One day more than a year ago, an 8-year-old named Andrew told his parents he wanted to learn to do long division. His dad, Tim Sylvester, looked up a YouTube video explaining the basic steps and began working with him through simple problems on a whiteboard in their house. A half-hour later, the child was dividing two-digit numbers into 20-digit numbers.

“He was ecstatic, running around,” said Sylvester, describing the moment as a “math high.” Several months later, when Andrew’s third-grade class at a public school in Santa Cruz, California, began tackling long division, the boy had it down cold and read a book on his own during the lessons.

Sylvester and his wife, Barbara Meister, wanted to keep Andrew, who is profoundly gifted with numbers, engaged and learning. So last fall, the father launched the Santa Cruz Math Circle: a six-week enrichment program that brought in mathematicians for two hours on Sunday afternoons to explore and discuss fascinating numerical puzzles and concepts with fourth- through eighth-graders. Math circles are an Eastern European and Russian tradition that spread across the U.S. in the last two decades. The goal wasn’t to plod through the standard formulas in preparation for a test, but to provide a stimulating and interactive environment in which Andrew and other kids with a knack for numbers could experience “math that was challenging and fun at the same time,” said Sylvester, a software engineer who works in Silicon Valley.

The new course in Santa Cruz has been “a huge success,” said Evelyn Strauss, whose 10-year-old son was an avid participant. “There’s a wide range of kids who are really enjoying the program.” Most students go through elementary school thinking that mathematics is only about adding and subtracting and arithmetic, “but there’s this whole world of math that’s not typically covered in school and that’s really interesting,” she said. Math circle reveals that world to kids.

Creating a math circle takes a lot of energy and planning — from finding dynamic instructors to booking classroom space — but it can be well worth the effort if no similar enrichment opportunities are available nearby. Parents might consider forming a circle when their children come home from school saying they’re bored with the level of mathematics that’s being taught, Sylvester said. “If your kid comes back and wants more challenging math, or if they love math or they love puzzles, then start one.”

One Sunday last November, the fledgling program was underway at a community center, with around 19 boys and seven girls mulling over a game called Conway’s checkers. The visiting instructor that day was Zvezdelina Stankova, a Mills College professor and director of the Berkeley Math Circle. Each student was given a sheet of paper with a grid of squares — divided in half by a thick line — and a pile of pennies to serve as checkers. Beginning at one end of the board, and given particular rules for checker-jumping and a theoretically endless supply of coins, Stankova asked: How far could the coins be advanced?

Students experimented with jumping the checkers to the third row on the other side of the line, and then the fourth, and Stankova demonstrated the winning solutions. What about the fifth row? Many kids became so engrossed, chattering among themselves, they didn’t want to stop to hear the next part of the lesson. “No touching the coins!” Stankova finally admonished. “It’s impossible to reach the fifth row,” she said, briefly explaining that the reason had to do with the quadratic equation and the Golden Ratio.

Stankova grew up in Bulgaria, where it was common for children to attend circles in math, physics, chemistry and poetry. “Kids went to the math circle because they loved what was happening in class and they wanted more of it,” she said. By contrast, in the U.S., “most of the kids come to the math circle because they don’t like what they see in school and they’re looking for something else.”

**A COMFORTABLE PLACE FOR MATH GEEKS**

About three years ago, Sylvester wanted to take Andrew to a math circle after hearing of the programs at Berkeley as well as Stanford, but those courses are popular and hard to get into. He and his wife have fostered their son’s passion for numbers in various ways, including online courses and a summer camp in mathematics, as well as math competitions. Although UC Santa Cruz was offering a monthly math circle at one point, that course was put on hold when the professor who ran it went on sabbatical.

Last summer, Sylvester decided to start a circle through the X Academy, a nonprofit that he founded to offer enrichment activities, and he reached out to math professor Paul Zeitz, co-founder of the San Francisco Math Circle. Zeitz promptly volunteered to give an introductory math circle session in Santa Cruz and connected him with Stankova and other willing expert instructors.“I thought I’d be lucky to get five kids in a room on a Sunday afternoon,” Sylvester said. To his surprise, 50 students and their parents signed up to attend the free kickoff session by Zeitz. Through an application process, the Santa Cruz Math Circle ended up with around 25 regular attendees for the rest of the fall course; registration cost $75, but scholarships were available.

Sylvester ran the course with help from X Academy board members (including Meister) and other parent volunteers, who assisted with promotion and setup. Meeting the needs of students from fourth through eighth grades in a single class proved too wide of a spread in ability and maturity to manage, so the software engineer brought in an extra instructor and split the students into two tracks; but younger kids such as Andrew who were ready for the math of the upper-level group could move up.

For many attendees, being in a place where they could dive into math and interact with other number-loving geeks was a welcome shift from the standard school environment, where the subject isn’t exactly popular. The circle “creates this community where it’s safe to come and talk about math,” said teacher Nicholas Bugayong of Rolling Hills Middle School in Watsonville, who volunteered to drive four eighth-grade math students, all from Latino families, to the Sunday sessions. Some of these pupils were otherwise unable to make the 17-mile trek — their parents couldn’t bring them — or to afford the course without X Academy scholarships.

“Other people don’t like math and they say it’s a waste of time, but I think it’s interesting to learn things that you don’t learn in school,” said Rolling Hills student Monica Alvarez, 13. Listening to the other students as they volunteered different ideas and strategies for tackling challenging problems “helps you to learn other ways to solve a puzzle,” she added.

As for Andrew Sylvester, now age 10, the program prompted some more “math highs,” such as when he got into thinking about the Conway’s checkers conundrum. And the lessons spurred lively discussions. During one pizza break, he stayed behind in the classroom with two other boys who are also profoundly gifted at math. “They were bent over a piece of paper scribbling, in animated conversation,” recalled Strauss, whose son was part of the confab.

“I just like learning the math and arguing with my friends about math stuff — for example, stuff like if one over infinity equals zero,” said 10-year-old Olin Ottemann-Strauss.

**MATH CIRCLE STARTUP LESSONS**

The Santa Cruz Math Circle resumes this spring. A major challenge will be continuing to arrange for high-quality instructors, Tim Sylvester said. While ideal teachers include math professors or scientists or engineers with doctorates in math, not all such experts have the experience or classroom management skills to instruct young pupils in a dynamic way. But Sylvester hopes to eventually join forces with UC Santa Cruz’s math circle.

Even if there is no nearby college to collaborate with, anyone interested in launching a math circle program can find helpful, step-by-step resources on the National Association of Math Circles website, including lesson plans from the book, *Circle in a Box*. Teachers at some Bay Area schools have created their own programs, Sylvester noted, such as Nueva School in Hillsborough.

Sustaining a circle does require adequate financing, which can be an ongoing challenge if registration fees are to remain affordable. The Santa Cruz program recently got a donation from Cisco Systems, where Sylvester used to work, and parents can make contributions to X Academy through its website. The academy has also applied for a seed grant from the Berkeley-based nonprofit Mathematical Sciences Research Institute, which has funded many math circles over the years. But the institute is currently reviewing that grant program and has ended its support of the San Francisco Math Circle.

While many students often question why they need to learn algebra or calculus that they might not use later in life, Sylvester sees the ability to figure out tough math problems as being an essential life skill.

“One of the reasons you learn math is for the thought process and the problem-solving process,” he said. “That’s why you do it.”

*Correction: An earlier version of this story misidentified Barbara Meister as Barbara Sylvester. It also stated that Andrew Sylvester’s third grade lessons in long division were at a private school, which is incorrect. He was attending a public school. *

One day, Adam Holman decided he was fed up with trying to cram knowledge into the brains of the high school students he taught. They weren’t grasping the physics he was teaching at the level he knew they were capable of, so he decided to change up his teaching style. It wasn’t that his students didn’t care about achieving — he taught at high performing, affluent schools where students knew they needed high grades to get into good colleges. They argued for every point to make sure their grades were as high as possible, but were they learning?

“I felt I had to remove all the barriers I could on my end before I could ask my kids to meet me halfway,” Holman said. The first thing he did was move to standards-based grading. He told his students to show him they’d learned the material, it didn’t matter how long it took them.

“The kids realized this made sense,” Holman said. He taught physics and math at Anderson High School in Austin, before moving on to become a vice-principal. His students were mostly well-off, high achievers, and they knew how to play the game to get the grades they needed. But Holman found when he changed the grading policy, students worried about grades less and focused more on working together to understand the material.

“It turned my students into classmates and collaborators because I didn’t have a system in place to deny the collaboration,” Holman said. His students stopped copying homework. There was no curve that guaranteed some kids would be at the bottom. Instead, the class moved at its regular pace, but if a student persisted at a topic until they could show they understood it, Holman would give them credit. “It turned the kids on my side,” Holman said. “I was there to help them learn.”

**BUILDING TRUST**

Holman didn’t just change his grading policies. He also changed his teaching style to focus on inquiry, good questions and independent discovery. Starting off, he knew juniors and seniors weren’t used to learning that way, so first he had to build trust with them so they’d understand why he was asking so much of them.

At the start of each class period Holman and his students did icebreakers and read and discussed articles about how human brains learn best. Holman knew he was asking students to be vulnerable with one another–to share their misperceptions about math and physics–and so he spent precious class time working to make sure students trusted one another and him.

The class read Timothy Slater’s article, “When Is a Good Day Teaching a Bad Thing?” which discusses the unspoken contract that can exist between teachers and students by which a teacher will pass a student as long as he or she doesn’t make trouble. Students recognized their own experience of education in the article. “It wasn’t meant to be a bash on teachers, but just to say we are aware that teaching is really complex,” Holman said. “It’s really difficult and sometimes we don’t know how to handle kids.”

Holman also asked students to read “Sermons For Grumpy Campers,” by Richard Felder, a graduate level professor who never lectured. In it, Felder describes his students grumbling that they hated group work and that it was his job to teach them, not the other way around. Holman’s students said the complaints sounded like they came from kindergarteners or themselves and were amazed to find out the complainers were graduate level engineering students.

Talking about these issues openly validated the inevitable complaints of students and helped them buy into the new approach. If an article was a little harder, Holman would use it as differentiated instruction, asking his best readers to take it on and summarize it for the class.

“It wasn’t perfect and it didn’t turn my kids into all physics majors, but for the kids who were on the border, it made a difference,” Holman said. Discussing their learning with them, switching grading policies and assigning more inquiry-based, hands on lessons all helped Holman’s students feel he trusted and respected them. And they rose to the challenge. “I think the kids were just waiting to be let loose and to be treated like adults,” Holman said.

**STUDENT RESPONSES**

Most of the students responded well to the new teaching style, Holman said, but he was most touched by his struggling math class. “I saw that my kids had been told they were stupid and failures, but I saw so much potential in them,” Holman said. They’ve never been given the time to master a concept through multiple tries. So when Holman opened his door to help them after school and during lunch for as long as it took, many seized the opportunity.

Holman remember one struggling math student, Isabel, particularly well. She was taking algebra, convinced she was terrible at math. But when the grading policy was changed and she had a little more time to work on units that were difficult for her, she became a top student in the class. “She said, ‘for the first time in my life I’m trying to learn everything instead of just get a 70 [percent],’” Holman said.

“Students clearly learned in Mr. Holman’s class, and he never pushed fear,” wrote a former student, Kate Nunke, in an email. She described the rest of her high school experience as one long fear fest: “Fear of not getting into college, fear of not passing, fear of disappointing parents, fear of looking like a fool in front of your peers,” the list goes on. But Nunke says Holman’s teaching style jolted students into thinking about their learning in a new way.

“I think many students didn’t realize that they could learn without a textbook or without step by step instruction,” Nunke wrote. “At times I felt that Mr. Holman’s physics class was the hardest class ever because I didn’t get a step-by-step instruction. We are used to being handed the answer, thus not necessarily learning, just being told.”

Nunke said she’s been thinking a lot about Holman’s approach now that she has graduated and is taking a gap year in which she spent a semester at an outdoor education school focused entirely on experiential learning. “A lot of the teaching that Mr. Holman did, now that I think back to it, was teaching his students how to ask questions and investigate by themselves,” she wrote.

**TEACHERS RESIST WHAT WORKS**

Despite his success, Holman has had a hard time convincing other teachers to try some of his more progressive approaches. He became a vice-principal to spread and support the instructional practices he believes work, modeling lessons and pushing teachers to step out of their comfort zone.

“We know how kids learn; we know what classes should look like, and yet our classes look almost the opposite,” Holman said. He says there’s a particular deficit in math, where teachers and parents expect things to be taught the way they learned them. Not everyone has experienced good math instruction themselves, Holman said, so they can’t even begin to conceptualize a new way of doing it. “Imagine explaining color to someone who has never seen it,” Holman said. “You have to show them, you have to model it.”

But all of these approaches require taking a leap of faith and many teachers don’t feel they have that luxury. Teachers often complain that more progressive approaches like this suck up time and they can’t cover everything in the jam-packed curriculum. These arguments are excuses, Holman said. He said he never covered every single topic in the curriculum, but he did delve deeply into the ones he saw as most important.

**HOLMAN’S READING LIST**

For those interested in building metacognitive moments into the day, here are the articles Holman found to be useful and more or less reading-level appropriate for his high school students.

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**By Anya Kamenetz, NPR**

Little children are big news this week, as the White House holds a summit on early childhood education December 10. The President wants every four year old to go to preschool, but the new Congress is unlikely to foot that bill.

Since last year, more than 30 states have expanded access to preschool. But there’s still a lack of evidence about exactly what kinds of interventions are most effective in those crucial early years.

In New York City, an ambitious, $25 million dollar study is collecting evidence on the best way to raise outcomes for kids in poverty. Their hunch is that it may begin with math.

**Time That Counts**

“One! Two! Three! Four! Five!”

Gayle Conigliaro’s preschool class are jumping as they count, to get the feeling of the numbers into their bodies–a concept called “embodied cognition.”

P.S. 43 is located in Far Rockaway, Queens, just steps from the ocean. The area is still recovering from Hurricane Sandy. But now it’s been chosen as one of 69 high-poverty sites around New York City for a research study to test whether stronger math teaching can make all the difference for young kids. The study is funded by the Robin Hood Foundation, which is dedicated to ending poverty in New York. Pamela Morris, with research group MDRC, is lead investigator.

“MDRC and the Robin Hood Foundation developed a partnership really with a broad goal,” she says, “Which is, they want to change the trajectories of low income children. And to do so by focusing on preschool.”

There’s plenty of evidence on the long-term importance of preschool. But why math? Morris says that a 2013 study by Greg Duncan, at the University of California, Irvine, showed that math knowledge at the beginning of elementary school was the single most powerful predictor determining whether a student would graduate from high school and attend college. “We think math might be sort of a lever to improve outcomes for kids longer term,” Morris says.But there’s a real lack of math learning in pre-K. In one study, in fact, just 58 seconds out of a five-hour preschool day was spent on math activities. Part of the problem, says Doug Clements, at the University of Denver, is that “Most teachers, of course, have been through our United States mathematics education, so they tend to think of math as just skills. They tend to think of it as a quiet activity.”

Clements is the creator of Building Blocks, the math curriculum being tested in this new study. Building Blocks is designed to be just the opposite: engaging, exciting, and loud. “We want kids running around the classroom and bumping into mathematics at every turn.”

At P.S. 43, math games, toys, and activities are woven through the entire day. At transition time, the teacher asks the students to line up and touches their heads with the “counting wand.” At circle time, fittingly, the children talk about shapes. Just a few months into the school year, they observe correctly that a geometric shape must be a “closed figure” and that a square is “a special rectangle.”

“How do you know it’s a circle?” asks the teacher. “Because it goes round and round,” says one girl with a bear barrette in her hair.

When Ms. Conigliaro asks, “how do you know,” she’s asking the kids to think about their own thinking. That’s a skill called metacognition. Explaining your reasoning out loud also develops verbal ability.

At choice time, besides the play-dough and pattern blocks, there are computer games matched to Building Blocks that keep track of each student’s progress. And two children play a game called Number Match (“Is three more than two? How do you know?”) as a teacher watches. The teacher is keeping notes of each child’s level of understanding. The idea of developmental paths, or “trajectories of understanding,” is a core concept in Building Blocks.

“There are reliable levels of thinking through which kids pass on their way to achieving a certain understanding in mathematics,” Clements says. For example, children go from simply chanting “onetwothreefourfive,” to separating out each number word, to associating a number word with a given amount, to knowing that when you stop counting, the last number tells you “how many.”

Also in the classroom is a coach from Bank Street College of Education, who comes every other week to help the teacher put Building Blocks into practice. This is important to the study design. The coaches ensure that the curriculum is being implemented. Pamela Morris says, “Often we ask teachers what curriculum they’re delivering and we find it’s a book on their bookshelf collecting dust.”

The study will follow up with these students and a control group all the way through the third grade. They’ll be directly assessing their math and reading abilities and looking at their grades and test scores later on. Morris is curious whether working on math will enhance the children’s ability to self-regulate, inhibit impulses, pay attention appropriately and hold important concepts in working memory. This is a group of skills known as executive functioning. For example, if the teacher says “clap and count to five,” will you be able to stop clapping before you get to six?

But Conigliaro, a 24-year veteran teacher, is already convinced of the value of this curriculum.

“I just feel like the aha moment. This is what teaching should be. Where’s the literacy program?” she says. “We would just like it to be a research based program so we can give our kids the best.” She says the kids’ progress amazes her every day.

Copyright 2014 NPR. To see more, visit http://www.npr.org/.

]]>A good portion of the adult population hates math, and a lot of people believe they aren’t good at it so they avoid it completely. Those perceptions often come from their experiences learning math in school, which may not have been positive. In her Atlantic article Jessica Lahey writes about a Cornell professor who takes special pride in teaching non-math majors to appreciate numbers. He does it with an inquiry-based, hands-on approach that would likely work for kids learning math for the first time too. Lahey writes:

“Twelve years of compulsory education in mathematics leaves us with a populace that is proud to announce they cannot balance their checkbook, when they would never share that they were illiterate. What we are doing — the way we are doing it — results in an enormous sector of the population that hates mathematics. The current system disenfranchises so many students.”

]]>Many of us tend to align ourselves with either numbers or words. We’re either math brains or we’re reading brains.

In college, my fellow English majors joked about how none of us could long-divide to save our lives, while our friends in engineering groaned about the fact that Lit 101 was a graduation requirement.

But it turns out that about half the genes that influence a child’s math ability also seem to influence reading ability, according to a study published in the journal *Nature Communications*.

“You’d think that cognitively what’s going on with math and reading is very different,” says Robert Plomin, a behavioral geneticist at Kings College London, and one of the authors of the study. “Actually, people who are good at reading, you can bet, are pretty good at math too.”

The researchers looked at 2,800 pairs of 12-year-old British twins who were part of the larger Twins Early Development Study. Some pairs were very nearly genetically identical; the other pairs were fraternal twins, meaning they are the same age and shared a quite similar early environment, but are no more genetically similar than other siblings.

The scientists assessed each child’s math and reading skills based on standardized tests. To gauge how genes influenced the students’ aptitude, the researchers compared the test results of twin siblings as well as the results of unrelated children.

The researchers also analyzed the participants’ DNA, in hopes of turning up a particular gene or set of genes shared by people with high math or reading ability — genes that were, perhaps, missing in people with low abilities. (Some earlier, smaller studies had suggested such highly influential gene variants might exist). But no particular gene or sets of genes emerged. That may be because a lot — maybe thousands — of genes may be involved in helping to shape these abilities, Plomin says.

What the study did find was that children’s reading ability and math ability seem to be related — and much of that relationship can be explained by genetics.

The research also showed that genes can’t explain everything about our abilities, Plomin says. “These genetic propensities are like little nudges,” he says. Slight variations in your genes may nudge you to read more for pleasure. “And that can snowball,” Plomin says.

These kids who like reading may spend more time at the library or may ask their parents to buy them more books — and all of that practice reading will push their skills even further.

Other kids may find reading to be a bit harder due to genetics, Plomin says. “It’s not that the child just isn’t motivated, or that he’s just not trying hard enough.” But with some extra encouragement and support, these children can become good readers as well.

Environmental factors may also explain why, among genetically identical twins, one may prefer math while the other prefers reading, Plomin says. One twin may end up with a really good math teacher, while the other doesn’t. Or one may fall ill, and that may set her back.

Right now, we don’t have all the answers, Plomin says. “I wish I knew what some of the genes are,” he says. That would allow scientists to learn more about how we each learn best.

“What’s going to be needed is very large samples of people to be able to isolate these genes,” says Douglas Detterman, an emeritus professor of psychology at Case Western Reserve University and editor of the journal *Intelligence*. Detterman, who wasn’t involved in this study, says scientists would likely have to look at the DNA of millions of people in order to start figuring out which genes affect our academic aptitudes.

It’s a daunting task, he says, “but I think it’ll happen faster than we expect.” As we learn more about the influence of genetics on learning, we’ll be able to more easily figure out which learning environment works best for each child.

Here, teachers are a bit like farmers, Detterman says. And children are a bit like corn. “You have corn plants that do well in certain environments, and don’t in others. And the farmer’s job is to get the corn plants into the right soil.”

Educators have been talking about changing the traditional way of teaching math for a long time, but nothing seems to change. Elizabeth Green’s New York Times Magazine article digs into why it has been so hard for U.S schools to effectively implement changes to math pedagogy, and just how far American students have fallen behind as a result. A lot of it comes down to ensuring teachers are comfortable with the new methods, she writes:

“In fact, efforts to introduce a better way of teaching math stretch back to the 1800s. The story is the same every time: a big, excited push, followed by mass confusion and then a return to conventional practices. The trouble always starts when teachers are told to put innovative ideas into practice without much guidance on how to do it. In the hands of unprepared teachers, the reforms turn to nonsense, perplexing students more than helping them.”

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