七天課堂：學(xué)習(xí)數(shù)學(xué)的錯(cuò)誤方法.doc

學(xué)習(xí)數(shù)學(xué)的錯(cuò)誤方法！ HERES an apparent paradox: Most Americanshave taken high school mathematics, including geometry and algebra, yet anational survey found that 82 percent of adults could not compute the cost of acarpet when told its dimensions and square-yard price. The Organization forEconomic Cooperation and Development recently tested adults in 24 countries onbasic “numeracy” skills. Typical questions involved odometer readings andproduce sell-by tags. The United States ended an embarrassing 22nd, behindEstonia and Cyprus. We should be doing better. Is more mathematics the answer?這里有個(gè)明顯的悖論：多數(shù)美國(guó)人都學(xué)過(guò)高中數(shù)學(xué)，里面包括了幾何和代數(shù)，然而一份國(guó)家調(diào)查發(fā)現(xiàn)，當(dāng)告訴他們毯子的尺寸和每平方碼的價(jià)格時(shí)，百分之八十二的成年人都算不出一張?zhí)鹤踊硕嗌馘X。經(jīng)濟(jì)合作與發(fā)展組織最近對(duì)24個(gè)國(guó)家的成年人的基本“識(shí)數(shù)”能力進(jìn)行了測(cè)試。典型問(wèn)題包括里程表的讀數(shù)和保質(zhì)期標(biāo)簽的制作。美國(guó)最終取得了令人尷尬的第二十二名，在愛(ài)沙尼亞和塞浦路斯之后。我們應(yīng)該做得更好。更多的數(shù)學(xué)課會(huì)解決這個(gè)問(wèn)題嗎？In fact, whats needed is a different kindof proficiency, one that is hardly taught at all. The Mathematical Associationof America calls it “quantitative literacy.” I prefer the O.E.C.D.s“numeracy,” suggesting an affinity with reading and writing.實(shí)際上，我們需要掌握的是另一種能力，這種能力我們基本沒(méi)有想到過(guò)。美國(guó)數(shù)學(xué)協(xié)會(huì)把這稱為“量化讀寫能力?！蔽腋矚gO.E.C.D.的“識(shí)數(shù)，”暗示著它與閱讀和書寫的相似之處。Calculus and higher math have a place, ofcourse, but its not in most peoples everyday lives. What citizens do need isto be comforp reading graphs and charts and adept at calculating simplefigures in their heads. Ours has become a quantitative century, and we mustmaster its language. Decimals and ratios are now as crucial as nouns and verbs.微積分和高等數(shù)學(xué)當(dāng)然有一定的地位，但它們并不在大多數(shù)人日常生活里并不常見。民眾們真正需要的是對(duì)閱讀圖表和圖形感到舒服并且能夠?qū)?jiǎn)單的數(shù)字熟練進(jìn)行心算。我們處在一個(gè)量化的世紀(jì)，而我們必須掌握這個(gè)世紀(jì)的語(yǔ)言。小數(shù)點(diǎn)和分?jǐn)?shù)如今就像名詞和動(dòng)詞一樣關(guān)鍵。It sounds simple but its not easy. I teachthese skills in an undergraduate class I call Numeracy 101, for which the onlyprerequisite is middle school arithmetic. Even so, students tell me they findthe assignments as demanding as rational exponents and linear inequalities.這聽起來(lái)簡(jiǎn)單但并不容易。我在本科生課程上教授這些技能，我把這課程稱為“識(shí)數(shù)101，”這門課唯一的預(yù)備知識(shí)為中學(xué)算術(shù)。即使這樣，學(xué)生們告訴我，他們發(fā)現(xiàn)課堂作業(yè)就像有理指數(shù)和線性不等式一樣難搞。Im sometimes told that what Im proposingis already being covered in statistics courses, which have growing enrollmentsboth in high schools and colleges. In 2015, nearly 200,000 students were takingadvanced placement classes in statistics, over three times the number a dozenyears ago. This might suggest we are on the way to creating a statisticallysophisticated citizenry.時(shí)不時(shí)有人跟我說(shuō)，我提議的東西現(xiàn)在已經(jīng)涵蓋在統(tǒng)計(jì)課程里了，而統(tǒng)計(jì)課程在高中和大學(xué)的報(bào)名人數(shù)也是越來(lái)越多。在2015年，近乎200,000的學(xué)生參加了統(tǒng)計(jì)學(xué)的大學(xué)欲修課程，比十幾年前多了三倍有余。這個(gè)或許在暗示中我們已經(jīng)在培養(yǎng)著精通統(tǒng)計(jì)學(xué)的民眾了。So I sat in on several advanced placementclasses, in Michigan and New York. I thought they would focus on what could becalled “citizen statistics.” By this I mean coping with the numbers thatsuffuse our personal and public lives like figures cited on incomedistribution, climate change or whether cellphones can damage your brain.Whats needed is a facility for sensing symptoms of bias, questionable samplesand dubious sources of data.于是我在密歇根和紐約旁聽了幾節(jié)大學(xué)預(yù)修課程。我想他們會(huì)把注意力放在一種能被成為“市民統(tǒng)計(jì)學(xué)”的知識(shí)上。我指的是那些用于應(yīng)付充斥在我們個(gè)人和公共生活中的數(shù)字的知識(shí)-比如講到收入分配，氣候改變，抑或是手機(jī)會(huì)不會(huì)損害你的大腦時(shí)會(huì)引用的那些數(shù)字。我們需要的是一種能力，讓我們能感知到偏見的特點(diǎn)，值得質(zhì)疑的樣本，和令人懷疑的數(shù)據(jù)來(lái)源。My expectations were wholly misplaced. TheA.P. syllabus is practically a research seminar for dissertation candidates.Some typical assignments: binomial random variables, least-square regressionlines, pooled sample standard errors. Many students fall by the wayside. Itsnot just the difficulty of the classes. They cant see how such formulasconnect with the lives theyll be leading. Fewer than a third of those enrolledin 2015 got grades high enough to receive credit at selective colleges.我的期待完全落空了。這個(gè)A.P.課程實(shí)際上是論文候選人的研究研討會(huì)。一些典型的作業(yè)：二項(xiàng)式隨機(jī)變量，最小二乘回歸直線，合并樣本標(biāo)準(zhǔn)差。許多學(xué)生退出了。這并不僅僅是因?yàn)檎n程的難度。他們看不到這些公式如何會(huì)和他們將要過(guò)的生活聯(lián)系起來(lái)。2015年參加課程的學(xué)生中只有少于三分之一得到了足夠高的分?jǐn)?shù)去換取自己選擇的學(xué)校的學(xué)分。Something similar occurred when theCarnegie Foundation for the Advancement of Teaching created a statistics coursefor 19 community colleges in 2012. It was advertised as an alternative toremedial algebra, with its sadistic attrition rates. In Statways, as it wascalled, here is some of what students were asked to master: chi-square test forhomogeneity in two-way ps, line multiple representation of exponentialmodels. Even with small classes and extra support, almost half of the studentsgot Ds or Fs or dropped the class.當(dāng)卡內(nèi)基教學(xué)促進(jìn)基金會(huì)于2012年為19間社區(qū)大學(xué)開辦了統(tǒng)計(jì)學(xué)課程時(shí)，相似的事情發(fā)生了。人們把它標(biāo)榜為用于補(bǔ)救代數(shù)學(xué)的替代品，但其學(xué)生流失率慘不忍睹。在這個(gè)被稱為Statways的地方，學(xué)生被要求掌握的知識(shí)里有這么一些：雙向表卡方檢驗(yàn)同質(zhì)性，直線指數(shù)模型的多個(gè)表示方法（注:我完全看不懂這句.）。即使是在小規(guī)模的班級(jí)和擁有額外支持的情況下，幾乎半數(shù)學(xué)生得到D或F的成績(jī)或者直接退出了課程。The Carnegie and A.P. courses were designedby research professors, who seem to take the view that statistics must be doneat their level or not at all. They also know that citizen statistics is not theroute to promotions. In the same vein, mathematics faculties at both highschools and colleges dismiss numeracy as dumbing down or demeaning. In fact,figuring out the real world deciphering corporate profits or what a healthplan will cost isnt all that easy.卡內(nèi)基和A.P.課程都是由研究教授涉及的，他們看起來(lái)認(rèn)為統(tǒng)計(jì)學(xué)只存在于他們那個(gè)層次。他們同樣知道，公民統(tǒng)計(jì)學(xué)并非晉升之道。同樣，高中和大學(xué)的數(shù)學(xué)教師蔑視識(shí)數(shù)，把這看成淺顯的或者是有辱身份的。事實(shí)上，理解這個(gè)真實(shí)的世界-算出公司受益或者一份健康計(jì)劃的成本-不總是那么簡(jiǎn)單。So what kinds of questions do I ask mystudents?那么我會(huì)問(wèn)我的學(xué)生什么問(wèn)題呢？One exercise focuses on visualizing data. Ihave the class prepare a report on how many households in the United Stateshave telephones, land and cell. After studying census data, they focus on two:Connecticut and Arkansas, with respective ownerships of 98.9 percent and 94.6percent. They are told they have to choose one of the following charts torepresent the numbers, and defend their choice.一項(xiàng)訓(xùn)練專注于可視化數(shù)據(jù)。我讓全班同學(xué)準(zhǔn)備一份報(bào)告，內(nèi)容是全美有多少戶家庭擁有電話，包括座機(jī)和手機(jī)。在研究過(guò)人口調(diào)查數(shù)據(jù)之后，他們把注意力集中在兩個(gè)地方上，康涅狄格州和阿肯色州，在這兩個(gè)地方，電話的擁有率為百分之98.9和94.6。他們被告知，他們需要在以下的表格中選擇一份去表現(xiàn)出這些數(shù)字，并且要為他們的選擇做辯護(hù)。The first chart suggests a much biggerdifference, but is misleading because the bars are arbitrarily scaled toexaggerate that difference.第一份表格顯示了更大的差距，但這有誤導(dǎo)性，因?yàn)槟切┲浑S意設(shè)定大小而夸大了差距。I also ask them to discern and analyzechanging trends. Each January, the National Center for Health Statisticsreleases its hefty “Births: Final Data.” Its rates and ratios range from theages of parents to methods of delivery. I ask students to scan these columns,looking for patterns. They found, for example, that women in Nebraska areaveraging 2.2 children, while Vermonts ratio is 1.6. Any theories?我同樣告訴他們?nèi)ケ鎰e和分析變化的趨勢(shì)。每年一月，國(guó)家健康統(tǒng)計(jì)中心都會(huì)發(fā)布其沉重的“出生：最終數(shù)據(jù)。“由雙親年齡到接生方法，各種數(shù)據(jù)的變化速率和比值。我要求學(xué)生去細(xì)看這些欄，去尋找規(guī)律。比如，他們發(fā)現(xiàn)，內(nèi)布拉斯加州的女性平均擁有2.2個(gè)孩子，然而佛蒙特州的比例是1.6。有什么理論可以解釋？Other ps focus on changes over time.Fertility rates for white and black women in 1989 stood at 60.5 and 84.8 perthousand, a discernible difference. By 2014, they were 59.5 and 64.5, a muchsmaller gap. Theres a story here about how black women are reconfiguring theirlives.別的表格則專注于隨時(shí)間的變化。1989年白人女性和黑人女性的生育率分別為每千人60.5和84.8，一個(gè)引人注意的差別。在2014年之前，這兩個(gè)量的數(shù)值是59.5和64.5，差距變小了。這其中有一段關(guān)于黑人女性如何重塑她們生活的故事。Finally, we talk about how math can help usthink about reorganizing the world around us in ways that make more sense. Forexample, theres probably nothing more cumbersome than how we measure time: Howquickly can you compute 17 percent of a week, calibrated in hours (or minutes,or seconds)? So our class undertook to decimalize time.最后，我們談?wù)摿藬?shù)學(xué)如何能幫助我們?nèi)ニ伎几幸饬x地去重組我們身邊的世界。比如，世界上可能什么東西比計(jì)量時(shí)間更艱難了。你怎樣才能快速算出一周的百分之十七，以小時(shí)為單位（或分鐘，或秒）？然后我們班開始對(duì)時(shí)間進(jìn)行十進(jìn)制化。Imagine if we had a 10-day week, each dayconsisting of 10 hours each. The class debated whether to adopt a three-dayweekend, or to locate an “off-day” in midweek. Since a decimal week would have100 hours, 17 percent is a flat 17 hours no calculator required. You have tothink both numerically and creatively if you want to, say, chuck out ourcurrent health care system and model the finances of a single-payer plan.想象一下，我們一周有十天，每天十個(gè)小時(shí)。這個(gè)班討論要不要采用每周三天制，或在每周中間放置一個(gè)“休息日?！币?yàn)橐粋€(gè)十進(jìn)制周會(huì)有100個(gè)小時(shí)，百分之十七就是整整十七個(gè)小時(shí)-不需要計(jì)算器。你必須考慮數(shù)字，如果你愿意的話，還可以天馬行空，比如說(shuō)，否決掉我們?nèi)缃竦尼t(yī)保系統(tǒng)并為只有一人付款的醫(yī)保計(jì)劃的財(cái)政建模。Mathematicians often allude to “the law ofthe excluded middle” (a proposition must be true or false). The same phrasecould be applied to a phenomenon in our own backyard. We teach arithmetic quitewell in early grades, so that most people cando addition through division. We then send students straight togeometry and algebra, on a sequence ending w