什么数学基础才能学微积分（微积分是高中数学的顶峰）

译者的几句话：我不是教育学背景的人，看大部分国人写的教育学论文感觉索然无味，与实际基本不符不说，还缺少实证研究。下面翻译了一篇美国人的分析文章给那些想做教育框架的人看看。在我看来拿着苏联教学大纲框定教学内容的做法不是愚蠢就是懒政！这篇文章紧盯着大数据，要求改革中学数学教学！

微积分是高中数学的顶峰？也许是时候改变一下了

By Sarah D. Sparks

May 22, 2018

For more than 30 years, calculus has been seen as the pinnacle of high school math—essential for careers in the hard sciences, and an explicit or unspoken prerequisite for top-tier colleges.

30多年来，微积分一直被视为高中数学的顶峰，对于从事硬科学职业来说是必不可少的，对于顶级大学来说，微积分是一个或明或暗的入学先决条件。

But now, math and science professionals are beginning to question how helpful current high school calculus courses really are for advanced science fields. The ubiquitous use of data in everything from physics and finance to politics and education is helping to build momentum for a new path in high school math—one emphasizing statistics and data literacy over calculus.

但是现在，数学和科学专业人士开始质疑，当前的高中微积分课程对高等科学领域有多大帮助。从物理、金融到政治和教育，无所不在地使用数据，有助于为高中数学新道路建立动力，即强调统计和数据素养而不仅仅是微积分。

"We increasingly understand the world around us through data: gene expression, identifying new planets in distant solar systems, and everything in between," said Randy Kochevar, a senior research scientist at the Education Development Center, an international nonprofit that works with education officials. Statistics and data analysis, he said, "is fundamental to many of the things we do routinely, not just as scientists but as professionals."

“我们通过数据越来越了解我们周围的世界：基因表达，识别遥远太阳系中的新行星，以及两者之间的一切，”教育发展中心（一家与教育官员合作的国际非营利组织）的高级研究科学家兰迪·科切瓦说。他说，统计和数据分析“是我们日常工作的基础，不仅仅是作为科学家，更是作为专业人员。”

He and other experts are still debating the best way to integrate a new approach in an already crowded high school curriculum. One of the most difficult philosophical challenges: how to prevent a statistics path from replicating the severe tracking and equity problems that have long existed in classical mathematics.

他和其他专家仍在讨论将新方法融入已经拥挤的高中课程中的最佳方法。最困难的方案上的挑战之一是：如何防止统计路径重导传统数学中长期存在的严重路径依赖和均衡内容的覆辙。

"There's a sense that calculus is up here and statistics is a step below," said Dan Chase, a secondary mathematics teacher at Carolina Day School in North Carolina, adding that he often struggles to suggest to students that, "if you are interested in engineering, that might be a good reason to go to calculus, but if you are interested in business or the humanities or social sciences, there are different paths you might go, even if you are a top-achieving math student."

北卡罗莱纳州卡罗莱纳日学校的中学数学教师丹·蔡斯（dan chase）说：“有一种感觉，微积分在这里，而统计数据在是接下来才进行的一个内容。”他还补充说，他经常努力向学生们建议：“如果你对工程学感兴趣，这可能是学习微积分的一个很好的理由，但是如果你对商业感兴趣。”或者人文科学或社会科学，你可能会走不同的道路，即使你是成绩最好的数学学生。”

On face value, new expectations for students already seem to be moving toward statistics. Both the Common Core State Standards, on which many states' math requirements are based, and the Next Generation Science Standards call for teaching data analysis and statistics, both on their own and in the process of learning other concepts.

从表面上看，对学生的新期望似乎已经转向了统计。许多州的数学要求，依据共同核心州标准和下一代科学标准都要求教学数据分析和统计，无论是独立的还是在学习其他概念的过程中。

But Kochevar warned: "There's a huge disconnect; if you look closely at the science standards, they are expecting students to have tremendous faculty with using data by middle school, but if you look at the courses, it's really not clear where those skills are supposed to be filled."

但科切瓦警告说：“这有一个巨大的脱节；如果你仔细观察科学标准，他们期望学生在中学时拥有大量的使用数据的能力，但是如果你看课程，就真的不清楚这些技能应该在哪里被获得了。”

Both sets of standards need more integration of data and statistics, he and others argue, because they were developed in the early years of the big data boom. Studies tracking data worldwide through the years have found people produced 1.5 exabytes of new data in 1999—or roughly 250 megabytes of data for every person alive—but by 2011, when states were adopting and implementing the math standards, people produced more than 14 exabytes a year. Today, people worldwide produce 2.5 exabytes of data every day, and the total data have doubled every two years.

他和其他人认为，这两套标准都需要更多的数据和统计数据集成，因为这两套是在大数据繁荣的早期开发的。多年来跟踪全球数据的研究发现，1999年，人们产生了1.5艾字节（exabyte）的新数据，或者说，对于每个活着的人来说，大约250兆字节的数据，但是到2011年，当各州采用和实施数学标准时，人们每年产生的数据超过14艾字节。今天，全世界的人们每天产生2.5exabytes的数据，而且总数据每两年翻一番。

Ironically, the rapid expansion of big data and statistics use in the broader society and economy comes at the same time American students seem to be struggling with those concepts. From 2007 to 2017, 4th and 8th students' scores on the National Assessment of Educational Progress in mathematics fell significantly on problems related to data analysis, statistics, and probability—a decline that helped drive overall dips on the math test in 2017.

具有讽刺意味的是，在更广泛的社会和经济中，大数据和统计数据的使用迅速增长，与此同时，美国学生似乎正在与这些概念作斗争。从2007年到2017年，全国数学教育进步评估的第4和第8项学生成绩即数据分析、统计和概率相关问题上显著下降，这助推了2017年数学考试成绩的整体下降。

In part, experts say, that's because statistics and data analysis have traditionally taken a back seat to calculus in high school math, and most students already have difficulty completing the classical path.

"The idea that statistics is hard is grounded in that fact that if you took statistics 10 years ago, you had to take calculus first, and the statistics used formal probability ... with theorems that built on calculus," said Uri Treisman, a mathematics professor and the executive director of the Charles A. Dana Center at the University of Texas at Austin. He's been working with K-12 and university systems to develop a statistics pathway as an alternative to classical calculus.

It's an idea that others have pushed back on, by situating a high school statistics pathway as either advanced material only suitable for students who have already passed calculus—or a less-rigorous path for students who can't hack it in classical math.

专家们说，部分原因是统计和数据分析传统上是高中数学微积分的之后才学，而且大多数学生已经很难完成经典路径。

“统计学很难的观点是基于这样一个事实：如果你10年级前学过统计学，你必须先学微积分，而统计学使用形式概率……数学教授、奥斯汀德克萨斯大学查尔斯达纳中心执行主任乌里特雷斯曼说。他一直在与K-12和大学系统合作，开发一种统计学途径，作为经典微积分的替代。

这是一个高于其它的想法，通过设置一个高中的统计路径，这要么是先进的材料，只适合那些已经通过微积分的学生，或是一个不太严格的路径，给不能用黑客的手段肢解经典数学的学生。

"Any time you have multiple pathways, the advantaged will capitalize on one and that will become the 'real' one," Treisman said. "If we are going to create data science pathways, they had better be anchored in things that lead to upward social mobility and have a rigor to them. We have to make sure new pathways have at least equal status as the traditional one—and ensure everyone has access to them. If we allow [statistics and data] to be the easy or weaker path, we relinquish the commitment to equity we started with."

特雷斯曼说：“只要你有多条路，优势者就会利用其中的一条，这将成为‘真正的’一条。”如果我们要创造数据科学的途径，它们最好被锚定在导致向上社会流动的事物中，并且对它们有严格的要求。我们必须确保新的途径至少与传统的途径具有同等的地位，并确保每个人都能接触到它们。如果我们允许[统计和数据]成为一条简单或较弱的道路，我们就放弃了对我们开始投资的的承诺。”

Mixed Signals in Calculus

For a picture of how severe that inequity can get, one only has to look at calculus.

Until about 1980, calculus was seen as a higher education course, primarily for those interested in mathematics, physics, or other hard sciences, and only about 30,000 high school students took the course. That began to change when school reformers glommed onto calculus as an early example of a rigorous, college-preparatory course, said David Bressoud, a mathematics professor at Macalester College and a former president of the Mathematical Association of America, who has examined the evolution of calculus studies.

"The more schools did this, the greater the expectation that they would do it" from parents, and district leaders—and in particular from colleges and universities, Bressoud said. "It's not just math majors or engineering majors; this has become an accepted requirement for admission to top universities. You are not going to get into Duke if you haven't taken calculus, even if you plan to major in French literature."

Today, some 800,000 students nationwide take calculus in high school, about 15 percent of all high schoolers, and nearly 150,000 take the course before 11th grade. Calculus classes have been and remain disproportionately white and Asian, with other student groups less likely to attend schools that offer calculus or the early prerequisites (like middle school algebra) needed to gain access to the course.

For example, in 2015-16, black students were 9 percentage points less likely than their white peers to attend a high school that offered calculus and half as likely to take the class if they attended a school that offered it. And if black students did get into a class, their teachers were also less likely to be certified to teach calculus than those of white students, according to an Education Week Research Center analysis of federal civil rights data.

要想了解这种不平等有多严重，我们只需要看微积分。

直到1980年，微积分被视为一门高等教育课程，主要面向那些对数学、物理或其他硬科学感兴趣的人，只有大约30000名中学生参加了这门课程。麦克莱斯特学院的数学教授、美国数学协会前主席大卫·布列索德（David Bressoud）研究了微积分学的发展，他说，当学校改革者把微积分作为一门严谨的大学预备课程的早期范例时，情况开始发生了变化。

布列索德说：“学校这样做的次数越多，家长、地区领导，尤其是大学领导对他们这样做的期望就越大。”这不仅是数学专业或工程专业，而且已经成为顶尖大学录取的普遍要求。如果你没有学微积分，即使你打算主修法国文学，你也不会进入杜克大学。”

今天，全国大约有80万学生在高中学习微积分，约占所有高中生的15%，近15万人在11年级之前学习微积分。微积分课程一直是和保持着不成比例的白人和亚裔，其他学生群体不太可能上提供微积分或获得该课程所需的早期先决条件（如中学代数）的学校。

例如，在2015-16学年，黑人学生上微积分高中的可能性比白人学生低9个百分点，而上微积分学校的可能性只有白人学生的一半。根据一个教育周研究中心对联邦民权数据的分析，如果黑人学生真的上了课，他们的老师比白人学生更不可能被认证教授微积分。

And despite the rapid growth of calculus as a gold standard, university calculus experts argue it is a much weaker sign that a student is actually prepared for postsecondary math in the science fields than it appears.

In fact, a new report by the Mathematics Association of America and the National Council of Teachers of Mathematics found many students who took Advanced Placement Calculus AB still ended up retaking calculus in college—and 250,000 students end up needing to take even lower-level courses, like precalculus or algebra.

In the end, the report found taking calculus in high school was associated with only a 5 percentage point increase on average in calculus scores in college—from 75 percent to 80 percent. Rather, the best predictor of earning a B or better in college calculus was a student earning no less than As in high school Algebra 1 and 2 and geometry.

So if high school calculus isn't the best indicator of a student prepared for college-level math, what does it signify in college admissions? In a word: Money.

More than half of students who take calculus in high school come from families with a household income above $100,000 a year, according to a study this month in the Journal for Research in Mathematics Education. By contrast, only 15 percent of middle-income students and 7 percent of those in the poorest 25 percent of families take the course.

"Math is even more important to upward mobility now than it was 20 or 30 years ago, because ... it's seen as related to your general ability to solve problems quickly," Treisman said, adding that as a result, "there's general anxiety and panic about equity issues for anything new, even though the current [calculus] pathway is a burial ground for students of color."

尽管微积分作为一个黄金标准学习人数快速增长，大学微积分专家认为，这仅仅是个比较弱的迹象，表明一个学生在实际上是为科学领域，来准备高等数学的。

事实上，美国数学协会和国家数学教师委员会的一份新报告发现，许多上了高级微积分AB的学生最终还是在大学里重修了微积分，25万学生最终需要上更低级的课程，如初等几何或代数。

最后，报告发现在高中学习微积分与大学微积分平均分数从75%提高到80%只有5个百分点的关联。相反，在大学微积分中获得B或更好成绩的最佳预测因素是学生的高中代数1和2以及几何的成绩为A。

所以，如果高中微积分不是一个学生为大学水平数学准备的最好指标，那么它在大学招生中意味着什么呢？一句话：钱。

根据本月《数学教育研究》杂志上的一项研究，在高中学习微积分的学生中，一半以上来自家庭年收入在10万美元以上的家庭。相比之下，只有15%的中等收入学生和7%最贫困的25%家庭的学生选择了这门课程。

“现在数学对向上流动的重要性比20或30年前更为重要，因为……Treisman说：“这被认为与你快速解决问题的一般能力有关，”他补充道，“因此，人们普遍对任何新事物的公平问题感到焦虑和恐慌，尽管当前的（微积分）途径是有色人种学生的安葬地。”

Forging a New Path

Statistics and data literacy advocates hope diversifying the field of interesting and rigorous math courses could broaden students' path to STEM and other careers. As of 2017, the U.S. Bureau of Labor Statistics estimations showed that jobs that require data literacy and statistics are among the 10 fastest-growing occupations in the country.

"We have two paths forward," said William Finzer, a senior scientist at the Concord Consortium, which works with school districts to improve their math curricula. "The easier one—like the path computer science took—is to develop a course or a subject area and get schools to give it time. ... The problem of that is, it doesn't spread the opportunity very widely. It becomes concentrated in the small group of kids who elect to take the course—and it's just one more subject to take."

统计和数据素养倡导者希望将有趣和严格的数学课程领域多样化，可以拓宽学生的STEM和其他职业道路。截至2017年，美国劳工统计局（U.S.Bureau of Labor Statistics）的估计显示，需要数据识别和统计的工作是美国增长最快的10个职业之一。

“我们有两条前进的道路，”康科德协会的高级科学家威廉·芬泽说，该协会与学区合作，以提高他们的数学课程。像计算机科学这样简单的方法是开发一门课程或一个学科领域，让学校给它时间。……问题是，它并没有很广泛地传播机会。它集中在选择参加课程的一小部分孩子身上，而这是另外一个主题。”

Progression for Statistics and Data

EDC’s Oceans of Data Institute is building learning progressions for statistics and data literacy at different grades. Randy Kochevar, who directs the institute, said they are based on the acronym CLIP, meaning students learn how to use:

Complex, multi-variable data (“We’re not just looking at hours of sunlight and heights of bean plants,” he said);

Larger data sets than students need to answer any one question, so they are forced to sort and understand relevance;

Interactively accessed data, rather than sample graphs just written out on paper; and

Professionally collected data that forces students to think about how and why it was collected—and what biases may exist in the samples.

Source: Oceans of Data Institute

Finzer instead envisions a more holistic approach in which at least one class a year—be it math, biology, or even civics or history—asks students to grapple with making sense of large data sets. Such an approach, he said, "would make a huge difference, because it would mean when you came out of high school, data would not be foreign to you."

EDC's Oceans of Data Institute is building learning progressions for statistics and data literacy at different grades. The progression would include concepts in statistics and data literacy, but also computer science—to be able to use common programming and tools used by data professionals—and more philosophical concepts, such as the ethical use of statistics and privacy protections.

Education Week Researcher Alex Harwin contributed to this report.

EDC的海洋数据研究所正在为不同级别的统计数据和数据文化建立学习进度。该研究所的负责人兰迪·科切沃（Randy Kochevar）说，他们是基于缩写词clip，这意味着学生可以学习如何使用：

复杂的、多变量的数据（“我们不只是看日照时间和豆类植物的高度，”他说）；

比学生回答任何一个问题所需的数据集更大，因此他们必须对相关性进行分类和理解；

以交互方式访问数据，而不是以书面形式写出的示例图；以及

专业收集的数据迫使学生思考如何和为什么收集数据以及样本中可能存在哪些偏差。

相反，芬泽设想一种更为全面的方法，即每年至少有一节课，无论是数学、生物学，甚至是公民或历史课，都要求学生努力理解大数据集。他说，这种方法“将产生巨大的影响，因为这意味着当你高中毕业时，数据对你来说并不陌生。”

EDC的海洋数据研究所正在为不同级别的统计数据和数据识字建立学习进度。这一进程将包括统计学和数据识字方面的概念，但也包括计算机科学，以便能够使用数据专业人员使用的通用编程和工具，以及更多的哲学概念，例如统计的道德使用和隐私保护。

教育周刊研究员亚历克斯·哈温对此报告做出了贡献。

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