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.

FullSizeRender

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:

FullSizeRender-5

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:

FullSizeRender-4

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.”

Transition plan

Two state-run schools sharing a Memphis building will now share a principal, too

PHOTO: Laura Faith Kebede
A moving van sits outside of Georgian Hills Achievement Elementary School. The state-run school relocated to a different ASD school because of roof damage, and the two schools will now share a principal.

When a state-run school had to relocate at the beginning of the school year because of a leaky roof, officials with the Achievement School District hoped students would soon be back.

That’s not going to happen, though. Georgian Hills Achievement Elementary School will stay at its temporary home at nearby Frayser Achievement Elementary through the school year. And besides sharing a building, they will share a principal.

ASD officials announced this week that Yolanda Dandridge will take the lead of Frayser Elementary, in addition to her role as principal at Georgian Hills.

Verna Ruffin, chief of academics for the ASD, said the decision was made after Jessica Tang resigned as principal of Frayser Elementary. Assistant principals at both schools will now report to Dandridge.

Georgian Hills relocated to Frayser Elementary, two miles away, in August. Ruffin said the goal is still for Georgian Hills’ 258 students to move back into their original building after the school year ends and the building is fully repaired. At that point, the ASD either will promote the assistant principal already serving at the 207-student Frayser or bring in a new leader.

Georgian Hills, under Dandridge’s leadership, recently came off of Tennessee’s list of lowest-performing “priority schools.” That success was a deciding factor, Ruffin said, in asking Dandridge to take responsibility temporarily for both schools.

“This is not how we originally planned it, of course, but it’s an example of taking something that could be a big challenge, putting our heads together, and creating a plan that benefits two schools,” Ruffin said. “Frayser teachers will get to collaborate with an outstanding principal.”

Both schools, like most in the ASD, have struggled with enrollment. Early this school year, the student count at Georgian Hills was down by 20 percent and Frayser’s had fallen 30 percent.

But Ruffin said there have since been enrollment gains and that, in the case of Georgian Hills, a dip was expected with an abrupt change in location.

“It’s a matter of adjusting to put two schools in one location,” she said. “The opportunity to unify around one school leader in the building will hopefully make it a smoother rest of the year for everyone.”

How I Teach

‘All our dreams are on his shoulders.’ The stories that motivate a bilingual teacher

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.

It had been a tiring day of parent-teacher conferences for Amanda Duncan, a sixth-grade dual language literacy teacher at Foster Elementary in Arvada. Then the last family of the evening snapped things back into perspective.

The mother had walked two miles through the snow with her sixth-grade son, pushing the baby in a stroller. She told Duncan that she and her husband hadn’t been able to pursue their education, but wanted something different for their son.

“Will you make sure he stays successful?” the mother asked Duncan. “All our dreams are on his shoulders.”

Duncan, who was named the 2017 Bilingual Teacher of the Year by the Colorado Association for Bilingual Education, talked to Chalkbeat about why that conversation was a valuable reminder about the role teachers play in shaping the future.

This interview has been condensed and lightly edited.

Amanda Duncan

Why did you become a teacher?
I really didn’t set out to become a teacher. I was always interested in other cultures and languages, and in college I considered majoring in anthropology, then public health. Finally, I settled on Latin American Studies. During my last semester, we had to write a mini-thesis about any topic that interested us, and I chose bilingual education — a controversial political topic at that time (still is!).

After graduating, I began working in a middle school English as a Second Language classroom as a paraprofessional, and realized that it felt right being in a school in a diverse setting. I enrolled in a one-year program to get a teaching license and haven’t looked back. I love how teaching makes you an integral part of the community. It allows you to create change in the world by helping students realize how valuable and unique they are.

What does your classroom look like?
My classroom has charts made by the students or me in English and Spanish. There is a lot of intentional use of color to denote important language frames or highlight vocabulary. Since everyone in the room is a second language-learner at some point during the day, I try to provide lots of visual support both with pictures and key words so they have something to latch on to if they are unsure of some word meanings.

For years I was always saying, “It’s Spanish time, I don’t want to hear English right now!” But over time, I had to accept that there is no way to turn off the other language in your brain. So as long as we are using one language as a springboard for understanding or creating in the other, I find value in letting kids use both languages at the same time. It is a natural thing our brains do anyway!

It is a fine line, though. Since we are an immersion program, it is vital that we really hold students to a high standard of production in their second language. But each child builds their second language differently, just like any type of learning. Some need to rely more heavily on their first language to avoid being overwhelmed by the second language. Others are very bilingual already and need to be reminded to continually use Spanish in an academic setting. Even the youngest students know that English is the language of power, and it can easily dominate even in a Spanish immersion classroom. The teachers at my school work tirelessly to constantly lift up the value and beauty of Spanish, so that students will undertake the extra effort of learning through two languages.

I couldn’t teach without my __________. Why?
Students. Hahaha! No, but really, the students are what keep me coming back day after day. Even after 20 years of working with children, they continue to surprise me with their creativity and initiative. I am constantly inspired by the stories they and their families share with me, their daily struggles and perseverance.

At times I feel exhausted by this job, but then I think of how hard many of our families work and the obstacles they are facing, and I am humbled. Teachers have a tremendous responsibility to give our students the best preparation possible so they can be successful in this country. Our community is counting on us and we cannot let them down.

What is one of your favorite lessons to teach?
One of my favorite units to teach is the personal narrative. I love helping students see that their regular lives contain incredible stories. Sometimes they are heartbreakingly sad. Other times they are ridiculously funny. I love helping them learn techniques to take a seemingly regular moment and create a terrific piece of writing. We do this by studying mentor texts (including those by former students) and I demonstrate my own thinking as I write my own story in front of them. I also incorporate drama into the writing process. and we have learned techniques to help students really immerse themselves in their memories. It is powerful to watch students create a “freeze” of their memory — making themselves into a statue that shows the feelings and the moment — then write. They also interview each other to dig even deeper into that moment in time. Their writing improves exponentially and they are so proud of the results.

How do you respond when a student doesn’t understand your lesson?
When a student doesn’t understand, I first have to figure out why. Were they listening? Is there a misconception? Was my lesson confusing? Did it not meet their learning style? Are they distracted by other things going on in their life? My response depends on what the root cause is for not understanding.

How do you get your class’s attention if students are talking or off task?
Any time a teacher redirects a class or a student, it’s a million times more powerful if she gives the reason why they are being redirected. For example, “Table 3, please focus your talk on the lesson. If you talk about other things right now, you will lose your train of thought about the lesson and you won’t produce your best quality of work when it’s time to write.” or “So-and-so, if you are talking while I am giving directions you will not know what to do, and if you don’t know what to do you will not learn this critical skill that will help you be successful in middle school and beyond.” Kids need to know that the rules are not about the teacher having control. Rules are there to protect and enhance the learning environment for everyone’s benefit.

How do you get to know your students and build relationships with them? What questions do you ask or what actions do you take?
I think the first step to building relationships with your students is having a genuine curiosity for who each one really is. You need to laugh at the annoying traits that kids exhibit at different stages of life, maybe roll your eyes about it with colleagues later, but also just enjoy watching the kids figure out who they are and who they want to be. My favorite strategy is paraphrasing what students say. It’s helpful when they are upset, or when the problem they are having is confusing or convoluted (No! Not in 6th grade!). It lets students know you “get them”. I also think it’s super helpful to tell them about a time you struggled with a similar issue, and explain how you learned to deal with it. It helps them feel connected. And if you haven’t experienced something like they have, just really saying with your whole heart, “Wow, that sounds so hard to deal with. I am not even sure what to say, but know that I am with you and I am thinking good thoughts for you.” It helps kids feel less alone.

Tell us about a memorable time — good or bad — when contact with a student’s family changed your perspective or approach.
One story that sticks with me is from a couple of years ago. It was the second night of parent teacher conferences, near the end of the evening. I was tired, feeling good because conferences had gone well, but also feeling overworked and exhausted. It’s easy to feel a little sorry for yourself.

It was a February evening in Colorado, so it was very cold and snowy. I just wanted to be home and curl up on the couch with my family. I peeked into the hall to see if my 7:30 p.m. appointment had arrived, and there was my student Oscar with his mom and his baby sister in the stroller. Their cheeks were rosy-pink and they were unwrapping themselves from their many jackets. They had walked to conferences from their home about two miles away in snowy 20 degree weather.

Oscar’s mom and I introduced ourselves, and she apologized for missing the fall conference, but that was right when her daughter was born.

“Ms. Amanda,” she began in Spanish, “I just want to know if Oscar is doing well in school. Is he respectful to you and his classmates? Does he work hard?” I assured her that Oscar was a model student. “You see, Ms. Amanda, Oscar is the hope of our family. My husband and I weren’t able to study as much as we would have liked to. And now, we work so hard. I work all day and my husband cares for the baby, then I come home and he takes our one car to work at night. But we want a different life for Oscar. Will you make sure he stays successful? All our dreams are on his shoulders.”

This particular story sticks with me because of the cold night, because of the baby, and because I was feeling sorry for myself right before this conversation. But we all hear stories like this all the time. They remind us that we hold in our hands the future of many families, generations even. We cannot let our community down.

What are you reading for enjoyment?
“My Grandmother Asked Me to Tell You She’s Sorry” by Fredrik Backman

What’s the best advice you ever received?
My mom was also a teacher and she always reminds me to take time for myself and my family. There is always more that you can do as a teacher. Don’t get too bogged down in trying to make every single lesson perfect. You need to say, “That is enough for today,” and let it go. Guess what? The sun will come up tomorrow and your students will still learn plenty. Especially if you are able to come back fresh and be willing to give your heart to them.