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

How Memphis students came face to face with the painful history in their school’s backyard

PHOTO: Laura Faith Kebede/Chalkbeat
Students at Memphis Grizzlies Preparatory Charter School examine a historical marker meant to share a more complete story of Nathan Bedford Forrest, a Confederate general, slave trader, and early leader of the Ku Klux Klan.

A few yards across from the parking lot of an all-boys Memphis school lies a small, tree-lined courtyard, where a class of eighth-graders studies a large historical marker.

The new marker tells them that children were sold as slaves in this spot. An older, nearby marker had failed to tell the whole story — Nathan Bedford Forrest, the subject of the marker, made Memphis a hub of the slave trade near that busy downtown corner.

The boys at Memphis Grizzlies Preparatory Charter School, who are nearly all black, earlier learned about the painful legacy while watching an episode of “America Divided,” a documentary series featuring celebrities exploring inequality across the nation. The episode featured the Memphis campaign that eventually removed a nearby statue of Forrest, a Confederate general, slave trader, and early leader of the Ku Klux Klan.

PHOTO: Laura Faith Kebede/Chalkbeat
Memphis Grizzlies Preparatory Charter School students visit county commissioners Tami Sawyer and Van Turner (back row), who were key in the city’s campaign to remove Confederate statues.

Being so close to such upsetting history that only recently has bubbled to the surface of public display was a lot for eighth-grader Joseph Jones to take in.

“I can’t believe that this history is right outside our school,” he said. “I think I barely know what happened in Memphis. So many things that I don’t know, that I need to know, and that I want to know, happened in this very city.”

Lately, Memphis — along with several other cities in the South — has been grappling with how tell the complete stories of historical figures who many felt were war heroes, but who also contributed to the enslavement of black Americans.

The new and larger historical marker about Forrest was erected on the 50th anniversary of the assassination of civil rights icon Martin Luther King Jr. He was killed two miles from the all-boys charter school. City leaders vowed and succeeded in taking down Forrest’s statue, which had loomed downtown for more than 100 years, before honoring King’s legacy.

From the archives: Meet the Memphis educator leading the charge to take down her city’s Confederate monuments

PHOTO: Laura Faith Kebede/Chalkbeat
Teacher Tim Green travels to several schools to teach students how to express their emotions in a healthy way.

Tim Green, a teacher at Grizzlies Prep, prompted discussion among students about how to use their newfound knowledge about history in positive ways to improve their city. It’s part of Green’s larger effort to teach students how to express their feelings on difficult topics.

Recently, that has meant delving into the city’s racial history, which is fraught with tragedy that has not been fully reckoned with in public discourse.

“Me forming this class was a way to talk about some of things we deal with and one of those things is our past,” he recently told students. “As African-American men — and African-Americans in general — we don’t have a clear understanding of where we came from.”

Students said their initial feelings after watching the documentary included anger, sadness, and fear.

“It kind of makes me feel scared because you know your parents and your teachers say history repeats itself,” said student Sean Crump. “So, we never know if it’s going to happen again because some things that people said are going to stop have came back. … That makes me feel scared of when I grow up.”

Watching the documentary was the first part of the class’ history exploration before meeting with two key people who organized the removal of Forrest’s statue at the county government building about two blocks from the school.

The documentary shows actor Jussie Smollett interviewing the younger brother of Jesse Lee Bond, a 20-year-old black sharecropper. He was killed in a Memphis suburb in 1939 after asking for a receipt at a store owned by a prominent white family that depended on the credit balances of its black customers.

The brother, Charlie Morris, said he was ready to go on a revengeful rampage after learning his brother had been shot, castrated, dragged by a tractor, and staked to the bottom of a nearby river. He said he still carried the trauma, but doesn’t carry the hate he felt nearly 80 years ago.

PHOTO: Laura Faith Kebede/Chalkbeat
Students watch as Charlie Morris, the brother of lynching victim Jesse Lee Bond, explains the racism that led to his brother’s murder in an episode of “America Divided.”

“The first step to equal justice is love. Where there’s no love, you can forget the rest,” he said in the documentary. Morris died in June at the age of 98.

After the episode ended, the class was silent for a moment. The first question from a student: Were the people who lynched Jesse Lee Bond ever convicted?

The answer made one student gasp in shock, but was predictable for anyone familiar with the history of lynching in America during the Jim Crow era that legalized racial segregation. Two men were charged and tried for Bonds’ murder, but were quickly acquitted.

Students seamlessly tied the past to racism and violence they see in their city today.

“I feel sad because there’s lynchings and people — mostly white Americans — they know there’s lynchings and they know what the Confederacy did to cause those,” said student Tristan Ficklen.

“I would have felt the same too because all he asked for was a receipt,” student Jireh Joyner said of Morris’ initial reaction to Bond’s death. “And now these days there’s still police brutality and it’s hurtful.”

In the Classroom

How an Indianapolis district became a national model for teacher leadership

PHOTO: Shaina Cavazos
Amy Peddie, a teacher at Southport High School, helps a student on an assignment in her class.

Kelly Wilber had been teaching in Perry Township for about seven years when the school district rolled out a new approach to teacher evaluation, mentorship, and coaching — and she felt the change almost immediately.

“I felt like I was a good teacher before,” Wilber recalled. “I mean, I studied all the things in the books, and we had professional development.”

But when the district started using the new approach, the TAP System, “we found the answer of what we needed to do to help our students grow,” said Wilber, who teaches fifth grade at Southport Elementary School.

The TAP System was developed as a strategy for improving instruction, and it is popular in Indiana, where state policymakers have encouraged schools to adopt the system. Perry Township has used it for seven years, and the district has become something of a poster child for the National Institute for Excellence in Teaching, the group behind TAP. On Thursday, the nonprofit recognized Perry Township schools with the organization’s first National Award of Excellence for Educator Effectiveness, which came with a $50,000 prize.

TAP relies on mentors and teacher leaders who are paid stipends to coach their colleagues — a tactic that’s becoming popular among schools as a way to allow experienced teachers to take on more responsibility without entirely leaving the classroom. Each week, groups of teachers meet with master teachers who work with them on strategies they can use in the classroom, like how to tackle word problems or use manipulatives in math.

The model also has guidance on common problems teachers encounter. In the first year of TAP, for example, Wilber had a student who said he wasn’t interested in school or homework and told her, “I’m only here because my brother came here, and I like to do what my brother does,” she recalled.

Wilber began trying techniques that TAP recommended, like using his name during model lessons and having him read the learning objectives. Soon, he was raising his hand in class.

“I felt like I knew what I needed to do because we had so much training and support,” Wilber said.

Perry Township has an unusual set of challenges. Nearly three-quarters of students are poor enough to get subsidized meals. About 25 percent of students are English language learners, and many of them are refugees fleeing religious persecution in Burma.

There is not much outside research on whether TAP improves student test scores. A 2012 study of the results in Chicago found that the program did not raise test scores, but it increased teacher retention. TAP’s developer has disputed the validity of the study, saying the TAP program was not properly implemented in Chicago schools.

But in Perry Township, educators say the approach is helping improve student results.

“If you want to make a difference with kids who are in poverty as well as have a lot of cultural differences, this format and this foundation is the best thing that you can utilize,” Superintendent Patrick Mapes said.

Joe Horvath, a master teacher at Southport High School, said his role is the same as coaches in other districts. Instead of having his own classroom, he is in charge of training 28 other teachers. One day a week, he meets with those teachers in groups. The rest of the week, he observes teachers in their classes, gives feedback, and models lessons.

“We are all on the same level,” Horvath said. “It’s not like I am their boss in any way shape or form. This is just something that allows us to continue to give a peer-to-peer feedback thing that I think is kind of missing sometimes.”