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.”

In the Classroom

Carranza aims to speed up anti-bias training for educators, calling it a “cornerstone” to school improvement

PHOTO: Christina Veiga
Chancellor Richard Carranza, bottom right, joined New York City principals and superintendents for an anti-bias training in Brooklyn.

After bending fluorescent pipe cleaners into loopy and angular shapes, a group of about 100 New York City principals and superintendents paired up for a chat. Their assignment: to recount their childhood aspirations of what they wanted to be when they grew up.

This was no arts and crafts class — and no ice breaker, either. The Wednesday morning session at Brooklyn Law School was an example of anti-bias training that the education department will now require for every employee who works with students across the country’s largest school system.

After committing $23 million to the work this year, Chancellor Richard Carranza announced at the session that the trainings will be mandatory, and that the city aims to speed up how quickly they happen. The goal is to compress the original four-year roll out to two.

“It’s about us as a community saying we want to change systems so that it privileges all of our students in New York City,” Carranza said. “The evidence right now, I will tell you my friends, is that not all students are being served well.”

Advocates had long agitated for the training, citing disparate rates in school discipline for black and Hispanic students, and high-profile incidents of schools accused of teaching racist lessons in the classroom. They argue that teachers need to be better equipped to serve diverse students as the city moves forward with plans to integrate its starkly segregated schools.

“We have to make school environments the most welcoming places possible for our young people. That includes adults doing personal work,” said Natasha Capers, a coordinator for Coalition for Educational Justice, a parent organization that lobbied for the training.  

Their advocacy has gotten a boost since Carranza became schools chancellor in April, bringing an approach that is bolder and more frank than his predecessor when it comes to addressing the system’s racial inequities. On Wednesday, he spent more than an hour participating in the training session just like the other school leaders, calling it “God’s work.”

“This is going to penetrate everything we do,” he said.

Wednesday’s session was lead by experts from the Perception Institute, a research and training organization, and Safe Places for the Advancement of Community and Equity (SPACEs), which provides leadership training. The pipe cleaners helped bring to life a metaphor about “bending” expectations for what educators might learn throughout the day. The one-on-one conversations were a way to “interrupt” stereotypical assumptions about other people by having sustained conversations with them, said trainer Dushaw Hockett.

“This isn’t some touchy-feely, get-to-know-you exercise,” he said.  

There is some evidence that, when done right, anti-bias trainings can work — and improve outcomes for students. But there is also research that shows it can often be ineffective.

Carranza said the city is committed to doing the work for the long-term, with the trainings designed to be ongoing and build on each other. He also said the department will keep an eye on measures such as student attendance and whether teachers report improvements in school climate to gauge whether it’s having an impact.

“This is going to be one of those cornerstone pieces in terms of, how are we going to continue to transform this immense system to really, truly serve all students?” he said. “This is going to be something that’s not going to fall off the radar. We’re going to keep pushing.”

outside the box

How one Chicago principal is leaning on data to help black boys

PHOTO: Adeshina Emmanuel

Test scores were rising at Fuller Elementary School when Marilyn McCottrell took over in 2016. Yet troubling trends loomed behind the numbers.

“A lot of growth has been made,” said McCottrell, Fuller’s third principal in six years. “But that growth is not equal among students.”

She’s talking about black boys.

Black girls had driven most of Fuller’s academic improvement since the 2012-13 school year, when Chicago Public Schools handed management of the Bronzeville school over to the non-profit Academy for Urban School Leadership, which replaced the staff and principal in a turnaround effort. Black boys had improved much slower. They got most of the school’s Ds and Fs, and were much less likely than girls to meet or approach expectations for college readiness on state tests.

PARCC Scores

Last school year, McCottrell and her staff crunched the data and made changes at Fuller to shorten the gaps between boys and girls. The stakes are high. Black boys, especially those from low-income households, are more prone than their sisters to falling behind in school and running into the juvenile criminal justice system. As adults, they are more likely to be arrested, imprisoned, or chronically unemployed. McCottrell believes what Fuller did, starting with painstakingly crunching data at the school, classroom and individual levels, could help other schools do better for black boys.

But she wants to be clear about something: Black boys don’t need to be “saved.”

“They need to be respected and appreciated for the differences and the unique gifts that they bring to the educational experience,” she said.

Black boys
PHOTO: Adeshina Emmanuel
Fuller Elementary School students (from left)Tyrese Robinson-Guy, Terrell Johnson, and Jasean Waters at a community garden in Bronzeville.

 

Fuller, a Level 1 school in good standing, occupies the corner of St. Lawrence Avenue and 42nd Street in Bronzeville. Nearly all of its 370 students are black and come from economically disadvantaged households. About half of the teachers are white, and about half are African-American. When CPS turned over management of Fuller, it was seeking to lift up a school that had been on academic probation five consecutive years. Fuller still has far to go. In 2017, only 10 percent of Fuller students were ready for the next level compared to 26 percent across the school district and 34 percent across the state. Growth has been above average, but, as McCottrell said, that growth hasn’t been equal.

PARCC Scores

Last August, McCottrell arrived at Fuller for a training session for teachers bearing handouts packed with data on black boys’ grades and test scores. Middle school reading teacher Arlicia McClain was shocked to see the stark disparities.

“It made me buck up and say I need to talk to these students,” she said. “I need to know what is going on that is preventing them from improving. Is it me? Is it something going on with them individually? Is it something they are missing?”

Girls’ math scores had increased by 193 percent compared with 90 percent for boys since the turnaround effort began in the 2012-13 school year. The gender performance gap was even more striking in reading, where black girls’ scores jumped 140 percent compared with 31 percent for boys. 

As McClain and other teachers reflected on the numbers, they recounted their  own experiences in the classroom. For example, they could all name which students were removed from class the most for disciplinary reasons, and nearly all were black boys.

Arlicia McClain
PHOTO: Courtesy of Arlicia McClain
Fuller Elementary School teacher Arlicia McClain.

McClain realized she tended to call on black girls more in class.

McClain, African-American herself, wondered if she was favoring girls or failing to challenge boys enough, and how that could affect their learning. She resolved to push black boys more during her second year at Fuller. 

She also left the session with another big take-away: A lot of boys who wouldn’t participate in classroom-wide sessions engaged more in small groups. Wedding the data to her realizations has helped the young teacher come up with tailored approaches for struggling students.

“Look at them as individuals who want to learn, but who sometimes need the individualized attention to do that,” McClain said. “If you really are about the progression of black youth, you’re going to need to be individual-focused, and you’re going to need the data to do it.”

In the 2016-17 school year, for the subjects of English language arts and math, about 70 percent of all Ds and Fs at Fuller went to black boys.

In the first quarter of last school year, McCottrell and her staff revised Fuller’s grading policies in hopes of addressing the disparity.

They switched to what McCottrell called “a more equitable grading scale,” where the lowest a student could score is a 50, adopted a “no-opt out policy” for homework, so children who failed to turn in their homework by deadline wouldn’t automatically get a zero and had to make up assignments, and allowed students to redo certain parts of failed tests and quizzes after reteaching.

By the end of the first quarter, the numbers of Ds and Fs had decreased by nearly half.

But black boys were still getting about the same percent of them as before.

So McCottrell decided to go in for a closer look.

“The numbers only tell part of the story,” she said.

McCottrell ate with boys in the lunchroom. She played flag football with them at recess. She sat with them in class, assisted their teachers, and taught her own lessons across grades and subjects.

She talked to the boys — and listened.

Jasean Waters, a black boy

Jasean Waters, 13, said he found it hard to focus on his school work.

Some distractions come from inside the classroom, like the bullies Jasean’s run into. Other distractions live in the world outside Fuller, like the gun violence whose victims are overwhelmingly black males.

“It’s a big struggle for us,” he said. “There’s a lot of people dying around here, so we gotta watch our backs, and when we’re walking home we feel like we’re unsafe, so we just focus on us being safe. It’s hard to focus on school.”

Boredom is another issue. Jasean said that he does well in math, but struggles sometimes with reading, and that his interest wanes with the lack of characters and authors he can relate to in school texts. That sounded familiar to McCottrell.  When she spoke with boys, she heard that school amounted to a seven-hour suppression of their personalities, interests, and voices — especially in reading and English classes, where black voices and black writers were missing.

“When kids have to pick a book for independent reading, they don’t relate to the characters in those classroom libraries,” she said. “It’s really hard coming to a class everyday when nothing relates to you.”

Parcc Scores

McCottrell decided to teach an optional African-American literature class every Friday during a weekly “intervention time” for students needing help in reading and math  About 17 boys showed up on the first day and read excerpts from Ralph Ellison’s “Invisible Man,” whose protagonist proclaims, “I am invisible, understand, simply because people refuse to see me.”

McCottrell said many of the boys could expertly analyze the Harlem Renaissance classic, because they related to the idea of not being heard, seen, or understood for who they really are. The students offered examples like the portrayal of black men in the media.

“Many of them were saying things like, ‘I’m not a gangbanger, but this is what people think I am, because I’m dark or because I’m tall,’” she said. “They talked about it in the context of their teachers not knowing who they are.”

The class soon doubled as word of mouth drew others in. Jasean, a C student at the start of the class, joined them. He said he learned things he hadn’t been introduced to before. He read about segregation, speeches by Martin Luther King, and books like “Bud, Not Buddy,” about a 10-year-old black orphan during the Great Depression.

He said he rededicated himself to doing 100 minutes of reading a night and by the end of last school year earned an A in reading. He said he raises his hand to ask and answer questions in class more.

“It feels good,” he said.

Jasean’s grandmother, local school council member Regina Waters, praised McCottrell’s hands-on approach with students and her efforts to build one-on-one relationships with the boys.

“She’s upfront with the kids, and she knows all the kids by name which is unusual in the short time she’s been there,” Waters said.

McCottrell
PHOTO: Adeshina Emmanuel
Marilyn McCottrell

Fuller’s boys closed the gap with girls in several ways over last school year.

They went from getting 70 percent of the Ds and Fs in English and math to 60 percent. In 2016-17, 46 percent of boys compared with 55 percent of girls were on track, meaning they earned a C or higher in reading and math and had an attendance rate of at least 95 percent. In 2017-18, the percentage of boys on track increased by 23 percentage points compared to 19 points for girls. But sitting in her office at Fuller one day earlier this summer, McCottrell admitted something about her efforts for black boys.

“Nothing is solved,” she said.

Despite some progress last school year, when the 2018-19 school year starts, black boys at Fuller will still lag behind black girls. Forces outside of education like poverty, mass incarceration, and racial discrimination will continue to disadvantage black youth in ways that manifest in classrooms, where they land heaviest on black boys.

The odds aren’t yet even for black boys at McCottrell’s school, or at most schools across America. However, McCottrell believes that educators learned a lot that they can build on down the line.

Next year, McCottrell said she’s urging teachers to incorporate more of the black experience and black voices into lesson plans and to increase small-group instruction.

Teachers are having more data conferences with McCottrell and with each other to guide instruction and target specific students’ needs. McCottrell is also promoting more social-emotional learning techniques and restorative practices rather than punitive approaches to discipline, and incorporating cultural awareness and bias training into teachers’ professional development.

Marlene Aponte, the Academy for Urban School Leadership’s director of coaching,  said that in some ways Fuller’s story resembles other schools’ in the years after turnarounds. After focusing on rigorous instruction and ambitious growth targets,“we’re starting to really hone in on some of the pieces that we may have overlooked, such as gender bias, gender equity, access in equity,” she said.

McCottrell wants her boys to have the tools to succeed. She knows there are some issues that her school won’t be able to solve.

But it’s a start.