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.”

First Person

I’m a Chicago teacher who has watched many Javions fall through the cracks. Here’s what would help.

PHOTO: Adeshina Emmanuel
A mural inside the Community Youth Development Institute.

As a Chicago Public School teacher and librarian for the past 15 years, I’ve seen many students fall through the cracks.

I remember the young man who sat in my class at the beginning of his senior year, eager to learn. By the end he was failing, having missed over 30 days of school for reasons unknown to me. Or the sophomore girl who transferred out after displaying behavior and academic problems. I saw her years later on the Chicago Tribune’s mugshot webpage.

Both of these students came to mind when I read Adeshina Emmanuel’s story about Javion Grayer, a 16-year-old Chicago student who reads at the second grade level.

What struck me about Javion’s story was that his educational experience was as disjointed as his home life. He switched schools several times, and his family wished someone could help him. But many of the schools he attended did not provide adequate special education services or reading specialists.

This is not a problem unique to Javion. In Chicago Public Schools, a district that prides itself on its variety of choices for families, Javion’s story is an illustration of how little all of those schools can have in common and how little they work together. The result can be students getting a patchwork education within one school district, with little horizontal alignment among schools that serve the same grades, and little vertical alignment among its K-8 schools and high schools.

At Lindblom High School, where I am a teacher librarian, we see transfer students who want to come to Lindblom in search of something better. I watch many transfer students who come with grass-is-greener hopes struggle because of differences in curriculum, instruction, expectations, and school culture. Lindblom is a selective enrollment high school, and many that transfer in are not coming from other selective enrollment schools, which may exacerbate this. But Javion’s experience was probably somewhat similar, watching schools right next to one another operate in very different ways.

Elementary schools that once acted as feeder schools to neighborhood high schools are now feeder schools for every high school in the city. That leaves elementary schools not knowing much about the expectations of the high schools their students will go on to attend, and high schools not knowing exactly what students learned during their elementary experience.

When I taught freshman English for a couple of school years, not once did I know any information about the curriculum my incoming students had used. I didn’t even know what elementary school they came from, let alone their specific struggles or strengths. Think about how much more effective our high schools would be if we actually had that information.

Javion also missed out on something that many CPS children are missing — access to a reading specialist. Reading specialists have been disappearing in the last decade thanks to student-based budgeting, where schools receive budgets based on their enrollment and principals decide how to allocate their funds. If their school’s enrollment is declining, principals often decide to forego a reading specialist to save a teaching position. This does not mean that those positions are not needed; on the contrary, they are very much needed in schools that serve high-needs students like Javion. Those are also often the schools declining in enrollment.

When I started teaching, I experienced the benefits of a reading specialist myself. I was not prepared to teach reading, as much of my teacher education taught me to become a teacher like the ones I had in high school — ones that assigned texts, held discussions, and gave feedback on essays. I knew little of what to do when students didn’t or couldn’t read the assigned text. I quickly discovered that holding a discussion on a chapter that only a few students have read or understood didn’t make for an effective classroom.

It was my school’s reading specialist who taught me how to help students access unfamiliar texts and incorporate a slew of strategies into my classroom.  She also let me know that it’s OK to use class time for reading, and that my students would benefit from me reading aloud to the class, too. My students become better readers, and many of my classroom management issues disappeared, too.

In 2019, to help students like Javion, the district should prioritize addressing both problems. Chicago Public Schools needs to make sure students are learning the same basic reading skills in every school, and that high schools understand what’s being taught to younger students and vice versa.

The district also needs reading specialists to return. I wish that Javion had attended a school where such a specialist could have helped his teachers and worked one-on-one with students who are grade levels behind in reading. He would have been noticed; he would have been helped. I have that same wish for students and teachers across the district.

Gina Caneva is a 15-year Chicago Public Schools veteran who works as a teacher-librarian and writing center director at Lindblom Math and Science Academy. She is a National Board Certified teacher and Teach Plus Teaching Policy Fellowship alum. She is also a certified reading specialist. Follow her on Twitter @GinaCaneva.

This Colorado history teacher sees the effects of immigration policy every day — in her worried students

PHOTO: Kelly Cvanciger
Kelly Cvanciger, at left, poses with students from her AP government class last year.

Here, in a feature we call How I Teach, we ask educators who’ve been recognized for their work how they approach their jobs. You can see other pieces in the series here.

Some of Kelly Cvanciger’s students at Bear Creek High School in Lakewood live with constant worry — about the possibility of deportation, arrest, or being separated from their families. They are immigrants legally residing in the U.S. through the federal Deferred Action for Childhood Arrivals program, an Obama-era initiative that the Trump administration has sought to end.

“Trying to convince kids to actively participate in their education and learn geography when they face an uncertain future cripples our education system,” said Cvanciger, a history teacher. “It’s just not fair.”

Cvanciger, who was one of six finalists for the 2019 Colorado Teacher of the Year award, talked about how current immigration policies distract students from their studies, why she moved her desk to the back of the classroom, and what she learned from her son, who has autism spectrum disorder.

This interview has been condensed and lightly edited.

Was there a moment when you decided to become a teacher?

I have had some amazing teachers in my life who inspired me as a student. In elementary school it was my fifth grade teacher, Mr. Strong. In high school, it was my French teacher, Ms. Nasvitis. In college, it was Dr. Harry Swanhart. They made me fall in love with education and never want to miss a day. I would cry if my parents told me I was too sick to attend school. I thought teaching was the coolest job because they made teaching look so fun and loved their students. While I temporarily flirted with majoring in veterinary medicine in college, my love of history lured me back to teaching and by my second year of college I was sure that it was my future. I have not looked back since.

Has having a child on the autism spectrum shaped your approach to teaching? If so, how?

Having a son diagnosed with autism opened up an entire new world in terms of understanding how education needs to look different for individual students. My son has taught me that too many people know little to nothing about autism spectrum disorder and that includes the vast majority of teachers who instruct students diagnosed with the disorder. Most do not understand sensory triggers and how students with autism learn. Every child with autism spectrum disorder is unique and their learning styles are more varied than the average mainstream learner.

Because of my son, I have become a better teacher. I taught for 13 years before he was born. He has opened my eyes to learning obstacles that I was otherwise not tuned into, especially for children with developmental disabilities who do not learn from the archaic model of “sage on the stage” teaching.

How do you get to know your students?

I constantly talk to my students and ask them questions so I can cater my examples in class to topics that they can relate to in their lives. Making a connection with each student is important so that they know people care about their lives. Most students are very willing to talk to teachers about their life, family situation, and goals for their future. Some people discredit the relationship-building piece in a classroom, but I believe fostering positive relationships is really the start to opening students up to a world of learning.

Tell us about a favorite lesson to teach. Where did the idea come from?

I love to start teaching with Hammurabi’s Law Code. It is an insightful look into the social and political ideals of the ancient river valley civilization of Mesopotamia. The reason it still has relevance today is that many of the social rules that existed related to birth rights, marriage, gender roles, and societal norms have changed little in over 3,000 years. It provides a useful glimpse into how inflexible social hierarchies really are and defines a foundation for right and wrong for millennium within those societies.

What object would you be helpless without during the school day?

This sounds so simple, but I have a pen in my hand the entire time, and write notes everywhere in my room. It is crucial to document everything in education down to the smallest detail and I always find that I could have taught a lesson differently so I write it down as I am teaching. I have entire notebooks full of “fixes” from years of teaching the same subjects. I always feel an urgent need to record thoughts before they slip away. A pen also helps when teaching as I can make comments specific to each child while students are working.

What’s something happening in the community that affects what goes on inside your class?

I have a lot of students that have “Deferred Action for Childhood Arrivals status” in my classes. The news affects my students on a daily basis. They worry about being deported, being arrested, and being separated from their families. It is impossible to end their worries with our existing immigration policies. Trying to convince kids to actively participate in their education and learn geography when they face an uncertain future cripples our education system. It’s just not fair. We have to find a way to teach students so that they understand their role in society and how they can change their future.

What part of your job is most difficult?

The most difficult part of my job is talking to students who have difficulties in their family situations. It breaks my heart when students come to school and face significant hardships at home through no fault of their own. For students, this presents an obstacle to learning and is a challenge to overcome.

What was your biggest misconception that you initially brought to teaching?

I used to think that all students wanted to learn and come to school since I always wanted to go to school to learn. But when I became a teacher I realized that was the exception, not the norm. Some students grow up with a conception of school as a necessary evil, something they are forced to do because the law says so or their parents say so. Not all students want to learn (at least the subjects they are offered in school), they have too many obstacles to learn, and most dread coming to school for a variety of reasons. No matter how hard I try to get every student to love history, some just never will, and that was a tough lesson.

What are you reading for enjoyment?

I have always been a reader, since I was a little girl. I find that it is the only way to calm my brain in the evenings. I only read two types of books: historical fiction and historical non-fiction. Reading allows me to keep up in my field. Right now, I am reading “The Gulag Archipelago” by Aleksandr Solzhenitsyn and “Stalin: The Court of the Red Tsar” by Simon Sebag Montefiore. Both are challenging reads in their own right, and discuss polarizing views on Stalin’s purges of Soviet-era Russia.

What’s the best advice you’ve received about teaching?

I was reading a study about a decade ago that came to the conclusion that your teacher desk should be at the back of the room. The study concluded that students have a fear of approaching teachers who place their desk at the front of the room because it creates a psychological barrier between the student and teacher, making the student feel inferior to the teacher as a human being. I immediately moved my desk to the back of the room. This was an eye-opening change as students wanted to talk to me more often as I navigated the classroom rather than coming to my desk with questions. By removing the barrier, I built closer relationships between myself and my students.