A Better Way to Teach Math
By DAVID BORNSTEIN
Fixes looks at solutions to social problems and why they work.Tags:
Imagine if someone at a dinner party casually announced, “I’m illiterate.” It would never happen, of course; the shame would be too great. But it’s not unusual to hear a successful adult say, “I can’t do math.” That’s because we think of math ability as something we’re born with, as if there’s a “math gene” that yo either inherit or you don’t.
School experiences appear to bear this out. In every math class I’ve taken, there have been slow kids, average kids and whiz kids. It never occurred to me that this hierarchy might be avoidable. No doubt, math comes more easily to some people than to others. But the question is: Can we improve the methods we use to teach math in schools — so that everyone develops proficiency?
Looking at current math achievement levels in the United States, this goal might seem out of reach. But the experience of some educators in Canada and England, using a curriculum called Jump Math, suggests that we seriously underestimate the potential of most students and teachers.
Peter BreggJohn Mighton teaching a grade five class at Brock Junior Public School in Toronto.Children come into school with differences in background knowledge, confidence, ability to stay on task and, in the case of math, quickness. In school, those advantages can get multiplied rather than evened out. One reason, says Mighton, is that teaching methods are not aligned with what cognitive science tells us about the brain and how learning happens.
In particular, math teachers often fail to make sufficient allowances for the limitations of working memory and the fact that we all need extensive practice to gain mastery in just about anything. Children who struggle in math usually have difficulty remembering math facts, handling word problems and doing multi-step arithmetic (pdf). Despite the widespread support for “problem-based” or “discovery-based” learning, studies indicate that current teaching approaches underestimate the amount of explicit guidance, “scaffolding” and practice children need to consolidate new concepts. Asking children to make their own discoveries before they solidify the basics is like asking them to compose songs on guitar before they can form a C chord.
Mighton, who is also an award-winning playwright and author of a fascinating book called “The Myth of Ability,” developed Jump over more than a decade while working as a math tutor in Toronto, where he gained a reputation as a kind of math miracle worker. Many students were sent to him because they had severe learning disabilities (a number have gone on to do university-level math). Mighton found that to be effective he often had to break things down into minute steps and assess each student’s understanding at each micro-level before moving on.
Take the example of positive and negative integers, which confuse many kids. Given a seemingly straightforward question like, “What is -7 + 5?”, many will end up guessing. One way to break it down, explains Mighton, would be to say: “Imagine you’re playing a game for money and you lost seven dollars and gained five. Don’t give me a number. Just tell me: Is that a good day or a bad day?”
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Courtesy of Mary Jane MoreauThis graph shows the percentile rankings of Mary Jane Moreau’s grade 5 class in 2006, which was before she taught JUMP curriculum, and her grade 6 class after a year of JUMP work. CLICK TO ENLARGEMighton saw that if he approached teaching this way, he could virtually guarantee that every student would experience success. In turn, the children’s math anxiety diminished. As they grew more confident, they grew excited, and they began requesting harder challenges. “More than anything, kids love success,” he says, “and they love getting to higher levels, like in a video game.”
As the children experienced repeated success, it seemed to Mighton that their brains actually began to work more efficiently. Sometimes adding one more drop of knowledge led to a leap in understanding. One day, a child would be struggling; the next day she would solve a problem that was harder than anything she’d previously handled. Mighton saw that if you provided painstaking guidance, children would make their own discoveries. That’s why he calls his approach “guided discovery.”
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Courtesy of Mary Jane MoreauThis graph shows the percentile rankings of Mary Jane Moreau’s grade 5 class in 2008, which was before she taught them JUMP curriculum, and her grade 6 class in 2009, after a year of JUMP work. CLICK TO ENLARGEWhen math is taught this way, surprising things happen.
Consider some of Jump’s results. It’s been used for four years in the public schools in Lambeth, one of the most economically depressed boroughs of London, England. Teachers placed into Jump the students who were struggling most in math. Among the 353 students who entered the program in fifth grade, only 12 percent began at grade level. Most were at least two grade levels behind and the vast majority were not expected to pass England’s grade six (KS2) national tests. But 60 percent did.
In rural Ontario, Jump was recently evaluated in a randomized controlled study involving 29 teachers and about 300 fifth-grade students (controlled studies of math programs are rare). Researchers from Toronto’s Hospital for Sick Children and the Ontario Institute for Studies in Education arranged for a control group of teachers to use their district’s standard curriculum while another group used Jump. Each set of teachers was given two days of training relevant to the materials they would be using.
In five months, researchers found substantial differences in learning. The Jump group achieved more than double the academic growth in core mathematical competencies evaluated using a well known set of standardized tests. (The study has not yet been published.) “Kids have to make pretty substantial gains in order to see this kind of difference,” explained Tracy Solomon, a developmental psychologist in the Research Institute at the Hospital for Sick Children who is the study’s lead author. “It’s impressive over a five-month period.”
Solomon believes that the key to Jump’s effectiveness is the way it “breaks math down to its component parts and builds it back up.” And she notes that this “flies in the face of the way math is typically taught.”
Interviewing teachers and principals, I have heard numerous stories of results like these. At times, they seem hard to reconcile with our assumptions about math. Isabel Grant, principal of the General Wolfe Elementary School, in Vancouver, British Columbia, has seen Jump produce impressive results in two schools where it has been used by a variety of teachers. Schools in British Columbia evaluate students based on whether they meet expectations for learning outcomes. “Teachers who used Jump were suddenly finding that they had all of their kids in the ‘fully meeting expectations’ category,” Grant told me. “It was such a foreign experience. It doesn’t typically happen when we’re teaching science or language arts. And they were kind of at a loss. ‘What do we do about this?’”
Another example is Mary Jane Moreau, who teaches at the Mabin School, an independent school in Toronto that does not screen students based on academic ability. Moreau, an experienced educator, dabbled with Jump for a year and started to see progress among her students, so she decided to immerse herself in the philosophy. “I was used to getting a bell curve in the past,” she told me, “but what I started seeing was all the kids getting between 90 and 100 percent on tests, and within months, they were all getting between 95 and 100 percent.”
She decided to see if the results would transfer to the standardized Test of Mathematical Abilities. Moreau teaches the same cohort of kids in fifth and sixth grades. Each September, for four years, the students wrote the test. From 2006 to 2007, the class percentile average jumped from 66 percent to 92 percent. From 2008 to 2009, with a new cohort, it increased from 54 percent to 98 percent.
Notably, the bell curve of the students’ scores shifted to the right and narrowed — which is to say that the performance differences between the “slow” kids and the “whiz” kids began to fade away. Moreau encouraged her sixth-grade students to enroll in the Mathematica Pythagoras contest, which attracts only five percent of Canadian students, most of whom would be deemed “gifted” in math. All but one did. For each group, 14 out of 17 students beat the contest average.
Moreau is a dedicated teacher — and she has the benefit of small classes — but, even so, she hadn’t seen results like this before. And it troubled her to think of students she had taught who didn’t have the opportunity to learn math this way. “When I think about what we’ve been doing for years when we could have been doing something else,” she told me, “I feel like I have to run so hard on this because I’m coming to the end of my career. But if I don’t help to change attitudes, I’ll feel like a criminal.”
Jump is a modest outfit. Mighton has a staff of 10 to create materials and conduct teacher trainings. With decisions about math curriculum highly politicized, it’s difficult for a small group to influence the debate. Big textbook companies and paid math consultants have a big say — and big investments — in what gets used.
It will take independent-minded educators to use Jump and see if its results can be replicated in more classrooms and schools. It’s hard to imagine what society might look like if we could undermine the math hierarchies that get solidified in grade school. These patterns tend to play out across society, often in negative ways. Wasn’t it the whiz kids who invented financial derivatives and subprime mortgages? And how many adults got themselves into hot water with their mortgages because, at bottom, they didn’t really understand the risks?
Even deeper, for children, math looms large; there’s something about doing well in math that makes kids feel they are smart in everything. In that sense, math can be a powerful tool to promote social justice. “When you have all the kids in a class succeeding in a subject, you see that they’re competing against the problem, not one another,” says Mighton. “It’s like they’re climbing a mountain together. You see a very healthy kind of competition. And it makes kids more generous to one another. Math can save us.”
On Friday, I’ll reply to comments, explain how Jump has helped one teacher to conquer her own math fears, and I’ll get into some more details about how the program works — including the vital role of bonus questions.
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Fixes explores solutions to major social problems. Each week, it examines creative initiatives that can tell us about the difference between success and failure. It is written by David Bornstein, author of “How to Change the World,” and founder of dowser.org, and Tina Rosenberg, contributing writer for The New York Times magazine and author of the forthcoming “Join the Club: How Peer Pressure Can Transform the World.”
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283 Readers' Comments
However, it is the method of instruction that is truly remarkable. They use something called, oh i'm not sure what they call it nowadays, but it is better known as a Socratic method: asking questions, posing examples and counterexamples, to get at the truth. Rather than a speaking down from some mountain of authority, they encourage the children to discover the beauty of mathematics themselves, by guided inquiry. Seeing a S.E.E.D class in action is to witness a poetry of education, and it is a gift i am grateful to have had the privilege to have observed myself, and a marker by which all instruction i have experienced since has been judged and found wanting.
It is my belief that the only reason education was not revolutionized in this country, by this program, was people playing politics and being foolish and absurd with funding.
Being an old child helper, I have never found yet a child who does not understand math. I have found many who do not understand the math taught by the school.
Math can be made so easy and fun. There is really very little which is not obvious.
Sometimes they learn with their feet. They stand in the stairs halfway and walk five stairs up and seven stairs down, and that's it.
Yet I have been amazed at their capacity to forget on Tuesday what they understood on Monday.
And even more amazed that one day, after a long stuggle, they came and everything was there in their head and they could nto understand why they had had problems before.
However a Caveat; without pain the neurons will not align properly in the head.
Myself I can manage the multiplications up to seven times five, then I gave up in my toddler days.
Firstly when I was in the fifth and sixth grades (with mainly the same group of students), on almost all math (really arithmetic) tests I received 100% -- if not, it may have been 95 or 98%%. A fellow student, John rarely passed. Math seemed so easy and fun to me -- so, I couldn't understand why everyone in the class (including, John) didn't realize how rational and enjoyable math was. Really!
In college, I took calculus. The professor (probably a graduate student) who was a young French woman and she certainly knew her "stuff." In fact, she would race through 50 minutes worth of material in (at times) in about a half hour. Having to both listen to her speed-lecturing and also having to take notes, I found that when I came home, I had to carefully review the material before gaining full comprehension of the material.
Here's the cause of the problem with her (though I am aware that in America, the word "problem" has been replaced by "challenge." But, this represented a problem -- OK? This young woman knew the material so well, she no longer was teaching undergraduates -- rather she was using a level and kind of discourse which was appropriate for her peers -- other professors.
So, if a student is having problems with a course -- usually math or science -- I often recommend that they be helped or tutored by a fellow student who is more likely to understand the difficulties one of their peers might be experiencing. (The helper had just learned it themselves and was aware of the pitfalls.)
I love math -- especially probability theory (yes -- the normal curve, included) and statistics. My affection stems from the realization of its utility when applied to many different areas of subjects. (E.g., economics, biology, psychology, education, physics, chemistry, sociology. and... so many other areas of endeavor.)
I have also taught these areas of math in college and to doctoral students. If math is introduced as an abstract subject... Well, there can be problems with some students.
But, when say, the normal curve is understood as a frequency distribution found in many parts of the natural world... And, even when one is using research and measurement in the (soft) social sciences, even baseball stats. Well, that is a way to capture the interest and gain comprehension in many.
One can then see the world in an entirely useful and far clearer way...
Two more anecdotes:
When my wife and I were in Jamaica on our honeymoon, we hooked up with a cab driver (Sidney) and we used him for our entire two week stay. We were talking... I asked him how many siblings he had. He indicated that he was one of ten boys and that he had no sisters.
I sat in the back (with my wife) and for a few seconds said nothing. Then I said, 'You know, Sidney the probability of only boys resulting from 10 births is one out of 1024. Wow!" [It's 2 to the 10th power.] My wife poked me (rightfully) in my ribs. That's the result of knowing too much math and having too little common sense.
Speaking of the ubiquitous normal curve: when someone tells me that their IQ is say, 130, I always ask: "On which test?" That's because different tests have diverse norming samples, means and standard deviations. But if they say the WAIS (Wechsler Adult Intelligent Test), immediately I respond: "Gee according to those exam results, you're score is higher than 98% of the population."
[Admitedly, only math lovers and a few others will understand this example.]
So, no one learns to drive a car with a modicum of proficency abstactly -- say from just lectures. It's when you actually get behind the wheel... And, drive and are reinforced by your growing proficency and its great utility (B.F.Skinner's Operant Conditioning) will you have fully mastered this art.
The same (to some degree), with mathematics.
Pencils for calculators, repetition of basics, instruction over innovation, and practice, practice, practice. Then...maybe...synthesis and application will follow.
I volunteer as a high-school tutor for an organisation called Ikamva Youth (http://ikamvayouth.org/) which is a volunteer driven programme to supplement the poor education the learners (the last 3 years of schooling) receive at the township schools they go to. Maths is a problem subject for many students, and even with successes of Ikamva Youth many students still struggle with maths
Is there anyway volunteer tutors could be trained or taught (even if there is just a booklet to read) how to use the Jump Maths' techniques to assist the learners? Or is this for teachers only?
Is Jump Maths only for younger learners or could it be used for older learners who are finishing school?
1. You have to make it "fun". No one talks about making it fun to learn grammar. You learn it. Period.
2. It's genetic (so we can't do much about it if students can't learn math). There is almost certainly a genetic component at the very high end, but good teaching and hard work are most of the battle to get to a pretty high level of competence in math and science.
3. Related to 2 above is the belief that girls are genetically inferior in math. And that they 'learn differently' from boys. These ideas are ancient notions and mostly down right dangerous.
These ideas are bias or left-overs from the excitement of rediscovering Mendel (there will be a gene for everything). But it turns out that the best genetics available today can not explain even the genetic basis of a simple trait like height; furthermore, not a single gene for genius or for math ability has yet been found by human gene hunters after spending billions of dollars (and they have looked). It's time to accept the fact that hard work, repetition, pounding away at fundamental skills and practice practice practice can make many more if not most people into functioning mathematicians and scientists (with genuinely useful job skills).
Wonderful article!
I will try is out on a on a junior high class. In visual art you get the gifted along with the learning disabilities all in one class. This should be interesting.
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