The quadratic equation has befuddled high school students for decades.
An intimidating problem fraught with square roots and X-variables, the methods for its solution involve countless steps that are difficult to remember, tedious to implement, and often leave students empty-handed. The quadratic equation is also something of a threshold; students who don’t master it generally find themselves near the terminus of their math education.
For those students, 16-year-old Elizabeth Seagle, unlikely math pioneer, might be a godsend.
It’s 10:30 a.m. on a Wednesday morning. The carpeted hallways of West Bloomfield High School are quiet — but not for long.
Classroom doors burst open and students stream out. A gangly teen whizzes by, arms raised high above his baseball cap in triumph. A spike-haired boy zigzags down the hallway delivering party fliers to a lucky few. A flock of girls brushes past, clad in identical Tiffany necklaces and sporting the kind of newly bronzed skin that, at this time of year, can only be acquired from an electrical bulb. The hallway is abuzz; soon even the ceilings are reverberating with sound. “Na-na-na-na, na-na-na-na, hey-hey, goodbye,” chant the voices streaming in from the loudspeaker overhead. Summer vacation is officially under way.
A petite blonde maneuvers through the heavy traffic. Outfitted in a pink tank top, navy gym shorts and a well-loved pair of black Converse sneakers, 11th-grader Lizzie Seagle appears flushed.
“I just came from weight training,” chirps the tiny athlete. “This is probably the hoochiest thing I’ve worn in my entire life, but I’m trying to stay cool.”
The comment seems a bit absurd in light of some of her scantily clad peers, but then again, so does her humble reaction to this semester’s whirlwind of events.
Lizzie, it appears, has pioneered a new way of factoring quadratic equations. Or so she thinks. Lizzie and her math teacher, Jennifer Newman, are on a quest to find out if her discovery is, indeed, just that or whether she’s simply stumbled across a more obscure method of factorization.
Striding toward the math wing, Lizzie is greeted by friendly hellos every few feet. Has she become more popular now that she’s earned the status of a semi-celebrity around school? “No,” Lizzie giggles in high-pitched embarrassment. “I just know a lot of people.”
But it’s hard to believe her list of admirers hasn’t grown in the past few months, and might be growing even longer if the math gods approve of her epiphany.
The Lizzie Method
The quadratic equation is a tricky formula that is solved to discover the intercepts of a parabola — the points where a line meets the X-axis of a graph. In practical mathematics, it can be used to figure out problems involving objects in motion (a diver hitting water) or numbers that fluctuate (a sum of money gathering interest). Students are usually taught the theory in 10th or 11th grade, but most math teachers find themselves reviewing the problems month after month, hoping that their instruction will “stick.”
Modern textbooks outline three methods for solving quadratic equations: The first involves plugging numbers into the quadratic formula, a complex and tricky device that students try to avoid whenever possible. The other two methods — the “split method” and “guess and check” — provide shortcuts, but only for equations that factor; that is, equations that yield real numbers, as opposed to imaginary ones. Even so, recommended “shortcuts” utilize lengthy computation that can involve plugging in and testing strings of numbers before finding the one that fits. These supposed time-savers can become even more frustrating when, after minutes of ciphering, a student realizes the given equation does not factor and must be solved using method No. 1, the intimidating quadratic formula.
Lizzie had worked on quadratic equations all year and found solving them to be a cinch. So in January when Newman was struggling to teach the “split” and “guess and check” methods to her senior-level math class once again, Lizzie decided to help out.
“I noticed that everyone in class was having trouble,” Lizzie explains, “So I just raised my hand and waited for Mrs. Newman to call on me.”
Lizzie promptly demonstrated to the class the method she had been using since September. It was a reliable, no-nonsense device that involved a short, three-step process. It was also a device that no one had ever heard of before.
“Mrs. Newman tried it out a couple times,” Lizzie recalls, “and I actually remember specifically, she looked up and said, ‘That’s frickin’ amazing,’ which was pretty funny. The whole class just kind of looked at me with their mouths open and were, like, ‘How’d you do that?’”
The Lizzie Method, as Newman dubs it, is unlike anything she’s seen in her 17 years of solving quadratic equations. (Newman has been teaching math at WBHS for seven years, but solving quadratic equations since age 13.) One might compare what the Lizzie Method has done to the standard forms of factorization as akin to what e-mail did to letter writing: made it practically obsolete.
“It’s a quick, sniper method for factoring,” Newman says. “It’s one shot, one kill, you know immediately if something is factorable or not, you don’t get lost in it, like you do in the other two methods. It’s just nice.”
Newman claims the Lizzie Method not only works, but can also help students solve problems faster and with less chance of error because of the reduction of steps.
“I’ve just gone bonkers over this method,” the teacher says. “Other students have had aha’s and good ideas, but not anything that changed the way I do something. This changed the way that I do factoring.”
Newman’s excitement over the discovery led her to submit Lizzie’s name for recognition in the “Who’s Who Among American High School Students” publication. She also nominated her for the West Bloomfield Youth Assistance Award, which Lizzie won in May. In her letter of recommendation to the school board, Newman wrote, “This method will soon find its way into the notebooks of future math students everywhere.”
Newman has already introduced the process to the students in her other classes and expects that most of the school’s math teachers will integrate the technique into their lesson plans next year. Meanwhile WBHS senior Chris Yaldo has written a mathematical proof to validate Lizzie’s shortcut. So far, Newman says, the math holds up.
Newman’s next effort could catapult the discovery into the global mathematics mainstream. She plans to submit a piece for publication to the National Council of Teachers of Mathematics, which, with about 100,000 members, is the world’s largest math education organization. Citing the work of her two students, Lizzie and Chris, Newman hopes to prove the validity and value of the Lizzie Method. If she succeeds, the Lizzie Method could make its way to high schools across the country.
“Mrs. Newman’s really been pulling the whole thing together, and making sure people hear about it,” Lizzie says. “Like, I wouldn’t have realized it was such a big deal if she hadn’t [pointed it out]. She really made me feel good about the whole thing.”
Asked about the method’s eponymous title, Lizzie blushes and says Newman is responsible for that as well.
“I wouldn’t name it ‘the Lizzie Method’ myself,” she says. “I’d probably think of some official-sounding name.”
Lizzie sits cross-legged on a kitchen stool in her West Bloomfield home, waiting for her long, straight hair to dry before friends pick her up for a movie. They’re going to see Disney’s Finding Nemo.
She confesses that she’s not the most studious pupil.
“I feel kind of guilty sometimes,” she concedes. “I don’t study for tests, but I guess I just remember things. Things kind of stick in my head when I learn them.”
Lizzie says she wants to earn good grades, but textbooks don’t rule her life. She says her early education helped her become a critical thinker; she spent grades 1-8 at Farmington Hills’ Hillel Day School. The school’s dual curriculum — taught in both English and Hebrew — was tough to balance, but prepared her well for the accelerated classes she’s been taking at WBHS. Lizzie acquired her current 3.9 GPA with relative ease.
“She just aces everything,” Newman says. “[On tests] it’s very rare if she doesn’t get all of the points that you could possibly get: The regular problems and the extra credit, she just gets them all.”
Although Lizzie’s teachers might see her as a prodigy — Newman calls her “a teacher’s dream” — Lizzie claims that her discovery of the unique method of factorization was a fluke.
Last September, when Newman was conducting a review lesson for quadratic equations, Lizzie, like most of her classmates, felt lost. She had forgotten how to solve the problems over the summer and couldn’t figure out how Newman had derived the answers listed on the review sheet. Pencil in hand, Lizzie forged ahead of Newman’s laborious explanation to see if she could figure things out for herself.
“I just started playing around with the numbers,” Lizzie remembers, “and trying to see what I could do with them until I ended up getting the answers that were [on the answer key].” When she found a way that worked, Lizzie figured the lesson had been learned.
She continued to use her alternative method for the remainder of the semester, but had no idea that she was on to anything new until she spoke up in class in January.
Even then, she says, “I didn’t realize it was that big of a deal. … It was just kind of funny because I had been doing it for so long.”
Lizzie demonstrates the quickie process on a tiny pad of paper, about the size of her palm. The computation barely fills the sheet. After the lesson she explains in her childlike warble, “I used logic to mess around with the numbers and everything, so it wasn’t completely by accident.” Then, in the straightforward tone one would expect from a math whiz, she reasons, “It just made sense to me that if you multiply by [the leading coefficient] in the beginning, you would divide by it in the end.” Of course. “It was just this instinct I had.”
A second opinion
Berkley High School sophomore Adam Bonnington remembers the day in class when one of his peers, a friend of Lizzie’s, demonstrated the process on the blackboard.
“Everyone was awestruck,” he recalls. Students got up from their desks and crowded around the paper holding the formula. The revelation incited “a bunch of commotion.”
Although news of the process didn’t make its way to BHS until the week before final exams, Adam plans to use the nifty shortcut in his math class next year.
“It doesn’t involve having to use any formula,” he remarks. “You just follow the steps and then you get your answer.”
BHS math teacher Paul Yowchuang is equally enthused. “I’ve never seen anything like it,” he says. Yowchuang won’t be teaching math next year, but he plans to pass the formula on to his departmental colleagues. “It saves thinking time and busy-work time” and it appears to be a completely valid form of factorization, he says.
Although he’s never met Lizzie, Yowchuang is impressed. “I think it’s exceptional and I think it’s amazing that someone of that age came up with it.”
BHS math teacher Steven Weingarden agrees. “No one’s ever, from what I’ve encountered, come up with anything close to this.”
But are heavy-hitter math experts just as certain? The Lizzie Method may be new to area high schools, but if it’s a technique that’s been used before, Newman’s efforts to publish might prove a waste.
From her home in West Bloomfield, Lizzie ruminates, “The thing I kind of worry about secretly all the time is what if someone’s already discovered it, but no one’s said anything yet.”
A mathematics professor at the University of Michigan believes that that’s exactly the case. Responding to an e-mail inquiry, the professor writes, “This method was the standard method taught when I was in school, and I suspect that in many parts of the USA it is still the standard method.”
The professor asked to remain anonymous, saying he didn’t wish to rain on anyone’s parade.
Professors elsewhere believe that Lizzie has, indeed, found something new.
Peter Lappan, a mathematics professor at Michigan State University, says Lizzie’s technique for factoring is an original approach.
“I think it’s quite an observation for someone to make,” Lappan states, explaining how Lizzie has taken existing mathematical concepts and put them together in a way that is “mathematically sound.” While he maintains that the traditional approach may be the more efficient route when using large numbers, the Lizzie Method is a quicker alternative to solving the average high school quadratic equation, where the figures are relatively small.
“It’s something that teachers and students ought to be aware of,” he states.
His wife, Glenda Lappan, a university distinguished professor at MSU, is a past president of the National Council of Teachers of Mathematics, the organization to which Newman will be submitting her work. In an e-mail she states that Lizzie’s process is, indeed, valid. “… [T]he student is correct. This is a method that works well for small numbers,” she wrote. “The student deserves congratulations on a very fine piece of work.”
Lowell J. Hansen, the chairman of the department of mathematics at Wayne State University, says what’s most impressive is that the idea came from a 16-year-old high school student. “I think it’s terrific,” he says.
But whether Lizzie’s approach is completely original, Hansen concedes, is another story. “I haven’t seen this in textbooks,” he says, but he adds that the math seems so simple that he wouldn’t be surprised if someone’s done it before.
Pioneer or not, for the time being Lizzie is enjoying the notoriety associated with her mathematical marvel. Some kids who hear her name ask if she’s “Lizzie, the factoring Lizzie?” but even more students have made the connection on their own; she commonly hears: “Oh, you’re Lizzie.”
Her entourage of adult fans has offered sobriquets. Newman refers to her as “The Factoring Guru” and her friends’ parents call her “The Math Whiz.”
While Lizzie’s mother, Mimi Seagle, is glad that the finding has given her introverted daughter a dose of much-deserved attention, Lizzie maintains that her greatest joy is the knowledge that she’s helping others.
“I’ve had a few friends come up to me and be like, ‘I used your method on the test and it was so much easier.’ That’s, like, the greatest reward so far, out of everything.”
Still, she admits, “I’d be really, really excited if it turned out to be the first time that anyone’s really done this, or at least has had it proved mathematically and all that stuff.”
While she’s hardly a stuffy math geek, Lizzie’s not a high-maintenance fashionista either. Her bedroom reveals a style completely unique.
Odds and ends are fastened to the bright orange walls with shreds of blue electrical tape: photos, drawings, and a giant “Classroom Rules” sign are among the artful clutter. A tie-dyed comforter lies tidily across the bed and an orange TV, circa 1972 (“It was my parents’”), is perched atop the dresser. Posters that suggest an obsession with music complete the statement.
Looking over images of Nirvana and Green Day, Lizzie comments on her comparatively larger collection of Blink 182 paraphernalia.
“Oh, goodness,” she sighs. “I had this whole phase in eighth grade where I practically worshiped them. I still appreciate them. I love them because they were one of my first, more recent, music loves. I don’t really listen to them that much anymore, but I don’t really want to take the posters down. Lately I’ve been a lot more into classic rock, or just more mellow music like Dave Matthews.”
Lizzie plays the drums and guitar and many of her friends are musicians. Yet she describes herself as mainly a “drama kid.” Working as a stage technician since ninth grade, Lizzie’s been involved in almost every aspect of production, from managing props to producing one-act plays. She’s also spent a fair amount of time in front of the footlights, but says her biggest thrill comes from seeing things she’s made being used on stage.
The natural designer loves to create artwork in other mediums as well; her newest passion is photography. In her introductory class at school, one of her black-and-white images won a “Photo of the Month” award.
This past semester Lizzie volunteered as a “Link Student” for one hour of every school day, helping special needs kids by assisting them with their homework and sharing their cafeteria table at lunch.
“What I love about it is, you just make them feel really good when you help them,” she reflects. “Like, one of the girls that I helped, I see her in the hall all the time, and she’ll come up to me and give me a hug. They’re so sweet.”
Lizzie’s teacher and parents cite her generosity as a chief characteristic. When asked to describe her star student in class, Newman first begins with a personal accolade. “Lizzie, first and foremost, is just a kind person. She’s just a nice person for other students to be around in class. She’s very giving of her time and energy to help someone understand if they ask for it, but yet she doesn’t boast about how smart she is either, so she’s very humble. People feel very comfortable coming to her.”
Lizzie’s mother, a psychologist, describes her daughter in much the same way. “She’s the kindest, sweetest person,” she says, adding that making the math discovery is in character. “She was always an ‘I want to do it myself, I have to do it myself’ kind of kid.”
Lizzie would like to continue helping out as she gets older, perhaps by joining the Peace Corps, but says she doesn’t follow politics all that closely. She’ll occasionally watch the evening news, but says generally, “I try to avoid heavy, heated debates.
“I’m sure at some point I’ll become a lot more involved, but right now I’m a teenager.”
Lizzie says her immediate goal is to make learning easier for her classmates, a realm in which she feels she can really make a difference. But she’s still reluctant to explain her mathematical invention to her mother.
“I don’t speak math,” Mimi Seagle admits. “[Lizzie] won’t even show me. She says, ‘Mom, there’s no way that you’re even going to be able to understand.’ But I once said to her, ‘When you have the time, I’m going to make you sit down with me and maybe I’ll understand how to do a quadratic equation for the first time in my life.’”
Lizzie’s father, Peter, a businessman, was made privy to the information right away. “Peter is a math whiz,” his wife declares, explaining where their daughter gets her mathematical genes. Not only does he understand the concept, she says, “He thinks it’s really cool.”
The next frontier
If the Lizzie Method is really all it’s cracked up to be, one mystery is why it’s gone undetected for so long.
“…[M]ath is one of those things where people don’t really question it so much because everyone’s taught, ‘Well, this is the formula; you have to use it,’” Lizzie says. “Sometimes I think it’s just so simple that people might have overlooked it.”
Lizzie’s method is, indeed, simplistic. Which could be why educators working in high schools, as opposed to colleges, have expressed the most interest thus far.
Figuring a faster way to crack a quadratic equation doesn’t reveal anything new about the nature of parabolas or mathematics in general, but it could still provide challenged high school students with an easy alternative to lengthy computation.
Says Newman, “I don’t think it’s so much a new mathematical concept as much as it is a more concise way to do what we’ve already known about. It’s not something that’s a new, novel idea for mathematicians or mathematics as a whole, as a science, but it’s definitely something that could help more students factor.”
In response to the comment made by the U-M professor, Newman agrees, “It might have been discovered somewhere else in the world at another time.”
But that, in Newman’s mind, doesn’t diminish Lizzie’s achievement. “She created it on her own and it takes a lot of mathematical insight to be able to create something like that,” Newman says. “Even if she discovered something somebody else did, she did discover it for herself. And it’s huge. Like, you dream about it as a math teacher.”
Though Lizzie might be Newman’s model student, the teenager’s feelings toward math are not exactly reciprocal. Asked whether she sees a future for herself in the field, Lizzie says, “I like doing math, but it’s not really something that I’m interested in doing for the rest of my life.”
Ideally, the 16-year-old would like to attend U-M like older sister Olivia, 19, to study education or psychology. “I’m definitely big on working with people.”
Newman had this notion all along. “I don’t think she’s interested in math, which is fine. You know, I see her going far in whatever she chooses. I know she’s very artistic, but there’s a lot of mathematics wrapped up in art, even if you don’t see it.”Ronit Feldman is a Metro Times editorial intern. E-mail [email protected]