Discovering Math Prodigies
by Elke Town
How Do Teachers Compare to Other Workers?
by David W. Livingstone and Fab Antonelli
Questioning the Data
by Brian Jamieson
Students for John Mighton's fledgling mathematics-tutoring program back in the 1990s – 15 of them – were selected by the principal of a neighbourhood school. Mighton's friends volunteered as tutors. His first student was a Grade 6 girl, Lisa, who could not count by twos.
As Lisa grew increasingly nervous during the course of the lesson, Mighton decided to encourage her by telling her she was brilliant, and found that she began to focus and make progress. The experience taught Mighton that praise could be fundamental to teaching math.
“Sheer quickness of mind is secondary to learning,” Mighton says. “Don't forget that Einstein, who we think of as a math genius, wasn't quick in math either.”
Although the notion of genius as inborn persists in the popular imagination, Mighton points out that we simply can't know which student in a class might be the math genius. It could be any one of them.
“Most people never get to see how brilliant they can be,” says Tom, the meek teaching assistant played by Mighton in the film Good Will Hunting. “They don't find teachers that believe in them. They get convinced they're stupid.”
The genius in the film, played by Matt Damon, came in the guise of a night janitor who sets his broom aside to write the proof for a complex math problem.
As a boy, Mighton was interested in both mathematics and writing, but after reading his older sister's psychology book he felt discouraged about ever entering either profession. The book convinced him that to be successful in either field he would have needed to be born a prodigy.
Mighton himself showed no outstanding early talent for either math or creative writing. At university he saw his marks in math decline, and finally dropped it altogether when he received a D in Calculus for the Life Sciences. Meanwhile, the B he received in creative writing, while seemingly a good mark, was the lowest in the class.
A year after graduating from university, Mighton read a book of letters by American poet Sylvia Plath. He discovered that Plath had taught herself to write by learning everything she possibly could about poetic metre and form, and then writing dozens of imitations of poems she admired.
Inspired by Plath's determination Mighton set about learning to write, and 10 years later he received a Governor General's Award for his work as a playwright.
Mighton has come to the conclusion that the important qualities for learning are diligence, the love of beauty, a willingness to question conventional wisdom and a desire to make connections between things not usually found together.
“Luck plays a role too,” he adds. “It's the unpredictable element that changes everything.”
With his accomplishment as a playwright under his belt, though still unable to make a living solely in that field, Mighton persuaded a tutorial agency to hire him. Subsequently he began tutoring in his home for a local school. In the process of tutoring and having to reduce explanations to simple steps, he realized that he could take up mathematics again. He returned to university and, despite failing tests from time to time, completed his doctorate while continuing to work with JUMP.
He now holds a PhD in graph theory, is an adjunct professor of mathematics at the University of Toronto, a fellow of the Fields Institute for Mathematical Research and an award-winning playwright. In addition he consulted on the 1997 film Good Will Hunting, in which he both acted and scripted his own lines.
Still, Mighton remains active in JUMP, sometimes volunteering as much as 50 to 60 hours a week – teaching, working on curriculum development and professional development for teachers, fundraising and fulfilling various speaking engagements.
JUMP long ago moved out of Mighton's apartment and is now a registered charity with a mission to work “within the education systems to create a nationally and globally numerate society.”
The idea of the program is that familiarity with and understanding of math basics, combined with encouragement, gives students a sense of mastery, of being skilled, of knowing and living a subject. From a mastery of basics, students can then progress to ever more complicated problems and ideas, using inquiry and experimentation to find answers. The missing link between repetition and discovery, Mighton might say, is confidence. It's impossible to make a leap before you learn the basics and can break problems down to the smallest of steps.
One benefit of the JUMP program, he adds, “is to demonstrate that kids can do more than we expect.” But Mighton knew that teachers were under serviced – without the materials and training they needed to help students make intellectual leaps.
JUMP trains and supports teachers and connects them with volunteer tutors who are trained to assist in the classroom or work one-on-one with students.
Curriculum-based workbooks cover Grades 3 to 8. For newcomers to JUMP math, it all starts with fractions. “The fractions units are a huge confidence builder,” Mighton says. “When students have success in fractions, it's much easier for them to develop problem solving and number skills in preparation for the grade-specific materials.”
Some educators have been critical of the JUMP method because they think it is based on rote learning. But Mighton argues that this is a serious misapprehension. “I understand what it means to learn and teach math conceptually and I always encourage my students to make discoveries and extend their knowledge to new cases by themselves,” he says. “I also have a realistic idea of what discovery means in mathematics. I know from my own work as a student and researcher that discoveries in mathematics are often made in tiny, painstaking steps.”
For Mighton, mathematical theories have direct implications in the lives of teachers and students. Chaos theory, for example, states that in nature complex systems show emergent behavior and will do things you cannot predict. The systems will make leaps, very much in the way that one drop of new knowledge can make a child who appears to be at the limit of learning take a surprising step forward.
“Intellectual abilities,” he says, “can emerge from small advances.”
Mixing it up
Mighton is wary of overdependence on investigative or discovery approaches to learning. “How you teach kids is a complex issue,” he says. “You can't force them to discover everything on their own; that mistakes the ends of education with the means. And for discovery to be effective, the steps that lead to it have to be minuscule.
“We want to encourage independent thinkers and train kids to be motivated,” he says.
JUMP is based on the idea that “guided discovery” is a more effective way of learning and that teachers are largely responsible for its success, says Mighton.
Peter Atto is a Mild Intellectual Disability teacher at Queen Victoria Public School in a downtown Toronto neighborhood. His class has 10 students with a range of abilities working at Grade 4, 5 and 6 levels. He teaches from JUMP workbooks two to three times a week and JUMP tutors come in twice a week.
“Most of the kids are working on their own learning program,” says Atto, “and go through the JUMP books at their own speed. The students respond really well to JUMP because they love the extra attention and they can celebrate small successes. The best thing is that they come to believe they can do math. And they can.”
Atto also teaches the different strands of math, including number sense, geometry, patterning, algebra, measurement and data management. “The great thing about JUMP is that there are no failures. If you go at the right speed and there is enough repetition, all students can experience success in their own way.”
At Stonecrest Elementary School in Woodlawn, 35 students from Grades 4 through 6 currently take part in once-a-week lunch-hour tutoring sessions with volunteer tutors trained through the Ottawa Centre for Research and Innovation.
Stonecrest teacher Mary Storms says JUMP helps because “what they're learning in JUMP relates to the rest of math they're doing.”
“The kids get really excited about the fractions units, where they immediately see their own progress,” she says.
Maureen McRae, who has taught at Rose Avenue Junior PS since 1993, is now the JUMP co-ordinator for the school. Like Atto, she believes that the program can effect change. “It helps students make sense of the world,” says McRae. “JUMP is fantastic for kids because it builds confidence that really helps them – especially those in the lower half of the class. They develop a sense of commitment to the learning process. ”
Rose Avenue introduced JUMP in 2001 after teachers there read about it in The Globe and Mail. Since then the school has expanded the program to over 50 one-on-one sessions – in the morning prior to classes with adult tutors and after school with secondary school students from Branksome Hall. Teachers direct students to JUMP tutoring when they need help in mastering the basics.
“Teachers are often hard pressed to find a way to help individual kids,” says McRae. “But when JUMP kids come back into the classroom with better skills, the level of the whole class is raised. Kids are then working at a higher standard and are better prepared for EQAO tests. They get better results because they have more confidence.”
McRae says that she asks the children and volunteer tutors to make a commitment to contact each other if they are unable to attend a session. “This instills a sense of responsibility on both sides.”
“The tutors come from all walks of life,” she adds. “We have everything from presidents of corporations to high school students.” McRae feels that the one-on-one aspect of JUMP tutoring allows for total focus and concentration. “It builds in complexity and lets students move at their own speed and level of ability. And, of course, they're not compared with others as they're working.”
Another Rose Avenue teacher, Megan Chadwick, incorporated JUMP into her Grade 5 classes in 2004 and 2005. She currently teaches the Home School Program in the morning and ESL in the afternoon. Many of her students have serious gaps in their education and she finds JUMP a valuable aid.
“Both the students and their parents really love JUMP,” she says, noting that parents use the books at home. “And some parents ask me to make sure that their kids' next teacher also uses JUMP.”
Of course, as Storms says, “results depend on each child and their efforts,” but the payoffs go beyond math. “JUMP helps them learn how to communicate their ideas.”
In The Myth of Ability Mighton writes, “In my experience, children want two things in order to be happy at school: to exercise their minds and to show off. Mathematics may well be the subject in which children can most easily satisfy both of these desires.”
John Mighton is the author of The Myth of Ability: Nurturing Mathematical Talent in Every Child (House of Anansi Press, 2003); his award-winning play, Half Life, was remounted at the Canadian Stage theatre in Toronto in January 2007.
Brock pilot study
JUMP collaborated with Brock University early in 2005 to assess JUMP's effect on 160 students in Grades 3 to 6.
A test was administered to provide a baseline measure of the students' math ability prior to the JUMP math program. Following the program a version of the same test was administered. All participating students showed post-JUMP improvement.
Selected summary of the results:
Independent OISE study
An OISE study in 2004 examined 14 classrooms in four elementary schools in Toronto over a four-week period. At the study's conclusion, a survey of teachers was administered by JUMP staff with the help of 10 interns from OISE. The resulting report is titled JUMP for Joy! The Impact of JUMP on Student Math Anxiety.
The teachers indicated that the JUMP program had a positive effect on student math confidence in the following areas:
In 2005 OISE questioned students concerning changes in their math confidence. Pre- and post-JUMP-program surveys were administered to 120 Grades 3 and 4 students who had never previously worked with JUMP. The surveys covered six different classrooms at three different schools.
This study found that 74 students out of 116 identified a difference between JUMP math and regular math curriculum. Of these, 88 per cent either favoured JUMP to regular math or liked both programs.
Students noticed a change after participating in the JUMP program and most responded positively, with 57.4 per cent of respondents to the post-survey saying they were good at math and 61.7 per cent saying that they did not worry about doing math in class.
Analysis of quantitative and qualitative data shows that there is “a very significant relationship between students' math confidence and the JUMP program.”
Rachel de Vries, research co-ordinator at JUMP, points out that a comparison of EQAO scores pre- and post-JUMP are not necessarily the best way to measure the program's effectiveness. Year-to-year changes in a school's score do not reflect the progress of individual students because it is a different set of students each year.
Nevertheless, a trend toward improved scores at schools using this method has been noted.
JUMP is designed as a supplement to Ontario's mandated math curriculum to help weaker students rise to the level of those who are thriving.
There are three JUMP programs: the in-class program, one-on-one and an alternative program that is currently used at the York Detention Centre. All materials are developed in collaboration with teachers and pretested by teachers.
The program and workbooks are currently available only in English, but JUMP staff hope to develop French-language versions when they are able to secure the resources.
For information on JUMP math workbooks, teachers' manuals, teachers' professional development opportunities or tutor training programs visit www.jumpmath.org or call 416-596-7716.