Trigonometry, the study of the lengths and angles of triangles, sends most modern high schoolers scurrying to their cellphones to look up angles, sines, and cosines. Now, a fresh look at a 3700-year-old clay tablet suggests that Babylonian mathematicians not only developed the first trig table, beating the Greeks to the punch by more than 1000 years, but that they also figured out an entirely new way to look at the subject. However, other experts on the clay tablet, known as Plimpton 322 (P322), say the new work is speculative at best.
Consisting of four columns and 15 rows of numbers inscribed in cuneiform, the famous P322 tablet was discovered in the early 1900s in what is now southern Iraq by archaeologist, antiquities dealer, and diplomat Edgar Banks, the inspiration for the fictional character Indiana Jones.
Now stored at Columbia University, the tablet first garnered attention in the 1940s, when historians recognized that its cuneiform inscriptions contain a series of numbers echoing the Pythagorean theorem, which explains the relationship of the lengths of the sides of a right triangle. (The theorem: The square of the hypotenuse equals the sum of the square of the other two sides.) But why ancient scribes generated and sorted these numbers in the first place has been debated for decades.
Mathematician Daniel Mansfield of the University of New South Wales (UNSW) in Sydney was developing a course for high school math teachers in Australia when he came across an image of P322. Intrigued, he teamed up with UNSW mathematician Norman Wildberger to study it. “It took me 2 years of looking at this [tablet] and saying ‘I’m sure it’s trig, I’m sure it’s trig, but how?’” Mansfield says. The familiar sines, cosines, and angles used by Greek astronomers and modern-day high schoolers were completely missing. Instead, each entry includes information on two sides of a right triangle: the ratio of the short side to the long side and the ratio of the short side to the diagonal, or hypotenuse.
Mansfield realized that the information he needed was in missing pieces of P322 that had been reconstructed by other researchers. “Those two ratios from the reconstruction really made P322 into a clean and easy-to-use trigonometric table,” he says. He and Wildberger concluded that the Babylonians expressed trigonometry in terms of exact ratios of the lengths of the sides of right triangles, rather than by angles, using their base 60 form of mathematics, they report today in Historia Mathematica. “This is a whole different way of looking at trigonometry,” Mansfield says. “We prefer sines and cosines … but we have to really get outside our own culture to see from their perspective to be able to understand it.”
If the new interpretation is right, P322 would not only contain the earliest evidence of trigonometry, but it would also represent an exact form of the mathematical discipline, rather than the approximations that estimated numerical values for sines and cosines provide, notes Mathieu Ossendrijver, a historian of ancient science at Humboldt University in Berlin. The table, he says, contains exact values of the sides for a range of right triangles. That means that—as for modern trigonometric tables—someone using the known ratio of two sides can use information in the tablet to find the ratios of the two other sides.
What’s still lacking is proof that the Babylonians did in fact use this table, or others like it, for solving problems in the manner suggested in the new paper, Ossendrijver says. And science historian Jöran Friberg, retired from the Chalmers University of Technology in Sweden, blasts the idea. The Babylonians “knew NOTHING about ratios of sides!” he wrote in an email to Science. He maintains that P322 is “a table of parameters needed for the composition of school texts and, [only] incidentally, a table of right triangles with whole numbers as sides.” But Mansfield and Wildberger contend that the Babylonians, expert surveyors, could have used their tables to construct palaces, temples, and canals.
Mathematical historian Christine Proust of the French National Center for Scientific Research in Paris, an expert on the tablet, calls the team’s hypothesis “a very seductive idea.” But she points out that no known Babylonian texts suggest that the tablet was used to solve or understand right triangles. The hypothesis is “mathematically robust, but for the time being, it is highly speculative,” she says. A thorough search of other Babylonian mathematical tablets may yet prove their hypothesis, Ossendrijver says. “But that is really an open question at the moment.”
Logan Prickett had been in a coma for nearly 12 days. The 13-year-old from Ohatchee, Alabama, had gone in for a routine MRI but soon slipped into unconsciousness, a rare allergic reaction to the contrast agents in his MRI. His family thought he would never wake up. When he did, he couldn’t walk, he lost many fine motor skills, and he couldn’t raise his voice above a whisper. He was also almost completely blind. For an active teen like Prickett, the changes were devastating. But his mind was still the same.
But when he returned to school a year later, he faced a slew of obstacles to learning advanced math. He could no longer read textbooks, and his limited range of motion prevented him from using Braille’s math equivalent. When in 2014 he enrolled in Auburn University at Montgomery (AUM) in Alabama as a psychology major, his frustrations peaked: How was he supposed to learn the precalculus and advanced statistics he would need to earn his degree and get into graduate school?
Before Prickett’s first year began, he met Ann Gully, a science, technology, engineering, and math (STEM) tutoring coordinator at AUM. She wanted to do everything she could to help him, and she started by making sandpaper cutouts of numbers and symbols, hoping Prickett might be able to literally feel his way through the math problems. But after several unsuccessful attempts, she realized it was going to take more. “We were just going to have to describe this math to him, and describe it in a way that he wouldn’t get lost in the weeds.”
Most blind students solve math by touch. The Nemeth Code—a version of Braille for math that codes for everything from fractions to trigonometric functions—is the “gold standard” for math education among visually impaired students, Gulley says. But students with limited mobility like Prickett have to use other methods, including software that converts equations into speech. But these programs were designed to turn regular text into speech—not walk students through complex math problems.
So Gulley and Prickett decided to invent their own system. They reached out to Jordan Price, an undergraduate studying ecological population models who worked as a math tutor. Together, the three started a process of trial and error: Price would first describe the “landscape” of an algebra problem, laying out the main objective. He would then break it down into auditory “chunks,” each of which was a definition, a value, or some other factor related to the problem. Prickett could interrupt at any time to ask questions.
When Prickett was satisfied that he understood the equation, he could tell Price how to simplify it or solve for smaller parts. With each change, Price would read back the newly edited problem. Though solving problems this way takes a lot of time, Prickett says the difference between doing math with and without it is “like night and day.” Breaking larger, complex problems into smaller pieces gives him time to create a mental picture of the problem he’s working through.
Their efforts paid off: In 2015, Prickett earned an A in his college algebra class. That got the team wondering whether other students could benefit. In 2015 they formed the Logan Project and started tailoring their teaching method—called process-driven math (PDM)—for students with other impairments, including dyslexia and dysgraphia, a learning disorder associated with impaired handwriting. When their first group of students showed noticeable improvements in math comprehension and problem solving, they reached out to Rice University in Houston, Texas, to expand the program. Last month, the joint team was awarded a National Science Foundation (NSF) grant of $591,622 for a 2-year research project involving 300 students, with and without disabilities, at seven institutions, including three colleges for the blind.
I don’t want anyone to get the impression that we think we’ve found the equivalent of the fountain of youth and we’re going to solve all math problems.
But not everyone is thrilled about the new method. “No professionally trained individual working in the blindness field would ever adopt this process,” says Gaylen Kapperman, who teaches math education for the visually impaired at Northern Illinois University in DeKalb. That’s because the average brain can hold only seven or so discrete items in short-term memory at once. Trying to do advanced processing using only sound would overtax that load, he says. The new method works well for Prickett because it was, in essence, designed for him. “This is not going to work for blind kids across the nation,” Kapperman says.
That’s the very reason Gulley and her team applied for the NSF grant: They want to build a larger body of evidence on how PDM might work for a wide range of learners. L. Penny Rosenblum, an educational researcher and professor of practice at the University of Arizona in Tucson, applauds the team’s efforts. She says the current lack of resources for nonvisual students gives the Logan Project great potential.
Prickett says he hopes that as the project expands, other students won’t have to struggle like he did. The eventual goal, says Gulley, is to use student feedback to refine the program so that it can be used more widely—perhaps even through computer software for teachers and students. “I don’t want anyone to get the impression that we think we’ve found the equivalent of the fountain of youth and we’re going to solve all math problems,” she says. “We are looking to create additional tools to students who need barriers reduced to mathematics.”
The type of information you will need are based on 1. prior research, 2. current research and 3.discussions on the future of the topic, you have selected.