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.

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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:

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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:

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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.”

YOUNG ADVOCATES

New program aims to make advocates out of Memphis high schoolers

PHOTO: Campaign for School Equity
Students discuss advocacy topics during their session at Fairley High School, one of 10 schools in Shelby County participating in the program.

When it comes to conversations about education policy, students are often the least heard.

But amplifying young voices is the goal of a new program launched by two Memphis-based advocacy groups, Campaign for School Equity and Latino Memphis.

“I joined the group because of things that are going on around school, and I believe that we as leaders can change it,” said Angel Smith, 16, a senior at Hillcrest High School, one of 10 schools in the program. “I want to change how our school does discipline … and learn why some schools have more money than others.”

Many students feel powerless to improve conditions at their schools, said Katie Martin, who will oversee the program as advocacy manager for Campaign for School Equity. “It is so exciting to help them discover their own voices and realize that they can have a direct impact on the issues that matter to them,” she said.

About 100 high school students from Fairley, Martin Luther King Preparatory, Hillcrest, Trezevant and Southwest Early College High will take a monthly class on topics ranging from advocacy strategies to political campaign development.

Beginning in November, high-schoolers from Cordova, Wooddale, White Station, Kingsbury, and Southwind will also have classes at their schools.

Mendell Grinter, executive director of Campaign for School Equity, said students have already expressed interest in pushing for better school facilities and more discipline practices based on restorative justice.

The goal is for students to help shape Campaign for School Equity’s legislative platform and run their own school-based advocacy campaigns. In December, students will vote on priorities for the upcoming legislative season, Grinter said.

Students will take courses on research, writing opinion pieces, advocacy methods and campaign development. They also will meet with their local representatives, such as Memphis City Councilwoman Patrice Robinson, who will speak with Hillcrest High students in late October.

Campaign for School Equity is funding the program, and students were selected based on their interest and school recommendations.

Grinter said the program marks a shift in his group’s priorities. Formerly known as the Tennessee chapter of the Black Alliance for Educational Options, Campaign for School Equity has focused on promoting school choice for black families and engaging Memphis clergy around education.

“There are programs in Memphis to reach parents and community members and get them involved with advocacy, but not really students,” Grinter said. “We’re really going to double down on creating that space.”

Latino Memphis is an advocacy group for the city’s Hispanic and Latino communities and is working with Campaign for School Equity to include Latino students. 

expansion plans

Here are the next districts where New York City will start offering preschool for 3-year-olds

PHOTO: Christina Veiga
Schools Chancellor Carmen Fariña, left, and Mayor Bill de Blasio, center, visited a "Mommy and Me" class in District 27 in Queens, where the city is set to expand 3-K For All.

New York City officials on Tuesday announced which school districts are next in line for free pre-K for 3-year-olds, identifying East Harlem and the eastern neighborhoods of Queens for expansion of the program.

Building on its popular universal pre-K program for 4-year-olds, the city this year began serving even younger students with “3-K For All” in two high-needs school districts. Mayor Bill de Blasio has said he wants to make 3-K available to every family who wants it by 2021.

“Our education system all over the country had it backwards for too long,” de Blasio said at a press conference. “We are recognizing we have to reach kids younger and more deeply if we’re going to be able to give them the foundation they need.”

But making preschool available to all of the city’s 3-year-olds will require an infusion of $700 million from the state or federal governments. In the meantime, de Blasio said the city can afford to expand to eight districts, at a cost of $180 million of city money a year.

Funding isn’t the only obstacle the city faces to make 3-K available universally. De Blasio warned that finding the room for an estimated 60,000 students will be a challenge. Space constraints were a major factor in picking the next districts for expansion, he said.

“I have to tell you, this will take a lot of work,” he said, calling it “even harder” than the breakneck rollout of pre-K for all 4-year-olds. “We’re building something brand new.”

De Blasio, a Democrat who is running for re-election in November, has made expansion of early childhood education a cornerstone of his administration. The city kicked off its efforts this September in District 7 in the South Bronx, and District 23 in Brownsville, Brooklyn. More than 2,000 families applied for those seats, and 84 percent of those living in the pilot districts got an offer for enrollment, according to city figures.

According to the timeline released Thursday, the rollout will continue next school year in District 4 in Manhattan, which includes East Harlem; and District 27 in Queens, which includes Broad Channel, Howard Beach, Ozone Park and Rockaways.

By the 2019 – 2020 school year, the city plans to launch 3-K in the Bronx’s District 9, which includes the Grand Concourse, Highbridge and Morrisania neighborhoods; and District 31, which spans all of Staten Island.

The 2020 – 2021 school year would see the addition of District 19 in Brooklyn, which includes East New York; and District 29 in Queens, which includes Cambria Heights, Hollis, Laurelton, Queens Village, Springfield Gardens and St. Albans.

With all those districts up and running, the city expects to serve 15,000 students.

Admission to the city’s pre-K programs is determined by lottery. Families don’t have to live in the district where 3-K is being offered to apply for a seat, though preference will be given to students who do. With every expansion, the city expects it will take two years for each district to have enough seats for every district family who wants one.