The Benefits of Natural Math

natural math, exploratory math, hands-on learning,

images: public-domain-image.com

Math as it’s used by the vast majority of people around the world is actually applied math. It’s directly related to how we work and play in our everyday lives. In other words it’s useful, interesting, even fun.

We now know babies as young as five months old show a strong understanding of certain mathematical principles. Their comprehension continues to advance almost entirely through hands-on experience. Math is implicit in play, music, art, dancing, make-believe, building and taking apart, cooking, and other everyday activities. Only after a child has a strong storehouse of direct experience, which includes the ability to visualize, can he or she readily grasp more abstract mathematical concepts. As Einstein said, “If I can’t picture it, I can’t understand it.”

As parents, we believe we’re providing a more direct route to success when we begin math (and other academic) instruction at a young age. Typically we do this with structured enrichment programs, educational iPad games, academic preschools, and other forms of adult-directed early education. Unfortunately we’re overlooking how children actually learn.

Real learning has to do with curiosity, exploration, and body-based activities. Recent studies with four-year-olds found, “Direct instruction really can limit young children’s learning.” Direct instruction also limits a child’s creativity, problem solving, and openness to ideas beyond the situation at hand. Studies show kids readily understand math when they develop a “number sense,” the ability to use numbers flexibly. This doesn’t come from memorization but instead from relaxed, enjoyable exploratory work with math concepts. In fact, math experts tell us methods such as flash cards, timed tests, and repetitive worksheets are not only unhelpful, but damaging. Teaching math in ways that are disconnected from a child’s life is like teaching music theory without letting them plunk piano keys, or instructing them in the principles of sketching without supplying paper or crayons. It simply makes no sense.

One study followed children from age three to age 10. The most statistically significant predictors of math achievement had very little to do with instruction. Instead the top factors were the mother’s own educational achievements and a high quality home learning environment. That sort of home environment included activities like being read to, going to the library, playing with numbers, painting and drawing, learning letters and numbers, singing and chanting rhymes. These positive effects were as significant for low-income children as they were for high income children.

There’s another key difference between kids who excel at math and kids who don’t. It’s not intelligence. Instead it’s related to what researcher Carol Dweck terms a growth-mindset. Dweck says we adopt certain self-perceptions early on. Some of us have a fixed mindset. We believe our intelligence is static. Successes confirm this belief in our inherent ability, mistakes threaten it. People with a fixed mindset may avoid challenges and reject higher goals for fear of disproving their inherent talent or intelligence.  People with a growth mindset, on the other hand, understand that intelligence and ability are built through practice. People with this outlook are more likely to embrace new challenges and recognize that mistakes provide valuable learning experience. (For more on this, read about the inverse power of praise.)

Rather than narrowing math education to equations on the board (or worksheet or computer screen) we can allow mathematics to stay as alive as it is when used in play, in work, in the excitement of exploration we call curiosity. Math happens as kids move, discuss, and yes, argue among themselves as they try to find the best way to construct a fort, set up a Rube Goldberg machine, keep score in a made-up game, divvy out equal portions of pizza, choreograph a comedy skit, map out a scavenger hunt, decide whose turn it is to walk the dog, or any number of other playful possibilities. These math-y experiences provide instant feedback. For example, it’s obvious cardboard tubes intended to make a racing chute for toy cars don’t fit together unless cut at corresponding angles. Think again, try again, and voila, it works!

As kids get more and more experience solving real world challenges, they not only begin to develop greater mathematical mastery, they’re also strengthening the ability to look at things from different angles, work collaboratively, apply logic, learn from mistakes, and think creatively. Hands-on math experience and an understanding of oneself as capable of finding answers— these are the portals to enjoying and understanding computational math.

Unfortunately we don’t have a big data pool of students who learn math without conventional instruction. This fosters circular reasoning. We assume structured math instruction is essential, the earlier the better, and if young people don’t master what’s taught exactly as it’s taught we conclude they need more math instruction. (“Insanity: doing the same thing over and over again and expecting different results.”)

But there are inspiring examples of students who aren’t formally instructed yet master the subject matter easily, naturally, when they’re ready.

1. The experiment done over 85 years ago by Louis Benezet showed how elementary school children can blossom when they’re free of structured math instruction.

2. Homeschooling and unschooling families around the world devote much less time to formal mathematics instruction. Studies indicate their children grow up to succeed in college, careers, and life with greater self-reliance and focus than their schooled peers. Interestingly, two different surveys of grown unschoolers showed that a much higher number of them work in STEM careers than schooled adults. The samples were small but intriguing. More proof? Many of our greatest science, technology, engineering, and mathematics contributors have already emerged from the homeschool community.

3. Democratic schools where children are free to spend their time as they choose without required classes, grades, or tests. As teacher Daniel Greenberg wrote in a chapter titled “And ‘Rithmetic” in his book Free at Last, a group of students at the Sudbury Valley School approached him saying they wanted to learn arithmetic. He tried to dissuade them, explaining that they’d need to meet twice a week for hour and a half each session, plus do homework. The students agreed. In the school library, Greenberg found a math book written in 1898 that was perfect in its simplicity. Memorization, exercises, and quizzes were not ordinarily part of the school day for these students, but they arrived on time, did their homework, and took part eagerly. Greenberg reflects, “In twenty weeks, after twenty contact hours, they had covered it all. Six year’s worth. Every one of them knew the material cold.” A week later he described what he regarded as a miracle to a friend, Alan White, who had worked as a math specialist in public schools. White wasn’t surprised. He said, “…everyone knows that the subject matter itself isn’t that hard. What’s hard, virtually impossible, is beating it into the heads of youngsters who hate every step. The only way we have a ghost of a chance is to hammer away at the stuff bit by bit every day for years. Even then it does not work. Most of the sixth graders are mathematical illiterates. Give me a kid who wants to learn the stuff—well, twenty hours or so makes sense.”

We know all too well that students can be educated for the test, yet not understand how to apply that information. They can recite multiplication tables without knowing when and how to use multiplication itself in the real world. Rote learning doesn’t build proficiency let alone nurture the sort of delight that lures students to higher, ever more abstract math.

Conventional math education may also limit our concept of what math can do. As Stanford mathematician Keith Devlin notes in a post titled “Most Math Problems Do Not Have a Unique Right Answer,”

One of the most widely held misconceptions about mathematics is that a math problem has a unique correct answer…

Having earned my living as a mathematician for over 40 years, I can assure you that the belief is false. In addition to my university research, I have done mathematical work for the U. S. Intelligence Community, the U.S. Army, private defense contractors, and a number of for-profit companies. In not one of those projects was I paid to find “the right answer.” No one thought for one moment that there could be such a thing.

So what is the origin of those false beliefs? It’s hardly a mystery. People form that misconception because of their experience at school. In school mathematics, students are only exposed to problems that (a) are well defined, (b) have a unique correct answer, and (c) whose answer can be obtained with a few lines of calculation.

Interestingly, people who rely on mental computation every day demonstrate the sort of adroitness that doesn’t fit into our models of math competence. In a New York Times article titled “Why Do Americans Stink at Math?” author Elizabeth Green (who defines the term “unschooled” as people who have little formal education) writes,

Observing workers at a Baltimore dairy factory in the ‘80s, the psychologist Sylvia Scribner noted that even basic tasks required an extensive amount of math. For instance, many of the workers charged with loading quarts and gallons of milk into crates had no more than a sixth-grade education. But they were able to do math, in order to assemble their loads efficiently, that was “equivalent to shifting between different base systems of numbers.” Throughout these mental calculations, errors were “virtually nonexistent.” And yet when these workers were out sick and the dairy’s better-educated office workers filled in for them, productivity declined.

The unschooled may have been more capable of complex math than people who were specifically taught it, but in the context of school, they were stymied by math they already knew. Studies of children in Brazil, who helped support their families by roaming the streets selling roasted peanuts and coconuts, showed that the children routinely solved complex problems in their heads to calculate a bill or make change. When cognitive scientists presented the children with the very same problem, however, this time with pen and paper, they stumbled. A 12-year-old boy who accurately computed the price of four coconuts at 35 cruzeiros each was later given the problem on paper. Incorrectly using the multiplication method he was taught in school, he came up with the wrong answer. Similarly, when Scribner gave her dairy workers tests using the language of math class, their scores averaged around 64 percent. The cognitive-science research suggested a startling cause of Americans’ innumeracy: school.

And Keith Devlin explains in The Math Gene that we’re schooled to express math in formal terms, but that’s not necessary for most of us—no matter what careers we choose. People who rely on mental math in their everyday lives are shown to have an accuracy rate around 98 percent, yet when they’re challenged to do the same math symbolically their performance is closer to 37 percent.

We have the idea that memorizing, practicing, and testing is the only way to higher achievement. It’s hard to imagine why we still believe that when studies show that high test scores in school don’t correlate with adult accomplishments (but do line up with interpersonal immaturity).

There are all sorts of ways to advance mathematical understanding. That includes, but isn’t limited to, traditional curricula. It’s time to broaden our approach. Let’s offer the next generation a more intrinsically fascinating, more applied relationship to math. Let’s foster analytical and critical thinking skills across all fields. The future is waiting.

This article is one in a series of three on natural math. 

Math Instruction versus Natural Math: Benezet’s Experiment. What happened when formal math instruction was eliminated? 

Natural Math: 100+ Activities and Resources. Finding and learning from math in daily life. 

Portions of this article are excerpted from Free Range Learning.

Math Instruction versus Natural Math: Benezet’s Example

Louis Benezet, natural math,

1930’s classroom (forestpark4.wikidot.com)

Children are intrinsically eager and able to learn. If we step back from our limiting preconceptions about education, we discover learning flourishes when we facilitate it rather than try to advance it through force, intimidation, and coercion.

Over 85 years ago a pioneering educator proved that delaying formal instruction, in this case of mathematics, benefits children in wonderfully unexpected ways. Louis P. Benezet, superintendent of the Manchester, New Hampshire schools, advocated the postponement of systematic instruction in math until after sixth grade. Benezet wrote,

I feel that it is all nonsense to take eight years to get children thru the ordinary arithmetic assignment of the elementary schools. What possible needs has a ten-year-old child for knowledge of long division? The whole subject of arithmetic could be postponed until the seventh year of school, and it could be mastered in two years’ study by any normal child.

While developing this rationale, Benezet spoke with eighth-grade students. He noted they had difficulties putting their ideas into English and could not explain simple mathematical reasoning. This was not only in his district; he found the same results with fourteen-year-old students in Indiana and Wisconsin. Benezet didn’t blame the children or teachers, he blamed introducing formal equations too early.  So he began an experiment, abandoning traditional arithmetic instruction below the seventh grade.

In the fall of 1929 I made up my mind to try the experiment of abandoning all formal instruction in arithmetic below the seventh grade and concentrating on teaching the children to read, to reason, and to recite – my new Three R’s. And by reciting I did not mean giving back, verbatim, the words of the teacher or of the textbook. I meant speaking the English language.

To start, he picked out five classrooms, choosing those districts where most students were from immigrant homes and the parents spoke little English. Benezet knew that in other districts the parents with greater English skills and higher education would have vehemently objected, ending the experiment before it started.

In the experimental classrooms, children were exposed to what we’d call naturally occurring math. They learned how to tell time and keep track of the date on the calendar. The students played with toy money, took part in games using numbers, and when dimension terms such as “half” or “double” or “narrower” or “wider” came up incidentally, they were discussed. Instead of math, the emphasis was on language and composition. As Benezet describes these children,

They reported on books that they had read, on incidents which they had seen, on visits that they had made. They told the stories of movies that they had attended and they made up romances on the spur of the moment. It was refreshing to go into one of these rooms. A happy and joyous spirit pervaded them. The children were no longer under the restraint of learning multiplication tables or struggling with long division.

At the end of the first school year, Benezet reported that the contrast between the experimental and traditionally taught students was remarkable. When he visited classrooms to ask children about what they were reading, he described the traditionally taught students as “hesitant, embarrassed and diffident. In one fourth grade I could not find a single child who would admit that he had committed the sin of reading.” Students in the experimental classrooms were eager to talk about what they’d been reading. In those rooms, an hour’s discussion went by with still more children eager to talk.

Benezet hung a reproduction of a well-known painting in the classrooms and asked children to write down anything the art inspired. Another obvious contrast appeared. When he showed the ten best papers from each room to the city’s seventh-grade teachers, they noted that one set of papers showed much greater maturity and command of the language. They observed that the first set of papers had a total of 40 adjectives such as nice, pretty, blue, green, and cold. The second set of papers had 128 adjectives, including magnificent, awe-inspiring, unique, and majestic. When asked to guess which district the papers came from, each teacher assumed that the students who wrote the better papers were from schools where the parents spoke English in the home. In fact, it was the opposite. Those students who wrote the most masterfully were from his experimental classes.

Yet another difference was apparent. It was something that Benezet had anticipated. He explained, “For some years I had noted that the effect of the early introduction of arithmetic had been to dull and almost chloroform the child’s reasoning faculties.” At the end of that first year, he went from classroom to classroom and asked children the same mathematical story problem. The traditionally taught students grabbed at numbers but came up with few correct results, while the experimental students reasoned out correct answers eagerly, despite having minimal exposure to formal math.

Based on these successes, the experiment expanded. By 1932, half of the third- to fifth-grade classes in the city operated under the experimental program. Due to pressure from some school principals, children in the experimental classrooms were back to learning from a math book in the second half of sixth grade. All sixth-grade children were tested. By spring of that year all the classes tested equally. When the final tests were given at the end of the school year, one of the experimental groups led the city. In other words, those children exposed to traditional math curricula for only part of the sixth-grade year had mastered the same skills as those who had spent years on drills, times tables, and exams.

In 1936, the Journal of the National Education Association published the final article by Benezet. His results showed the clear benefits of replacing formal math instruction with naturally occurring math while putting a greater emphasis on reading, writing, and reasoning. The journal called on educators to consider similar changes.

As we know, schools went in the opposite direction.

Louis Paul Bénézet

Louis Paul Bénézet

This article is an excerpt from Free Range Learning. (Next post, the extraordinary benefits of emphasizing natural math over math instruction.)

Emphasis On Testing Cheats Everyone

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SAT language scores aren’t near the levels seen in the early 1970’s.  Some test-takers resort to cheating, most recently seven teens from Great Neck, NY who hired an impersonator to stand in (well, sit in) for them. The company that administers the SAT estimates cheating, mostly by collaboration, occurs in only one-tenth of 1 percent of the 2.25 million students who take the test annually. That seems insignificant, but it underscores a larger problem—our test-obsessed educational system.

E. D. Hirsch Jr., author of The Making of Americans: Democracy and Our Schools wrote in a recent New York Times op-ed that declining verbal scores have to do with enduring school days stripped of “substantial and coherent lessons concerning the human and natural worlds…” He goes on to note,

The most credible analyses have shown that the chief causes were not demographics or TV watching, but vast curricular changes, especially in the critical early grades. In the decades before the Great Verbal Decline, a content-rich elementary school experience evolved into a content-light, skills-based, test-centered approach.

I don’t agree with Hirsch’s basic stance that a core curriculum be taught in every U.S. school but he’s got a point. It’s not a new point. Learning has taken a hit heavy hit from the emphasis on standardized tests. The zombifying effect on schools, teachers and kids brought by high stakes testing isn’t pretty.

Even in the best districts, attaining those all-important numbers eliminates opportunities for innovation and time to work with students’ interests. Less stellar districts see their schools under test-heavy siege charged with getting results or getting eliminated.  This drive also shapes the kind of material students see, relentlessly preparing them to reach higher for the Almighty Score while giving scant attention to more complex yet essential skills for higher learning like critical thinking, creativity, initiative, and persistence.

Why this emphasis on testing? We assume policy-makers know what they’re doing. Surely they haven’t been restructuring education based on bare numbers unless they have substantial proven results. Greater competiveness on the world market or at least greater individual success?

Nope.

Here are some eye-opening facts from my book:

National test scores

It’s widely assumed that national test score rankings are vitally important indicators of a country’s future. To improve those rankings, national core standards are imposed with more frequent assessments to determine student achievement (meaning more testing).

Do test scores actually make a difference to a nation’s future?

Results from international mathematics and science tests from a fifty-year period were compared to future economic competitiveness by those countries in a study by Christopher H. Tienken. Across all indicators he could find minimal evidence that students’ high test scores produce value for their countries. He concluded that higher student test scores were unrelated to any factors consistently predictive of a developed country’s growth and competitiveness.

In another such analysis, Keith Baker, a former researcher for the U.S. Department of Education, examined achievement studies across the world to see if they reflected the success of participating nations. Using numerous comparisons, including national wealth, degree of democracy, economic growth and even happiness, Baker found no association between test scores and the success of advanced countries. Merely average test scores were correlated with successful nations while top test scores were not. Baker explains, “In short, the higher a nation’s test score 40 years ago, the worse its economic performance . . .”   He goes on to speculate whether testing [or forms of education emphasizing testing] itself may be damaging to a nation’s future.

Individual test scores

What about individual success?

Educational reformer Alfie Kohn explains, “Research has repeatedly classified kids on the basis of whether they tend to be deep or shallow thinkers, and, for elementary, middle, and high school students, a positive correlation has been found between shallow thinking and how well kids do on standardized tests. So an individual student’s high test scores are not usually a good sign.”

Why do we push standardized tests if it has been demonstrated that the results are counterproductive? Well, we’ve been told that this is the price children must pay in order to achieve success. This is profound evidence of societal shallow thinking, because the evidence doesn’t stack up.

Research shows that high test scores in school don’t correlate with later accomplishments in adulthood such as career advancement or social leadership.

We’ve known this for a long time. Back in 1985, the research seeking to link academic success with later success was examined. It was appropriated titled “Do grades and tests predict adult accomplishment?

The conclusion?

No.

The criteria for academic success isn’t a direct line to lifetime success. Studies show that grades and test scores do not necessarily correlate to later accomplishments in such areas as social leadership, the arts, or the sciences. Grades and tests only do a good job at predicting how well youth will do in subsequent academic grades and tests. They are not good predictors of success in real-life problem solving or career advancement.

Kids certainly cheat because they haven’t prepared for the test, but they also may cheat because they simply see the whole testing game as a farce.  A survey of 43,000 high school students showed that 59 percent admitted cheating on a test during the past year.

What can we do?

Some of us homeschool.

Some parents submit compelling letters telling schools they will not permit their children to be tested, part of a larger opt out movement.

Some are heartened by the ever-growing list of four year colleges that don’t require the ACT or SAT for admission.

Do you think high-stakes testing cheats our kids?