Many math teachers around the country have adjusted their expectations for students as a result of the Common Core State Standards. But a pilot professional-development program is going above and beyond the new benchmarks by teaching small groups of elementary teachers in three states to teach a math skill that’s typically been reserved for high school and college students.
The three-year project, known as “Immersion” and funded by a $1.3 million grant from the National Science Foundation, is focused on mathematical modeling.
Young children start using physical models in mathematics as soon as they can count. But mathematical modeling is something different and more complex: It’s the process of taking an open-ended, multifaceted situation, often from life or the workplace, and using math to solve it.
“Sometimes making a mathematical argument helps you make decisions that seem too big or messy to understand,” explains Rachel Levy, an associate professor of mathematics at Harvey Mudd College in Claremont, Calif., who is leading the Immersion project.
The idea is to get young students seeing how mathematics can be applied to everyday life—in essence, an extension of the common standards’ push for critical thinking and application.
Here’s an example: A group of students wants to buy pizza for a class party. The teacher asks them to consider the different ways they might select a pizza place (cost, taste, proximity to school, etc.), and to come up with a mathematical argument to justify which pizza place is the best. The students then create a method, or model, that other classes could use for deciding on a pizza place as well.
Unlike much of what students learn in math class, these big, messy problems tend to have multiple entry points and no single right answer.
“Traditionally, students are sort of told and shown everything they’re supposed to learn—solve this kind of problem this way, and so on,” said Martin Simon, a mathematics education professor at New York University, who is not involved with the NSF project. “But mathematical modeling is a very different kind of process. ... It’s engaging students in the process of thinking about a situation and trying to find ways to mathematize that situation.”
Because mathematical modeling requires higher-level thinking, decisionmaking, and synthesis of various skills, it’s not generally taught to younger pupils. In fact, according to Simon, it hasn’t historically been part of K-12 teaching at all.
Common-Core Connections
The common-core standards have changed that, though, by mentioning modeling in several ways. For example, one of the eight Standards for Mathematical Practice—the overarching standards that describe the behaviors of “proficient” math learners—says that students should “model with mathematics.”
Teachers participating in the Immersion program created a video to illustrate for other elementary teachers and students what mathematical modeling looks like.
Produced and copyrighted 2015 by the IMMERSION PROGRAM at Harvey Mudd College
According to William McCallum, a lead writer of the common-core math standards, that statement was intended to encompass conceptual mathematical modeling (and not just using physical models). Children can begin doing simple word problems, which he calls “baby modeling,” as young as kindergarten.
“But the full modeling cycle involves some powers of discrimination and decisionmaking that we don’t normally attribute to kids of that age,” McCallum said in an email. Under the common core, students aren’t expected to go through the full modeling cycle until they get to high school.
There’s no doubt the Immersion program, conducted through university-district partnerships, is ambitious in introducing modeling lessons to elementary-level teachers. But Simon said getting students thinking earlier about how to justify choices with mathematics is important.
“That’s how students learn what mathematics is really about,” he said, “instead of thinking math is about doing 50 long-division problems and then going to high school and [having teachers] trying to change their minds.”
Over the summer, two dozen K-6 teachers from the Fairfax County school district in Virginia gathered at George Mason University for a week’s worth of training on mathematical modeling.
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 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 start with a big, messy, real-world problem?
• 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?
• Are the students communicating with someone who cares about the solution?
• Have they explained if/when their answer makes sense?
• Have they tested their model/solution and revised if necessary?
SOURCE: Rachel Levy, Harvey Mudd College
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 prototype—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.
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 confused 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.”
