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*.

Thank you. Some really interesting concepts here that I will be exploring further.

We homeschool and my wife and I were discussing this issue just last night. We keep asking our son to keep up with the school curriculum in maths just ‘because’.

We have not ourselves used almost any of the concepts taught, ever – despite me being an accountant and programmer and my wife an audiologist.

We certainly have never needed to prove the similarity of two triangles, which is what we were explaining last night. So as far as we could see, he was learning it to pass the required module test and then could comfortably forget it until he needs to force his child to learn it for their module test. Seems pretty pointless.

I get the importance of maths and do use advanced maths sometimes for business analysis (regression analysis mainly), but each time I do I just learn what I need to know in 15-20 minutes and it is almost never difficult. As the Sudbury experience showed, it is just the process used in school that is difficult as it has no value at all.

So much food for thought here – thanks again.

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Interestingly two different surveys of grown unschoolers showed a much higher number of them work in STEM careers than schooled adults. The samples were small but intriguing.

http://www.homeschoolretrospective.com/what-are-they-doing-now-the-answers-2/

http://blogs.kqed.org/mindshift/2014/09/how-do-unschoolers-turn-out/

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Really interesting. Not surprising though at all as in my 20+ years of owning a software firm I have not once asked to see someones maths scores, as they are simply not relevant at all to programming – admittedly other STEM jobs require more maths, but even then it is usually stats, which is simple to learn if you are interested.

For me, their passion for the job completely trumps everything. Without the love of whatever bit of STEM interests them it is impossible to keep up with the constant changes, but with that passion people can learn anything.

I have never, and never will, employ someone as a programmer who tells me they studied IT because they heard the work pays well and who does’t love technology.

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I was wondering if you could talk a little bit about a child who is being homeschooled and wants to go into the sciences as a career – how would you suggest structuring math, since it is likely to be important to his college education? This is a gifted child who has a wonderful memory and very capable problem-solving skills, but is resistant to learning things that aren’t important to him.

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There’s so much to talk about this topic that a whole book couldn’t cover it!

I understand your concerns. I hope the emphasis in this post isn’t on at all about ditching math. Instead I hope it shows how important it is to fling our concept of math wide open by using it naturally and eagerly in our lives.

I don’t know how old your child is, but I’ll give you examples of how more natural math worked for two of my kids, both boys. Most weeks of their homeschooling lives they did, indeed, do a small amount of structured math although nothing close to the hours and hours they’d be expected to put in on a school or homeschool math curriculum. Sometimes they did worksheets I made up with silly problems related to their lives, sometime various types of math workbooks (I kept trying different approaches), sometimes hands-on challenges that had math hidden in them. Later on in their teen years one of them used an online math program Aleks.com, then took a few community college courses for a boost. Overall, they spent very little time on math. I worried that I was not preparing them well for a math-related future.

But most of their math smarts came from hands-on interests, much of it of their own devising. They used math to make chores on our house and farm easier, to do all sorts of projects like audio sound system repair and woodworking, to alter recipes, to win arguments, to design and build their own model planes and rockets, to stay current with their science/tech interests, to restore a vintage car, to keep up with experts on forums, to come up with projects for their science club, and so on.

One of these boys had attended school up to third grade, where he was diagnosed with dyscalculia—-basically a math-related learning disability. I can attest, he simply could not memorize math facts and had to recalculate the problem each time. All forms of math instruction at school as well as at home went slowly, with much resistance. I despaired of his ever achieving any real success in any field requiring math, hoping his many math-y experiences were some help. He went on to college, wowing admissions with his many independent (entirely self-motivated projects), and did wonderfully there. His only problem in college was being interested in too many subject areas! He now works full-time in a field that requires him to calculate loads and stresses, estimate and order materials, design structures, and efficiently run a team of co-workers. He’s very happy, loves the math in his work, and is planning to go back to college to study physics.

The other boy homeschooled from 5 years old on. He was good at math but had so many other competing interests that he just didn’t spend much time on it. When he announced he wanted to become an engineer I was distraught, sure that he was hugely behind in math. I knew he’d be in college with kids who’d been taking upper level AP math for years. He’s now a senior majoring in mechanical engineering and an all A student. His professors tell him all the time how far ahead he is from so much real world math experience. He’s on course to graduate into a high-paying career.

These are only two examples, I realize, but it’s always illuminating to talk to parents of grown homeschoolers and grown homeschoolers themselves. The now-adult homeschooled kids we’ve known for years are flourishing in a wide range of careers, no matter if they unschooled or homeschooled. I sum up the crux of our experience in the post with this sentence, “Hands-on math experience and an understanding of oneself as capable of finding answers— these are the portals to enjoying and understanding computational math.”

For more posts about how we’ve approached math and related subjects, here are some links.

Help Kids Learn about Business and Finance: 60 Resources https://lauragraceweldon.com/2014/03/06/help-kids-learn-about-business-finance-60-resources/

Getting Science on Everything

https://lauragraceweldon.com/2013/06/19/getting-science-on-everything/

Benefits of Special Interests Groups

https://lauragraceweldon.com/2012/05/08/benefits-of-special-interest-groups/

We Don’t Need No Age Segregation

https://lauragraceweldon.com/2012/11/07/we-dont-need-no-age-segregation/

Successful Teen Homeschooling

https://lauragraceweldon.com/2012/05/24/successful-teen-homeschooling-two-vital-factors/

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First, I want to say an enormous THANK YOU for taking the time to write back. I read Free Range Learning two years ago, and it completely changed my outlook on how to ‘do’ homeschool, giving me the confidence to try something different. The results have been *amazing*!

The one subject that’s always made me nervous is math, because it just seems like trying to learn it away from paper won’t work, specifically when it comes to things like regrouping, long division, times tables, et cetera. I hate making decisions out of fear, but that’s definitely what’s been happening.

On the flip side, I’m keenly aware that trying to force-teach doesn’t work – the material doesn’t click, because there’s little-to-no interest.

I’ve had to come up with inventive ideas, like you talked about above. Right now we’re doing 10-minute math 3x per week (the timer tells him exactly how long he’ll have to work, and he likes that). He also has a daily “math vitamin” that I write based on his current interests.

He enjoys doing problems out loud, and he’s really good at it, probably because it feels like a game. But I’m never sure it’s ‘enough’, especially because he’s talking about careers like physics, chemistry, and astronomy, and I’ve seen the math that goes into those. I don’t see how we’ll be able to manage algebra and above without a pencil and paper, and that’s where the nerves kick in.

All that to say this: what you wrote about your son who is studying mechanical engineering really bolstered me. I’ve read your posts on math for a while now, especially the ones about the class at the democratic school that was able to cover all the basics in short amount of time. Maybe that’s what I need to remember: that when it’s needed, he’ll find a way to learn it.

It’s tough to loosen the need for control, but I’m going to repeat the mantra “Why not?” until it sticks.

Thank you again so much for the helpful advice and links!

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Daily math vitamin—love it!

Here’s one more quick mention of what worked for us (just to broaden the range of math learning possibilities). Another of my homeschooled boys went far beyond my math ability quite early. When he was 15 we hired a high school math teacher to moonlight as his tutor. The tutor was thrilled to have an enthusiastic and friendly student. He sat at the kitchen table with my son once a week to work together. They rapidly accelerated through calculus and trig, not only with plenty of real teaching backed up with homework but also with brain twisters, games, math hacks, and lots of laughter. The teacher loved the freedom to bring all sorts of new ideas to their lessons, the freedom he didn’t have at school. Each week this gentleman stayed well past the time we paid him to stay, This continued for a little over a year. The tutor told us it was the best experience of his teaching career.

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That is a TERRIFIC idea! It takes some of the pressure off trying to get him into it at an earlier age and instead wait until he’s ready and willing.

Thank you! 🙂

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As a homeschooling mother of three, I was drawn to this post. Thank you! I will be sharing this with my husband. 🙂

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Great, informative article, thank you.

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Thank you so much for this piece and your last one. My children are all teenagers now and this pretty much jives with the way we have handled math, but often more out of a sense of loss on my part. Several times I have thought that maybe I was failing them by not teaching them “math on paper” which is what they’ve come to call it. I’ve got one in college now and the math has been the most stressful part for her. So much to think about. Wonderful articles. I will bookmark and return. I am looking forward to the next installment.

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Much of what we’re talking about here has to do with readiness and interests, but it also has to do with a society that overvalues test-able math skills without recognizing the value of applied math skills.

We did “math on paper” learning as homeschoolers, although avoiding much of the busywork imposed by school. If your daughter is struggling with math in college, I hope she takes heart knowing that some of our most accomplished mathematicians, physicists, and engineers came to math rather late.

http://mathoverflow.net/questions/3591/mathematicians-who-were-late-learners-list

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