Across Connecticut, teachers are stumped by math. Their lessons just don’t seem to be getting through, leaving more than half of all elementary-school students behind grade level.
One fifth-grade teacher in New Haven has hit on a formula that might show the rest of Connecticut how to solve the problem.
New Haven, at least, is preparing a “paradigm shift” in the direction that educator, Kelly Inga, has forged.
Last school year, the students in Inga’s fifth-grade class at Quinnipiac Real World Math STEM School, a K-5 school in the Heights, showed more growth in their math scores than any other group of kids in the state.
Almost of all of them demonstrated more than a year’s worth of learning in math, despite severely high needs. Of the 50 students in her class, 46 were poor enough to qualify for free lunch, 11 were still learning English, and 8 were in need of special-education accommodations.
Despite those obstacles, their math scores skyrocketed. Before they even reached middle school, the average fifth-grader leaving Inga’s class was scoring higher than the average eighth-grader at 18 other elementary schools in New Haven.
The Connecticut State Department of Education reported that the average student in Inga’s class last year maxed out their targeted growth. That’s a feat that no other class in the state could beat, though 13 other classes tied it, largely in wealthy areas like Westport, Fairfield, Old Lyme and Farmington.
So, how did Inga do it?
Just as at New Haven’s other top-performing elementary schools, where high-needs students keep pace with their peers, Inga relied on a form of personalized learning that meets students where they’re at, then challenges them to take on ambitious problems.
In what’s known as the “Math Workshop” model, Inga briefly walks through a mini-lesson at the board. Then, the bulk of class is spent letting students work in small groups on a series of increasingly difficult equations, as she floats around the room, nudging them along. The class finishes with a brief review of what they’ve learned, prompting “thinking about what it is that their thinking and putting it into word format,” said Nicholas Perrone, Quinnipiac’s principal.
Inga compares the model to teaching someone how to ride a bike.
“I can give you all the steps on how to ride a bike, which is what I feel like my mini-lessons are. But unless you’re actually out there practicing and trying it, you’re never going to learn how to use them,” she said. “The word problems and the real-life application is kind of like the bike. You’ve got the tools and the skills; I’ve taught you how to ride it. Now, show me you can do it.”
Inga took the training wheels off last week with some tough word problems that required the students to choose the right mathematical operation at each intersection of a multi-step problem.
Before diving in, the students took a seat on the floor at the front of classroom.
Inga wrote out a word problem on the board. It described how many hours two people spend playing videogames. But she left out any hint of what the students needed to solve. That’s because she didn’t want them to instantly plug in a formula. Instead, she asked them to just figure out what information they did have.
Then, Inga gave them an actual problem to solve.
This one asked about how many markers a girl had before she gave a bunch to her friends. It required the students to work backwards, using multiplication to figure out how many markers her pals received. Then, the students couldn’t forget the last step of adding that total back to how many she’d kept.
Inga asked students to draw blocks to walk through the problem — a visual aid that makes the math real. Then, after crunching the numbers, Inga had everyone write out an explanation of how they’d solved the problem on a worksheet.
After that, Inga let the students go off on their own.
She divided the class into groups, ranging from three to five students, that could work on three problems. They started at varying difficulty levels, and they escalated with each successive problem. But she let them know all of them would be tough.
“There’s not going to be any easy problems in here,” she told them.
Inga said she put some “cusp kids” into groups with students who are ahead. She said that makes her job easier, creating “more experts in the room,” and it pushes students to challenge themselves.
“They’re able to remind one another and share ideas together,” Perrone said. “The traditional model has students working on these independent ideas by themselves. They’re going to either crash and burn or excel, but either way, they’re going to do it by themselves. This method that Ms. Inga uses allows to build off one another’s ideas.”
Some students, assisted by Kayla Dickinson, a special-education teacher, worked on one-step word problems. They started, for instance, by adding 447 chicken sandwiches to 46 turkey sandwiches to find the total number a chef made.
Hold on. “What’s a Chef Andrew?” one girl at the table asked.
Another pause. “We do multiplication,” suggested Justin, one of the boys at the table.
Dickinson walked them through the problem again, pointing to the context clues. The question asked for the number of sandwiches “in all,” a good sign that addition was needed, she said. How many “times more than” usually signaled multiplication, she added.
The students each wrote out 447+46 in erasable marker right on the surface of the tables, and added the digits together. They remembered to carry the one.
“I like this,” Justin exclaimed at one point.
Others worked on more advanced problems. One group finished up, for instance, by figuring out how much water was in three containers, with only two clues about a container’s 87-gallon capacity and the comparative size of the other containers.
At a middle table, Jaquin and Linda (pictured above and below) puzzled through how much a family earned annually. The problem said that a girl earned $720 a month and that her father earned three times as much.
Jaquin started off by drawing four blocks, each representing $720. He knew he could add them together to figure out how much the family made each month, or he could multiply them as a shortcut.
But Inga asked them to make sure they looked at what the problem wanted them to solve. “What would that tell me?” she asked, as the two students arrived at $2,880. “Let’s go back and read the question.”
Linda realized they needed to multiply the family’s monthly income again by 12.
It took the group another couple minutes to get to the right answer: $34,560.
Jaquin got to that number quickly, but then he didn’t stop.
Racing along, he added an additional $720, thinking he hadn’t put in the girl’s salary. Meanwhile, Linda didn’t line up her digits for the multiplication, resulting in a carrying error.
“What’d you get?” Jaquin asked Linda.
They both compared numbers. Different. Hmm.
They erased their work and started over. On a second try, they both got it right.
Inga’s workshop aligns with what we now know about how students learn, an education expert said.
“In a traditional classroom, it’s typically a one-way conversation: all of the questions come from the teacher, and there are a set of clean answers to that question. It’s a very teacher-driven discussion,” said Nancy Songer, the dean of Drexel University’s School of Education.
“That’s no longer the pattern,” she went on. “Kids are asking more questions, and multiple people are providing responses to each other’s work. There might even be multiple correct solutions, as opposed to that very structured, teacher-initiated conversation.”
The problem-solving in Inga’s workshop model likely encourages deeper critical thinking, Songer added. That might be because it taps into two learning theories, she suggested.
Inga’s word problems put otherwise-abstract problems into real context, in what’s known as “situated learning.” Rather than “just doing formulas,” those problems feel familiar and useful, helping students to retain the information better with a “memory hook,” Songer said.
Their difficulty also challenges students to surprise themselves. By tailoring the work for each group, Inga can create problems that “stretch” students into the “zone of proximal development.”
“The idea [behind this theory] is that people learn best when the materially is slightly outside of what they think they can do by themselves,” Songer said. “Their brains are engaged; they find it intriguing. They do it best with others, using different abilities and voices in the group to get that slightly higher goal.”
After they solve these tough problems, students get the added benefit of knowing they’re up for a challenge, empowering them to take on harder problems, Songer added.
Quinnipiac’s math proficiency scores don’t look much different from the rest of New Haven’s. In fact, Quinnipiac came under state scrutiny in 2016 because of the achievement gap in math scores.
“Our math scores had only one direction to go, and that was up,” said Grace Nathman, the former principal who transferred to Celentano School this year.
Under her leadership, the school completed a turnaround. The state removed Quinnipiac’s “focus school” status at the first chance it had, saying the school had demonstrated progress, even though proficiency still lags.
Last year at Quinnipiac, students, on average, made it about three-quarters of the way to their growth target — a figure that represents how much student test scores need to rise for them to get on grade level within five years if they’re behind or stay slightly ahead if they’re already meeting benchmarks. The school had the third-highest growth rate in the district, behind Worthington Hooker and Strong Schools.
In Inga’s classroom, the fifth-graders did even better. Her students, on average, cleared 100 percent of their growth targets. That means that most of students actually exceeded a year’s worth of learning or more.
Nathman said that the Quinnipiac’s faculty made a deliberate effort to improve math instruction, starting with the “Math Workshop” model.
“We looked closely at the data and followed the Workshop model to a T,” she said. That allowed the teacher and math coach “to offer regular, targeted small-group instruction,” she added. “During the students’ independent work, the teacher made certain they were exposed to in-depth problems that mirrored the [Smarter Balanced Assessment] performance tasks.”
They also rolled out two computer programs that made the instruction more tactile.
Nathman added that these initiatives “are only as effective as the teacher implementing them.” She said that Inga saw so much success because she “pulled data constantly and instructed the students based on their needs.”
The rest of the district could soon be following suit.
Last year in New Haven, just one in five elementary-school students tested on grade level in math. The majority came in far behind, scoring at the lowest level.
Faced with those low scores, Kenneth Mathews, the district’s math supervisor, said there’s a “paradigm shift” underway in how math is taught in New Haven. Teachers can’t rely on equations and variables anymore. Instead, they’re now told to start with something concrete, even if it’s just blocks, to depict what’s happening. After that, they move to pictures and then finally to the conceptual.
“They can’t teach in the way they were taught math. Math was presented in the abstract; it was all about developing skills. Now, it’s so much deeper than that,” he said.
Across the district, supervisors are trying to blur the once-rigid divisions between subjects, using more science texts in writing classes and more word problems in math class. That flexibility opens up students’ problem-solving abilities, whether for a standardized-test question or a real-world issue.
“In a lot of classrooms, the students get very bored of just doing the same problem over and over,” said Perrone, who took over as principal after Nathman. “They learn how to complete and solve that problem, but the moment anything else is a little bit different — a different sign, an extra place value, decimals or fractions — it throws this curveball at them. They go back to, ‘You’ve got to show me how to do this again, and I’ll copy.’
“What I like that Ms. Inga does is allow for the struggle to happen. That’s a very important piece of learning,” he added. “If answers or ideas are given, our students are reliant on an adult. We don’t want that for problem-solving. She makes a point to put it on them by asking questions and holding them accountable. She’s encouraging that struggle so that they grow.”