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Education for the Future: Learning and Teaching for Sustainable Development in Education

Blending Pedagogy: Equipping Student Teachers to Foster Transversal Competencies in Future-oriented Education Provisionally Accepted

  • 1 Department of Education, Faculty of Educational Sciences, University of Helsinki, Finland

The final, formatted version of the article will be published soon.

Blended teaching and learning, combining online and face-to-face instruction, and shared reflection are gaining in popularity worldwide and present evolving challenges in the field of teacher training and education. There is also a growing need to focus on transversal competencies such as critical thinking and collaboration. This study is positioned at the intersection of blended education and transversal competencies in the context of a blended ECEC teacher-training program (1000+) at the University of Helsinki. Blended education is a novel approach to training teachers, and there is a desire to explore how such an approach supports the acquisition of transversal competencies and whether the associated methods offer something essential for the development of teacher training. The aim is to explore what transversal competencies this teacher-training program supports for future teachers, and how students reflect on their learning experiences. The data consist of documents from teacher-education curricula and essays from the students on the 1000+ program. They were content-analyzed from a scoping perspective. Students' experiences of studying enhanced the achievement of generic goals in teacher education, such as to develop critical and reflective thinking, interaction competence, collaboration skills, and independent and collective expertise. We highlight the importance of teacher development in preparing for education in the future during the teacher training. Emphasizing professional development, we challenge the conventional teaching paradigm by introducing a holistic approach.

Keywords: blended teacher training, Transversal competencies, future of education, Teacher Education, early childhood education

Received: 19 Jan 2024; Accepted: 15 May 2024.

Copyright: © 2024 Niemi, Kangas and Köngäs. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: Dr. Laura H. Niemi, Department of Education, Faculty of Educational Sciences, University of Helsinki, Helsinki, 00014, Uusimaa, Finland

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Reflection: Dartmouth Essays That Worked

One writer looks back on her admissions process in light of the dartmouth’s new book, “50 dartmouth application essays that worked.”.

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Five years ago, I began my Common Application essay with the following sentence: “To quote Ferris Bueller, ‘Life moves pretty fast. If you don’t stop and look around once in a while, you could miss it.’ I don’t intend to miss my life.” Half a decade later, those words still ring true. 

Any college was taking a chance when they admitted a girl who quoted a film famous for encouraging students to play hooky. Not only did I open with this line, I hammered the point home as I described my disillusionment with valuing academic learning over personal experience — I was done running on the high school hamster wheel. When people ask me what essay got me into Dartmouth, I usually respond, “An essay about having a really fun summer.” While those words are true, there’s a deeper moral to the story — as incredible as Dartmouth’s academic resources are, and as academically rigorous as my high school had been, I wanted to learn outside of the classroom, to learn by doing, to learn from my friends. “As much as I value my academic identity and as far as my passion for learning goes, my interpersonal relationships teach me just as much,” I wrote.

I was honestly surprised when Dartmouth accepted me in April 2020. I had been deferred early decision, and the last student who had gotten into Dartmouth from my public Florida high school was a cross-country recruit in 2016. Like tens of thousands of high school seniors, I had the grades, test scores and extracurriculars, but I was full of self-doubt. I was also completely burnt out. Was I really “Ivy League material”? I certainly didn’t feel it. Looking over my statistics, I was just another data point. Not Ellie Anderson, but applicant 8,677. 

My “Why Dartmouth?”  and supplement essays allowed me to make my case. I crafted three versions of the former, and I could have kept going. I labored over my words carefully, drafting response after response, but it was challenging only having 250 words to respond, in some form, to a prompt that every Dartmouth applicant has read: “It is, Sir … a small college. And yet, there are those who love it!” Other than the encouraging words and flamboyant edits from my high school English teacher, I didn’t know if they were any good. Where to begin …

I would have loved to understand what makes an admissions essay compelling when I was in the throes of applying to college. Recently, The Dartmouth published “50 Dartmouth Application Essays That Worked,” a compilation of successful admissions essays. Looking through this collection, I felt like I was stepping back into my 17-year-old self. The selection includes essays featuring many of the qualities Dartmouth seems to be looking for in its students, or at least those I’ve found in my friends: compassion, curiosity, humility and a collaborative spirit. 

The book opens with essays about environment and nature before progressing to the expected categories: academic interest, arts, heritage, identity, sports and, of course, “miscellaneous.” 

A few stories grabbed me for their honesty, especially one that begins, “I have a complicated relationship with the truth.” I was hooked — it was real and raw. Her father suffers from bipolar disorder even though, to the outside world, nothing appears to be wrong. She has a secret too — she’s seeing a girl. How is one supposed to apply to college when their entire world is being torn apart, “standing in the middle of the bridge and setting fire to both ends,” as she says. But she learns a valuable lesson — to live her own truth, not anyone else’s. 

When I was applying to colleges, I was given the following advice: “Don’t make your admissions essay a sob story.” But this essay certainly isn’t a pity party, which proves you can be honest and address your difficulties in the span of a few hundred words. These kinds of essays instead place their writers’ most beautiful strengths and flaws on full display.

Another such essay begins, “My feet live in infamy.” Yes, you can write your Common Application essay about your gnarled and calloused feet. Although the story begins with an anecdote of “ugly” feet, it becomes so much more — a toe-centric reflection. As the writer’s skin became thicker, she found her voice as well. She comes out of her shell in high school, learning to speak up after several tumultuous adolescent years as an introvert. By the end, she’s finally ready to bear her infamous feet and use her voice.

A deep current of intellectual curiosity runs across the essays, too. I laughed when I read a story about an applicant playing Super Mario Bros on a childhood road trip. The writer makes an in-game blunder, sending Mario hurdling into a turtle. “It was then that the terrible realization that curled my six-year-old toes hit me: Mario would return to play again, but when I die, I will not,” they said. What could have been a decade-long existential spiral instead drove the writer to philosophy and math, where they found solace in understanding the world rather than cowering at the unknown. 

These writers are brave — both for sharing their stories to the black-box admissions panel and for allowing us readers a peek years later. On a campus where we often interact in passing “Hey, what’s up”-isms, reading the diverse selection of essays has grounded me once more in an understanding of what makes Dartmouth, Dartmouth. Students here are radically courageous in their quests for knowledge, acts of kindness and pursuits of greatness. In these essays, 650 words no longer looks limiting but becomes the etchings of a beautiful cohort.

After re-reading my own essay alongside those published, it struck me. As a 17-year-old sending off a piece of yourself to a nebulous online portal, it can be difficult to envision your future — your story is a moment in time caught in between all that you’ve been and all that you hope to become. I’m asking myself this question again as I look forward to my senior year at Dartmouth and re-read my ambitions and fears from the essay I penned in 2019. It’s been a lot of laughing at my naïvete, cringing at a heavy-handed application of adjectives and finding pride in my values.

Not only is this book a tool for Dartmouth applicants, but it’s a time capsule from the Class of 2023 to the Class of 2026, whose essays are included. This is who we were at 17. Looking back at my essay, so much has changed between now and then. How could it have not? But I see the seeds of who I’ve become in my essay, like an incantation: “I learn to understand others and to understand myself.”

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Better Siri is coming: what Apple’s research says about its AI plans

Apple hasn’t talked too much about ai so far — but it’s been working on stuff. a lot of stuff..

By David Pierce , editor-at-large and Vergecast co-host with over a decade of experience covering consumer tech. Previously, at Protocol, The Wall Street Journal, and Wired.

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The Apple logo with a little AI sparkle.

It would be easy to think that Apple is late to the game on AI. Since late 2022, when ChatGPT took the world by storm, most of Apple’s competitors have fallen over themselves to catch up. While Apple has certainly talked about AI and even released some products with AI in mind, it seemed to be dipping a toe in rather than diving in headfirst.

But over the last few months, rumors and reports have suggested that Apple has, in fact, just been biding its time, waiting to make its move. There have been reports in recent weeks that Apple is talking to both OpenAI and Google about powering some of its AI features, and the company has also been working on its own model, called Ajax .

If you look through Apple’s published AI research, a picture starts to develop of how Apple’s approach to AI might come to life. Now, obviously, making product assumptions based on research papers is a deeply inexact science — the line from research to store shelves is windy and full of potholes. But you can at least get a sense of what the company is thinking about — and how its AI features might work when Apple starts to talk about them at its annual developer conference, WWDC, in June.

Smaller, more efficient models

I suspect you and I are hoping for the same thing here: Better Siri. And it looks very much like Better Siri is coming! There’s an assumption in a lot of Apple’s research (and in a lot of the tech industry, the world, and everywhere) that large language models will immediately make virtual assistants better and smarter. For Apple, getting to Better Siri means making those models as fast as possible — and making sure they’re everywhere.

In iOS 18, Apple plans to have all its AI features running on an on-device, fully offline model, Bloomberg recently reported . It’s tough to build a good multipurpose model even when you have a network of data centers and thousands of state-of-the-art GPUs — it’s drastically harder to do it with only the guts inside your smartphone. So Apple’s having to get creative.

In a paper called “ LLM in a flash: Efficient Large Language Model Inference with Limited Memory ” (all these papers have really boring titles but are really interesting, I promise!), researchers devised a system for storing a model’s data, which is usually stored on your device’s RAM, on the SSD instead. “We have demonstrated the ability to run LLMs up to twice the size of available DRAM [on the SSD],” the researchers wrote, “achieving an acceleration in inference speed by 4-5x compared to traditional loading methods in CPU, and 20-25x in GPU.” By taking advantage of the most inexpensive and available storage on your device, they found, the models can run faster and more efficiently. 

Apple’s researchers also created a system called EELBERT that can essentially compress an LLM into a much smaller size without making it meaningfully worse. Their compressed take on Google’s Bert model was 15 times smaller — only 1.2 megabytes — and saw only a 4 percent reduction in quality. It did come with some latency tradeoffs, though.

In general, Apple is pushing to solve a core tension in the model world: the bigger a model gets, the better and more useful it can be, but also the more unwieldy, power-hungry, and slow it can become. Like so many others, the company is trying to find the right balance between all those things while also looking for a way to have it all.

Siri, but good

A lot of what we talk about when we talk about AI products is virtual assistants — assistants that know things, that can remind us of things, that can answer questions, and get stuff done on our behalf. So it’s not exactly shocking that a lot of Apple’s AI research boils down to a single question: what if Siri was really, really, really good?

A group of Apple researchers has been working on a way to use Siri without needing to use a wake word at all; instead of listening for “Hey Siri” or “Siri,” the device might be able to simply intuit whether you’re talking to it. “This problem is significantly more challenging than voice trigger detection,” the researchers did acknowledge, “since there might not be a leading trigger phrase that marks the beginning of a voice command.” That might be why another group of researchers developed a system to more accurately detect wake words . Another paper trained a model to better understand rare words, which are often not well understood by assistants.

In both cases, the appeal of an LLM is that it can, in theory, process much more information much more quickly. In the wake-word paper, for instance, the researchers found that by not trying to discard all unnecessary sound but, instead, feeding it all to the model and letting it process what does and doesn’t matter, the wake word worked far more reliably.

Once Siri hears you, Apple’s doing a bunch of work to make sure it understands and communicates better. In one paper, it developed a system called STEER (which stands for Semantic Turn Extension-Expansion Recognition, so we’ll go with STEER) that aims to improve your back-and-forth communication with an assistant by trying to figure out when you’re asking a follow-up question and when you’re asking a new one. In another, it uses LLMs to better understand “ambiguous queries” to figure out what you mean no matter how you say it. “In uncertain circumstances,” they wrote, “intelligent conversational agents may need to take the initiative to reduce their uncertainty by asking good questions proactively, thereby solving problems more effectively.” Another paper aims to help with that, too: researchers used LLMs to make assistants less verbose and more understandable when they’re generating answers.

A series of images depicting collaborative AI editing of a photo.

AI in health, image editors, in your Memojis

Whenever Apple does talk publicly about AI, it tends to focus less on raw technological might and more on the day-to-day stuff AI can actually do for you. So, while there’s a lot of focus on Siri — especially as Apple looks to compete with devices like the Humane AI Pin, the Rabbit R1, and Google’s ongoing smashing of Gemini into all of Android — there are plenty of other ways Apple seems to see AI being useful.

One obvious place for Apple to focus is on health: LLMs could, in theory, help wade through the oceans of biometric data collected by your various devices and help you make sense of it all. So, Apple has been researching how to collect and collate all of your motion data, how to use gait recognition and your headphones to identify you, and how to track and understand your heart rate data. Apple also created and released “the largest multi-device multi-location sensor-based human activity dataset” available after collecting data from 50 participants with multiple on-body sensors.

Apple also seems to imagine AI as a creative tool. For one paper, researchers interviewed a bunch of animators, designers, and engineers and built a system called Keyframer that “enable[s] users to iteratively construct and refine generated designs.” Instead of typing in a prompt and getting an image, then typing another prompt to get another image, you start with a prompt but then get a toolkit to tweak and refine parts of the image to your liking. You could imagine this kind of back-and-forth artistic process showing up anywhere from the Memoji creator to some of Apple’s more professional artistic tools.

In another paper , Apple describes a tool called MGIE that lets you edit an image just by describing the edits you want to make. (“Make the sky more blue,” “make my face less weird,” “add some rocks,” that sort of thing.) “Instead of brief but ambiguous guidance, MGIE derives explicit visual-aware intention and leads to reasonable image editing,” the researchers wrote. Its initial experiments weren’t perfect, but they were impressive.

We might even get some AI in Apple Music: for a paper called “ Resource-constrained Stereo Singing Voice Cancellation ,” researchers explored ways to separate voices from instruments in songs — which could come in handy if Apple wants to give people tools to, say, remix songs the way you can on TikTok or Instagram.

An image showing the Ferret-UI AI system from Apple.

Over time, I’d bet this is the kind of stuff you’ll see Apple lean into, especially on iOS. Some of it Apple will build into its own apps; some it will offer to third-party developers as APIs. (The recent Journaling Suggestions feature is probably a good guide to how that might work.) Apple has always trumpeted its hardware capabilities, particularly compared to your average Android device; pairing all that horsepower with on-device, privacy-focused AI could be a big differentiator.

But if you want to see the biggest, most ambitious AI thing going at Apple, you need to know about Ferret . Ferret is a multi-modal large language model that can take instructions, focus on something specific you’ve circled or otherwise selected, and understand the world around it. It’s designed for the now-normal AI use case of asking a device about the world around you, but it might also be able to understand what’s on your screen. In the Ferret paper, researchers show that it could help you navigate apps, answer questions about App Store ratings, describe what you’re looking at, and more. This has really exciting implications for accessibility but could also completely change the way you use your phone — and your Vision Pro and / or smart glasses someday.

We’re getting way ahead of ourselves here, but you can imagine how this would work with some of the other stuff Apple is working on. A Siri that can understand what you want, paired with a device that can see and understand everything that’s happening on your display, is a phone that can literally use itself. Apple wouldn’t need deep integrations with everything; it could simply run the apps and tap the right buttons automatically. 

Again, all this is just research, and for all of it to work well starting this spring would be a legitimately unheard-of technical achievement. (I mean, you’ve tried chatbots — you know they’re not great.) But I’d bet you anything we’re going to get some big AI announcements at WWDC. Apple CEO Tim Cook even teased as much in February, and basically promised it on this week’s earnings call. And two things are very clear: Apple is very much in the AI race, and it might amount to a total overhaul of the iPhone. Heck, you might even start willingly using Siri! And that would be quite the accomplishment.

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What I’ve Learned From My Students’ College Essays

The genre is often maligned for being formulaic and melodramatic, but it’s more important than you think.

An illustration of a high school student with blue hair, dreaming of what to write in their college essay.

By Nell Freudenberger

Most high school seniors approach the college essay with dread. Either their upbringing hasn’t supplied them with several hundred words of adversity, or worse, they’re afraid that packaging the genuine trauma they’ve experienced is the only way to secure their future. The college counselor at the Brooklyn high school where I’m a writing tutor advises against trauma porn. “Keep it brief , ” she says, “and show how you rose above it.”

I started volunteering in New York City schools in my 20s, before I had kids of my own. At the time, I liked hanging out with teenagers, whom I sometimes had more interesting conversations with than I did my peers. Often I worked with students who spoke English as a second language or who used slang in their writing, and at first I was hung up on grammar. Should I correct any deviation from “standard English” to appeal to some Wizard of Oz behind the curtains of a college admissions office? Or should I encourage students to write the way they speak, in pursuit of an authentic voice, that most elusive of literary qualities?

In fact, I was missing the point. One of many lessons the students have taught me is to let the story dictate the voice of the essay. A few years ago, I worked with a boy who claimed to have nothing to write about. His life had been ordinary, he said; nothing had happened to him. I asked if he wanted to try writing about a family member, his favorite school subject, a summer job? He glanced at his phone, his posture and expression suggesting that he’d rather be anywhere but in front of a computer with me. “Hobbies?” I suggested, without much hope. He gave me a shy glance. “I like to box,” he said.

I’ve had this experience with reluctant writers again and again — when a topic clicks with a student, an essay can unfurl spontaneously. Of course the primary goal of a college essay is to help its author get an education that leads to a career. Changes in testing policies and financial aid have made applying to college more confusing than ever, but essays have remained basically the same. I would argue that they’re much more than an onerous task or rote exercise, and that unlike standardized tests they are infinitely variable and sometimes beautiful. College essays also provide an opportunity to learn precision, clarity and the process of working toward the truth through multiple revisions.

When a topic clicks with a student, an essay can unfurl spontaneously.

Even if writing doesn’t end up being fundamental to their future professions, students learn to choose language carefully and to be suspicious of the first words that come to mind. Especially now, as college students shoulder so much of the country’s ethical responsibility for war with their protest movement, essay writing teaches prospective students an increasingly urgent lesson: that choosing their own words over ready-made phrases is the only reliable way to ensure they’re thinking for themselves.

Teenagers are ideal writers for several reasons. They’re usually free of preconceptions about writing, and they tend not to use self-consciously ‘‘literary’’ language. They’re allergic to hypocrisy and are generally unfiltered: They overshare, ask personal questions and call you out for microaggressions as well as less egregious (but still mortifying) verbal errors, such as referring to weed as ‘‘pot.’’ Most important, they have yet to put down their best stories in a finished form.

I can imagine an essay taking a risk and distinguishing itself formally — a poem or a one-act play — but most kids use a more straightforward model: a hook followed by a narrative built around “small moments” that lead to a concluding lesson or aspiration for the future. I never get tired of working with students on these essays because each one is different, and the short, rigid form sometimes makes an emotional story even more powerful. Before I read Javier Zamora’s wrenching “Solito,” I worked with a student who had been transported by a coyote into the U.S. and was reunited with his mother in the parking lot of a big-box store. I don’t remember whether this essay focused on specific skills or coping mechanisms that he gained from his ordeal. I remember only the bliss of the parent-and-child reunion in that uninspiring setting. If I were making a case to an admissions officer, I would suggest that simply being able to convey that experience demonstrates the kind of resilience that any college should admire.

The essays that have stayed with me over the years don’t follow a pattern. There are some narratives on very predictable topics — living up to the expectations of immigrant parents, or suffering from depression in 2020 — that are moving because of the attention with which the student describes the experience. One girl determined to become an engineer while watching her father build furniture from scraps after work; a boy, grieving for his mother during lockdown, began taking pictures of the sky.

If, as Lorrie Moore said, “a short story is a love affair; a novel is a marriage,” what is a college essay? Every once in a while I sit down next to a student and start reading, and I have to suppress my excitement, because there on the Google Doc in front of me is a real writer’s voice. One of the first students I ever worked with wrote about falling in love with another girl in dance class, the absolute magic of watching her move and the terror in the conflict between her feelings and the instruction of her religious middle school. She made me think that college essays are less like love than limerence: one-sided, obsessive, idiosyncratic but profound, the first draft of the most personal story their writers will ever tell.

Nell Freudenberger’s novel “The Limits” was published by Knopf last month. She volunteers through the PEN America Writers in the Schools program.

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Book Review: Memoirist Lilly Dancyger’s penetrating essays explore the power of female friendships

This cover image released by Dial Press shows "First Love" by Lilly Dancyger. (Dial Press via AP)

This cover image released by Dial Press shows “First Love” by Lilly Dancyger. (Dial Press via AP)

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Who means more to you — your friends or your lovers? In a vivid, thoughtful and nuanced collection of essays, Lilly Dancyger explores the powerful role that female friendships played in her chaotic upbringing marked by her parents’ heroin use and her father’s untimely death when she was only 12.

“First Love: Essays on Friendship” begins with a beautiful paean to her cousin Sabina, who was raped and murdered at age 20 on her way home from a club. As little kids, their older relatives used to call them Snow White and Rose Red after the Grimm’s fairy tale, “two sisters who are not rivals or foils, but simply love each other.”

That simple, uncomplicated love would become the template for a series of subsequent relationships with girls and women that helped her survive her self-destructive adolescence and provided unconditional support as she scrambled to create a new identity as a “hypercompetent” writer, teacher and editor. “It’s true that I’ve never been satisfied with friendships that stay on the surface. That my friends are my family, my truest beloveds, each relationship a world of its own,” she writes in the title essay “First Love.”

The collection stands out not just for its elegant, unadorned writing but also for the way she effortlessly pivots between personal history and spot-on cultural criticism that both comments on and critiques the way that girls and women have been portrayed — and have portrayed themselves — in the media, including on online platforms like Tumblr and Instagram.

This cover image released by Norton shows "This Strange Eventful History" by Claire Messud. (Norton via AP)

For instance, she examines the 1994 Peter Jackson film, “Heavenly Creatures,” based on the true story of two teenage girls who bludgeoned to death one of their mothers. And in the essay “Sad Girls,” about the suicide of a close friend, she analyzes the allure of self-destructive figures like Sylvia Plath and Janis Joplin to a certain type of teen, including herself, who wallows in sadness and wants to make sure “the world knew we were in pain.”

In the last essay, “On Murder Memoirs,” Dancyger considers the runaway popularity of true crime stories as she tries to explain her decision not to attend the trial of the man charged with killing her cousin — even though she was trained as a journalist and wrote a well-regarded book about her late father that relied on investigative reporting. “When I finally sat down to write about Sabina, the story that came out was not about murder at all,” she says. “It was a love story.”

Readers can be thankful that it did.

AP book reviews: https://apnews.com/hub/book-reviews

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Collection  29 March 2022

2021 Top 25 COVID-19 Articles

The 25 most downloaded  Nature Communications  articles* on COVID-19 published in 2021 illustrate the collaborative efforts of the international community to combat the ongoing pandemic. These papers highlight valuable research into the biology of coronavirus infection, its detection, treatment as well as into vaccine development and the epidemiology of the disease.

Browse all Top 25 subject area collections  here .

*Data obtained from SN Insights (based on Digital Science's Dimensions) and normalised to account for articles published later in the year.

Microscopic view of 3D spherical viruses

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Watch CBS News

Teens come up with trigonometry proof for Pythagorean Theorem, a problem that stumped math world for centuries

By Bill Whitaker

May 5, 2024 / 7:00 PM EDT / CBS News

As the school year ends, many students will be only too happy to see math classes in their rearview mirrors. It may seem to some of us non-mathematicians that geometry and trigonometry were created by the Greeks as a form of torture, so imagine our amazement when we heard two high school seniors had proved a mathematical puzzle that was thought to be impossible for 2,000 years. 

We met Calcea Johnson and Ne'Kiya Jackson at their all-girls Catholic high school in New Orleans. We expected to find two mathematical prodigies.

Instead, we found at St. Mary's Academy , all students are told their possibilities are boundless.

Come Mardi Gras season, New Orleans is alive with colorful parades, replete with floats, and beads, and high school marching bands.

In a city where uniqueness is celebrated, St. Mary's stands out – with young African American women playing trombones and tubas, twirling batons and dancing - doing it all, which defines St. Mary's, students told us.

Junior Christina Blazio says the school instills in them they have the ability to accomplish anything. 

Christina Blazio: That is kinda a standard here. So we aim very high - like, our aim is excellence for all students. 

The private Catholic elementary and high school sits behind the Sisters of the Holy Family Convent in New Orleans East. The academy was started by an African American nun for young Black women just after the Civil War. The church still supports the school with the help of alumni.

In December 2022, seniors Ne'Kiya Jackson and Calcea Johnson were working on a school-wide math contest that came with a cash prize.

Ne'Kiya Jackson and Calcea Johnson

Ne'Kiya Jackson: I was motivated because there was a monetary incentive.

Calcea Johnson: 'Cause I was like, "$500 is a lot of money. So I-- I would like to at least try."

Both were staring down the thorny bonus question.

Bill Whitaker: So tell me, what was this bonus question?

Calcea Johnson: It was to create a new proof of the Pythagorean Theorem. And it kind of gave you a few guidelines on how would you start a proof.

The seniors were familiar with the Pythagorean Theorem, a fundamental principle of geometry. You may remember it from high school: a² + b² = c². In plain English, when you know the length of two sides of a right triangle, you can figure out the length of the third.

Both had studied geometry and some trigonometry, and both told us math was not easy. What no one told  them  was there had been more than 300 documented proofs of the Pythagorean Theorem using algebra and geometry, but for 2,000 years a proof using trigonometry was thought to be impossible, … and that was the bonus question facing them.

Bill Whitaker: When you looked at the question did you think, "Boy, this is hard"?

Ne'Kiya Jackson: Yeah. 

Bill Whitaker: What motivated you to say, "Well, I'm going to try this"?

Calcea Johnson: I think I was like, "I started something. I need to finish it." 

Bill Whitaker: So you just kept on going.

Calcea Johnson: Yeah.

For two months that winter, they spent almost all their free time working on the proof.

CeCe Johnson: She was like, "Mom, this is a little bit too much."

CeCe and Cal Johnson are Calcea's parents.

CeCe Johnson:   So then I started looking at what she really was doing. And it was pages and pages and pages of, like, over 20 or 30 pages for this one problem.

Cal Johnson: Yeah, the garbage can was full of papers, which she would, you know, work out the problems and-- if that didn't work she would ball it up, throw it in the trash. 

Bill Whitaker: Did you look at the problem? 

Neliska Jackson is Ne'Kiya's mother.

Neliska Jackson: Personally I did not. 'Cause most of the time I don't understand what she's doing (laughter).

Michelle Blouin Williams: What if we did this, what if I write this? Does this help? ax² plus ….

Their math teacher, Michelle Blouin Williams, initiated the math contest.

Michelle Blouin Williams

Bill Whitaker: And did you think anyone would solve it?

Michelle Blouin Williams: Well, I wasn't necessarily looking for a solve. So, no, I didn't—

Bill Whitaker: What were you looking for?

Michelle Blouin Williams: I was just looking for some ingenuity, you know—

Calcea and Ne'Kiya delivered on that! They tried to explain their groundbreaking work to 60 Minutes. Calcea's proof is appropriately titled the Waffle Cone.

Calcea Johnson: So to start the proof, we start with just a regular right triangle where the angle in the corner is 90°. And the two angles are alpha and beta.

Bill Whitaker: Uh-huh

Calcea Johnson: So then what we do next is we draw a second congruent, which means they're equal in size. But then we start creating similar but smaller right triangles going in a pattern like this. And then it continues for infinity. And eventually it creates this larger waffle cone shape.

Calcea Johnson: Am I going a little too—

Bill Whitaker: You've been beyond me since the beginning. (laughter) 

Bill Whitaker: So how did you figure out the proof?

Ne'Kiya Jackson: Okay. So you have a right triangle, 90° angle, alpha and beta.

Bill Whitaker: Then what did you do?

Bill Whitaker with Calcea Johnson and Ne'Kiya Jackson

Ne'Kiya Jackson: Okay, I have a right triangle inside of the circle. And I have a perpendicular bisector at OP to divide the triangle to make that small right triangle. And that's basically what I used for the proof. That's the proof.

Bill Whitaker: That's what I call amazing.

Ne'Kiya Jackson: Well, thank you.

There had been one other documented proof of the theorem using trigonometry by mathematician Jason Zimba in 2009 – one in 2,000 years. Now it seems Ne'Kiya and Calcea have joined perhaps the most exclusive club in mathematics. 

Bill Whitaker: So you both independently came up with proof that only used trigonometry.

Ne'Kiya Jackson: Yes.

Bill Whitaker: So are you math geniuses?

Calcea Johnson: I think that's a stretch. 

Bill Whitaker: If not genius, you're really smart at math.

Ne'Kiya Jackson: Not at all. (laugh) 

To document Calcea and Ne'Kiya's work, math teachers at St. Mary's submitted their proofs to an American Mathematical Society conference in Atlanta in March 2023.

Ne'Kiya Jackson: Well, our teacher approached us and was like, "Hey, you might be able to actually present this," I was like, "Are you joking?" But she wasn't. So we went. I got up there. We presented and it went well, and it blew up.

Bill Whitaker: It blew up.

Calcea Johnson: Yeah. 

Ne'Kiya Jackson: It blew up.

Bill Whitaker: Yeah. What was the blowup like?

Calcea Johnson: Insane, unexpected, crazy, honestly.

It took millenia to prove, but just a minute for word of their accomplishment to go around the world. They got a write-up in South Korea and a shout-out from former first lady Michelle Obama, a commendation from the governor and keys to the city of New Orleans. 

Bill Whitaker: Why do you think so many people found what you did to be so impressive?

Ne'Kiya Jackson: Probably because we're African American, one. And we're also women. So I think-- oh, and our age. Of course our ages probably played a big part.

Bill Whitaker: So you think people were surprised that young African American women, could do such a thing?

Calcea Johnson: Yeah, definitely.

Ne'Kiya Jackson: I'd like to actually be celebrated for what it is. Like, it's a great mathematical achievement.

Achievement, that's a word you hear often around St. Mary's academy. Calcea and Ne'Kiya follow a long line of barrier-breaking graduates. 

The late queen of Creole cooking, Leah Chase , was an alum. so was the first African-American female New Orleans police chief, Michelle Woodfork …

And judge for the Fifth Circuit Court of Appeals, Dana Douglas. Math teacher Michelle Blouin Williams told us Calcea and Ne'Kiya are typical St. Mary's students.  

Bill Whitaker: They're not unicorns.

Michelle Blouin Williams: Oh, no no. If they are unicorns, then every single lady that has matriculated through this school is a beautiful, Black unicorn.

Pamela Rogers: You're good?

Pamela Rogers, St. Mary's president and interim principal, told us the students hear that message from the moment they walk in the door.

St. Mary's Academy president and interim principal Pamela Rogers

Pamela Rogers: We believe all students can succeed, all students can learn. It does not matter the environment that you live in. 

Bill Whitaker: So when word went out that two of your students had solved this almost impossible math problem, were they universally applauded?

Pamela Rogers: In this community, they were greatly applauded. Across the country, there were many naysayers.

Bill Whitaker: What were they saying?

Pamela Rogers: They were saying, "Oh, they could not have done it. African Americans don't have the brains to do it." Of course, we sheltered our girls from that. But we absolutely did not expect it to come in the volume that it came.  

Bill Whitaker: And after such a wonderful achievement.

Pamela Rogers: People-- have a vision of who can be successful. And-- to some people, it is not always an African American female. And to us, it's always an African American female.

Gloria Ladson-Billings: What we know is when teachers lay out some expectations that say, "You can do this," kids will work as hard as they can to do it.

Gloria Ladson-Billings, professor emeritus at the University of Wisconsin, has studied how best to teach African American students. She told us an encouraging teacher can change a life.

Bill Whitaker: And what's the difference, say, between having a teacher like that and a whole school dedicated to the excellence of these students?

Gloria Ladson-Billings: So a whole school is almost like being in Heaven. 

Bill Whitaker: What do you mean by that?

Bill Whitaker and Gloria Ladson-Billings

Gloria Ladson-Billings: Many of our young people have their ceilings lowered, that somewhere around fourth or fifth grade, their thoughts are, "I'm not going to be anything special." What I think is probably happening at St. Mary's is young women come in as, perhaps, ninth graders and are told, "Here's what we expect to happen. And here's how we're going to help you get there."

At St. Mary's, half the students get scholarships, subsidized by fundraising to defray the $8,000 a year tuition. Here, there's no test to get in, but expectations are high and rules are strict: no cellphones, modest skirts, hair must be its natural color.

Students Rayah Siddiq, Summer Forde, Carissa Washington, Tatum Williams and Christina Blazio told us they appreciate the rules and rigor.

Rayah Siddiq: Especially the standards that they set for us. They're very high. And I don't think that's ever going to change.

Bill Whitaker: So is there a heart, a philosophy, an essence to St. Mary's?

Summer Forde: The sisterhood—

Carissa Washington: Sisterhood.

Tatum Williams: Sisterhood.

Bill Whitaker: The sisterhood?

Voices: Yes.

Bill Whitaker: And you don't mean the nuns. You mean-- (laughter)

Christina Blazio: I mean, yeah. The community—

Bill Whitaker: So when you're here, there's just no question that you're going to go on to college.

Rayah Siddiq: College is all they talk about. (laughter) 

Pamela Rogers: … and Arizona State University (Cheering)

Principal Rogers announces to her 615 students the colleges where every senior has been accepted.

Bill Whitaker: So for 17 years, you've had a 100% graduation rate—

Pamela Rogers: Yes.

Bill Whitaker: --and a 100% college acceptance rate?

Pamela Rogers: That's correct.

Last year when Ne'Kiya and Calcea graduated, all their classmates went to college and got scholarships. Ne'Kiya got a full ride to the pharmacy school at Xavier University in New Orleans. Calcea, the class valedictorian, is studying environmental engineering at Louisiana State University.

Bill Whitaker: So wait a minute. Neither one of you is going to pursue a career in math?

Both: No. (laugh)

Calcea Johnson: I may take up a minor in math. But I don't want that to be my job job.

Ne'Kiya Jackson: Yeah. People might expect too much out of me if (laugh) I become a mathematician. (laugh)

But math is not completely in their rear-view mirrors. This spring they submitted their high school proofs for final peer review and publication … and are still working on further proofs of the Pythagorean Theorem. Since their first two …

Calcea Johnson: We found five. And then we found a general format that could potentially produce at least five additional proofs.

Bill Whitaker: And you're not math geniuses?

Bill Whitaker: I'm not buying it. (laughs)

Produced by Sara Kuzmarov. Associate producer, Mariah B. Campbell. Edited by Daniel J. Glucksman.

Bill Whitaker

Bill Whitaker is an award-winning journalist and 60 Minutes correspondent who has covered major news stories, domestically and across the globe, for more than four decades with CBS News.

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    3.3 million articles on ScienceDirect are open access. Articles published open access are peer-reviewed and made freely available for everyone to read, download and reuse in line with the user license displayed on the article. ScienceDirect is the world's leading source for scientific, technical, and medical research.

  9. JSTOR Home

    Harness the power of visual materials—explore more than 3 million images now on JSTOR. Enhance your scholarly research with underground newspapers, magazines, and journals. Explore collections in the arts, sciences, and literature from the world's leading museums, archives, and scholars. JSTOR is a digital library of academic journals ...

  10. ResearchGate

    Access 160+ million publications and connect with 25+ million researchers. Join for free and gain visibility by uploading your research.

  11. Research

    News about Research, including commentary and archival articles published in The New York Times.

  12. Wiley Online Library

    One of the largest and most authoritative collections of online journals, books, and research resources, covering life, health, social, and physical sciences.

  13. Academia.edu

    Work faster and smarter with advanced research discovery tools. Search the full text and citations of our millions of papers. Download groups of related papers to jumpstart your research. Save time with detailed summaries and search alerts. Advanced Search. PDF Packages of 37 papers.

  14. 110553 PDFs

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  16. Connected Papers

    Get a visual overview of a new academic field. Enter a typical paper and we'll build you a graph of similar papers in the field. Explore and build more graphs for interesting papers that you find - soon you'll have a real, visual understanding of the trends, popular works and dynamics of the field you're interested in.

  17. Publications

    Publications. Our teams aspire to make discoveries that impact everyone, and core to our approach is sharing our research and tools to fuel progress in the field. Google publishes hundreds of research papers each year. Publishing our work enables us to collaborate and share ideas with, as well as learn from, the broader scientific community.

  18. Journal Top 100

    Journal Top 100 - 2022. This collection highlights our most downloaded* research papers published in 2022. Featuring authors from around the world, these papers highlight valuable research from an ...

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    Marie Ng. Maigeng Zhou. The study's findings support the increased risk of premature mortality associated with low education, particularly in women and urban populations. The considerable number of deaths attributed to educational inequality underscores the necessity for more effective and targeted public health interventions.

  20. Research Article

    Research articles represent the ultimate, final product of a scientific study. You should assume that your published work will be indefinitely available for anyone to access. • Research articles always consist of a title, abstract, introduction, materials and methods, results, discussion, and references sections, and many include a supplemental materials section. There are strategies for ...

  21. Frontiers

    The aim is to explore what transversal competencies this teacher-training program supports for future teachers, and how students reflect on their learning experiences. The data consist of documents from teacher-education curricula and essays from the students on the 1000+ program. They were content-analyzed from a scoping perspective.

  22. Flood of Fake Science Forces Multiple Journal Closures

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  23. Reflection: Dartmouth Essays That Worked

    Recently, The Dartmouth published "50 Dartmouth Application Essays That Worked," a compilation of successful admissions essays. Looking through this collection, I felt like I was stepping back into my 17-year-old self. The selection includes essays featuring many of the qualities Dartmouth seems to be looking for in its students, or at ...

  24. Journal Top 100

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  25. Apple's AI research suggests features are coming for Siri, artists, and

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  27. Book Review: Memoirist Lilly Dancyger's penetrating essays explore the

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  28. 2021 Top 25 COVID-19 Articles

    The 25 most downloaded Nature Communications articles* on COVID-19 published in 2021 illustrate the collaborative efforts of the international community to combat the ongoing pandemic.These papers ...

  29. Teens come up with trigonometry proof for Pythagorean Theorem, a

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