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In my previous post, I argued that in addition to teaching mathematics for its own sake, we should also teach mathematics so that students learn to value diversity, see mathematics in their lives and cultural backgrounds, and analyze and critique social issues and injustices. These learn-see-analyze purposes require connecting mathematics to real-world contexts, which is emphasized in the Common Core’s fourth Standard for Mathematical Practice: Model with mathematics. What does it look like to connect mathematics to real-world contexts? I see two general approaches.

NCTM’s *Principles to Actions**: Ensuring Mathematical Success for All* provides an example of using the real world as a **stepping-stone** for thinking about mathematical concepts. It describes a teacher engaging students with real-world problems involving proportional relationships; see the task below (NCTM 2014, p. 31):

Although this task is grounded in an out-of-school context, it is not a genuine dilemma that most students are likely to face outside the classroom either now or in their future lives. I am not advocating against this type of problem—these problems serve an important role in teaching mathematics. This real-world context is *familiar enough* and *imaginable* to students and can therefore serve as a stepping-stone for thinking about important mathematical concepts, like scaling up proportional relationships. This is similar to ideas from Realistic Mathematics Education (see here and here).

Another approach uses more **authentic** real-world contexts. These are either genuine problems that arise outside the classroom for which mathematics is useful or they are social issues that students can learn more about through mathematical analysis. Consider the following examples:

- How can we redesign a neighborhood park that burned down?

(Adapted from Turner et al. 2009)

- How unequal is the distribution of income in the United States?

(Adapted from Felton, Simic-Muller, and Menéndez 2012)

Problems like these are harder to use for several reasons. First, these problems are often open-ended and ill-defined. Although it is crucial for students to learn how to deal with messy real-world contexts, they will rarely encounter them in the classroom. Second, because of the nature of these open-ended problems, it is much harder to anticipate what mathematics students will use. The problems above can be approached with a range of mathematics, which is important for seeing the interconnected nature of mathematics. However, these examples can cause some teachers to shy away in an era of increased pressure to address particular standards in their lessons. Finally, because of the problems’ open-ended nature, students sometimes find approaches to these problems that use little or no important mathematics.

Despite these difficulties, I hope that teachers will integrate authentic real-world contexts in their classrooms. These contexts are crucial for engaging students in mathematical modeling and for preparing students to use mathematics beyond the classroom. Keep in mind that the problem must be authentic *and* that the teacher must encourage students to draw on their real-world knowledge and experiences and approach the task authentically.

Please share your thoughts below. What are your experiences with authentic real-world contexts? What concerns do you have? What opportunities do you see?

**References**

Felton, Mathew D., Ksenija Simic-Muller, and José María Menéndez. 2012. “ ‘Math Isn’t Just Numbers or Algorithms’: Mathematics for Social Justice in Preservice K–8 Content Courses.” In*Mathematics Teacher Education in the Public Interest: Equity and Social Justice*, edited by Laura J. Jacobsen, Jean Mistele, and Bharath Sriraman, pp. 231–52. Charlotte, NC: Information Age Publishing.

National Council of Teachers of Mathematics (NCTM). 2014. *Principles to Actions: Ensuring Mathematical Success for All.* Reston, VA: NCTM.

Smith, Margaret S., Edward A. Silver, Mary Kay Stein, Melissa Boston, and Marjorie A. Henningsen. 2005. *Improving Instruction in Rational Numbers and Proportionality: Using Cases to Transform Mathematics Teaching and Learning. *Vol. 1. New York: Teachers College Press.

Turner, Erin E., Maura Varley Gutiérrez, Ksenija Simic-Muller, and Javier Díez-Palomar. 2009. “ ‘Everything Is Math in the Whole World’: Integrating Critical and Community Knowledge in Authentic Mathematical Investigations with Elementary Latina/o Students.” *Mathematical Thinking and Learning* 11 (3) (July 8): 136–57.

Mathew Felton is an assistant professor of mathematics education in the department of mathematics at the University of Arizona and will be starting in the department of teacher education at Ohio University this fall. He is a coauthor of *Connecting the NCTM Process Standards and the CCSSM Practices*. His research focuses on supporting current and future teachers in connecting mathematics to real-world contexts and on teachers’ views of issues of equity, diversity, and social justice in mathematics education.

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