Although she has settled into a life of teaching undergraduate students and working on her own research, Manes still cares deeply about K-12 education. To stay connected to teachers in that world she helped start a Math Teachers’ Circle in Honolulu. The circle meets once a month and invites math teachers from all grade levels to get together and work on fun, challenging math alongside research mathematicians.

“I try to bring that creativity and joy and excitement and discovery piece into the Math Teachers’ Circle and hope it trickles into the classroom,” Manes said. Unlike other professional development opportunities, the focus of these circles is not on lesson plans or pedagogy. Most of the time is spent working on and discussing a problem that the facilitators bring, with the hope that teachers will rediscover what they love about math and how it feels to be a learner.

This issue is personal to Manes, who wanted to be a scientist as a kid precisely because she likes solving problems. Stories about how science had improved the world were exciting to her, even if the science they were doing in school wasn’t. That wasn’t true of her math classes. She grew up thinking math was about procedures applied correctly to get a right answer — something she was good at — but she didn’t associate the discipline with discovery. It was only when she got to college and took higher level mathematics that she realized how exciting it could be.

Manes hopes Math Teachers’ Circles can help K-12 classroom teachers experience the fun of working on a challenging problem collaboratively, of being confused but continuing to struggle through, of ultimately having that feeling of discovery.

“In the end we do want to transform their experience and their students’ experiences,” Manes said. The Honolulu circle is just one of over a hundred all over the United States. Every circle runs a little differently depending on its context, but in Honolulu, where weeknight traffic is terrible, Manes has found that Saturday mornings work best. Teachers and mathematicians get together and work on math for several hours and then spend some time discussing the experience and how it might apply to the classroom. Manes has arranged it so teachers can get professional development credit for participating.

“For me, it’s a lot of listening, wandering around seeing what people are doing, having a sense of the room and then knowing what I want people to get out of the session,” Manes said. While groups are often working on the task from different directions or entering at different levels of understanding, Manes will often stop work if there’s something she wants to make sure all participants notice before the time is up.

“One thing I’ve learned from Math Teachers’ Circles is watching mathematicians who I have tremendous respect for make errors and be corrected and be OK with that,” said Heather Danforth. She’s co-director of curriculum at Helios School, an independent school for gifted kids in the San Francisco Bay Area. She has always thought of herself as a reader and a writer, not a math person, but when she started teaching elementary school she decided to take some classes to brush up on her math skills. That’s when she found Math Teachers’ Circles.

“It was this opportunity to engage with math in this really engaging, exciting endeavor of trying to figure out problems and maybe not always reach a solution,” Danforth said. Participating helped her revise the narrative she held about her math abilities, which was largely based on her experience of being slow with multiplication tables in third grade.

“Most mathematicians don’t really care how fast you can do your multiplication tables,” Danforth said. And more importantly, identifying herself as a mathematician and experiencing what that means, helped her think of math as primarily about problem solving. “That new definition of math allows more people to be good at it,” she said.

Danforth now leads the math teachers at her school in math circles as part of their regular professional development and they also carve out time on Fridays for students to engage in circles as well. “A well run math circle leaves everyone feeling capable,” she said. “It’s not that everyone finishes at the same place, because you don’t. But everyone has something they can engage with in a meaningful way.”

Danforth thinks of math circles as an opportunity to experience what it means to be a mathematician, whereas math class is more learning about math. She compares it to learning scales versus playing music. The scales are important, but the music is what people love, and what motivates them to continue to work at the scales. She believes that if students never experience the fun, exciting side of math problem solving, free of pressures to get the right answer in a specific amount of time, then they may never choose to pursue math in the future.

A common theme among teachers who have participated in Math Teachers’ Circles is that by placing themselves in the position of learner they are able to empathize with their students more. Many teachers felt initially intimidated to do math with professional mathematicians, as well as other K-12 teachers who may have more advanced skills.

“I felt the entire range of emotions because I was with other teachers who had different background experiences,” said Sara Good, a seventh grade math teacher outside of Cleveland, Ohio. She said she fell in and out of confidence throughout her first circle, an interesting experience since so many math teachers love the subject because it’s easy for them.

Good used to be a district math coach before cost cutting in her district landed her back in the classroom this year. She is struggling to create the vibrant community of problem solvers that she knows would be best for student learning and finds that attending Math Teachers’ Circles rejuvenates her. Participating reminds her of effective questioning strategies and helps connect her with other math teachers who want to bring a sense of wonder and discovery back to math classrooms.

Good says it’s “easy to feel like you’re off in your own pedagogical corner” a lot of the time, but the math circles remind her she’s part of a community and that playing with math is fun if it’s set up right. She also knows many of her students think math is far from fun, largely because of the way it has been presented to them in school.

Math Teachers’ Circles have become more popular in the past five years as teachers in states that have adopted the Common Core work to understand the Mathematical Practices that undergird the math they teach.

“A lot of teachers weren’t familiar with thinking about math that way,” said Brianna Donaldson, Director of Special Projects at the American Institute of Mathematics (AIM). Her organization supported the first Math Teachers’ Circle in 2006 and has helped educators around the country as they start their own.

“Each circle is intended to be a real partnership between teachers and mathematicians,” Donaldson said. And while it may seem like research mathematicians wouldn’t want to do math with K-12 teachers, the reality is that often they learn a lot about how to teach undergraduates through these circles. Everyone participating in the circle is learning the difficult lesson to “help less.”

“Learning how to be less helpful can be really challenging, but a lot of times facilitators say it has a big effect on their teaching,” Donaldson said. “It really changes how they see what the learners in whatever environment can do and what they’re capable of.”

In her research, Manes often works on the same problem for years, methodically trying different problem solving strategies to a thorny challenge that no one in the world has solved yet. That process can sometimes shake her confidence and she likes interacting with other math-lovers around fun problems as a way to remind her of her capabilities and passion for the subject.

“I would not want one of these research-only jobs where you never teach,” Manes said. “When I get stuck in research and I can go into a class and lead activities and answer questions and guide people to help them understand things, I feel really reenergized. It gives me confidence to go back to my research.” Engaging in this way reminds her that getting stuck is part of the process, and coaching other people through those emotions serve the dual purpose of reminding herself to stick with it too.

And while it’s hard to draw a straight line between the experiences teachers get in Math Teachers’ Circles and their approaches to the classroom, Manes said that many of her participants do report that it has changed how they think about teaching. They say that they’ve realized they need to give students more think time, that they focus on discourse around the mathematics more, that they assign groups to work on open-ended problems, and that they’re more open to trying new things in the classroom.

The American Institute of Mathematics is excited about how popular Math Teachers’ Circles have become and hope that soon there will be a circle within driving distance of every teacher in the country. They also hope that within five years, between five and ten percent of math teachers will be participating in a circle. They’re already supporting the creation of Math Teachers’ Circle networks within states, with Montana and Ohio closest to achieving statewide coverage.

“What we found is the more teachers go to Math Teachers Circles the more they see math is about problem solving,” Donaldson said. “And this problem solving view of math is highly predictive of really productive mindsets, like growth mindset and belief in grit, that if you persist at something you’re going to make progress. And that’s an important part of doing well at something.”

]]>“At first we had no idea what it meant,” Thomson said, but as the researchers explained cognitive science studies showing the power of spatial reasoning in the early grades they were gradually convinced that it was worth trying. Early elementary teachers like Thomson in select Rainy River District schools began using Math For Young Children lessons designed by the researchers.

The lessons focus on specific spatial reasoning skills like mental rotation, visual spatial reasoning, and spatial vocabulary all done in a playful, exploratory style that is developmentally appropriate for students ages four to eight.

“On day one of our professional development, we would work with kids and directly show how these ideas play out in classrooms or with kids,” said Zachary Hawes, a doctoral candidate in the Numerical Cognition Laboratory at the University of Western Ontario. He is one of the Math For Young Children researchers along with Joan Moss, Cathy Bruce, Bev Caswell, and Tara Flynn. Since 2011, these researchers led by Moss and Bruce have been conducting research at several sites around Ontario. They felt including students in the professional development trainings would help give teachers a chance to see the lessons in action and help them imagine how they could bring them back to their classrooms.*

“We would take those lessons and games and think about what else we could do with these. How could we extend it, what could we try?” Thomson said. She and her colleagues would take the lessons researchers developed in a lab and try them out in their classrooms, returning to the next professional learning session with feedback and examples of how they’d modified or extended activities.

“Everything is supposed to be exploratory and it comes from the kids,” Thomson said. She noted they particularly love pattern blocks, which are like puzzles to them and tend to calm them down. She doesn’t ever lecture her students on how to use the spatial reasoning tools, but rather sets kids a challenge and lets them figure out how to put the blocks together. Often she’ll lead them in one group activity and then leave the materials out around the room so kids can play with them during free time as well.

A favorite lesson is the “magic key” activity where she puts on a witch hat and explains to her kindergarteners that a witch has hidden a treasure behind a door, casting a spell to lock it. She then gives them a set of pentominoes which contains five squares, and tells students their job is to find as many ways to combine the squares with one full side touching as they can. The more combinations they find the better their chances are of locating the key.

“They discover within the half hour that there are 12 of these keys and we can’t make more,” Thomson said. As she and her colleagues experimented with spatial reasoning activities like this one, they were consistently amazed at how much more young students could do than they expected. And because the activities largely deal with manipulating shapes, practicing mental rotations and talking about positional language, kids who struggle with more traditional numeracy exercises were shining.

“We started making sure we labeled this as math,” Thomson said. Before, kids thought math was just numbers, but when she worked to broaden the definition to include spatial reasoning tasks and toys they suddenly started to really enjoy math time, often choosing to play with materials during choice time.

Thomson said she was so impressed with the results she was getting that she focused almost exclusively on spatial reasoning, neglecting other kindergarten concepts like patterning and numeracy. That made her a little nervous, so she was surprised and delighted when her students still performed well on those more traditional math concepts by the end of the year. That direct experience of success validated the research the Math For Young Children team presented.

“It was good for me to see how important it was,” Thomson said. She’s now pulling the spatial reasoning tasks in more, connecting numeracy concepts like the number line to spatial and geometry concepts. She’s has students use blocks on number lines to help them understand the concept of magnitude, for example.

Cristol Bailey also began using spatial reasoning in her classes several years ago. At that time she taught at a rural school with a high First Nations population. Bailey taught special education, but many of the students were underachieving even without that categorization. She says she was skeptical of spatial reasoning, but it was a “seeing is believing situation” for her.

“The lower achieving kids had such a high degree of success with these activities and showed strengths that more standardized number sense lesson plans would never have brought out,” Bailey said. “For them to be successful in math — and successful to the degree they were — was mind boggling.”

She began to see her entire math program through a spatial and geometry lens. Even when students were doing number sense activities she would encourage them to gesture with their hands or visualize the number line. She found often kids didn’t have the language to describe spatial positioning, but as they used their hands to gesture they began to find the words.

“We went into it with a sort of learning trajectory in mind and most of the time they far surpassed what we thought they’d be capable of,” Bailey said. She now teaches Grade 2 students, most of whom have had spatial reasoning lessons since kindergarten. They’ve mastered many of the tasks, but she still finds more difficult ones to grow their skills. One favorite is the “hole punch symmetry challenge,” in which students imagine punching a hole in a folded up piece of paper. As the paper unfolds, where will the holes be?

“It is my struggling paper and pencil kids who nailed it right off the bat, which was really surprising and great because I was not expecting that,” she said.

In Ontario, students take an important standardized test in Grade 3 called the EQAO that determines whether they are on grade level. That means that even in Grade 2 there’s pressure to cover a broad array of topics and anxiety that kids won’t be ready. Teachers go over diagnostic data at divisional meetings, creating lessons to target concepts and skills that students haven’t mastered. Bailey has noticed that students often struggle with tasks that involve spatial sense, a further indicator to her that spatial reasoning should be the norm in every early elementary classroom.

While she still uses spatial reasoning in her Grade 2 classroom, Bailey admits that without the support of colleagues working to adapt the materials to this grade level it’s more of a challenge. She thinks her experience with the Math For Young Children team and curriculum has changed her teaching forever, but wishes it was more of a priority even as kids get older. Perhaps just as important, the experience of working with math researchers and colleagues to refine lessons has her thinking about going back to school for another degree on how to better teach math.

**MATH FOR YOUNG CHILDREN**

There’s a well-known rift between those who believe the only type of developmentally appropriate early childhood education is a play-based one, and those concerned that relying solely on any learning that comes out of play could put students coming from impoverished backgrounds at a disadvantage. Research has shown that students from lower socioeconomic groups enter school with significantly less mathematical knowledge, and it is difficult to overcome that gap without intentional mathematics programming. But, at the same time, traditional teacher-led instruction often isn’t developmentally appropriate for five-year-olds.

“This project started as a way to show young children engaged in rigorous mathematics in ways that were play,” said Joan Moss, Associate Professor Emerita at the University of Toronto’s Ontario Institute for Studies in Education. She stresses that while math learning doesn’t only emerge from play, as some insist, the activities are still developmentally appropriate because they are presented playfully; students have lots of choice, there are many entry points, and while there are right answers, teachers build a culture in which getting a wrong answer isn’t bad.

For example, in the “quick image activity” the teacher flashes a complex pattern made out of pattern blocks. Students see it for a very brief time and then try to recreate it themselves. After working for a bit they get to see the original image again and make fixes to their original attempts. Hawes and Moss say a lot of learning happens in the fixing.

In addition to demonstrating that well-trained teachers can teach math concepts in developmentally appropriate and playful ways, the Math For Young Children project has been an experiment in a more collaborative type of professional development. The university researchers are working alongside classroom teachers to fine tune lessons and evaluate how well they work. The Rainy River School Board teachers who were the first participants kept logs of when they used spatial reasoning activities, how long they took, and the tweaks they made. They brought feedback from the classroom back to researchers, and used a lesson study approach to improving the lessons together.

Joan Moss says this collaborative model of professional development, featuring teachers working alongside researchers to build quality activities grounded in research and classroom practice has been thrilling and a huge part of the program’s success. Teachers agree: “To be able to get together with people with that much math knowledge, it was an amazing experience,” Cristol Bailey said.

“It changed my teaching in the fact that I think of myself as a teacher-researcher, as they call us,” Thomson added. She now approaches every classroom activity as a mini experiment, tweaking and adjusting along the way. “I’m a lot more reflective in what I’m doing and what I put out there. It’s a neat lens to look through.”

The University of Toronto team evaluated the Math For Young Children program as it was being implemented in the Rainy River schools. Since teachers of the experimental group were engaged in inquiry-based professional development with researchers around spatial reasoning, the control group’s teachers also had interaction with researchers on a different topic. This was meant to make the groups more similar in exposure, but with different focuses.

After the first year, students in the experimental group made significant gains on assessments of geometry, spatial reasoning and numerical skills compared to the control group. In the second year, researchers decided to test students on the KeyMath measures, which are used to assess school-based mathematical concepts and skills. Students in the experimental group showed significant gains on those more traditional measures as well (a paper with these findings will soon be published in Cognition and Instruction). The first class of students will take the EQAO this year, and researchers hope they will show increased learning over peers in the rest of the province.

The Ontario Ministry of Education is interested in spreading the spatial reasoning work that the researchers started. Hawes and Flynn wrote a document titled “Paying Attention to Spatial Reasoning” that the ministry distributed to educators across the district.** Individual school boards are also showing interest in training and implementation.

*The article has been updated to note that Cathy Bruce helped lead the Math For Young Children research, which is taking place in several locations around Ontario.

**The article has been updated to include Tara Flynn’s contribution to the document. We regret these errors.

By Joan Moss, Catherine D. Bruce, Bev Caswell, Tara Flynn, and Zachary Hawes

Our journey began when we conducted an extensive literature review at the outset of the project (Bruce, Flynn, & Moss, 2012) and learned about the crucial importance of spatial reasoning. This theme was consistent across many research disciplines, including biology, cognitive sciences, psychology, developmental sciences, education, as well as educational neuroscience—an emerging transdisciplinary ﬁeld which sits at the intersection of these other disciplines and aims for a collaborative approach in which educational theory and practice are informed by new ﬁndings in the cognitive sciences, and vice versa (Fisher, 2009). We also learned—and have experienced in our careers as mathematics educators and researchers—that spatial reasoning is a curiously unacknowledged and neglected area of the curriculum. During our involvement with the M4YC project, we have become more and more convinced of reasons why we should pay attention to spatial reasoning in early years mathematics. Below we offer our Top Five reasons why, as educators, we should care about spatial thinking when we plan, observe, and assess mathematics in our classrooms.

**1. Spatial reasoning and mathematical thinking are intimately linked.**

There are numerous research studies that demonstrate the relationship between spatial reasoning and what we typically think of as mathematical ability. For example, one research study found that the quality of block play at four years of age was a predictor of high school mathematics achievement (Wolfgang et al., 2001). Another study found a relationship between young children’s construction skills (such as playing with jigsaw puzzles and blocks) and strong number sense and success in solving mathematical word problems (Nath & Szücs, 2014). In fact, as Mix and Cheng (2012) report, “The relation between spatial ability and mathematics is so well established that it no longer makes sense to ask whether they are related” (p. 206). Researchers have underlined that the link between spatial reasoning and math is so strong that it is “almost as if they are one and the same thing” (Dehaene, 1997, p. 125). Reﬂecting on the strength of this relationship, others have noted that “spatial instruction will have a two-for-one effect” that yields beneﬁts in mathematics as well as the spatial domain (Verdine, Golinkoff, Hirsh-Pasek, & Newcombe, 2013, p. 13). Of course, the practices of mathematicians also beneﬁt from spatial reasoning; many mathematicians stress that their work relies strongly on visual and spatial representations and forms of understanding (Whiteley, Sinclair, & Davis, 2015).

We can see how the various strands of mathematics are inherently spatial. Think about what happens when we compare the area of two polygons, such as a rhombus and a rectangle. To be successful, we can draw on spatial strategies such as composition and decomposition of 2D shapes, mental rotation, and visualization. In fact, research shows that spatial reasoning is linked to performance within many strands of mathematics including: basic magnitude and counting skills (Thompson, Nuerk, Moeller, & Cohen Kadosh, 2013), mental arithmetic (Kyttälä & Lehto, 2008), word problems (Hegarty & Kozhevnikov, 1999), algebra (Tolar, Lederberg, & Fletcher, 2009), calculus (Sorby, Casey, Veurink, & Dulaney, 2013), and advanced mathematics (Wei, Yuan, Chen, & Zhou, 2012).

In one of the ﬁrst studies of its kind to show speciﬁc links between spatial and mathematical skills, Cheng and Mix (2013) assessed children in both spatial and math skills. Children were randomly assigned to one of two groups: one group engaged in spatial training involving mental rotations, and the other group spent the equivalent amount of time working on crossword puzzles. Both groups of children completed pre- and post-tests involving a range of math and spatial skills. Children in the spatial training group outperformed those in the crossword puzzle group, demonstrating signiﬁcant improvements in their calculation skills.

In another study, Verdine, Irwin, Golinkoff, and Hirsh-Pasek (2014) found that a child’s spatial skill at age three was a reliable predictor of the child’s grasp of number concepts such as more, less, equal, and second, as well as overall number knowledge skills. Taken together, research suggests that spatial instruction offers a potentially powerful means of supporting children’s mathematical thinking and learning.

2. Spatial reasoning can be improved. Education matters!

Spatial reasoning is malleable; that is, it can be improved. Spatial reasoning is not a biologically determined cognitive trait as was once thought to be the case. A recent meta-analysis of 217 studies, representing more than two decades of research on spatial training, found that a variety of activities improve spatial reasoning across all age groups (Uttal et al., 2013). Not only did the authors ﬁnd that spatial training led to improvements on spatial tasks closely related to the training task, but improvements were also seen on other types of tasks that were not part of the training. More research is needed to discover how and why this is the case. In the meantime, the ﬁnding that spatial ability can be improved at any age has massive implications for educators, particularly given that spatial reasoning is proving to be an important domain with strong connections to mathematical achievement.

**3. Spatial thinking is an important predictor of achievement in STEM careers.**

Research shows that spatial thinking is an important predictor of achievement in the STEM disciplines—science, technology, engineering, and mathematics (Wai, Lubinski, & Benbow, 2009). Sometimes these are called “STEAM” to reﬂect the inclusion of the arts. In addition, recent research indicates that early attention to developing children’s spatial thinking increases achievement in math and science and can promote skill and interest in future careers in STEM disciplines (Newcombe, 2010). Currently, many countries are concerned by the low numbers of post-secondary students, particularly female students, entering these disciplines. For example, a 2013 report found that fewer than 50 percent of Canadian secondary school students were graduating with senior-level STEM credits, while 70 percent of the highest-paying jobs require expertise in these disciplines (Let’s Talk Science [with Amgen Canada Inc.], 2013).

Geometry spans mathematics and science and plays a central role in disciplines such as surveying, astronomy, chemistry and physics, biology, geography and geology, art and architecture (Wai, Lubinski, & Benbow, 2009).

4. Spatial reasoning is currently an underserved area of mathematics instruction.

The National Council of Teachers of Mathematics recommends that at least 50 percent of mathematics instruction focus on spatial reasoning (National Council of Teachers of Mathematics [NCTM], 2006, 2010). Despite calls to bring geometry and spatial thinking to the forefront of early math curricula, local and international studies reveal that geometry and spatial sense receive the least amount of attention in early years math (Bruce, Flynn, & Moss, 2012; Sarama & Clements, 2009a), making it an underserved area of mathematics instruction. Spatial thinking is important in many areas of mathematics and beyond; most subjects in school—art, geography, science, language, and physical education to name a few—rely on at least some aspects of spatial thinking. Yet spatial reasoning itself is rarely, if ever, paid explicit attention. The National Research Council (2006) has highlighted this as a “major blind spot” in education and calls on educators and researchers to pay attention to spatial reasoning. Otherwise, the Council warns, spatial reasoning “will remain locked in a curious educational twilight zone: extensively relied on across the K–12 curriculum but not explicitly and systematically instructed in any part of the curriculum” (p. 7). Geometry and spatial reasoning in the early years typically focus on having children label and sort shapes (Clements, 2004), yet cognitive science and educational research, including the M4YC research, shows us that young children are capable of—and interested in—more dynamic and complex spatial thinking.

**5. Spatial reasoning provides multiple entry points and equitable access to mathematics.**

Many educators in our research classrooms have found that a focus on spatial reasoning provides multiple entry points for children to explore mathematics in an accessible and inclusive way. In fact, many educators have reported to us that, through using the activities that now appear in this book, they have been able to see their students in a new light. This, in turn, gives children the opportunity to participate in the mathematics and to contribute to mathematical discussions in the classroom, building their identities as mathematicians. For example, educators have found that some children who may be struggling in the area of number sense may excel in the area of spatial reasoning. For most children, a spatial approach enhances their developing sense of number. According to Baroody, Lai, and Mix (2006), “Most individual differences [in math ability] are probably due to the lack of opportunity” (p. 200). When we focus on spatial reasoning, we highlight and invite the diverse strengths that children bring to school (Flynn and Hawes, 2014).

*Joan Moss is an Associate Professor in the Department of Applied Psychology and Human Development at the Dr. Eric Jackman Institute of Child Study at the Ontario Institute for Studies in Education of the University of Toronto.*

*Catherine D. Bruce is a Professor and Dean of the School of Education and Professional Learning and Director of the Centre for Teaching and Learning at Trent University.*

*Bev Caswell is the Director of the Robertson Program for Inquiry-Based Teaching in Mathematics and Science at the Dr. Eric Jackman Institute of Child Study and Assistant Professor, Teaching Stream at the Ontario Institute for Studies in Education of the University of Toronto.*

*Tara Flynn is an educator, author, and editor, and Project Manager and Research Officer for Dr. Cathy Bruce at the Trent University School of Education and Professional Learning.*

*Zachary Hawes is a Ph.D. candidate in the Numerical Cognition Laboratory at the University of Western Ontario. Prior to this, he completed his M.A. and teacher training at the University of Toronto’s Dr. Eric Jackman Institute of Child Study.*

**MOTIVATION AND ENGAGEMENT**

Motivating students is a perennially difficult aspect of teaching, so it’s no wonder that there is robust interest in the neuroscience behind motivation. Researchers found that when test subjects could see how their brains were reacting to different motivational strategies on MRI images, they got better using successful approaches. But they also found it exhausting. While not yet applicable to the classroom setting, this neuroscience does offer educators insights into strategies that did and didn’t work, as well as how tiring the process can be.

On a more practical note, an article featuring 20 tips to engage even the most seemingly reluctant students also grabbed readers’ attention. No teaching approach is going to reach every student, so teachers need lots of strategies. When teachers have many ways to present information, to offer varying points of entry, and know how to demonstrate concepts from multiple viewpoints, they can better serve the different needs of their students.

**SELF-REGULATION AND TRAUMA**

Increasingly, teachers are being asked to do far more than deliver content, and that shift requires a new set of strategies and a compassionate approach to the job. Often educators are looking for guidance on how they can help kids improve self-control and behavior, as well as address their social and emotional needs.

Managing the behavior of 30 kids in an enclosed space is one of the most difficult aspects of teaching, so it’s no surprise that no teacher knows exactly how to respond to every situation. Yet acting out is a form of communication that can easily be misinterpreted as intentional disobedience or malice. That’s why tips to de-escalate situations with anxious or defiant students, presented by an experienced behavior analyst, was so helpful to educators.

Similarly, more and more educators are beginning to realize how much trauma their students have endured and how their behavior is often a symptom of those experiences. Educators are gravitating to workshops on how to teach with a trauma-informed lens, and are seeking support as they deal with the taxing work of educating children who are suffering intensely.

One school turned to a program that combines mindfulness and education about the brain to deal with residual trauma from a school fire, as well as the daily trauma of poverty that many students experience. The program has helped shift the culture of the school into a more positive place for students and staff with mindfulness baked into most school processes.

Early research on mindfulness has found that practices like focusing on one’s breath or intentionally showing gratitude can positively influence executive functioning skills that are also crucial for focusing in class, organizing work and many other cognitive functions. The importance of self-control on life outcomes has been well documented by psychologists, research that educators are now taking advantage of in classrooms.

**DEEPENING TEACHING PRACTICE**

Alongside discussions about how to instill character, improve school climate and motivate students to do their best work, educators are also continually trying to hone their craft, learning from research about the most effective ways to pull the best thinking out of every child. Often the articles that stimulate the most excitement and debate are not about specific curriculum or tools, but instead grapple with how to improve students’ metacognition. Researchers at Harvard have studied educators who focus on “teaching for understanding” for several years and have narrowed in on some practices that help improve the depth of student thinking.

In math classrooms a similar discussion is raging, with many math teachers looking for strategies to provide multiple entry points into the underlying conceptual topics in the curriculum. At the same time, most math curricula are stuffed with so many standards that teachers struggle to cover them all well. Math teachers are balancing trying to both prepare students for tests and give them the space and time to explore the foundations of math, a key practice to future math success.

**CAN PARENTS BE TOO INVOLVED?**

Parents are crucial partners for teachers in the academic and social development of children. Many parents take that responsibility seriously, reading up on how they can prepare their kids for academic success through the myriad of small interactions that happen daily. But the obsession with doing everything right is taking a toll on parents and may not be that great for kids either.

Teachers at the K-12 and university level are beginning to notice a worrying trend of overinvolvement from parents — while well-intended, it is actually depriving kids of crucial learning experiences. Parents, too, are noticing this tendency in themselves and are trying to pull back, with varying levels of success.

Reporting about education so often comes down to examining how humans interact with one another. Many of the themes that caught MindShift readers’ attention this year deal with how a bureaucratic system filled with well-intentioned people can nurture the whole child, paying attention to their academic minds, of course, but also recognizing that success in life rests on so much more. The trajectory of a life is a complicated interplay of opportunity, psychology, mentors and skills. The parents and teachers that help young people down this path have a very difficult job, but it can ultimately be one of the most rewarding ones, too.

]]>In a Scientific American article, Stanford education professor Jo Boaler and Pablo Zoido, the Education Lead Specialist at the Inter-American Development Bank, explain that students reported three main strategies for learning math: memorizing algorithms, relating new topics to those already learned, and routinely evaluating learning and focusing on areas not yet learned. Boaler and Zoido draw this conclusion:

In every country, the memorizers turned out to be the lowest achievers, and countries with high numbers of them—the U.S. was in the top third—also had the highest proportion of teens doing poorly on the PISA math assessment. Further analysis showed that memorizers were approximately half a year behind students who used relational and self-monitoring strategies. In no country were memorizers in the highest-achieving group, and in some high-achieving economies, the differences between memorizers and other students were substantial. In France and Japan, for example, pupils who combined self-monitoring and relational strategies outscored students using memorization by more than a year’s worth of schooling.

The U.S. actually had more memorizers than South Korea, long thought to be the paradigm of rote learning. Why? Because American schools routinely present mathematics procedurally, as sets of steps to memorize and apply. Many teachers, faced with long lists of content to cover to satisfy state and federal requirements, worry that students do not have enough time to explore math topics in depth. Others simply teach as they were taught. And few have the opportunity to stay current with what research shows about how kids learn math best: as an open, conceptual, inquiry-based subject.

Boaler and Zoido go on to recommend that math teachers focus on presenting students with visual, engaging tasks that let students grapple with the problem, test out various strategies, and thus gain a deeper understanding of core concepts. They point to research showing that students who solve problems by memorizing algorithms use a completely different part of the brain than those who work out the problem with various strategies. They posit that if the U.S. wants to improve the math abilities of its young people, it must heed the research and switch approaches.

Countries like Canada, Estonia, Germany and Hong Kong emerged as leaders in math education from the 2015 PISA results. Not only do students in these countries score well, but the gaps between rich and poor students are much smaller.

## Why Math Education in the U.S. Doesn’t Add Up

In December the Program for International Student Assessment (PISA) will announce the latest results from the tests it administers every three years to hundreds of thousands of 15-year-olds around the world. In the last round, the U.S. posted average scores in reading and science but performed well below other developed nations in math, ranking 36 out of 65 countries.

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“As he talked about students feeling that they don’t really belong, I had an epiphany,” Good said. She realized the discomfort she’d felt studying mathematics had nothing to do with her ability or qualifications and everything to do with a vague sense that she didn’t belong in a field dominated by men. Stereotype threat is a term coined by psychologists Joshua Aronson and Claude Steele. They found that pervasive cultural stereotypes that marginalize groups, like “girls aren’t good at math,” create a threatening environment and affects academic achievement.

Good was so fascinated by how powerful psychological forces can be on learning, including her own, that she switched fields again to study social psychology, and she ended up working closely with Carol Dweck for several years when Dweck’s growth mindset work was in its early stages and not yet well-known among educators. Good now works at a psychology professor at Baruch College.* Originally, Dweck and Good hypothesized that believing intelligence is flexible — what we now call a growth mindset — could protect students from stereotype threat, an inherently fixed idea.

“If students are first really encouraged and taught to believe in brain plasticity, our hypothesis was that they could be protected,” Good said. While that hypothesis was shown to be true, Dweck and Good also began to uncover forces that seemed to undermine individual mindsets.

“What we found was that students’ perception of what’s going on in their learning environments are often more important than their own beliefs,” Good said. In other words, if a classroom climate is one of fixed ability, it will override a student’s own beliefs about his brain plasticity. This effect was even more pronounced when stereotype threat was present. Students were less likely to feel belonging and were less likely to engage with content. That, in turn, led to lower achievement and lower grades.

“When you are looking at a long-term trajectory that’s when the culture really becomes much more important,” Good said — especially in certain fields of study, like math and science, where stereotype threat exists and traditional classroom structures favor a laddered approach to learning that screens out the unworthy and is inherently sending fixed mindset messages.

**APPLYING GROWTH MINDSET
**

A recent Education Week Research Center survey of 600 K-12 teachers nationwide found that over three-quarters of respondents felt “familiar” or “very familiar” with growth mindset as a concept, and nearly all reported feeling it had a positive potential for teaching and learning. A large portion of respondents also connected growth mindset with a range of positive outcomes and behaviors, but only 20 percent felt strongly that they themselves were good at cultivating a growth mindset in their students. Still fewer had confidence in their colleagues and administrators.

The gap between awareness of growth mindset as a good thing to incorporate into the classroom and the confidence to actually do so, especially in specific courses, may be why Carol Dweck and others are warning that growth mindset has been misinterpreted, sometimes to ill effect.

“People are much more likely to fall back on negative stereotypes with a naive understanding [of growth mindset],” Good said. She described a study one of her graduate students recently completed that tested teachers’ perspectives on student success. The graduate student gave one group of teachers an article to read that could be described as a “pop culture” understanding of growth mindset. The other group read a paper explaining that the most important way to increase student learning is for teachers to be reflective on their own pedagogical practices. Then that group reflected on new approaches they might try to help a struggling student.

Teachers who received only a broad brush understanding of growth mindset were less likely to reflect on their practice and more likely to shift blame back onto the struggling student for not having a growth mindset. Author, and critic of many traditional education practices like grades and standardized testing, Alfie Kohn, has also written about this danger.

Often teachers take away two messages from growth mindset articles or trainings: Effort is important and mistakes should be celebrated. But when applied simplistically, both these takeaways can be damaging. For example, for a student who is trying hard, but not achieving success, being told to try harder could be demoralizing. And celebrating mistakes without taking time to reflect on new strategies to try again doesn’t lead to the same learning gains.

Confusion about growth mindset and traditional structure of many classrooms are particularly apparent in math class, and to some extent science as well. As a former mathematician turned social psychologist with a deep interest in helping marginalized groups succeed and feel welcome in science, technology, engineering and math fields, Good has some specific ideas about how growth mindset could be incorporated into the fabric of math class.

**CULTURE**

The first big obstacle is embedded in American culture. Somehow it has become acceptable to brag about not being “a math person.” Good says that has to stop, especially when that type of math anxiety is coming from teachers and parents. “It’s almost like an infection model where the class fixates on that anxiety and is infected as well,” Good said.

**PROBLEMS WITH ERRORS**

One concrete mathematical teaching strategy that inherently promotes a growth mindset is to present students with worked-out problems that have errors. Students follow the thinking in the problem, identify the mistakes and rework them. “Embedded in that worked example is a lovely opportunity to talk about growth mindset and mistakes and process,” Good said.

THINK LIKE A MATHEMATICIAN

School math has become almost entirely about demonstrating how to solve a problem, rather than actually engaging in the kind of problem-solving that is at the heart of what professional mathematicians do. In other subject areas teachers encourage students to “think like historians” or to become writers. In those disciplines students create their own variations on expert texts and are encouraged to become practitioners. Not so in math. Good said the discussion around math should be about pushing through challenge, the same way real mathematicians do every day.

**RETHINK ASSESSMENTS**

One of the biggest ways math teachers can embed a growth mindset into the structure and environment of class is to change the role of assessment. Rather than taking tests whose scores accumulate into a final grade, students should get credit for returning to problems they didn’t get right, recognizing their mistakes and reworking the problems. Growth over the course of the year should be rewarded. Students shouldn’t be penalized in their final grade for doing poorly at the beginning of the year if they worked hard to learn the material over time. Assessments send very clear mindset messages that are far more powerful than anything a teacher says about growth mindset.

“Yes, we have to give assessments,” Good said, “Yes, we have to give grades. But when teachers say this grade doesn’t mark you or indicate what you are capable of in the long term, it shifts the whole meaning of the assessment for students,” Good said.

She favors a mastery approach that allows students to go back, relearn concepts that they got wrong and earn points for that work, in part because it ensures students actually learn the material before moving on, but also because it is important for teaching a growth mindset. It shows the teacher has high expectations, but believes the students can succeed and will provide support as they work to understand.

“This is where assessment can drive learning, but only if you go back and look at what you did and learn from it,” Good said.

**HELPFUL FEEDBACK**

Feedback is one of the most effective ways to help a student grow, but teachers must be mindful that students will always receive critical feedback through the lens of their stereotype threat. Human brains are also wired to pay more attention to negative inputs than positive ones. When teachers couch feedback with assurances that they will continue to hold the student to high standards and that they know he can get there, it helps protect him from the stereotype.

On the flip side, teachers who have fixed mindsets themselves are more likely to give comforting feedback meant to make the student feel better. Comments like, “It’s OK, let’s look at where you do have strengths,” are meant well, but communicate a fixed mindset to the student. “Things we do for students to boost their self-esteem actually have these ironic effects of making students feel you don’t believe in them,” Good said.

RETHINK ADVANCEMENT

Good sees the current practice of looking at math learning as a ladder with progressively more difficult rungs as a detrimental approach. It encourages teachers to act as gatekeepers to higher- level classes, funneling the “smart” kids into advanced courses and keeping out those who struggle. That in turn communicates low expectations and a fixed mindset about students’ abilities. Good said there should be multiple entry points, as opposed to a linear progression.

**PREPARE EVERYONE**

Growth mindsets are often discussed in relationship to kids who struggle, but the concept is just as relevant to kids who breeze through the material. Telling those kids they are smart is not setting them up for success later when they do struggle. For Good, that struggle didn’t come until graduate school, but she distinctly remembers feeling “not smart anymore” because she was struggling. Math teachers need to give high achievers opportunities to struggle and persevere early and often so the experience is not foreign to them.

Embedded in all of this growth mindset work is a general culture shift around how math is taught and who can excel at it. It’s no surprise that teachers are struggling to integrate growth mindset into their teaching practice because every child is different. When it comes to perceptions of intelligence, belonging and whether a teacher cares, many factors come into play. Most teachers were educated in math classrooms with fixed mindset messages, as were most parents, so shifting the culture of classrooms and schools is work that takes time and incremental changes. But when teachers commit to that work, the shift is possible.

* *Catherine Good is currently on leave from Baruch College, serving as Senior Research Scientist at the organization Turnaround for Children.*

“If a student has a solid understanding of fractions and precisely what they represent, they’re likely to perform much better with algebra,” said Valorie Salimpoor, a researcher at the Rotman Research Institute in Toronto, during an edWeb presentation on the neuroscience of fractions. She thinks educators have an opportunity to leverage what researchers know about brain science to ensure students learn fractions well, but also admits that learning math is cognitively taxing.

Learning a new math concept takes a toll on the brain not only because of the new math concepts, but also because students must recruit many parts of the brain to solve any problem. For example, students need visuospatial and auditory working memory when solving a fractions problem, and they must focus attention, inhibit distractions, order tasks, recall information from long term memory and integrate new concepts into an old schema. There’s a lot of mental processing going on when learning math, so understanding how careful brain-based instruction can prime the brain for new learning becomes extra important.

**OPTIMIZE INFORMATION PROCESSING**

When a person learns something new she forms a connection between two previously unconnected neurons. If that connection is weak, the new learning can easily be lost through forgetting. But the stronger the pathway and the more ways a person has learned the information, the more likely it will become encoded into long term memory.

To improve a student’s information processing around fractions neuroscience tells us teachers should both present information and give students ways to interact with it, in a variety of ways. For example, explaining how to add fractions only uses an auditory pathway. Showing fractions on a line graph or as a pie chart are two other ways of presenting the same information. Asking students to visualize parts of the whole and show fractions using manipulatives are two more ways.

“Every time you are visualizing this in a different way, you are recruiting different neurons and neural connections,” Salimpoor said. And she says active learning through problem solving or manipulation is a whole different ballpark neurally than passively listening, partly because even if a student looks like she is listening she still may not be paying attention.

“In one case you are passively absorbing information and in the other case you have to initiate the motor sequences and carrying them through,” Salimpoor said. When students are actively learning they are using the frontal lobe to determine what needs to happen next and to organize the information, as well as the motor regions of the brain to carry it out.

“The more areas we recruit, the more elaborate this network becomes,” Salimpoor said. That’s important because later when the student tries to recall the new information and pings one node in the network, the entire system is primed to help retrieve the information. It’s also important for students to continue recalling that information in order to strengthen the network, even if a concept was introduced in many ways initially. Ultimately for something to be learned well it needs to be integrated into the student’s general schema, which happens through practice.

**STRENGTHENING NEURAL NETWORKS**

Practicing a new concept is one way good way to strengthen a neural network, but if students only practice using one problem solving strategy, that practice is leading to memorizing, not deep conceptual knowledge. To strengthen the whole network and by doing so deepening understanding, they must practice using various ways of visualizing and solving the same problem.

“It’s most important to create problems where students have to initiate and come up with their own solutions,” Salimpoor said. “This makes a big difference. You can never have too much of that.”

Neurochemicals like dopamine can also create shortcuts to encoding information. The brain releases dopamine in response to novelty or when a person is anticipating something and doesn’t know what’s going to happen. When a student is emotionally invested in the learning or finds it intrinsically motivating, dopamine is also present. And dopamine helps strengthen neural networks.

Salimpoor uses the dopamine trick whenever she wants her toddler son to learn something new. She knows he loves trains, so if she can fold whatever new information she wants him to learn within a train narrative she knows he will be paying attention, intrinsically motivated and releasing dopamine that will help hardwire the new concepts. Rewards can also be a way to get at dopamine, but Salimpoor warns external rewards are never going to be as powerful as getting students to personally care about the learning.

**GETTING AT CONCEPTUAL UNDERSTANDING**

One of the trickiest things about helping students develop a deep conceptual understanding of a topic like fractions is that each student is coming into the learning experience not only with different levels of math knowledge, but also with different levels of working memory, executive functioning skills, ability to pay attention, and all the other non-content skills related to learning.

One way a teacher can use brain science to help students get at the deeper concepts is to relieve the pressure on students working memories as they are learning the new information. Salimpoor says working memory is a big challenge for many children. “If you can’t hold all of this new information in your mind, you can’t really process it,” she said. And if the information isn’t getting processed, it isn’t getting integrated into the large schema a child holds in his head of how things work.

“While some students might be very skilled at working memory, the ones who aren’t as good really suffer because they can’t take in all that information and process it, so they just tune out,” Salimpoor said. Teachers can be aware of this and try to break concepts down into the cognitive elements, always being mindful of how many pieces of new information the children need to hold in their minds at the same time to solve a problem.

When possible teachers can give students supports, like visuals representing the fractions, to work with as they are introduced to a concept. When the student has the visual, he doesn’t have to hold the symbolic representation of the fraction in his working memory as he figure out how to add the two together. After the concept has been introduced, teachers can slowly remove those scaffolds. Writing information down is another way student can offload some of what would be stored in working memory.

Working memory is one challenge, but it is very hard for teachers to identify all the specific ways students differ from one another cognitively. This is where technology can help.

Salimpoor has helped design a videogame focused on adding, subtracting, multiplying and dividing fractions — the skills any student should have mastered by the end of sixth grade — that puts all her neuroscience expertise for learning into the game mechanics. Called Fog Stone Island and produced by Cignition, the game is free and available online to teachers and students. Salimpoor said the game tracks the intangible information like working memory ability and executive functioning that a teacher would have a hard time identifying.

Salimpoor and her Cignition colleagues know full well that there are a lot of commercial games promising cognitive training that don’t deliver. There are also a lot of math games that are essentially procedural math disguised in a game form. The Fog Stone Island designers wanted to move away from both these models to develop a game that uses intrinsic and external rewards, offers multiple pathways to understand fractions, gives working memory support at the beginning of a task and slowly takes it away, and is situated within a context in which fractions would actually be used.

“We’ve tried to think of real life scenarios when you’d use math,” Salimpoor said. So, for example, in one corner of the immersive Fog Stone Island world, players use raw materials to build structures, a bit like Minecraft. The bricks have different lengths and the player must add like and unlike denominators to build a wall. Early in the concept the game provides a digital sketchbook — essentially virtual working memory — for students to use.

“Students can understand what they need to do in the situation without taxing their other cognitive abilities,” Salimpoor said. As they progress through the game they will gradually take over those working memory functions within the greater problem. Salimpoor finds it a little odd that she has ended up working on a videogame, but she wants good brain-based practices to be embedded in teaching and found that videogames were a far easier way to affect the many mental processes recruited when solving math problems.

Because Fog Stone Island is an immersive game, players can wander between the house building area, a farm plot and other zones that work on different elements of fractions. “The reason we wanted to keep it all in one game is it helps with investment,” Salimpoor said. Success in one area can lead to success in another area, and the different ways of working with fractions are integrated so students can develop a deep understanding of how the concepts are connected.

Salimpoor said the great thing about the game is that it adapts to the player’s level not just of math knowledge, but also working memory and executive functioning needs too. But Salimpoor is proud of how hard she and the designers worked to build those support skills into the game in a natural way that makes sense for the game. For example, rather than playing a silly side game that requires a player to repeat a sequence forward and then backwards — a common working memory exercise — Fog Stone Island may require a player to remember five items in the service of building part of his world. Now there’s an intrinsic motivation to build up working memory.

“In the video game we can do that so in depth,” Salimpoor said. “And really understand where each child is in each of these areas.” She says the latest iteration of the game is getting good reviews from students, who want to play even when it isn’t required for class. And very preliminary research results indicate that students are deepening their fractions knowledge outside of the game too, although those effects are still being studied.

]]>Helping students develop growth mindsets is made even trickier because mindsets about learning can change depending on context. And unfortunately math class is a time when many students have preconceived notions about their abilities. Many adults, including teachers, grew up receiving negative messages about their math ability and can unintentionally pass on unhelpful messages to students through casual words or actions.

That’s why it’s impressive that educators at Two Rivers Charter School in Washington, D.C. recognized a culture of math fear among the staff and worked hard to change teachers’ relationships to math as part of their broader strategy to improve math achievement. The school’s Director of Curriculum and Instruction, Jeff Heyck-Williams, described their efforts in an Education Week article:

In August of 2010, we started by listening deeply to our teachers’ math stories. We recognized that if we didn’t start with their learning first, we would never be able to approach the kinds of mindset shifts necessary to impact the learning of students. Teachers—even the art teacher and the pre-school teacher—wrote their math stories, sharing their deepest feelings about math and the people and experiences that led them to those beliefs. Over 65% of the stories that teachers told were negative. When they were students, our teachers had been given messages like “girls aren’t good at math,” “it is OK if you don’t get this, you won’t need it once you get out of school anyway,” and “math is either something you get, or you don’t get.” These messages were pervasive and came from teachers with an affinity towards math as well as teachers who couldn’t stand math. By acknowledging these messages, we brought them to the surface and made teachers aware of the messages that they were explicitly and too often implicitly sending to kids about math.

By starting with the mindsets of teachers, and recognizing that each person has her own mathematical history, Two Rivers was able to empower teachers to deepen their math skills. This professional development in turn helped teachers feel capable of teaching in problem-based ways that stretch student thinking.

*Learning and Loving Math: A Problem-Based Approach from Two Rivers Public Charter School on Vimeo.*

The Two Rivers Charter School example is a good reminder how so often the culture of particular schools and the attitudes of the adults in the building affect efforts to improve academic outcomes. It also shows that with a concerted effort, the teachers at this school flipped the script on their own math stories, learning the math they teach more deeply, while simultaneously becoming better mentors and guides to students.

## How We Got Teachers to Love Math-And Improved Our Math Scores

A few years ago, I left an interview with an elementary school candidate deeply concerned. She was looking to work with us at Two Rivers, a Preschool – 8th Grade urban public charter school in Washington, D.C. In many ways she was the perfect candidate. She had skills in literacy instruction and classroom management.

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A functional MRI study of 17 people blind since birth found that areas of visual cortex became active when the participants were asked to solve algebra problems, a team from Johns Hopkins reports in the *Proceedings of the National Academy of Sciences*.

“And as the equations get harder and harder, activity in these areas goes up in a blind person,” says Marina Bedny, an author of the study and an assistant professor in the department of psychological and brain sciences at Johns Hopkins University.

In 19 sighted people doing the same problems, visual areas of the brain showed no increase in activity.

“That really suggests that yes, blind individuals appear to be doing math with their visual cortex,” Bedny says.

The findings, published online Friday, challenge the idea that brain tissue intended for one function is limited to tasks that are closely related.

“To see that this structure can be reused for something very different is very surprising,” says Melissa Libertus, an assistant professor of psychology at the University of Pittsburgh. “It shows us how plastic our brain is, how flexible it is.”

Earlier research found that visual cortex could be rewired to process information from other senses, like hearing and touch. But Bedny wanted to know whether this area of the brain could do something radically different, something that had nothing to do with the senses.

So she picked algebra.

During the experiment, both blind and sighted participants were asked to solve algebra problems. “So they would hear something like: 12 minus 3 equals x, and 4 minus 2 equals x,” Bedny says. “And they’d have to say whether x had the same value in those two equations.”

In both blind and sighted people, two brain areas associated with number processing became active. But only blind participants had increased activity in areas usually reserved for vision.

The result suggests the brain can rewire visual cortex to do just about anything, Bedny says. And if that’s true, she says, it could lead to new treatments for people who’ve had a stroke or other injury that has damaged one part of the brain.

Drugs or even mental exercises might help a patient “use a different part of your brain to do the same function,” Bedny says. “And that would be really exciting.”

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

]]>Satisfaction and engagement may not be the most common feelings among students studying introductory calculus. According to Jo Boaler, a professor of math education at Stanford, roughly 50 percent of the population feels anxious about math. That emotional discomfort often begins in elementary school, lingering over students’ later encounters with algebra and geometry, and tainting the subject with apprehension—or outright loathing.

Professor Mary Helen Immordino-Yang, associate professor of education, psychology, and neuroscience at the University of Southern California has explored how emotions are tied to learning. “Emotions are a piece of thinking,” she told me; “we think of anything because our emotions push us that way.” Even subjects widely considered to be outside the realm of emotion, like math, evoke powerful feelings among those studying it, which can then propel or thwart further learning.

Is there a way to separate negative emotions from the subject, so that more students experience math with a sense of satisfaction and pleasure? Immordino-Yang believes so. “It’s not about making math ‘fun’,” she added; games and prizes tend to be quick fixes. Instead, it’s about encouraging the sense of accomplishment that comes from deep understanding of difficult concepts. “It’s about making it satisfying, interesting, and fulfilling.”

Adam Leaman, who teaches variations of algebra, trigonometry and calculus to high schoolers in Summit, N.J., said that a sense of awe about mathematics drew him to the subject beginning with algebra 2. “There’s something satisfying about knowing there’s an answer and knowing I have the ability to get it,” he said. Today, he sees the same pattern with his students: they are most engaged when they’re figuring out hard problems.

There are several ways teachers can replace student fretfulness over math with a sense of appreciation.

**Be clear about why understanding math concepts matters**. Kids who believe that they must simply endure algebra and calculus until they’re through with school—and that the actual learning is pointless because they’ll never use it again—should be reminded why understanding mathematical concepts is valuable. Most importantly, being able to comprehend a “symbolic, representative system,” Immordino-Yang says, teaches the brain how to think theoretically and logically. “Learning how to think abstractly is a useful ability in all aspects of life,” Immordino-Yang said. In fact, people who have studied complex math in high school tend to have better life outcomes, she said. Teachers who share this information may persuade reluctant math-learners to stay engaged.

**Assign projects that help kids see math’s usefulness**. Students are more apt to participate if they see a practical application to their studies. “This goes beyond learning how to balance a checkbook,” Immordino-Yang adds. By studying how fast and far vegetable oil spreads on tissue paper, for example, students can learn not only about the math concept of direct variation, but also about how oil spills are measured. Sharing stories from the news where math understanding is featured in the narrative—in a story about price fixing, say, or one on climbing interest rates—also can help students see its usefulness in the real world. Learner.org, a free educational resource from the Annenberg Center, provides such practical lesson plans for math at all levels, including the oil spill example.

**Discuss mathematical role models, and share how their ideas have changed the world.** Because math is the foundation of so many other fields—physics, engineering, finance, astronomy, among others—history teems with mathematical virtuosos whose creativity and curiosity shaped the modern world. In addition to the usual suspects of math icons, including Pythagoras, Rene Descartes, and Ptolemy, more contemporary role models might spark student appreciation for the subject. There are many: Alan Turing, who’s code-breaking during World War II helped defeat the Axis powers; Ada Lovelace, who created the Analytical Engine, which presaged modern programming; even Nate Silver, a popular mathematician who uses statistical forecasting to predict outcomes in Major League Baseball and political elections. The humbler discoveries made by annual recipients of the Presidential Early Career Awards for Science and Engineering might also inspire.

**Strive to minimize the sources of fear.** Math anxiety stifles clear thinking. At those moments when students most need to marshal their intellectual resources—during a test, say, or when called up before class to work a problem—those who fear the subject are apt to panic and shut down. Sian Beilock, a professor of psychology at the University of Chicago, and author of *Choke: What the Secrets of the Brain Reveal About Getting it Right When You Have To*, describes the anxious over-reaction to high pressure situations as a “malfunction of the prefrontal cortex.” For nervous students who feel pressure to perform well on a test, that worry causes them to execute beneath their skill level. “Anxiety is robbing you of working memory,” Immordino-Yang explained. “You’re wasting your thought powers,” she added. Math phobic kids need help from teachers to lessen their fear.

Keeping the classroom “kid-centric,” Immordino-Yang said, can help. Teachers who act as facilitators, or resident experts, rather than omniscient instructors, invite students to explore without fear of messing up in front of an authority. Freeing up fellow students to explain problems also allows for more personalized instruction. During Jacqui Young’s happy year studying pre-calc, she worked with peers who struggled to keep up with the teacher’s pace. “I think it was better because I’d be working one-on-one or two-on-one with my classmates,” she said. Another way to tamp down dread is to set up class in a roundtable and encourage student-led give-and-take. Known formally as the Harkness Method of teaching, this collaborative approach to learning may be especially useful in math subjects.

Returning to older math processes and ideas when introducing new material also works to lessen anxiety about the new learning. While teaching synthetic division to his Algebra 2 class, for example, Adam Leaman reminds students that this “new” concept is a cousin of the factoring they did in Algebra 1. “They have something from the past to draw reference from when tackling this new subject,” he said. Bringing up old material this way also helps students who might have struggled when they learned it the first time. Leaman said that some kids groan when he brings up factoring, but that they often end up understanding it better when going through it a second time, and in relation to a different concept. “Even if they have the perception that they’re not good at it, when we come back to it I have students say, ‘I get it now’.”

]]>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. Rather than asking the questions, she’s listening to her students push one 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.

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