Education Week - September 30, 2015 - S5

EDUCATION WEEK SEPTEMBER 30, 2015
Professional Development > www.edweek.org/go/pd-report
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The group was one of three such cohorts, with public school teachers
in Bozeman, Mont., and Pomona, Calif., also meeting that month at
local universities for similar training. All the teachers had to apply and
be selected to take part in the planned yearlong program.
Padmanabhan Seshaiyer and Jennifer Suh, both mathematics professors at George Mason, opened the session by asking the teachers to
brainstorm the kinds of problems they've had to solve in their own lives
recently. On poster paper, the teachers wrote queries such as:
Is it worth driving farther to get cheaper gas?
Should I fix up my house before I sell it? What rooms should I do?
We're driving to Atlanta with a 2-year-old and a beagle-when is the
best time to leave?
The exercise served to get the teachers thinking about how often they
encounter problems that might benefit from mathematical modeling.
The participants then tried a problem together. They looked at a larger
room in the building that had been split into two smaller classrooms,
with a temporary wall and door connecting them. They worked on figuring out how the new configuration of the room would affect the amount
of time it takes people to exit in the event of an emergency.
As they worked, Seshaiyer asked the teachers to identify their
"assumptions and constraints," words typically used in engineering
classes, referring to the factors they believe to be true and those that
limit their solutions. For example, an assumption might be that the
doors to the hallway could not be moved. The number of doors and their
positions would be constraints.
One teacher suggested testing the problem with a protoype-putting
marbles into a box designed like the room and tilting it to see how they
exit with and without the extra wall.
Through the Immersion project, teachers will meet periodically
throughout the school year in groups of four or five to do "lesson
study"-a collaborative teaching-improvement process with origins in
Japan. The groups will choose a modeling problem, devise lesson plans
around it, and predict how students will solve the problem, as well as
the kinds of mistakes they'll make. After doing the modeling problem
with their classes, the team will meet again to debrief on what worked
and didn't work with students.
"Having all those heads together to anticipate what kids will do will be a
benefit," said Brian Kent, a math-resource teacher at Weyanoke Elementary School in Fairfax who took part in the professional development.

ARE MY STUDENTS ENGAGED
IN MATHEMATICAL MODELING?
The process of taking a real-world situation and
modeling it mathematically is a new concept for many
elementary educators. Rachel Levy, who directs the NSF's
Immersion project, says teachers can ask themselves the
following questions to ensure they're on the right track
when doing a classroom modeling activity.

Did the students
ask questions
and then make
assumptions to define
the problem?

Are they
choosing
mathematical
tools to solve the
problem?

Are they using
the mathematical
tools to solve the
problem?

CONTENT-KNOWLEDGE CONCERNS
Spencer Jamieson, an elementary-mathematics specialist for Fairfax
County, emphasized that modeling is just one of many tactics teachers
will use in the classroom-and that students still do need to learn basic
computation and other math skills. Jamieson, who is the project liaison for his district, recommends elementary teachers "start off small,"
perhaps doing just one modeling problem with students per quarter.
But mathematical modeling is not an easy concept to understand in
itself, much less to explain to early learners.
Ensuring that elementary-math teachers-most of whom are generalists-have the content knowledge necessary to teach mathematical modeling is "a huge concern," said Simon of New York University,
"and we have not yet addressed elementary-math teachers' content
knowledge in any effective way."
"On the other hand," he said, "I would not say let's do more impoverished math teaching because of their more limited background."
Experts caution that mathematical-modeling problems for the classroom should be chosen with care.
"You really need to anticipate how students are going to enter into
the modeling problem, and you need to be very clear on what are the
mathematical targeted goals you want kids to learn and develop,"
said Karen Koellner, an associate professor at Hunter College in New
York City who teaches courses in mathematics education. (While not
involved in the Immersion project, Koellner did similar work providing professional development in modeling for middle school teachers
in the early 2000s.)
Teachers also need to know how to get students who stray during the
modeling process back on track, said Koellner. "What are the probes
and questions you're going to ask that don't direct kids in a particular
way but cause them to rethink what they're doing?" she said. When the
teacher sees students going in the wrong direction, he or she should ask
questions that cause them "a little cognitive dissonance."
That sort of questioning takes preplanning and an upfront analysis
of where misconceptions may occur, experts say.
During the workshop at George Mason, Levy, the project leader,
showed how difficult modeling can be in the moment. She posed a
student solution to the exit problem that seemed logical, but was
mathematically unsound.
Once the teachers discovered the mistake-the student had con-

Did the students
start with a big,
messy, real-world
problem?

Are the
students
communicating
with someone who
cares about the
solution?
Have they
explained if/
when their answer
makes sense?

SOURCE: Rachel Levy, Harvey Mudd College

fused fractions and ratios-a long conversation ensued about whether
it was worth explaining what ratios were if the student hadn't
reached that standard. The teachers ultimately couldn't come to
a consensus on how to respond.
"It's hard, right? I don't want to pretend that it's easy," Levy
told the group. "Elementary students are capable of proving conundrums that are incredibly difficult to untangle. But that's
why mathematical modeling can lead to incredibly rich conversations
about what makes sense and why." l
Coverage of "deeper learning" that will prepare students with the skills and knowledge needed to succeed in a rapidly changing world is supported in part by a grant
from the William and Flora Hewlett Foundation, at www.hewlett.org. Education
Week retains sole editorial control over the content of this coverage.

Have they
tested their
model/solution
and revised if
necessary?

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Education Week - September 30, 2015

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