“I was a math teacher, and I’ll be honest,” Keeler said, “I didn’t teach it to be creative.” She always felt pressure to move more quickly through the curriculum. Every day brought a new topic, whether or not students had deeply understood what came before. When Keeler read Boaler’s book, *Mathematical Mindsets*, she saw herself as a young student in much of what Boaler described. With tears in her eyes, she told a group of educators at the International Society for Technology in Education (ISTE) that since fourth grade she secretly thought she was dumb because she couldn’t pass timed math tests. Boaler’s message that fast is not the same thing as smart was liberating to her as a person and as a math teacher.

“When we work on a math problem, any type of problem, there are five different pathways in the brain that light up and are working,” Boaler said by video call at the same presentation. “Two of them are visual.” She argues that much of traditional math teaching focuses on numerical representations, teachers demonstrating procedures, and memorization, when it would be more effective to try to strengthen connections between the various parts of the brain needed when working on math.

“That comes about by showing information in different ways,” Boaler said. Representations of math problems using words, images and numbers each use different parts of the brain, so the concept gets hardwired in a neural network drawing on multiple brain faculties instead of one numerical pathway.

“The least likely way of helping kids have those brain connections is having kids sit and listen to lectures,” Boaler said. That doesn’t mean all math classes need to be project-based or that direct instruction is always bad, but when lecture is the default classroom mode, it doesn’t require students to use their brains to make sense of the new ideas.

Boaler’s website YouCubed has many activities to help teachers learn to open up the exploration of math from one of closed questions with a right and wrong answer, to one where different ways of seeing and articulating math are valued. When teachers ask students to explain why their thinking makes sense, students are forced to articulate their thought process, how it compares and contrasts to ideas peers have shared, and in doing so may help the teacher identify any misconceptions.

A simple example of opening up math in this way starts with a closed question: Divide one by two-thirds. But rather than asking students to apply a rule, ask students to come up with a visual proof. “What happens is the kids have these amazing discussions with different visual proofs, and it’s such a great way of taking a very closed question and opening it up,” Boaler said.

As a math teacher Alice Keeler loves the ideas on YouCubed and readily admits most of them can be done without technology. However, Keeler sees many ways that technology could enhance the visual and collaboration elements of the work, so she has adapted several YouCubed activities for the Google Suite. While Keeler spent 14 years in the classroom, she now has her own consulting business and teaches at California State University Fresno. She also co-wrote two books on using Google Classroom with Libbi Miller: *50 Things You Can Do with Google Classroom* and *50 Things To Go Further With Google Classroom.*

“It’s not about being digital and it’s not about being paperless,” Keeler said. “That doesn’t make learning better. But collaboration does.” She likes doing open-ended math activities in Google Slides because each student can play with visual representations, give feedback to peers, and receive ongoing feedback from the teacher. She usually makes blank slides and gives editing power to students.

“I ask each student to add their own slides explaining how they did it, how they visualized it, and we’re all doing it together in the Google Slides,” Keeler said. She’s found that when students can see how a peer visualized the problem, they then reflect on different approaches. She also values her ability to comment in real time with students because it becomes a conversation, not a static comment on returned work that the student may or may not look at again.

“I can have conversations with them around the ideas and help them to develop their thinking rather than just marking things right and wrong,” Keeler said. A math teacher who isn’t using G-Suite in class could also have these kinds of formative conversations by circling the room and talking with students working in groups, but Keeler likes using the technology because she can easily see how each individual is thinking about the problem. And students can interact with one another’s ideas, even when they aren’t physically in her class.

Keeler often tells students not to delete mistakes from the slides, instead telling them to duplicate the slide and keep working. That way she can see the progression of their thinking. This also helps students to see how far they’ve come.

A popular YouCubed problem asks students to take exactly four 4s and use any combination of operations to come up with the numbers 1-20. Keeler often does this in Google Slides, where each slide is a place for students to show how they combined four 4s to get “1” and then on the next slide the work for “2,” etc.

She likes working in Google Slides because students can add media or even do work on paper and upload an image. This gives different types of learners options. Students with disabilities or who benefit from speech-to-text help can also participate using EquatIO, a Chrome add-on that has voice typing capabilities, as well as handwriting recognition. EquatIO used to be g(Math), and now also makes it possible to use math symbols in slides and other Google apps.

Another popular YouCubed activity asks students to visualize division by divvying up a pan of brownies equally among friends. Keeler does this activity in a spreadsheet, and often asks students to create their own brownie pans — their own problems — in the next tab. “It allows them to experiment and play,” she said.

Keeler has become something of an evangelist for technology in math classrooms, learning how to set up conditional statements and even simple code in Google Sheets to aid her purposes (she also shares these ideas regularly on Twitter, including activity templates). Over time her teaching evolved and by the time she left the K-12 classroom she had upended some of the practices she once considered fundamental, like assigning homework.

One of the most controversial ideas in math education revolves around how, when and how much students should practice. Many teachers believe it is important for students to do homework so they can practice new concepts learned in class. Boaler agrees that practice is important, but doesn’t think that requires doing the same type of rote problem over and over. Boaler explained this to her daughter’s teacher and was pleasantly surprised at how she used the feedback. After their discussion, the teacher started giving students four problems to practice the calculations and then asked them to represent the concept some other way. They could write a story, make a drawing or come up with something else. The key was showing their knowledge in different ways.

Many elementary school educators are willing to consider that homework is not necessary for the young learners they teach, but far fewer high school teachers agree. Keeler taught Algebra and AP Statistics when she was in the classroom. She found that “the only kids who did the homework were the ones who didn’t need to,” so she stopped assigning homework. “It didn’t make a lick of difference” in terms of achievement, she said, but kids started enjoying class more. When she eliminated homework, Keeler found she had much more positive relationships with students and parents, a benefit that far outweighed what she called the “marginal gains of more rote practice.”

Ultimately both Keeler and Boaler hope that by making math a subject that’s about ideas, discussion, differing viewpoints and visual representations, students will learn they can not only do math, but excel at it. Too many students don’t feel that way now, which is why teachers are beginning to see the need for a new approach.

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As teachers try to improve how they teach math by applying numeracy, inquiry-based learning, productive failure and complex instruction, the idea of how to become better math teachers is gaining a wider audience. But Zager writes in her book, “We moved right into a new way to teach math, without addressing teachers’ personal histories with and understanding of mathematics.”

Zager traveled around the country observing and interviewing outstanding math teachers, and recently published a comprehensive book that invites teachers to reconsider how they think about and teach mathematics. In *Becoming the Math Teacher You Wish You’d Had: Ideas and Strategies from Vibrant Classrooms*, Zager strives to motivate teachers to replace the procedural and uninventive methods of ordinary math instruction with approaches that celebrate pure mathematics, with all its creativity, intuition and risk-taking.

And what is it true mathematicians do that the teachers should try to emulate? Mathematicians take risks, make mistakes, demand precision, rise to challenges, ask questions, connect ideas, use intuition, reason and prove—habits of mind that can be taught and learned in math classes. Zager introduces us to several exemplary teachers who find a way to do that, offering readers practical techniques to build the kind of classroom that embraces true mathematics. Zager’s book is divided into 13 chapters, each of which explores a different characteristic of a mathematical mind, and then follows up with stories of teachers who work to instill these qualities in their students.

“These teachers aren’t superheroes who were just born this way,” Zager cautions. Rather, they are mere mortals, all working with different populations of kids, who have honed their practice over the years through professional development, coaching and teacher inquiry. What unites them is a common desire to be more effective at their work.

One such teacher, Debbie Nichols, teaches first- and second-graders in a rural New England town. She wants her students to ask questions so she consciously finds ways to draw out her students to put their questions at the center of the class. One way she does this is by inviting students to work together to come up with questions, as she did in a class about shapes. “I would like to hear some of your questions so that we can figure out what we want to investigate!” she told her class. The students’ numerous questions were profound in their simplicity; they wondered, for example, “Are shapes the same all around the world?” and “Are shapes fragile?”

Nichols then asked students to select the four questions that would help them learn best, and continued to provide guidance and instruction as they probed deeper. Recognizing that students’ understanding of shapes was too limited to answer the questions they posed, she provided students with geometric materials like blocks and tiles to help them make sense of shapes. Midway through the unit, and then at the end, Nichols asked students again to think what new questions they might have. This focus on student questions, Zager writes, ignites their curiosity and spirit of inquiry, which are essential features of mathematical thinking.

Another teacher, Jen Clerkin Muhammad, encourages her fourth-grade students to make connections between various concepts. Among other reasons, connecting one idea to another builds on what’s already understood, establishes links between apparently unrelated subjects, and helps students apply mathematics to the real world. Muhammad recognizes that seeing problems solved in multiple ways promotes deeper understanding, and expects students to explore many representations of the same problem. In one class, for example, Muhammad asked students to draw a picture of a multiplication problem—if Darlene picked four apples, and Juan picked four times as many, how many does Juan have?– and then to walk around and examine others’ representations. “Where do you see the four times as many in this representation?” Muhammad asked her students. Seeing varied perspectives on the same problem enhances student understanding of the essential concept and builds new connections.

Shawn Towle, who teaches eighth grade in suburban Portland, prods his students to debate their ideas. Such verbal give-and-take is inherent in mathematical discovery, and Towle challenges his students to take positions and then defend them. In one project, Towle gave students a math problem involving a spinning game and asked them to consider whether the problem is fair or unfair, and why. He then divided the class into like-minded groups and asked each group to clarify their reasoning. Students moved back and forth between groups, and then debated each other one-on-one. This student engagement in mathematical disputes, defending and then abandoning or sticking to their reasoning, resembles the way many high-level mathematical discoveries are made: through a blend of solitary work, collaboration and disagreement.

In some ways, these classes represent what Zager wishes she had in school.

Zager had her own experience in middle school when she felt humiliated in her algebra class for questioning the possible outcomes of a math sequence. In her book, she surmises that her teacher responded in a closed-off manner because he was emulating how he had been taught. In Zager’s case, though, the experience didn’t deter her from math, but encouraged her to find the root of that reaction and the anxious feelings she saw in the teachers she mentored.

Many teachers suffer from the same math anxiety and general discomfort with the subject as their students. Thus, changing the way they think about and teach math won’t come quickly or without work. “It requires cognitive and emotional work to relearn and have a better way to teach mathematics,” Zager said. But just like their students, teachers benefit when they approach their own work with a growth mindset.

Teachers like Shawn Towle should inspire others who want to shake up the way they teach mathematics but who feel reluctant or afraid, Zager said. Having taught using traditional methods for 14 years, Towle took part in professional development that opened his eyes to new pedagogical tools, which he then shared with colleagues. He piloted a program called Connected Mathematics Project, and went on from there to find more ways of returning creativity and wonder to his math classes.

“He’s a single teacher who has had a positive impact on countless teachers and students in his school,” Zager said. “He gives me such hope.”

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“I thought,* No way. *Babies probably can’t count, and they certainly don’t count the way that we do,” she says. But the seed was planted, and vanMarle started down her path of study. The person who made that flyer, Karen Wynn, became her mentor and they have since co-published several studies together.

I spoke with vanMarle, an associate professor at the University of Missouri whose research focuses on children’s early cognitive development, to find out what she’s been up to lately. The interview that follows has been edited and condensed.

**So, what’s all this about being born mathematicians? **

In my lab, we are particularly interested in numerical development and understanding of objects — how the early number skills of young infants, possibly even newborns, get built upon to develop a uniquely human capacity for symbolic math.

The roots of those abilities and those skills seem to come from an endowment that is evolutionarily ancient and that we share with most other species.

**In other words, we’ve evolved to know math — along with almost every other animal. How did you become interested in this?**

I’ve always been fascinated with the idea that you can have this sophisticated knowledge — at least the foundations of it — in place, very early on. And we know now that it’s very broadly available across animal species. Species as different from humans as fish: Guppies are sensitive to numbers in the environment. Of course, primates are. Salamanders. Various insects. It’s this basic ability that helps animals navigate their environment. I mean, literally, navigate the environment by calculating angles and distances and so forth. It helps them choose the greater amount of food if they’re choosing between two quantities. It shows up in foraging contexts all the time.

So I’ve gotten interested in how these early abilities might provide a foundation for these much more sophisticated abilities that humans grasp pretty ubiquitously. If you’re exposed to math and counting, all humans will get it to a degree. Some more easily than others, of course, we all experience that. But the capacity is certainly available.

**What’s been the focus of your most recent research?**

Being literate with numbers and math is becoming increasingly important in modern society — perhaps even more important than literacy, which was the focus of a lot of educational initiatives for so many years.

We know now that numeracy at the end of high school is a really strong and important predictor of an individual’s economic and occupational success. We also know from many, many different studies — including those conducted by my MU colleague, David Geary — that kids who start school behind their peers in math tend to stay behind. And the gap widens over the course of their schooling.

Our project is trying to get at what early predictors we can uncover that will tell us who might be at risk for being behind their peers when they enter kindergarten. We’re taking what we know and going back a couple steps to see if we can identify kids at risk in the hopes of creating some interventions that can catch them up before school entry and put them on a much more positive path.

**How exactly do you study something like that? **

We followed kids through two years of preschool and assessed a really broad range of quantitative skills. Because when you talk about math achievement and number knowledge, it’s not a single solitary construct.

Over the two years of preschool, we gave them 12 different tasks — twice a year. Some were symbolic: being able to recite the Arabic numerals or the verbal count list. Others were tapping these earlier, emerging non-symbolic skills: being able to estimate which of two sets of dots is bigger, being able to keep track of additions and subtractions that happen in the environment. Skills like that are building on these evolutionarily ancient core capacities.

**So which of those actually predict math achievement? **

Out of those 12 different skills, there’s really one or two that matter most. When we followed up with these kids in kindergarten and first grade, their ability to estimate quantities — this ancient ability — seems to be really important. And also their ability to engage in cardinal reasoning i.e. knowing that the number three — when you see it on a page or hear someone say “three” — that it means exactly three, which is really at the root of our ability to count.

This cardinality, in particular, seems to be the most important skill that we can measure at a very young age and then predict whether kids are going to be succeeding in a much broader assessment of math achievement when they enter kindergarten.

**Will this have an effect on what kids learn in preschool?**

Well, we hope so. If you look at preschool curricula — kids who are getting structured instruction in math early on — it’s really trying to tap these different skills. But when you have a lot of different things you’re trying to teach, you don’t go into depth with them, right? You’re just trying to touch on all of them at once.

Our research points to the possibility that it might be more effective for early education if you focus on these core skills that seem to matter the most for developing symbolic knowledge. We’re currently running a pilot study — an intervention that targets this ability.

**What does that intervention look like? **

Children count and create sets. We use ice cube trays to count some number of objects. We say, ‘Can you put six items into this tray?’ And then we point out very interactively where they make mistakes and try to reinforce rules.

**Has it been effective?**

It’s too early to say. We are currently inputting the data and analyzing it so I don’t have the punchline for you, unfortunately. But we’re hopeful it will be effective.

It’s the kind of thing that parents and early educators can engage in with children. It’s possible to even create an app that would allow kids to make sets on an iPad. Of course, that’s way down the road for us. But that’s kind of where we’re headed — getting an intervention that works. We know how to identify which kids are likely to be at risk so the logical next step is to figure out a way to help them.

**Your research points out that parents aren’t engaging their kids in number-learning nearly enough at home. What should parents be doing?**

There are any number of opportunities (no pun intended) to point out numbers to your toddler. When you hand them two crackers, you can place them on the table, count them (“one, two!” “two cookies!”) as they watch. That simple interaction reinforces two of the most important rules of counting — one-to-one correspondence (labeling each item exactly once, maybe pointing as you do) and cardinality (in this case, repeating the last number to signify it stands for the total number in the set). Parents can also engage children by asking them to judge the ordinality of numbers: “I have two crackers and you have three! Who has more, you or me?”

Cooking is another common activity where children can get exposed to amounts and the relationships between amounts.

I think everyday situations present parents with lots of opportunities to help children learn the meanings of numbers and the relationships between the numbers.

Copyright 2017 NPR. To see more, visit http://www.npr.org/.

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It is also the single most failed course in community colleges across the country. So if you’re not a STEM major (science, technology, engineering, math), why even study algebra?

That’s the argument Eloy Ortiz Oakley, chancellor of the California community college system, made today in an interview with NPR’s Robert Siegel.

At American community colleges, 60 percent of those enrolled are required to take at least one math course. Most — nearly 80 percent — never complete that requirement.

Oakley is among a growing number of educators who view intermediate algebra as an obstacle to students obtaining their credentials — particularly in fields that require no higher level math skills.

Their thinking has led to initiatives like Community College Pathways, which strays away from abstract algebra to engage students in real-world math applications.

What follows is an edited version of Siegel’s Q&A with Oakley.

**What are you proposing?**

What we’re proposing is to take an honest look at what our requirements are and why we even have them. So, for example, we have a number of courses of study and majors that do not require algebra. We want to take a look at other math pathways, look at the research that’s been done across the country and consider math pathways that are actually relevant to the coursework that the student is pursuing.

**You are facing pressure to increase graduation rates — only 48 percent graduate from California community colleges with an associate’s degree or transfer to a four-year institution within six years. As we’ve said, passing college algebra is a major barrier to graduation. But is this the easy way out? Just strike the algebra requirement to increase graduation rates instead of teaching math more effectively?**

I hear that a lot and unfortunately nothing could be farther from the truth. Somewhere along the lines, since the 1950s, we decided that the only measure of a student’s ability to reason or to do some sort of quantitative measure is algebra. What we’re saying is we want as rigorous a course as possible to determine a student’s ability to succeed, but it should be relevant to their course of study. There are other math courses that we could introduce that tell us a lot more about our students.

**Do you buy the argument that there are just some forms of reasoning — whether it’s graphing functions or solving quadratic equations that involve a mental discipline — that may never be actually used literally on the job, but may improve the way young people think?**

There’s an argument to be made that much of what we ask students to learn prepares them to be just better human beings, allows them to have reasoning skills. But again, the question becomes: What data do we have that suggests algebra is that course? Are there other ways that we can introduce reasoning skills that more directly relate to what a student’s experience in life is and really helps them in their program of study or career of choice?

**A lot of students in California community colleges are hoping to prepare for a four-year college. What are you hearing from the four-year institutions? Are they at ease with you dropping the requirement? Or would they then make the students take the same algebra course they’re not taking at community college?**

This question is being raised at all levels of higher education — the university level as well as the community college level. There’s a great body of research that’s informing this discussion, much of it coming from some of our top universities, like the Dana Center at the University of Texas, or the Carnegie Foundation. So there’s a lot of research behind this and I think more and more of our public and private university partners are delving into this question of what is the right level of math depending on which major a student is pursuing.

**And there are people writing about concepts of numeracy that may be different from what people have been teaching all this time. Do you have in mind a curriculum that would be more useful than intermediate algebra?**

We are piloting different math pathways within our community colleges. We’re working with our university partners at CSU and the UC, trying to ensure that we can align these courses to best prepare our students to succeed in majors. And if you think about it, you think about the use of statistics not only for a social science major but for every U.S. citizen. This is a skill that we should have all of our students have with them because this affects them in their daily life.

**Are you at all disappointed that the high schools who are sending students to California’s community colleges are not already teaching their students these algebra skills before they graduate?**

Certainly, these questions come up in K-12 education, but if we consider who really drives K-12 education — that is our four-year university system. By creating requirements, we ensure that K-12 has to align with those requirements. So as long as algebra is the defining math course, K-12 will have to teach it.

**Bob Moses , the civil rights activist, ****started the Algebra Project, teaching concepts of algebra to black students in the South. He ****saw the teaching of math as a continuation of the civil rights struggle. **

**Rates of failure in algebra are higher for minority groups than they are for white students. Why do you think that is? Do you think a different curriculum would have less disparate results by ethnic or racial group?**

First of all, we’ve seen in the data from many of the pilots across the country that are using alternative math pathways — that are just as rigorous as an algebra course — we’ve seen much greater success for students because many of these students can relate to these different kinds of math depending on which program of study they’re in. They can see how it works in their daily life and how it’s going to work in their career.

The second thing I’d say is yes, this is a civil rights issue, but this is also something that plagues all Americans — particularly low-income Americans. If you think about all the underemployed or unemployed Americans in this country who cannot connect to a job in this economy — which is unforgiving of those students who don’t have a credential — the biggest barrier for them is this algebra requirement. It’s what has kept them from achieving a credential.

**Do you risk a negative form of tracking? Depriving a student of the possibility of saying in community college: “Wow, that quadratic equation is the most interesting thing I’ve ever seen. I think I’m going to do more stuff like this.”**

We’re certainly not saying that we’re going to commit students to lower levels of math or different kinds of math. What we’re saying is we want more students to have math skills that allow them to keep moving forward. We want to build bridges between the kinds of math pathways we’re talking about that will allow them to continue into STEM majors. We don’t want to limit students.

The last thing I’d say is that we are already tracking students. We are already relegating students to a life of below livable wage standards. So we’ve already done so, whether intentionally or unintentionally.

Copyright 2017 NPR. To see more, visit http://www.npr.org/.

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Stanford Mathematics Education Professor Jo Boaler is championing a dramatic shift in how many math teachers approach instruction. Rather than focusing on the algorithms and procedures that make mathematics feel like a lock-step process — with one right way of solving problems — Boaler encourages teachers to embrace the visual aspects of math. She encourages teachers to ask students to grapple with open-ended problems, to share ideas and to see math as a creative endeavor. She works with students every summer and says that when students are in a math environment that doesn’t focus on performance, speed, procedures, and right and wrong answers they thrive. They even begin to change their perceptions of whether they can or can’t do math.

Solving The Math Problem (Subtitles) from YouCubed on Vimeo.

In an opinion piece for The Hechinger Report, Boaler lays out five ways teachers can approach instruction differently. She points out that many students experience math anxiety, which is negatively related to performance. While psychology research has pointed to smaller interventions to lower anxiety before tests or to help students combat stereotype threat, Boaler says those measures fall short. She writes:

Widespread, prevalent among women and hugely damaging, math anxiety is prompted in the early years when timed tests are given in classrooms and it snowballs from there. Psychologists’ recommendations — including counseling and words to repeat before a test — severely miss the mark. The only way to turn our nation around is to change the way we teach and view math. The problems that we have now include these:

First, math is often taught as a performance subject. Ask your children what their role is in math class, and they are very likely to say it is to get questions correct. They do not say this about other subjects. More than any other subject math is about tests, grades, homework and competitions.

Check out Boaler’s recommendations to change the math teaching paradigm in the U.S.

## OPINION: It’s time to stop the clock on math anxiety. Here’s the latest research on how – The Hechinger Report

Our future depends on mathematical thinking, but math trauma extends across our country – and the world – due to the ineffective ways the subject is often taught in classrooms, as a narrow set of procedures that students are expected to reproduce at high speed.

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Pattern recognition is a fundamental part of mathematics and kindergarteners are not too young to notice, compare and describe simple patterns. In this video, kindergarten teacher Donella Oleston describes how she had to back up and explain to these young learners what it means to “explain your thinking,” because at first students would only answer, “My brain told me so.” With practice, she says students have gotten to deeper levels of noticing, moving past the obvious and picking out more abstract similarities and differences between two pattern sets.

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In this next puzzle, viewers are cast as intrepid secret spies, tasked with deactivating a death ray. It’s also an interesting introduction to visual models and graph theory. The answer explanation starts at 1:04 in the video.

Who can resist trying to solve a brain teaser that Albert Einstein supposedly wrote? This problem seems pretty complicated at first, but it could be a great way to give students an opportunity to sift through the information given and start making sense of it. The video explicitly talks about some effective problem solving strategies like trial and error that can help students develop their logical intuition. And, while this is a silly problem about a stolen fish, multiple variable equations require a similar type of logic.

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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.”

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“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.

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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.*

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**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.

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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.*

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“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.

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