More than memorization

Teachers adding critical thinking skills to math curriculum


After 11 years of teaching, Cheryl Arnett witnessed a breakthrough recently in her students' math comprehension.

"Look," she said excitedly, flipping through scores of worksheets.

On one side of the page Arnett's first-grade Sunset Elementary students had filled in rows of addition problems, such as two plus three. But on the flip side, students had tackled a few more difficult math problems like 71 plus 25, and many of the answers were correct.

"These first-graders are making gains beyond any first-grade class I've had," Arnett said ecstatically.

"It's not new math," she explained. "It's getting students to share their thoughts on how they solve problems so other kids can see the possibilities."

The success of Arnett's class is spurred in part by some new ways she's learned to teach math. It involves letting students grasp the process of doing math instead of always pushing for memorization of math facts.

Pushing for change

Arnett's pilot program fits into a primary goal of the Moffat County School District to increase students' math scores.

According to the results of the 2003 Colorado Student Assessment Program (CSAP), wide swaths of students in Moffat County Schools are performing below the state average in math.

It's one reason why district officials created a math specialist position this year.

In the job, Cassia McDiffett, a former Moffat County high school math teacher of eight years, travels between district schools providing instruction to math teacher leaders on how to instruct students in alternative ways of thinking about math.

"When you tell people what you're doing they always say the same thing, 'I wish I had been taught like that,'" McDiffett said. "I think it's important to give kids the strategy to work out problems. Even kids that say they hate math can learn to do this."

The basis for the program goes like this: Instead of pressing kids to solely memorize math problems, teachers are also asking students to explain how they arrive at conclusions. While strict memorization of facts is key to figuring out a number of math problems, it doesn't always provide a solid basis when student encounter an unfamiliar math problem, said Assistant Superintendent Joel Sheridan.

"What we want is kids to understand what division is and not just by memorizing a bunch of numbers," he said. "If they've done that and you ask them to divide18 by five, they think it can't be done. I want division to mean something to them."

Sheridan estimated that only two-thirds of students comprehend math who have been taught in the traditional "skill and drill" method. The district has stopped giving timed math tests, which requires students to fill out a series of similar math problems at a hectic pace.

"We're not going away from the basics," Sheridan said. "We just want to balance it out a little bit."

Time to learn

To provide a hands-on example of the district's new approach, two Moffat County High School students were asked to solve a story problem in front of the Board of the Education Thursday afternoon.

Junior Alicia Thompson and sophomore Chad Eitel were charged with determining the total number of digits and how many of each a hotel owner would have to buy to place numbers on each of 100 rooms.

They hadn't seen the problem in advance, McDiffett said.

The students took about 15 minutes to figure it out, bouncing ideas off one another. They developed a graph to solve the problem instead of manually counting up all the digits.

After the presentation, the students talked about their progress in school this year.

"At the beginning of the year it would have taken (our class) about week to figure this out," Thompson said. "It's like we have ways to figure it out now."

At the start of the year Eitel said he'd produce math problems with the correct answers but teachers would ask to see his work.

"They wanted to see how I worked it out," he said. "It made it a little harder at first."

But educators said the extra time for students to grasp a math concept is well worth it.

Sheridan spoke from the example provided by the high school students.

"If a teacher told them how to do it, it could be done in five minutes," he said. "It's not to say we're going to turn them loose, but we can't be thinking for them."

Earlier in the day McDiffett noted another example of a student thinking conceptually about math.

A first-grader, who likely hadn't been versed in multiplication tables, informed McDiffett that classmates would create a total of 40 shapes during an elementary geometry lesson. The young student came to that reasoning after hearing that students would glue twenty shapes each onto two pages.

"If kids understand the concept they can learn to do things differently," McDiffett said.

Raising the bar

In some respects, the district has come a long way in helping students succeed in math.

But a lot of work still needs to be done, said parent Christine Balderson.

When the family moved to town five years ago, Balderson noted the district's lack of emphasis on math compared to what she was used to at the Denver schools.

Her daughter, Stephanie, now in seventh grade at Craig Middle School is having an easier time with the subject.

"I do notice that it's getting better," Balderson said. "But I feel like they (Denver schools) pushed math more."

These days, Balderson's son, a third-grader at Sunset Elementary School, is progressing at a rapid rate in math.

"It just floors me," she said. "He's bringing home multiplication, division and story problems. The work makes him feel he can do it, but it seems pretty detailed to me."

District officials have admitted they've concentrated more resources on reading and writing comprehension in the past.

While those efforts continue to reflect in higher marks on the state-required Colorado Student Assessment Program tests, math scores haven't yet crested above the state average.

Students at Moffat County's Elementary schools aren't tested for math skills on CSAP tests.

Moffat County High School students scored 11 percent below the state average on CSAP tests last year while Craig Intermediate School students scored 13 percent below the state average.

That kind of data makes Balderson nervous.

"That is a real concern for me," she said. "I know that not all students are struggling with math but unfortunately some of them are."

Results from the district's latest effort to revamp its math teaching strategies may appear as soon as the middle of next year, McDiffett said.

"We're moving in the right direction," she said. "Some people say you can see changes as soon as one is made, but it may be a little harder than that to predict."

In another move, the district's board of education is also expected to adopt new math curriculum next month.

They are also acutely aware that future students will have to meet new requirements if they plan to enter Colorado's four-year colleges upon graduation. That requires students to enroll in core math classes at the high school level, which may require a stricter math focus at the district's elementary and middle schools.

But eliminating the apathetic nature many have about math may be the first hurdle to helping students succeed.

"We're fighting a pretty big battle," McDiffett said. "People have no problem saying that they're no good in math but it's unacceptable to say you can't read."

Amy Hatten can be reached at 824-7031 or


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