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.”

How I Teach

This Memphis teacher went viral for holding ‘class’ on Facebook Live during a snow day

PHOTO: Caroline Bauman
Candous Brown teaches one of her 12th-grade English classes at Raleigh-Egypt High School. Brown has been teaching in Memphis for 10 years.

How do teachers captivate their students? Here, in a feature we call How I Teach, we ask great educators how they approach their jobs. You can see other pieces in this series here.

When a week of snow days brought Candous Brown’s 12th-grade English class to a wintry halt, her students convinced her to take her lesson live on Facebook.

So wearing pajamas and with occasional photobombs by her 10-year-old son, Brown sat down at her laptop and convened an impromptu class with about 40 students from Raleigh-Egypt High School in Memphis. Some participants were actually previous students who decided to drop in.

“I’m so proud of y’all for actually wanting to do this,” she said at the outset, complimenting her students for their resourcefulness, ingenuity, and good use of technology.

The 33-year-old teacher has a knack for engaging her students where they are. That means frequently tapping into their love of music to grow their passion for literature.

“Why wouldn’t we focus on that?” she asks rhetorically.

During Black History Month, for instance, Brown pairs excerpts of Martin Luther King Jr.’s 1964 book “Why We Can’t Wait” with freedom songs from the documentary “Soundtrack for a Revolution.”

“I want them to know how music was utilized during the civil rights movement,” she said. “ In many instances, it was the thing that kept people motivated and unified.”

Chalkbeat spoke recently with Brown about teaching on Facebook Live and how she builds relationships with her students every day. (Her answers have been lightly edited for clarity.).

Why did you become a teacher?

I have always enjoyed literature and reading so it fit that I would be an English teacher.  As a student, my teachers would use me as a peer tutor.  I assisted classmates with their assignments and they would tell me I’d make a great teacher.  Of course, I would reject the idea; but looking back on it, they were leading me in the right direction.

What is one of your favorite lessons to teach? How did you come up with the idea?

I love teaching anything Shakespeare. But more recently, my favorite lesson has been to teach “The Hero’s Journey.” We were reading Beowulf and I wanted the students to trace Beowulf’s journey into the hero that we know him to be in today’s culture. When I first start the unit, I have them think of heros within their own lives. Or times when they felt like they were the hero in a situation. I want them to be able to connect this hero’s journey to themselves.  We read the text, participated in class discussion, did an analytical comparison of the movie and the text.  The students loved it.

Recently, you received national attention for holding class via Facebook Live during a snow day. Why was it important to make instructional time happen during that long break? How do you instill excitement for learning in your students?

That was actually my very first time going live. I was so nervous. I didn’t want to say something foolish and have the entire virtual world see my flub. I got up that morning, planned for some anticipated misconceptions, and went for it.

My students were the ones who set everything up. They asked if I’d be willing to do the lesson and, of course, I couldn’t say no when they were willing to do the work. I told them about my apprehensions and then one student used a phrase that I tell them when they are afraid to try something new: “First time for everything.” At that moment, I knew I had to do it. It was important to make it happen because they wanted it to happen. I always tell them that they cannot wait to be within the confines of a school to learn.

It pleased my soul that they were still attempting to do the work without me and that they trusted me enough to reach out. I think when they see me get excited or passionate about certain topics, it resonates with them.

How do you respond when a student doesn’t understand your lesson?

Students tend to shut down when they don’t understand a lesson. Then, they state the infamous sentence: “I don’t get it.” I force them to think about the lesson and target the source of confusion. They have to be able to explain the problem to me before I help them. More often than not, their own explanation of the misconception helps them figure out the issue on their own. Also, they know that I am a last resort.  They will ask a peer or neighbor before they ask me because they know I will make them explain everything they know before I will help. It forces them to explore their own understanding of the concept.

How do you get your class’s attention if students are talking or off task?

I usually don’t have to say or do much. My facial expressions do the talking for me.  Once the kids see my face, they tell each other to get it together before I start fussing.  Apparently, the last thing they want to hear from me is fussing.

PHOTO: Caroline Bauman
Brown says her facial expressions can do the talking for her when her students get off track.

How do you get to know your students and build relationships with them? What questions do you ask or what actions do you take?

At the beginning of the school term, my students complete an “Interest Survey.” I participate with them and allow them to ask me questions. I figure if I’m asking them questions about their lives outside of the classroom, they should be allowed to ask the same of me, within reason. When the surveys are done, I file them. No one will see their answers but me. When appropriate, I incorporate things I learn about them into the lessons to make them more relatable. In that way, they know that I am paying attention and it opens the floor to them so that they know I am trustworthy and truly have their best interest at heart. I never demean them for the things they reveal and I don’t shy away from tough conversations. My door stays open to them unless I’m grading or planning.

What’s the best advice you ever received as a teacher?

To remember why I’m in the classroom. Sometimes, the classroom can be daunting and overwhelming. I have my students, I’m the single mother of a 10-year-old son and, on top of that, I’m working toward a master’s degree. I could easily get discouraged. But if I remember why I’m there, it becomes manageable. I am there to serve my students. I am there to lead my students. Those two things are never lost upon me.

school rules

Arkansas passed a law banning suspensions for truancy. Then it was largely ignored.

PHOTO: Andrea Chu

What if an education law passed, but nobody followed it?

That appears to be the bizarre situation in Arkansas, which in 2013 enacted a straightforward law banning out-of-school suspensions for truancy.

But three years later, nearly 1,100 students were still suspended for not showing up to school. Many Arkansas schools were simply not complying with the law, according to a new study.

What happened? It’s not entirely clear, but a communication breakdown may be to blame. The study notes that schools didn’t hear explicitly from the Arkansas Department of Education about the new law until January 2017.

The state disputes this — kind of — pointing to 2014 and 2015 memos, though neither actually mentions the rule change or acceptable penalties for truancy. A department spokesperson said the memos’ “regulatory authority” include the law banning suspensions.

“While [the department] does not track every phone call or correspondence, in general we have ongoing communication with educators, schools, districts and education service cooperatives,” said the spokesperson, Kimberly Friedman.

What’s clear is that only some Arkansas schools changed their practices. In the 2012-13 school year, about 14 percent of truancy cases resulted in out-of-school suspensions, and by 2015-16 that had dipped to 9 percent. It’s not clear whether that drop was due to the law.

(Notably, nearly 2 percent of truancy cases in 2015-16 resulted in corporal punishment, which remains legal in Arkansas public schools despite efforts by the federal government to eliminate the practice.)

The study, which was published last week in the peer-reviewed Peabody Journal of Education, also found that schools serving more students of color were less likely to have followed the law.

Schools with 10 percent more black students than average were about 5 percentage points less likely to eliminate suspensions for truancy. That finding underscores concerns from discipline reform advocates about the disproportionate effect suspensions have on students of color.

“The types of schools that the state was likely intending to impact … were also the types of schools that failed to comply,” researcher Kaitlin Anderson of Michigan State University wrote.

Although pointing to an outlier case, the paper highlights a key challenge of changing school discipline rules: laws and mandates are no guarantee of real change. That’s especially true if educators don’t believe in the changes, schools aren’t given the resources to change, there’s no enforcement of new guidelines — or if schools don’t know that rules have changed at all.

“You might expect [suspensions for truancy] to go down to 0 percent, but that would be if all schools knew about the law, were able to comply with the law, and wanted to comply with the law,” said Anderson.

It’s not the first study to highlight the challenges of instituting, and tracking, school discipline changes. After Philadelphia banned suspensions for certain lower-level offenses, more than three-quarters of schools did not fully comply, another recent paper found. In Washington, D.C., an investigation found that some schools simply didn’t report all out-of-school suspensions amid the district’s efforts to cut down on exclusionary discipline.

In other cases, though, policy changes are leading to fewer suspensions, at least according to official numbers. Los Angeles and New York City, for instance, have reported substantial drops in out-of-school suspensions in recent years.

A slide from research presented to the Arkansas Board of Education in February 2016. ISS refers to in-school suspensions, and OSS refers to out-of-school suspension.

In Arkansas, the back and forth over the new findings began in February 2016, when the researchers presented preliminary findings to the Arkansas State Board of Education. They reminded board members that suspensions for truancy were illegal and noted that “over 100 districts were still doing this as of 2014–15.”

Nearly a year later, in January 2017, the state commissioner of education issued a brief memo, which said that “State Board members requested the department remind districts” of the ban.

Friedman said there wasn’t data on whether schools are complying with the law this year, since schools don’t submit discipline reports to the state until June.

Arkansas now has another chance to tackle the challenge of implementing a new discipline policy. Just last year, the state passed a law prohibiting most out-of-school suspensions in in elementary school.

Anderson said that it makes sense for state leaders to engage local district and school officials more when trying to change how schools do business. “Having some of those conversations is going to be more productive in the long run rather than trying to just set a hand-offs, high-level policy,” she said.