Article: Best Way to do Math

A Better Way to Teach Math - NYTimes.com

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A Better Way to Teach Math

By DAVID BORNSTEIN

Fixes looks at solutions to social problems and why they work.

Tags: elementary school, john mighton, math, teaching Is it possible to eliminate the bell curve in math class?

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 you 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.

“Almost every kid — and I mean virtually every kid — can learn math at a very high level, to the point where they could do university level math courses,” explains John Mighton, the founder of Jump Math, a nonprofit organization whose curriculum is in use in classrooms serving 65,000 children from grades one through eight, and by 20,000 children at home. “If you ask why that’s not happening, it’s because very early in school many kids get the idea that they’re not in the smart group, especially in math. We kind of force a choice on them: to decide that either they’re dumb or math is dumb.”

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.

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4/20/2011

A Better Way to Teach Math - NYTimes.com

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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?”

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 ENLARGE Separating this step from the calculation makes it easier for kids to understand what the numbers mean. Teachers tell me that when they begin using Jump they are surprised to discover that what they were teaching as one step may contain as many as seven micro steps. Breaking things down this finely allows a teacher to identify the specific point at which a student may need help. “No step is too small to ignore,” Mighton says. “Math is like a ladder. If you miss a step, sometimes you can’t go on. And then you start losing your confidence and then the hierarchies develop. It’s all interconnected.”

Mighton 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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4/20/2011

A Better Way to Teach Math - NYTimes.com

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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 ENLARGE The foundation of the process is building confidence, which Mighton believes should be the first goal of a math teacher. Confidence begets attention, which begets rich learning. “I’ve never met a teacher who will tell you that a student doesn’t need to be confident to excel in school,” explains Mighton. “But I’ve never seen a math curriculum that follows the implications of that idea rigorously.” Math is well-suited to build confidence.

Teachers can reduce things to tiny steps, gauge the size of each step to the student and raise the bar incrementally.

When 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?’”

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A Better Way to Teach Math - NYTimes.com

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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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A Better Way to Teach Math - NYTimes.com

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TD Baltimore April 18th, 2011 10:55 pm Thank you so much for this article. I moved to the US in the 9th grade and that's where my math learning stopped. In Brazil it had been my favorite subject and the one that I was strongest in so it was completely devastating that I was no longer able to learn it. I got through quadratic equations when I left Brazil and my ability to solve everything up to quadratic equations is strong but I have no idea how to do anything that came after that. When I would complain to my mother about my math classes, I'd always say that I couldn't understand why they wanted us to figure out how to solve the problem and explain it back to them before they taught us how to solve the problems. It seemed like the teachers were trying to make math harder than it really was. In Brazil they would give us problems and teach us the formulas to solve them and we would spend time learning how to use and adapt the formulas. In the US they made us figure out the formulas by "thinking critically" about the problems. I never understood how you can expect someone to think critically about something that they don't have a solid foundation in. It's absolutely pathetic that math is so difficult in this country when it's one of the easiest, most straight forward subjects that we have in school.

Recommend Recommended by 184 Readers 2. harold NY April 19th, 2011 8:10 am Other methods of teaching that break subject matter down into very small steps and give learners a continuous feeling of success are also very effective. The Suzuki violin method when properly taught, for example, gets excellent results, for just about every learner even those seemingly not musically "talented". Yet it too is very upsetting to old school music teachers, who feel it must be a trick of some kind.

Recommend Recommended by 45 Readers 3.

KL Portland April 19th, 2011 8:11 am

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A Better Way to Teach Math - NYTimes.com

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I would suggest that anyone interested in this topic take a look at the organization called "Project S.E.E.D" that was started in the 1960's by a man named William Johntz. They have been teaching advanced mathematics to elementary school children ever since. By advanced mathematics, i mean some topics introduced in undergrad programs at universities. It turns out that some of that material is, by virtue of its abstraction and internal logic, no more difficult than say, oh, learning positive and negative numbers for the first time (some examples are basic group theory, number theory, series, etc). 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.

Recommend Recommended by 30 Readers 4. why worry today Tokyo, Japan April 19th, 2011 8:11 am The article states that "all" kids can learn math but then tells us very little about the makeup of the test groups.

What are their racial, ethnic and socioeconomic constituents? Are the pilot schools public or private? Without that kind of information it's impossible for the reader to decide if the claims are credible or if it's just another educational fad/snake oil.

Recommend Recommended by 23 Readers 5.

Xanthan Gum Germany April 19th, 2011 8:11 am How can the jump program be used at the college level?

Recommend Recommended by 17 Readers 6. jdriver linn grove, IA April 19th, 2011 8:11 am This sounds like great stuff. Teach us to fish. Tell us more about the program and make give us the info we need to connect with the Jump people.

Recommend Recommended by 20 Readers 7.

Jeanne Bernish Cincinnati April 19th, 2011 8:12 am I am very interested in the control group study for math. The data is certainly impressive!

Recommend Recommended by 3 Readers 8.

Tynan Kelly Beirut, Lebanon April 19th, 2011 8:12 am This is wonderful to hear! There still may be students who pick up on these microsteps faster (i.e. don't need to be shown them) but I agree that everyone can understand more complicated forms of math. Still, there is a hierarchy of mathematical understanding and comprehension, some people can just "see" high levels of math (this technique shows that anyone can see math). There will always be a higher class of mathematicians, those proving the unimaginable; but for the rest of us mathematical understanding is much more ascertainable and homogenized.

Recommend Recommended by 14 Readers 9.

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A Better Way to Teach Math - NYTimes.com

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Ambabelle Paris April 19th, 2011 9:03 am Paris, drought and sunshine. 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.

Recommend Recommended by 11 Readers 10.

Mikee Anderson, CA April 19th, 2011 9:03 am First we make math difficult for girls. Then we brand everyone who can't do all the problems in their head as stupid. Finally we invent vocabulary that completely separates the subject from any real world relationships or connections. There seems to be a conspiracy to make it a much harder subject than necessary for the perverse purpose of separating people into categories of smart and dumb. Example: what are math 'facts?' And who let Mr. Bell throw a curve?

Recommend Recommended by 16 Readers 11.

David Chowes New York City April 19th, 2011 9:04 am Two personal anecdotes: 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

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A Better Way to Teach Math - NYTimes.com

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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.

Recommend Recommended by 24 Readers 12.

Josh P Rego Park, NY April 19th, 2011 9:04 am Yet the Mayor wants to lay off thousands of teachers...

Recommend Recommended by 19 Readers 13.

Tropiclightning Charles City, VA April 19th, 2011 9:04 am Is it really revelatory that intense, structured attention to a single subject or set of processes presented in a consistent and pyramidal fashion will improve comprehension in that area?Pencils for calculators, repetition of basics, instruction over innovation, and practice, practice, practice. Then...maybe...synthesis and application will follow.

Recommend Recommended by 23 Readers 14.

S Suttner Johannesburg, South Africa April 19th, 2011 9:05 am I noticed that you said you will reply to comments on Friday, so here goes: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 mathsIs 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?

Recommend Recommended by 17 Readers 15.

MKO Albany April 19th, 2011 9:06 am My mother was a sixth grade teacher in the 1960's and 70's and I remember her complaints that so many of her students thought that math was "magic" -- they would randomly assign a numerical answer to a story problem and hope that by luck it would be the correct one. I hope JUMP works as advertised and that teachers are willing and able to implement it. Success in school can be so important and I do agree that it is a tool for social justice -"I am smart" translates to "I can do anything" -- and we need that attitude as a society.

Recommend Recommended by 13 Readers 16.

Michele Lebanon, Indiana April 19th, 2011 9:06 am I forwarded this on to our school district's director of curriculum and the administrator who oversees math teaching in grades 6-8. I completely agree with the assertion that a good teacher instills confidence--I have seen my daughter suffer from a teacher who has made her believe she's not good at math, though she's always performed well in class and on standardized tests. This program seems designed to give students many chances to succeed. Thank you!

Recommend Recommended by 13 Readers 17.

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Drew Andersen Gozo, Malta April 19th, 2011 9:06 am I have gone through life hating math so much because it was alys my worst subject in school. I grew up as an an Army brat and changed scholls at least every three years and sometimes more often depending on where my parents wanted to live. My life was a chos and my school grades suffered because of this business of constantly changing schools. I almost failed seventh grade because the schools there were so far ahead of the States in math.

Fortunately, the two years of Catholic schooling that I had in 2nd and 3rd grade had made me so excellent in all my other subjects, especially reading that I was able to excel and make it through school with only a so-called "math block"!

Recommend Recommended by 3 Readers 18.

DanRN NYC April 19th, 2011 9:06 am The notion that being smart in math implies being smart in everything is interesting. The converse is also interesting. What is unique to math, is that even those of us not particularly talented in math understand its primacy in reality.Not grasping a logically implied conclusion makes one feel "blind" in a sense, because one knows that the logic is unassailable. You are failing to grasp something that is provable. It cannot be written off to lack of taste or apprececiation. It is a failure to understand, and that never feels good.

Recommend Recommended by 15 Readers 19.

Nancy New York April 19th, 2011 9:06 am Bravo! It's time (way past) that people agree that learning to do math is as important as learning to read. And no harder. There are multiple myths about teaching math and science to children (and adults for that matter). Here are a few I've observed: 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!

Recommend Recommended by 33 Readers 20.

MKM New Jersey April 19th, 2011 9:06 am I'd like to know how it differs from Kumon - will you be addressing that, as well?

Recommend Recommended by 5 Readers 21.

Greatstuff1 Los Angeles April 19th, 2011 9:06 am I teach visual arts and I am not a fan of every new thing that is thrown at teachers every year. But, this sounds great. When I read the example of the -7 + 5 and instead of a numerical answer, I was asked if I lost seven dollars in the morning but found five dollars in the afternoon, would I be having a good day or a bad day? The light bulb went off and I thought, why didn't my teacher put it this way?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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4/20/2011