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How can teachers help all students become successful in mathematics?

It is a deceptively complicated question鈥攐ne 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 鈥渟uccessful鈥� in math actually means, according to results from a new EdWeek Research Center survey.

Math is rising on education policymakers鈥� agenda, thanks to debates over new California guidelines in the subject, a historical drop in students鈥� test scores during the pandemic, and鈥攊nspired by the 鈥渟cience of reading鈥� movement鈥攊nterest in how evidence about how students learn can inform math teaching and learning.

As part of 澳门跑狗论坛鈥檚 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鈥攁nd 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鈥� philosophies about K-12 math education and how they hope future teachers will teach.

鈥淲e are in a very interesting time in the world where we鈥檙e determining what counts as mathematics,鈥� said Cathery Yeh, an assistant professor in STEM Education at the University of Texas at Austin. 鈥淭here鈥檚 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.鈥�

Many of the university instructors and math education experts 澳门跑狗论坛 spoke with for this story assert that certain narratives undergird math education鈥攁nd that the survey results demonstrate how those assumptions translate into practice. But they didn鈥檛 agree on what those narratives were.

Some argued that the field as a whole is too focused on skill with procedures and algorithms鈥攐thers that there鈥檚 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鈥攐n instructors鈥� 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.

鈥淭his is really getting at some of those different theoretical approaches that we see in the mathematical education space,鈥� 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 鈥榩uzzling things out鈥�

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鈥檚 OK to let students struggle and puzzle things out, and discussion in math class should focus on students鈥� ideas and approaches鈥攏o 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.

鈥淲e definitely should value that students are struggling; that鈥檚 when the brain is getting a really good workout,鈥� said Jo Boaler, a professor of education at Stanford University who studies students鈥� math mindset and promoting equity in math classrooms.

鈥淲hen you teach methods and rules first, often students aren鈥檛 able to think conceptually, because they鈥檙e so focused on those methods and rules,鈥� 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.

鈥淢ost of us were taught math in a very procedural way. Here鈥檚 the math, here鈥檚 the problem, here鈥檚 how to solve it, there鈥檚 no other way,鈥� said Wiggins. 鈥淚t鈥檚 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鈥檚 in front of them.鈥�

Other postsecondary educators offered a different perspective: Puzzling through problems only pays benefits, they said, if students have enough background knowledge in what they鈥檙e 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鈥檚 perimeter. But the students aren鈥檛 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鈥檛 realize that they were performing the wrong operation.

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

鈥淏ut 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],鈥� Wu said. 鈥淢ath is about reasoning. It鈥檚 about finding solutions to problems. If it鈥檚 reasonably difficult, then they can benefit from it.鈥�

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 鈥渃hampioning a lot of the exact same things.鈥�

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鈥檙e 鈥渧ery inefficient, and hard to achieve.鈥�

鈥淚鈥檝e definitely come around on that there鈥檚 some really solid research on explicit instruction, and the need to make things explicit for students,鈥� he said.

A solid foundation sets students up for problem-solving success, he said. 鈥淚f they get frustrated by it, if they don鈥檛 get the thing that you wanted them to learn from it, that鈥檚 kind of a terrifying thought,鈥� he said. 鈥淚 think there鈥檚 a place for problem-solving, and I think it鈥檚 after students have some knowledge of something that might be helpful on the problem.鈥�

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鈥檚 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 12x9.

But efficiency and reducing students鈥� cognitive load aren鈥檛 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.

鈥淚 think people don鈥檛 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.鈥�

For example, McNeil said, a lot of K-12 teachers use partners of 10. They build students鈥� 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鈥攁 skill that sets them up for algebraic thinking down the line, McNeil said.

鈥淲hen you are organizing facts practice in this way to create connections鈥攕uch that when you activate one, the others are also activated鈥攊f you do that beyond the 10s, it starts to create a network,鈥� she said.

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

Most math and math education instructors鈥�78 percent鈥攁greed 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.

鈥淲ho鈥檚 traditionally been considered 鈥榞ood at math鈥� are students who are able to perform computations and procedures with speed and accuracy. Those are the ones who tend to get labeled, 鈥楾hat鈥檚 a good math student,鈥欌�� Brown said.

K-12 teachers who reviewed the survey findings took a middle road鈥攖hey 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. 鈥淵ou have to have your students work on things that are conceptual problems, but it builds鈥� from fact fluency, Warren said.

Instructors鈥� perspectives on curriculum and course sequencing

Instructional philosophy is just one ingredient in classroom practice鈥攚hat 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鈥檛 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. 鈥淭hat is setting your teachers up for failure, and just a lot of unnecessary work,鈥� he said.

There can be value in starting with a common program, he said鈥攊t 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.

鈥淎bility grouping is just never a good idea,鈥� she said. 鈥淲hat usually happens in these [mixed-ability] classrooms is collaborative discussion between students, and the different abilities of students lift each other up.鈥�

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

鈥淭his is a lot of the reason why you don鈥檛 see a lot of African Americans and Latinos and students experiencing poverty in this field,鈥� Wiggins said of STEM professions. 鈥淭hey鈥檙e saying there should be some way to block you out of calculus. But calculus is the only way you鈥檙e going to get into these fields.鈥�

Not all high schools offer calculus, and the data on which students have access to the course : 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 鈥渉igh鈥� and 鈥渓ow鈥� 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.

鈥淭he thing that we know about intervention is that earlier support is better,鈥� 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鈥� perspectives on teaching methods to some of the deeper, thornier issues about a changing world.

鈥淲hat 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鈥檝e had a pandemic where data literacy has become a problem?鈥� she asked.

鈥淲hat is math?鈥� she continued. 鈥淲hat places does math take place? What does it mean to be successful in math?鈥�

Data analysis for this article was provided by the EdWeek Research Center. Learn more about the center鈥檚 work.