By Graham Fletcher,
posted April 11, 2016 –
When
the Common Core State Standards (CCSS) were released in 2010, a heavy emphasis was
placed on the eight Standards for Mathematical Practice (SMP). In fact, more
emphasis was probably placed on the SMP than the content standards themselves. Through
personal experience, I have learned that for all the emphasis placed on the SMP,
the modeling with mathematics standard has proved most elusive in its
understanding.
What Is Not Modeling with Mathematics
As
elementary school teachers, we have misinterpreted the term model to mean simply the use of
manipulatives, a misunderstanding that is causing our students to miss the mark
when it comes to modeling with mathematics. The questions then arise, What is
modeling with mathematics? and How can we make a cognizant effort to be sure it
is taking place in our elementary school classrooms?
In
short, the use of manipulatives does not ensure that modeling with mathematics is
taking place. If the mathematics is not contextualized, modeling with
mathematics cannot exist. Modeling with mathematics does not mean, “I do, we
do, you do.”
Many
times, elementary school problem solving engages students in a frontloaded process;
that is, all the information required to solve a problem is freely handed over
to students. Traditional word problems clearly specify the “givens,” the “goals,”
and the “legal” solution steps for students; the interpretation processes for
the child have been minimized or eliminated (English 2004, p. 207). When this
happens, students tend to locate the quantities in the problem, identify a keyword,
apply the operation, and call it day. Modeling activities avoid this pitfall
because they ask that both the teacher and student to approach the mathematics
differently.
What Is Modeling with Mathematics
Mathematical
modeling is used to interpret realworld situations in mathematical formats
(English, Fox, and Watters 2005). In elementary school, this could be as simple
as writing an equation to represent a contextualized problem. But in looking at
the CCSS high school modeling strand, I suggest that modeling requires more
than just a simple equation. Here are three actions that begin to ensure our
students are engaged in modeling:

Identify the problem, or pose a question.
 Make an estimate.
 Identify the variables needed to solve, and
answer the problem or question posed.
I
rarely observe these actions taking place in an elementary classroom. We
usually give students the questions to solve, refrain from giving them
opportunities to estimate, and almost always give the information they need to
solve the problem or question. Mathematical modeling emphasizes the fact that “thinking
mathematically” is about interpreting situations mathematically at least as
much as it is about computing (Lesh and Lehrer 2003, p. 111). So how can
we get students to begin thinking mathematically and engage in the components
required when modeling with mathematics?
ThreeAct Tasks
In
2010, Dan Meyer introduced the world to threeact tasks, and the result of his
work has empowered teachers to become more effective storytellers on the path to
mathematical modeling.
Act 1—Posing the
Question
Watch the video.
Just as students would if they were to watch this video, begin to mathematize
what you’re seeing. Begin to notice and wonder things.
You
have now just identified the problem or posed a question. For the purpose of
this blog post, we’ll stick with the question, How many Whoppers are in the
jar?
Your homework
Estimate
how many Whoppers are in the jar, and post your guess in the comment section. Be
sure to justify your reasoning. In addition to making an estimate, to determine
how many Whoppers are inside the jar, be sure to identify the information (aka variables) that you will need. This
information will be provided in our next post (check back on April 25, 2016).
The
lack of mathematical modeling in elementary schools is a cause for concern
because it limits our students’ ability to reason algebraically and to see the
everyday usefulness of mathematics. By simply posing a video in place of a
traditional word problem, we are engaging in the process of mathematical
modeling.
Your Turn
Join
us next time when we look to wrap up this modeling problem and share the
benefits of including threeact tasks into your teaching repertoire. Until
then, we want to hear from you! Post your comments below or share your thoughts
on Twitter @TCM_at_NCTM using #TCMtalk.
References
Common Core State
Standards Initiative (CCSSI). 2010. Common Core State Standards for Mathematics
(CCSSM). Washington, DC: National Governors Association Center for Best Practices
and the Council of Chief State School Officers. http://www.corestandards.org/wpcontent/uploads/Math_Standards.pdf
English, Lyn. 2004. “Mathematical
Modeling in the Primary School.” In Mathematics
Education for the Third Millennium: TOWARD 2010, Proceedings of the 27th Annual
Conference of the Mathematics Education Research Group of Australasia, Vol.
1, pp. 207–14. Sydney, Australia: MERGA.
English, Lyn,
Jillian Fox, and James Watters. 2005. “Problem Posing and Solving with Mathematical
Modeling.” Teaching Children Mathematics
12 (October): 156–63.
Lesh, Richard, and
Richard Lehrer. 2003. “Models and Modeling Perspectives on the Development of Students
and Teachers.” Mathematical Thinking and
Learning 5 (2 and 3): 109–29.
Meyer, Dan. 2015. “Missing
the Promise of Mathematical Modeling.” Mathematics
Teacher 108 (April): 578–83.
Graham Fletcher has worked in education for
over ten years as a classroom teacher, math coach, and district math
specialist. He graduated from the University of Georgia, where he earned his
specialist degree in Math Education. Fletcher’s passion for conceptual
understanding through problembased lessons has led him to present
internationally and throughout the United States. He continues to be an
advocate for best practice and a change agent for elementary school mathematics.