ETHETE, Wyo. — On one of the last days of school at Wyoming Indian Elementary School in the spring, Cheryl Williams is playing a game of “8-plus” with Atsa, a first-grader with a round, reflective face under a fuzzy thatch of black hair. These sessions are often playtime for Atsa, but they are always serious for Williams, a teacher who works one-on-one with young children to jump-start their stalled mathematical thinking.

The game is simple. Atsa throws a die with the numerals four through nine on its six faces.

“Seven,” he says.

Next, Atsa must calculate 8+7. Solving such an equation might seem simple, but Williams watches to see how he addresses the problem. It’s the boy’s strategy, not his answer, that will tell her what kind of progress he’s making.

Before he started working with Williams, Atsa would have approached the 8+7 problem by looking around for objects to count. If he didn’t have enough fingers, he may have counted objects lying on the table. Educators might reason he did the math this way because he lacked what the Common Core standards call “number sense,” the notion that eight is composed of smaller numbers like 4+4 or 5+3. Without the ability to “compose” and “decompose” numbers, experts say, children struggle with large quantities and fall behind on the path to understanding division, decimals, fractions, and percentages.

Atsa and Williams have grown close after spending half an hour together every day for a semester. They lean in over the small table like collaborators, if not playmates. He looks at her for approval. She snakes her arm around the back of his chair as he solves the equation on a wooden rack that holds two rows of beads, half of them red and the other half white. The apparatus, which looks a bit like an abacus, is designed to help children understand that numbers from 11 to 19 are made up of tens and ones—a concept integral to number sense, as laid out in the Common Core first-grade standards.

A bead rack known as a rekenrek—that’s Dutch for calculating rack. Children use this device to express numbers between 11 and 19 as a sum of 10 and another number. (Ron Feemster / Hechinger)

Atsa counts out eight of the 10 beads on the upper rod, sliding them to one side. Then he moves seven beads into place on the lower rod. From there, in an effort to see if Atsa can form numbers from tens and ones, she instructs him to swap the beads. He adds two to the first row and, in turn, takes two away from the one below: 10 on top and five on the bottom.

“How many is eight and seven?” Williams asks.

Atsa considers the bead rack: “15,” he answers.  

Dozens of exercises like this one, used in individual sessions and in classrooms at schools like Wyoming Indian, seem as though they were designed to teach to the Common Core standards, which have been rolled out in most states.

But they weren’t. This teaching approach, called “Strength in Number,” was developed a decade before work even started on Common Core. First People’s Center for Education, a small nonprofit based in Sheridan, Wyoming, trains teachers at poor, mostly Native American schools throughout the West, like this one on the vast Wind River Indian Reservation. And although the sample size is small—Wyoming Indian Elementary enrolls about 350 students including prekindergarten—the program’s results have been striking.

Before 2006, when First People’s came to Wyoming Indian Elementary, less than a quarter of  the school’s third-graders were rated “proficient” or higher on the state’s standardized math test. Since then, more than 60 percent have achieved this benchmark annually (with the exception of one year after the state increased the test’s rigor). Amid widespread debates over the Common Core’s long-term benefits, a handful of isolated, American Indian schools may offer an example of how a Common Core approach can work.

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Atsa did not come to Williams because he failed a quiz or a standardized test. He was selected for this “math intervention” based on teacher recommendations and a diagnostic interview midway through first grade. Because Williams must spend so much time with each student, she must choose her candidates carefully. Only one in 10 first-graders gets an intervention, but school officials say that many more children would benefit if they had the money to hire and train more teachers for this role. When Williams interviewed Atsa, she asked him to solve problems with some of her math toys. He wasn’t on track to meet first-grade standards, said Williams, adding that she identified two issues in particular that were holding him back: “arithmetical strategies” and “number structure.” These Strength in Number terms also happen to align closely with the Common Core.

Williams made video recordings of every moment of every exercise with Atsa, as she does with every child. She spends hours each week replaying the lessons, looking for clues about how a child is thinking and solving problems.

Over the course of the semester, Williams took Atsa through a series of increasingly sophisticated strategies for approaching numbers. Williams used toys like the bead rack, popsicle sticks in bundles of tens and ones, number tapes to show the order of numbers, and a host of other hands-on tools designed to help kids understand arithmetic. When Atsa did well, she took the toy away and asked him to work the problem in his head. If a concept proved troublesome, or a new application was confusing, Williams went back to the toys and started over. At the end of the semester, Williams sent Atsa on to second grade, confident that he would do well.

Getting the right answers on test questions isn’t enough for First People’s. Like many of today’s Common Core practitioners, Strength in Number teachers require students to explain their work. For very young children in reservation schools, this can be a problem. Social workers and teachers on the Wind River Indian Reservation estimate that 5-year-olds in their community arrive at school with a vocabulary of 3,000 words—about 2,000 less than average for that age. Even as first-graders, many lack the vocabulary to explain their reasoning.

“That’s when you can go back and watch the video and see if there is any finger stuff going on, any head bobbing stuff, any tapping going on,” Williams said. Those actions are evidence of counting. That’s a sure clue, Williams says, that the child does not yet understand something important—and that the teacher needs to diagnose the problem.  

First People’s has been taking its approach to numeracy into Native schools in Wyoming, Montana, Washington, and Alaska since 1999. Craig Dougherty, the superintendent of the Sheridan, Wyoming, school district, founded the organization after spending much of his early teaching career at Native schools in Alaska. “I could see the low achievement in those schools,” he said. “And I knew that Indian kids were as smart as any other kids. I concluded that we needed to do something about how they were taught.”

So Dougherty created First People’s to train teachers to teach math in a more natural, orderly way compared to rote memorization.

La Vina Witt, a second-grade teacher at St. Stephens School, shows students popsicle sticks to represent a three-digit number: two bundles of 100, four bundles of 10 and eight individual sticks make 248.

La Vina Witt, a second-grade teacher at St. Stephens School, shows students popsicle sticks to represent a three-digit number: two bundles of 100, four bundles of 10 and eight individual sticks make 248. (Ron Feemster / Hechinger)

By the mid-2000s, First People’s had become a nonprofit and attracted funding from foundations and the federal government. U.S. Senator Mike Enzi, who went to high school in Sheridan, earmarked funding in the federal education budget for First People’s. Over the years, this funding—which had totaled nearly $1 million by 2008—allowed the program to expand and send a traveling cadre of educators to teach Strength in Number for free to more than two dozen schools.

This funding mechanism, however, was controversial. Lawmakers often demanded earmarks to pay for programs in their districts in exchange for supporting a colleague’s bill. And while this loosened congressional gridlock, critics argued it enabled favoritism. The House eventually banned earmarks, which caused First People’s federal-funding stream to dry up. Consequently, the center began charging for its services, and many schools dropped out. Today, though some schools continue to use First People’s methods, many of them don’t have the money to implement Strength in Number as it was designed, and only nine schools still receive the center’s services, administrators say. A Wyoming state budget proposal to set aside $285,000 for First People’s died in a legislative committee this spring.

“One problem we have now is that many schools cannot afford to train all of the teachers, and they can’t afford a ‘math interventionist,’” said Lucinda “Petey” MacCarty, a longtime First People’s educator.

La Vina Witt, a second-grade teacher and former math interventionist at Wyoming’s St. Stephens Indian School, which is operated by the Bureau of Indian Affairs, learned the Strength in Number approach to teaching math in 2009. “It totally flipped my world upside down,” said Witt, a veteran educator who had previously taught middle-school math at non-Native schools in South Dakota. “The philosophy is so different.”

Now she concentrates on “allowing the students to do more discovery,” continued Witt, who also likened the Strength in Number philosophy to the Common Core. “And giving them the time to do that.”

Aside from emphasizing discovery and deep comprehension, however, the Common Core standards also require first- and second-graders to tackle concepts much more difficult than those previously expected of kids their age. Witt has read complaints about the standards requiring 7-year-olds to add and subtract numbers up to 1,000, and to count by hundreds or thousands—expectations that critics say are meant for 8-year-olds. Still, Witt, who uses many of the techniques that Williams used with Asta, argues that if the concepts are presented correctly much younger children can work with the bigger numbers; in fact, she, too, has also pushed her students to master those more rigorous skills.

Test scores rose at St. Stephens after teachers began using the First People’s system, much as they had at the Wyoming Indian Elementary several years earlier. By 2013, roughly two-thirds of its students were rated proficient or higher, compared to just a fourth in 2009, according to school-reported data.

At both Wyoming Indian and St. Stephens, however, a significant number of students fall short of proficiency on the PAWS (Proficiency Assessments for Wyoming Students) test even in the best of years.

But Witt and her colleagues say they rely on a much different measure of mastery, one that relies on qualitative, day-to-day assessments of progress in the classroom. “If they are applying what they learn to a totally different kind of problem than what you are doing in class, and can carry that over and be able to explain what they are doing, then you know they’ve got it,” Witt said. According to Witt and others, while some of the students make procedural errors on typical standardized-test questions—finding the lowest common denominator to add fractions, for example—their conceptual understanding allows them to assess and explain fractional diagrams that sometimes baffle better standardized-test takers.

“PAWS doesn’t ask you to explain,” Witt said. “If they would give kids credit for being able to explain a strategy and recognize that it’s the most efficient strategy, instead of just [giving credit for] a right answer, the scores would look different. They would say, ‘Yeah, these kids do understand math.’”

Perhaps ironically, third-grade math scores at Wyoming Indian Elementary dropped sharply after the state aligned its test to the Common Core standards—an exam considered to be much more rigorous. This past school year, the rate of those performing at proficient or higher dropped to 7 percent. Although similar trends were seen statewide, the drop was sharper at Wyoming Indian.  

“A lot of the math test is really a reading test,” said Owen St. Clair, the former principal of Wyoming Indian Elementary who now serves as the district’s superintendent. “The wording of the problems is difficult. If our kids aren’t reading at a high level, it’s hard for them to take that test.” But St. Clair acknowledges that the reading issue isn’t an excuse: “We’re making strides in math,” he said. “I think we’ve hit the low point and are on the way back.”

Most of the time, incorrect answers in class help Witt and Williams see what their students need to work on next. Near the end of her session with Atsa, Williams challenged him to think “in fives” about a new situation. The exercise culminated in the test that Witt says proves mastery: Can the student apply a concept in a new way and explain what he did?

After a short practice session in counting forward and backward by fives, Williams asked Atsa to create a rectangle of tiles by building rows of five. When he reached three rows of five, Atsa was able to look at the array and see that it contained 15 tiles. He also answered correctly that he had created six rows of five tiles. But when Williams asked Atsa how many tiles he had altogether, he started to count by twos instead of fives. He understood the strategy, but he could not apply it to the new task.

The result is not failure, according to Williams. It is simply a clear diagnosis of the limits to Atsa’s comprehension at this stage.

“We try to look at what he knows and build from there,” she said. “This is how we find out what he needs to work on next.”

This story was produced in collaboration with The Hechinger Report.