | X | X | X | X | ||
| MP1 Make sense of problems and persevere in solving them. | X | X | X | X | |||
| MP2 Reason abstractly and quantitatively. | X | X | X | X | X | ||
| MP3 Construct viable arguments and critique the reasoning of others. | X | X | X | X | |||
| MP4 Model with mathematics. | X | X | X | X | X | X | |
| MP6 Attend to precision. | X | X | X | X | |||
Note. Adapted from National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010, and National Research Council, 2013.
aSpecify the Problem
bMake Assumptions and Define Variables
cMath Model
dCreate Physical Model
eCommunicate Solutions
fPresent Final Project
The use of mathematical modeling around an authentic task—designing an informed plan of action to decrease the school’s environmental footprint—provided organic opportunities for connections across disciplines. However, this does not happen without careful teacher planning and scaffolding. Ms. Curria did important front-work, identifying grade-level standards that the projects could feasibly address. Then, responsive planning came into play: (a) scaffolding groups’ thinking as they defined their foci, (b) asking probing questions to scaffold groups as they defined their variables, and (c) sourcing appropriate texts and data in response to group’s goals. Teacher scaffolding was critical to supporting groups’ development and refinement of the mathematical model and communicating results.
CONCLUSION
Much like Turner and Font Strawhun’s (2007) student-led mathematics investigation into their school’s overcrowding, Ms. Curria’s unit was a place-based investigation that balanced student autonomy to chart their journey toward environmental stewardship with teacher scaffolding to center inquiries on targeted grade-level mathematics standards. While many teachers express concerns about relinquishing control and managing variability, Ms. Curria demonstrates how achieving a balance of both openness and scaffolding can foster an authentic grade-level inquiry in which student engagement and ownership is through the roof (and the solar panels too).
It is important to note that Ms. Curria was operating within an administrative climate that supported teacher autonomy and prioritized teaching practices that develop student agency. Supportive administration and/or creative teachers can find ways to navigate mandates while making room for responsive teacher practices. Whether teachers modify their curriculum to incorporate integrated science and mathematics projects with relevance to students, or choose the path of least resistance following packaged curriculum materials, every choice reflects power; no choice is neutral (Felton, 2010). Teacher/school/district curricular choices impact students’ opportunities for knowledge and empowerment. By making space for student choice in their sustainability inquiries and facilitating their application of mathematics and science content standards in authentic situations, Ms. Curria unlocked the motivation and joy of mathematical modeling inquiries. _
REFERENCES
Felton, M. D. (2010). News and views: Is math politically neutral? Teaching Children Mathematics, 17(2), 60–63. https://doi.org/10.5951/TCM.17.2.0060
Fry, K., & English, L. D. (2023). How big is a leaf? Mathematical modeling through STEM inquiry. Mathematics Teacher: Learning and Teaching PK–12, 116(2), 99–107. https://doi.org/10.5951/MTLT.2022.0219
Garfunkel, S., & Montgomery, M. (Eds.). (2019). GAIMME: Guidelines for assessment and instruction in mathematical modeling education (2nd ed.). COMAP and SIAM. https://www.comap.com/resources/free-resources/gaimme
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Alicia Curria, she/her, aliciacurria@gmail.com, teaches fourth grade for Milford Public Schools at John F. Kennedy Elementary School in Milford, CT. She recently completed her certificate of advanced studies in STEAM and enjoys integrating the disciplines to create meaningful thinking and learning experiences for her students.
Lindsay Keazer, she/her, keazerl@sacredheart.edu, teaches mathematics education courses to prospective and practicing teachers at Sacred Heart University in Fairfield, CT. She studies teacher learning of pedagogies that further culturally responsive student-centered math and STEAM in K-12.
Darcy Ronan, she/her, ronand@sacredheart.edu, teaches science education courses to prospective and practicing teachers at Sacred Heart University in Fairfield, CT. She studies teacher identity and curriculum integration and serves as program director for STEAM+CS education.
