After photographing and mounting their pictures on the wall in numerical order, the students sat on the floor with their sketchbooks and began to draw and talk. “I had expected them to learn something about number composition,” James said, “but I didn’t expect the remarkable observations they began to have about the photographs.” For example, when one girl looked at a picture of two red scissors and three blue scissors (2+3=5), she noticed that the direction of the handles gave rise to a *new* number sentence: 4 scissors pointing left + 1 scissor pointing right = 5 scissors.

James, who recently published a paper about creativity in the classroom, said moments like these remind her that “creativity is not fluff or an add-on, but is instead an essential part of what it means to be a mathematician.” In fact, she believes creativity is the key to helping her students become confident and skilled mathematical thinkers.

Heather Hill, a professor at the Harvard Graduate School of Education, encourages teachers to make room for creativity in the math classroom “because there’s heaps of evidence that kids are naturally very creative when it comes to mathematics.” In the same way that kids create their own stories or make up songs, “kids will invent their own methods for solving mathematics problems, even problems that are sometimes very complex.”

In math education, said Hill, creativity is defined as “kids having their own ideas about how mathematics works and being able to work to verify that those ideas are correct.” As it turns out, she noted, these are the same traits that are recognized and celebrated in advanced mathematics. When elementary teachers encourage students to ask questions, make observations, and tackle problems in inventive ways, they create an environment that supports creative mathematical thinking.

Here are some ways to tap into that creativity:

**Encourage Students to Question and Observe
**“Asking mathematical questions is a form of creativity,” said Hill. Kids love to figure out how things work, so when teachers present a new concept, they should also build in time for students to make observations and ask questions. James uses prompts such as, “What do you notice about this [shape, number, story, or design]?” or “How else could we use [addition, graphing, or sorting] in the classroom?” to help students build these habits.

Pose Open-Ended Questions

**Engage in Rich Conversation
**One-on-one conversations help students articulate and extend their thought processes. As James circulates through the room, she uses prompts such as “Tell me about that; How did you think of that?; and What steps did you take?” to get kids talking. “I encourage students to share their thinking, and in turn I am open to the unexpected strategy,” according to James. “I am willing to say, ‘Wow, I never thought about that before.’”

**Apply Skills to New Contexts
**During one lesson, James asked her kindergartners to write a number sentence and then invent a story based on that sentence. Students depicted their story in three ways: as an illustration, as a written sentence, and as a number sentence. James was surprised to find that a few kids who zoomed through their math facts really struggled to complete this task. “They wanted to give me a number sentence without a story,” said James. Being asked to manipulate and view numbers in this way “caused them a bit of internal conflict.” To help them through the process, James said she just sat with them — wondering out loud and asking questions — until they found their footing.

An activity like this is effective, said Hill, because it posed a question that “stretched kids outside of their comfort zone and called on them to think and invent.” James was asking her students to* contextualize, *which is “a core mathematical practice.” When young children are given opportunities to apply their math skills to novel situations, they take steps toward becoming confident and creative mathematical thinkers.

**How Parents Can Help**

Parents also play a key role in nurturing a child’s mathematical mind. They can help kids discover the math that is embedded in our daily experiences. “Anything you can make into a math problem is a win,” said Hill, “because it shows the child how useful math can be, and gives them some practice in applying their own thinking to math problems.”

James and Hill offered these strategies for parents:

**Look for Patterns
**Be on the lookout for patterns and sequences. For example, said James, a parent could make a plate with one piece of cheese, two tomatoes, three carrots, and four grapes and then ask, “Did you notice what I did with your lunch?” Simple activities such as sorting toys, setting the table, or going on a nature walk can provide opportunities too look for and create color, size, number, and shape patterns. These activities also hone observational skills.

**Leave Math Notes
**James suggests leaving little, unexpected math messages around the house such as, “Did you eat more pretzels or raisins? How many more?” or “How many different routes can you take to get from the kitchen to the bathroom?” Kids will likely start to leave notes for you to respond to, as well. She also recommends putting a number on a big sheet of paper and leaving it up for a few days, letting everyone in the family add something they know about that number. For example, for the number ten, someone might draw ten fingers while another might write 10 + 2, two less than a dozen, the square root of one hundred, or the names of ten friends.

**Have Math Chats
**Take time each week to talk about math with your kids in the same way you might talk about letters and stories. “Ask

When parents and educators model creative engagement with mathematics, children come to see math as more than simply a set of facts and operations. “We want our students to become mathematical thinkers, not mathematical machines,” said James. “Even in kindergarten, I want to shape people who love solving problems creatively and who have the skills they need to someday change the world.”

]]>Then through the miracle of mathematics instruction I was back in a low Algebra track by 9^{th} grade and limped along through terrible math classes until my senior year in high school. In 12^{th} grade, I enrolled in a course called, “Math for Liberal Arts.” Today this course might be called, “Math for Dummies Who Still Intend to Go to College.” I remember my teacher welcoming us and saying, “Now, let’s see if I can teach you all the stuff my colleagues were supposed to have taught you.”

This led to two observations:

- Mr. O’Connor knew there was something terribly wrong with math education in his school.
- I looked around the room and realized that most of my classmates had been in Unified Math with me in 7
^{th}grade. These lifeless souls identified as mathematically gifted six years ago were now in the “Math for Dummies Who Still Intend to Go to College” class. If this occurred to me, I wondered why none of the smart adults in the school or district had observed this destructive pattern?

Two things I learned in school between 7^{th} and 12^{th} grade kept me sane. I learned to program computers and compose music. I was actually quite good at both and felt confident thinking symbolically. However, majoring in computer science was a path closed to me since I wasn’t good at (school) math – or so I was told.

I began teaching children in 1982 and teachers in 1983. I was 18-19 years old at the time. While teaching others to program, I saw them engage with powerful mathematical ideas in ways they had never experienced before. Often, within a few minutes of working on a personally meaningful programming project, kids and teachers alike would experience mathematical epiphanies in which they learned “more math” than during their entire schooling.

In the words of Seymour Papert, “They were being mathematicians rather than being taught math.”

Teaching kids to program in Logo exposed me to Papert’s “Mathland,” a place inside of computing where one could learn to be a mathematician as casually as one would learn French by living in France, as opposed to being taught French in a New Jersey high school class for forty-three minutes per day.

I met Seymour Papert in 1985 and had the great privilege of working with him for the next 20+ years.

Papert was a great mathematician with a couple of doctorates in the subject. He was the expert Jean Piaget called upon to help him understand how children construct mathematical knowledge. Papert then went on to be a pioneer in artificial intelligence and that work returned him to thinking about thinking. This time, Papert thought that if young children could teach a computer to think (via programming), they would become better thinkers themselves. With Cynthia Solomon and Wally Feurzig, Papert invented the first programming language for children, called Logo. That was in 1968.

What makes Papert so extraordinary is that despite being a gifted mathematician he possesses the awareness and empathy required to notice that not everyone feels the same way about mathematics or their mathematical ability as he does. His life’s work was dedicated to a notion he first expressed in the 1960s. Instead of teaching children a math they hate, why not offer them a mathematics they can love?

As an active member of what was known as the Logo community, I met mathematicians who loved messing about with mathematics in a way completely foreign to my secondary math teachers. I also met gifted educators who made all sorts of mathematics accessible to children in new and exciting ways. I fell in love with branches of mathematics I would never have been taught in school *and I understood them.*Computer programming was an onramp to intellectual empowerment; math class was a life sentence.

It became clear to me that there is no discipline where there exists a wider gap than the crevasse between the subject and the teaching of that subject than between the beauty, power, wonder, and utility of mathematics and what kids get in school – math.

Papert has accused school math of “killing something I love.”

Marvin Minsky said that what’s taught in school doesn’t even deserve to be called mathematics, perhaps it should just be called “Ma.”

One of our speakers, Conrad Wolfram, says that every discipline is faced with the choice between teaching the mechanics of today and the essence of the subject. Wolfram estimates that schools spend 80% of their time and effort teaching hand calculations at the expense of mathematics. That may be a generous evaluation.

Over the years, I’ve gotten to know gifted mathematicians like Brian Silverman, David Thornburg, Seymour Papert, Marvin Minsky, and Alan Kay. I’ve even spent a few hours chatting with two of the world’s most preeminent mathematicians, John Conway and Stephen Wolfram. In each instance, I found (real) mathematicians to embody the same soul, wit, passion, creativity, and kindness found in the jazz musicians I adore. More significantly, math teachers often made me feel stupid; mathematicians never did.

*You can read the remainder of this post at the Reinventing Mathematics Education blog, which originally published this piece. Gary Stager is the founder of the Constructing Modern Knowledge summer institute for educators. **He will lead a day-long symposium in Los Angeles on January 4th to explore Reinventing Mathematics Education. **Dr. Stager’s latest book, Invent To Learn – Making, Tinkering, and Engineering in the Classroom was published in May 2013 by Constructing Modern Knowledge Press.*

How did this happen?

Little teaches at Martin Luther King Jr Middle School in Berkeley, California, where classes like sewing, woodshop, and metal shop — what she calls “practical ways of learning math” — are no longer offered; tight budgets and renewed emphasis on academic learning have eliminated them. But Little couldn’t bear to subject already disengaged students to yet another ho-hum class of multiplication tables and long division.

Instead, she took a gamble and brought some materials to school for her students to play with: a sewing kit, the 3-D doodler she’d just been given, her son’s marble-run set and a MaKey MaKey device she knew nothing about, donated by a friend.

To Little’s surprise, the students dove in. They put themselves in their own groups based on personal interest, and worked together to grapple with these mysterious tools. Little herself was unaware of the MaKey MaKey instrument. “I’m afraid of maker-type technology,” she told me. But she imagined that she and the students could figure out together how to use the alien circuit board to interesting effect.

“I didn’t know how to do it, but I could teach them how to learn,” she said.

Inspired by an online investigation, one group of students decided to build a banana piano. This meant they needed to program the computer — Little’s laptop — to the MaKey MaKey circuit board, which the students were able to do. Then, by trial and error, the 12-year-olds learned how to build a full circuit: They attached one set of wires from the circuit board to six bananas (borrowed from lunch), and another connector to the laptop.

Little gets fuzzy explaining exactly how the banana piano worked. “I could not build one,” she said. But the students understood, and one day they brought the unorthodox instrument to the entire regular math class.

“They played a song for everyone, and everyone went wild,” Little said. “’We want to come to math support!’” she recalled the students saying.

Little then invited all her math students to attend the twice-weekly optional remediation classes where she’d first introduced the practical tools. “They all opted for extra math,” Little said.

Intrigued by the bananas, one group worked with lemons, this time completing the circuit by holding hands. The electricity ran through every kid in the class to the last one, who could then “play” the lemon bongos. Kids with a more literary bent wrote a book and set music to it; they rigged the MaKey MaKey device to play exciting music during adventurous passages and dirge music during sad ones. Other students flocked to the sewing, 3-D pen and marble maker, including one child who cross-stitched the emblem of the San Francisco Giants as a gift for his grandmother.

**Student Empowerment **

The learning didn’t stop when supplies finally dwindled. Rather than solicit parents for money, Little turned the need for materials into an immersion in marketing and sales. They would sell pencils, the children decided, but at what price? They debated strategies: If we sell 60 pencils at $2 per pencil we’ll reach our goal. But who will spend that much on a pencil? Maybe 50 cents per pencil this week, and 75 cents every day thereafter.

“The kids were totally in control of how the price would vary and how it would affect profit,” Little said. In the end, the children earned $120, enough to resupply the 3-D doodler, buy more circuits and restock the sewing basket.

Little is astonished by the changes she saw among her students. By making the classroom hands-on, she upended the traditional social hierarchies: Kids who might have been ostracized for being deficient in math were suddenly valuable when their strengths — like problem-solving or brainstorming — became visible and needed.

Likewise, the stigma attached to remediation classes, and even to math itself, disappeared. Everyone wanted to join in. In addition, kids abandoned their usual roles: Some who traditionally sat quietly and waited for direction began to take charge, and others who claimed to hate reading devoured turgid manuals. And Little herself became more of a facilitator than an instructor, helping kids find what they needed but not spoon-feeding them information.

“Teachers are no longer the holders of all knowledge,” she said. Rather, the students themselves, having discovered on their own how to program fruit to play a tune, developed unexpected confidence.

“They became bold and self-directed when they realized I did not have the answers,” Little wrote about the experience. “I became a curious and excited partner in their discoveries.” She shared her findings at the FabLearn conference at Stanford and wrote a report about the results.

**Meeting Standards**

When school began in September, Little brought back the MaKey MaKey contraption and other tools to her classroom, this time introducing them to all her students from the start. She remains ebullient about the possibilities, and encourages her math-teaching peers to give it a try.

At a minimum, teaching this way satisfies the Common Core requirements for students to solve problems and understand concepts, she said.

“Math teachers who have only paper and pencil at their disposal, find some of these important standards very difficult to illuminate,” according to Little. “When students must work in groups to complete a real project, all of these mathematical standards come into play. Instead of being told, ‘Your calculations are wrong,’ students experience a real setback in their creation and must problem solve to get it working.”

She discovered that this approach stimulates children to learn, helping them to understand and use math in ways they’ve never considered. For example, Little uses cross-stitch to develop understanding of math.

“Cross-stitch is like creating art with pixels,” according to Little. “You cannot actually make a curve but can approximate one by using a stair step method. Distance from the piece creates the illusion of a smooth curve.” She said students were excited by this discovery and were giving feedback on one another’s projects.

Little had hesitation over these new maker tools, but she saw in them the same qualities that made sewing and wood shop classes a practical way of learning math, back when they were available. Both require planning, visualization and precise measurements. Educators may not feel fully prepared to start these hands-on projects, but Little says not to worry about lesson plans or about not fully understanding how it all works. “Just start,” she said. “Get some supplies and go. Don’t be afraid!”

]]>“He makes everything fun like a game,” says one student. “That really helps us understand how everything works.”

Before hearing about the academic benefits from the teacher, take a look at a video students recently created.

Below is the “Soul Pancake” video featuring math teacher Robert MacCarthy showing how his students worked together to make the songs.

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Jennifer Green did a nontraditional program back in the 1990s. She got five weeks of training in things like “introduction to classroom management” and “introduction to planning.” Then she was a teacher, in a huge, struggling high school.

“I would come in in the morning. I would close the door,” she says. “I would struggle through the day. I would cry three times a week after my third period, which was my most challenging group of students. I would dust myself off. I would tell my fourth period class that I had terrible allergies and that’s why my eyes were so red.”

She says she got no help. The first time an administrator came to check on her it was January — and the administrator just needed to know if she had enough textbooks.

Green and Bankhead both wanted to become great teachers. But the system didn’t seem set up to help them do it.

There’s a pervasive American myth that good teachers are born, not made, and that good teachers have a set of inborn traits that naturally blossom as they figure the job out on their own. To get better teachers, the theory goes, schools need to find more people with those traits. The other myth is that teaching is easy — the work involves children and the content is pretty basic, so it must be easy.

“Teaching is complex work that people actually have to be taught to do,” says Deborah Loewenberg Ball, dean of the School of Education at the University of Michigan. Ball spent years as an elementary school teacher and was always praised for being a “natural,” but she says teaching never came easily. She worked hard at her job.

Now, she’s trying to dramatically change teacher training to focus on the specific knowledge and skills that teachers need to effectively help students. Understanding math and knowing how to teach it are two separate skills. And understanding how to teach math well doesn’t come naturally.

People who want to be teachers “deserve to learn how to do this work well,” Ball says. “And the children that they teach particularly deserve to have those teachers taught.”

**Professionalizing Teaching**

Ball and her colleagues at the University of Michigan have started treating teacher preparation like any other profession. That means identifying the core set of skills, techniques and knowledge required by an entry-level employee in that field. To be a plumber, for example, one needs to know how to vent a sanitary drainage system. To be a pilot, one needs to know how to do a crosswind approach and landing. And one would have to prove one can do these things to get licensed.

“This is true primarily at least across occupations and professions where people’s safety is at risk,” Ball says. “And I do think it’s of great concern that we don’t as a culture appear to think that children are at risk when we don’t execute that same kind of responsibility” when it comes to training teachers.

There’s a lot of resistance to the idea of defining a core set of skills and knowledge that teachers should know before they start teaching, Ball says. It goes back to the belief that the ability to teach is a personal trait, dependent on individual style and talent. But Ball isn’t advocating that teachers give up their personality. She’s just trying to ensure every new teacher has the right skills for the job.

**High-leverage teaching practices**

About 10 years ago, Ball and her colleagues at the University of Michigan decided to try to identify what that core set of teaching would look like.

Tim Boerst, chair of the Elementary Teacher Education program, says the question they asked themselves was this:

“When a teacher goes out into the field, what are they routinely going to be needing to do? And how are those routines, those particular practices, really important in the learning of students? Because there are all kinds of things that teachers routinely do. Which are the ones that we’re going to be picking that we really think advance the learning of academic subject matter?”

They got a bunch of teachers and researchers together and came up with a list of the things they thought all beginning teachers should know how to do. Their list had 84 things on it. That was clearly too many. They needed a set of skills they could actually teach in their two-year program, so they whittled their list down to 19 skills and gave them a name: high-leverage teaching practices.

The list includes skills like these:

*Leading a whole-class discussion.*“In instructionally productive discussions, the teacher and a wide range of students contribute orally, listen actively, and respond to and learn from others’ contributions.”*Designing a sequence of lessons toward a specific learning goal.*“Teachers design and sequence lessons with an eye toward providing opportunities for student inquiry and discovery and include opportunities for students to practice and master foundational concepts and skills before moving on to more advanced ones.”*Eliciting and interpreting individual students’ thinking.*“To do this effectively, a teacher draws out a student’s thinking through carefully chosen questions and tasks and considers and checks alternative interpretations of the student’s ideas and methods.”*Analyzing instruction for the purpose of improving it.*“Learning to teach is an ongoing process that requires regular analysis of instruction and its effectiveness.”

**Helping Teachers Learn Those Skills**

Teaching teachers is particularly difficult because everyone has some experience of either being a student or teaching something informally. And those prior experiences shape ideas about what education should look like.

When a student comes into the teacher preparation program at Michigan, faculty want to know what beliefs and skills students are bringing with them. Professors then tailor the curriculum to focus on the things students don’t know. They also work hard to help pre-service teachers unlearn habits or beliefs they picked up from their own years as children in school that are *not* productive ways to help kids learn.

To figure out what incoming students already know about teaching, Michigan faculty asks them do a role-playing exercise where they actually do some teaching. The pre-service teachers are given a piece of paper with a math problem on it. The paper also includes an answer. Here’s one of the problems the Michigan students are given, with an answer a student might actually have given.

The Michigan students get a few minutes to look at the problem. Then they sit down with a graduate student or professor who plays the role of the kid who came up with the answer 83. The pre-service teacher’s goal is to find out what the student did to produce that answer, and why. The entire teaching moment is recorded on video.

The point of this simulation is to see how well the Michigan student can elicit and interpret student thinking. That’s one of the high-leverage practices, and it’s hard to do. Even if the pre-service teacher can figure out what the student did, it’s really hard to leave space for the student to explain his or her own thinking.

Often, rather than eliciting the kid’s thinking, the “teacher” tells the kid what she thinks the kid was thinking, says Boerst. He calls it “filling in student thinking.”

“And that happens in classrooms all the time,” he says. “Teachers make assumptions about what kids are thinking. Kids don’t really know how to say otherwise or maybe aren’t inclined to say otherwise. Like, ‘Yeah, that’s what I was thinking ’cause I don’t really want to say what I was thinking.’”

Boerst says this can lead teachers to think kids understand the material when they don’t.

By watching and coding these simulated assessments, Boerst and his colleagues have found that half of the students coming into the elementary teacher prep program at Michigan do this “filling in of student thinking.”

Teaching students out of this habit is one of the goals of the Michigan teacher prep program. But just reading or talking about the fact that you shouldn’t do this as a teacher isn’t enough, says Boerst. People have to practice doing it a different way.

Teacher preparation in the United States hasn’t been focused enough on practice, says Ball. Traditionally, students in teacher prep programs spend a lot of time reading and talking about teaching.

“The assignments in the past were much more reflection, analysis,” Ball says. “In some sense, we could have been misled by people getting good grades for writing well. And, although it may sound a little too extreme, I think we’re more interested now in whether they can do it well, not how well they can talk about it.”

At Michigan, students are continually recording themselves as they practice teaching, and then watching the video and analyzing it. When teachers encounter a difficult moment in the classroom, like a misconception that they aren’t sure how to debunk, they have a tendency to just get through it and try not to think about it again.

“Many of us have had that experience of, ‘OK, phew, that’s over, I don’t have to do it again,’ ” says Betsy Davis, a professor in the elementary education program. That’s exactly what Michigan is trying to train its pre-service teachers not to do. Instead, the program tries to instill reflection for the purpose of improvement into everything, especially mistakes.

“By having the interns watch their own video of their teaching really carefully, they see things or they hear themselves saying things that don’t make sense or that are missed opportunities,” says Davis. “And that’s one of the things we ask them to highlight in their videos: What did you miss the chance to do that if you were doing this over you would do?”

**A New Approach to Student Teaching**

But is all this video recording necessary when most teacher preparation programs include a student teaching component? Many people expect student-teachers will learn practical skills from the veteran educators with whom they are paired.

The problem with pinning all the practical experience on student teaching is that the quality of those experiences varies widely. Even more shocking, data collected in the mid-2000s showed that more than 20 percent of first-year teachers had no student teaching experience at all. Forty-two percent of science teachers did no student teaching.

Remember Jasmine Bankhead, the student teacher who was left alone in the classroom on her very first day? That’s not so uncommon. Student teachers are either given too much responsibility, or they’re not given enough; they make copies or do recess duty. Or they just sit and watch the teacher teach. They might see really effective teaching, and they might not.

All of these things were happening when students at Michigan went out into the field for their student teaching experience. It was always a scramble to find classrooms to send them to. There was no consistency. Students “were actually starting to pick up some negative practices from the field,” says Elizabeth Moje, an associate dean at the Michigan School of Education who helps oversee the student teaching program.

Moje wanted her students to see teachers who were really good at things like eliciting student thinking and leading class discussions. She needed a way to send only students to observe teachers the university knew were very effective. So that’s what they did.

Now, rather than sending the students out to dozens of schools all over the Detroit metropolitan area, Michigan rotates its students in groups to just a few different classrooms in a few different schools. It’s similar to the way medical students rotate through different specialties during their training. There’s a lot Michigan has borrowed from the medical field. In fact, all the pre-service teachers are now called interns.

When interns visit classrooms, they check in with the teacher and then work in small groups or one-on-one with students. A Michigan adviser is also in the class, observing as interns work with students. If one of the interns struggles, the advisor can jump in with real-time feedback.

For example, in a discussion about the causes of the Civil War, one intern repeatedly asked questions hoping to solicit a particular response from the middle school students that just wasn’t coming. There were lots of awkward pauses as the intern waited for the kids to pick up on his train of thought.

Adviser Rebecca Gadd was observing. When the intern had tried a few different questions with no luck, she stopped the discussion and pulled the intern aside to give him some tips.

“OK, so what I would suspect is that the way that this is explained is a little bit abstract,” she says, referring to the reading assignment.

“So you need to think, are you going to ask or are you going to explain?” She recommends that he stop asking the students questions because they clearly missed the point in the reading. It’s time to explain it to them. Just tell them what you want them to know.

Gadd is a former middle school teacher. She wishes her training had included this kind of guided practice. Teachers can go through their entire training — their entire careers even — without anyone taking them aside and offering in-the-moment feedback. She says Michigan got the idea for doing this from medical training.

“When aspiring doctors are practicing with patients, medical educators don’t wait until they’ve killed the patient to intervene and say, ‘You should have done this differently,’ ” she says. “Instead, they intervene in the moment and say, ‘OK, we need to be doing this.’ ”

**Becoming a Teacher**

Michigan students in the secondary teacher education program spend two semesters in classrooms, observing and working with kids one-on-one or in groups. The idea is a gradual assumption of responsibility.

They don’t actually do what most people think of as student teaching until their third semester. That’s when they’re promoted from intern to resident, and they actually get to take charge and teach the class.

Grace Tesfae is in her semester-long residency, getting ready to graduate from Michigan in a few months. She’s excited about having her own classroom, but also scared to be on her own. “I feel like I’ll be ready when the time comes,” Tesfae says, sounding a bit uncertain.

Michigan doesn’t have much data yet on whether the new approach is working. It’s not even clear what kind of data would provide a meaningful measure of what Michigan is trying to do. They could look at test scores of students in their graduates’ classrooms. That would tell them something. But Michigan wants to know if its teachers can do things like elicit and interpret student thinking and lead class discussions. Test scores don’t tell you that.

Michigan does have its interns repeat the simulation they did at the beginning of the program, where they tried to figure out how a kid was thinking while solving a math problem. By the end of their first year in the program, most interns are no longer filling in, rather than eliciting, a student’s thinking. The Michigan interns show progress on other elements of the 19 high-leverage teaching practices, too.

Deborah Ball admits that those 19 practices are just a first bet for changing teacher education. “These aren’t necessarily the end, but they are the best bets we had,” Ball said. “And we have to have a systematic way of revising those.”

Bottom line though, from Ball’s point of view, is that the teaching profession needs to come to some kind of common understanding about the skills that are required to enter the profession. And just like plumbers and pilots, new teachers should have to demonstrate they have these skills.

Ball has started an organization to try to develop new licensing assessments for people who want to be teachers and to work with teacher preparation programs across the country to develop common approaches to professional training. It’s a big job. The U.S. Department of Education projects that by 2020, the United States will need nearly 430,000 new teachers a year.

Ball’s ultimate goal is to make sure every first–year teacher in the United States is what she calls “a well-started beginner.” That’s what she and her colleagues are aiming for at Michigan.

“We’re really eyeing the first year, honestly,” she says. “Really, the goal is that kids wouldn’t have first-year teachers who are completely underprepared, that it wouldn’t be true anymore that you could just end up with a teacher who, this is her year to have a wreck year.”

Ball feels particular urgency about this question because in the United States, it’s poor kids who are most likely to get first-year teachers. Ball says that to improve education for all kids, and especially for poor kids, first-year teachers have to be much better prepared.

**Emily Hanford** is an education correspondent for American RadioWorks, the national documentary unit of American Public Media. Check out the American RadioWorks website for a more in-depth version of this article. You can also read other articles about teacher learning and listen to the accompanying radio documentary program. American RadioWorks hosts a weekly education podcast available here.

After six years in the classroom and a doctorate in math education from Stanford, Meyer is now the Chief Academic Officer at Desmos. But his blog has long been a go-to resource for teachers looking for different ways to inspire what he calls “patient problem-solving” in students. He says to help students develop mathematical reasoning, teachers need to focus less on the computation side of math, and more on helping kids determine which pieces of information are useful, which are extraneous and how it all fits into the real world.

“Students come into our classrooms with loads of experiences mathematically that we don’t anticipate,” Meyer said. “We need to draw those out of students and build on those.” His strategy is to ditch textbooks that lay out problems in neat, simple ways that easily fit into predetermined formulas, and instead get students talking about a problem that draws on their intuition. For example, he will show students a picture of two hills. One hill is taller and the other is longer. The he’ll ask, “Which hill is steeper?” It’s a deceptively simple question that provokes a big discussion because there are good arguments for both answers. Out of that discussion students develop an understanding of slope.

Problems like this one engage even the most math-intimidated student because it starts from what Meyer’s calls “the level playing field of intuition.” All students have seen hills and can talk about what they are seeing. Gradually the math enters the conversation to make it easier to discuss. The math serves the conversation, not the other way around, Meyer said.

“In a good argument kids walk away feeling like they’re a thinking thing,” Meyers said. That may seem obvious, but so much of teaching math through a computational lens asks students to find the right equation and plug in numbers. It doesn’t ask them to be big thinkers; but it’s precisely the experience of grappling with a problem that sparks curiosity, motivates students and develops the patient problem-solving that is so lacking in much of the population.

Watch Meyer’s funny and enlightening Tedx talk for a more comprehensive explanation of how he teaches.

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Well … there’s an app for that. Tons of them, in fact. And a study published today in the journal *Science* suggests that at least one of them works pretty well for elementary school children and math-anxious parents.

A team from the University of Chicago used a demographically diverse group of first-graders and their parents — nearly 600 in all — across a wide swath of Chicago. One group got to use an iPad app called Bedtime Math, built by a nonprofit with the same name. (The app is also available for Android, but we’re told most used the iPad version) The no-frills app uses stories and sound effects to present kids with math problems that they can solve with their parents.

The control group was given a reading app with similar stories but no math problems to solve. The results at the end of the school year?

I reached out to University of Chicago psychology professor Sian L. Beilock, one of the paper’s lead authors, to find out more.

**I read to my child all the time. But I don’t read bedtime math stories. ****After reading your study, maybe I should?
**

Our study suggests that doing Bedtime Math with your kids can help advance their math achievement over the school year, and this might be especially important for parents who are a little bit nervous about their own math ability.

**That’s me! How big an increase and what kind of improvement did you see when kids used this iPad app?**

We compared kids who used the Bedtime Math app that involved reading stories and doing math problems with their parents to kids who did a very similar app that didn’t have the math content. We showed that when kids frequently used the app with their parents, those who used the math app were three months ahead in terms of math achievement relative to kids who just did the reading app.

**Your team found that the app worked even better for children whose parents tend to be a bit anxious or uncomfortable with math?**

Many adults in the U.S. and around the world profess to be uncomfortable or anxious about math. Oftentimes dealing with your kid around math can be a nerve-wracking experience — whether it’s homework or just talking about it. We found that doing this Bedtime Math app with kids was especially beneficial for those kids whose parents tended to be the most nervous about math. In essence, these kids grew significantly throughout the course of the year and looked like kids whose parents weren’t anxious about math by school year’s end.

**And you saw improvement even in children who used the app with parents as little as once a week?**

Yes, it was somewhat surprising to us that such little use would have such important benefits. One of the ideas is that we think that when parents get comfortable with talking with their kids about math — it doesn’t have to be complex math problems, it could be anything from shapes to even counting — they likely engage in math talk even when they’re not using the app. And we know that parents who talk more with their kids about math — whether you’re counting out the number of cookies or counting the minutes to bedtime — those kids tend to achieve at higher rates in math.

**Bottom line for you: A little bit of math can go a long way, at least in terms of this one study’s findings? **

That’s exactly what we’re showing.

**There are a lot of apps out there. Why’d you choose this app in particular? What was special about it? **

There is certainly a billion dollar education app industry out there. What we’ve realized in our initial work is that a lot of it isn’t based on research. It’s unclear what the benefits are. In fact, there has been some research that shows that apps with lots of bells and whistles can actually be detrimental to kids’ learning because it distracts them. We base our investigations on learning science.

We’ve shown that, when parents interact with their kids and talk with them about math, that really impacts what kids learn. We were interested in this because it really is a no-frills app, an easy way for parents to interact with their kids, to talk with their kids about math. It’s not an app that they use by themselves. And we thought that that potentially had promise in terms of what math knowledge kids gained.

**I admit I’m kind of a math-anxious parent. But when doing stuff like woodworking, I try to incorporate a little geometry and basic measurement whenever I can. “Hey, let’s measure this again! Twenty-four inches — how many feet is that?” It’s a fun way to sneak a little bit of math in. **

And to realize that math is part of everything we do, and math is not something scary or that one should be anxious about. And it’s really healthy to try to incorporate that into daily life. And often, as you said, parents think about reading bedtime stories, but there is a place for thinking also about bedtime math.

**Culturally and socially, it seems we don’t think about math as integral a part of parenting as reading. And few adults would say, “I’m not so good at reading.” But many people say, “I’m not so good at math.” And somehow that’s socially acceptable.**

Yes, in my book,* Choke, *where I talk about stress and performance, I mention how you don’t hear people walking around bragging that they’re not good at reading. But very intelligent people brag about not being good at math. And it turns out that that anxiety and social acceptability has implications for our nation’s success in math and science fields. And it’s really important that we as parents and teachers and adults try to convey to our kids that math is something that’s (a) enjoyable and (b) learned. You’re not born a math person or not; it’s something that’s acquired. And every time we talk about it and we integrate it into our daily lives, children may see the importance of it and that math is not something to be fearful of.

**Where do you think some of that math anxiety comes from?
**

Math anxiety comes likely from lots of different places. Previous work that my group has done shows that teachers who tend to be anxious about math affect their kids’ perceptions of math and what they learn across the school year. We also know that when parents are anxious about math they can transfer that to their kids, especially when they’re helping a lot with math homework. We tend to point to the schools to be the source for math knowledge. But kids spend lots of time outside of school and get lots of information from parents and from other adults. So being cognizant of how we talk about math and how we integrate it into our daily lives is important — both inside and outside the classroom.

**Did you see any improvements in the parents’ math ability by any chance [laughs]?**

Ha, well, that’s a really interesting question. We are just looking into those questions now. You can imagine that for parents who have a fear of math or less than optimal math training, it might take more than one school year to move the needle for them. But we are seeing improvements with their kids. And that’s a first step. And we will be looking (in future studies) at how parents think about math, how they do in math, and most specifically their attitudes when interacting with their kids.

**So there is hope for me?**

There is hope for all of us! And, as you said, integrating these sorts of counting and math activities into daily routines is a great way to socialize both kids and their parents to the benefits of math.

Copyright 2015 NPR. To see more, visit http://www.npr.org/.

]]>It was 2013. She was the new principal of the O’Keeffe School of Excellence, an elementary school on Chicago’s South Side that had been struggling for years. Finally, the school district had taken dramatic action by firing the principal, the staff and all the teachers.

That’s when Bankhead was hired. Her job was to turn a failing school into a successful one, with all the same kids, but an entirely new teaching staff that she got to choose.

Bankhead had a very clear idea about what kind of teaching she wanted to see at her school. She calls it “inquiry-based” teaching. It’s an approach, supported by research, that begins by posing questions to students rather than presenting them with facts or knowledge. It’s the opposite of the way she was taught.

“My teachers stood in the front and talked,” she says. “And that was it.”

To help the teachers at O’Keeffe learn how to do inquiry-based teaching, she gave them training. Lots of training. She set up workshops and sent them to professional development days.

But, it wasn’t working. She and her administrative team would visit classrooms, hoping to see all this great inquiry-based teaching. What they saw instead were a lot of teachers standing at the front of the room, talking. The teachers were learning about inquiry-based teaching at the workshops, but they didn’t know how to actually *do* it when they got back to their classrooms. So they fell back on what they remembered about how their teachers taught, says Bankhead.

This is a common complaint about the traditional approach to teacher professional development in the United States. Teachers go to workshops and professional development days where they might get great new ideas about teaching. But when they get back to their classrooms and try to put those ideas into practice, all kinds of questions come up. And the expert who led the workshop isn’t there to help. Often, there’s no one to turn to for help.

Teachers in the United States have been expected to go into their classrooms, shut their doors, and figure things out on their own.

Bankhead and her administrative team realized the typical American approach wasn’t going to work if they wanted to dramatically change teaching at their school. One of the O’Keeffe assistant principals had recently learned about an approach to professional development called “lesson study” in a class taught by a Japanese professor. They decided to get in touch with the professor, see if he could help them.

**Bringing Lesson Study to Chicago**

Akihiko Takahashi is a professor of math education at DePaul University. Before that, he was an elementary school teacher in Japan. He first came to the United States in the early 1990s looking for all the great approaches to teaching math that he and his colleagues in Japan had learned about from American researchers. When he couldn’t find these approaches being used in classrooms, he soon realized why: There was no lesson study in the United States.

Lesson study is a form of professional development Japanese teachers use to help them improve and to incorporate new ideas and methods into their teaching.

“If there’s no lesson study,” Takahashi says, “how can teachers learn how to improve instruction?”

Here’s how lesson study works.

A group of teachers comes together and identifies a teaching problem they want to solve. Maybe their students are struggling with adding fractions.

Next, the teachers do some research on *why* students struggle with adding fractions. They read the latest education literature and look at lessons other teachers have tried. Typically they have an “outside adviser.” This person is usually an expert or researcher who does not work at the school but who’s invited to advise the group and help them with things like identifying articles and studies to read.

After they’ve done the research, the teachers design a lesson plan together. The lesson plan is like their hypothesis: If we teach this lesson in this way, we think students will understand fractions better.

Then, one of the teachers teaches the lesson to students, and the other teachers in the group observe. Often other teachers in the school will come watch, and sometimes educators from other schools too. It’s called a public research lesson.

During the public research lesson, the observers don’t focus on the teacher; they focus on the students. How are the students reacting to the lesson? What are they understanding or misunderstanding? The purpose is to improve the lesson, not to critique the teacher.

In the United States, we tend to think that improving education is about improving teachers – recruiting better ones, firing bad ones.

But the Japanese think about improving teaching. It’s a very different idea, says James Hiebert, an education researcher at the University of Delaware who has written about lesson study.

“Everything we do in the U.S. is focused on the effectiveness of the individual,” Hiebert says. “Is this teacher effective? Not, are the methods they’re using effective, and could they use other methods?”

Hiebert says to improve education in the United States, we need to shift from thinking about how to improve *teachers* to thinking about how to improve *teaching*. Lesson study is one way to do that, he says.

**Lesson Study at O’Keeffe**

Akihiko Takahashi now helps run an organization called Lesson Study Alliance that helps American teachers, mostly in Chicago, learn lesson study. One of the schools is O’Keeffe.

I visited O’Keeffe in January 2015 to talk with teachers about their experience with lesson study and to see a public research lesson.

One of the first things to understand about lesson study is that it’s a long process, kind of the opposite of the one-day workshop American teachers are used to. Teachers come together to identify a problem they want to solve. Then they spend months doing research and planning a lesson.

I spent most of my time at O’Keeffe with a group of three teachers who had been working together as part of a lesson study group since the previous summer. Angela Flores and Melissa Warner teach third grade. Wanna Allen teaches fourth grade math and science.

When they first came together to identify the teaching problem they wanted to solve, they had several things on their mind. One, they knew the overall goal for the school was for teachers to work on inquiry-based teaching. Two, they were thinking about the Common Core. That’s a set of new education standards that lay out what kids should know and be able to do in each grade. Teachers at O’Keeffe – and across the country – are still figuring out how to teach the standards. Lesson study, they thought, would be a good way to do that.

“I’d rather struggle together than struggle by myself,” says Flores. She liked the idea of lesson study right away.

Flores, Warner and Allen decided to plan a math lesson that would focus on the third grade Common Core math standards for geometry. They noticed that kids often struggled with understanding how to find the area of a shape. Memorizing the formula “length times width” wasn’t a problem for many of them, but they didn’t seem to understand what the formula meant. If they were asked to find the area of an odd shape – a parallelogram or a few rectangles put together – kids often had no idea where to begin.

It took months of planning and consultation to come up with a lesson plan.

“It’s a lot of meeting after school,” says Warner.

That gets a laugh from her colleagues. They don’t get paid for this extra time. Their principal, Bankhead, does arrange for subs to come in occasionally to free them up for planning. But for the most part, doing lesson study requires teachers to be willing to work at night and on weekends.

“The pay is in the results,” says Allen. “You’re getting better as a teacher.”

Warner says lesson study has helped her think about teaching in a new way.

“It was about me before,” she says. “It was like, these are the things I’m going to teach you, and this is my end result.”

She was more focused on whether kids could demonstrate what they’d learned on an assignment or a test. She was less aware of how kids were actually learning.

Lesson study helps you “get into new habits as a thinker, and as an instructor,” Warner says. “And I see such a difference in my kids because of it. I feel like in the past, if my kids got an unfamiliar problem, they would just shut down, not know what to do. Now everyone’s creating a solution, and then we’re ready to talk about it.”

Lesson study is a welcome change from the old way of doing professional development, Warner says. It’s no longer “you going back to your classroom and stumbling around with an idea.”

Now, she says, there’s someone to give you feedback and say, try it this way.

“It’s turned my practice around.”

**Results**

Teachers at O’Keeffe haven’t been doing lesson study long enough to know what kind of impact it’s having on student learning. Other schools in Chicago that have been doing lesson study have seen test score growth, but there’s no way to know for sure whether that’s because of lesson study.

There is some evidence that lesson study improves teaching. A recent review of research on professional development in the United States looked at 643 studies on approaches to improving math teaching. Only two of the approaches were found to have positive effects on students’ math proficiency. One of them was lesson study.

Jasmine Bankhead, the principal at O’Keeffe, believes lesson study is working at her school.

“I’m seeing much better teaching, and there’s an attitude in the building that we’re all in this together,” she says. “That’s what we needed here. I know that as I plan and budget that I have to make room for this type of collaboration in my school, so that my teachers can continue to grow.”

Catherine Lewis, an American researcher who has been helping teachers in the United States learn lesson study for 15 years, says she recently asked one of the teachers she’d been working with, what’s the biggest change with lesson study?

She says the teacher told her, “The talk around the water cooler has really changed. We used to hide it when we had a failure. And everybody has failures in teaching. But we used to hide them. And now, we’re perfectly comfortable saying, ‘You know, I don’t have a good way of teaching division with remainders. What do you do? Can I come see it in your classroom?’”

It’s hard to know how many teachers in the United States are doing lesson study. There’s no official count. Lewis estimates thousands of teachers are doing it.

There’s even a whole state that’s trying it: Florida, which got a federal grant in 2010 to encourage its schools to adopt lesson study.

But lesson study can be challenging in American schools. There are practical challenges, like finding time for teachers to plan together and watch each other teach. Japanese teachers have this kind of time built into their work schedule.

And there are cultural challenges. The organizing principle behind Japanese lesson study is that the best ideas for improving education come from teachers. It’s a bottom up kind of approach.

In the United States, education improvement tends to be top-down.

“The American approach has been to write and distribute reform documents and ask teachers to implement those recommendations,” says Hiebert.

Lesson study flips the script. It’s one of the reasons so many American teachers who try lesson study like it. But it’s also why lesson study can be a fragile enterprise in the United States. There are plenty of stories about educators who start lesson study, then a new principal comes in with a different idea about how to do things, and lesson study falls apart.

Another challenge for lesson study in American schools is the fact that it’s a long and intensive process.

“We are so addicted to quick fixes,” says Hiebert. “If it doesn’t fix things in two years, it’s not worth it.”

We have this attitude about teachers too, he says. Research shows that teachers in the United States improve the most early in their careers, but after about three to five years in the classroom, they’re about as good as they’re going to get. If you’re not a great teacher after a few years, you might as well quit or be fired. That’s the thinking in the United States anyway.

But in Japan, you’re not considered an expert teacher until you’ve been in the classroom for at least 10 years. The Japanese take teacher learning seriously, Hiebert says. They believe teachers will improve if they work in a system that values improvement.

The United States needs that kind of system, he says.

“We have an education system that is always reforming, but not always improving.”

*Emily Hanford is an education correspondent for American RadioWorks, the national documentary unit of American Public Media. Check out the American RadioWorks website for a more in-depth version of this article. You can also read other articles about teacher learning and listen to the accompanying radio documentary program. American RadioWorks hosts a weekly education podcast available here. *

Pixar is constantly solving new technical challenges that allow its artists, designers and storytellers a broader range of movement and texture in the movies they make. Now the company is teaming up with Khan Academy to use examples like DeRose’s discovery of surface representation to show students how the math and science they’re learning in school is applied by Pixar animators.

Khan Academy is best known for its modular videos explaining various curriculum topics that students can use to better understand and practice a concept. But, like many other groups working on reaching large numbers of people through online video lessons, its content producers have discovered that lots of people stop watching partway through.

When Pixar started looking around for a distribution partner, Khan Academy content producer Brit Cruise got excited that this partnership, now known as Pixar In A Box, might keep people interested in the content longer.

“The place we intersected was this need to pull people in,” Cruise said. “I wanted to do better, but that required a paradigm shift.”

The Pixar In A Box lessons start with a technical problem that animators face and work into the math from there. In each video a real Pixar animator lays out the technical problem, and then students get to experiment with interactive elements to better understand the problem. Gradually the video works towards a more explicit explanation of the math involved, and by the end the student is calculating to solve the actual problems faced at Pixar.

For example, in the character modeling lesson, based on the surface representation work DeRose pioneered, students learn about weighted averages. An animator lays out the problem DeRose faced and then students get a chance to play with 2-D and 3-D shapes, manipulating different functions to create midway points and move them in ways that might smooth the shape. As they play, they begin to intimately understand the challenge.

The lesson then turns to an explanation of why weighted averages help create the smoothing effect needed to make skin look more real. Students then return to the same tool they used before, but the math behind it is exposed, in this case simple algebra.

“They have this more interactive intuition lesson,” Cruise said. “They’re not just calculating.”

He is also working to add hands-on activities to enhance the video lessons. Right now only two of the 12 modules have an interactive lesson, but Cruise is working with a group of teachers to develop others that can be done simply, in 40 minutes, with cheap materials.

“We really are trying to intersect art and math, and you don’t often see those things represented really well in hands-on activities,” Cruise said. The hands-on activities are meant to push both the artistic elements and the math concepts a little further, challenging students to continue applying the information to more complex situations.

The emphasis on art isn’t a coincidence. While the video lessons are currently about only math topics, Pixar In a Box producers are working on science lessons now (mainly computer science) and hope to make others for the humanities side of Pixar’s work as well.

“At Pixar, art and technology go hand in hand,” said Elyse Klaidman, director of Pixar University, an internal department that encourages Pixar employees to continue their learning. In school, math and science often seem completely divorced from the humanities, but at Pixar the storytellers, artists and sculptors must work hand in hand with computer scientists like DeRose, tasked with figuring out how to animate those ideas.

Klaidman said the creative demands of stories push the technical innovations that allow the studio to represent those ideas, but just as frequently a new technical innovation will spur the storytellers and artists to dream up new things.

“It’s the interplay between the two sides that’s essential to what we do,” Klaidman said.

The Pixar In A Box videos also do a good job of taking viewers inside the world of Pixar, into the offices and studios of real employees. And kids like feeling on the inside. “One of the things we found from eighth-graders was they were hungry for more of the Pixar personality,” Klaidman said. “At that point we were focused on the content.”

The videos show the people behind careers many kids have never considered. Students liked meeting real Pixar employees and getting a sense of who they are and where they work. This feedback prompted the content producers to add a “Getting to Know” section at the end of the videos, where viewers learn about the backgrounds of Pixar professionals and how they landed their jobs. The people featured in these interviews are intentionally diverse to help kids see themselves in many kinds of jobs.

While some teachers are already getting excited about the Pixar in a Box lessons, it’s worth noting they were designed for the individual user, who isn’t necessarily a student in a public school. Cruise was clear that in order to make the videos feel authentic, they wanted to start from real technical problems Pixar has solved and explain the math behind them. Not all those problems are explicitly related to the Common Core, although some are. Every module has a lesson guide, which maps the lesson to Common Core standards when applicable.

Instead, these videos are meant to delight, to ask people to be creators as well as learners, and to push users to finish online lessons. Cruise said he hopes teachers might consider assigning the video lesson at home so class time could be used for the hands-on activities and a deeper dive.

“I honestly think you don’t actually start learning until that stage,” Cruise said. “I’m a huge proponent of hands-on learning.”

All the materials are free and the initial lessons are often suitable for all ages. The follow-up lessons are more grade- and standard-specific and range from fourth grade through high school.

]]>In his TED Talk, mathematician Conrad Wolfram argues much of this angst is about how well students can compute by hand, not how well they understand math. He breaks math down into four steps:

1. Pose the right question about an issue

2. Change that real world scenario into a math formulation

3. Compute

4. Take the math formulation and turn it back into a real world scenario to verify it

Math as we teach it now, he argues, is mostly about step 3 – computation, the one part of the whole process that can be automated. However, Wolfram would like to see more attention paid to the other three crucial steps, which he says play a much larger role in enhancing both practical and conceptual understandings of math. He’d like to see students demonstrate understanding of the process and procedures of math through computer coding instead of computation. In this talk, he takes on some of the most common arguments for the current way of teaching math, including the fear that computers will “dumb down” students’ understanding of mathematics.

A study conducted by researchers at the Human Performance Lab at the University of Chicago has found that if parents have high levels of math anxiety and give a lot of homework help (more than two to three times a week), they can have a negative impact on their child’s math achievement and increase their child’s math anxiety.

“This is one of the first studies to look at how math anxiety is produced,” said Erin Maloney, lead author on the study. Researchers have already established that a teacher’s math anxiety negatively affects student performance over the course of the year and leads female students to endorse negative stereotypes about girls’ math abilities. For this study, Maloney and her co-authors Gerardo Ramirez, Elizabeth A. Gunderson, Susan C. Levine and Sian L. Beilock statistically controlled for teachers’ math anxiety and content knowledge, along with the socioeconomic status of the children.

“Kids as young as first grade really do report high anxiety about math and that anxiety is linked to their math achievement,” Maloney said. The study found a caregiver’s math anxiety negatively impacted both male and female children equally. And, the effects were bigger the more anxious a parent was and the more often they helped with homework. While not a study of female caregivers in particular, 89 percent of primary caregivers in this study were women.

The study was done with a large sample size of children in first and second grade that was designed to be representative of the Illinois state population. Students in the study completed tests measuring math, reading and math anxiety in the first 12 weeks of school. They were tested again in the last 8 weeks of school. Parents were given a 25-question survey that asked questions about how they feel in various real-world situations requiring math, like tipping at a restaurant.

“When parents are really high in math anxiety, their children, both boys and girls, learn less math over the school year and become more anxious over the school year,” Maloney said, but only in cases where the primary care provider helped with homework frequently.

The effects of math anxiety on achievement are well documented. Stanford researchers did brain scans of children solving math problems. In math-anxious children, the part of the brain tasked with handling negative emotions was overactive, whereas the math problem solving parts of the brain were diminished. The children’s fear was interfering with their ability to use problem solving skills.

Sian Beilock and her colleagues at the Human Performance Lab posited that math anxiety was taking up students’ working memory, impeding their ability to succeed on tests. They asked students to write about their math anxiety before a test. By offloading those negative emotions first, the intervention has produced encouraging results in the classroom.

But what are parents to do with this new information?

“It’s really not enough for us as educators to just say ‘get involved,’” Maloney said. She believes researchers and educators need to develop better tools for parents to understand the math their children are working on so they can help appropriately. Unfamiliar teaching methods heighten the math anxiety that’s already found in many American adults.

]]>In his funny TED talk, Spanish mathematician Eduardo Saenz de Cabezon says most mathematicians either defend math by arguing it exists all around us or attack the questioner by pointing out that math is beautiful and doesn’t need a reason to exist, like poetry. But Saenz de Cabezon says the real reason math is worth studying is that it’s forever. A proved theorem is more permanent than any diamond.

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This struggle may come from a fundamental misunderstanding about the discipline and how it should be taught.

That’s the stance David Wees has arrived at after more than 20 years of teaching at many different kinds of schools all over the world. It has taken a long time, but Wees has stopped labeling student work with the word “mistake” and has started paying attention to what he can learn about how students are thinking, based on the work (right or wrong) they produce.

“I want to know the ways that they are thinking rather than the ways they are making mistakes,” said Wees, who now works as a formative assessment specialist in mathematics for New Visions for Public Schools, an organization supporting public school teachers in New York City. “My interpretation that they’re making a mistake is a judgment and usually ends my thinking about what they are doing.”

In that situation, it’s extremely tempting to tell the student where he or she went “wrong” and move on. But what does the student learn in that scenario? Not much, beyond how to memorize computational formulas, said Wees.

“My goal is for them to become the truthmakers,” Wees said. “I’m trying to build a mathematical community where something is true when everyone agrees it’s true.” To do that, he asks students to talk through mathematical ideas, struggle with them and give one another feedback. “A major goal of math classrooms should be to develop people who look for evidence and try to prove that things are true or not true,” Wees said. “You can do that at any age”

Fundamentally, Wees wants to increase the amount of thinking “at the edge of their knowledge” that students do. “There’s lots of evidence that what we think about is what we know later,” he said. “I want to increase the amount of thinking going on in math class.”

Wees points out that while practice is important, students are repeating an action with which they are at least a little familiar.

He wants students to struggle in the zone of proximal development, where they don’t quite understand yet but aren’t frustrated. When working in New York public schools, Wees found if he gave students problems to solve that allowed for different points of entry, all students could struggle together. One student might be more advanced than another, but if each could access some element of the problem, they discussed it together and either relearned core concepts or were exposed to more advanced ones.

For example, Wees asked his students to solve the Seven Bridges of Konigsberg problem. It goes like this: A river flows through the middle of Konigsberg, forming an island in the middle and then separating into two branches. The citizens of Konigsberg have built seven bridges to get from place to place. The people wondered if they could walk around the city in such a way that they would cross each bridge once and only once.

“The kids understood the problem and virtually all attacked it,” Wees said. “Some kids worked on it for weeks.” Wees posted it in the hallway and at one point almost all the ninth-graders were working on the problem. Students got tired of carefully drawing the bridges, river and city over and over, so they naturally began to abstract the map into something that looked like a graph.

No student solved the problem — in fact, the mathematician Leonhard Euler proved it was impossible. Wees showed his students Euler’s proof, and pointed out how similar their graphing was to his. Wees said kids were a little mad when they discovered there was no answer, but they enjoyed the experience and along the way realized that learning is about the process.

“Over time I tended to embed projects of various kinds because at the time I was thinking I needed to get them interested,” Wees said. “They weren’t interested directly in the mathematics itself because they’d experienced so much failure, so I was trying to get them excited.”

Slowly throughout his career, Wees began to see that projects could be more than just excitement builders — they could be the vehicle for teaching content and the assessment. And the range of mathematical ideas was much broader than he thought if he used his imagination.

“The range of mathematical ideas the kids struggled with were pretty wide,” Wees said. After working in inner-city schools, Canadian schools and international schools for expat kids in London and Bangkok, Wees has come to the conclusion that all kids make the same kinds of mistakes.

“It was clear to me that the mistakes in some cases were a function of the mathematics and the way kids think about the math, rather than whether the kid is rich or poor,” he said.

**MATHEMATICIAN’S LAMENT**

Over the course of his career, through trial and error, Wees came to see what Paul Lockhart describes in his essay, “The Mathematician’s Lament”:

By concentrating on what, and leaving out why, mathematics is reduced to an empty shell. The art is not in the “truth” but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity— to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you deny them mathematics itself. So no, I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes.

**KIDS ASK THREE KINDS OF QUESTIONS**

When doing his master’s in education technology and the pedagogy around it, Wees learned to categorize the three kinds of questions students ask and changed his teaching practice entirely. Kids ask questions: 1) to find out if they did the problem right; 2) because the teacher is standing near them and they can, and; 3) occasionally they ask “I wonder what if” questions, which show they are thinking about the math. Wees took to not answering the first two kinds of questions and encouraging the third.

“I went from really trying to answer questions and support them in that way, to really trying to think of questions that would support them to learn it themselves,” Wees said. He found himself often asking the same question, whether a student had gotten the problem right or wrong. He’d ask them to explain their answer or how they could check to see if they were right or wrong.

“I became better at having a poker face so I wasn’t communicating whether they were right or wrong,” Wees laughed. When students asked questions because he was nearby, he deferred them to their peers, who often explained the math quite well.

**THE TIME FACTOR**

Many math teachers will say a community of learners like Wees describes is a fairytale classroom with no time constraints and no standards to cover. They say their jobs depend on covering all the topics on the test and helping students correct their errors, not taking days to uncover the thinking behind that error. Wees acknowledges the limitations that many math teachers struggle with, but points out the way most people teach math now doesn’t work, so it could be considered a waste of time anyway.

“Whatever time people are putting in to teach mathematics is kind of wasted in many cases,” Wees said. “Are [students] learning anything that they can transfer, that they can use in other contexts? If they’re not doing these things, then I don’t know what they’ve learned.”

He points out students often did very well on the New York Regents test when teachers focused on teaching specific kinds of problems, but whether kids learned the full range of mathematics possible that year is another thing entirely.

Beyond time limitations, a broader problem is that many math teachers know only one way to solve the problems they teach. Even professional development often focuses on breadth instead of depth, with the result that many teachers carry the same fundamental gaps in math understanding as their students.

“We have generations of math-phobia,” said Laura Thomas, director of the Antioch Center for School Renewal. “A lot of teachers who teach math are second- and third-generation math-phobic, so our system is really calculation-based as opposed to applying in context.”

Thomas said it takes a person with deep understanding of both math and project-based pedagogy and coaching to effectively lead students through what is often a very messy process requiring students to use problem-solving skills to figure out solutions, rather than being told what skills to apply.

Wees is frustrated at how linear math learning has become. “The standards are a list of things the kids are supposed to do, not a list of things you have to teach,” Wees said.

In other words, many standards can be embedded in a problem so that students are exposed to lots of ideas in different ways. When teachers focus on clusters of standards as opposed to individual ones, “that kid who doesn’t get one idea on Thursday is going to get 10 or 12 other ways of looking at the idea in the unit,” Wees said.

For example, a teacher might give students this math problem: “I’m traveling 50 mph. How far will I have driven in 10 minutes?” This problem does not confuse students. They know what they are being asked and in discussing it they could hit many standards — multiplication, number lines, writing down possible solutions to think it through and fractions, to name a few.

“The kids get exposed to all of the standards every day in different ways,” Wees said. And more importantly, they’re having to think through the standards every day, leading to a deeper level of learning.

“You really have to understand math is a range of ideas and not individual standards,” Wees said.

When teachers are comfortable teaching in this more complex style, they are able to offer the multiple points of entry that allow for differentiation to take place — but in community, not isolation. If students are segmented out to learn only with the students “at their level,” some students will be in danger of never moving past fractions.

]]>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.

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