Building on this background research, Stanford education researchers tested a teaching strategy explicitly focused on using symmetry to teach integers to fourth-graders. They wanted to see if recruiting the visual symmetry parts of the brain would improve students’ facility and understanding of the concept. Their findings, published in “Cognition and Instruction” in May, indicate that teaching with symmetry could have a big impact not just on students’ understanding of integers, but also on more advanced concepts that go well beyond the scope of instruction as well.

“Finding a fundamental discovery in neuroscience about how the brain is processing these numbers led us to realize the instruction needs to change,” said Daniel Schwartz, director of Stanford’s AAALab and one of the authors of the study.

When people are asked to find the distance between a negative integer and a positive integer oriented in a blank space, they are able to do so much more quickly when the numbers are spaced around zero. For example, people can more quickly calculate the difference between -2 and 3 than they can -5 and 3. Furthermore, an fMRI scan of the brain while this calculation is taking place reveals that visual symmetry is in play.

In math instruction “nobody had been focusing on the significance of zero, which is the reflection point,” Schwartz said. There are two common instructional methods used to teach integers in current curricula: the first uses distances on a number line, asking students to jump forward and backward between integers. The other uses the idea of cancellation, where students are taught that positive and negative numbers represent opposite quantities that cancel each other out.

The AAALab researchers split students into three groups and used these two accepted instructional models to teach integers to two of the groups. The third group used a custom-built manipulative that bends at zero so students could visually see the symmetry. All the groups played with hands-on manipulatives for a while to get the hang of the idea and then practiced problems using the condition they had been taught on computer games to increase their facility. Schwartz called this practice time “increasing the dosage.” At the end, researchers gave them all the same test.

Students in all three conditions performed well on the straightforward symbolic problems on the test, the core of the material that had been taught. Schwartz said that’s good because it shows that all the instruction given was good, and variables like poor teaching did not influence the results.

“The difference showed up on bigger problems, harder problems, where you really had to start organizing the numbers in your head,” Schwartz said. “You couldn’t just do fact retrieval.”

For example, one question asked: In a Tug of War game, the red team has five people and the blue team has three people. The blue team thought that was unfair, so they added four more players to their team. How many players does the red team now need to make the teams even? Another question asked about negative and positive fractions. Even without understanding fractions, students correctly found the mirrored version.

“On these problems where you have to learn new things, suddenly you see that symmetry stuff showing up and it’s basically because kids had a much better structure to organize new ideas and new information,” Schwartz explained.

And, even better in his mind, the test included a few questions that would be hard to solve using the symmetry method the test group learned. As Schwartz hoped, the students who were using the symmetry methods struggled with those problems. He said that’s good because it proves that the intervention improved their knowledge in a specific way, not just generally. The prowess the symmetry condition students showed on more difficult tasks and even pre-algebraic tasks was not a halo effect.

“It really showed [the intervention] improved them on many things we really care about and made them worse at something that we don’t care about,” Schwartz said.

The study points to a clear instructional method that could improve student learning not just on simple integer problems, but on many other concepts using symmetry. Students often struggle with the idea of equivalence, but recruiting symmetry could provide the structure they need to grasp this concept that is so central to algebra.

The next step will be for someone to go out and build manipulatives that bend at 0, like the ones Schwartz’s lab fashioned for this experiment. He thinks it’s possible a computer program could simulate this idea as well, but it would need to make sure the student’s eyes were following the hands and focusing on the point of inflection.

]]>But the real question is–can you figure out when Cheryl’s birthday is?

]]>Teachers at the public magnet school Science Leadership Academy use a project-based inquiry model of teaching in an effort to connect all subjects to students’ lives. Examining social justice issues by the numbers has proven to be one strong way teachers can connect student passions to math.

In one project, groups of three or four students were responsible for a written mathematical analysis of their topic, two visual representations of the data, an engaging public service announcement video explaining the data and a list of recommendations for how the issue could be addressed.

“The biggest part of this was finding this information and saying, ‘Now what do I do with it?’” said Zack, a junior at Science Leadership Academy who did this project in his sophomore year.

Zack’s group examined incarceration rates in the United States, with each group member looking at an aspect of the issue, like educational attainment or geographical location of prisoners. As they each researched their own part, they kept a shared Google doc with information they were finding, sharing relevant research with one another when appropriate.

One of the hardest parts of the assignment was taking research and framing it in ways that would be useful for their claims, Zack said. Without that step they couldn’t be sure they were accurately comparing different numbers.

“It turns out that just five states in the South account for 20 percent of the country’s total prisoners,” Zack said, a disproportionately high number for the population of that part of the country. When his research led him to that point, Zack decided to hone in on those five states to make his case.

“Simplifying the information made the info more digestible and applicable,” he said.

Zack’s group also found that the majority of prisoners are high school dropouts, most are living under the poverty line, and 33 percent of the nation’s black males will be incarcerated during their lifetime. The group recommended the government look at issues of bias within the criminal justice system based on this data.

“This was taking concepts we’ve learned, making them more complex or advanced, and seeing real world application for the math,” Zack said. “It’s important for getting students into math because you hear every day kids asking, ‘When will we ever use this?’”

Demonstrating mathematical concepts, like central tendency or odds, and probability, suddenly felt very real to students.

“We wanted to give students a lot of room to have choice,” said math teacher Brad Latimer of a project in his algebra II class.

Students chose social justice-themed issues that interested them and then used research and data analysis to prove how the topic connected to social justice.

Students had to document specific mathematical concepts laid out by their teachers in the assignment. While this was a group project, the assignment clearly states individuals are responsible for analyzing an aspect of the data in terms of central tendency. The assignments reads, “This should include a focus on mean, median, mode, range, quartiles, and IQR (interquartile range), and should also include at least two original percentage-based statements about your data.”

**SOCIAL JUSTICE AND STATISTICS**

Statistics is arguably one of the most useful math disciplines, since citizens encounter numbers proving claims everyday in the news and as justification for various political policies. That also makes studying statistics a powerful vehicle for interdisciplinary learning.

SLA’s statistics and “Science and Society” teachers teamed up to examine the differences between organic and non-organic foods, as well as their cost and prevalence in different parts of the city. Students learned about the science behind different growing methods and how they affect nutritional qualities of food in science, while doing a statistical analysis of food availability in Philadelphia.

Students in groups of three to five visited grocery stores, sometimes of the same brand, in different zip codes throughout Philadelphia. No student in the class could go to the same store.

“It increased our analysis because we had more data from around the city,” explained Adam, an SLA senior.

They had to look for and note the prices of the organic and non-organic versions of different food items their teacher, Mark Miles, had selected. Students took selfies of themselves in the stores to prove that they’d actually gone. Each student was responsible for calculating and interpreting the 5-number summaries and IQRs, and means and standard deviations. They also had to draw and interpret box plots and histograms for all the group’s prices together, non-organic prices, organic prices and the difference between non-organic and organic prices.

“Growing up in Philadelphia, there were a lot more stores with non-organic because it’s cheaper,” Adam said.

He went on to note that after learning about the nutritional value of organics he felt it was unfair that poor people in his city didn’t even have access to products that might improve their health.

Even students who struggled in math were engaged in these social justice-oriented projects because teachers were careful to build in authentic choices that allowed students to investigate an area of interest.

“A lot of kids who struggle in math don’t see the relevance or they say they don’t care about that application,” math teacher Erin Giorgio said.

She found that even the kids who say they hate math grab onto these projects, and the best part is that their research leads them to ask lots of questions as they grapple with their data.

**OTHER SOCIAL JUSTICE IDEAS**

There are lots of ways to make math applicable to problems in the real world, but it takes creativity on the part of teachers and students. Giorgio will sometimes ask her students to analyze attendance data in Philadelphia based on the kind of school students attend: magnet, charter or neighborhood. As they notice things like the fact that attendance is much higher at magnet schools, they start asking more questions and talk about the reasons why that trend holds true.

“The end game is to get kids to recognize that math is important in their life,” Giorgio said.

Other ideas might include using physics to engineer a product that helps someone else or using geometry to investigate architecture in different neighborhoods or acreage of vacant lots.

The power of investigating social justice issues by the numbers lies in high school students’ passion for changing the world. Adolescents are becoming aware of their place within the wider world and many want to have a positive impact on it. Understanding how math will help them do that only makes them more prepared.

]]>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.

- “From Degrading to De-Grading,” by Alfie Kohn
- “Sermons For Grumpy Campers,” Richard Felder
- “When Is a Good Day of Teaching a Bad Thing?,” by Timothy Slater
- “Navigating the Bumpy Road to Student-Centered Instruction,” by Richard Felder
- “Minimizing resistance to inquiry-oriented science instruction: The importance of climate setting,” by Carl J. Wenning
- “Well, Duh!” — Ten Obvious Truths That We Shouldn’t Be Ignoring,” by Alfie Kohn
- “Opinion: Why TEAL Works: 10 Years Ago MIT Had a Physics Problem. TEAL Fixed It,” by Ryan Normandin

]]>

**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/.

]]>“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.”

“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.”

]]>Prodigies in piano or dance can study at schools like Juilliard to develop their musical or performing arts talent. By contrast, nothing like Juilliard exists for children who show great promise at math. But an ambitious experiment will soon change that: In fall 2015, a small, independent school that’s exclusively tailored for math whizzes will open in downtown San Francisco.

Designers of the new, non-profit Proof School intend to provide mathematically gifted youth an intensive and complete education in grades 6-12 that typical schools can’t muster. The pupils will learn advanced areas of math, such as number theory topics that a university math major or graduate student might tackle. They’ll work on math research projects, and engage in community service through math tutoring.

“They’re going to be involved in math in a really different way, a really exciting and dynamic way,” said Sam Vandervelde, who is leaving his math professorship at St. Lawrence University in New York to become the new school’s dean of mathematical sciences.

Proof School will initially open with roughly 45 children in three grades, with plans to grow to around 250 students in a decade. Getting in won’t be easy, but the school’s mission is to serve the needs of “math kids” in the Bay Area — ranging from high-IQ wunderkind types to students who participate in math competitions or math circles, to children who love to play with numbers. “What we want is kids who are passionate about math,” said Paul Zeitz, school co-founder and chair of mathematics at the University of San Francisco.

The new school takes its inspiration from math circles, an Eastern European and Russian tradition that spread to the U.S. starting in the 1990s. These weekly extracurricular clubs bring youngsters together with a mathematician who guides them in exploring numerical ideas and concepts in depth. It’s often a highly interactive conversation, with the kids avidly chiming in with questions and thoughts.

For kids who live and breathe for numbers, the experience can be transformative, as Ian Brown of Marin County, Calif., can attest. In 2011, he began taking his 10-year-old son, Nico, to local math circles. Nico hadn’t been happy or thriving in his public elementary school, because “he wasn’t finding kids in his classes who understood what he was going on about when he was talking about higher mathematics,” Brown said. But math circle changed everything. “Not only did the lights go on, but the heart went on,” he said.

About a year later, Nico joined an advanced, invitation-only math circle for a half dozen students that was led by Zeitz. “Here they all are, for two hours once a week, joyful, joyful, joyful,” Brown recalled. One day in January 2013, as he watched the group animatedly discussing how many ways there are to color a cube with two colors, he turned to another student’s father, Dennis Leary, and marveled: “Look at these guys, they’re thrilled to be working together. Why don’t we do this all day long — and every day?”

Brown wanted to build a school for kids like his son “that they feel is really meant for them.” One conversation led to another and to the birth of Proof School, with him, Leary, and Zeitz as co-founders. To jumpstart it, Brown left his job as a language-arts teacher and dean at a private school for gifted and talented youth where his son, now 13, currently attends seventh grade.

While San Francisco has several high-caliber schools, including Lowell High School, it lacks specialized science schools such as Stuyvesant High School in New York City or the North Carolina School of Science and Mathematics in Durham. But Proof School won’t be like any school out there, anywhere, Zeitz said. Not only will its student body be different — they’ll all have exceptional math ability — but so will its teachers.

At a traditional school, a teacher in a top-notch math classroom might take students on the intellectual equivalent of a strenuous hike that brings them to top of the hill. But as Zeitz put it, “what they don’t realize is that they’re in this incredible mountain range, which they can’t see because their teacher doesn’t know how to get them to put on a hang glider and jump off the cliff and see the entire topography at once.” Proof School teachers will ideally have math Ph.D.s and the deep expertise to do that, he said.

As in math-circle style, the curriculum will emphasize working on and communicating about interesting math problems. Because one of Proof School’s guiding principles or “axioms” is not to waste their pupils’ time, the kids will be spared the unchallenging busy work or mind-numbing exercises that are common in standard schools, Zeitz said.

Every afternoon, students will spend two-and-a-half to three hours learning mathematical sciences, including computer science. Following an unconventional block curriculum structure, the academic year will be broken into six blocks of math instruction that each immerse the entire school in a single topic (such as problem solving or algebra) for five weeks straight. For each topic, kids will be placed into 10 to 12 different tiers by their skill level, Vandervelde said, which allows a lot of flexibility in meeting their individual needs.

“We’ll sort kids into groups based on what they’re ready for,” he said, not by age or grade. Some off-the-charts precocious students will be able to take on very advanced problems at the level of the U.S.A. Mathematical Olympiad, and “we’re going to be ready for them too,” said Vandervelde, who, like Zeitz, competed in the International Mathematical Olympiad as a teenager. “We want to develop and nurture every one of those kids and bring them along as far as they are capable of going.”

Recruiting girls to the school is a high priority, Zeitz said, noting that many young girls are enthusiastic about math but often drop out in their interest between sixth and ninth grades. “We would like to fight that trend as much as possible,” he said.

Beyond numbers, the school will offer a full education, with non-math courses in English, history, languages, and science all scheduled in the mornings in a traditional grade-level manner. Proof School’s teaching style will also draw upon blended learning methods that make use of technology in the classroom, as well as inquiry-based learning practices. Because classroom facility space will initially be limited, the founders plan to tap nearby educational resources: Students might go to the Exploratorium for hands-on science learning, to the Museum of the African Diaspora for history, and to TechShop for 21st century shop class.

Since some math kids are not exactly social butterflies when it comes to people skills, the school’s guiding axioms also make a point of teaching students how to engage with and navigate the world around them. “We will work as hard on social-emotional intelligence and communication skills — writing and public speaking — as we will on anything else,” Zeitz said.

Zeitz and his colleagues have much work ahead to make all the prime factors of their creative ideas, logistical plans, and hiring goals — which includes finding a charismatic humanities dean who “is able to stand up to math nerds,” he said — add up to an equation for success. They’re getting ready to launch an early admissions program and give “a day in the life” school preview this summer. Currently in fundraising mode, the founders hope to secure at least $1 million in order to keep the private tuition as low as possible and provide ample scholarships and financial aid.

To make the school accessible to math kids around the Bay Area, the campus will be located near public transit, most likely in San Francisco’s South Financial District area. The founders also plan to share their math curriculum and resources with the world in an open-source way, which will include hosting math talks and events for the public.

Many families in Silicon Valley have expressed strong interest in Proof School, but other reactions have ranged from initial skepticism to some concerns that the school will be elitist. “We’re not going to be elitist but we will be elite,” Brown said. “We’re not going to be snobby. We’re simply taking kids who operate at this [intellectual] level and putting them together with their peers, which they haven’t had in the past. And many have suffered for it.”

His own son, for example, is leaving his private middle school after this academic year because he has no math peers there, Brown said. If all goes well, after a gap year of homeschooling, the plan is to start Nico in ninth grade at Proof School in September 2015. “Oh, he can’t wait!” Brown said.

]]>*Part 8 of MindShift’s Guide to Games and Learning.*

Keith Devlin is a well-known mathematician and the author of many popular math books. He is co-founder and Executive Director of Stanford University’s Human-Sciences and Technologies Advanced Research Institute and is well known as the “NPR Math Guy.” He’s also a big fan of using video games as a teaching tool and the founder of an education technology company called BrainQuake.

Devlin believes the future demands a substantial change in the way we think about math education. “So many people in the U.S. have never experienced good mathematics teaching. They have a totally false impression of what mathematics is,” he says.

In this conversation, he explains why we now need different mathematical skills than we once did, and points out that the math curriculum of the 20th century did not equip today’s adults to mentor children in the math skills of the 21st century.

Devlin argues that video games are the perfect tool for teaching math. He also sheds some light on those Common Core math problems that are so controversial.

**Jordan Shapiro: In 2011, you published a book entitled ***Mathematics Education For A New Era: Video Games As A Medium For Learning.* W**hat inspired you to write that book? Do you play video games?**

**Keith Devlin: **I not only play video games, I actually wrote one, way back in the early 1980s when personal home computers were just starting to appear. I wrote a math game where you use trigonometry to find buried pirates’ treasure on a desert island. I did it for use in the math class at the local elementary school my daughters attended. When I saw how my daughters were totally engaged by video games (professional ones, much better than mine!), I recognized at once their potential in mathematics education.

I wrote the book to try to give teachers some sense of what it would take to build a good video game for mathematics learning and to explain to game designers what it would take to embed *good* mathematics learning into a game. I was not thinking about building a video game myself (back then).

**JS: In the book, you argue that video games are the best way to teach math to middle school kids. We’ve spoken about this before and you’ve talked about the importance of learning through doing. Why is context so important? And how can thinking of video games as math simulations help to move the typical math class from an product based pedagogy to a process based pedagogy?**

**KD: **Video games are an ideal medium for developing mathematical thinking. It’s a classic case where a technology that renders old skills less relevant actually provides their replacement. Many people outside of the worlds of mathematics (particularly math as used in the world) and mathematics education are simply not aware of the dramatic shift that both disciplines have undergone as a result of digital technologies.

When today’s parents were going through schools, the main focus in mathematics was on mastery of a collection of standard procedures for solving well-defined problems that have unique right answers. If you did well at that, you were pretty well guaranteed a good job. Learning mathematics had been that way for several thousand years. Math textbooks were essentially recipe books.

But now all those math recipes have been coded into devices, some of which we carry round in our pockets. Suddenly, in a single generation, mastery of the procedural math skills that had ruled supreme for three thousand years has become largely irrelevant. In my book, I used the term “mathematical thinking” to distinguish the kind of math that is relevant today from the (procedural) “mathematics” most people are familiar with.

The skill that is in great demand today, and will continue to grow, is the ability to take a novel problem, possibly not well-defined, and likely not having a single “right” answer, and make progress on it, in some cases (but not all!) “solving” it (whatever that turns out to mean). The problems we need mathematics for today come in a messy, real-world context, and part of making progress is to figure out just what you need from that context.

Books and lectures (in a room or on a screen) are useful *resources* for mathematical thinking, but they are no longer at the heart of math learning. The only way to acquire mathematical thinking ability is by a process of exploration – lots of trial-and-error and reflection. This is exactly what video games can deliver. They can provide small scale simulations of the kinds of open-ended, context-influenced, project-based, problem solving that is at such a premium in today’s world.

**JS: I wrote about your game Wuzzit Trouble ****earlier in this series****. I also sometimes show it when I’m doing workshops or public speaking. I’ve used your metaphor often: learning games should work like musical instruments. What does it mean to create a mathematical instrument? Can we do the equivalent for other subjects?**

**KD: **Yes, it’s fortunate that the word “play” applies both to musical instruments and video games. Everyone knows that the best way to learn an instrument is to start to play it. We don’t ask someone to learn to read music before they sit at a piano or pick up a guitar. But that’s exactly what we do in mathematics! The reason we do is that through most of its history, mathematics did not have any instruments. Video games can provide math instruments you can use to learn mathematics. Take a look at the twenty-minute video of the talk I gave recently. In it, I use *Wuzzit Troubl*e to explain how this can be done.

We can do the same kind of thing in other subjects, but many of them already have instruments (e.g. laboratory equipment in the natural sciences). Before video game technologies, however, all we had for math were paper and pencil, augmented with ruler and compass for geometry.

**JS: So, are you saying that kids don’t need to learn how to scribble numbers and equations on paper? Does that mean the symbolic Hindu-Arabic representation of mathematics is obsolete? **

**Keith: **No, kids need to master Hindu-Arithmetic *representation* of numbers far more today than in the past. What they don’t need to do – and I think this is what you’re getting at with your question – is train themselves to do long computations, as was necessary when I was a child. No one calculates that way any more! What they (we) need in today’s world is a deeper understanding of how and why Hindu-Arabic arithmetic works.

In all four offerings of my mathematical thinking MOOC to date, I have had as students, engineers with years of experience who suddenly found themselves out of a job when their employers replaced them with software systems (or sometimes overseas outsource services). Those engineers are now having to retool to learn this other skill of creative problem solving – *mathematical thinking*.

Incidentally, the Common Core State Standards that are so much in the news were designed to guide the shift to this new kind of mathematics, as I’ve tried to explain before. Many of the parents who object to the basic goal of Common Core appear do so because they too have not realized how much the world has changed in its demands for mathematical skills. (There is a lot to argue about when it comes to the implementation of the Core, but that’s a separate issue.)

**JS: In the past, you and I have discussed the “Symbol Barrier.” Now I think I misunderstood the concept. I always thought you meant that Hindu-Arabic representation was an obstacle to mathematical thinking. But clearly that’s not what you meant. Explain the “Symbol Barrier” and how video games can help to circumvent it.**

**Keith: **The Symbol Barrier is the name I gave to a phenomenon first noticed in the early 1990s. If people need to use basic computational mathematics in their everyday lives or careers, they quickly pick it up and are able to perform at a 98 percent accuracy level. That’s not just some people, it’s ALL people (apart from around 3 percent who have a brain condition called dyscalculia). But if you ask those same people to solve the same math problems presented in a traditional, school, paper-and-pencil test format, their performance levels drops from 98 percent accuracy to a mere 37 percent correct. Those people (all of us) don’t have an inherent math problem, they (we) have a language problem! That’s the Symbol Barrier.

The Symbol Barrier is a problem – i.e. a barrier to entry into mathematical thinking – because at the moment, learning mathematics and being tested in mathematics is all done by way of the symbolic representation! It’s as if we taught and tested people’s musical ability by instructing them in musical notation and testing how well they could write music using that notation. We don’t do that. We ask them to sit down and play an instrument!

Same with driving. Would you prefer to be driven by someone who had just passed the written part of the driving test, or would you want to know they had passed the road test?

Or an airline pilot. Which gives you more confidence, someone who has flown successfully in a simulator or a trainee pilot who has just passed a written test on how to fly a plane?

Likewise for mathematics. I would feel much more confident in the arithmetical ability of someone who had scored highly on my own game *Wuzzit Trouble* than someone who had simply learned how to line up numbers in columns and apply the standard algorithms of arithmetic. The latter just requires rule following, the former makes you *think*.

**JS: What other types of learning products are you working on at your company, BrainQuake? Which mathematical concepts are you trying to tackle? **

**KD: **Our basic platform is mobile games, though we intend to bring out Web-based PC versions for classroom use. The user data we obtained from the release of *Wuzzit Trouble*, last fall was as good, if not better, than we had hoped. We were ranked highly in many general game categories, not just educational games, and we were appealing to people of all ages.

Once we knew *Wuzzit Trouble* was succeeding as a general category video game, it made sense to develop that side more, so as to appeal to the widest possible audience. So we have been focusing on embedding that basic gears mechanic into a richer video game.

However, to be honest, *Wuzzit Trouble* is barely classified as a game. I would (and did) call it an entertaining and engaging math-based mobile app. To reach a much wider audience, we need to incorporate more of the elements of the highly successful video games that attract tens of millions of players. Teaming up with John Romero (the legendary creator of *Doom* and *Quake*) as our chief Game designer provided us an exciting opportunity to go after that elusive big audience.

We are also developing the assessment side. Students, parents, and teachers – and adult player-learners – want to know how well they are progressing in terms of math learning. Video-game learning provides extremely powerful mechanisms to track learning – potentially far better than the existing methods used in standardized tests. Doing that requires the development of powerful algorithms that take the raw player data (which goes right down to individual finger actions on the screen) and use it to infer the player’s thought processes behind those actions, so we can see *how* a player is trying to solve the problem. This will take the old idea of “show your working” to a whole different level.

**JS: This sounds great. Should we be trying to do this for every part of the mathematics curriculum? **

**KD: **BrainQuake is absolutely *not* obsessed with trying to achieve “full curriculum coverage,” whatever that might mean. To my mind it makes no sense to try to use video games to provide learning for parts of mathematics where an alternative, interactive representation does not add anything significantly new.

Also, I think it will always be largely up to a good teacher to help learners master the symbolic approach necessary for more advanced mathematics. I think that people who see technology as a way to eliminate the need for good classroom teachers fundamentally misunderstand what it takes to help someone learn how to think a different way – in my case the mathematical way. Technology can help. It can help in significant ways. But it cannot replace a good teacher.

**JS: What do teachers and parents need to know about the future of game-based learning, the future of pedagogy, and the future of education in general?**

**KD: **Game-based learning *is* the future. Games are just simulators with an internal incentive structure (often dopamine based). That means they tap into the way humans, and all living creatures, are hard-wired to learn: *by doing*.

But people are not machines. We are social creatures. If you want a child to develop into a useful citizen who can have a good life and contribute to society, you need to develop that child fully as a human being. That requires good parenting and great teaching. Doing it right requires close integration of the technologies with the human interactions.

*The MindShift Guide to Games and Learning is made possible through the generous support of the Joan Ganz Cooney Center and is a project of the Games and Learning Publishing Council.*

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In an article for the *New York Times* Motherlode blog, MindShift contributor and author Annie Murphy Paul explains why the perception that U.S. students are bad at math might indicate schools aren’t challenging students enough.

“‘Calculations kids are forced to do are often so developmentally inappropriate, the experience amounts to torture, she says. They also miss the essential point—that mathematics is fundamentally about patterns and structures, rather than ‘little manipulations of numbers,’ as she puts it. It’s akin to budding filmmakers learning first about costumes, lighting and other technical aspects, rather than about crafting meaningful stories.”

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