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教数学的错误方法

  • 日期:2016-06-16   点击:   作者:admin   来源:未知  

The first chart suggests a much bigger difference, but is misleading because the bars are arbitrarily scaled to exaggerate that difference.

最后,我们谈论了数学如何能帮助我们去思考更有意义地去重组我们身边的世界。比如,世界上可能什么东西比计量时间更艰难了,8438.com大丰收心水论坛。你怎样才能快速算出一周的百分之十七,以小时为单位(或分钟,或秒)?然后我们班开始对时间进行十进制化。

我的期待完全落空了。这个A.P.课程实际上是论文候选人的研究研讨会。一些典型的作业:二项式随机变量,最小二乘回归直线,合并样本标准差。许多学生退出了。这并不仅仅是因为课程的难度。他们看不到这些公式如何会和他们将要过的生活联系起来。2015年参加课程的学生中只有少于三分之一得到了足够高的分数去换取自己选择的学校的学分。

实际上,结果通常是那些X项和Y项会让你难以灵巧处理生活中的数字。

当卡内基教学促进基金会于2012年为19间社区大学开办了统计学课程时,相似的事情发生了。人们把它标榜为用于补救代数学的替代品,但其学生流失率惨不忍睹。在这个被称为Statways的地方,学生被要求掌握的知识里有这么一些:双向表卡方检验同质性,直线指数模型的多个表示方法(注:我完全看不懂这句..)。即使是在小规模的班级和拥有额外支持的情况下,几乎半数学生得到D或F的成绩或者直接退出了课程。

I’m sometimes told that what I’m proposing is already being covered in statistics courses, which have growing enrollments both in high schools and colleges. In 2015, nearly 200,000 students were taking advanced placement classes in statistics, over three times the number a dozen years ago. This might suggest we are on the way to creating a statistically sophisticated citizenry.

卡内基和A.P.课程都是由研究教授涉及的,他们看起来认为统计学只存在于他们那个层次,8438.com大丰收心水论坛。他们同样知道,公民统计学并非晋升之道。同样,高中和大学的数学教师蔑视识数,把这看成浅显的或者是有辱身份的。事实上,理解这个真实的世界---算出公司受益或者一份健康计划的成本---不总是那么简单。

My expectations were wholly misplaced. The A.P. syllabus is practically a research seminar for dissertation candidates. Some typical assignments: binomial random variables, least-square regression lines, pooled sample standard errors. Many students fall by the wayside. It’s not just the difficulty of the classes. They can’t see how such formulas connect with the lives they’ll be leading. Fewer than a third of those enrolled in 2015 got grades high enough to receive credit at selective colleges.

The assumption that all this math will make us more numerically adept is flawed. Deborah Hughes-Hallett, a mathematician at the University of Arizona, found that “advanced training in mathematics does not necessarily ensure high levels of quantitative literacy.” Perhaps this is because in the real world, we constantly settle for estimates, whereas mathematics ? see the SAT ? demands that you get the answer precisely right.

我同样告诉他们去辨别和分析变化的趋势。每年一月,国家健康统计中心都会发布其沉重的“出生:最终数据。“由双亲年龄到接生方法,各种数据的变化速率和比值,8438.com大丰收心水论坛。我要求学生去细看这些栏,去寻找规律。比如,他们发现,内布拉斯加州的女性平均拥有2.2个孩子,然而佛蒙特州的比例是1.6。有什么理论可以解释?

微积分和高等数学当然有一定的地位,但它们并不在大多数人日常生活里并不常见。民众们真正需要的是对阅读图表和图形感到舒服并且能够对简单的数字熟练进行心算。我们处在一个量化的世纪,而我们必须掌握这个世纪的语言。小数点和分数如今就像名词和动词一样关键。

时不时有人跟我说,我提议的东西现在已经涵盖在统计课程里了,而统计课程在高中和大学

Something similar occurred when the Carnegie Foundation for the Advancement of Teaching created a statistics course for 19 community colleges in 2012. It was advertised as an alternative to remedial algebra, with its sadistic attrition rates. In Statways, as it was called, here is some of what students were asked to master: chi-square test for homogeneity in two-way tables, line multiple representation of exponential models. Even with small classes and extra support, almost half of the students got D’s or F’s or dropped the class.

It sounds simple but it’s not easy. I teach these skills in an undergraduate class I call Numeracy 101, for which the only prerequisite is middle school arithmetic. Even so, students tell me they find the assignments as demanding as rational exponents and linear inequalities.

I also ask them to discern and analyze changing trends. Each January, the National Center for Health Statistics releases its hefty “Births: Final Data.” Its rates and ratios range from the ages of parents to methods of delivery. I ask students to scan these columns, looking for patterns. They found, for example, that women in Nebraska are averaging 2.2 children, while Vermont’s ratio is 1.6. Any theories?

别的表格则专注于随时间的变化。1989年白人女性和黑人女性的生育率分别为每千人60.5和84.8,一个引人注意的差别。在2014年之前,这两个量的数值是59.5和64.5,差距变小了。这其中有一段关于黑人女性如何重塑她们生活的故事。

Finally, we talk about how math can help us think about reorganizing the world around us in ways that make more sense. For example, there’s probably nothing more cumbersome than how we measure time: How quickly can you compute 17 percent of a week, calibrated in hours (or minutes, or seconds)? So our class undertook to decimalize time.

Calculus and higher math have a place, of course, but it’s not in most people’s everyday lives. What citizens do need is to be comfortable reading graphs and charts and adept at calculating simple figures in their heads. Ours has become a quantitative century, and we must master its language. Decimals and ratios are now as crucial as nouns and verbs.

So I sat in on several advanced placement classes, in Michigan and New York. I thought they would focus on what could be called “citizen statistics.” By this I mean coping with the numbers that suffuse our personal and public lives ? like figures cited on income distribution, climate change or whether cellphones can damage your brain. What’s needed is a facility for sensing symptoms of bias, questionable samples and dubious sources of data.

于是我在密歇根和纽约旁听了几节大学预修课程。我想他们会把注意力放在一种能被成为“市民统计学”的知识上。我指的是那些用于应付充斥在我们个人和公共生活中的数字的知识---比如讲到收入分配,气候改变,抑或是手机会不会损害你的大脑时会引用的那些数字。我们需要的是一种能力,让我们能感知到偏见的特点,值得质疑的样本,和令人怀疑的数据来源。

Mathematicians often allude to “the law of the excluded middle” (a proposition must be true or false). The same phrase could be applied to a phenomenon in our own backyard. We teach arithmetic quite well in early grades, so that most people can do addition through division. We then send students straight to geometry and algebra, on a sequence ending with calculus. Some thrive throughout this progression, but too many are left behind.

这里有个明显的悖论:多数美国人都学过高中数学,里面包括了几何和代数,然而一份国家调查发现,当告诉他们毯子的尺寸和每平方码的价格时,百分之八十二的成年人都算不出一张毯子花了多少钱。经济合作与发展组织最近对24个国家的成年人的基本“识数”能力进行了测试。典型问题包括里程表的读数和保质期标签的制作。美国最终取得了令人尴尬的第二十二名,在爱沙尼亚和塞浦路斯之后。我们应该做得更好。更多的数学课会解决这个问题吗?

那么我会问我的学生什么问题呢?

The Carnegie and A.P. courses were designed by research professors, who seem to take the view that statistics must be done at their level or not at all. They also know that citizen statistics is not the route to promotions. In the same vein, mathematics faculties at both high schools and colleges dismiss numeracy as dumbing down or demeaning. In fact, figuring out the real world ? deciphering corporate profits or what a health plan will cost ? isn’t all that easy.

这个数学会让我们更擅长算术的想法是错的。亚利桑那州立大学的数学家Deborah Hughes-Hallett发现“数学方面的高级训练不一定就会成就高水平的量化读写能力。”也许这是因为在真实世界,我们一直都安于估算,然而数学---看看SAT---要求你给出精准的答案。

这听起来简单但并不容易。我在本科生课程上教授这些技能,我把这课程称为“识数101,”这门课唯一的预备知识为中学算术。即使这样,学生们告诉我,他们发现课堂作业就像有理指数和线性不等式一样难搞。

一项训练专注于可视化数据。我让全班同学准备一份报告,内容是全美有多少户家庭拥有电话,包括座机和手机。在研究过人口调查数据之后,他们把注意力集中在两个地方上,康涅狄格州和阿肯色州,在这两个地方,电话的拥有率为百分之98.9和94.6。他们被告知,他们需要在以下的表格中选择一份去表现出这些数字,并且要为他们的选择做辩护。

想象一下,我们一周有十天,每天十个小时。这个班讨论要不要采用每周三天制,或在每周中间放置一个“休息日。”因为一个十进制周会有100个小时,百分之十七就是整整十七个小时---不需要计算器。你必须考虑数字,如果你愿意的话,还可以天马行空,比如说,否决掉我们如今的医保系统并为只有一人付款的医保计划的财政建模。

So what kinds of questions do I ask my students?

Other tables focus on changes over time. Fertility rates for white and black women in 1989 stood at 60.5 and 84.8 per thousand, a discernible difference. By 2014, they were 59.5 and 64.5, a much smaller gap. There’s a story here about how black women are reconfiguring their lives.


In fact, what’s needed is a different kind of proficiency, one that is hardly taught at all. The Mathematical Association of America calls it “quantitative literacy.” I prefer the O.E.C.D.’s “numeracy,” suggesting an affinity with reading and writing.

的报名人数也是越来越多。在2015年,近乎200,000的学生参加了统计学的大学欲修课程,比十几年前多了三倍有余。这个或许在暗示中我们已经在培养着精通统计学的民众了。

数学通常不经意地涉及“排中之法”(一项主张必须是正确或错误的)。同样的句子对于我们后院的一个现象上也同样适用。我们在每个年级都把算术教得很好,所以大多数人都可以做加减法。我们后来让学生直接接触几何学和代数学,按顺序一直到微积分。一些学生在整个过程都做得很好,但有太多的学生吊车尾了。

HERE’S an apparent paradox: Most Americans have taken high school mathematics, including geometry and algebra, yet a national survey found that 82 percent of adults could not compute the cost of a carpet when told its dimensions and square-yard price. The Organization for Economic Cooperation and Development recently tested adults in 24 countries on basic “numeracy” skills. Typical questions involved odometer readings and produce sell-by tags. The United States ended an embarrassing 22nd, behind Estonia and Cyprus. We should be doing better. Is more mathematics the answer?

Indeed, it often turns out that all those X’s and Y’s can inhibit becoming deft with everyday digits.

第一份表格显示了更大的差距,但这有误导性,因为那些柱被随意设定大小而夸大了差距。

Imagine if we had a 10-day week, each day consisting of 10 hours each. The class debated whether to adopt a three-day weekend, or to locate an “off-day” in midweek. Since a decimal week would have 100 hours, 17 percent is a flat 17 hours ? no calculator required. You have to think both numerically and creatively if you want to, say, chuck out our current health care system and model the finances of a single-payer plan.

One exercise focuses on visualizing data. I have the class prepare a report on how many households in the United States have telephones, land and cell. After studying census data, they focus on two: Connecticut and Arkansas, with respective ownerships of 98.9 percent and 94.6 percent. They are told they have to choose one of the following charts to represent the numbers, and defend their choice.

实际上,我们需要掌握的是另一种能力,这种能力我们基本没有想到过。美国数学协会把这称为“量化读写能力。”我更喜欢O.E.C.D.的“识数,”暗示着它与阅读和书写的相似之处。