In the Classroom

With new learning strategies, kids tackle higher-level math

PHOTO: Shaina Cavazos
Natalie Shaw checks subtraction and addition problems with her second-graders at IPS School 61. The school is part of the district's pilot in racial equity training.

Pia Hansen has a message for teachers and parents: math has changed.

Or, to be more specific, math teaching has changed. The new methods, she told a room full of math teachers in Indianapolis last month, are good for helping more kids understand how math works.

But sometimes it’s up to students and teachers to help parents get it, too.

Hansen’s session on teaching kids the building blocks for solving math problems at the national conference National Council of Teachers of Mathematics in late October at the Indianapolis Convention Center drew a crowded room of teachers who came to learn techniques to communicate math concepts visually with hand-held “number racks,” by having kids draw pictures to explain their answers and simply by using more precise language.

In Indiana, where new academic standards now call for students to demonstrate better mastery of math through a deeper understanding of the reasons that lead them to choose a particularly strategy to solve a problem, the ideas are especially useful.

The new standards, which detail what children must know, call for students to not just learn facts, but understand how to get answers. They learn the intuition behind borrowing in subtraction problems or figure out why an author made certain choices when writing a book. This higher level thinking and analysis helps kids be better prepared to go to college or the workforce, educators say.

Hansen, a former math teacher now with Oregon nonprofit The Math Learning Center, said it’s about time math was taught more like English, where memorization takes a back seat to understanding meaning.

“It’s not rote memorization,” Hansen said. “It’s all about thinking strategies.”

New strategies seek higher-level thinking

For some of her parents, who may have learned their basic math a generation ago, Natalie Merz’s second-grade math lesson might look pretty unfamiliar.

The long worksheets of stacked numbers to add, subtract, multiply or divide are gone. Students in her class at Indianapolis Public School 61 work on fewer problems at a time, working to explain how they came to their answers.

And although a math worksheet even five years ago would probably have a strict time limit — how many can you answer in one minute? — this activity had no such pressure.

But giving fewer problems and more time lets students work at their own paces and allows them develop better problem-solving skills. Rushing through timed tests, Hansen said, makes it harder for a struggling student to discover problem-solving strategies that work best for them. That can mean they actually learn less math and feel more frustration with the subject.

As she moved from group to group, Merz made gentle suggestions to her students who weren’t going far enough to explain why.

“Don’t try to do it in your head,” she told one group. “Draw a picture.”

Some students still counted on their fingers or borrowed to solve a subtraction problem and then went back to illustrate it. But most of them followed the directions: they drew a picture and wrote down the answer.

A correct “picture” next to the equation looked something like this, with tally marks visually representing numbers in the “tens” place and circles representing those in the “ones” place.

Merz was reinforcing the concept that students must recognize which numbers are “tens” and which are “ones” to fully grasp the concept of place value in addition and subtraction.

In schools strained by poverty, where children come from families with limited resources, students often struggle to articulate how they got an answer, teachers said. Teachers have to work to bridge the gap with wealthier students, where extra reading, study or academic conversation at home can help prepare kids to better explain what they mean, because the benefit of understanding how they solved a problem doesn’t just end in second grade.

“My fiance does computer programming, and he has to understand the ‘why’ logic behind what he does works,” Merz said. “There’s a process behind those jobs. I think a lot of other countries have been doing that reasoning-based math a lot more. Especially with math, it builds so much. If you don’t understand math addition, you don’t get multiplication, division, algebra or calculus.”

But when kids learn math a new way, it can make it harder for their parents not just to help them with schoolwork, but to even follow the logic themselves.

An emerging parent-child divide

At the conference, Hansen told a true story that illustrated this problem.

A father and daughter she knows were working together on the problem 17 times 99, she said. The father believed his approach was best — multiplying 17 by 99 on paper the way he learned to do it:

The daughter tried to solve it differently using a strategy called “grouping.” Her approach would break numbers down and re-group them in ways that can make the problem easy to solve quickly.

The daughter thought it was easier to multiply 100 times 17. Then, she told her dad, she would solve that equation by taking away one “group” to get the answer to the original problem.

“A group of 99?” he asked, puzzled.

“No,” she said in a matter-of-fact tone, “A group of 17.”

She calculated 17 times 100 to be 1,700 and then subtracted one group of 17 to find the correct answer: 1,683.

But her dad needed more explanation. So she drew him a picture.

The daughter illustrated her answer by drawing a grid with 17 rows and 100 columns. Altogether, the grid had 1,700 squares. When she subtracted one row of 17 squares, or taking away one group of 17 as she had said before, what was left was 1,683 boxes.

“Yeah,” he told his daughter. “I get it now.”

Visuals, Hansen said, such as number lines or grids, can help students to more quickly develop comfort with numbers and lead them to understand why a problem-solving process was used in the first place. The daughter not only knew how to get the answer, but she clearly understood the concepts behind multiplication — well enough to teach them to her father.

“Give them strategies,” Hansen told the teachers, “then drill facts.”

A different way of thinking about numbers

IPS’ curriculum team has both the new standards, and the new thinking about math, in mind when it advocates for the new strategies.

Curriculum coaches Nick Meyer and Eric Beebe believe if students learn to work through math strategies without help, it won’t just benefit them in school and in college but also better prepare them to consider the high tech jobs of the future.

“Understanding the relevance increases student engagement, but it also helps students be more successful because they can make connections from math to the everyday world,” Beebe said. “It also kind of opens the doors for them to understand what careers are associated with math and how math drives so much of what happens around us.”

But to get to there, kids have to master the basics that many adults take for granted.

Some teachers call the adult approach of doing math inside the head “mental math.” But relying on such a strategy without knowing the reasoning behind it can slow a child’s progress toward understanding.

Consider the problem 9 + 7, Hansen said. This is a problem the entire room of teachers could all do in their heads.

But when Hansen asked teachers to explain their answers, they gave a variety of different methods, but all used the same concept: grouping.

One volunteer wanted to make 9 into a 10 to make adding easier. So she split seven in two parts — a one and a six — then took the one and added it to nine. Now she had 10 and six, which add easily to make 16.

Another volunteer saw instinctively that borrowing could work the other way. She split nine in two parts — making an eight and one — then took the one to add with seven. Now she had eight plus eight, which she thought was easier to calculate to the same right answer: 16.

Hansen drew out the solutions and projected them on a screen. They looked something like this, with the arrows indicating how the broken down numbers were combined with the other to get the answer:

This strategy, at its core, shows the kind of thinking higher-level math the new standards encourage students to employ. Many adults use them instinctively. But young children must be taught how to understand numbers that way, or they are likely to resort to counting by ones or memorizing, Hansen said.

“I don’t want you to promote one-by-one counting,” Hansen told the teachers. “I want (students) to think in chunks and groups . . . (visuals) that support that one-by-one counting are the death of us.”

If the methods to solve the new problems seem complicated, it’s because they are, she said. The goal is to help kids reach a higher standard of academic reasoning. The standard algorithm — numbers stacked on top of each other with a plus or minus sign — can lead to the right answer. But it doesn’t get at the understanding behind the math — it takes more effort and thinking for students to explain why that was the best way to solve the problem.

“We could do the algorithm we learned,” Hansen said. “But we wouldn’t be able to justify. Now, standards ask students to justify that.”

‘It feels like losing a family’

This Memphis poetry team is the best in the state. But they will scatter as their school closes.

PHOTO: Caroline Bauman
The GRAD Academy poetry team, from left to right: Olivia Randle, ShuKyra Harris, Alesha Griggs, Belle Edgeston, Timothy Moore, MarQuita Henderson, Zakyah Harris.

MarQuita Henderson had a vision for how her senior year of high school at GRAD Academy Memphis was going to go.

The 11th-grader was going to continue leading her school’s award-winning poetry team, which she believes changed her life. She was going to graduate with her best friends. She was already working on a poem to perform at graduation.

But all that changed in January, when GRAD Academy announced it was closing the charter school in South Memphis in June because of high costs and low enrollment. The school enrolled 468 students this year in a school built for 2,000. GRAD opened in 2013 as part of the Achievement School District, a state-run district tasked with turning around low-performing schools.

In a city with too many schools and too few students, school closures have been common in  Memphis, mostly because of low enrollment and poor academic performance. At least 21 schools have closed since 2012 in the local district, Shelby County Schools. Over the past year, four schools in the state-run district have announced closures.

“It’s hard to think about us not being together next year after we spent so much time thinking about being seniors together,” MarQuita, 17, told Chalkbeat. “But I think, at least I did poetry here. I have a new confidence in myself. There’s a voice in me that wasn’t there before.”

MarQuita is one of six students on GRAD’s poetry team, which was founded three years ago and is led by Timothy Moore, a creative writing teacher. The group was named the best high school poetry team in Tennessee this month by Southern Word, a statewide poetry competition.

The team has become incredibly close knit, they said. They have traveled outside of Memphis for poetry competitions, spent hours editing each other’s work, and doing homework together. They lean on each other if they are having a bad day, need some support, or just want to hang out.

“I didn’t really know anyone on the team when I joined this year, I just knew being on the poetry team had been my dream all of high school,” said Alesha Griggs, 16. “But now, it’s like I can’t imagine not knowing these girls. And we’ll lean on each other now more than ever, because we’re going to new schools where we don’t know anyone else.”

Moore, who has taught at GRAD for four years, tries to make sure the conversations around school closures include the voices of those most affected — the students.

“As a team, we’ve been able to work through some of the anger and hurt that came with the announcement our school was closing,” Moore said. “We’ve had a space to do that. So many students don’t. But I still worry, will another teacher look after them next year? Did I do enough for them?”

The six friends will be split between three high schools — Hillcrest, Middle College, and Craigmont.

Most of them live in the neighborhood surrounding GRAD Academy, where school closures are all too familiar. The school is housed in the former South Side High School building, which was converted into a middle school and then closed in 2015 by Shelby County Schools.

“I was at South Side Middle School when it closed,” MarQuita said. “So when I heard GRAD was closing, my first thought was, is this our fault again? It feels like losing a family.”

Unlike in many school closures, GRAD Academy officials said they weren’t closing the school because of floundering academics. It has the greatest percentage of ASD high school students scoring on grade level, according to state data from 2017.  But “higher-than-projected transportation and facilities costs” were cited by GRAD officials as the main reasons to close.

For poetry member Belle Edgeston, that reasoning wasn’t enough.

“The reason, that it was such a business decision… still bothers me,” said Belle, 17. “We were the future 12th-grade class. That meant something to us, especially in being able to mentor younger kids in poetry.”

All six poetry members said being on the team has had a significant impact on their lives  — especially under Moore’s leadership.

“This is my first year with a 4.0 GPA,” said Olivia Randle, 16. “I didn’t think that was possible. But I also would have never dreamed of us winning state, or of getting to travel for poetry. Mr. Moore made us think more of ourselves.”

Tamala Boyd-Shaw, the executive director of GRAD Memphis, said she’s proud of the confidence and experiences students have gained as part of Moore’s team.

“The students’ poems are often about struggles they endure as students in their own communities and families,” Boyd-Shaw said. “It’s allowed them to reflect and be proud, not just of what they’re saying, but of who they are. My hope is that all of our students land in schools next year that gives them opportunities like this.”

The girls hope to keep practicing together next school year, even though they know scheduling will be hard. Moore said he was hopeful they will keep competing, either as individuals or as a team.

“We’re going to become masters of group apps and Skype,” Moore said. “But I know we’re really going to miss writing as a family together after class. It’s funny, I’m a 36-year-old man, and I’m surprised at how much they taught me. They helped me find my own voice.”

Watch students Kyla Lewis and Olivia Randle perform “Systematical Fear:”

First Person

As historians and New York City educators, here’s what we hope teachers hear in the city’s new anti-bias training

PHOTO: Christina Veiga

New York City Schools Chancellor Richard Carranza and Mayor Bill de Blasio just committed $23 million over the next four years to support anti-bias education for the city’s teachers. After a year in which a white teacher stepped on a student during a lesson on slavery and white parents used blackface images in their PTA publicity, it’s a necessary first step.

But what exactly will the $23 million pay for? The devil is in the details.

As current and former New York City teachers, and as historians and educators working in the city today, we call for the education department to base its anti-bias program in an understanding of the history of racism in the nation and in this city. We also hope that the program recognizes and builds upon the work of the city’s anti-racist teachers.

Chancellor Carranza has promised that the program will emphasize training on “implicit bias” and “culturally responsive pedagogy.” These are valuable, but insufficient. Workshops on implicit bias may help educators evaluate and change split-second, yet consequential, decisions they make every day. They may help teachers interrogate, for example, what decisions lead to disproportionately high rates of suspension for black children as early as pre-K, or lower rates of referrals to gifted programs for black students by white teachers.

But U.S. racism is not only split-second and individual. It is centuries deep, collective, and institutional. Done poorly, implicit bias training might shift disproportionate blame for unequal educational resources and outcomes onto the shoulders of classroom teachers.

Anti-bias education should lead teachers not only to address racism as an individual matter, but to perceive and struggle against its institutional and structural forms. Structural racism shapes the lives of students, families, and communities, and the classrooms in which teachers work: whether teachers find sufficient resources in their classrooms, how segregated their schools are, how often their students are stopped by police, and how much wealth the families they serve hold. Without attending to the history that has created these inequities, anti-bias education might continue the long American tradition of pretending that racism rooted in capitalism and institutional power can be solved by adjusting individual attitudes and behaviors.

We have experienced teacher professional development that takes this approach. Before moving to New York, Adam taught in Portland, Oregon and participated in several anti-bias trainings that presented racism as a problem to be solved through individual reflection and behaviors within the classroom. While many anti-racist teachers initially approached these meetings excited to discuss the larger forces that shape teaching students of color in the whitest city in America, they grew increasingly frustrated as they were encouraged to focus only on “what they could control.”

Similarly, at his very first professional development meeting as a first-year teacher of sixth grade in Harlem, Brian remembers being told by his principal that neither the conditions of students’ home lives nor conditions of the school in which he worked were within teachers’ power to change, and were therefore off-limits for discussion. The only thing he could control, the principal said, was his attitude towards his students.

But his students were extremely eager to talk about those conditions. For example, the process of gentrification in Harlem emerged repeatedly in classroom conversations. Even if teachers can’t immediately stop a process like gentrification, surely it is essential for both teachers and their students to learn to think about conditions they see around them as products of history — and therefore as something that can change.

While conversations about individual attitudes and classroom practices are important, they are insufficient to tackle racism. Particularly in one of the most segregated school districts in America, taking a historical perspective matters.

How do public school teachers understand the growth of racial and financial inequality in New York City? Consciously or otherwise, do they lean on tired but still powerful ideas that poverty reflects a failure of individual will, or a cultural deficit? Encountering the history of state-sponsored racism and inequality makes those ideas untenable.

Every New York City teacher should understand what a redlining map is. These maps helped the federal government subsidize mid-twentieth century white suburbanization while barring African American families from the suburbs and the wealth they helped generate. These maps helped shape the city, the metropolitan region, and its schools – including the wealth or poverty of students that teachers see in their classrooms. This is but one example of how history can help educators ground their understanding of their schools and students in fact rather than (often racist) mythology.

And how well do New York City educators know and teach the histories of the communities they serve? Those histories are rich sources of narratives about how New Yorkers have imagined their freedom and struggled for it, often by advocating for education. Every New York City teacher should know that the largest protest of the Civil Rights Movement took place not in Washington D.C., not in the deep South, but right here. On February 3, 1964, nearly half a million students stayed out of school and marched through the city’s streets, demanding desegregation and fully funded public schools. Every New York City teacher should know about Evelina Antonetty, a Puerto Rico-born, East Harlem-raised advocate who organized her fellow Bronx parents to press for some of the city’s first attempts at bilingual education and just treatment for language minority students in school.

Even if they don’t teach history or social studies, educators can see in the 1964 boycott and in Antonetty’s story prompts to approach parents as allies, to see communities as funds of knowledge and energy to connect to and build from. The chancellor’s initiative can be an opportunity to help teachers uncover and reflect on these histories.

Ansley first taught at a small high school in central Harlem, in a building that earlier housed Junior High School 136. J.H.S. 136 was one of three Harlem schools where in 1958 black parents protested segregation and inequality by withdrawing their children from school – risking imprisonment for violating truancy laws. The protest helped build momentum for later educational activism – and demonstrated black Harlem mothers’ deep commitment to securing powerful education for their children.

Although she taught in the same school – perhaps even the same classroom – where boycotting students had studied, Ansley didn’t know about this history until a few years after she left the school. Since learning about it, she has often reflected on the missed opportunities. How could the story of this “Harlem Nine” boycott have helped her students learn about their community’s history and interrogate the inequalities that still shaped their school? What could this story of parent activism have meant for how Ansley thought about and worked with her students’ parents?

Today, teaching future teachers, Ansley strives to convey the value of local and community history in her classes. One new teacher, now working in the Bronx, commented that her own learning about local history “taught me that we should not only think of schools as places of learning. They also are important places of community.”

The history of racism and of freedom struggles needs to be part of any New York City students’ learning as well as that of their teachers. Some of the $23 million should support the work of local anti-racist educators, such as those who spearheaded the Black Lives Matter Week of Action last February, in developing materials that help teach about this history. These efforts align with the chancellor’s pledge for culturally responsive education. And they offer ways to recognize and build on the knowledge of New York City’s community organizations and anti-racist education networks.

Attitudes matter, and educators – like everyone – can learn from the psychology of bias and stereotype. But historical ignorance or misrepresentation has fed racism, and history can be a tool in its undoing.

That would be a good $23 million investment for New York and all of its children.

Ansley Erickson is an associate professor of history and education at Teachers College, Columbia University and a former New York City high school teacher.

Brian Jones is the associate director of education at the Schomburg Center for Research in Black Culture of the New York Public Library and a former New York City elementary school teacher.

Adam Sanchez is a teacher at Harvest Collegiate High School in New York City and an organizer and curriculum writer with the Zinn Education Project.