How does a revolution start? Sometimes, it’s a simple question. For Sarah Powell, an associate professor of special education at the University of Texas at Austin, the question was this math problem: Donna and Natasha folded 96 paper cranes. Donna folded 25 paper cranes. How many did Natasha fold?
In a study, Powell posed that question to children at the end of third grade, when they should have been able to answer it easily. Instead, most couldn’t solve it. One underlined 11 words in the question but didn’t attempt any math. Another jotted down the number 96 and gave up. A few wrote down random numbers that had nothing to do with the problem. More than half the students added the numbers 96 and 25 together. Only two children out of 15 she showed me got the correct answer: 96-25=71.
“I could send you hundreds of these,” Powell said. “It’s heartbreaking. How did we let it get to this? These are kids that just get passed from one grade level to the next. You shouldn’t let a kid get to fourth grade if they can’t add 12 plus 13. I see it as a huge equity issue. It is totally unfair what we are doing to these kids.”
In early 2020, at an academic conference just before the pandemic hit, Powell commiserated with other experts in special education and students who struggle in math. They shared her frustrations about the state of math education in America. A majority of students weren’t mastering the subject, according to the National Assessment of Educational Progress (NAEP), a test that tracks academic achievement. The most recent test had shown that 60 percent of fourth graders and 67 percent of eighth graders failed to score at or above the proficient level for their grade.
The researchers begrudged the lack of attention math gets in schools compared to reading, but suddenly their rival discipline – reading – was providing a role model for action. At the time, the debate over why schools ignored the science of reading dominated education news. Parents and teachers were pressuring schools to switch to a phonics heavy curriculum in kindergarten and first grade. “We have a science of math just like there is a science of reading,” said Powell. “They are analogous. We just need to get people going.”
The special education community was also instrumental in drawing the public’s attention to the science of reading. In that case, it was parents of children with dyslexia who were clamoring for change in how schools taught reading. This time, in math, special education researchers are taking the lead.
In its most extreme version, this new math movement revives an old fight between advocates of teacher-led instruction of step-by-step procedures against those who favor student discovery and a conceptual understanding of math. It also raises new questions about what makes for good evidence in math education and pits well-designed quantitative studies of achievement gains against qualitative studies of people’s attitudes about math and why more women and people of color don’t enter STEM (science, technology, engineering and math) fields.
At the urging of her colleagues, Powell emerged as a founder of the nascent movement. In December 2020, she invited dozens of like-minded education researchers to the first science of math Zoom meeting. They believed the research showed that teaching math properly in the early grades would drastically cut the number of children struggling with the subject who might need special interventions to catch up.
The researchers continued to meet almost every month during 2021 as their campaign gained momentum. They launched a website, an advocacy group and an auxiliary group for teachers. A Science of Math Facebook group, which they started in December 2020, now has more than 21,000 members. More than 150 education professionals, ranging from teachers to education professors, have added their names to a public list of supporters. One of their leaders held a Science of Math event in Pennsylvania in 2022 and is planning another in 2023.
Their first public assault on the status quo came in August 2022, when Powell and two of her science of math collaborators – Elizabeth Hughes of Penn State and Corey Peltier at the University of Oklahoma – published a paper titled Myths that Undermine Math Teaching. They took direct aim at some of the teaching practices recommended by the influential National Council of Teachers of Mathematics (NCTM), and Jo Boaler, a controversial professor of math education at Stanford University who has a large and devoted following of math teachers.
Citing 115 research studies to back up their views, Powell and her co-authors attacked what they described as common misconceptions about teaching math. They said it’s not essential to make sure children understand mathematical concepts before they are taught calculations. They insisted that algorithms, efficient ways of solving problems quickly, such as long division, aren’t harmful. They said that inquiry-based learning, where teachers encourage students to discover answers for themselves, is often not the best way to teach while explicit, direct instruction usually is. Forcing students to struggle with problems that they not only don’t know how to solve, but also haven’t mastered the tools needed to do so, isn’t helpful. Timed tests? It’s important, the researchers said, for students to master their sums and multiplication tables in order to free up the brain’s working memory to learn more complicated concepts. Periodic timed tests help teachers measure whether students are building speed and accuracy.
Powell says she and 13 other organizers have been volunteering their time to the cause and their group hasn’t taken any money from outside organizations or foundations. Powell’s own research is primarily funded by the U.S. Department of Education and the National Science Foundation.
The group is not arguing for a return to old-fashioned rote instruction, Powell says. She’s an advocate of active hands-on learning with tactile objects, what educators call “manipulatives.” But she says that research shows that children learn best when new topics begin with direct explanations from teachers who teach procedures and formulas alongside concepts. Then students practice mastering them. She isn’t opposed to inquiry learning, but she says it’s very hard to teach this way and the appropriate time is after children have mastered multiple strategies and have the tools to think through different possibilities.
The pushback has already begun. In an interview, Stanford’s Boaler says the myths article is wrong because Powell and her colleagues “cherry picked” the research and “dangerous” because it will lead teachers in the wrong direction. And she questions why special education experts should determine what constitutes the science of math. She points out that there are no mathematics experts in Powell’s group.
In January 2023, the NCTM, the math teachers group, reiterated its opposition to the rote memorization of math facts, such as multiplication tables. But that group’s president, Kevin Dykema, said the timing was a “fluke” and not in response to the science of math movement. Still, Dykema said he was “concerned” about the group and their disregard of rival research that shows kids are turned off by math when it’s taught as a boring set of procedures.
“I worry that the science of math is so focused on rote memorization,” said Dykema, a middle school math teacher. “I know a lot of students see math as very meaningless. They think that math is a bunch of isolated skills that need to be memorized, and they don’t see any value in learning it.” A session on the science of math debate is currently being planned for the annual meeting of the NCTM in October 2023.
Behind the scenes, officials at state education agencies and education trade associations from Colorado to North Carolina are asking questions. Powell said she’s already received a positive response from the Kansas State Department of Education. Meanwhile, opponents are privately circulating drafts of rebuttals to the “myths” paper. The homepage of the North Carolina chapter of the Association of Mathematics Teacher Educators says that the organization is working to understand the research behind the science of math movement and how to respond. “More information is coming soon!” the site promises.
Jon Star, a prominent professor of math education at Harvard Graduate School of Education, says that the science of math isn’t as clear as the science of reading and that there’s much we still don’t understand about the best ways to teach the subject. He also points out that we really don’t know that much about how math is taught around the country. Although Powell’s paper discusses the shortcomings of progressive ideas about emphasizing conceptual math and not drilling math facts, it’s unclear if that’s what’s actually going on in classrooms and whether those practices are to blame for poor math performance.
It seems clear that we may be heading for a new battle in the math wars, which have been raging on and off in American schools for decades. And that makes one veteran of these battles weary.
“I go into this with some ennui,” said Deborah Loewenberg Ball of the University of Michigan Ann Arbor, another prominent professor of math education. “But this is a very important conversation to me.” She said that in order to come up with the most effective approach for teaching math, we need to agree on the goals of math instruction. Do we want kids to be able to compute accurately? Yes, but not everyone agrees that this should be the main goal of mathematics education. “The public needs to understand that the goals of math education are contested,” she said. Merely invoking the word “science” doesn’t resolve that debate, Ball said.
I’m fascinated with this science of math group and what it has to say. In the coming weeks and months, I’ll be digging into the research on math instruction and what newer studies tell us about these old debates on procedures, concepts, multiplication tables, how to cultivate number sense, add fractions and solve word problems. I’m eager to see how it all adds up.
This story about the science of math was written by Jill Barshay and produced by The Hechinger Report, a nonprofit, independent news organization focused on inequality and innovation in education. Sign up for Proof Points and other Hechinger newsletters.
Jill, your article begins with “How does a revolution start? Sometimes, it’s a simple question. For Sarah Powell, an associate professor of special education at the University of Texas at Austin, the question was this math problem: Donna and Natasha folded 96 paper cranes. Donna folded 25 paper cranes. How many did Natasha fold?”
A not so simple question, depending on the children’s prior experiences. If they are used to working together on projects, they might easily have assumed that the 96 paper cranes were a joint effort, and then Donna, by herself also folded 25 additional cranes…leaving no way to figure out how many Natasha might have folded by herself.
Perhaps less confusing questions and a prior sequence of story problems would have enable greater success. Nuffield Maths from England provided a powerful math basis (e.g. Graphs Leading to Algebra, etc.). Might be worth checking it out.
Thank you for the article on the “Science of math teaching.” I was unaware of this math educational movement and want to learn more about it. I am a remedial reading teacher (Orton-Gillingham OG) reassigned to teach remedial math for the past ten years. I have noticed similar problems in teaching math as in reading education. There does not seem to be much structured, sequential, “hands-on” instruction to allow the students to understand the basics of the subject. To improve my students’ understanding of math, I have applied many of the principles advocated by the OG researchers and practitioners. Thank you for this article and your many other elementary education articles.
There are two issues I’d like to bring up.
The first is how we are defining math improvement. If an assessment measures rote memorization and/or application of mysterious procedures, and measures breadth over depth, then of course it will reinforce the idea that we should keep teaching that way. Then, the researchers can say, “see, it’s better to teach procedures and algorithms.”
That’s like measuring a healthy diet by only looking at how much weight loss it promotes. Sure, weight loss can be a good thing (like algorithms are a good thing for dealing with difficult calculations). But only focusing on weight loss might yield an incredibly unhealthy diet, just as focusing on algorithms might yield a very unhealthy relationship with true mathematical reasoning.
Most of us teachers have seen, over and over, that the same small group of students do well at math (and seem to like it), while the majority either just “get by” or, worse, reject math and mathematical reasoning and say things like, “I guess I’m not a math person”.
It may be harder to teach conceptual math. And, it may be much harder to assess whether students are able to think mathematically and apply that thinking in the real world. But I think we can figure all of that out when we accept that a major change is needed.
The second issue is about math curriculum in general. Why is it that we focus so much on abstract algebraic thinking, trigonometry, and a sort of “all roads lead to calculus” mindset rather than a focus on statistics, correlation / causation thinking, and financial literacy? Most people will never factor polynomials, calculate cosines, or use derivatives in their work or personal lives. But literally everyone will make financial decisions and literally everyone will deal with statistics in some way (for example, political decisions, news, analyzing what school to send their kids to…). Why aren’t we making math curriculum decisions that create more capacity for true mathematical reasoning that can be applied to the real world in a concrete way? Ironically, that strategy of real-world math might help so-called “non-math students” embrace math, and therefore be up for the challenge of factoring polynomials, converting graphs into linear equations, and/or finding sine, cosine, and tangents.
Instead, for most students, math is a mysterious set of rules and procedures that yield an answer. Oh, and to check that answer, go ahead and apply a different set of mysterious procedures. It’s insane! How can anyone think this is good? That’s not the type of math activities mathematicians do. That’s not what scientists do when they think about how to collect and analyze data. That’s not what stock analysts do. Accountants…ok maybe. But why shouldn’t our students get the chance to have the same experience as the scientists and mathematicians who created those awesome algorithms and theorems? Then, when they know why the algorithms or theorems work, they can embrace those procedures for the efficiency they bring to harder calculations.
This shift in strategy wouldn’t preclude students from taking Calculus. People often want to use “but…” statements. Like, “well, I like this idea, BUT when will they have time to do pre-calculus?” First off, there should always be options for students who love so-called “higher math”. Even so, I would argue that even the high-achieving students would benefit tremendously from more statistics and financial literacy, which are chalk full of critical thinking and mathematical application. In other words, even the current “winners” of the math game would benefit from a math reasoning approach.
I realize that another common reaction to these ideas is a statement like: “but to get into a good college, they need to at least get to pre-calculus in high school,” or, “well, the system just isn’t set up for this”. Right, that is the structure we have, but if we all agree that structure could be improved, then changes can start to happen. Sea changes are slow, but rely on all of us being honest about what we see in our classrooms, in our own children struggling with math, and with the lack of mathematical reasoning we see out in the world.
Or, we can just sit on our hands and keep the status quo, and keep wondering why the same silent majority dislikes math, why they often don’t use math reasoning effectively in their personal lives, and why so many say, “I’m just not a math person”. And those of us who are “good at math” can keep promoting more of the same instead of being open to a more holistic, authentic, mathematical reasoning approach.
Jill, in your article covering debate over the science of teaching math a sample problem about paper crane folding was presented as being a question third graders could not answer and showed fear in their response attempts such as underlining parts of the problem only to leave the question unanswered.
Please allow me to say I think that third graders could not solve this problem because it is an algebra question. When I went to school eons ago algebra was reserved for middle school. I think the question writer was likely unaware they were asking third graders to do algebra without prior instruction in algebra? I would love to know more about this.
The correct algebraic rewriting of the question posed to the third graders is not 96-25=71. A correct interpretation of the question is
96= 25 + x
In this instance to solve for x it is appropriate to subtract 25 from 96 to obtain an answer.
I teach physics to high school students and I think that looking ahead to the skills needed for physics down the road it would be of great help to understand why 96-25=75 is not the same thing as 96= 25 + x and that the latter is what was being asked.
It seems like this discussion is framed as whether teachers should focus on math concepts or rote procedures. I say neither. The focus should be on word problems. Yes, the dreaded word problems. I remember in school everybody groaned when the homework assignment included word problems. My own kids used to complain similarly. They always got mad when I told them the word problems are the entire point. If you go through 12 years of math and you can’t solve the word problems, you really did waste your time. Apart from actual, professional mathematicians the only use anybody is ever going to get out of math is building models of the real world to answer real world problems. That starts with Donna and Natasha and goes all the way up to spacecraft trajectories and and quantum mechanics. You probably only need calculus if you’re going into a STEM field. But even in everyday life there are plenty of uses for algebra, geometry, trigonometry and statistics — if you know how to use them.
I try not to take any educational movement as gospel, but what I like about the Science of Math is that it pushes back against the tremendous amount of group-think going on in progressive education. Constructivism, inquiry-based learning, a deemphasis of procedures, discovery learning, productive struggle, etc. have all completely taken over modern educational practices – to the point where explicit instruction or teaching algorithms has become taboo.
The researchers behind the Science of Math aren’t arguing that there is no place for inquiry-based learning, conceptual understanding, or growth mindset. They simply argue that concepts and procedures should be taught together, that students should be explicitly taught problem solving skills and then pushed to apply and synthesize those skills, by having students explore complex tasks ALONE will not yield deeper understanding. It’s amazing how people think this means going back to rote memorization and straight computation 100% of the time. This is the kind of all-or-nothing group think that plagues every aspect of political debate in this country and prevents real progress. People think there is one right team and they are on it. People think there is one right way to teach, but the reality of teaching human beings is much more complicated.
Also, let’s talk about equity. As math instruction around the country has deemphasized procedural fluency, many students that are successful in math are getting explicit instruction and procedural practice from their parents or tutors outside of school. By not providing that type of instruction and practice at school, we are actually widening the privilege gap for students with parents who have not worked with them or who were not able to afford outside help. Students whose parents have taught them procedural fluency at home while they have gotten conceptual instruction at school have benefitted from progressive educational movements. However, for the students who have only received the conceptual instruction at school and nothing else, the learning gap has widened.
Also, is Blooms Taxonomy completely dead? According to Bloom, higher levels of learning are dependent on having attained prerequisite knowledge and skills at the lower levels. I feel like lately we have been asking kids to demonstrate knowledge at the upper levels of the hierarchy but they are missing major pieces of the lower levels because they have been deemphasized for years. Just my perspective.
This debate is useless. Neither method is good for grade schoolers. Research shows that if you don’t teach math explicitly or directly (while allowing for some light exploration of math concepts as they arise in other subjects and lessons) for Pre-K through 4th grade, that a 5th grader can learn and master K through 5th math in 20 contact hours of direct instruction.
Up until 5th grade, we are working against their brains trying to teach them math. If we wait until 5th grade, we are working with their brain, they are actually interested, they actually retain it, and it happens fast.
End of debate.
Thanks for this article. Pity you missed this article: Designing mathematics standards in agreement with science.
Here’s the abstract (the article is open access)
To learn mathematics, historically students had no choice but to memorize fundamental facts and apply memorized algorithms. Since 1995 in the US, all states have adopted standards to govern K-12 mathematics instruction, and in most, standards have de-emphasized memorization and emphasized reasoning based on concepts. This change assumed the brain could reason in mathematics withoutrelying on memorized knowledge. Scientists who study the brain have recently verified this assumption was mistaken. Due to stringent limitations
in working memory (where the brain solves problems), mathematical problem-solving of any complexity requires applying well-memorized facts and procedures. A decade after the implementation of standards in most states, US young adults ranked last in testing in mathematics among 22 nations. Changes are proposed to state K-12 standards, which recent scientific research suggests would substantially improve student mathematics achievement.
https://www.iejme.com/download/designing-mathematics-standards-in-agreement-with-science-13179.pdf