He writes:

For the right teacher, Twitter is the best ambient, low-intensity professional development and community you’ll find. Maybe Twitter isn’t as good for development or community as a high-intensity, three-year program located at your school site. But if you want to get your brain spinning on an interesting problem of practice in the amount of time it takes you to tap an app,

Twitter is the only game in town.

But here’s the problem: It’s takes a lot of courage to put yourself out there, to broadcast your opinions and to ask for input. And just because you put yourself out there, it doesn’t always mean you’ll get a response. Hence Meyer’s new website: Lonely Math Teachers, in which he curates those lonely tweets that have gone unanswered.

Interesting looking at my early #MTBoS tweets. Most of the time I got no response at all. I wonder why I kept at it.

— David Butler (@DavidKButlerUoA) July 29, 2017

This conundrum brings us to an important question: As an educator, how do you overcome the shortcomings of social media and network with other innovative teachers beyond the Internet?

For answers, we turned to various educators, who shared tips on building camaraderie and tapping into existing math communities. These tips and strategies include both online and offline approaches, many of which can be applied to any subject, not just math.

**Find mentors and collaborators in your own school**

“Does your school allow for teachers to build communities ad hoc?” That’s the first question you should ask yourself, said José Luis Vilson, a math teacher and author of *This Is Not A Test: A New Narrative on Race, Class, and Education**.* “Generally, high performing countries spend way more time planning that they do actually working in the classrooms,” he said.

When Vilson was just starting out as a classroom teacher, he turned to his peers to build upon his own skills and create camaraderie. “I leaned heavily on visiting all of my colleagues two to three times a week,” he said. He would spend time observing and then ask follow-up questions in free periods: “’How did you do that?’ ‘What did you do?’ ‘Can I take notes on that?’ I started becoming a conglomerate of all the teachers I’ve seen,” he said.

Ultimately, educators don’t all have to teach the same way but those classroom experiences can help you innovate and tailor your curriculum in a way that makes sense for you and your students.

“Obviously you hear what to do and what not to do—but mainly it’s about what actually fits into your teacher persona,” he said. “We don’t all have to do the same thing. We are going to teach in a way that is responsive to the students but also what’s responsive to what we envision as a teacher.”

**Curate a Twitter feed filled with diverse voices to broaden perspective **

Lybrya Kebreab is a math coach at Westlake Middle School in Oakland, California. She says Twitter is where she found her people. “It was good to see [through Twitter] I wasn’t the only person to see the potential of education, and specifically math, to empower students and teachers.”

But it took her awhile to actually join in on the conversation. At first, she just observed. “You don’t have to say anything to anyone but at least get on and watch and listen to other perspectives for as long as you need to. The whole point is to keep growing.”

That being said, it’s easy for social media platforms to become an echo chamber filled with similar voices. Kebreab‘s advice is to curate a diverse feed filled with educators from different backgrounds and expertise. “People who have done different jobs have a much wider and broader sense of the big picture [of educational systems]. But then people who have only been in one place have a very deep knowledge base—so both of those perspectives are needed,” she said. “Our upbringing and experiences have a huge impact on how we see things.”

Kebreab spent many years as a classroom teacher, and she said Twitter helped her expand her craft. She used the platform as a way to gather new ideas and inspiration around math curriculum.

“Anytime I did a math lesson that I really wanted to go well and I wanted different perspectives on, I would just put the idea on Twitter and people would give feedback and specific ideas on how to make it better,” said Kebreab. “Every time I did that, they saw the lesson from a perspective I didn’t see.” She recalls a time when she heard feedback from a teacher in the Midwest: “That perspective the teacher in Nebraska gave me from Twitter might be similar to the perspective of a kid sitting in my room. But I can’t see it with my California perspective—so I am able to reach more children with more eyes on the lesson.”

**Share your point of view, and encourage your peers to do the same **

If you feel comfortable on Twitter, one way to build upon those relationships is to share your experiences on a personal blog. Just last month, Kebreab started a WordPress blog. And in her first post, she wrote about why play has become so important to her while teaching math: “I’m obsessed with creating joy, laughter and warm memories of in-depth explorations with equations, shapes, patterns, graphs, conjectures, arguments and counterarguments, and most importantly, connections, in the rooms for which I have been blessed with stewardship,” she writes.

She kicked off her blog with a topic she’s been exploring for some time and she says that’s a key part of building a community and generating conversation: having a niche. **“**I’ve been incessantly encouraged to start blogging by my Twitter friends and by my friends in my immediate math circle. They said I should write, so I wrote,” she said. “I wanted to write something I was passionate about. And I’m passionate about kids being kids and analyzing consequences for the way adults impose their adult worries onto kids in this culture of high testing.”

Thus far, Kebreab says all conversations have been respectful, even when there are disagreements or “Twitter fights” as she calls them. “We have to engage the other side or else we aren’t really pushing ourselves as much as we could,” she said. Ultimately, she said, the more you talk about what you care about, the more people will come to you for information and support on that specific topic. “You start to be known for whatever your pocket is, and the more you expose yourself to the people on Twitter, the more people want to interact with you,” she said. “The more you put yourself out there, the more people come to support you.”

1st session of the day. #MidSchoolMath2018 #iteachmath #mtbos Developing Risk-Takers!Happy to see @TracyZager's beautiful face #BecomingMath pic.twitter.com/qRfwNwIFDd

— Lybrya Kebreab (@LybryaKebreab) March 2, 2018

**Tap into existing math networks online and in person**

There are collaborators who have been working to create and cultivate online and in-person spaces for math educators. Tracy Zager is one of those people. She’s a longtime math coach and author of the book *Becoming the Math Teacher You Wish You Had.*

She encourages math educators to use hashtags on social media like #iteachmath, #mtbos, #elemmathchat, #msmathchat, and #swdmathchat. This helps raise the visibility of posts, and it also provides relevant streams of conversation that you can follow. ** **She says to look for “little teeny moments to send something out.” For example: “If I am in a classroom and a kid says or does something interesting, I will take a picture of their work or jot down their thinking, and then tweet with a math teacher twitter hashtag like #mtbos, #iteachmath, or #elemmathchat and ask, *What would you ask this student? *or *What problem should we pose next? *Within a couple hours, the teacher and I will have new ideas to think through together,” Zager said. “I can pull all these colleagues into my classrooms with me. I find reaching out to teachers online enriches my conversations and relationships with teachers in person.”

I always take test questions structured like this as a dare. CANNOT?! We'll see about that. CAN TOO.

In other words, I hate this question design. pic.twitter.com/oNSWKaXyuL

— Tracy Johnston Zager (@TracyZager) January 25, 2018

For additional experiences outside of social media, Zager points to teacher organized opportunities like the Global Math Department’s free weekly webinar, as well as the in-person conference, Twitter Math Camp.

And then there’s the annual game night for math teachers that Zager and a few of her friends dreamt up. The last one took place in San Antonio and was sponsored by the National Council of of Teachers of Mathematics. (You can follow the tweets at #MTBoSGameNight.) She says those in person events can deepen relationships and foster emotional connections with other math innovators. And oftentimes, those relationships will sustain over time, not only in person but also online.

“The internet is awash with stuff. There’s so much out there and much of it is terrible, so I think that is part of the reason to find a community,” she said. “It helps us find the good stuff more quickly because there is just too much to sift through alone. We can help each other develop good taste in what choices to make and what resources to use in thoughtful ways.”

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While it’s easy to envision using math picture books in elementary school classrooms, literature for older grades poses a bigger challenge. Can reading fit into the curriculum as the books get longer and the math gets more complex?

Bezaire thinks it can, and so does another teacher, Sam Shah, of Brooklyn. The two occupy opposite ends of the secondary math spectrum — seventh-grade pre-algebra and 12th-grade calculus, respectively — and both have found ways to strengthen student engagement through reading.

**Novel study in pre-algebra**

During Bezaire’s *Curious Incident *unit, each period begins with a typical 20-minute math lesson, followed by a 15-minute discussion of the previous day’s reading. For the rest of the 55-minute period, he reads a new chapter aloud. As he reads, Bezaire often pauses to dig deeper into the story’s math. Sometimes, the concepts align directly with the day’s pre-algebra lesson. For example, on the day when they learn about prime numbers, the class also reads why the main character, Christopher, chose primes as his system for labeling chapters.

“The literary hook for this lesson is strong, and kids are really into learning more about primes thanks to the context of the story,” said Bezaire. “The lessons don’t always line up this nicely, but so much of what Christopher writes about regarding mathematics is about flexibility with numbers that it’s a really nice match.”

After class students complete written reflections about the book, with different types of questions serving multiple pedagogical goals.

Mathematical questions, which often relate to puzzling out the novel’s two mysteries, allow students to practice problem-solving strategies in a context with more buy-in than the usual practice worksheets. They also encourage deeper thinking about the reasoning behind a math strategy. For example, after students test Christopher’s method for mental math with large multiplication, they are asked how easy or hard it was and when it might be most useful.

Questions related to plot and language, Bezaire said, help his less confident math students. “Students who more easily self-identify as ‘English types’ immediately get a little more comfortable in math class if they experience those types of (literary) questions regularly.”

Others questions invite students to connect personally with the text. For instance, one question asks students to share the meaning of their name. Another asks them to consider how it might affect their interactions if they could not read facial expressions, like Christopher, who has autism. These questions allow Bezaire to learn about his students in ways that equations cannot. They also improve students’ patience and understanding with each other, he said.

If any MS math teacher is interested in trying my curriculum for the novel The Curious Incident of the Dog in the Night-Time (https://t.co/FVg9TA4aDC) I have a set of 20 novels I can send you. I’ve accumulated way too many extras and would love to pass them on. #mtbos pic.twitter.com/m9pmFnLJ9a

— Joel Bezaire (@joelbezaire) January 9, 2018

“Context and prior knowledge are critical components in fostering comprehension, regardless of the topic,” according to Faith Wallace, co-author of *Teaching Math Through Reading*. Reading literature is one of several ways to build that context and background knowledge. “When math is integral to the story students can learn the concepts in a natural way, become inquisitive, engage in thoughtful conversation, and more,” said Wallace.

The combination of mathematical, literary and personal reflection, along with students’ genuine interest in the story, leads to higher student engagement, Bezaire said. Year after year, his students who previously kept quiet raised their hands during discussions of *The Curious Incident*. “Very often, this continues into the rest of the year after our novel study is done,” he said. “Often when middle school students get momentum in a certain class, they are hardy enough to allow it to continue even when the unit of study changes.”

**Calculus Book Club**

Two years ago, when a schedule change created extra periods in Sam Shah’s multivariable calculus course, he instituted a class book club. The students started with the satirical science fiction novel *Flatland* by Edwin Abbott and later read several nonfiction texts about mathematicians and mathematical ideas. Book club meetings took place during a block of 30 to 40 minutes a few times per month, with a rotating pair of students leading each discussion.

In the first year, Shah said, his biggest challenge was deciding how much to chime into the discussions. It’s as important to create a relaxed atmosphere for the meetings as it is to keep students focused on the text, he said.

During the second year of book club, he made an effort to intervene less often. “When I reflected upon what my goal was … it wasn’t to teach kids how to do close readings and feel like drudgery. The readings were picked to inspire kids to think, to have strong feelings about, to be curious about something mathematical that showed up, to see and think about math differently.”

That’s what the readings did.

“What I appreciated most is how humanizing our conversations were of mathematics — in terms of who was doing it — and how much curiosity students brought to the mathematical ideas they were exposed to.”

One student, for example, used *The Calculus of Friendship* by Steven Strogatz as the model for her final course project, in which she explored her identity and mathematical experiences using calculus concepts.

Although another schedule change this year forced Shah to drop book club from calculus, he has continued the club outside class with interested students.

“If you have the time, it’s a magical way to get kids to see mathematics through a number of different lenses,” he said.

Shah created a list of titles that would work well in a math book club, and Bezaire’s curriculum for *The Curious Incident of the Dog in the Night-Time *is available on his website. MindShift asked the two teachers to share their advice for educators who want to try incorporating literature into their own math classrooms.

**Tips for ***Curious Incident*

- Lay the groundwork for the unit by communicating with parents, other teachers, administrators, students before diving in. There may be resistance: the book has been removed from some schools because of profanity.
- Stock up on throat lozenges or tea with honey. “I speak out loud in front of classes for a living, and it’s still a stretch for me to read the book out loud four times per day,” Bezaire said.
- Pre-read each day’s reading passage to find teachable moments: clues, red herrings and math worth expanding on. Make notes in your copy of the novel to make things easier the second year.

**Tips for math book club**

- Bring snacks and drinks. Make the class feel like it is embarking upon something special and different.
- Don’t grade anything. Let it be fun and non-stressful.
- Don’t talk too much.
- Have all kids come with two to four discussion points to share at the start of each book club. This allows everyone to know what others found interesting and see if there were any topics that the discussion leaders should definitely address.
- Plan and lead the first book club to set an example of the structure and style.

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“Mathematics is very creative and playful and joyful,” says Kirsten Bohl, a spokeswoman for the Mathematical Sciences Research Institute. “Books connect with that sense of wonder and imagination and creativity.”

To spotlight such books, MSRI created the Mathical Book Prize in 2015. Each year a panel of librarians, teachers, mathematicians and early childhood experts selects winners and honor books in five age categories. This year’s picks brings the full Mathical list to more than 50 titles that cross genres and formats, including picture books, graphic novels, biographies, and young adult novels.

What matters most, according to Jordan Ellenberg, co-chair of the selection committee, is that the books succeed in communicating mathematical ideas or problems and also succeed as great books.

“Anybody can stick a lot of math in a book,” says Ellenberg, a University of Wisconsin professor of math who also holds a master’s degree in creative writing. “If it’s a bad book, it’s not going to interest kids.”

The committee also looks for books touching on a range of interests. “A lot of kids don’t think of themselves as math people, but the intention of the award is to help kids understand that math is for everybody,” says Bohl.

Below are 10 books on the Mathical list, including this year’s winners.

**Baby Goes to Market**

Written by Atinuke, illustrated by Angela Brooksbank

Pre-K

2018 Mathical prize winner

At a Nigerian outdoor market, Mama shops while Baby attracts edible gifts from the vendors. Baby eats one of each treat before adding the rest to the basket. When the basket gets heavy, Mama hurries home to feed “Poor Baby!” who’s “not had one single thing to eat.” According to Ellenberg, the math in a children’s book need not be announced in the title, but children should be able to recognize that it’s happening. For example, Fran Wilson, one of the other judges, shared that second-graders in her classroom pointed out the subtraction at play in *Baby Goes to Market *without her prompting. “That’s a sign of a real winning book,” says Wilson.

**Have You Seen My Dragon?** By Steve Light

Pre-K

2015 Mathical prize winner

A child narrator searches high and low for his pet in this multilayered counting book. Children will enjoy spotting the roving dragon amid a richly detailed, pen-and-ink cityscape. They will also practice counting via a set of objects featured in spot color on each spread: two hot dogs, five water towers, 12 pigeons, and so on — up to the 20 lanterns in a final scene where the child and dragon are reunited.

**Sheep Won’t Sleep: Counting by 2’s, 5’s, and 10’s**

Written by Judy Cox, illustrated by Nina Cuneo

Grades K-2

2018 Mathical prize winner

Clarissa tries all the usual tricks for falling asleep, including counting sheep. When that fails, the sheep suggest counting alpacas — this time in pairs. When that also fails, the alpacas recommend counting llamas — by fives. As the trend continues, Clarissa’s room fills up with woolly animals in bold colors and patterns. The comical storyline and bright illustrations will engage early elementary schoolers as they practice advanced counting skills.

**Absolutely One Thing ***
*By Lauren Child

Grades K-2

2017 Mathical prize winner

Big brother Charlie guides little sister Lola through the rules of numbers as they trek to the store to pick out a toy. Younger readers will enjoy correcting Lola’s faulty number sense, while older readers can join Charlie in embedded calculations, such as: how many stickers does Lola have left after sticking five on the pavement, three on a tree, two on her shoes, one on her brother and one on the dog? (Answer: zero.) Although the pages are chock full of digits, a well-paced plot, sibling humor and funky illustrations make for a breezy read.

**A Hundred Billion Trillion Stars***
*Written by Seth Fishman, illustrated by Isabel Greenberg

Grades 3-5

How do you convey the immensity of numbers like three hundred seventy billion billion? Writing 37 followed by 19 zeros likely won’t do the job. Instead, this book uses awe-inducing facts from the cosmos (a hundred billion trillion stars in the universe) and the ground beneath our feet (10 quadrillion ants on Earth) to explore scale and estimation. According to Ellenberg, literature can’t replace classroom instruction in the practice of math but books like this one can spark a “dizzy excitement” that’s also important to the learning process. “You can tell a kid there’s no largest number,” he says. “If they can feel it, that’s a different story.”

**Secret Coders #1: Get with the Program***
*Written by Gene Luen Yang, illustrated by Mike Holmes

Grades 3-5

2016 Mathical prize winner

Hopper knows there’s something strange about her new school. When she and basketball star Eni team up to find out what, things turn out weirder than she imagined. Readers are introduced to the principles of programming through logic puzzles, a robotic turtle and creepy birds in this adventure-filled graphic novel from the 2016 National Ambassador for Young People’s Literature. The book is the first in a series. Mathical also offers an educator tip sheet for *Secret Coders.*

**Giant Pumpkin Suite
**By Melanie Heuiser Hill

Grades 6-8

2018 Mathical honor book*

In addition to presenting calculations, figures, and puzzles, another way for a book to be mathematical is by portraying characters who love and practice math. In *Giant Pumpkin Suite*, math and cello enthusiast Rose deals with literal and figurative growing pains as she joins her shorter, nonmusical twin brother in a project to grow a record-breaking pumpkin for the state fair. “We want kids to see mathematics but we also want kids to see mathematicians, because math is in the end a human activity,” says Ellenberg.

**Really Big Numbers**

By Richard Evan Schwartz

Grades 6-8

2015 Mathical prize winner

With frank, funny narration Schwartz takes readers on a numerical journey that begins at one and proceeds through mind-bogglingly large figures — but still ends very far from infinity. Complex ideas are tackled through everyday objects and patterns in whimsical, geometric drawings. Unanswered questions on some spreads will encourage readers to keep pondering and calculating after putting the book down. Mathical also provides an educator tip sheet for this book.

**Hidden Figures: The American Dream and the Untold Story of the Black Women Mathematicians Who Helped Win the Space Race**

By Margot Lee Shetterly

Grades 9-12

2018 Mathical honor book*

Better known as one 2017’s hit movies, Hidden Figures tells the inspiring story of African-African women “computers” whose brains and grit helped launch NASA astronauts into orbit. With more depth than the film, this book illuminates the interconnections between math, careers and social history. A young readers’ edition and a picture book version also are available.

**Genius: The Game**

By Leopoldo Gout

Grades 9-12

2017 Mathical prize winner

Three online friends and tech prodigies from diverse parts of the world connect in real life during a high-stakes global tech challenge. Together they embark on a quest not only to win “the Game” but also to solve their individual problems: Rex is searching for his missing brother, Tunde is on the run from a military warlord, and Painted Wolf must conceal her identity after exposing corrupt Chinese officials. Told in realistic, rotating voices and filled with suspensful plot twists, this book will grip teenage readers regardless of their prior interest in coding or programming.

**The Mathical selection committee did not name winners for middle or high school grades this year. According to Bohl, that decision reflects the lack of frontlist titles that were submitted by publishers for the older age groups.*

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“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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