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Home arrow Mathematics arrow Modeling mathematical ideas: developing strategic competence in elementary and middle school


The traffic-jam problem clearly gives the students an opportunity to engage in communication, collaboration, critical thinking, and creative problem solving, the four pillars of twenty-first-century skills. The problem allowed the students to begin the work they were assigned without any bias or opinions about what is or is not expected from the task. The misconceptions and misrepresentation that arose in the process helped them to get a better insight. Such student insights often open up new problem-solving strategies or even questions overlooked by the teacher.

The process of mathematical modeling provided students with opportunities to develop their twenty-first-century skills (P21) Learning and Innovation Skills which emphasize the 4 Cs: communication, collaboration, creativity, and critical thinking (see Table 3.1).

Mathematical modeling challenges students to use their mathematics communication as they collaborate on a problem. Students also use creative problem solving and critical thinking as they work to identify variables and make assumptions to build a mathematical model. In one of our STEM lesson studies, we had students designing a package that was environmentally friendly (Suh, Seshaiyer, Moore, Green, Jewell, & Rice, 2013).

In the project, Being an Environmentally Friendly Package Engineer, students were given objects for which they needed to design a package. They were given a large

Table 3.1 Twenty-first-century skills (Partnership for 21st Century Skills, 2011)

Critical Thinking and Creativity

Reason Effectively/Use Systems Thinking

1. Use various types of reasoning (inductive, deductive, etc.) as appropriate to the situation

Make Judgments and Decisions

  • 1. Effectively analyze and evaluate evidence, arguments, claims, and beliefs
  • 2. Analyze and evaluate major alternative points of view

Solve Problems

  • 1. Solve different kinds of unfamiliar problems in both conventional and innovative ways
  • 2. Identify and ask significant questions that clarify various points of view and lead to better solutions

Work Creatively with Others

1. Demonstrate originality and inventiveness in work and understand the real-world limits to adopting new ideas

Collaborate with Others

1. Demonstrate ability to work effectively and respectfully with diverse teams. Assume shared responsibility for collaborative work, and value the individual contributions made by each team member


  • 1. Articulate thoughts and ideas effectively using oral, written, and nonverbal communication skills in a variety of forms and contexts;
  • 2. Use communication for a range of purposes (e.g., to inform, instruct, motivate, and persuade) (Partnership for 21st Century Skills, 2010)

graph paper to create a prototype. One group created a “4 x 4 x 4” cube and then realized that they had a lot of wasted space. The student decided to redesign the package into a “4 x 4 x 2” rectangular prism in order to use more space, to be cost-efficient, and more eco-friendly (see Figure 3.8). To evaluate their design, students were asked to calculate the volume for their packages and the surface area.

The design phase involved an iterative cycle. Students created their prototype after discussion and then altered and modified the design based on the following factors: efficiency of space, economical, eco-friendly, or appealing design. An evaluation criterion was to decide which packaging design was most efficient. Students had to negotiate as they collaborated on their design. In the 5/6 upper-grade classrooms, we were able to provide the same scenario to explore packages but added more creative designs including cylinders and triangular prisms.

An interesting discussion that the class had was to think about what determines the best packaging container. If you want a package for something without a particular shape of its own, and minimize the amount of packaging material used, a sphere is the most efficient because it has the lowest surface-to-volume ratio of any geometric solid. In addition, round containers are generally used for any application requiring great strength. However, if all the packages were spheres like a soccer ball, packing multiple sphere packages would create empty space between containers. So if you are trying to pack something into multiple containers, the sphere is not the most efficient shape. Instead, a rectangular prism or cube might be the most efficient. This was a

Table 3.2 Prompts for students to self-assess and peer-assess after a problem-solving task

Assessing your Twenty-first-century Learning Skills in Mathematics

Prompts to assess your 4 Cs contribution

Critical Thinking:

How did you solve this problem in new ways linking what you know?


What new approaches did you consider to solve this problem or did you invent a strategy that was efficient?


Did you share thoughts, questions, and solutions?


How did you work together to each a goal, using your knowledge, talents, and skills?


Peer group

great discussion to have about why most real-world packaging is box-shaped rather than spherical.

Assessing student skills and knowledge is essential to guide learning and provide feedback to both students and teachers on how well they are doing in reaching desired twenty-first-century learning goals (Trilling & Fadel, 2009). Some of the ways to assess students’ twenty-first-century skills are to use a simple assessment rubric (see Table 3.2). Self-assessing one’s own contribution to group projects can help develop students’ self-monitoring skills and accountability in group collaboration.

The packaging iterative design phases

Figure 3.8 The packaging iterative design phases.

Think about it!

How does the mathematical modeling process prepare our students to be creative problem solvers for the 21st century?

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