On March 24th, 2023, St. Mary’s Academy New Orleans Students Calcea Johnson and Ne’Kiya Jackson announced they came up with a breakthrough in math history: they solved the Pythagorean Theorem, but in a novel way. Now, for anyone to have achieves another new way of solving the Pythagorean Theorem is news worthy, but for two African American teenagers to do it is should be a headline story in a World media obsessed with the crazy idea that if you’re young and black, you’re always doing something wrong.
Calcea Johnson and Ne’Kiya Jackson are to be celebrated, but first, we have to get the confution over what they did out of the way, to even see its true meaning. Contrary to the story that:
Calcea Johnson and Ne’Kiya Jackson, high schoolers at St. Mary’s Academy, have proven Pythagoras’ Theorem without circular logic, according to WWL-TV. The mathematical discovery is said to have not been proven by mathematicians for 2,000 years.
The truth is that the Pythagorean theorem has been solved many ways. Johnson and Jackson discovered another way that was not thought possible. But, what’s the Pythagorean theorem, to start with? Well, when I learned it in school, it was the idea that in a given right triangle, the square one can form from the right line is equal in size to adding the squares formed by bottom line of that right triangle, and the slanted line on the left side of the same right triangle. Or it looks like this:
Now, if you were paying attention in school, you recognized that diagram, even if you didn’t really understand why it was important. Well, that mathematical theorem has been proven many ways using different methods. But all of those approaches involved using some form of the diagram above. The idea that the Pythagorean theorem without a diagram was considered not possible. Have a look at this seven-year-old example seen in Stack Exchange of thought before Johnson and Jackson came along:
A: There are some proofs of Pythagoras theorem which don’t even require high school maths to understand, but they all are using shapes to prove of the theorem. However, I am trying to find some proofs of Pythagoras theorem that don’t use shapes in their proofs, for example a purely algebraic proof. Besides, they would still be easy enough to teach for students with high school level of knowledge in mathematics.
My question: Prove that for ?2+?2=?2
there are infinity many triples in ℤ+ fit into the equation? – which requires high school maths knowledge to understand.
Reponses:
1. You’re trying to prove a geometrical theorem without using geometry?
2. How do you define a right-angled triangle in an analytic/algebraic way?
And the examples claiming to do so were observed as actually having nothing to do with the Pythagorean theorem.
Finally, another observer wrote “You could try to use something like this, (using Arc length
which is the distance between two points along a section of a curve) but as you’ll see, it would be a circular argument, because in the derivation of these results the Pythagorean theorem is already used. You could just Define arc-length to be the linked expression, but then I think you’ll loose your high school students.” But it’s at the place regarding the use of “ a circular argument” where Johnson and Jackson’s finding takes flight, because they proved a circular argument could work. That stands in stark contrast to the age old idea that “There are no trigonometric proofs, because all the fundamental formulae of trigonometry are themselves based upon the truth of the Pythagorean theorem.”
Here’s the research duo’s abstract (https://meetings.ams.org/math/spring2023se/meetingapp.cgi/Paper/23621):
In the 2000 years since trigonometry was discovered it’s always been assumed that any alleged proof of Pythagoras’s Theorem based on trigonometry must be circular. In fact, in the book containing the largest known collection of proofs (The Pythagorean Proposition by Elisha Loomis) the author flatly states that “There are no trigonometric proofs, because all the fundamental formulae of trigonometry are themselves based upon the truth of the Pythagorean Theorem.” But that isn’t quite true: in our lecture we present a new proof of Pythagoras’s Theorem which is based on a fundamental result in trigonometry—the Law of Sines—and we show that the proof is independent of the Pythagorean trig identity \sin^2x + \cos^2x = 1.
Now, I have not seen Calcea Johnson and Ne’Kiya Jackson’s research paper to give you a full presentation of what they showed to the American Mathematical Society. But what they presented was impressive enough for the American Mathematical Society to encourage them to present their findings in a peer-reviewed journal. But basically, the pair is saying that the Pythagorean theorem can be solved using the Law of Sines, and since the Law of Sines is part of trigonometry, then Johnson and Jackson have, indeed, used a circular argument. And that is a breakthrough if peer review of their work confirms that is what they’ve done.
But the question is why is the Pythagorean theorem important?
The Pythagorean theorem is about finding the shortest distance between two points, but it’s also a story of how to think about building structures. Understanding the Pythagorean theorem is required in engineering course work, which in turn leads to the development of buildings, engines, and bridges. It is used to calculate how steep a slope is. It’s employed in basic survey work. And it’s a basic part of flying an airplane, where a pilot can use the distance from the airport, and the altitude of the plane, to determine when to start the decent to land (something computers do much faster, today).
The Law of Sines has a place in our world beyond theory because it’s about solving for an unknown. Mathcentre UK says “To solve a triangle is to find the lengths of each of its sides and all its angles. The sine rule is used when we are given either a) two angles and one side, or b) two sides and a non-included angle. The cosine rule is used when we are given either a) three sides or b) two sides and the included angle.” Remember that all sides of a triangle must add up to 180 degrees.
I had that stuff abut the Pythagorean theorem and the Law of Sines and Cosines drilled into my head at Skyline High School and Bret Harte Junior High in Oakland back between 1975 and 1980. But since I wasn’t into proving or disproving theorems, it never occured to me to really think about what I was being told. But consider that it never occured to most humans.
So, this story is really one of two African American young female teens daring to think critically about what they’re told and then challenge conventional wisdom. It’s one thing for anyone at any age to do that, but for two Black female teenagers to do it is a sign that the mental scars of racism and slavery passed down from generation to generation are slowly fading to a point where we as Black Americans don’t have to fear retribution for going against the status quo in areas commonly belived to be “for white people” like math and science. From that perspective folks, what Calcea Johnson and Ne’Kiya Jackson have done is really a celebration of American social progress that we all should be proud of. It’s not that it’s “Ok to think or talk like a white person”, which, I might add, is OK, but that it’s OK to be yourself.
For me, I have been told I “talk like a white person” or “think I’m smarter than anyone else” and I have got that message more from black folks in America than anyone else, and I’m black. But I’ve got it from white Americans too. I remember when I wanted to run the Oakland Economic Development Department after serving at Economic Advisor to Oakland Mayor Elihu Harris. For some reason, Elihu picked Bill Claggett, an economic consultant who was white (and who’s a friend today, but it is what it was) . I was really hurt and wanted to leave the City of Oakland. But then City Manager Robert Bobb, who’s title was changed to City Administrator after Jerry Brown won the 1998 Oakland Mayoral Election, wanted me to stick around, and so told me to go and talk to Claggett.
So, I did. I met Bill for lunch in early December of 1998, at Max’s Opera Cafe in Oakland City Center. The lunch did not go well. Basically, Bill Claggett told me that “I talked like I think I know everything”. My response was that I talked like that since I was six years old, and I wasn’t changing. So, after lunch I went back to my office with the idea of leaving. I happened to run into Robert Bobb, and told him what happened and Bill said. Mr. Bobb’s response was “You do talk like you think you know everything. You’re young. You’re gifted. You’re black.” On those words alone, I elected to stay and work for the City of Oakland under Robert Bobb. Eventually, I headed the effort to bring the Super Bowl to Oakland. Out of 11 cities, Oakland emerged to be one of the three finalists NFL Cities with Jacksonville and Miami. And even though we lost on the third ballot, that Oakland got that far shocked everyone. And Bill Claggett, who said I talked like I think I know everything, put $80,000 into the Oakland-Alameda County Sports Commission I created from scratch. And sadly, Jerry Brown, who was upset that I wrote an email to my board saying he wasn’t comfortable with me as a black person who talked authoritatively, basically proved he wasn’t by firing me.
We as a nation – black white and in between – have a long way to go in embracing smart black people. To this day, I’m told, especially in the South, that I talk like a white person or that I think I know everything. Calcea Johnson and Ne’Kiya Jackson’s work the field of mathematics and being allowed to present it and advance it as Black teenagers, is a major American social advancement that should not be taken for granted. To me, it’s on a par with Barack Obama becoming America’s first black president.
Stay tuned.