Universities Are Teaching Competing Math Philosophies to Future Teachers. Why That Matters

How can teachers help all students become successful in mathematics?

It is a deceptively complicated question—one that invites different ideas in the field about how best to prepare students for complex problem-solving, and one that surfaces disagreements about what being “successful” in math actually means, according to results from a new EdWeek Research Center survey.

Math is rising on education policymakers’ agenda, thanks to debates over new California guidelines in the subject, a historical drop in students’ test scores during the pandemic, and—inspired by the “science of reading” movement—interest in how evidence about how students learn can inform math teaching and learning.

As part of Education Week’s new line of math coverage, the EdWeek Research Center surveyed a nationally representative sample of 373 postsecondary math and math education instructors in June and July of this year. Of this group, 126 taught only math courses, 142 taught only math education courses (typically as part of teacher education programs), and 105 taught both.

Our goal was to dig into what teachers learn in their preparation programs—and how that shapes what happens in K-12 classrooms nationwide. (The Bill & Melinda Gates Foundation provided financial support for the survey. EdWeek designed the survey instrument and maintains sole control over articles informed by the results.)

This second of three stories on the results explores university instructors’ philosophies about K-12 math education and how they hope future teachers will teach.

“We are in a very interesting time in the world where we’re determining what counts as mathematics,” said Cathery Yeh, an assistant professor in STEM Education at the University of Texas at Austin. “There’s a general agreement that all children should have access to mathematics, and that we want them to do well. And what we consider mathematics, and what it means to do math, may differ.”

Many of the university instructors and math education experts Education Week spoke with for this story assert that certain narratives undergird math education—and that the survey results demonstrate how those assumptions translate into practice. But they didn’t agree on what those narratives were.

Some argued that the field as a whole is too focused on skill with procedures and algorithms—others that there’s an overemphasis on conceptual understanding to the detriment of foundational skills.

Some claimed that math education must change in order to serve future students, while others argued that we already have solid evidence on what works to improve student learning.

In essence, the results show widespread division in the field—on instructors’ perspectives on explicit instruction, the importance of math facts, inquiry-based learning, and grouping students by ability.

This dissonance has implications for future teachers and their students, as educators will likely enter the field without a shared core of principles to guide their instruction.

“This is really getting at some of those different theoretical approaches that we see in the mathematical education space,” said Sarah Powell, an associate professor of special education who studies math instruction at the University of Texas at Austin.

Explicit instruction less favored than ‘puzzling things out’

The survey asked math and math education instructors to respond to questions about their philosophies on K-12 math education.

A majority of instructors agreed that it’s OK to let students struggle and puzzle things out, and discussion in math class should focus on students’ ideas and approaches—no matter whether their answers are correct or incorrect. But there was more of a split in responses for questions about teacher modeling and student memorization.

There was also a divide in whether K-12 teachers should focus explicitly on teaching rules and procedures. (Explicit teaching generally has the teacher model how to solve a problem and then guide students through examples.) Seventy percent disagreed that teachers should do so, while 30 percent agreed.

Some math education figures were happy to see that most instructors favored discussion and puzzling things out, and were less supportive of explicit teaching.

“We definitely should value that students are struggling; that’s when the brain is getting a really good workout,” said Jo Boaler, a professor of education at Stanford University who studies students’ math mindset and promoting equity in math classrooms.

“When you teach methods and rules first, often students aren’t able to think conceptually, because they’re so focused on those methods and rules,” Boaler said.

A discussion-based approach gives students the opportunity to understand the subject more deeply, said Afi Wiggins, the interim managing director of the Dana Center, a math research and technical assistance organization at the University of Texas at Austin.

“Most of us were taught math in a very procedural way. Here’s the math, here’s the problem, here’s how to solve it, there’s no other way,” said Wiggins. “It’s more about process and getting to the right answer than it is about helping students think critically and creatively about a problem that needs to be solved that’s in front of them.”

Other postsecondary educators offered a different perspective: Puzzling through problems only pays benefits, they said, if students have enough background knowledge in what they’re expected to do.

Xiuwen Wu, an associate professor of special education at National Louis University in Wheeling, Ill., talked about a video she shows to her math education students.

In the video, which depicts an elementary lesson on perimeter and area, the teacher asks students to use one-inch tiles to calculate a shape’s perimeter. But the students aren’t clear on the difference between perimeter and area, and some fill in the shape with tiles and then count those. In effect, it shows, the students didn’t realize that they were performing the wrong operation.

“If the students within that group lack prior knowledge about perimeter and area, the definitions, it’s a dangerous struggle because they would be frustrated easily,” Wu said. That kind of struggle, without a clear goal, can make kids feel defeated, and the exercise futile, she added.

“But if the students have enough prior knowledge, and they can use the activity to connect to that prior knowledge, I think it could [be beneficial],” Wu said. “Math is about reasoning. It’s about finding solutions to problems. If it’s reasonably difficult, then they can benefit from it.”

Brian Bushart, a 4th grade teacher in West Irondequoit Schools in New York, remembers learning about these kinds of hands-on, sensemaking activities in his preservice program in the early 2000s. To him, it seems like universities are still “championing a lot of the exact same things.”

However, some of his own beliefs have since shifted, Bushart said. Sometimes, these activities are effective ways to solidify and connect student knowledge. But other times, he said, they’re “very inefficient, and hard to achieve.”

“I’ve definitely come around on that there’s some really solid research on explicit instruction, and the need to make things explicit for students,” he said.

A solid foundation sets students up for problem-solving success, he said. “If they get frustrated by it, if they don’t get the thing that you wanted them to learn from it, that’s kind of a terrifying thought,” he said. “I think there’s a place for problem-solving, and I think it’s after students have some knowledge of something that might be helpful on the problem.”

Fact fluency: Is it necessary or nice to have?

Another divide among survey respondents surfaced in fact fluency, the ability to quickly recall basic arithmetic facts.

Cognitive science researchers say that fact fluency is important, because it frees up brain space for students to work through more complicated, multi-step problems. It’s harder for a student to work out a system of equations, for example, if they also have to spend time and effort figuring out 56/7 or 12×9.

But efficiency and reducing students’ cognitive load aren’t the only reasons that math facts are important, said Nicole McNeil, a professor of cognitive psychology who studies math learning at the University of Notre Dame.

“I think people don’t appreciate, necessarily, the idea that if you become fluent in the math facts in a well-organized way, that the connections made there actually create conceptual knowledge.”

For example, McNeil said, a lot of K-12 teachers use partners of 10. They build students’ understanding of 10 using equivalent values: 4+6 = 10, 5+5 =10, so 4+6 = 5+5. That helps students think about the relationships between different sets of numbers—a skill that sets them up for algebraic thinking down the line, McNeil said.

“When you are organizing facts practice in this way to create connections—such that when you activate one, the others are also activated—if you do that beyond the 10s, it starts to create a network,” she said.

But not all university educators agreed that fact fluency is essential.

Most math and math education instructors—78 percent—agreed that K-12 teachers should teach relevant skills and facts at the same time as they teach problem-solving. But there was a divide between instructors who felt that fact fluency was essential, and those who thought it was helpful, but not essential.

Too much focus on memorizing facts can often mean less value is placed on engaging deeply with mathematical concepts, said Kyndall Brown, the executive director of the California Mathematics Project, a professional development network. It can also create a narrow understanding of what success in the subject means, he said.

“Who’s traditionally been considered ‘good at math’ are students who are able to perform computations and procedures with speed and accuracy. Those are the ones who tend to get labeled, ‘That’s a good math student,’” Brown said.

K-12 teachers who reviewed the survey findings took a middle road—they agreed that fact fluency was important, in part because it forms a foundation for more conceptual problem-solving.

Tara Warren, a math instructional coach in the Santa Monica-Malibu Unified school district, said she leans more toward thinking that fact fluency is essential. “You have to have your students work on things that are conceptual problems, but it builds” from fact fluency, Warren said.

Instructors’ perspectives on curriculum and course sequencing

Instructional philosophy is just one ingredient in classroom practice—what materials teachers use, and how courses are structured and sequenced, also play a role.

When asked about curriculum materials, most math education instructors thought that teachers should have some flexibility in choosing resources. Fifty-two percent said that all teachers should use the same core math instructional program, but pick and choose parts of it to use as needed, while 31 percent said that teachers should select, assemble, and/or create their own curriculum. Seventeen percent said that teachers should follow a core curriculum closely.

Their preference toward individual choice aligns with what most K-12 teachers actually do. Previous surveys have shown that most teachers use a combination of materials, supplementing and creating resources when they feel their district-provided options don’t serve student needs.

But Bushart, the New York state teacher, was surprised to see that almost a third of instructors expected teachers to create their own curriculum. “That is setting your teachers up for failure, and just a lot of unnecessary work,” he said.

There can be value in starting with a common program, he said—it provides a foundation for new teachers, and it makes it more likely that students will see familiar terms and strategies used across classes and grade levels.

The survey also asked about course progressions and ability grouping, a process in which students are divided by their proficiency level and some are exposed to material that is more advanced.

When asked about who should take calculus in high school, 42 percent of math and math education instructors said that only students who have taken prerequisite courses and have earned a certain grade and/or a certain score on a math test should be able to take the course.

Thirty-seven percent said that anyone who wants to take the course should be able to, but that it should not be required.

These findings on ability grouping and calculus concerned Wiggins, of the Dana Center, who favors mixed-ability groups.

“Ability grouping is just never a good idea,” she said. “What usually happens in these [mixed-ability] classrooms is collaborative discussion between students, and the different abilities of students lift each other up.”

She was disappointed to see that many instructors thought there should be restrictions on which students can take calculus.

“This is a lot of the reason why you don’t see a lot of African Americans and Latinos and students experiencing poverty in this field,” Wiggins said of STEM professions. “They’re saying there should be some way to block you out of calculus. But calculus is the only way you’re going to get into these fields.”

Not all high schools offer calculus, and the data on which students have access to the course show inequities: White and Asian students are more likely to attend schools that offer calculus than their Black and Latino peers.

Powell, the special education professor at the University of Texas at Austin, thought that many of the survey respondents probably imagined ability grouping as “high” and “low” classes. But there are other types of ability grouping. For example, she said, working with small groups of students to intervene on discrete topics can have significant benefits. That kind of grouping does have a place in schools, she said.

“The thing that we know about intervention is that earlier support is better,” Powell said.

Asking the right questions

For all the focus on philosophy, some math educators say they have even more essential questions about how math is taught.

Yeh would have wanted to see more questions in the survey that went beyond instructors’ perspectives on teaching methods to some of the deeper, thornier issues about a changing world.

“What are the key things that our society needs now, now that we have phones that can calculate thousands of calculations in a second, and that we’ve had a pandemic where data literacy has become a problem?” she asked.

“What is math?” she continued. “What places does math take place? What does it mean to be successful in math?”