Is 2/7 larger than 4/11?

That’s the question the middle school class was struggling to answer. Fractions hadn’t really connected with the students, says John Barclay, a teacher in Richmond Public Schools in Virginia. The concept just wasn’t intuitive.

But one student piped up: She’d noticed that if you figure out how much you’d have to add to the numerator to get a whole number, then you can tell which fraction is larger.

That really wasn’t the rule that she was being taught. But Barclay — a former Virginia math educator of the year — thought back to his own experience with a discouraging middle school teacher and decided to think through what the student was saying rather than dismiss it. The student’s rule brought her to the right answer: 4/11 is larger. “I was like, ‘Oh, shit. Is that brilliant?’” Barclay says.

The student’s shortcut turned out to be unreliable and could have sent her to the wrong answer in some cases. But that wasn’t immediately clear. It takes critical thinking and a sense for the numbers to even understand how or why a student’s approach might be wrong, Barclay says.

This isn’t unusual: Students often get weird concepts of math, developing logical-seeming routes for answering questions, Barclay says. It can be tempting to fall back on procedural rules, particularly since students’ strange alternative rules can be time-consuming to explore. But thinking through many of students’ alternative thought patterns, so crucial to relaying math concepts, is becoming more difficult, according to Barclay. He feels like he’s given increasingly less leeway in how to implement the curricula he receives from his district, even when students are not fully understanding key concepts.

Barclay is a friend of mine, and his insight about the learning that is — or isn’t — happening in his classroom helped to shape my thinking about a question that’s been on my mind since I started reporting about recent efforts to change the way that math is taught.

Although it’s not as obviously contentious as say, social studies, educators and researchers (not to mention students and families) have strong feelings about math instruction (remember New Math? Or even newer math?) and competing ideas about why so many kids struggle with this core subject.

So I’ve been wondering: What do we know about the science behind good math education? And what’s preventing that knowledge from making it into classrooms where students are falling behind?

## Linear Path, Nonlinear Progress

There’s a lot riding on students’ ability to grasp fractions, one of the first real exposures they get to an abstract math concept. Since math classes progress in a mostly linear way, students have to get fractions to set them up for algebra; and how they do in algebra will likely influence whether they even get to try for advanced courses like calculus, a traditional weed-out metric for lucrative science, technology, engineering and math (STEM) careers.

But many people believe that math instruction in the U.S. isn’t working.

That’s led to a growing interest in changing how it’s done, with political fights over how to shake things up and large investments into improving math curriculum. And the latest national assessment scores in 4th and 8th grade were historically low, a troubling early sign that more students may get stuck when they encounter more advanced mathematics in high school and college, possibly deterring them from pursuing STEM studies.

But the finer points of instructional science or political brawls aren’t the only things decelerating student learning. Poor teacher support and inadequate training, researchers suggest, may be contributing.

The conversation around math instruction suggests that not that much is really known about how to teach K-12 math. Actually, an understanding of how to teach math effectively has become more refined in recent years.

Teaching students math means developing their problem-solving, thinking and reasoning skills, says Tammy Baumann, a vice president of academic services for academic assessment firm NWEA, which was recently acquired by publisher Houghton Mifflin Harcourt. It means moving students from understanding a “concrete” math concept, like adding or counting, to grasping abstract concepts, like a series of formulas for performing math.

While it was initially thought that students learn this linearly — progressing neatly from understanding a concept to using it to having a procedural set of rules — it’s messier than that. Research from the last three to five years shows that it’s more iterative, Baumann says. Students have to keep working things out again and again, picking up pieces of the concept and fluency as they go. And it varies for every student.

But that research may not be filtering into classrooms. Partly, that’s because it’s not just students, but also teachers, who sometimes struggle with math.

We’re not preparing teachers well in math, especially at the elementary level, says Yasemin Copur-Gencturk, an associate professor of education at the University of Southern California, whose research focuses on teacher education. She’s found that teachers have high levels of math anxiety. It’s common for her to hear, “Oh, you know, I decided to be an elementary school teacher because I don't want to teach math.” That mindset reflects a broader cultural anxiety, and when teachers don’t like the subject it hinders student achievement.

“I don't want to sound like I'm blaming teachers, but, unfortunately, that's a well-known problem,” Copur-Gencturk says.

It’s more common to focus on literacy than numeracy methodologies in teacher training programs, even though the two are deeply connected, Baumann, of NWEA, says. But worse, teachers seem to struggle with the conceptual understanding of math themselves, Baumann adds, pointing back to the broader cultural anxiety and inadequate teacher development.

“There's a huge need to make sure that the early elementary teachers have really deep and strong knowledge of math, but also of how to teach it,” says Kyndall Brown, executive director for the California Mathematics Project at the University of California, Los Angeles.

We’re starting to see elements of effective math instruction make it into state education frameworks, he says. For example, cognitively-guided instruction is mentioned in the draft of the California math framework. But focusing on developing a sense for numbers really isn’t the way most people have learned math, he adds.

## Treading Water

Even teachers who feel they have a strong sense for numbers may not feel empowered to focus their time on ensuring students make the cognitive journey toward strong numeracy, especially in lower-performing schools. Those schools, Brown says, really focus on regular testing and assessment. It ends up pressuring teachers to force students to stay on track with a generic schedule rather than engage students in critical thinking, which takes time. It raises equity concerns, Brown says, because this is more common in the lowest-performing schools with large numbers of students of color.

Barclay, of Richmond, reports feeling this crush. He feels like his school district takes a top-down managerial style, which leaves him little room for connecting more deeply with individual students.

Barclay’s frustration doesn’t surprise some researchers. The quality of math curricula has improved over the past decade or so, Baumann says. But, even with better curriculum, districts still give mandates to teachers. The focus is more on showing teachers how to implement curriculum rather than on growing teachers’ depth of knowledge or pedagogy, she says. The result is that teachers may not be getting better at laying out math concepts for students, or uncovering what students really know about math.

And the curriculum isn’t always good.

Elementary math textbooks claim to be aligned with Common Core standards, but they’re sometimes not, Copur-Gencturk says. In those cases, textbooks can treat math as a series of disconnected and isolated concepts, which makes comprehending math concepts difficult and can even spread math misconceptions.

“I have a daughter who's in fifth grade right now. And the way that the textbook they are using is displaying the math is heartbreaking, because it doesn't give students a clearer picture of how math ideas are connected,” Copur-Gencturk says.

Realities like this can force teachers to heavily modify materials so they can use them, she says: “It's an unacceptable amount of work for teachers.”

What should be done?

For Brown, the answer is teacher training, which he says was vital to his own career. That means development that takes place at the school site, during the school day, and that is focused on the curriculum teachers are using and the students that they are teaching, Brown explains. He believes that would allow teachers to collaborate and learn from each other in a meaningful way that actually supports them.

“I definitely think we need to invest a lot more money in high-quality professional development,” Brown says.