One after another, these young women, who had all graduated from an urban high school serving many kids living in poverty, described how math class made them feel safe, heard and able to express their ideas without fear.

“I felt like they cared for me,” said Martha Hernandez, who graduated in 2002 and is now a social worker. “They cared for my education and they wanted me to succeed.”

Hernandez was designated an English language learner in high school and was the first in her family to go to college. She loved her math classes so much that almost 15 years later, in the NCTM session, she held out physical examples of her work as she cried about the impact the non-traditional math program at Railside High had on her confidence and future success.

“It changed what math meant,” said Maria Velazquez, who now studies education policy at the University of Wisconsin. “It was a process and it required other people. It wasn’t just you and your work and not talking.”

Before high school, these young women, like many students in the U.S., experienced math as lecture, sitting at desks quietly. Many believed they weren’t good at math because they didn’t understand or compute quickly. But the math program at Railside High changed that for each of these women, showing them their strengths and allowing them to bring all of themselves to the pursuit of mathematics.

But what was so different about how these women learned math in high school? How did their math teachers form bonds so strong that years later they were attending students’ weddings in Mexico?

The answer: Complex Instruction. This pedagogy is not specific to math and has been in the literature for decades, originally researched by Elizabeth Cohen and Rachel Lotan at Stanford University. Teachers at Railside High discovered the methodology when they were undergoing an accreditation review and were told they needed to drastically change something to improve their results. The ultimatum prompted teachers to try something different — heterogeneous classes, high expectations for all students and, above all, approaching math with an eye to students’ strengths.

The three main tenets of Complex Instruction are that learning should have multi-ability access points, norms and roles that support interdependency between students, and attention to status and accountability for learning. In most Complex Instruction classrooms the majority of class time is spent with students working in groups of four on a rich task that has multiple entry points and ways it could be solved. If one student can solve the problem in his or her head, it’s not a rich task.

Each student in the group has a role: team captain, resource manager, recorder-reporter and facilitator. While these roles might sound cheesy to some students, they are important for helping groups to work equitably, ensuring that every group member has a crucial and intellectual task. The roles help students learn how to effectively participate and, because each role is necessary to solve a task, everyone must share their ideas.

“Participation leads to more learning because learning is a socially constructed activity,” said Lisa Jilk, program director of Reculturing Math Departments for Excellence & Equity, part of the Mathematics Education Project at the University of Washington. Jilk taught at Railside High, and when she left to get her doctorate she studied how and why Complex Instruction worked for so many students from various backgrounds. Now she’s dedicated to helping other math departments around the country “reculture” themselves to think about what learners bring to math that will help them, rather than only about the information they are missing.

When the three tenets of Complex Instruction are all working together simultaneously it can feel like a magical experience. But getting there takes a lot of work. When Jilk starts training teachers, one of the first things that must be discussed is the idea of status in the classroom and how to break that down. Teaching with Complex Instruction is intimately tied to research in educational psychology, which says that to succeed students need more than content knowledge — they need to see themselves as efficacious learners.

That is particularly hard in math, where many students believe they are dumb or incapable because of past math learning experiences. To combat that, a core part of Complex Instruction is to teach with a strengths-based approach, rather than only seeing student deficits.

“Every person who walks through our doors has mathematical strengths,” Jilk said. “They also have mathematical needs or weaknesses, things they have yet to learn. So we need each other.”

High School CI math teachers doing math and planning collaboratively. @SFUSDMath pic.twitter.com/O3ieqEsYEJ

— SFComplexInstruction (@SFComplex_Instr) February 3, 2016

The Complex Instruction model works because when students work in groups to grapple with a rich math task (Jilk says College Prep Math is a good place to look), they are each encouraged to bring their full personality and ways of seeing math to the task. The teacher’s job is to observe what’s going on within groups and assign status when she sees a great idea, technique or way of thinking.

“You definitely can’t fake these moments,” said Yuka Walton, a seventh- and eighth-grade math teacher at James Denman Middle School in San Francisco. “You can’t assign competence or publicly acknowledge kids for things that aren’t meaningful because then it feels super fake.” Kids are great at detecting inauthentic praise, which ends up sounding condescending.

But when a teacher recognizes competence in students who don’t often feel like they have much status as a math learner, it can make a huge difference. Walton remembers one student, Alexis, who would often push the limits in class and consistently referred to herself as bad at math. One day in group work, Walton’s Complex Instruction coach noticed that Alexis was using a really smart, unique technique to organize the numbers in the problem, and her method was propelling her group’s thinking forward. Walton publicly acknowledged how smart that specific technique was and why it was adding value to the group. From then on, the whole class started calling that technique the “Alexis Method.”

“It helped her feel ownership over her own learning and her own smartness and power,” Walton said. Over time, Alexis built an identity as a math person, and as she had more confidence in her ability to contribute to her group, other students started assigning her status on their own by asking her for help. In order for teachers to assign competence well, they need to be open to many ways of solving the problem and many kinds of “smartness.”

Tracy Thompson teaches math at George Washington High School in San Francisco. Her math department was one of the first in the district to take on Complex Instruction seven years ago, before San Francisco made the decision to detrack math classes through sophomore year of high school. When Thompson started trying this approach, she had a group of juniors taking a class called “Applied Math,” an alternative to Algebra II that mostly low-performing math students chose to take. The class counted for graduation credit, but many students couldn’t wait to finish and be done with their math requirements.

By the end of that year, students had changed their tune. “Most of the kids that were juniors told me on their own that they wanted to go to Algebra II now,” Thompson said. Even though these students came from 10 years of school where they felt bad at math, with one year of strengths-based instruction that focused on kids working together to figure out interesting problems, they wanted to take on more challenging math.

Both Thompson and Walton were clear that this is difficult work and that it doesn’t happen overnight. It can be overwhelming for teachers to balance all the elements: designing or choosing a rich task for every lesson, monitoring status issues, holding students accountable to the norms and roles of group work, and not helping too much when students struggle. It doesn’t always go perfectly. But both teachers say they’d never go back to teaching any other way.

“The most important thing is it makes you see so much more clearly,” Thompson said. “Even though things aren’t perfect, it gives me these tools to work with and it just becomes part of the lesson planning process.” Now, when a student is unengaged in the lesson she doesn’t assume he’s lazy. Instead, she tries to find ways to make the classroom a dynamic, comfortable place for him to share his ideas and to participate.

“It has expanded my thinking about what makes you smart at math,” Thompson said. “It’s really helped me understand that there are different strengths that people have and that also the fastest calculator is not the best math student always.” Thompson now teaches both Algebra II (which all juniors take) and Calculus BC, one of the few tracked classes for high achievers. She says she has more trouble getting her calculus students to explain their thinking because they believe the best students are godlike and don’t push on their thinking.

**RECULTURING MATH DEPARTMENTS**

San Francisco has been training teachers in Complex Instruction for seven years. The district started by focusing on high schools, bringing in cohorts of teachers who worked at the same school in order to build a community that could collaborate on this difficult and transformative work.

“We’re broadening this idea of smart,” said Angela Torres, high school math content specialist for SFUSD. She and Ho Nguyen have championed the Complex Instruction program within the district, slowly broadening its reach as teachers heard about the program and expressed interest.

“We literally have to reculture these spaces so we are providing people with a new message and a new narrative about what they bring, the strengths and smartness they bring, and redefine what they’re capable of,” Jilk said.

Just a few years after San Francisco began dabbling in Complex Instruction, California adopted Common Core standards, which require more focus on the conceptual underpinnings of math, explaining thinking and reasoning, and less focus on procedural quickness. The SFUSD math department responded to the new standards by inviting teacher leaders to help them write the new math curriculum, pilot test it and offer feedback. They’re still iterating on that work, but the result has been a more engaged math team throughout the district, and more interest in strategies like Complex Instruction that can help teachers get students where they need to go.

“It took us really about four years to really understand what it takes,” Nguyen said. “And it wasn’t just about teacher change. It was really about reculturing the math department. We had to go through our own struggles.” SFUSD teachers have received training from Lisa Jilk’s organization, including classroom coaching.

The district has also been working to build up its own capacity to coach teachers through Complex Instruction so they can continue sustaining and broadening the program’s reach throughout the district. Coaches watch teachers as they teach and often provide on-the-spot feedback when they notice a student displaying a strength that the teacher missed. The coach will often nudge the teacher to acknowledge that student, sometimes to the whole class, as a way of breaking down some of the status issues in the classroom.

Torres and Nguyen have strategically tried to build teams of teachers at school sites who have incubated the ideas and continue pushing each other. As with students, teachers each have their own strengths and issues of status. Working together to develop rich math tasks, align assessments and discuss strategies has helped them experience the kind of learning environment they are trying to create. And there are meetings to connect educators across the district doing Complex Instruction, as well as a “video club” to practice identifying and assigning competence to different students.

“When grading we see students are able to think in this critical way that they weren’t able to do before,” Walton said. She used to teach in a district that used direct instruction, a type of teaching that came naturally to her. But she noticed that her students struggled as soon as a problem involved something that had not been explicitly taught.

“After doing Complex Instruction, it didn’t matter how complicated the problem was. Even if kids hadn’t seen it before, they would dive right in and get started,” Walton said. Even better, “you see these moments where these kids who before were so discouraged, brighten up and engage and feel more empowered. It has made it so much more meaningful.”

All the teachers and coaches involved in Complex Instruction stress that like any other truly transformative teaching practice, getting good takes time. For this style of pedagogy to work well all three elements of the program must be in place and functioning simultaneously. Teachers have to have high expectations for all students, and a real belief that each learner is coming to the experience of learning math with strengths, not just gaps in learning. It takes time to get good at listening for authentic moments of brilliance in student work, and to help students create the interdependence on one another necessary for strong group work.

“If you do only one thing, and that is to create opportunities for kids to leverage their strengths in your classroom activities and then name those strengths for them, if you can create those strengths for them, you will already be changing things for most kids in ways that are otherwise not possible,” said Jilk.

And when it all starts to come together, and every student is in the “sweet spot,” it’s like magic. That’s when students start to feel the connection and recognition that the graduates of Railside High were so grateful to have experienced.

]]>In this first video about prisoners’ hats the problem set-up ends at 1:35, so stop the video there if you want kids to work on the problem before learning how to solve it.

In this zombie bridge problem the set-up ends at 2:00.

The riddle of the 100 green-eyed logicians ends at 1:53.

Don’t miss an episode of *Stories Teachers Share*.

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]]>But turning the difficult experience of failure into a positive isn’t as easy as telling students to change their mindsets; it takes careful lesson design, a strong classroom culture and an instructor trained in getting results from small failures so his or her students succeed when it matters.

Manu Kapur has been studying what he calls “productive failure” for most of his career, attempting to turn the general advice to “learn from mistakes” into a clearly defined, specific pedagogical design process that yields strong learning results. Now a professor of psychological studies at the Education University of Hong Kong, Kapur has conducted both quasi-experimental and randomized controlled trials on how teaching through productive failure measures up to both direct instruction as well as more constructivist problem-solving approaches.

For Kapur, productive failure is not just a maxim about persisting through challenges; it’s an effective teaching strategy that enables students to not only do well on short term measures of knowledge, like tests, but also affords better conceptual understanding, creative thinking, and helps students to transfer learning to novel situations.

“Learning from failure is a very intuitive and compelling idea that’s been around for ages, but teachers may not know how to use it,” Kapur said. He has run enough experiments both in lab settings and in real classrooms to have a fairly good idea of how to structure lessons that include failure up front, followed by consolidation of understanding through instruction.

The general idea is to develop tasks that students will not be able to solve, but require them to call upon their preexisting knowledge to try to solve the problem. That knowledge can be of the subject itself, as well as the informal insights students bring from their lives. The students will inevitably fail — as the teacher expects them to — but that failure is framed as part of learning and so is not seen as shameful. This process primes students’ brains to learn the new concept from their instructor after the initial failure.

“It is failure-based activation of knowledge to prepare them to learn,” Kapur said.

It might seem like this process would frustrate kids until they stop trying, but Kapur’s studies found that instead of feeling bad about their inability to solve the problem, students’ interest in the concept spiked. “I think that’s a great place to get students to before we teach them something,” Kapur said.

After students experience failure in their own discovery and problem solving process, the teacher facilitates a discussion that highlights various student attempts and teaches the new concept, consolidating students’ understanding of the processes required to complete the task.

**PRINCIPLES OF PRODUCTIVE FAILURE LESSON DESIGN**

- Tasks must be challenging enough to engage learners, but not so challenging they give up.
- Tasks must have multiple ideas, solutions or ways to solve so that students generate a multitude of ideas. It cannot be a closed task with only one path to finding a correct answer.
- The task must activate prior knowledge, and not just formal learning from a previous lesson. “If you design a task where a student only displays their prior class learning it’s not good because then you aren’t tapping into their intuitive reasoning,” Kapur said. Intuitive reasoning is a big part of how students transfer knowledge to new situations.
- While the task should activate knowledge, it should be designed so that the knowledge students have is not sufficient to solve the problem. They should hit a roadblock that they can’t get around. “It makes the child aware of what he or she knows, and the limits of what he or she knows, and that creates a motivation to figure out what it is they need to know to solve this problem,” Kapur said.
- It helps if that task as an “affective draw,” in that it’s related to something students care about or is concerns something with which they identify.

Kapur has tested productive failure teaching strategies with students of varying abilities in Singapore and has found it to work with all students, regardless of ability. “Initial pre-existing conditions between students do not predict how much they learn,” Kapur said. “How they solve the initial problem is what predicts how much they learn.”

Singapore tracks students into ability-based schools after primary school, which makes it easy to conduct research that compares low, middle and high achievers. However, Kapur has also tested productive failure in Indian schools in which students were not grouped by ability. He saw good results there as well. “The task is open enough that kids from different abilities can work together,” Kapur said.

Part of Kapur’s research has been to show that teaching with productive failure doesn’t harm students’ ability to perform on tests, but does improve knowledge transfer and conceptual understanding. In the process he’s discovered an interesting element of creative thinking in math that appears to disprove the generally held notion that students need basic content knowledge before they can move on to more creative uses of the information.

“We’ve found that creativity actually suffers if you teach kids something too early,” Kapur said. When students who have been taught with direct instruction are later asked to generate as many ways of solving the problem as they can, many can’t go beyond the method they have already been taught.

“They were locked into that way of thinking,” Kapur said. “When we start with generating or exploring we find that students still learn the material later on, but the knowledge was more flexible.” This finding tells Kapur that creativity is itself a function of how students’ acquire information.

**SINGAPORE TAKES IT TO SCALE**

Kapur’s research on productive failure has convinced Singapore’s Ministry of Education to use the pedagogical model for the statistical portion of it’s A-level curriculum. Statistics make up about one third of the Cambridge A-level exam, Kapur said. All university-track junior-college students in Singapore are in school to pass that exam (junior-college in Singapore is like high school in the US).

Although Singapore’s education system is very test-based, its Ministry of Education is interested in research-proven pedagogical approaches that lead to lasting learning beyond the test. “There is a very strong policy emphasis on changing how we teach,” Kapur said of Singapore. “Just because there are tests does not mean we can’t teach in ways that lead to very deep learning while doing well on the tests.”

Kapur was able to show that productive failure worked well with students at the least prestigious of Singapore’s 20 junior colleges, which provided a compelling proof of concept to scale up to all students studying for the Cambridge A-levels. Kapur and his team have designed a curriculum of tasks that use productive failure, and are training Singapore’s teachers in the method.

The concept is new to many Singaporean teachers and Kapur says the first part of his training focuses on helping teachers understand the problems with direct instruction. He uses the analogy of watching a film. The average viewer focuses on plot, and perhaps pays some attention to acting ability or cinematography. When a director watches the same film, on the other hand, she is likely noticing nuances of camera placement, shot selection, and much more. That’s the difference between what a novice sees and what an expert sees.

“No matter how engaging, entertaining or logically structured the new information is, the novice by definition is not going to see the same thing as the expert in the presentation,” Kapur said. He works to help teachers understand the flawed assumption that students will understand after a concept has been told to them, explaining that direct instruction doesn’t prime students’ brains to process the new information.

“We won’t make the assumption that you’re prepared to learn yet; what we will do is activate your formal and informal knowledge systems,” Kapur said.

The teacher training program also focuses on improving teachers’ content knowledge. Working with student ideas and misconceptions requires the instructor have a deep understanding of the subject matter. Finally, Kapur helps teachers improve on important pedagogical aspects of this model like facilitating group work and consolidating ideas after students have grappled with a problem and failed.

“Your job as a teacher is to first prepare them, to give them the proverbial eyes to be able to see what is important, and then show them what is important in interesting and engaging ways,” Kapur said.

Singapore’s Ministry of Education has agreed to give Kapur’s team four years to build teachers’ capacity in this new style of teaching before evaluating its effectiveness. Kapur sees this as a huge gift, knowing that the effectiveness of any program lies in its implementation and that it takes time to get people up to speed.

]]>Stanford professor Jo Boaler writes in The Atlantic about the neurological benefits of using fingers and how it can contribute to advanced thinking in higher math.

Stopping students from using their fingers when they count could, according to the new brain research, be akin to halting their mathematical development. Fingers are probably one of our most useful visual aids, and the finger area of our brain is used well into adulthood. The need for and importance of finger perception could even be the reason that pianists, and other musicians, often display higher mathematical understanding than people who don’t learn a musical instrument.

Boaler has developed research and curriculum to support a more engaging way to teach math by applying visual thinking, numeracy and growth mindset. Her program, YouCubed, at Stanford University, helps students and teachers get past roadblocks to learning math. Math anxiety has been well-documented as an obstruction to learning math. By drawing attention to these disparities and rethinking how math is taught, Boaler is creating a wider path for students, and adults, to develop a love of math.

It is hardly surprising that students so often feel that math is inaccessible and uninteresting when they are plunged into a world of abstraction and numbers in classrooms. Students are made to memorize math facts, and plough through worksheets of numbers, with few visual or creative representations of math, often because of policy directives and faulty curriculum guides. The Common Core standards for kindergarten through eighth grade pay more attention to visual work than many previous sets of learning benchmarks, but their high-school content commits teachers to numerical and abstract thinking. And where the Common Core does encourage visual work, it’s usually encouraged as a prelude to the development of abstract ideas rather than a tool for seeing and extending mathematical ideas and strengthening important brain networks.

## Using Fingers to Count in Math Class Is Not ‘Babyish’

Evidence from brain science suggests that far from being “babyish,” the technique is essential for mathematical achievement. Please consider disabling it for our site, or supporting our work in one of these ways Subscribe Now > Stopping students from using their fingers when they count could, according to the new brain research, be akin to halting their mathematical development.

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Recently, Desmos has been building out its platform to offer customizable lessons. Led by Chief Academic Officer Dan Meyer, a former math teacher who left the classroom to pursue a PhD in math education, Desmos has been using its platform to model how technology could change pedagogy.

Desmos tries to harness the social nature of online interactions into meaningful math inquiry. Meyer says students love the internet because it’s a social place to share and create. And, a math classroom at its best is also a place where students are creating hypotheses, testing their thinking, critiquing each other’s work and discussing how and why mathematical laws work.

“Typically, online math platforms have no concept of the student in relationship to other students,” Meyer said in reference to “personalized” programs where students work through a set of problems or concepts “at their own pace,” but do so in a vacuum. Meyer argues this model doesn’t capture what’s powerful about a class full of students.

In contrast, Desmos allows teachers to make a series of slides with interactive elements. One slide might have a video of a glass filling with water, with a question asking students to graph it. When a student submits his graph, a Desmos default function then shows the student three other student answers and asks him to give feedback on the solutions. Teachers can shut off this function, but Desmos intentionally made it a default to encourage discussion.

“Our core assumption is that students need to be aware that there are other students in the class, and that refining process is part of every math lesson,” Meyer said. Teachers could also have students create their own glass filling at a specific rate and ask them to graph that. Or, maybe each student submits the problem they devised for other students to work on. Teachers can also display all the solutions, with or without student names, and ask the class to analyze each other’s strategies.

Meyer hopes to support teachers as they design lessons that have some common elements of good math instruction: Students are thinking beyond equations, the learning is social, processes are made visible and students get written feedback, often from peers. “Getting numerical or binary right-wrong feedback tends to make the student think about the self,” Meyer said. “Written feedback about the work tends to focus on the work itself.”

**HOW ARE TEACHERS USING DESMOS?**

Middle school math teacher Cathy Yenca started using Desmos when her district discovered the graphing calculator tool was going to be integrated on state tests in Texas, where she teaches. Wanting her students to be familiar with the tool, she started experimenting with it in the classroom and it has now become an instructional “necessity.” Yenca works at Hill Country Middle School, a public school in Austin. Her students all have iPads and Yenca is passionate about the power of tech in learning, but she hasn’t liked a lot of what’s out there for math specifically.

“When you come across something that’s not just skill, drill, kill, and is kinda rich, it gets your attention,” she said. She uses Desmos because it makes inquiry in math class easy.

For example, in one recent lesson with her algebra one students, Yenca was teaching transformations in quadratic functions. But she didn’t tell students that’s what they were studying. Instead, she set up a task where through exploration they were telling her the lesson by the end of class.

“Watching how they change a parameter and that instant feedback of what they just changed and how that impacts a graph, they’re hooked,” Yenca said. “When you have that reaction from middle schoolers around math, that’s a win.” She sees how peers influence one another’s thinking, and that even if not every student is on the right track the whole time, they are figuring it out together.

Yenca often takes a moment to show the whole class everyone else’s answers. This overlay would typically be what a teacher would use for formative assessment, to check for understanding, but Yenca said, “to me, keeping that for the teacher’s eyes only is a disservice.” She says learning this way requires building a classroom culture that values mistake making, but once that’s in place, so much rich learning comes out of students being able to see the trends in misperceptions. Together they discuss and untangle thinking until they’ve arrived at the math concept.

In a lesson with her eighth-graders on shapes in the coordinate plan, Yenca had students creating reflections and dilations on a Desmos graph. “Desmos is only going to do what you tell it to do, so if it does something you didn’t expect, you’ve got to figure it out,” Yenca said. Her students manipulate the variables and gradually come to an understanding.

Yenca says some of her colleagues think Desmos does too much for students, making graphing too easy. She says if rote graphing is the goal, then yes, Desmos does too much, but wonders if that’s the right goal. “If we are concerned that a graphing tool can graph for our kids, maybe we need to ask more of our kids,” Yenca said.

While Desmos is trying to make it easier for math teachers to incorporate these elements into classrooms, the platform doesn’t force the issue. The tool is completely open; teachers shape their own lessons within it, and could easily make something that looks essentially like a worksheet.

**REDEFINING CLASS**

Audrey McLaren teaches at a virtual school in Quebec. Her classes are all completely online, although they happen in real time, with students participating as they would in a brick-and-mortar classroom. Most of McLaren’s students live in rural places and their local schools don’t offer the courses she’s teaching in English.

“We’re not in the same physical space as our students,” McLaren said, “so we couldn’t see what they were doing until Desmos activity builder came along.” Now, she can pose a problem to students and then watch as each student tries to solve it. She can look at the class as a whole or zoom in and interact one-on-one with a student. McLaren thinks students are participating more than they would be in a normal class because every student has to do the work and share their thinking, whereas in a typical classroom (where she taught for 20 years) only about 10 percent of students raise their hands and participate in discussions.

“I try to design things so that after three to four slides, or questions, I stop everybody and within the online environment we put everything up on the board, classify the findings, and talk about which ones they agree with and why,” McLaren said. She uses the early slides to let students have a discovery period, where they’re playing with a concept, developing hypotheses and looking for patterns. “I want them to get an intuitive sense of what I want them to know; I don’t want to just tell them,” McLaren said.

She appreciates how Desmos will put forward a new kind of activity, let teachers play with it, and then open the tool so educators can build something similar. For example, Desmos has a few math “games,” but unlike many games that are basically practice with a prize at the end, the Desmos games make math knowledge central to completing the task.

In a Polygraph, for example, a pair of students might be given 16 graphs that all look different, but are all linear functions. One student chooses a graph and her partner has to guess which graph has been chosen by asking “yes” or “no” questions, a bit like the game Battleship. Students have to use math vocabulary and knowledge of terms like slope and y-intercepts to eliminate various graphs and zero-in on the correct choice. “You learn math from playing the game itself,” McLaren said.

At first Desmos created these Polygraph lessons around different common curricular topics, but now they’ve opened it up so teachers can make their own. She often uses Desmos in class as a way to explore a concept and then has her students watch a video at home to nail down the concept. But she also uses Desmos to deepen understanding in the middle of a lesson and as formative assessment at the end as well. Since she’s been using the Desmos activity builder for only a few months, McLaren doesn’t have any data to prove that teaching this way is improving math achievement. But anecdotally she’s confident it has increased participation, which should increase understanding, and she’s been impressed at how her students are discussing and writing about math with one another.

**TEACHER COMMUNITY**

In Bob Lochel’s Advanced Placement statistics classroom, getting technology to each student is a challenge. Even though he teaches in an affluent suburb of Philadelphia, his students don’t have one-to-one access and booking a computer lab can be a pain. So often Lochel relies on his students’ personal devices for access to Desmos. He’ll ask students to complete a few questions and then, like McLaren and Yenca, he often projects multiple student answers on the board as a jumping-off point for a discussion. Student are critiquing one another’s thinking. “That’s not the kind of thing we were asking before,” Lochel said.

One of his favorite parts of using Desmos is the community of math teachers that comes with it. Every educator spoke about the collaborative community of teachers sharing ideas using the hashtag #MTBoS. Many of the active educators in this community also write their own blogs, where they track the success and challenges of different lessons. Teachers can upload their lessons to Desmos as well, making it easy to find and use all or part of another teacher’s work for their own purposes. Lochel said often if he’s putting together a lesson and isn’t quite sure if it’s reaching the mark, he’ll put it out to the community for feedback.

Just like developing all good lessons, Lochel said it can be tricky to design a Desmos activity that both allows students to be creative and inspired, but also drives towards the ultimate goal for the class period. He appreciates the virtual community of educators that are helping him refine this skill. Lochel said when a lesson successfully allows students to arrive at their own conclusions, like the one he did on binomial distribution and how it’s linked to normal distribution, students understand in a much deeper way. Instead of telling them the rule, “this time they discovered the rule,” he said. And the buy-in that creating the rule engendered meant that they could also debunk the rule.

Desmos employees like Meyer, for their part, are constantly working with teachers to improve what the platform offers, while balancing a desire to seed good teaching practices. Meyer said while thousands of teacher lessons have been uploaded to Desmos, only a fraction are available through the search tool. Those are the lessons that he and his educator team have hand-polished, reaching out to the original author for permission, and re-releasing. He also looks at a random sample of teacher-created lessons every week and believes the quality has gone up over time.

The best part of Desmos for many educators, whether they are using it only for its graphing calculator capabilities or for these more involved, inquiry-based lessons, is that it’s free. That’s possible because Desmos licenses its calculator tool to curriculum and testing companies. The fees from that work fund the curriculum development and training work that Meyer does. He’s hopeful that before too long he and his team, in cooperation with teachers around the globe, will have developed what amounts to an Algebra I curriculum designed entirely out of low-floor, high-ceiling Web-based tasks like the ones described in this article.

“I was worried that we couldn’t figure out how to make “good” work in the market, but it’s been nice that we’ve found traction with paying customers,” Meyer said. “Part of that is the product and part of that is that there’s been a sea change in online math education.”

]]>- I have a quarter, a dime and a nickel. How much money DO I have?
- I have three coins. How much money COULD I have?

The first question is a basic arithmetic problem with one and only one right answer. You might find it on a multiple-choice test.

The second is an open-ended question with a number of different possible correct answers. It would lend itself to a wide-ranging debate over the details: Are these all American coins? Are any of them counterfeit? Do you have any bills?

Frankly, it’s a lot more interesting than the first.

Andrew Hacker is professor emeritus of political science at Queens College, City University of New York, and the author of several more-or-less contrarian books about education, some of them bestsellers.

His latest is called *The Math Myth: And Other STEM Delusions.* It poses many nagging, open-ended questions like the second example above, without a lot of neat, tied-up-with-a-bow answers like No. 1.

Hacker’s central argument is that advanced mathematics requirements, like algebra, trigonometry and calculus, are “a harsh and senseless hurdle” keeping far too many Americans from completing their educations and leading productive lives.

He also maintains that there is no proof for a STEM shortage or a skills gap; and that we should pursue “numeracy” in education rather than mathematics knowledge. And, furthermore, that we should teach numeracy in an active, engaged, social way, with more questions like No. 2.

**How do you define numeracy? **

Being agile with numbers. Regarding numbers as a second language. Reading a corporate report or a federal budget. This is not rocket science–it’s easy to do. Kids become numerate up through 5th or 6th grade.

**And what is the difference between numeracy and mathematics? **

There’s a firm line between arithmetic and mathematics. When we talk of quantitative skills, 97 percent of that is arithmetic. Mathematics is what starts in middle school or high school, with geometry, algebra, trigonometry, precalculus and calculus.

**Why are Americans apparently so bad at teaching and learning math?**

When I say most of it is badly taught, what I really mean is that most teachers just can’t really rouse enthusiasm for math among 90 percent of the students. Surely you’ve had such teachers.

**No comment. But lots of people have raised the alarm about this. Why isn’t the solution just to have math teachers, and students, work harder and do a better job?**

I’m saying: No, we don’t need that many people studying mathematics. We’re shooting ourselves in the foot. One in five people don’t graduate high school — this is one of the worst records of developed countries. And the chief academic reason is that they fail algebra — of course there are other nonacademic reasons, like prison and pregnancies. In our community colleges, 80 percent don’t get a college degree. The chief reason is that 70 percent fail remedial math. And even in our four-year colleges, 40 percent don’t get B.A.s [after 6 years]. And the biggest reason is they fail freshman math. We’re killing our kids. We’re destroying their futures because of this requirement. I think it’s outrageous and we’re doing a lot of harm.

**But what’s the alternative? Simply dumbing down the curriculum so everyone can pass?**

When I first wrote the article “Is Algebra Necessary?” in the *New York Times*, most of the letters I got were from people who love math, are good at math and believe everybody should have to do it whether they like it or not. And again and again they talk about how mathematics teaches rigor, it’s tough. There’s this whole discipline thing. It’s like as if math is an enforced number of pushups.

I’m not anti-math. It’s a grand human achievement up there with chess and crossword puzzles.

**But you don’t want everyone to have to master chess to get a high school diploma.**

I’m going to be very careful about what Andrew Hacker wants to be compulsory. What I would like is for math teachers starting in high school to make the subject so fascinating that kids will want to take it. In writing the book, I went out and sat in on two dozen math classes from Virginia to Michigan to Mississippi. In some of them — not too many — the teachers were so infectiously enthusiastic that the kids joined in. And I wish we could bottle what they do and spread it around.

**What about the need for more people with STEM skills?**

Well, we certainly need people who know how to do coding. When it comes to engineers, according to the Bureau of Labor Statistics, we’re producing all the engineers we need. The skills shortage is a myth. The chief shortage is getting people who will work for low wages. That’s why companies in California want to bring people in on H-1B visas who will live eight in a room and do coding for a small amount above minimum wage.

**What impact do you think the Common Core State Standards are having on math learning and teaching?**

They’re expecting everybody to get almost up to the SAT level in high school. Either there’s going to be massive failures, or the states will ratchet down the requirements.

**You taught your own alternative numeracy course at Queens College designed to make students more agile with numbers. How did you make the topic more appealing?**

I had 19 students. I broke them up in groups of three or four. Math is always highly individualized, but in the world of work we want people to work in teams. I’d give them exercises, like, ‘How would you decimalize time?’ It’s really cumbersome the way we do it — we have a 60-minute hour, a 24-hour day, a seven-day week. How would you make a 10-day week or a 10-month year? Six different teams can come up with six different answers to that question.

Copyright 2016 NPR. To see more, visit http://www.npr.org/.

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In a Washington Post article, Moriah Balingit details one such program in Fairfax County, VA. Grade level teachers said they learned a lot from the trained arts-integration specialists and have tried to incorporate some of the strategies on their own as well. Balingit writes than an external evaluation of the program found it to be success at improving achievement as well:

“Researcher Mengli Song said the students in the program did not necessarily learn additional math content but they did demonstrate a better grasp of the material. And the effect was comparable to other early-childhood interventions. ‘It’s not a huge effect, but it’s a non-trivial, notable effect,’ Song said.”

## Teachers are using theater and dance to teach math – and it’s working

The children puffed out their chests and mimicked drama teacher Melissa Richardson, rehearsing their big, booming “rhino voices.” “Giant steps, giant steps, big and bold!” the kindergartners yelled in unison in a classroom at Westlawn Elementary in Fairfax County.

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In The Atlantic, Peg Tyre describes what makes these extracurricular (and often expensive) math programs so successful. She found that almost universally teachers in these elite programs focus on conceptual math, not memorization or formulas. Students use their conceptual knowledge to explore, conjecture, predict and solve open-ended problems. And it’s fun because the focus is on solving a complicated, challenging problems, not on practicing one skill over and over. Tyre writes:

“Rifkin trains her teachers to expect challenging questions from students at every level, even from pupils as young as 5, so lessons toggle back and forth between the obvious and the mind-bendingly abstract. ‘The youngest ones, very naturally, their minds see math differently,’ she told me. ‘It is common that they can ask simple questions and then, in the next minute, a very complicated one. But if the teacher doesn’t know enough mathematics, she will answer the simple question and shut down the other, more difficult one. We want children to ask difficult questions, to engage so it is not boring, to be able to do algebra at an early age, sure, but also to see it for what it is: a tool for critical thinking. If their teachers can’t help them do this, well—’ Rifkin searched for the word that expressed her level of dismay. ‘It is a betrayal.'”

Tyre’s article shows that with good instruction, kids from every background can succeed at the highest levels. The big question remains, how can this type of teaching become the norm for all students from the earliest ages? Without some big systemic changes, deep math preparation could quickly become yet another barrier to economic equality in this country.

## The Math Revolution

Please consider disabling it for our site, or supporting our work in one of these ways Subscribe Now > On a sultry evening last July, a tall, soft-spoken 17-year-old named David Stoner and nearly 600 other math whizzes from all over the world sat huddled in small groups around wicker bistro tables, talking in low voices and obsessively refreshing the browsers on their laptops.

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Reading is an opportunity for you to learn about your child’s world. Young children (roughly ages 2–6) are often described as egocentric. They see the world from a limited perspective. But adults can be equally egocentric. They often do not understand what the world looks like from a child’s point of view. As you read *with — *and not just* to — *your child, you may learn that she interprets events differently from you, that she sees things in the story that you did not, and that she learns from the story in ways you did not expect. Reading *with* can provide a window into your child’s mind as well as clues to nurturing her thinking.

It is important for you to read storybooks that have math learning as their primary goal. Counting books and shape books are of this type. Of course, goals are different from quality. “Anno’s Counting Book” uses beautiful illustrations to pose the challenge of finding different numbers of objects. Other counting books are conventional and tedious.

Another type of storybook does not aim to teach math explicitly, but contains important mathematical ideas embedded within the story. Goldilocks sees that the Baby Bear’s bed is the smallest, and that Mama’s bed is bigger than Baby’s but smaller than Papa’s. Also, Baby Bear is smaller than Mama, who is in turn smaller than Papa. The beds are in increasing order of size, and so are the bears. The order is more complex than it initially appears: Mama is both bigger than Baby Bear and smaller than Papa Bear. Also, there is a simple correlation between the size of the bears and the size of the beds: the bigger the bear, the bigger the bed.

So, the story of Goldilocks and the Three Bears contains some fundamentally important math ideas, some of which children find difficult, about relative size, order and the relations between two sequences.

When reading books, it’s important to realize that math is a broad subject. Clearly, a counting book or a shape book describing circles and squares are both explicitly about math, in the sense of the kind of formal math we usually learn in school. Although not explicitly about school math, Goldilocks entails relatively complex math ideas— order and correlation. Other storybooks deal in an informal way with patterns, spatial relations, measurement, addition and subtraction, and division — all of which are “math.” Indeed, it would be hard to find a non-math storybook that does not include everyday math in this broad sense. In fact, ordinary storybooks may contain more interesting math than do explicit math storybooks (and textbooks, too!).

This in turn raises the question of the type of math you want your child to learn — school math or embedded math? The answer is both. Children need to memorize the counting words, but also need to know that their order specifies relative magnitude. They need to memorize 1, 2, 3, 4, but also need to know that 3 is a bigger number than 2 because it comes after 2, but it is also a smaller number than 4, because it comes after 3. Memorizing symbols is not enough, just as knowing the everyday story is not enough. Eventually, the child needs to know how the informal ideas provide the meaningful basis for the formal math.

Given all this, how can you read storybooks in such a way as to promote your child’s math learning?

Some do’s:

— Read books that you both find interesting, amusing and full of wonder, books that will grow the child’s budding love of reading. Bypass boring stories, even if you think they are “educational.” Enjoy the story!

— Talk with your child about the various ideas, including the math ideas. “Who is bigger, Mama Bear or Papa Bear? How do you know? Which bear gets the biggest bed? Why?”

— Use math language to describe and explain (“This is a square because it has four sides and they are all the same length.”) and encourage the child to put her ideas into words.

— Keep the child engaged in the book, for example, by asking her to point out certain things on a page. “Show me the biggest bear.” Or you can make the questions very open-ended by asking, “What do you see on this page? What is happening?”

— Think about your own experiences with math and whether you might unintentionally transmit any negative feelings about math to the child.

In the end, reading math storybooks and storybooks with embedded math can stimulate your child’s thinking, language and enjoyment. Reading can involve you and the child in an intellectual adventure in exploring mathematical ideas. Reading can help you bond with the child. Reading can provide a warm blanket for the child’s mathematical knowledge and provide insights into the child’s mind.

If you do it well, reading math storybooks can set the stage for meaningful math achievement in school during the years to come.

*Herbert P. Ginsburg, Ph.D., is the Jacob H. Schiff professor of psychology and education at Teachers College, Columbia University. He has drawn on cognitive developmental psychology to develop a mathematics curriculum (Big Math for Little Kids), storybooks for young children and tests of mathematical thinking.*

“They can assure themselves and don’t have to wait for the teacher to come around and say, ‘yeah, you got it.'” Saul said of the approach. She makes sure students have time to work independently before they share their strategies with one another, a time when they practice using math language and explaining their thinking. Meanwhile, Saul is rotating around the room, supporting students and pushing their thinking along. One of the most important parts, she says, is when she invites students to come to the front and share their solutions. This student-led solution time reinforces the class culture and helps students see one another as experts.

Algebra is another important area of math and is often seen as the gateway subject to higher math. While students may see algebra as a time to memorize equations, strong teachers know this is an incredibly important time to make sure students’ math reasoning is solid. In the video below, math coach Audra McPhillips explains how she leads eighth graders through the process of developing a conjecture about functions. She asks them to look for patterns and has intentionally given them three examples that have something in common (the rate of change) and a point of difference (the y-intercept), meant to push student thinking a little further.

McPhillips does very little telling students how to think, instead she lets them develop a conjecture that they believe to be true beyond the examples in front of them and requires them to explain why. Note, she doesn’t expect all students to write a conjecture by the end of the lesson, but she does have them fill out exit slips to record what they learned and how far they got as a quick reflection before they head to their next class.

**Believe in All of Your Students**

I have always known how important it is that students know their teacher believes in them; I knew this as a teacher and more recently became more acutely aware of it as a parent. When my daughter was five, she realized the teacher of her class in England was giving other students harder math problems, and she came home to me and asked why. When she realized that the teacher did not think she had potential—and sadly, this was true; the teacher had decided she had limited ability—her self-belief was shattered, and she developed a terribly fixed mindset that damaged her learning and confidence for a long time afterward. Now, some years later, after a lot of work from her parents and some wonderful teachers, she is transformed: she has a growth mindset and loves math. Despite the fact that the teacher never said to my daughter that she did not believe in her, she managed to communicate that message loud and clear, and this was understood by my daughter even at the young age of five.

The school that my daughter attended in England put students into ability groups in second grade, but they stopped this practice after reading the research evidence and learning about the strategies for teaching heterogeneous groups. After they made this change, the principal wrote to tell me it had transformed math classes and raised achievement across the school. If students are placed into ability groups, even if they have innocuous names such as the red and blue groups, students will know, and their mindsets will become more fixed. When children were put into ability groups in my daughter’s school, children from the lower groups came home saying “All the clever children have gone into another group now.” The messages the students received about their potential as learners in general (not just about math) were devastating for them. One of the first steps we need to take, as a nation, is to move away from outdated methods of fixed mindset grouping and communicate to all students that they can achieve.

The importance of students thinking their teacher believes in them was confirmed in a recent study that had an extremely powerful result (Cohen & Garcia, 2014). Hundreds of students were involved in this experimental study of high school English classes. All of the students wrote essays and received critical diagnostic feedback from their teachers, but half the students received a single extra sentence on the bottom of the feedback. The students who received the extra sentence achieved at significantly higher levels a year later, even though the teachers did not know who received the sentence and there were no other differences between the groups. It may seem incredible that one sentence could change students’ learning trajectories to the extent that they achieve at higher levels a year later, with no other change, but this was the extra sentence:

“I am giving you this feedback because I believe in you.”

Students who received this sentence scored at higher levels a year later. This effect was particularly significant for students of color, who often feel less valued by their teachers (Cohen & Garcia, 2014). I share this finding with teachers frequently, and they always fully understand its significance. I do not share the result in the hope that teachers will add this same sentence to all of their students’ work. That would lead students to think the sentence was not genuine, which would be counterproductive. I share it to emphasize the power of teachers’ words and the beliefs they hold about students, and to encourage teachers to instill positive belief messages at all times.

Teachers can communicate positive expectations to students by using encouraging words, and it is easy to do this with students who appear motivated, who learn easily, or who are quick. But it is even more important to communicate positive beliefs and expectations to students who are slow, appear unmotivated, or struggle. It is also important to realize that the speed at which students appear to grasp concepts is not indicative of their mathematics potential (Schwartz, 2001). As hard as it is, it is important to not have any preconceptions about who will work well on a math task in advance of their getting the task. We must be open at all times to any student’s working really well. Some students give the impression that math is a constant struggle for them, and they may ask a lot of questions or keep saying they are stuck, but they are just hiding their mathematics potential and are likely to be suffering from a fixed mindset. Some students have had bad math experiences and messages from a young age, or have not received opportunities for brain growth and learning that other students have, so they are at lower levels than other students, but this does not mean they cannot take off with good mathematics teaching, positive messages, and, perhaps most important, high expectations from their teacher. You can be the person who turns things around for them and liberates their learning path. It usually takes just one person—a person whom students will never forget.

*Jo Boaler is a Professor of Mathematics Education at Stanford University and co-founder of www.youcubed.org. Former roles have included being the Marie Curie Professor of Mathematics Education in England, and a mathematics teacher in London comprehensive schools. She is the author of eight books including What’s Math Got To Do With It?*

This story demonstrates how clearly kids understand that unlike their other courses, math is a performative subject, where their job is to come up with answers quickly. Boaler says that if this approach doesn’t change, the U.S. will always have weak math education.

“There’s a widespread myth that some people are math people and some people are not,” Boaler told a group of parents and educators gathered at the 2015 Innovative Learning Conference. “But it turns out there’s no such thing as a math brain.” Unfortunately, many parents, teachers and students believe this myth and it holds them up every day in their math learning.

“We live in a society with lots of kids who don’t believe they are good at math,” Boaler said at an Education Writers Association conference. “They’re put into low groups; they’re given low-level work and their pathway has been set.” But math education doesn’t have to look like this.

Neuroscience research is now showing a strong connection between the attitudes and beliefs students hold about themselves and their academic performance. That’s a departure from the long-held traditional view that academic success is based only on the quality of the teacher and curriculum. But researchers like Carol Dweck, Camille Farrington and David Yeager have shown repeatedly that small interventions to change attitudes about learning can have an outsized effect on performance.

Neuroscientists now know that the brain has the ability to grow and shrink. This was demonstrated in a study of taxi drivers in London who must memorize all the streets and landmarks in downtown London to earn a license. On average it takes people 12 tries to pass the test. Researchers found that the hippocampus of drivers studying for the test grew tremendously. But when those drivers retired, the brain shrank. Before this, no one knew the brain could grow and shrink like that.

“We now know that when you make a mistake in math, your brain grows,” Boaler said. Neuroscientists did MRI scans of students taking math tests and saw that when a student made a mistake a synapse fired, even if the student wasn’t aware of the mistake. “Your brain grows when you make a mistake, even if you’re not aware of it, because it’s a time when your brain is struggling,” Boaler said. “It’s the most important time for our brains.”

A second synapse fires if the student recognizes his mistake. If that thought is revisited, the initial synapse firing can become a brain pathway, which is good for learning. If the thought isn’t revisited, that synapse will wash away.

A recent study of students with math learning disabilities found in a scan that their brains did behave differently from kids without the disability. “What they saw was the brain lighting up in lots of different areas while working on math,” Boaler said. The children were recruiting parts of the brain not normally involved in math reasoning.

The researchers tutored the group of students with math disabilities for eight weeks using the methods Boaler recommends like visualizing math, discussing problems and writing about math. At the end of the eight weeks, they scanned their brains again and found that the brains of the test group looked just like the kids who did not have math disabilities. This study shows that all kids can learn math when taught effectively. Boaler estimates that only 2 to 3 percent of people have such significant learning disabilities that they can’t learn math at the highest levels.

People who learned math the traditional way often push back against visual representations of math. That kind of thinking represents a deep misunderstanding of how the brain works. “When you think visually about anything, different brain pathways light up than when we think numerically,” Boaler said. The more brain pathways a student engages on the same problem, the stronger the learning.

**GROWTH MINDSET AND MATH**

Increasingly, educators are buying into the compelling research showing that what students believe about themselves affects how their brains approach learning. Growth mindset is probably the best known aspect of this research, and many school leaders are trying to figure out how to implement growth mindset programs in their classrooms.

“More kids have a fixed mindset about math than anything else,” Boaler said. And it’s no coincidence that they feel this way. Teachers often believe their students can’t achieve at the highest levels, and in turn, students believe that about themselves. Plus, the tasks themselves communicate a fixed mindset.

“It is very difficult to have a growth mindset and to believe that you can grow or learn if you are constantly given short, closed questions with a right or wrong answer,” Boaler said. Instead, she recommends giving visual problems that provoke discussion and have multiple ways they could be solved.

She also says kids should not be grouped by ability or tracked into “advanced” or “remedial” groups. That common practice sends fixed mindset messages to students, both the “advanced” ones and the “low-performing” ones. Kids considered to be “gifted” suffer from ability grouping the most because they develop the ultimate fixed mindset. They become terrified that if they struggle they’ll no longer be considered smart.

Instead, mixed ability grouping can work if the tasks are open-ended and what Boaler calls “low-floor/high-ceiling” tasks that allow every student to participate, while allowing lots of space within the task for students to grow in their thinking.

Boaler has lots of example tasks on her website, YouCubed, and on the NRICH website.

**PUTTING IT INTO PRACTICE**

During the summer of 2015, Boaler invited 81 seventh- and eighth-graders from a low-income district near Stanford to come to a summer math camp focused on algebra concepts. She gave the students a pre-test and found that their abilities ranged from very low (getting 0 answers correct) to fairly high. Then, for 18 days she taught them math well.

The instructional program focused on mindset messages, was full of inquiry-based, low-floor/high-ceiling tasks, was visual and used mixed achievement groups. At the end of 18 days, when Boaler gave them another test they had improved on average by 50 percent.

“They improved because they changed their beliefs that they were not a math person to believing they were a math person,” Boaler said. After the course, students said they looked forward to math and saw math as a creative subject.

Administrators from the district came to observe partway through the camp and couldn’t tell who was a low achiever and who was a high achiever in the class. Boaler also makes it clear to the students in the workshop what she expects from them, and speed is not something she’s evaluating. Instead, they do norm building so that everyone knows how to appropriately work in groups, help one another and be supportive.

“If we don’t pay attention to those kinds of interactions, and kids are dominating, or thinking they’re smarter, then we’re really in trouble,” Boaler said.

Removing the time pressure from math is another important issue for Boaler. Neuroscience research out of Sian Beilock’s lab at the University of Chicago has shown that time pressure often blocks the brain’s working memory from functioning. This is particularly bad for kids with test anxiety.

“The irony of this is mathematicians are not fast with numbers,” Boaler said. “We value speed in math classrooms, but I’ve talked with lots of mathematicians who say they’re not fast at all.” But it is common for math teachers to call on the kids who get the answer quickly, reinforcing the idea for all students that rapidity is what matters.

**COMMON PUSHBACK**

Math education experts have been making the same case as Boaler for decades, and yet math education in the U.S. has not shifted much. Teachers often say they have to cover all the topics in the curriculum to prepare students for the tests they will be expected to pass, leaving them with no time for the kinds of open-ended, discussion-based math that Boaler advocates.

Boaler agrees with teachers that there is way too much to cover in the curriculum, especially because she finds much of it to be obsolete (don’t get her started on the textbooks themselves). “The most important thing we can give kids is to think quantitatively about the world and apply a mathematical lens to different situations,” she said.

In addition to teaching students, Boaler trains teachers in her methods. Often they go back to their classrooms and apply these theories, which means they aren’t covering every topic in the textbook, and yet their students do better on the standardized tests anyway. Boaler is not a fan of all the tests American students must take, but she says teaching math the right way deepens kids’ understanding of math in real ways that show up on tests, too.

Teachers and parents often push back against this kind of math. They wonder where memorization of math facts fits into the model, given the belief that kids must know their times tables to succeed in higher-level math. Boaler says that’s unnecessary. She is a math education teacher and has risen to high levels of math learning without ever learning her math facts. She has number fluency, knows how to manipulate numbers and understands concepts, but she doesn’t have her math facts memorized.

The Programme for International Student Assessment test (PISA), which is often used to compare achievement across countries, has a section about attitudes and beliefs. Those surveys show that kids who approach math as memorization are the lowest achievers in the world. “America has more memorizers than almost any country in the world,” Boaler said. The highest achievers are those who think about the big ideas and make connections.

Likewise, repetition of math tasks is not helpful to deep learning. The same kind of problem with different numbers does not improve understanding, Boaler said. What students really need is “productive practice,” approaching the problem from different directions, applying the ideas and explaining reasoning.

Boaler is on a mission to “revolutionize” how math is taught in the U.S. She has written several books to help teachers learn to teach with her methods, offers a free online course, and even gives away curriculum for teachers, students and parents on her YouCubed website. During one week at the start of the 2015 school year Boaler gave away five free math lessons, encouraging teachers to try this approach. She’s pleased that 100,000 schools tried the lessons, and teachers could see the difference in their students. A survey of students found that after the lessons and the growth mindset videos, 96 percent believed they should keep trying after making a mistake in math.

Boaler said a big problem is that math teachers themselves are math-traumatized. They came through a system very similar to the one in which they work. Elementary school teachers in particular often feel insecure about math.

“When they try math in these ways they get it, too,” Boaler said. “They can see this is much more valuable and enriching.”

]]>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.*