Many students cruise along just fine in math until fourth grade or so. Then, they hit a wall—fractions.

The wall is about to get taller. With mastery of the topic seen as a crucial stepping stone to progressing in math, federal standards are stepping up emphasis on fractions starting in third grade. National tests show nearly half of eighth-graders aren't able to put three fractions in order by size.

The government is funding new research on more effective ways to teach the often-dreaded subject. The new methods preface early rote learning of complicated fraction rules with more work on building a conceptual understanding of fractions. And instead of traditional pie charts, they rely more on tools like number lines, paper models and games putting fractions in context.

Ryan Spence, a technology-integration specialist for Propel Schools, a charter-school operator based in Pittsburgh, says fourth-graders can learn the basics quickly by playing a computer game with number lines, Battleship Numberline offered by BrainPOP, a New York City firm that creates animated educational resources. Kids use a fraction clue to try to bomb a battleship hidden between 0 and 1 along a number line, winning points for accuracy. After an hour playing the game in two classes he taught, all 40 students posted perfect scores on a test comparing fractions, says Mr. Spence, who taught fourth grade before taking his current job incorporating technology in the classroom.

Teachers typically introduce fractions in third grade, explaining denominators—the bottom half of the fraction—as equal parts of a whole. Students study drawings of pizzas cut into wedges and label the fractional parts as fourths or sixths. Lessons then move into memorizing step-by-step rules for adding, subtracting, multiplying and dividing fractions.

Ian Mitchell

Some children have trouble grasping what fractions measure. When two pizzas sit side-by-side, slices of one divided into sixths may not look that different from slices of another divided into fifths.

Fractions are especially confusing because they break rules third-graders have already learned. Whole numbers increase when multiplied, but fractions get smaller, for example. "Those are hard concepts" for children, says Lynn Fuchs, a professor of special education at Vanderbilt University.

Teachers using the new method wait to introduce problem-solving until after students understand what denominators and numerators mean, and how fractions compare to each other. Fraction bars and number lines are considered easier than circles for children to draw and divide into parts. They also let students line up fractions in a row and see the difference in size, something they can't do when dividing up a pie in the traditional approach.

Knowing how to place fractions on a number line in third grade is a better predictor of kids' fourth-grade fraction skills than calculation ability, working memory or the ability to pay attention, according to a recent study of 357 children headed by Nancy Jordan, an education professor at the University of Delaware, Newark's Center for Improving Learning of Fractions. The effect continues at least through fifth grade, based on recent research, Dr. Jordan says.

RaMiah Griffin, 10, mulls a fraction while using the game. Michael Henninger for The Wall Street Journal

Another recent study, led by Dr. Fuchs, shows working with number lines made a difference for struggling fourth-graders in 13 Nashville public schools. Ten-year-old Robert Robinson had trouble with fractions before he had tutoring last spring, but "he has made a lot of progress" and is now getting high grades in math, says his mother Shatika. Robert says that by answering more fractions questions correctly than his tutoring partner, he scored more touchdowns on a paper model of the Tennessee Titans football field. "I feel pretty happy about that," he says.

At Union County Middle School in Blairsville, Ga., teacher Donna Owens says she uses strategies from Carnegie Learning, a math-curriculum developer for middle and high school, among other sources, to help her students figure out different ways to solve problems.

If a student doesn't understand denominators, she has him imagine being invited to two birthday parties, one with seven guests and the other with three, then fold a strip of paper into sevenths and another one into thirds, providing a quick way to see that 1/3 is bigger. Using paper and kids' own drawings is a way of making it easy for them to practice many approaches to problem-solving.

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A child's knowledge of fractions in fifth grade predicts performance in high-school math classes, even after controlling for IQ, reading achievement, working memory, family income and education, and knowledge of whole numbers, according to a 2012 study led by Bob Siegler, a professor of cognitive psychology at Carnegie Mellon University.

The finding is based on long-term studies of a total of 4,276 students in the U.S. and Britain comparing their scores on math tests at ages 10 to 12, and again at ages 16 to 17, then controlling the results for differences in the children's intelligence-test scores and family background. Researchers believe the reason may be that to master advanced math, students must broaden their understanding of how different kinds of numbers relate to each other, and how they must apply different rules as needed when working with different kinds of numbers.

Ian Mitchell

"If you don't understand fractions, it's literally impossible for you to understand algebra, geometry, physics, statistics, chemistry," Dr. Siegler says. "It closes a lot of doors for children." New federal standards known as the Common Core, which are being implemented in most states, require students to be multiplying and dividing fractions by fifth grade.

Trouble with fractions is the most common reason parents seek math help for their fourth- and fifth-graders, says Larry Martinek, chief instructional officer of Mathnasium Learning Centers, a Los Angeles-based franchiser with 385 U.S. tutoring centers. Many students are confused by the terms often used to describe fractions, such as "common denominator," so tutors offer clearer, more concrete names.

Denominators, for example, are "the name of the fraction," rather than simply "the bottom number," Mr. Martinek says. This helps kids understand why they can't add ½ and 1/3 and get 2/5, he says. Tutors explain, "One apple plus one apple is two apples. One banana plus one banana is two bananas. But one apple plus one banana isn't two banapples."

Zach Bedell did well in math until fifth grade, when multiplying and dividing fractions started "giving him a really hard time," says his mother, Kim, of Martinez, Ga.With 30 students in his math class, the teacher didn't have time to answer all his questions.

Tutors at an Augusta, Ga., Mathnasium helped him understand what fractions meant and how the rules worked, says Ms. Bedell, and Zach, age 11, now scores 98% on average in an advanced sixth-grade math class.

Write to Sue Shellenbarger at sue.shellenbarger@wsj.com