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.