Best Practices for Mathematical Modeling
By Ellen Dairyko, UChicago STEM Education Curriculum Developer
Process and practice standards — standards that describe what it means to do mathematics — have been around for quite some time, including in NCTM’s Principles and Standards and even before that in various state standard documents. The Common Core State Standards (CCSS) adopted by many states since 2011 include eight Standards for Mathematical Practice (SMPs); many other states have adopted similar practice standards. While some of these standards are fairly straightforward, many of them can be complex, especially when considering what they should look like in an elementary school classroom. This can sometimes make it difficult for those of us who teach elementary grades to understand what our students should be doing to demonstrate that they are proficient with a practice standard and how we can support their development of these practices. One frequently misunderstood practice standard appears in the CCSS as Standard for Mathematical Practice 4 (SMP4): Model with mathematics.
Mathematical modeling is about representing a real-world problem situation in mathematical terms (e.g., representing the situation of three birds being joined by two more birds as 3 + 2, or representing the results of an opinion poll on a bar graph or pie chart), and then using the resulting representations to make sense of the situation or draw conclusions. To effectively and authentically engage in the practice of mathematical modeling, students must learn how to represent mathematical features of real world situations in models, which could be drawings, tables, symbols, numbers, expressions, equations, or diagrams. They must also learn how to use or manipulate their models to solve the real-world problem and to revisit the real-world situation to see if their answer makes sense. The cyclic process in which students go back and forth between the real-world situation and their mathematical model of that situation continues until a reasonable result is obtained.
Modeling with mathematics is closely related to Standard for Mathematical Practice 2 (SMP2): Reason abstractly and quantitatively. SMP2 emphasizes abstraction, including building mathematical representations of various sorts, making sense of those representations, and making connections among representations. SMP4, on the other hand, emphasizes using abstract representations in mathematical modeling and making connections between real-world situations and mathematical representations that model those situations.
The CCSS elaborates on mathematical modeling as follows: “Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace,” and the writers include an example from elementary school: “In early grades, this might be as simple as writing an addition equation to describe a situation.” In our experience, young students also frequently develop models by acting out real-world situations with manipulatives, drawing pictures, completing diagrams, or sketching open number lines. Different situations often lend themselves to different mathematical models. Teachers in early grades should be familiar with the wide variety of ways in which children can model with mathematics.
To help teachers better understand the mathematical practices, embed them into their everyday instruction, and recognize them in students’ work, math educators at UChicago STEM Education have developed what we call “Goals for Mathematical Practice” (GMPs) for each of the CCSS Standards for Mathematical Practice. The following are the two GMPs that unpack SMP4:
- Model real-world situations using graphs, drawings, tables, symbols, numbers, diagrams, and other representations.
- Use mathematical models to solve problems and answer questions.
These goals help clarify what it means to model with mathematics, but there are additional questions to consider. How does mathematical modeling progress across the elementary grades? What is the expectation for mathematical modeling in a 1st grade classroom and does that expectation change as students move through the grades? One way to answer these questions is to examine student work samples from challenging tasks at different grades that require mathematical modeling to solve.
Join us on Friday, April 27, from 8–9:15, at NCTM’s National Convention, in Washington DC, as we explore and discuss mathematical modeling. We will look at student-created models from a series of grade-level tasks to show how the practice of modeling with mathematics develops across elementary grades.
By Ellen Dairyko, with Sarah Burns, Andy Isaacs, and Debbie Leslie
Ellen, Sarah, Andy, and Debbie work at UChicago STEM Education, a center devoted to research and development that aims to support high quality STEM instruction and learning for all students. Ellen can be reached at email@example.com.