Model with Mathematics

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The Standard

The fourth standard of mathematical practice, model with mathematics, is not what it appears. The typical definitions of model include that of a verb, to show something, a noun, a small copy of something, and an adjective, remarkable. The connotation of the word model in this mathematical practice is none of these. Here, it means to take a situation and abstract the math from it in order to better understand it, or as Mike Flynn says in the book Beyond Answers, to “mathematize the world around them”. 

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In order to mathematize, students need to take a contextualized situation and create a mathematical model from it. In the primary grades, this does not come naturally. Students are building the foundation needed as they become more skilled mathematicians and it is important to introduce students to the idea of mathematizing their world, even if it is difficult. This can be as basic as asking students to write an equation based on a story as long as they get many opportunities to mathematize situations. The more practice we give students with dissecting the math out of a real world situation, the more practice they get modeling with mathematics.

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The Classroom

In the classroom, modeling with mathematics moves past literal drawings to represent problems as students begin to make more abstract drawings, even equations. For example, a student may draw literal dogs and cats to represent a problem about an animal shelter but may move toward using circles and squares to represent the dogs and cats instead. They may then move to just using tally marks or even numbers to make it fully abstract. Remember, this does not come naturally to students so they need consistent practice.

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Once students have become comfortable with mathematizing the world around them, they may start to use their models to make predictions or draw conclusions. They should constantly be asking themselves, as well as being asked, “Does this make sense?” Students may need to reconsider their model and revise it to get to an answer that makes sense or completely start over and this should be encouraged. Teachers can utilize 3 Act Math tasks to practice mathematizing and creating models.

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Question Stems

What number model could you construct to represent the problem? 

What math do you know that you could use to represent this situation?

What formula might apply in this situation? 

What are some ways to represent the quantities?

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Picture Books

Who Sank the Boat? By Pamela Allen

Goodnight Baseball by Michael Dahl and Christina E. Forshay

Hottest Coldest Highest Deepest by Steve Jenkins

The Lion’s Share by Matthew McElligott

How Many Seeds in a Pumpkin by Margaret McNamara

A Hundred Billion Trillion Stars by Seth Fishman

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Next Steps

Now that you’ve got a basic understanding of the fourth Standard of Mathematical Practice, there are many places that you can dive deeper to learn even more. First, the book series Mathematize It! includes a K-2, 3-5 and 6-8 version that provide resources and strategies to help students solve word problems by mathematizing them. The Illinois Early Learning Project has 4 graphic “Tip Sheets” to help encourage young children to mathematize and the Erikson Institute provides resources for teachers, families, and administrators to encourage students to mathematize the world around them. Brittany Hege, creator of Mix and Math, has made a video about why teaching students to mathematize is important. Finally, EQSTEMM (Advancing Equity and Strengthening Teaching through Elementary Mathematical Modeling) has different routines and assessments created as well as a planning template available.

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Sara VanDerWerf, MDE, will also be hosting a webinar for the fourth mathematical practice on December 4th, 2024 at 7:00 AM which you can find by registering here.

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If you’d like more information, support, or guidance on developing a better understanding of Mathematical Practice #1, please reach out to our Math Team here at Resource Training and Solutions.

Mindy Strom

Math Lead

Email: mstrom@resourcecoop-mn.gov

Phone: (612) 505-7997

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Megan Klaphake

Math Coach

Email: mklaphake@resourcecoop-mn.gov

Phone: (218) 770-0026

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References:

^{SanGiovanni, J. (2019). Using the mathematical practices effectively in the classroom. https://www.mheducation.com/unitas/school/explore/research/reveal-math-using-mathematical-practices-effectively-classroom.pdf} 

^{Illustrative Mathematics. (2014, February 12). Standards for mathematical practice: Commentary and elaborations for K–5. Tucson, AZ. Retrieved December 29, 2018 from http:// commoncoretools.me/wp-content/uploads/2014/02/Elaborations.pdf}

^{Flynn, M. (2017). Beyond answers: Exploring mathematical practices with young children. Stenhouse.}