Look For and Make Use of Structure

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The Standard

The seventh standard of mathematical practice, look for and make use of structure, asks students to identify the structure within the mathematics that they are doing and then use it to solve a problem. Mathematics is full of structure. For example, when learning about place value, students are learning about the structure of powers of ten. Another example is when a student can create a hexagon out of two trapezoids, they are using the structure of how geometric shapes are created. Understanding the different structures in mathematics is vital when solving problems.

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Students do not always understand the structures that they are using to solve problems; rather they just know that it has worked consistently, so they continue doing it. It is not about naming the structures within the mathematics, but rather being able to recognize  them and then put them into practice. For example, students may recognize that adding zero to a number doesn’t change its value at a young age but they won’t know this as a property of mathematics until later in their mathematical journey. However, students should be able to defend their generalizations about the structure that they are using through multiple representations such as verbally, pictorially and using an algorithm. It’s also important to note that not all students will be ready for all structures at the same time. When students have the ability to look for and make use of structure in mathematics, they have a greater understanding of mathematics.

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In order for teachers to notice, and capitalize on, when their students are using different mathematical structures, they themselves have to have a solid conceptual understanding of what they are teaching.

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The Classroom

In the classroom, looking for and making use of structure often comes up naturally while working on problems, or through number talks. Teachers can capitalize on this by noticing when students are using structure and asking guiding questions to help students recognize that what they are doing is important. However, teachers can also guide students toward a particular mathematical structure by encouraging purposeful exploration by designing specific tasks to help them see the structure. When planning a unit, teachers need to be on the lookout for specific structures they can point out. They can do that by asking themselves: “What ideas surface that are generalizable?”

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It is important that when students are exploring a new structure(s), the teacher is facilitating the learning rather than leading it. Students need to explore structures in math in order to create “rules” for themselves rather than being told the “rule”. When students own their generalization about a structure, they are able to utilize it.

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Question Stems

What do you notice when … ?

How does this relate to … ?

In what ways does this connect to ,,, ?

What have we learned before that is useful in solving this problem?

Why does this always work?

You say that it always works, what is the IT?

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Picture Books

Ada Twist, Scientist by Andrea Beaty

Stuck by Oliver Jeffers

Going Places by Peter and Paul Reynolds

The Boy and the Airplane by Mark Pett

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Next Steps

Now that you’ve got a basic understanding of the seventh Standard of Mathematical Practice, there are many places that you can dive deeper to learn even more. The elevated achievement group has created a learning progression that shows what looking for and making use of structure looks like from preschool through high school which can be found here. Build Math Minds podcast has a short episode focused on the seventh mathematical practice, which you can listen to at this link or anywhere you find your podcasts.

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Sara VanDerWerf, MDE, will also be hosting a webinar for the seventh mathematical practice on February 26th, 2025 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.}