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.”

deep cuts

New York City teachers don’t get paid maternity leave. Their paychecks prove it.

PHOTO: Emily James/Courtesy photo
Brooklyn high school teacher Emily James with her children.

Susan Hibdon opened her front door and saw nothing but white.

It was a day that would go down in tabloid headline history after schools Chancellor Carmen Fariña declared it “absolutely a beautiful day,” despite a forecast calling for 10 inches of snow. For Hibdon, a Brooklyn high school teacher, it was memorable for a different reason. It was exactly six weeks after she had given birth, which meant it was time to go back to the classroom.

She kissed her infant goodbye and headed into the wet February weather.

“If you want to pay your rent, you have to go right back to work,” she said. “That’s not just bad for the mother who just gave birth. That’s bad for everybody.”

New York City teachers have no paid maternity or family leave, a policy that takes a toll on teachers’ paychecks and creates deep gender inequity in an education workforce that is about 77 percent women.

Hibdon and fellow teacher and mother Emily James recently launched an online petition calling on the United Federation of Teachers to negotiate for paid leave, which is not included in any of the city’s contracts with unionized workers. Almost 78,000 people have signed on, and the women will present their request at the union’s executive board meeting on Monday.

“I think the irony of it sticks out to many people: These are women who are paid to raise children and they aren’t paid to raise their own children,” Hibdon said.

As it stands now, teachers who want to take paid time off after having a baby must use their sick days. The policy only applies to birth mothers, putting a strain on those who become parents through adoption or surrogacy, and fathers who want to take a leading role in the earliest moments of parenthood.

“We talk so much about parents being active in their child’s education,” said Rosie Frascella, a teacher who has also pushed for paid leave policies. “Well, let’s let teachers be active in their child’s education.”

For teachers, the policy packs a financial blow on multiple levels.

If a mother wants paid time off after giving birth, the only option is to use sick days. Women are limited to six weeks of sick time after a vaginal birth, and eight weeks after a C-section.

Teachers earn one sick day per school month. In order to save up for an eight-week leave, a teacher would have to work about four years without using any sick days.

Many women haven’t accrued that many days, so they can “borrow” sick days they haven’t yet earned. Teachers run into problems, though, if they actually get sick — or their children do — since they can only borrow up to 20 sick days. Once they hit that number, any additional time off is unpaid. And if a teacher leaves the education department, she must repay any sick days she borrowed.

Hidbon learned that the hard way. She has three children — and precious few sick days in the bank. Hidbon remembers a time that she completely lost her voice, but still had to go to work.

“No one could hear me. I had to conduct my entire class writing notes on the board,” she said. “I’m supposed to be teaching and I can’t do my job because of the way the system is set up — and my students are getting the short end of the stick.”

The crunch for sick time could lead to a financial blow later in a woman’s career. Teachers are allowed to accrue up to 200 sick days, and receive a payout for unused time when they retire. The city could not provide numbers for how many sick days men versus women retire with. But it makes sense that men would rack up far more since women with children are more likely to get stuck with a negative balance.

James, a Brookyln high school teacher and co-starter of the online petition, still has a negative balance of 16 sick days — almost three years after giving birth. The problem is compounded by the fact that women are more likely to take time off when a child is sick or there are other family obligations, a pattern that is seen in professions across the board.

“There were many times when I was so sick at work the kids were like, ‘Why are you here? Miss, go home,’” she said. “But it costs a lot of money to stay home.”

Even when women don’t have to borrow sick days, they can still lose financially. The city only allows women to use up to eight weeks of their banked time. Any additional days off are entirely unpaid.

Amy Arundell, a former director of personnel for the UFT, said many mothers stay home longer because of the federal Family and Medical Leave Act, which provides job protections for 12 weeks of leave.

“The people who don’t take 12 [weeks] obviously have real financial commitments” that make taking unpaid time off impossible, she said.

Women who take that time get hit with a double-punch to their salaries. Because of the way summer pay is calculated, unpaid time off results in a smaller summer paycheck, too. Arundell said the hit is usually equivalent to one paycheck.

Same sex-couples and those who become parents through surrogacy or adoption face many of the same financial setbacks, since only birth mothers are allowed to use sick time after having a baby.

After years on a waiting list, Seth Rader and his wife had only weeks’ notice that their adoptive baby was on the way. Since his wife was in grad school, the couple decided Rader would stay home with their new son — even though Rader, a Manhattan high school teacher, is the primary breadwinner at home.

“In a lot of ways, I’m much more bonded with him as a father, and him to me,” Rader said. “Are we really in a place where we want to discourage fathers from taking that role?”

At the time, the couple were saving for a down payment to buy a place of their own. After the expense of Rader taking off from work, they still are.

“I think all of this has to be affecting the sustainability of teaching,” he said. “If we create a system where people can’t imagine being teachers and parents at the same time, then that’s a loss.”

When it comes to the push for family leave, teachers have been left behind even as strides are made elsewhere. New York State recently passed a mandatory paid leave policy that will cover private employees. Last winter, Mayor Bill de Blasio signed a paid leave act for city employees.

But that benefit isn’t extended to workers with unions, like the United Federation of Teachers. Currently, no union in New York City has paid maternity leave, according to a city spokeswoman.

Teachers across the city are fighting to change that. The petition started by Hibdon and James calls on UFT President Michael Mulgrew to “fight for our teaching mothers.”

“They’re supposed to really care about what teachers are struggling with and they’re our voice,” James said. “I just wish that they would take this seriously.”

Both the city and the United Federation of Teachers say they have held talks to extend similar benefits to teachers. In an emailed statement, Mulgrew called family leave “an important issue for the UFT and its members.”

“In our talks so far, the city has failed to come up with a meaningful proposal,” he said.

In an article published in the UFT journal, which ran shortly after the city passed its parental leave policy, the union pointed out that gaining that benefit came at the cost of a scheduled raise for managers and fewer leave days for veteran employees.

According to the article, Mulgrew said he “looked forward to negotiations with the de Blasio administration for an appropriate way to expand parental benefits for UFT members.”

How I Teach

When a parent pushed for a lesson on Mayan history, this Colorado teacher invited her in to help

PHOTO: Megan Witucki
Megan Witucki, a teacher at Compass Montessori School in Wheat Ridge, works with students.

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.

Megan Witucki, an elementary teacher at Compass Montessori School, a charter school in Wheat Ridge, believes in the power of community experts.

That’s why when a parent pushed for a lesson on Mayan history, Witucki invited her in to speak about native cultures. Likewise, for a major end-of-year art project, Witucki brought in a local artist who shared her secrets with students.

Witucki talked to Chalkbeat about why she started tapping into community expertise and how the Montessori approach helps her get to know students and foster a culture of work.

Witucki is one of 20 educators selected for the state’s new Commissioner’s Teacher Cabinet. The group provides input to officials at the Colorado Department of Education on the impact of education policies in the classroom.

This interview has been condensed and lightly edited

Why did you become a teacher?
I became a teacher because I wanted to give back to my community in a meaningful and lasting way. I love working with children and I admire their sense of wonder and their infectious passion to learn. Each day provides me with the chance to empower a child as well as the opportunity to grow and learn myself.

What does your classroom look like?
I am one of two fully-trained certified Montessori teachers who guide the instruction of 33 first through third grade children in our multi-age, lower elementary classroom.

Our classroom is designed to foster choice. It is inviting, cozy, inspiring and engaging. Our classroom is not very big and we have to accommodate 33 little bodies. We also have a plethora of Montessori materials that need to be available to the students at all times. Rather than traditional desks we use individual lap tables, small group tables and work rugs that define the children’s work spaces.

We have space for the students to display their work on the walls and framed prints of art masterworks to inspire creativity. We have classroom plants to add natural beauty to the environment and provide students with hands-on experience during botany lessons.

Throughout our morning work period, one of us offers individual or small group lessons while the other monitors the children’s independent work and re-directs or guides when necessary.

Fill in the blank. I couldn’t teach without my __________. Why?
I couldn’t teach without the support of my school community. Montessori education is a team effort that requires the assistance of and support from the child, their parents, their peers, the co-teaching pair, staff, administration and the larger Compass Montessori community.

Co-teaching is an integral part of a larger network that cares for the children. This network functions like concentric circles that surround and support the child-learner. In the innermost circle is the child, driven by their inherent passions and intrinsic interests. Next, the parents and family, who support the child with their work and education. In the succeeding circle, the learner is given academic, social and emotional support by my co-teacher and me.

The following circle of support is offered by the child’s classmates who vary in age and provide the child with peer guidance as well as opportunities to mentor others and take on multiple leadership roles. This circle is surrounded by an involved administration and the larger staff support circle. Finally, comes the support circle of the greater Compass Montessori community of parents and extended families.

What is one of your favorite lessons to teach? How did you come up with the idea?
My favorite lesson to teach is assemblage art. In Montessori, we naturally integrate art and music into our regular classroom curriculum. The idea for assemblage art came to me when my co-teacher and I were inventorying our leftover art supplies and craft items from previous years’ projects. We had identified odds and ends that we wanted to find a use for when I stumbled upon another teacher’s Pinterest pin.

Our version of this project would be the culminating celebration of all of the year’s original artwork. We began by studying the work of the Russian-American artist Louise Nevelson, who is best known for her groundbreaking work with found art, later known as assemblage art. We discussed how this visionary saw the potential beauty in items discarded by others. The students had rich discussions about what art is and where it can be found. They concluded that sometimes art is where we least expect it.

I then invited our school chef Michelle Lundquist, whom everyone refers to as the “grandma” of our school. Michelle is also a talented local artist who specializes in assemblage art. She shared her inspirations with us and spoke about her artwork.

The children then spent a week collecting old or forgotten knickknacks, pieces from recycling bins, artifacts from the natural world and even items destined for the garbage. In addition, my co-teacher and I organized our miscellaneous craft items; we cut cardboard boxes into 8 x 8 canvases, and set out all of the materials for children to use in their art works.

We invited several parents into the classroom to help with the hot-gluing process and then dedicated a full three-hour morning work period to constructing the assemblages. After the pieces were glued, I spray-painted them monochromatic black, bronze, or gold according to each child’s choice. I was astonished at the strikingly beautiful creations! Though I had hoped to display the composite in the hallway, the children, duly proud of their pieces, insisted on taking them home.

How do you respond when a student doesn’t understand your lesson?
When a child does not understand the material I have presented, I will first attempt to evaluate why. Finding out why is key to understanding the solution.

The reason may be non-classroom related, like the child has not eaten breakfast or they are distressed by home issues. I will then attempt to remedy the situation as best I can. I may feed the child or offer the lesson at another time when they can better focus.

At times, lack of understanding is due to the level of the material being inappropriate. If so, I might go back to a previous lesson with the child to scaffold their knowledge base and better equip them for the more advanced concept. If the material is too easy and the child is bored, I might progress the child ahead to offer more challenge.

A child might need to see the concept presented in a way that better suits their individual learning style. Some children need to manipulate the material themselves; some need to draw the concept; some need to write it out; others need to move and fidget while they listen. We have access to the multitude of Montessori materials and accompanying curricula that allow us to teach all concepts kinesthetically when necessary.

How do you get your class’s attention if students are talking or off task?
Our classroom often looks like organized chaos. At any given time, there may be 33 children working on 33 different projects all in our small space. When I am tempted to stop the class because they look off task, I find it is best practice to wait a minute and observe. I often discover that my first impression of off-task behavior — loud, excited talking, movement — is actually educational in nature and can lead to great work. If allowed, the discussions the students have with one another are often the foundation of invaluable learning pathways and great peer-driven projects.

On the other hand, when the behavior is indeed off task, I will have one of the children ring our chime and then politely ask the class to adjust their behavior to better allow for focused, respectful work. We work hard with the children to create and foster a culture of work by providing varied opportunities for them to take ownership over their learning process.

How do you get to know your students and build relationships with them? What questions do you ask or what actions do you take?
Allowing children to take ownership of their learning through choice offers students the ability to show me what drives and motivates them. I then take that information to design an evolving individualized curriculum.

I also have the benefit of teaching all my students for three years. As a result I can carefully observe their educational, social, and emotional behaviors and choices. In three years I also regularly interact with the child’s family and often gain more valuable insights. With careful observation and purposeful interaction I am able to foster authentic relationships with my students and their families.

Tell us about a memorable time — good or bad — when contact with a student’s family changed your perspective or approach.
In my first year of teaching, I had my first lesson in the power of inviting parents and the community into my classroom. I was approached by a mother in my classroom who was passionate about Mayan history and wanted her child to have more exposure to the history of native cultures. I asked if she would be willing to volunteer a few times a month to share her knowledge with the whole class. She graciously agreed and provided a richly detailed program that I still use today.

Her contributions also inspired me to establish additional classroom-community partnerships with educators from the Jefferson County Indian Education Program and the Mayan dancers from the Denver Chicano Humanities and Arts Council. The experiences shared by community members enriched our classroom in a way that would never have been possible otherwise. This mom’s passion for Mayan culture taught me to seek input from all the resources available to me so I can teach most effectively.

What are you reading for enjoyment?
I just finished The Alchemist by Paulo Coelho. I found it to be inspiring, thought provoking and a great read for a teacher’s summer list!

What’s the best advice you ever received?

One of my many mentors once told me that a child comes to us with many hidden gifts and treasures, and it is our job as educators to guide and encourage that child to bring those gifts forth and share them with the world. As Maria Montessori once wrote, “Free the child’s potential and you will transform him into the world.”