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 an Indianapolis teacher is making fourth grade more like a video game

PHOTO: Dylan Peers McCoy
Guillermo Perez, right, finished nearly all of his assignments from their class game at home in his free time.

The tension was rising in Amanda Moore’s class. Fourth graders were facing off against a dragon-like Sea Raptal, and it was a close fight. Victory hung in the balance.

“What is the universal theme of our text?” asked Moore, calling on a boy to explain a story students had been reading in small groups. His answer — to treat others as you want to be treated — was correct, leading to the defeat of the monster, and causing the class to erupt in chatter and cheers.

All this excitement is because of “gamification,” a new approach Moore recently began using in her fourth-grade class at Chapelwood Elementary School in Wayne Township. With the help of an online platform called Classcraft, which allows students to inhabit characters, earn points, and complete quests, Moore designs adventures that entice students to practice math and reading skills.

Gamification is a growing trend in education that aims to use games to engage students in school work. Critics, though, raise concerns about students spending too much time on screens and the quality of the games. But games are becoming increasingly popular among teachers, and research suggests that games can improve student scores in subjects such as math and history.

Moore, who has taught at Chapelwood for a decade, learned about gamification recently while completing a master’s degree in curriculum and education technology at Ball State University. Since she started using games to teach in January, it has totally transformed the class, Moore said. Now, she is building positive relationships with students because she is playing games with them.

“We forget that kids are kids, and they want to play. And they are motivated by play, and they learn through play,” she said. “Gamification allows us to get back to that a little bit.”

PHOTO: Dylan Peers McCoy
Characters from an adventure that’s helping Chapelwood elementary school students master reading skills.

This week is spring break, so Moore is working with a group of students who are slightly below grade level in reading for intersession. When the week began, Moore told the students that they were on a magical boat that was shipwrecked. As a class, they must collect enough crystals for their ship to set sail again.

In part, the game is based online, and students can bring laptops from school and keep playing at home. There is an adventure map, and every student has a character. Students can earn points online by completing assignments where they practice making inferences and identifying themes, and Moore can see how they are progressing. But the game is also the backdrop for other work, and the class sometimes comes to a halt when students face random events, where they can win or lose points.

“It’s fun because you can learn while you are playing a game,” said Lilly Mata-Turcios, a student in the class.

Since Moore started using online gaming, students have been more engaged, and they’ve continued to do school work at home so they can win rewards such as new armor for their characters or pets, she said. The class has built a strong community because students have to work together to defeat monsters like the Sea Raptal, Moore said.

“It’s a model of what personalization can look like in a blended classroom,” said Michele Eaton, the district director of virtual and blended learning.

During most weeks, students spend about an hour each day completing math and reading assignments through Classcraft. Moore also works with small groups and does instruction with the whole class. But everything they do takes place against the backdrop of their adventure.

“I think it’s just a really powerful way to teach,” Moore said. “It is absolutely worth the time.”

thrown for a loop

Elementary school teachers sometimes follow a class of students from year to year. New research suggests that’s a good idea.

PHOTO: Denver Post file
Student Jaela Manzanares gets reading help from substitute teacher Colleen Rys in her third-grade class at Beach Court Elementary School in Denver.

When Kim Van Duzer, an elementary school teacher in Brooklyn, had a chance to follow her students from third to fourth grade the next school year, she jumped at the opportunity.

“It was such a positive experience,” she said. “One of the big advantages is starting in September hitting the ground running — you already know the kids and the things they did the previous year and the things they need to work on.”

Now, a new study seems to confirm Van Duzer’s experience. Students improve more on tests in their second year with the same teacher, it finds, and the benefits are largest for students of color.

Repeating teachers is “a beneficial and relatively low-cost policy that should be given due consideration,” write the researchers, Andrew Hill of Montana State University and Daniel Jones of the University of Southern Carolina.

The paper focuses on North Carolina students in grades 3 to 5 who had the same teacher two years in a row. That usually occurred not when a whole class repeated  with the same teacher — what’s often called “looping” — but with a small share of students ending up with the same teacher twice, for whatever reason.

How much did that second year with a teacher help? The overall effect was very small, enough to move an average student from about the 50th to the 51st percentile. But even this modest improvement is notable for several reasons.

First, it’s a policy that, at least in theory, doesn’t cost anything or require legislation to implement. Schools, if they choose to, could make looping a habit.

Second, the gains were larger for kids of color than for white students, suggesting that this could make a slight dent in longstanding test-score gaps.

Third, the students who saw the biggest gains had teachers who were lower performing overall, suggesting that having the same students twice may be particularly useful for helping teachers improve.

Fourth, it’s an idea that could affect a lot of students. Just being in a class where many peers were repeating with a teacher seemed to benefit kids who were new to the teacher, the study finds. The researchers think that could be because those teachers’ classroom environments improve during that second year with many of the same students.

That aligns with Van Duzer’s experience, when she had a handful of new students in her looped class. “The other kids were really welcoming to them, and they became fully integrated members of our class community,” she said.

Fifth, there may be other benefits not captured by test score gains. For Van Duzer, being able to pick up existing connections with students’ families was another perk. “It takes a school year to fully develop a relationship with kids and their parents — for everybody to get to know each other, to develop trust, to be able to speak really openly,” she said.

One important caveat: the study can’t prove that if looping were expanded, that the benefits would persist. Past research also isn’t much of a guide because there’s so little out there, but what exists is consistent with the latest study.

A recent analysis found students in rural China scored higher on tests as a result of the approach. Here in the U.S., the best evidence might come from what amounts to the reverse of the policy: having teachers of younger students focus on a single subject, and thus not have a single class of students. In Houston, this led to substantial drops in student test scores and attendance.

These studies suggest early grade teachers do better when they “specialize” in a small group of students, rather than a certain academic subject.

To Van Duzer, who now serves as a math coach at her school, having a firm understanding of what students learned the previous year is crucial and helps explain the findings.

“A lot of times when kids move into a new grade, the teachers are like, ‘You learned this last year!’ and the kids are like, ‘We did?’” she said. “But then if you say certain words … you remind them of certain experiences, like ‘Remember when we studied China and we talked about this?’ and then they’re like ‘Oh yeah, I do remember.’ But if you haven’t been there with them for those experiences, it’s harder to activate that knowledge.”