Text Box 3.3 A Math Happening 3c: The Traffic Jam

There is a 7-mile traffic jam. How many cars and trucks are on the road?

Mathematical Modeling through Unstructured Real-world Problems. This research lesson was focused on mathematical modeling with the process of connecting mathematical reasoning with real-world situations (see Appendix MMI Toolkit

3.2 Planning, Debriefing and 3.3 Evaluating Lesson Study). These types of problems are open ended and messy and require creativity and persistence. When student model such situations, they are applying mathematics in a way similar to how they would visualize this in reality.

Students, therefore, need to make genuine choices about what is essential, decide what specific content to apply, and finally decide if the solution is reasonable or useful. It provides an opportunity to develop and practice their twenty-first-century skills including collaboration, communication, critical thinking, and creativity. After teachers worked on the table tennis task, they wanted to experiment with more open-ended messy problems so they presented the following “traffic-jam problem.”

The teachers in this lesson study group felt that this problem met the higher-level demands (Doing Mathematics) in terms of a rich mathematical task (Smith & Stein, 2012). As we examine this one mathematical modeling problem, think about the process one needs to engage in that connects to the Mathematical Modeling Process from the CCSS for High School and SIAM math modeling process. Although this is an elementary task, can you see the parallel in the process a modeler will go through in solving this problem?

The teacher began the lesson showing a photo of a major traffic jam. Then stated the problem statement. There is a 7-mile traffic jam. How many cars and trucks are on the road? To help students make assumptions, define, and simplify the problem, the teacher asked students to think about all the questions that come to mind. Students shared the following open-ended questions:

  • 1. Does a four-lane highway consist of two lanes going in the same direction or four?
  • 2. What types of vehicles are on the highway? Eighteen wheelers, Smart Cars, Motorcycles, etc... .
  • 3. What is the ratio or percent of each type of vehicle?
  • 4. How much, if any, space is between each vehicle?

Text Box 3.4 Mathematical Modeling Phase

Pose the Problem Statement: Pose questions. Is it real world and does it require math 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/ information is necessary to answer your question?

(Bliss, Fowler & Galluzzo, 2014)

To answer these preliminary/founda- tional questions, the students (and teachers) had to find other resources (i.e., videos and traffic definitions). Students took a quick fieldtrip out to the parking lot to measure cars. Having some of this information provided students with a starting point, but quickly students realized that they had to make some assumptions to define and simplify this problem.

For example, they needed to think about the ratio of trucks to automobiles to motorcycles on a given stretch of highway and the average distance between vehicles during a traffic jam. The host teacher mentioned, “I know some questions that all of us have encountered in math class, such as this one, require students to recall and use ‘common knowledge’ that is not provided (i.e., how many feet are in a mile like knowing 5280 feet in a mile). However, how rare of a problem is it that expects, or requires students to do outside research on ‘uncommon knowledge,’ (or even better, allows students to set their own parameters or assumptions), before starting the calculations.”

Implement tasks that promote reasoning and problem solving. The teacher team chose this particular task and listed the following reasons for why they thought the problem was worthwhile. The task was open ended; modeled a real-life authentic situation, and many students (or most) have had some personal experience that would help them connect with, understand, or at least feel somewhat familiar with the task. Additionally, there was no specific procedure for solving the task; students would have to apply what they know; there was no answer or absolute solution, and students’ answers would be as good as their justification of their procedure and rationale.

Ultimately, the teachers chose the task because it met the criteria of being a proportional reasoning problem; higher-order thinking and twenty-first-century skills were required to work through the task; the task required students to present how they reached their answer, there were numerous variations in problem-solving strategies and modeling the original problem (e.g., considering restrictions, car space, ratio of cars to trucks, number of lanes); and the task links to and aligns with many curriculum standards, key skills, and concepts.

Elicit and use evidence of student thinking. The lesson gave an opportunity for the teachers in the lesson study not only to observe student learning but also to help enhance their pedagogical practices. The host teacher commented, “This idea of giving students a ‘real-world’ math problem was very inspiring. Not only does it reveal and improve students’ common sense and/or math sense, but also it provides opportunities for students to apply what they have learned during instruction to these new problems and see the different methods other classmates use to solve the same problems.

I was so impressed by this that I began to implement similar types of problems the last week of school with my fifth-grade students.” This collaborative study also gave a chance to analyze student learning through various factors. For instance, the participating teachers reported that most groups forgot to take into consideration space between each vehicle and that there were two lanes of traffic. They also noticed that one of the groups that worked most efficiently and presented very professionally fell into this category. Soon after they realized their mistake by way of another group commenting, they went back to correct part of their answer about the two lanes. They weren’t able to change their answer to take into account space in between each car, which would have been more challenging then multiply their answer by two (i.e., % increase, proportions). The work is illustrated in the Figure 3.6.

Supporting productive struggle. The teachers observed that the student group that was most productive and came up with the most accurate and justifiable answer still experienced a significant flaw in their interpretation of their calculations. For the second problem “How long would it take for the cars to clear out of the traffic jam?” the student group correctly found the number of minutes it would take [162 minutes]. However, the teachers noticed that the students in the group used long division to attempt to convert their time to hours. Again, their calculation of the hours was correct [2.7 hours]. The flaw was interpreting this decimal as “two hours and seventy minutes” instead of “two WHOLE hours and seven-TENTHS of an hour” (see Figure 3.7). As a result, they converted their interpretation to “Three hours and ten minutes,”

Students making assumptions about the number of cars in a mile. Source

Figure 3.6 Students making assumptions about the number of cars in a mile. Source: Authors.

Dividing hours and making sense of decimal. Source

Figure 3.7 Dividing hours and making sense of decimal. Source: Authors.

which they misrepresented again as 3.1 hours. It was curious why they did not ask, “How many hours are in 162 minutes (two),” and then figure out how many minutes remained (42)?

Teachers also observed that one student group struggled to understand the numbers that they were using throughout the entire class, which was ultimately revealed during their presentation. Although this group, with some help, performed the calculations correctly to determine how many vehicles were in the traffic jam [4022.8571 cars], there was one major problem: It was a decimal (“How can you have 0.8571 of a car?”). During their presentation, the students were asked if they were able to have “part” of a car. No one in the group understood what the teachers were talking

Text Box 3.5 Mathematical Modeling Phase

Build Solutions: Generate solutions.

Formulating a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables.

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?

Present and Justify the Reasoning for Your Solution.

(Bliss, Fowler & Galluzzo, 2014)

about. The teachers felt they understood that you could not have 0.8571 of a car driving around the highway; however, the teachers realized that the students in this group were all “lost in the numbers,” probably overwhelmed by this “rich task” that they never fully grasped. (This makes the host teacher wonder how many of my students are like this during math class, and/ or while working on math assessments or activities.)

The host teacher concluded, “I feel that this last group’s answer reveals some areas of neglect in my math instruction: checking your work, estimating precise numbers, asking questions when you do not understand something, and ultimately using common sense. These, skills just mentioned, are what all students need to succeed in the real world (compared with many of the math standards that we are required to teach). I feel these ‘rich tasks’ coupled with collaborative learning projects and presentations are an effective way to assess and improve students ‘essential math knowledge and skills.’”

Through this case study and professional development with teachers, we have seen models and modeling of mathematics expressed and developed in interrelated ways. Each of these approaches was shown to contribute toward developing strategic competence in both students and teachers. Through our professional development model, we were able to expand teachers’ understanding of modeling math ideas and provide them with practical means to do so.

The following reflection from the teacher who led the lesson study for the traffic- jam problem captures this. “I find it ironic that these ‘higher-level demanding’ problems are called ‘doing mathematics’ because you can’t start ‘doing the math’ until you research and/or agree on some assumptions, tasks that you would ‘TYPICALLY’ find in science and other subjects. The professional development experiences that I take back are redefining what math problems, math class, and ‘doing the math’ should encompass: not just plugging numbers into calculators and equations, but exploring and understanding the nature of the tasks, accessing relevant knowledge and experiences and making appropriate use of them while working throughout the task, and applying unpredictable and new approaches that aren’t explicitly suggested by the task. All of the descriptions/characteristics of ‘higher-level demanding problems (doing mathematics)’ have become relevant because I saw and heard for myself this process with my students while implementing the ‘traffic-jam’ task.”

This chapter offers a snapshot of what mathematical modeling for solving real- world problems looks like in an elementary and middle-grade classroom. Launching a unit with a motivating and intriguing mathematical modeling task can help students see how mathematics is an important tool we use to make decisions in our everyday life. Throughout the book, we share several benchmark problems that have been considered at multiple levels to engage students in conceptual learning and enhancing their understanding of the application of mathematics in the real world. Such problems provide rich mathematical and inquiry-based tasks that require the use of twenty-first-century skills, namely communication, collaboration, critical thinking, and creativity.

Think about it!

Describe the role of the teacher and the students engaged during mathematical modeling task. How might that role be similar or different than when engaged in a traditional mathematics classroom?

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