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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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IMAGES

  1. A Good Teacher Essay

    research paper on good teacher

  2. 219909134-ESSAY-Qualities-of-a-Good-Teacher.docx

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  3. Reasearch Ideas for High School Students

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  4. (PDF) How to be a good teacher

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  5. (PDF) Research in Teacher Education

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    research paper on good teacher

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COMMENTS

  1. Characteristics of a good teacher: A case study at ...

    The aim of this paper is to find out the characteristics of a good teacher in a specific course at a higher education institute from the perspectives and evaluation of the learners. This empirical ...

  2. A Review of the Literature on Teacher Effectiveness and Student

    A teacher's professional knowledge refers to their subject-matter knowledge, curricular knowledge, and pedagogical knowledge (Collinson 1999).This professional knowledge is influenced by the undergraduate degrees earned by a teacher, the college attended, graduate studies undertaken, and opportunities to engage with on-the job training, commonly referred to as professional development ...

  3. PDF Twelve Characteristics of an Effective Teacher

    in the essays and class discussions of what makes a good teacher: stu-dents emphasized the personal (qualitative) traits of memorable teach-ers rather than academic (quantitative) qualifications. Students seldom mentioned where teachers attended school, what degrees they held, or whether they had been named a "Teacher of the Year."

  4. Full article: Good teachers are always learning

    Equally, in the USA, Kincheloe (Citation 2003) advocated for teachers' research of their own practice as a 'path to (their own professional) empowerment' with potential to mitigate against mediocrity in education caused by 'top-down standards and the desecration of teachers' (p.5).He proposed that 'Teachers as researchers could develop and implement a curriculum connected to the ...

  5. PDF The Good Teacher: an Investigation of The Core-competencies and

    Martin Bernard, Jr. A research paper submitted in conformity with the requirements For the degree of Master of Teaching Department of Curriculum, Teaching and Learning Ontario Institute for Studies in Education of the University of Toronto. Martin Bernard, Jr. April 2015.

  6. PDF Teacher Quality and Student Achievment: Making the Most of Recent Research

    Teacher social capital, or sharing of information, vision, and trust, positively impacted observed instructional quality and school achievement in reading and mathematics (Leana & Pil, 2006). Kannapel & Clements (2005) found that collaborative decision making differentiated high- from low-performing elementary schools.

  7. Teacher well-being: A systematic review of the research literature from

    1. Introduction. Teacher well-being (TWB) is a crucial issue for schools and society. It is seen as relating to teaching effectiveness, student outcomes, and educational governance (Duckworth, Quinn, & Seligman, 2009; Sutton & Wheatley, 2003).High TWB has been shown to help schools—as organizations—stabilize their functioning and increase the commitment of staff members (Creemers & Reezigt ...

  8. PDF A Good Teacher in Every Classroom A Good Teacher in Every Classroom

    This paper excerpts a report based on the National Academy of Education's Preparing Teachers for a Changing World: What Teachers Should Learn and Be Able to Do,which summarizes research on learn-ing, teaching, and teacher education.1 A Good Teacher in Every Classroom 113

  9. PDF A Review of the Literature on Teacher Effectiveness and ...

    In a thorough review of the research on teacher effectiveness attributes, Rice (2003) found that the selectivity of undergraduate institution and the teacher ... personal research paper, or portfolio presentation) tended to prepare more effective teachers. Beyond the institution a teacher attends, the coursework they choose to take

  10. What Makes Good Teachers Good? A Cross-Case Analysis of the Connection

    What Makes Good Teachers Good? A Cross-Case Analysis of the Connection Between Teacher Effectiveness and Student Achievement ... Gomez E., Anderson M. C., Mendro R. L. (1998, April). Policy implications of long-term teacher effects on student achievement. Paper presented at the 1998 Annual Meeting of the American Educational Research ...

  11. (PDF) Teachers' professional learning and its impact on students

    teachers learn from and with other teachers, but it may lead to encul- turation and socialisation into existing practices rather than creative and transformative practices ( Attard Tonna & Shanks ...

  12. On the outcomes of teacher wellbeing: a systematic review of research

    To answer the first research question (RQ 1), key data were gathered from each research paper; these data are shown in Table 1. Most of the sources included in this review ... Satisfied teachers are good teachers: the association between teacher job satisfaction and instructional quality. Br. Educ. Res.

  13. [PDF] What Makes a Good Teacher

    Published 2017. Education. Universal Journal of Educational Research. The introduction of the new Teacher Career Model, the School Inspectorate and the Complex School Assessment imply the basic question "What makes a good teacher?". The scholars have been focusing on the issue for a long period of time. The target of the recent studies is ...

  14. 'A good teacher should…': exploring student perceptions of teaching

    3. Focus of study. The key aim of the paper is to identify what Rwandan secondary school students consider as a 'good teacher'. Using a qualitative approach, our research focus is on supporting students to think about and identify the multi-dimensionality of a 'good' teacher.

  15. Journal of Teacher Education: Sage Journals

    The mission of the Journal of Teacher Education, the flagship journal of AACTE, is to serve as a research forum for a diverse group of scholars invested in the preparation and continued support of teachers who can have a significant voice in discussions and decision-making. Issues covered include preparing teachers to effectively address the needs of marginalized youth; program design and ...

  16. PDF A Descriptive Analysis of Good Teaching and Good Teachers from the

    McEwan (2001) mentioned that good teachers are passionate and positive (see also Leibowitz, van Schalkwyk, Ruiters, Farmer & Adendorff, 2012). Moreover, research has suggested that good teachers care about their students (Arnon & Reichel, 2009; Azer, 2005; Beishuizen et al. 2001; Brown, Morehead & Smith, 2008; Läänemets, Kalamees-Ruubel &

  17. Full article: Teacher-Student Relationship Quality and Student

    Girls, grade 4, School A) and having good teachers (e.g. 'The teachers, there should be good teachers.' Boys School A, grade 8). By analysing the students' narratives, a good school was to a large extant equated with the teacher in terms of a process of the teacher as a promotor of a good school climate being constructed as a core ...

  18. PDF Effective Teacher Professional Development (research brief)

    Using this methodology, we found seven widely shared features of effective professional development. Such professional development: 1. Is content focused 2. Incorporates active learning utilizing adult learning theory 3. Supports collaboration, typically in job-embedded contexts 4.

  19. Research paper Towards "good" teaching —a narrative inquiry of a

    Globally and nationally, there is an increasing amount of research that addresses the issue of teacher professionalism. Originating in the belief that holds teachers' work life as social-situated activities and as 'stories to live by', this research uses life history approaches under an ecological theory lens, to elucidate Lynn - a veteran teacher's work life in an over-40-year span, and ...

  20. Barriers to Racial Equity for Teachers of Color and Indigenous Teachers

    This paper explores obstacles to recruiting and retaining teachers of color and Indigenous teachers (TOCIT) in California's schools. The researchers used a mix of qualitative and quantitative data collected over the course of one year from system leaders in teacher preparation, pre-service teachers, in-service teachers, and former teachers to better understand current policies and practices ...

  21. ERIC

    The burnout and attrition of teachers is a critical issue both in the United States and internationally. However, there is insufficient empirical research addressing these concerns among world language teachers. This paper reports the results of surveying high school world language teachers across all regions of the United States (N= 313) to investigate the relationship between three ...

  22. Welcome to the Purdue Online Writing Lab

    Teachers and trainers may use this material for in-class and out-of-class instruction. Mission The Purdue On-Campus Writing Lab and Purdue Online Writing Lab assist clients in their development as writers—no matter what their skill level—with on-campus consultations, online participation, and community engagement.

  23. PDF The positive and negative effects of teacher attitudes and behaviors on

    An effective and a good teacher is one who commands the content knowledge, knows the strengths of the student, encourages the students, allows them to express themselves, and knows how to use technology (Darling-Hammond, 2008). ... The research data were collected using a structured interview form. Specifically, this form asked

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    With personalized plans, practice tests and more, Khan Academy is good preparation for any test in the SAT Suite. Go to Khan Academy Preparing for the SAT From free test prep to a checklist of what to bring on test day, College Board provides everything you need to prepare.

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

    A high school teacher didn't expect a solution when she set a 2,000-year-old Pythagorean Theorem problem in front of her students. Then Calcea Johnson and Ne'Kiya Jackson stepped up to the challenge.

  26. 2024 AP Exam Dates

    AP Seminar and AP Research students to submit performance tasks as final and their presentations to be scored by their AP Seminar or AP Research teachers. AP Computer Science Principles students to submit their Create performance task as final. Late Testing . Occasionally, circumstances make it necessary for students to test late.