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

很多学生一直到四年级左右数学都学得不错，之后就会遇到一堵 ──分数。

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.

这堵 有越来越高的趋势。由于对分数的掌握被认为是数学进阶至关重要的一步，联邦政府的标准中从三年级开始加强了对分数的重视。全国考试结果显示，近半数八年级学生没有能力将三个分数按照大小进行排列。

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.

美国联邦政府正在资助新的项目，研究如何更有效地教授这门常令学生望而生畏的科目。新的方法是，在早期阶段让学生硬记复杂的分数计算规则，然后通过更多的学习建立对分数概念的理解。他们不用传统的扇形图，而是更依赖数轴、纸模和包含分数计算的游戏。

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.

匹兹堡特许学校经营机构Propel Schools技术整合专员莱恩・斯宾塞(Ryan Spence)说，四年级学生玩一种数轴的电脑游戏可以很快学会这些基础知识，这种游戏就是纽约市动画教育资源制作公司BrainPOP提供的战舰数轴(Battleship Numberline)。孩子们要用一个分数线索炸掉隐藏在数轴上0到1之间的一艘战舰，精准炸掉就能得分。斯宾塞说，在他教的两堂课中玩这款游戏一小时后，40名学生在比较分数大小的考试中都拿到了满分。在从事目前的将科技融入课堂的工作前，斯宾塞是四年级老师。

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.

老师们一般都在三年级引入分数，向学生解释分母表示整体被分成了多少等份。学生学习时将披萨图切成楔状，分成四等份或六等份。然后学习记忆分数加减乘除的详细规则。

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.

有些孩子难以理解分数测量的是什么。两块披萨放在一起时，分成六份的披萨和分成五份的披萨看起来可能没什么不同。

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.

分数尤其令人困惑的原因是它打破了三年级学生已经掌握的规则。比如整数相乘时会增大，但分数相乘会变小。范德堡大学(Vanderbilt University)特殊教育教授林恩・福克斯(Lynn Fuchs)说，对孩子们来说“那些是很难的概念”。

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.

采用新方法的老师们等到学生理解分母和分子的意义后，才会介绍解题方法以及如何比较分数大小。分数柱状图和数轴被认为比扇形图更容易画出来并分成等份。他们还会让学生把分数排成一排区分大小，而用传统方法划分扇形图则没法做到这一点。

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.

特拉华大学(University of Delaware)教育学教授、纽瓦克分数学习改善中心(Center for Improving Learning of Fractions)的南希・乔丹(Nancy Jordan)近期牵头进行了一项对357个孩子的研究。结果发现，相比计算能力、工作记忆力或集中精神的能力，三年级时对将分数放在数轴上的掌握程度，更能预测孩子四年级时的分数运用能力。乔丹说，最近的研究显示，这种影响至少会持续到五年级。

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.

另一项由福克斯牵头的近期研究显示，在纳什维尔13所公立学校，运用数轴对学习分数有困难的四年级学生很有效果。十岁的罗伯特・鲁滨逊(Robert Robinson)去年春天补习之前在分数学习上有问题，但他妈妈沙提卡(Shatika)说他“有了很大进步”，现在数学分数很高。罗伯特说，他答对的分数问题比补习伙伴多，所以在田纳西泰坦队(Tennessee Titans)的纸模游戏中他触地得分更多。他说：“我觉得很开心。”

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.

在佐治亚州布莱尔思维尔(Blairsville)的中学Union County Middle School，多娜・欧文斯(Donna Owens)老师说，她采用初高中数学课程开发商Carnegie Learning的教学策略等资源帮助学生找出不同的解题方法。

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.

如果有学生不懂分母，她就会让他想象被邀请参加两个生日派对，一个有七位客人，一个有三位客人，然后将两张纸条分别叠成七等份和三等份，马上就能看到三分之一比较大。用纸和孩子自己画的图能够很容易地让他们练习不同的解题方法。

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.

卡内基梅隆大学(Carnegie Mellon University)认知心理学教授鲍勃・希格勒(Bob Siegler)2012年领衔的一项研究发现，学生五年级时掌握的分数知识能够预测其高中数学课的表现，在排除智商、阅读成绩、工作记忆力、家庭收入和教育、整数知识等因素的影响后也是如此。

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.

这项发现是基于对美国和英国共4,276名学生的长期研究。这项研究对10至12岁学生的数学成绩进行比较，再对他们16到17岁间的数学成绩进行比较，然后排除智商测试分数及家庭背景差异的影响。研究人员认为，产生上述结果的原因可能在于，要掌握高级数学，学生们必须扩大对不同类型数字相关性的理解，以及如何在处理不同类型数字时根据需要采用不同规则。

'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.

希格勒说：“如果不懂分数，几乎就不可能理解代数、几何、物理、统计学、化学，很多扇门就被关上了。”被称为“共同核心”(Common Core)的最新联邦标准要求学生在五年级时掌握分数乘除法，目前大多数州都已经实施该标准。

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.

洛杉矶连锁辅导机构Mathnasium Learning Centers教学总监拉里・马丁内克(Larry Martinek)说，分数学不好是家长为自己上四年级和五年级的孩子寻求数学帮助最常见的原因。该辅导机构在全美有385家辅导中心。很多学生都对常用于形容分数的词汇感到不解，比如“同分母”，所以辅导老师会用更清晰更具体的词汇。

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 1/2 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.'

马丁内克说，比如分母就是“分数的名字”，而不是简单的“分数线下面的数字”。他说，这样有助于学生理解为什么不能把二分之一和三分之一相加得到五分之二。辅导老师会解释说：“一个苹果加一个苹果是两个苹果。一个香蕉加一个香蕉是两个香蕉。但一个苹果加一个香蕉不是两个苹果香蕉(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.

佐治亚州马丁内兹的扎克・比戴尔(Zach Bedell)在五年级之前数学都学得很好，他的妈妈金・比戴尔(Kim Bedell) 说，五年级以后分数乘除开始“让他学得很艰难”。因为班里有30名学生，老师没时间回答他所有的问题。

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.

比戴尔太太说，佐治亚州奥古斯塔Mathnasium补习中心的辅导老师帮他理解分数的意义以及计算规则， 在六年级高阶数学班里，11岁的扎克现在平均能得98分（满分100分）。