By Graham Fletcher, posted April 11, 2016 –

When the Common Core State Standards (CCSS) were released in 2010, a heavy emphasis was placed on the eight Standards for Mathematical Practice (SMP). In fact, more emphasis was probably placed on the SMP than the content standards themselves. Through personal experience, I have learned that for all the emphasis placed on the SMP, the modeling with mathematics standard has proved most elusive in its understanding.

What Is Not Modeling with Mathematics

As elementary school teachers, we have misinterpreted the term model to mean simply the use of manipulatives, a misunderstanding that is causing our students to miss the mark when it comes to modeling with mathematics. The questions then arise, What is modeling with mathematics? and How can we make a cognizant effort to be sure it is taking place in our elementary school classrooms?

In short, the use of manipulatives does not ensure that modeling with mathematics is taking place. If the mathematics is not contextualized, modeling with mathematics cannot exist. Modeling with mathematics does not mean, “I do, we do, you do.”

Many times, elementary school problem solving engages students in a front-loaded process; that is, all the information required to solve a problem is freely handed over to students. Traditional word problems clearly specify the “givens,” the “goals,” and the “legal” solution steps for students; the interpretation processes for the child have been minimized or eliminated (English 2004, p. 207). When this happens, students tend to locate the quantities in the problem, identify a keyword, apply the operation, and call it day. Modeling activities avoid this pitfall because they ask that both the teacher and student to approach the mathematics differently.

What Is Modeling with Mathematics

Mathematical modeling is used to interpret real-world situations in mathematical formats (English, Fox, and Watters 2005). In elementary school, this could be as simple as writing an equation to represent a contextualized problem. But in looking at the CCSS high school modeling strand, I suggest that modeling requires more than just a simple equation. Here are three actions that begin to ensure our students are engaged in modeling:

1. Identify the problem, or pose a question.
   
2. Make an estimate.
   
3. Identify the variables needed to solve, and answer the problem or question posed.
   

I rarely observe these actions taking place in an elementary classroom. We usually give students the questions to solve, refrain from giving them opportunities to estimate, and almost always give the information they need to solve the problem or question. Mathematical modeling emphasizes the fact that “thinking mathematically” is about interpreting situations mathematically at least as much as it is about computing (Lesh and Lehrer 2003, p. 111). So how can we get students to begin thinking mathematically and engage in the components required when modeling with mathematics?

Three-Act Tasks

In 2010, Dan Meyer introduced the world to three-act tasks, and the result of his work has empowered teachers to become more effective storytellers on the path to mathematical modeling.

Act 1—Posing the Question

Watch the video.

Just as students would if they were to watch this video, begin to mathematize what you’re seeing. Begin to notice and wonder things.

You have now just identified the problem or posed a question. For the purpose of this blog post, we’ll stick with the question, How many Whoppers are in the jar?

Your homework

Estimate how many Whoppers are in the jar, and post your guess in the comment section. Be sure to justify your reasoning. In addition to making an estimate, to determine how many Whoppers are inside the jar, be sure to identify the information (aka variables) that you will need. This information will be provided in our next post (check back on April 25, 2016).

The lack of mathematical modeling in elementary schools is a cause for concern because it limits our students’ ability to reason algebraically and to see the everyday usefulness of mathematics. By simply posing a video in place of a traditional word problem, we are engaging in the process of mathematical modeling.

Your Turn 

Join us next time when we look to wrap up this modeling problem and share the benefits of including three-act tasks into your teaching repertoire. Until then, we want to hear from you! Post your comments below or share your thoughts on Twitter @TCM_at_NCTM using #TCMtalk.

References

Common Core State Standards Initiative (CCSSI). 2010. Common Core State Standards for Mathematics (CCSSM). Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards.org/wp-content/uploads/Math_Standards.pdf

English, Lyn. 2004. “Mathematical Modeling in the Primary School.” In Mathematics Education for the Third Millennium: TOWARD 2010, Proceedings of the 27th Annual Conference of the Mathematics Education Research Group of Australasia, Vol. 1, pp. 207–14. Sydney, Australia: MERGA.

English, Lyn, Jillian Fox, and James Watters. 2005. “Problem Posing and Solving with Mathematical Modeling.” Teaching Children Mathematics 12 (October): 156–63.

Lesh, Richard, and Richard Lehrer. 2003. “Models and Modeling Perspectives on the Development of Students and Teachers.” Mathematical Thinking and Learning 5 (2 and 3): 109–29.

Meyer, Dan. 2015. “Missing the Promise of Mathematical Modeling.” Mathematics Teacher 108 (April): 578–83.

Graham Fletcher has worked in education for over ten years as a classroom teacher, math coach, and district math specialist. He graduated from the University of Georgia, where he earned his specialist degree in Math Education. Fletcher’s passion for conceptual understanding through problem-based lessons has led him to present internationally and throughout the United States. He continues to be an advocate for best practice and a change agent for elementary school mathematics.