Reason Abstractly and Quantitatively

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The Standard

The second mathematical practice is arguably one of the most difficult to understand and identify in a classroom without any prior understanding or background knowledge. This practice states that students should be able to reason abstractly and quantitatively by making sense of quantities and their relationships in problems. In layman's terms, the second mathematical practice is asking students to focus on their ability to make sense of numbers and use that understanding to develop flexible strategies to solve problems. 

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It also says that students should be able to decontextualize and contextualize problems interchangeably.  Context is very important when solving problems, especially in the younger grades where they need to lay the foundation of numbers and operations skills. Students need to be able to both read a story problem and turn it into an equation as well as take an equation and turn it into a story problem. Knowing how to read a problem, break it apart, and then represent it using manipulatives, pictures or numbers in order to solve is a key strategy students should be trying when utilizing this mathematical practice. 

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The quantitative reasoning strategies include understanding what the numbers represent in the problem and not just how to solve them or what operation to use. There also has to be an understanding that the operation that they used may not have been the most effective or efficient. Students need to be able to utilize what they know about numbers and reason whether or not their strategy to solve a problem will result in the correct answer.

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The Classroom

Now, how do you implement this into your classroom today? First, allow students to solve problems using their own methods using strategies that are grounded in their understanding. Ask students to picture equations in their head; what do they see? Begin connecting problems with pictures as well as connecting pictures to equations. Pair the “math language” of an equation with the “kid language” of a number sentence. You can pair the different languages throughout your entire math block by using the math vocabulary alongside the more kid friendly terminology. 

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In lower elementary, it is important to emphasize the ability to contextualize an abstract problem through visual representations and stories so that students can understand what the numbers represent. Preschool through second grade is laying the foundation for what the rest of mathematics education is built upon. Being able to contextualize and decontextualize is very important in both elementary school as well as in high school when students are constantly asking, “When will I use this in the real world?” Word of caution: don’t over contextualize a problem just so that students have the opportunity to decontextualize it. Instead, it should be a natural part of their learning within application problems.

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Question Stems

• What do the numbers used in the problem represent?
• How is _______ related to ________? 
• What properties might we use to find a solution? 
• How did you decide in this task that you needed to use...? Could we have used another operation or property to solve this task? Why or why not?

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Picture Books

• Not a Box by Antoinette Portis
• Walter’s Wonderful Web by Tim Hopgood
• The Little Old Lady Who Was Not Afraid of Anything by Linda Williams

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There are many books connected to decontextualized language! Here is a great way to think about four ways to include it in your work or share with families!

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Next Steps

Now that you have a basic understanding of the second mathematical practice, there are many places that you can go to dive deeper to learn even more. If you enjoy watching TED Talks, Maurice Ashley leads a talk where he proves that context is needed because the human mind is very logical and proceeds forward at the cost of accuracy. It is a short talk that will help with the understanding of why context is so important. The National Council of Teachers of Mathematics (NCTM) has a book all about how to connect the mathematical practices to the NCTM Process Standards and the chapter about the second mathematical practice also highlights a few key connections to the Number and Operations Standard as well as the Algebra Standard. Finally, the state of Ohio’s Department of Education has created some amazing handouts that highlight different ways each mathematical practice can be taught in each grade level. You can find the Kindergarten through Grade 5 handout here, grades 6 through 8 here and the high school handout here.

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Sara VanDerWerf, MDE, will also be hosting a webinar for the second mathematical practice on October 16th, 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.

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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} 

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^{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}

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