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Nearly three years ago, Elham Kazemi stood before teachers at Lake­ridge Elementary with more bad news about their students’ math skills.

The teachers at the Renton School District school already knew they had a big problem. The year before, fewer than 10 percent of the third- and fifth-graders passed state math tests, which put Lakeridge in the bottom 5 percent of elementary schools statewide.

But Kazemi, a University of Washington professor, had more distressing details.

When she and her colleagues asked students to solve a half-dozen word problems, most just added numbers together, even if the problem asked them to subtract or multiply.

Only a very few fifth-graders could solve problems with the skill that, in other schools, was common in third or even second grade.

Principal Jessica Calabrese, then new to the school, can still picture Kazemi in the school library that day, wearing a long cardigan and a look of pain. For her, that’s when the depth of the school’s troubles fully sank in.

“It wasn’t just some students struggling,” Calabrese said, “it was nearly all of them.”

With math skills that weak, she knew her students wouldn’t be ready for algebra by the eighth grade.

That, by itself, could keep them off the college track in high school, which in turn would almost certainly put the growing number of Washington’s well-paid technology jobs beyond their reach.

Some educators in her position double down on the basics, drilling students on a narrow set of key skills in the belief that it’s the best way to help them catch up.

But Calabrese and Lake­ridge teachers didn’t consider that route — seeing it as a short-term fix that probably would raise math scores quickly but could rob students of the deeper understanding they’d need to succeed in algebra and beyond.

They instead set out on what they figured would be a slower, but ultimately more effective path — one that focused on giving students a strong grasp of math concepts.

With help from Kazemi and her colleagues, the teachers have turned many math lessons into carefully guided conversations in which students explain their approaches, defend their reasoning and critique each other’s ideas.

The students have learned to say, “I respectfully disagree,” and, without embarrassment, “I want to revise my thinking.”

The approach lines up with the new Common Core standards for math that most states are starting to put in place. In some schools, parents and teachers are objecting, saying the Common Core is confusing and wastes time, requiring students to explore multiple ways to solve problems when all they need is one good one.

Such concerns haven’t surfaced at Lakeridge, perhaps because test scores are going up.

In just two years — much faster than anyone expected — Lakeridge’s performance on state math tests rose from the bottom 5 percent to somewhere near average. If pass rates keep going up, the school could become a model for how to raise math performance without the heavy dose of drill that’s often seen as a fix for low-income schools.

The new approach required considerable teacher training — eight full days starting in the 2011-12 school year, another six days the second year and four the third, with informal coaching sessions in between.

Veteran teachers have made adjustments that math coach Teresa Lind compares to lifelong golfers learning a new way to swing.

Lynn Simpson, who has worked at Lakeridge for 16 years as a teaching assistant and teacher, said having to teach in front of her peers and her principal was, at first, gut wrenching.

“You’re an adult and you figure that you know third-grade math,” she said, “But we didn’t know enough about how kids learn.”

Labeled as failing

Lakeridge Elementary sits high on the Renton School District’s West Hill, with a sweeping view of Lake Washington from the playground out back. Yet much of the surrounding neighborhood is not high rent.

The school has always had a significant number of students who qualify for the federal free- or reduced-price lunch program, but that rate rose to 88 percent a few years ago after a huge apartment complex went up nearby.

In 2010, Lakeridge’s pass rates in math sank to the point that the school ended up on a list of the state’s lowest-performing schools.

The label stung, but it also put Lakeridge in line for one of the large federal grants aimed at helping such schools rise up from the bottom.

When Calabrese arrived in spring 2011, the school already had won one of those grants, a three-year, $2.7 million award. To get it, teachers had proposed an improvement plan that included extending the school day by half an hour, adding five days to the school year, introducing a new student behavior program — and digging into how to improve instruction in math.

As a former assistant principal at nearby Dimmitt Middle School, Calabrese strongly supported a focus on math concepts. At Dimmitt, she’d seen far too many sixth-graders give up in math because they didn’t understand it.

“If math is already painful to you, if math is something that you’ve already decided you’re not good at,” she said, “how can we convince you you’re smart at all?”

Ideas from students

Math lessons at Lakeridge look different from what happens about 20 miles away at another high-poverty school, Gildo Rey Elementary in Auburn.

At Gildo Rey, where nearly every student passes state math tests, teachers usually demonstrate how problems should be solved, with students following along. Teachers fire questions at a fast pace and students shout back in unison.

That style — which draws heavily from what’s often called direct or explicit instruction — is backed by research showing it can help raise test scores, especially among low-income students.

In comparison, teachers at Lakeridge focus more on drawing ideas from students, an approach that’s also been shown to be effective.

They pose a problem, then give students a minute or two to consider how to solve it in their heads before calling on them to explain their approaches. They don’t move on when someone gets the right answer, but rather ask if anyone solved the problem a different way.

In Stephanie Latimer’s third-grade class, for example, she recently asked students whether 60 minus 45 was equal to 61 minus 46, then called on one of the three students who initially thought the equation was false.

That boy, named Ishmail, explained, step by step, his largely correct approach — how he figured that 60-45 was 15, and 60-46 was 14, and since 15 isn’t the same as 14, the two sides weren’t equal.

Latimer didn’t correct his goof, turning to the class to ask if anyone disagreed. A girl named Joselyn quietly said, “You forgot to add the one.”

“Oooh, plus one,” Ishmail said, seeing his error immediately.

Lakeridge teachers teach some math directly, explaining terms such as parallelogram and median that students would not come up with on their own.

They also make sure students memorize their multiplication tables, and eventually show them the standard methods for adding, subtracting, multiplying and dividing.

But first, they want students to understand why those approaches work.

Adjusting instruction

It might be easy to view Lakeridge and Gildo Rey as a microcosm of the so-called math wars, but that would be an oversimplification.

Though their emphases differ, both schools teach math concepts as well as math skills, not just one or the other.

If people from each side of the math wars sat in the back of a Lakeridge classroom, they’d agree they were seeing good, strong instruction, said Andrew Kelly, who directs school-improvement efforts at the state Office of Superintendent of Public Instruction.

What impressed him most, he said, was the close attention teachers pay to how well students grasp each lesson.

At Gildo Rey, teachers also adjust instruction daily based on what students learned — or didn’t learn — the day before.

Teachers at both schools also work to make instruction consistent from classroom to classroom and grade to grade and — perhaps most important — they meet regularly with one another and their principals to talk about how they can improve.

Teachers’ turn to learn

Such teamwork was part of Kazemi’s goal when Lake­ridge teachers hired the UW to work with them three years ago.

For 15 years, she had been working on ways to help teachers teach math and had come to believe they do best when they work together, trying out new techniques in their own classrooms.

She has led many of the full-day training sessions at Lakeridge, called math labs.

Groups of teachers, along with math coach Lind, Calabrese and often Kazemi or one of her colleagues, together plan a lesson, then try it out in two different classrooms, making adjustments as they go.

One day in May, it was the third-grade teachers’ turn, and as they filed into the first class, the adults told the students that it was their turn to learn, by listening closely to what students were thinking.

Then two of the teachers launched into a lesson about fractions, asking students to determine whether 2/8ths was greater than, less than or equal to 6/8ths.

The teachers at first thought the problem might be too easy, but quickly saw otherwise when a number of students insisted 2/8ths was bigger.

As one boy explained: “Cutting into sixths, it would be smaller … than if you cut into two eighths, it would be larger.”

The teachers paused for a couple of timeouts, talking among themselves about how to proceed.

It was Simpson who first realized many students thought they were supposed to figure out how big a piece they would get if they shared a sandwich or a pizza with two students, or with six. That’s the type of problem they had been doing just a few days before.

The teachers decided to help students visualize the fractions so they drew two rectangles on the board. Then they divided each into eight equal pieces, shading in six pieces on one rectangle and two on the other.

That did it.

“The size of the pieces are the same,” one student said, “but you get more pieces.”

“Don’t give up”

The math labs aren’t designed to give teachers a blueprint or a recipe, Kazemi said, but a set of principles that can be used with any math textbook.

Kazemi said the principles are grounded in research that started decades ago at the University of Wisconsin that helped map the progression students follow in learning math, similar to the one in reading.

Through other studies, the researchers found that teachers who understood this trajectory were more effective. They called their approach Cognitively Guided Instruction.

In one study in the late 1980s, researchers compared students from 40 first-grade teachers — 20 who received training in Cognitively Guided Instruction and 20 who attended general workshops on problem-solving. Students from the first group of teachers performed better on several types of math problems, and the same on the rest.

A new study is under way in Florida, aimed at replicating those results.

No one knows exactly how widely Cognitively Guided Instruction is used, since it’s not a copyrighted program, but it has become influential, said Diane Briars, president of the National Council of Teachers of Mathematics — the nation’s biggest professional group for math teachers.

At present, Arkansas and Iowa now have statewide programs to help teachers learn the approach, and the Teachers Development Group near Portland trains abut 4,000 teachers a year.

At Lakeridge, Simpson said the approach helped her see, for the first time, what to do when students stumbled.

For years, she would just reteach the material the same way, which often didn’t work.

Now she can better diagnose where students are on the math-learning trajectory, and what they need next.

“Students,” she said, “don’t give up like they used to.”

Choosing strategies

Lakeridge’s success has caught the attention of Washington STEM, a group that advocates for better and more instruction in science, technology, engineering and math.

The group sees Kazemi’s approach to training teachers as a model that might work in many schools.

To test that belief, the group gave Kazemi and her colleagues a grant for similar efforts in science and math in a handful of schools in Highline and at nearby Campbell Hill Elementary in the Renton School District. At Campbell Hill, they are trying shorter, more frequent training sessions as a way to lower the cost for schools.

Calabrese said the math training wasn’t all that expensive, just a fraction of the $2.7 million federal grant. She plans to keep the training going even after that grant is gone, at an estimated cost of about $130,000 a year, enough to pay a math coach and substitutes for four or five math and literacy labs a year.

When this year’s test results come out in late August, Calabrese expects Lake­ridge’s math passage rates to rise again — although she gauges progress more by what she sees in classrooms.

She tells the story of a quiet second-grader who, faced with a subtraction problem, politely rejected a classmate’s suggestion that she subtract 60 from a bigger number by jumping back 10 places on a number line six times.

The girl, saying she wanted a more efficient strategy, jumped back 50, then 10.

That’s success, said Calabrese, when a 7-year-old doesn’t just get the right answer, but is able to say she wants a more efficient strategy — and then finds one.

Seattle Times staff reporter Justin Mayo contributed to this report.

Linda Shaw: 206-464-2359 or On Twitter @LShawST