Engaging in Mathematical Modeling in the Elementary and Middle Grades
Text Box 3.1 A Math Happening 3a: Planning for the Table Tennis Championship
Next week, there will be a table tennis championship. Plan how to organize the league, so that the tournament will take the shortest possible time. Put all the information on a poster so that the players can easily understand what to do.
—Problem from the PRIMAS Website
MATHEMATICAL MODELING IN THE ELEMENTARY AND MIDDLE Grades: wHAT Are The BuILDING BLoCKS?
Mathematics educators have examined models and modeling in relation to problem solving with a body of evidence that suggests modeling activities are quite successful for most students (Lesh & Doerr, 2003; Lesh & Zawojewski, 2007; Schorr & Koellner-Clark, 2003; Schorr & Lesh, 2003; Zawojewski & Lesh, 2003). In Second Handbook of Research on Mathematics Teaching and Learning, Lesh and Zawojewski (2007) described that learning mathematics takes place through modeling, “Students begin their learning experience by developing conceptual systems (i.e., models) for making sense real-life situations where it is necessary to create, revise, or adapt a mathematical way of thinking using a mathematical model” (p. 783).
Traditionally, in many classrooms, applied problem-solving experiences do not come until the end of the unit when all skills are introduced and mastered. Lesh and Zawojewski (2007) with their research on model eliciting activities suggest the opposite, so that the “mathematical modeling drive the learning in the conventional curriculum and traditional story problems become a subset of the applied problems through which students learn mathematics” (p. 783). In their description below (Figure 3.1), when modeling is the approach to teaching mathematics concepts, “mathematical ideas and problem solving capabilities are co-developed during the problem-solving process.” The constructs’ processes and abilities are needed to solve “real-life problems (i.e., applied problems are assumed to be at intermediate stages of development, rather than ‘mastered’ prior to engaging in problem solving) (p. 783).”
Figure 3.1 Diagram from Lesh and Doerr (2003) work as appeared in Lesh and Zawojewski (2007).
Mathematical modeling may be thought of as an unstructured implicit problemsolving process. A typical mathematical modeling process entails introducing a problem with context, which promotes problem solving and requires a qualitative rather than quantitative analysis. Explicit problem solving often involves the gathering of information, determining what needs to be solved, and using well-known direct problem-solving methods (such as those that can either be seen as routine problems for a particular concept) to come to a solution. However, when the problem is unstructured (i.e., mathematical modeling), one is often required to employ implicit problemsolving methods (such as a creative nonroutine approach).
Mathematical modeling has a very distinct meaning and involves teachers and students mathematizing authentic situations, and requires the application of mathematics to unstructured real-life problem situations. Mathematical modeling has been encouraged and is often used in secondary mathematics courses where real-world applications are emphasized. However, we will introduce creative ways to engage teachers and students in the elementary and middle grades to use mathematical modeling and problem solving to promote twenty-first-century skills such as collaboration, communication, critical thinking and creativity. The ways we have decided to use the terms models and modeling are in no means exclusive or all inclusive to the ways mathematics educators and researchers have used them. Instead, the ways we incorporate and integrate these ideas and practices are ways we have found to be important places for mathematics teaching and learning as seen through our research lessons from our part lesson studies.
We discuss mathematical modeling as a process of connecting mathematics with real-world situations. As stated in the Common Core Standards for Mathematical
Figure 3.2 Four-step modeling cycle used to organize reasoning about mathematical modeling.
Source: (from Reasoning and Sense Making [NCTM, 2009]).
Modeling, “Real-world situations are not organized and labeled for analysis; formulating tractable models, representing such models, and analyzing them are appropriately a creative process.” These real-world problems tend to be messy and require multiple math concepts, a creative approach to math, and involves a cyclical process of revising and analyzing the model.
The figure above (Figure 3.2) outlines a cycle often used to organize reasoning in mathematical modeling (from Reasoning and Sense Making, NCTM, 2009). The steps identified in this four-step modeling cycle illustrates how a real-world problem can be translated to a mathematical model under specific assumptions, which can then be solved using appropriate mathematical tools. The solutions can then be interpreted in terms of the real world and in turn helps validate the predictive capability of the model proposed.
An important way to align our standards and teaching practices is to think about and unpack the ways we teach and learn mathematics across the learning progression. So let’s consider a high-school standard called mathematical modeling. As you read the description from the CCSSM document, think about ways this topic applies to elementary and middle grades? What would be the precursor to mathematical modeling in the earlier grades?
In our work with teachers, we used mathematical modeling process (see Figure 3.3) that we modified from the steps defined by the Society for Industrial and Applied Mathematics SIAM—Moody’s Mega Math Challenge—which is a national mathematical modeling contest for high-school students sponsored by The Moody’s Foundation (see http://m3challenge.siam.org/resources/modeling-handbook).   
- 4. Build Solutions: Generate solutions.
- 5. Analyze and Validate Conclusions: Does your solutions make sense? Now, take your solution and apply it to the real-world scenario. How does it fit? What do you want to revise?
- 6. Present and Justify the Reasoning for Your Solution. (Modified from the SIAM—Moody’s Mega Math Challenge website)
One of the teachers from our mathematical modeling project was so inspired by the professional development workshop that he decided to engage students in the beginning of the year with mathematical modeling. He began by creating a mathematical modeling bulletin board to reinforce his commitment to incorporating mathematical modeling as part of his instructional routine in his mathematical classroom.
Above his math tool shelf, his bulletin board displayed the six steps: P = Pose the Problem; A = Make Assumptions; V = Define the Variables; S = Get a Solution; A = Analyze the Model; R = Report the Results. He launched his school year with a mathematical modeling unit called Proposal for a Sport Stadium. His fourth-grade class was starting the year with place value to the millions and estimation and computation (Relates to CCSSM fourth grade: Students generalize their understanding of place value to 1,000,000, understanding the relative sizes of numbers in each place. Depending on the numbers and the context, they select and accurately apply appropriate methods to estimate or mentally calculate products).
Fourth-grade standard 4.1. The student will (a) identify orally and in writing the place value for each digit in a whole number expressed through millions; (b) compare two whole numbers expressed through millions, using symbols (>, <, or =); and (c) round whole numbers expressed through millions to the nearest thousands, ten thousands, and hundred thousand.
Figure 3.3 Math modeling process. Planning resources created based on math modeling process as described by Bliss, Fowler, and Galluzzo (2014).
Fourth-grade standard 4.2. The student will (a) read, write, represent, and identify decimals expressed through thousandths; (b) round decimals to the nearest whole number, tenth, and hundredth; (c) compare and order decimals; and (d) given a model, write the decimal, and fraction equivalents.
He posed the problem statement:
Text Box 3.2 A Math Happening 3b: Proposal for a Sport Stadium
The governor of your state wants to build a new sports stadium in one of the five big cities in your state. Propose a plan for the new sports stadium.
This problem tied in not only with the mathematics standards but with their social studies standards related to learning about their state. They also worked in their teams while enhancing their technology and research skills and their communication arts using Google map, Excel spreadsheet, and other Internet resources. In terms of mathematics, the teacher noted that mathematical modeling allowed for students to work with large numbers when researching about the capacity of stadiums and researching the top populated cities within their state and had no difficult making sense of the place value.
The number they were reading 6-9 digits made sense to them, and they could understand the magnitude of number because they could relate it to the capacity of a sports stadium. In addition, he was ecstatic that students came up with related mathematical ideas through this mathematical modeling task that would help him build meaning to other mathematics standards that he would teach throughout this year. In our debrief with the teacher, he listed many of the mathematics connections that this mathematical modeling task offered for now and for later in the year. For example, in discussing the cost of the stadium, materials used in construction, size, and cost per seat, students were working with decimal place value and computation with decimals. In financing a large project, students talked about loans, investments, wages, and running costs, which exposed them to percent and economics and financial literacy.
In terms of geometry and measurement, students were brainstorming shape, size, and considering scale as they planned for where they would build their stadium within the open land they found on Google map. In thinking about seating plans to maximize capacity, they thought about prime and composite numbers and realized composite numbers give you the most flexibility in arranging rows and columns. In considering the use of the stadium, they were researching and estimating the entry cost versus running costs, profitability, sale of food, and merchandise. Finally, for statistics, students analyzed the stadium use by age, gender, and nationality. Another related mathematical modeling task for the future would be to investigate and model the emergency evacuation time for the stadium. The teacher noted that the time investment in the stadium project was worthwhile and could be leveraged throughout the year to bring meaningful connections to this real-world scenario.
In addition, as we worked with teacher co-designers on lessons focused on mathematical modeling, we identified five critical norms (see Figure 3.4) needed in the classroom to ensure success. Teachers recognized that they needed to (a) choose a context that provided an authentic problem in which students can engage in the mathematics;
(b) provide appropriate scaffold to help students revise and refine their mathematical ideas during mathematical modeling; (c) build in opportunities and capacity to mathematize, predict, optimize, and or make decisions for the mathematical modeling process to be worthwhile; (d) both teacher and students needed to embrace the open- ended nature of the mathematical modeling task that had multiple solutions depending on the assumptions made; and finally, (e) both teacher and students needed to establish agreed upon socio-mathematical norms such as persevering through complexity and productive struggle.
Figure 3.4 Norms for promoting mathematical modeling. Source: Authors.
Think about it!
What mathematical modeling tasks could you incorporate into your curriculum that would elicit many of the mathematics standards you teach for your grade level? Share them with your colleagues.
-  Pose the Problem Statement: Pose questions. Is it real world and does it requiremath modeling? What mathematical questions come to mind?
-  Make Assumptions, Define, and Simplify: What assumptions do you make?What are the constraints that help you define and simplify the problem?
-  Consider the Variables: What variables will you consider? What data/informationis necessary to answer your question?