First, students who are enrolled in even one remedial course have a high chance of dropping out. According to a 2006 National Education Longitudinal study, the dropout rate in remedial courses is more than 70%, with only 28% of remedial students completing a degree after 8.5 years. Second, the extra money to pay for remediation is costing states billions: the Community College Research Center (CCRC) estimates that the national cost of providing these courses to all students is approximately $7 billion.

But according to Tom Bailey, who heads the CCRC at Teachers College, Columbia University, it’s the students who are paying the most. “It is students who probably have to bear the most significant costs,” he writes. “They must not only pay for the classes but also must delay their progress through college. Many students are discouraged when they find out that they are not eligible for college-level courses. This may explain the high ‘no-show’ rates among those referred to remediation.”

It is not an exaggeration, says Rachel Beattie, director of productive persistence at Carnegie Math Pathways, to say that one developmental course can derail an entire college career, and even the future beyond it. “People will keep coming back, because they’re really persistent. We see that many of our students have been enrolled in college for five or ten, even twenty years, they’re trying to get that math credit, but no luck,” she said. “We see a lot of unproductive persistence.” Part of Beattie’s job is to help mold unproductive persistence into something more fruitful, and that involves changing both the students’ mindset and how teachers teach developmental courses.

In 2010, Beattie and team launched the Carnegie Math Pathways at the Carnegie Foundation for the Advancement of Teaching, two developmental math courses which now operate on community college campuses in 19 states and strive to help students who need to remediate in math complete their courses but also do something more: introduce students to the “soft skills” they may be missing to help get them through college. Their primary focus: convincing students that they can learn. “About 2/3 of our students come in to the Carnegie Math Pathways with the belief that, no matter what they do, they are not ‘math people,’” Beattie said. “That there is this race of math people out there, and that they’re not one of them, and no matter how hard they try, or what strategies, it doesn’t matter because they’re never going to be one of those people.”

After interviewing researchers and teachers, Beattie and team found that if community college math professors could instill five “high-leverage” factors into students, they had a much higher chance of completing their math courses:

* Students believe they can learn,

* Students have social ties to peers during the course,

* They see the course has both short- and long-term value,

* They have the know-how, skills and habits to succeed,

* And finally, having faculty support students’ skills and mindset.

So Beattie teaches professors how to teach to students who truly believe they can’t learn. The two courses, a statistics course called Statway and the quantitative math course, Quantway, are developmental in the sense of what material is covered, but *how* the material is covered plays a big role. “Unlike in K-12, those of us who taught and teach in higher ed, we’re not always explicitly trained in how to teach,” Beattie said. “They know a lot about mathematics, a lot of them have PhDs in mathematics, but we show them how students learn, and how to promote mindsets and learning strategies is something that a lot of times they don’t have a bag of tricks for.”

So the first lesson for students in both courses is how the brain learns; professors also cover the research behind growth mindset. “You’re not wired from birth knowing how to do logarithms, that’s just not how the brain works,” said Beattie, who has a PhD in developmental psychology, and did a post-doc in cognitive neuroscience. “It’s actually quite plastic and changes based on the experiences that you have. So as you increase your knowledge you create more sophisticated connections between neurons in your brain, and you’re able to make connections to information better and in different ways, and this actually helps with experience.”

Understanding how the brain works goes hand in hand with another success strategy, which is tackling “belonging uncertainty,” in which students believe that they don’t belong in college, or in college-level math. Beattie also shares with professors how to build trust and a sense of community in classrooms—and sometimes the strategy for belonging is almost agonizingly simple, like noticing a student who is absent and contacting them to let them know they were missed.

In the six years since they began, Statway and Quantway have tripled the success rate of students (meaning they passed the course and could move on to college work) in half the time. While 6% of students complete traditional math pathway courses, the Carnegie classes show 50-60% of students earning their math credit in one year.

What Beattie hopes to show next is that the strategies students learn in the Math Pathways carries over to the rest of college. “There is some preliminary evidence that we are seeing from our colleges that our students are being successful in future classes,” she said. “Because many students end up transferring to four-year schools—which is really exciting—it can be kind of difficult to track them down, but we are getting some preliminary evidence of sustained success,” she said.

]]>Instead of asking all the questions, Franco says, “I’m going to see how you guys work by yourselves.” She’s circling the room, listening for understanding and interesting solving problem techniques, but rather than asking the questions, she’s listening to her students ask one questions to another to defend their problem solving.

“You came up with you own idea. It doesn’t have to be the same as mine as long as we get the same answer,” Franco said.

Understanding Fractions through Real-World Tasks from Teaching Channel on Vimeo.

]]>In this Teaching Channel video of Sarah Dietz’ second grade class, she uses a video clip about cookie monster to grab her students’ interest and get them questioning. The video presents a puzzle to students and Dietz makes sure to draw out their questions, honing in on the common theme (and the lesson for the day) based on their authentic questions. She’s also asking them to decide what information they need to solve their question, an important part of math in the real world that is often left out of traditional textbook problems. Then, she gives them time to work through the question they’ve posed using a model of the cookie package and their knowledge of various subtraction strategies.

“To [the students], it’s not a math lesson; it’s a puzzle that needs to be solved,” said Dietz. “It’s a problem they want to work out.”

When students share their answers at the end, Dietz asks them to use their work to explain their thinking and she leaves enough time for multiple examples of different strategies. Emphasizing that there are many acceptable ways to solve a problem can help students remain open to struggle and figuring things out in the ways that make sense to them.

In another Teaching Channel video about this three-step process called “Three-Act Tasks,” kindergarten teacher Kristin Alfonso says: “I love that Three-Act Tasks are usually just difficult enough that even if kids can figure out really quickly on the carpet, they still have to go back to their tables and show us, and be able to prove their thinking to us.”

Three-Act Tasks: Modeling Subtraction from Teaching Channel on Vimeo.

]]>But before she could help the student, her attention was called away by a disturbance on the other side of the room among her 34 students. When she turned back to the struggling student, he had solved his issue. Buljan hadn’t moved the camera the whole time, so she captured him figuring it out on his own.

“The whole time I was distracted and not talking to him at all, he was thinking and redesigning his problem,” Buljan said. At that moment she knew she needed to stop talking so much. She still provides support, but she’s changed the kinds of questions she asks. She used to ask what she calls “funneling questions,” prompts that lead the student where she wanted them to go, like, “what comes next.” Now she tries to ask focusing questions like, “how do we get started?” Or, “who do you know who’s already good at this that you could ask?”

“I have watched so much wrong counting it hurts,” Buljan said. “The urge to fix their thinking is so strong, but I just changed my own mindset for creating space for them to think.” She’s learned to think of this approach as “going at the pace of the learning,” a phrase she heard from Akiko Takahashi, an expert in lesson study.

Buljan no longer rushes to cover everything in the pacing guide. Instead, she spends as long as is necessary on the most fundamental structures of math, making sure students know those really well. By her logic, going slower at the beginning is more efficient because her students learn concepts like subtraction well once, rather than having to learn it again in third, fourth and fifth grade.

She applies the Teaching For Robust Understanding of Mathematics (TRU) framework in her classroom. TRU could feel like another “new” math program, but it’s actually a simple way to remember the things many good teachers already know. TRU is five basic dimensions that will sound familiar to most teachers: content, cognitive demand, equitable access to content, agency, authority and identity, and uses of assessment.

“I’ve organized things that the whole field knows, so that there’s a small enough number of things to keep in mind,” said Alan Schoenfeld, University of California Berkeley professor of education and mathematics, and the person behind the TRU framework. “The main virtue of TRU is not that I’m telling you anything new.”

The TRU Framework focuses on how students experience the math, not on what the teacher is doing. “Our framework says you should really be focusing on, ‘What does it feel like to be a student in that classroom?” Schoenfeld said. “What’s the experience from the point of view of the student? Because that’s what shapes who the student becomes.” And how a student feels about him or herself has everything to do with motivation, persistence and agency.

When Schoenfeld introduces the TRU Framework to teachers, he often shows three classroom videos and asks educators to make a list of all the behaviors they see happening. As a group they then categorize those observations into the five framework dimensions. In this way, teachers co-construct an outline of important elements in a classroom and can see that when they are present rigorous learning is happening.

“They provide a straightforward way to focus on and reflect on practice in a way that will really make a difference,” Schoenfeld said.

**TEACHING WITH TRU**

Buljan teaches second grade at Glassbrook Elementary school in Hayward, California. She’s been using the TRU framework for several years and finds it particularly helpful when her students are having difficulty with a concept. Thinking through the TRU framework lens helps her step back and focus on aspects like agency and authority.

For example, if two students get different answers, she might talk about the idea of proof, asking them to convince one another of their right answer. “That’s some of the richest growth, when they’re able to have that conversation,” Buljan said. Approaching their learning impasse through the TRU dimension of authority helped her to structure students’ conversations differently. Focusing on authority and agency in that moment led to student growth.

“You don’t ever just use one [element of the framework],” Buljan said. She has found the idea of cognitive demand particularly helpful in her diverse classroom, where students speak 11 different languages, and 90 percent qualify for free or reduced priced lunch. Buljan says teachers have a tendency to provide too much support to English language learners in their attempt to help them access the content.

“The danger we get into is you can scaffold so much that you pull the thinking out of it,” Buljan said. “So the trick is, how do we create access into certain kinds of problems, but still make it about the kids doing the proof and doing the thinking themselves?”

The TRU framework has kept Buljan focused on creating cognitive demand for all her students, regardless of language barriers or prior knowledge. She does that largely by focusing on the structure of problems, as opposed to one specific standard in the second grade curriculum. She wants her students to deeply understand fundamental mathematical structures like place value, or how to group and ungroup numbers, so she tries not to give students “rules” that will help with one kind of problem, but later could lead to confusion.

Subtraction is a good example. Some teachers tell students the “rule” is to always subtract the smaller number from the larger number. But that rule gets subverted when students start doing multidigit subtraction and they see each column of numbers as a free floating problem detached from the idea of place value.

Instead, Buljan gives students a lot of thinking time. “It is brutal,” she said. “Sticking with second graders long enough for them to push through all that confusion and get to a place where all those underlying structures are part of who they are is tough.” But, it’s also rewarding. When her students hit challenging problems, she tells them if it was easy, they would already know it. “They totally get it,” Buljan said. “They will say things like, ‘I’m really learning now,’”

Teaching with the TRU framework has also prompted Buljan to think creatively about how structures she uses for English instruction could be applied to math. She joked that in second grade, no one really cares about math, it’s all about reading, so there’s a lot of professional development around reading strategies. Buljan has adapted many of them for math.

For example, she’s applied writers and readers workshop to math workshop. She usually introduces a topic quickly, gives kids 30 minutes to think through the problem together, and then she does a quick wrap-up. There’s very little direct instruction in her math teaching. She also has adapted the idea of a “mentor text,” or in math a “mentor problem.”

“When you teach reading there’s this idea that I’m not teaching you to read this book, I’m using this text to teach you a strategy that you can use to read any book,” Buljan said. She’s used that same idea to teach kids math problem solving. She’s not teaching them how to do a certain type of problem, she’s trying to teach how this problem can help her students solve any problem. Instead of teaching rules of adding and subtracting, as a class students focus on describing the parts of the problem, what’s happening in the problem and how to talk about patterns in math.

Buljan said her second graders work on just six problems for half the year. The quantities will change and the items being added or subtracted change, but essentially the problem is the same. Within that familiarity, students are identifying parts of the number sentence, using location and quantity to describe patterns, determining what type of problem it is and moving forward with various strategies, but it all feels safe. And for second graders, small changes in the problem feel big – going from adding apples to adding stickers makes it a whole new problem, Buljan said.

To meet the equitable access to content part of the TRU framework for her many English language learners, Buljan has modified the common practice of using sentence frames. She felt a typical sentence frame like “ I thought __ because __” funneled student thinking too much. So now she just gives sentence stems like, “I noticed,” to help model how students can have an academic conversation. She then monitors how they are using those stems and gives immediate feedback.

**COACHING FOR TRU**

David Foster from the Silicon Valley Mathematics Initiative shot videos from the first several weeks of Buljan’s class and the last several weeks of class to show how she works to develop her students’ language, conversation ability, and classroom culture. Foster has been coaching math teachers for decades and he likes the TRU framework because it distills all the research into five easy-to-understand and recognize dimensions. He finds teachers often struggle to get kids to work productively in groups because they haven’t spent time at the beginning of the year developing a class culture of trust and collaboration. And they often aren’t giving students a worthwhile task that’s worth discussing in the first place. He says if teachers pull a problem out of the textbook only one in 90 times will it be a worthwhile problem.

Foster says many of the structures Buljan used intuitively are great strategies to home in on questions of equity, agency, authority and cognitive demand. Sentence frames help students get into the habit of defending their thinking with evidence. Number talks help teachers pinpoint exactly where student misperceptions lie and are a venue for students to practice talking about math. Roles and norms can help ensure every group member has an equitable role.

Good teachers are doing these things already. Foster has seen very effective math classrooms in almost every school he enters. His work is to help all math teachers improve the quality of their teaching, something everyone needs. He finds the teachers who are most effective in this process are the ones who are never completely satisfied. They are the ones who leave at the end of the day worrying about how to reach the one kid who is still struggling. That hunger to improve is a huge part of becoming more effective.

**OAKLAND STUDY**

Alan Schoenfeld, in collaboration with colleagues at the University of Michigan, recently received a National Science Foundation grant to develop tools to coach math teachers in effective classroom practices. Schoenfeld is focusing his side of the work on high school teachers in Oakland using TRU along with lesson study. The central office math coaches have found the framework useful as a point of departure for conversations with teachers.

“It creates a structure for someone to give feedback and engage in reflecting with that teacher on something the teacher is interested in working on,” said Barbara Shreve, Oakland Unified’s Secondary Math Coordinator. She’s also found it helps administrators on classroom rounds focus in on what they’re seeing.

“There have not been a huge number of spaces where teachers get to talk together about the meat of what happens day-to-day in the classroom,” Shreve said. She hopes conversations centered around TRU will give everyone the same point of departure and a useful language to move towards solutions. “Success is going to look like having a much more common language for talking about the successes and challenges we’re experiencing as educators,” she said.

Schoenfeld, for his part, hopes to use the research period to develop a set of tools that could help other districts conduct coaching and professional development around the five dimensions of TRU. In previous research funded by the Bill & Melinda Gates Foundation and verified by independent evaluators at UCLA’s National Center for Research on Evaluation, Standards, and Student Testing (CRESST), Schoenfeld found that teachers who were trained on the TRU framework and used it in their classrooms saw on average an improvement in student understanding that correlates to 4.6 months of additional learning.

“We documented changes in the teacher’s behavior over time because of the lessons and the support,” Schoenfeld said. Teachers stopped telling students what to do and instead got students to work the problems out for themselves. The structure of lessons forced teachers to teach differently. Schoenfeld hopes that if he can develop an effective toolkit, more districts can easily scale up their work on TRU.

]]>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.