National Science Foundation grant to help improve math skill development in children

Mary Wagner, a University of Dayton associate psychology professor, received a $702,643 National Science Foundation grant to help preschoolers improve their math skills in preparation for math instruction in elementary classrooms.

The five-year research grant will have Wagner working with Elizabeth Gunderson, an Indiana University associate professor of psychology. The full grant, totaling $2,489,778, will fund three studies to examine how children between the ages of three and five begin learning mathematical concepts, and how that information can be used to prevent them from falling behind when learning math at later stages of their schooling.

While any child can struggle with math, those difficulties can be exacerbated for children in under-resourced schools and communities, Wagner said. Her research is focused on helping young children from those environments — many of whom lack access to high-quality, early learning resources — be more prepared when they get to kindergarten, and beyond. Children from a range of socioeconomic backgrounds will be recruited to participate, including students in Head Start programs and preschools that participate in Dayton-Montgomery County Preschool Promise, a nonprofit that helps area preschools improve their programming.

Wagner’s past studies, along with other early childhood development research, show young children can have difficulty distinguishing between numerical and spatial concepts, or magnitudes. For example, if they see three elephants and six mice, they might say there are more elephants than mice because the elephants are larger in size.

“They’re just trying to figure out their world,” Wagner said. “As a developmental psychologist, it helps to better understand what processes they’re using. If we can identify those processes, it can give educators tools to help children better understand number concepts before they’re in a formal setting.”

Wagner found that children’s early ability to pay attention to numbers and not be distracted by size, color or shape — called flexible attention to magnitudes — helped predict future math achievement. Through the NSF grant, she’ll study whether interventions could improve young children’s flexible attention to magnitudes, and if that improvement will affect their future mathematics achievement. She’ll also study whether the children’s socioeconomic status impacts the development of that skill and future ability to succeed in math in school.

The NSF grant will build on Wagner’s past research, which included a grant from the National Institutes of Health in 2020, to study early development of mathematics skills in young children, particularly those from under-resourced backgrounds.

Wagner’s lab also supports research opportunities for UD undergraduate students. Wagner works with 15 to 20 undergraduates each year, including summer internships that allow those students to write institutional review board proposals and manuscripts, and help conduct tests working with young children. UD students get involved in data management as well, gaining a better understanding of what happens after data is collected from participants.

“Working in Dr. Wagner’s lab this summer has been an amazing experience,” said UD junior Deja Richardson. “Learning about research in psychology in my classes is one thing, but being able to apply it this summer on various tasks in the lab has been extremely beneficial. I have also been able to learn more about early childhood development by doing my own independent research with supportive guidance from Dr. Wagner. I am grateful for this opportunity and I am looking forward to utilizing it in my future career.”

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In California, a Math Problem: Does Data Science = Algebra II

After faculty protests and a debate over racial equity, the state’s public universities reconsider whether high school students can skip a foundational course.
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California is one of 17 states that now offer data science to high school students as an alternative to Algebra II.Jim Wilson/The New York Times

Since 2020, California has led a contentious experiment in high school math.

That year, public universities in the state — including Berkeley and U.C.L.A. — loosened their admissions criteria, telling high schools that they would consider applicants who had skipped Algebra II, a cornerstone of math instruction.

In its place, students could take data science — a mix of math, statistics and computer science without widely agreed upon high school standards. Allowing data science, the universities said, was an “equity issue” that could send more students to college. But it also raised concerns that some teenagers would be channeled into less challenging coursework, limiting their opportunities once they got there.

Now, the California experiment is under review.

On Wednesday, the State Board of Education voted to remove its endorsement of data science as a substitute for Algebra II as part of new guidelines for K-12 schools.

“We have to be careful and deliberate about ensuring rigor,” Linda Darling-Hammond, president of the state board, said before the vote.

A U.C. faculty committee — which controls admission requirements for the state’s entire public university system — announced on Wednesday that it will re-examine what high school courses, including data science, meet the standards for “advanced math.”

The turnabout in California reflects the national quandary over how to balance educational standards with racial and economic equity. Could data science draw students into higher-level math? Or will offering data science as an alternative to algebra divert students from obtaining the quantitative skills required for a range of careers? Should there be a workaround if higher math is blocking some students from attending college?

In California, hundreds of high schools across the state now offer data science courses. The ability to collect and assess data is a valuable life skill, which could benefit every student.

And California is one of 17 states that now offer data science to high school students in some form, and at least two states, Oregon and Ohio, offer it as an alternative to Algebra II, according to Zarek Drozda, the director of Data Science 4 Everyone, a philanthropy-backed organization based at the University of Chicago.

The push for data science is also complicated by the wide racial disparities in advanced math, especially in calculus, which is a prerequisite for most science and math majors. In 2019, 46 percent of Asian high school graduates nationally had completed calculus, compared with 18 percent of white students, 9 percent of Hispanic students and 6 percent of Black students, according to a 2022 study by the National Center for Education Statistics.

“Many educators are justifiably concerned that the calculus pathway institutionalizes racial inequities by decreasing the number of Black and Latino students in college,’’ Robert Gould, the author of a high school data science course, wrote in a 2021 article. Data science courses, he suggested, connect students’ everyday lives to their academic careers, “which one hopes will lead to a more diverse university enrollment.’’

But in a May 2022 letter to the U.C. faculty senate committee, eight Black faculty members argued that data science courses “harm students from such groups by steering them away from being prepared for STEM majors.”

Race isn’t the only issue. Hundreds of faculty members from the state’s public and private universities have signed an open letter expressing concern that substituting data science for Algebra II would lower academic standards. Offering a way around Algebra II, they said, deprives students of their best chance to absorb the mathematical principles increasingly central to many fields, including economics, biology and political science.

There was also dissent from the California State University System. Its academic senate stated in January that the shift “threatens to increase the number of students entering the CSU who are identified as needing extra support to succeed.”

But supporters have argued that data science is important for navigating an increasingly number-centric society and would help more students go to, and graduate from, college. Jo Boaler, a math education professor at Stanford who has been a vocal proponent of data science, argued in an opinion piece in The Los Angeles Times that Algebra II is largely irrelevant for many students: “When was the last time you divided a polynomial?”

Some faculty members said that, at the very least, students and parents should understand that high school data science won’t even qualify a student to take data science in college — because undergraduate data science classes require calculus.

“The messaging is very confusing,” Brian Conrad, a Stanford professor and director of undergraduate studies in math, said. “Who would think that taking a course in high school chemistry would not be useful for chemistry in college?”

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Cleo, the Mysterious Math Menace

In 2013 a new user named Cleo took an online math forum by storm with unproved answers. Today she’s an urban legend. But who was she?

Full Transcript

Anthony Bonato: It’s a bit of an urban legend in mathematics. There’s a sort of a romance to the story, in a way.

Allison Parshall: I’m Allison Parshall, and you’re listening to Science, Quickly. Today we’ve got an episode about a mysterious figure in the online math world. They disappeared years ago but are still sparking debate and speculation.

[CLIP: Show theme music]

Parshall: We all love a good puzzle. Some people have their crosswords. Some people play Sudoku. Other people are still doing Wordle.

But Ron Gordon, a patent agent and former physicist in Massachusetts, does hardcore calculus. Back in 2013, when our story takes place, he spent enough time on this online forum called Math Stack Exchange that it could have qualified as a full-time job.

Gordon: I was working my full- time job, and then I was on Stack Exchange. Plus, I had a family, too. I was having so much fun with it that I just didn’t even keep track of how many hours I was dedicating to it.

Parshall: The Mathematics Stack Exchange website is like Yahoo Answers, if the people on Yahoo Answers had graduate-level STEM degrees.

Now Ron has solved 2,954 math problems in his decade on Stack Exchange, but he’s most famous for his answer to one integral in particular. On November 11, 2013, a Stack Exchange user asked a question:

“I need help with this integral: the integral from negative one to one of one over x times the square root of one plus x over one minus x times the natural log of 2x squared plus 2x plus one, all divided by 2x squared minus 2x plus one, dx.”

Jay Cummings: Okay, that’s a crazy integral. And there are so many parts to it that, you know, one thing changes, any one of these one thing changes, and the answer is completely different.

Parshall: That’s Jay Cummings. He’s an associate professor of math at California State University, Sacramento. I’ve enlisted his help to figure out what the heck I’m looking at.

As much as solving integrals has haunted my nightmares since Calc II, the idea of an integral is actually pretty simple. Picture a line on a graph. Now imagine taking a colored pencil and shading in the area beneath that line, down to the bottom axis of the graph.

What we’re trying to find is the area of this colored region. For a straight line, this is super easy—it’s basic geometry. But the more complicated and curvy and weird your line gets, the more difficult it is to figure out the area underneath it. Now the integral in the November 11, 2013, post—that was difficult. The line on the graph looks like the spine of a long-necked dinosaur.

The original poster tried using a few computer programs, but none of them could give what’s called the “closed form” of the answer—that’s a precise and concise solution. Five minutes after it was posted, someone commented:

“Do you have any reason to believe there is a closed form for that horrid-looking thing?”

Gordon: And that was a very good question ... because it would save everybody a lot of time if someone said, “This thing is impossible. Forget it. There’s no way.”

Parshall: Then, four and a half hours after the original post, there’s an answer:

“I equals 4 pi times the arccotangent of the square root of the golden ratio.”

The answer came from a user named Cleo. It was a new account with only one previous answer. Cleo provided no notes, no proof, no explanation—just a single hyperlink over the symbol for the golden ratio, which takes you to a definition of the golden ratio. Oof.

Cummings: Which is such a ridiculous answer. It’s like you get this sense of “Am I dealing with a supercomputer here, a theorem-prover that has not been released yet? Did ChatGPT start back in 2012 with integral solving?”

Parshall: The Stack Exchange community, which always showed their work, erupted in arguments in the comments section. Here’s one:

“I defer to Hamming: ‘The purpose of computing is insight, not numbers.’ Unless the result itself is particularly illuminating, I do not agree that it is an answer.”

Parshall: That last comment came from Ron Gordon, the patent agent and former physicist, who didn’t see a whole lot of value in Cleo’s bare-bones answer.

Gordon: I think at the end of the day, the value of a website like Stack Exchange lies in what knowledge you can impart to people. And I think just the bare answer to the question, by itself, doesn’t have that much value.

But it affected my determination to come up with a final solution for sure. And I spent the better part of a weekend doing it, writing it up. Took me about half a legal pad to work through it.

Parshall: It turns out Cleo had been right. Ron posted the full answer, which immediately started collecting upvotes from community members. A lot of them were in awe of the techniques he’d used to solve the problem. It was eventually posted to the subreddit r/Math under the title “Master of Integration.”

Gordon: It’s insane. This is one thing I did 10 years ago. I think I have better answers in the Stack Exchange world than that one, believe it or not. But yeah, Cleo also, you know, I think hits a nerve, too, obviously.

Parshall: Cleo’s drive-by answer had unleashed madness on Math Stack Exchange. Between 2013 and 2015, she’d go on to do this 37 more times, often popping in unreasonably quickly to solve incredibly complex integration problems with fully formed answers. She did not show even an iota of her work. Then she’d disappear again into the ether.

Anthony Bonato Experts really are divided about Cleo. You know, it’s clearly someone who has a real mastery of integration techniques.... Like, she mentions these strange functions, like, I’ve never heard of.

Parshall: That’s Anthony Bonato. He’s a mathematician at Toronto Metropolitan University.

Bonato: Some people have speculated that maybe Cleo is Stephen Hawking—or was Stephen Hawking—or, you know, the late Maryam Mirzakhani, the Fields Medalist.

Parshall: Food for thought, I guess.

Cummings: Or is this, I don’t know, Terence Tao, you know, just relaxing in the evening?

Parshall: For the record, Terence Tao, sometimes described as one of the greatest living mathematicians, confirmed via e-mail that he was not, in fact, Cleo.

Cummings: Or is this a Ramanujan...? Is Cleo another math genius from southern India who just is doing this in their spare time?

Parshall: That genius he’s talking about, that’s Srinivasa Ramanujan, one of the most enigmatic figures in mathematics history. You might have heard of him—Dev Patel played him in a 2016 biopic called The Man Who Knew Infinity.

[CLIP: Dev Patel in The Man Who Knew Infinity: “We need proofs of your work.” “But they are right, sir.” “I hadn’t completed that proof; how do you know?” “I just do.”]

Parshall: He was born in Tamil Nadu in 1887, but he comes up a lot when you talk about Cleo.

Cummings: He had this intuitive feel for math that was ... frankly awe-inspiring.... He had no advanced math education. And yet, somehow, he came up with these incredible theorems.

Parshall: They seem to have struck the same nerve 100-some years apart.

Cummings: Because he didn’t include proofs. And that was sort of Ramanujan’s gift and curse. I mean, he was so, so talented, but he was never put into the educational box that says, “Here’s how you prove things; this is the path to take in order to do mathematics.”

Gordon: I think a lot of people who just hated being told, “Show your work, show your work, show your work...,” here’s someone flaunting not showing their work, and people are cheering behind that.

Parshall: But for Ron and for so many on Math Stack Exchange, all of the fun of their shared hobby is in showing your work. It’s not a dry explanation—it’s an adventure. Take Ron’s answer to that infamous 2013 integral.

Gordon: By the time I got to where I wanted it, it had like an eighth-degree polynomial in the denominator, which, under normal circumstances, would mean “No, you’re not going to be able to do this.” But it turned out that the polynomial had a lot of symmetry and I could then exploit that symmetry to deduce all the roots. I was able to reduce what I had to find from an eighth-degree polynomial to a quadratic, and from the quadratic, the golden ratio fell out.

Parshall: It turned out that Ron’s methods for solving the problem were compelling to a lot of people. His answer has earned almost 1,000 upvotes and is still shared around today.

Gordon: Do you ever watch The Big Bang Theory? There’s a scene where Sheldon has this big formula on his whiteboard and he goes, “Look at it. I feel like I just made a baby.” And I have to say, when he said that, I laughed so hard. Because there’s a lot of truth in that. When you come up with something that’s 4 pi arccotangent square root of phi, and you’ve derived it, you do feel like you created something.

Parshall: And Cleo created something, too, in her own way. But who she was, why she did it—nobody seems to know.

Parshall (tape): Do you have any personal thoughts on who Cleo is, what she does, why she does what she does?

Gordon: Absolutely not. I have no idea who Cleo is. In fact, a lot of the people I corresponded with and interacted with on the site, I know very little… I know very little of.

Parshall: Recently speculation has sparked back up again, thanks to a viral TikTok video about Cleo. Since then a user on Twitter has claimed to be Cleo but hasn’t offered any proof, and while some people are buying it, a lot of people aren’t. Whoever Cleo was, it seems that she was just very, very good at math—though some, like Bonato, suspect a computer might have been involved at some point.

Still, that doesn’t mean she was a bot, either. Computing ability for this kind of integration is still limited and would have been even more so in 2013.

Gordon: Given that the software couldn’t do these integrals, I doubt it. I’d be real curious to find out what she’s got her hands on.

Parshall: Cleo’s profile itself, which hasn’t been updated in seven years, tragically does not provide any clues. Today her bio reads:

“My real name is Cleo, I’m female. I have a medical condition that makes it very difficult for me to engage in conversations, or post long answers, sorry for that. I like math and do my best to be useful at this site, although I realize my answers might be not useful for everyone.”

But—but—I did wonder, “Has that always been her bio?” I thought I’d double-check so I went on the Internet Archive, pasted in her URL and clicked a snapshot that was taken in 2013 because, remember, kids, nothing on the internet is ever truly gone. And her bio was different back then. And guess who she quotes?

“‘While asleep, I had an unusual experience. There was a red screen formed by flowing blood, as it were. I was observing it. Suddenly a hand began to write on the screen. I became all attention. That hand wrote a number of elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writing.’ —Srinivasa Ramanujan”

Then Cleo wrote:

“Remember, you are not locked into a single axiom system. You may invent your own, whenever you wish—just use your intuition and imagination.”

[CLIP: Theme music]

Parshall: Science, Quickly is produced by Jeff DelViscio, Tulika Bose, Kelso Harper, and Carin Leong. Our theme music was composed by Dominic Smith.

Don’t forget to subscribe to Science, Quickly wherever you get your podcasts. For more in-depth science news and features, go to ScientificAmerican.com. And if you like the show, give us a rating or review!
For Scientific American’s Science, Quickly, I’m Allison Parshall.

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Building India into a powerhouse in maths

Despite its rich legacy and size, India doesn’t fare well on global benchmarks of maths. Here’s a road map on how to do it

Mathematics is the foundation of modern science and technology, and its role in critical areas of national security such as cryptography is even more singular. However, despite a rich historical legacy (Aryabhatta, Brahmagupta and Bhaskara, for instance) as well as renowned mathematicians in the 20th century (such as S Ramanujan, SN Bose, PC reagents Mahanobis and CR Rao), India has fared less well in this critical field in recent years.espite a rich historical legacy and renowned mathematicians in the 20th century (such as S Ramanujan, SN Bose, PC Mahalanobis and CR Rao), India has fared less well in this critical field in recent years(Wikimedia Commons)

A recent study evaluated the contributions of top mathematicians based on the Discipline H-index or D-index, which tracks each scientist’s published scholarly papers and citations in their specific discipline.

Among the top 50 mathematicians, only one is of Indian origin; among the top 100, four are of Indian origin. Among the top 500, 15 are of Indian origin, but just one is based in India. Among the top 2,332 mathematicians, just 17 are based in India (less than 1%).

India ranks 19th, lower than tiny countries such as Israel, Austria and Belgium. While there have been two Indian-origin winners of the Fields medal (the Nobel Prize of mathematics) – Akshay Venkatesh and Manjul Bhargava – their parents had emigrated to Australia and Canada, respectively. And while institutions such as the Chennai Institute of Mathematics, Indian Institute of Science, Indian Statistical Institute and Tata Institute of Fundamental Research, have excellent mathematicians, the pool is thin.

Given India’s size, this is deeply dismaying. There are two reasons why India should be doing much better in mathematics. First, among all Science, Technology, Engineering and Mathematics (STEM) fields, the last one is the least capital-intensive. There is relatively less need for capital investments in labs and equipment, which need to be constantly upgraded. Mathematics also does not require large running costs of elaborate support infrastructure such as lab technicians or costly reagents.

Furthermore, if STEM is a ladder of social mobility, mathematics is even more so. Merit is rarely unambiguous, since the nature of standards and the referees that enforce them markedly shape its perception. But some fields of human endeavour have more unambiguous markers of merit.

In mathematics (along with sports, chess, and music), quality cannot be easily gamed. The language of maths is universal — and the standards of merit are unambiguous. The great Indian mathematician, Ramanujan, became a byword in number theory and pure mathematics despite his poverty, weak English language abilities, and being a fish out of water amidst the dons of Cambridge University.

India has failed to produce another Ramanujan despite the language of mathematics being universal (and, hence, English language proficiency mattering much less), a population that has grown five-fold, and is much more educated (at least as measured by the percentage of population finishing high school). This speaks volumes about India’s education system. Indeed, the current system of education is expressly designed to polish the stones and dim the diamonds.

Brilliant mathematicians are well likely to perform weakly or even fail in some subjects that don’t interest them. But India’s education prefers well-rounded mediocrity to narrow brilliance.

One of India’s biggest security challenges going forward will be cybersecurity and at the heart of it will be cryptography, for which the country needs hundreds of young talents in pure mathematics. Currently, key national security agencies such as the National Technical Research Organisation (NTRO) face serious human capital challenges (the largest employer of mathematicians in the US is the National Security Agency or NSA).

But the security challenge for India is much greater. Like all armies, the Indian armed forces are well prepared for the last war. This may matter less under conditions of slow technological change. But when technological change is rapid – and even more when it occurs at the breakneck speed as is the case today – the structures and personnel need a serious rethink.

To take an example, the US set up a new Army Futures Command (AFC) in 2018 (with about 20,000 personnel) to develop the technologies and concepts that will enable its armed forces to stay abreast of the sheer range and speed of unprecedented disruptive technologies that are impacting warfare. AFC is charged with leveraging developments in areas such as robotics, quantum computing, hypersonics, directed energy and Artificial Intelligence pioneered by the private sector. The key personnel are PhDs, who work, unlike the army’s usual hierarchical ethos. Knowledge and expertise, not rank and polish, matter. Breaking from the insularity that characterises militaries, it deliberately listens to a range of external civilian advisers, regularly convening meetings with technology experts.

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International research award for Maths Professor

A Professor of Applied Mathematics at the University of Strathclyde has received an international award in recognition of her outstanding accomplishments in research.

Professor Apala Majumdar has been named as a recipient of a Friedrich Wilhelm Bessel Research Award from the Humboldt Foundation. It recognises her research in the mathematics of liquid crystals and partial differential equations.

The award will involve a period of exchange at the University of Wurzburg, one of Germany's oldest higher education institutions, where she will join a research group, led by Professor Dr Anja Schloemerkemper, on the mathematics and applications of complex materials.

The Alexander von Humboldt Foundation grants around 20 Friedrich Wilhelm Bessel Research Awards per year, funded by Germany’s Federal Ministry of Education and Research, annually to internationally-renowned academics. The awards are worth 45,000 euros each.

Professor Majumdar said: "I am absolutely delighted to receive this international award from the Humboldt Foundation. The Humboldt Foundation is known to support the best science and talented researchers from around the world.

"I will use the award to spend time in Professor Dr Anja Schloemerkemper's research group on the mathematics of materials science. We plan to work together on new mathematical theories for materials that combine liquid crystalline, magnetic and elastic properties and such materials could offer new prospects for designer materials technologies.

"I also plan to use this opportunity to build and drive a new global network of female applied mathematicians, and in doing so, Anja and I will help form a scientific community with a long-lasting global legacy. I much look forward to initiating these new connections in the spring of 2023."

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