教育部考試中心以正式文件明確提出在高考數(shù)學(xué)考題中要體現(xiàn)數(shù)學(xué)文化，這是一個(gè)明顯的信號(hào)：要求學(xué)生加強(qiáng)對(duì)數(shù)學(xué)文化知識(shí)的學(xué)習(xí)，自覺(jué)地、有針對(duì)性地重視對(duì)數(shù)學(xué)文化修養(yǎng)的提升。
本書(shū)創(chuàng)新性地從以數(shù)化人與人數(shù)學(xué)化兩個(gè)角度來(lái)談數(shù)學(xué)文化，對(duì)數(shù)學(xué)文化在數(shù)學(xué)歷史、數(shù)學(xué)精神、數(shù)學(xué)應(yīng)用、數(shù)學(xué)之美、數(shù)學(xué)語(yǔ)言、交匯拓展、數(shù)學(xué)游戲等各個(gè)主題上都作出了全面透徹的剖析和論述。除此之外，作者還結(jié)合歷年高考數(shù)學(xué)試題進(jìn)行具體的案例分析，以此讓高考生能夠精準(zhǔn)消化吸收高考中的各種數(shù)學(xué)文化知識(shí)。
對(duì)高中生而言，這是一部讓他們明確數(shù)學(xué)文化是什么怎么考怎么練，為備考數(shù)學(xué)文化提供清晰學(xué)習(xí)方向的紅寶書(shū)；對(duì)數(shù)學(xué)教學(xué)工作者或研究者而言，這是一部系統(tǒng)揭示命制背景、展示解題過(guò)程、凸顯數(shù)學(xué)文化價(jià)值的寶貴內(nèi)參書(shū)；對(duì)于廣大的數(shù)學(xué)文化愛(ài)好者而言，這又是一部讓人崇尚真理，凈化心靈，提升文化修養(yǎng)和文化品格的數(shù)學(xué)美書(shū)。
★為什么要學(xué)習(xí)數(shù)學(xué)文化？其實(shí)這個(gè)問(wèn)題也等同于為什么要學(xué)習(xí)數(shù)學(xué)。數(shù)學(xué)本身就是一種文化，所以這本書(shū)旨在幫助學(xué)生形成良好的數(shù)學(xué)思維能力，掌握解決問(wèn)題的方法，培養(yǎng)理性精神。
★現(xiàn)在的高考試題越來(lái)越注重考生解決問(wèn)題的能力，倘若只會(huì)刷題而不能理解題中真意將會(huì)寸步難行。本書(shū)以文化為內(nèi)核，將理論、方法、歷史、文化融為一體，知識(shí)性、趣味性和實(shí)操性完美結(jié)合。
★數(shù)學(xué)家顧沛高度贊賞的數(shù)學(xué)文化典范之作，中國(guó)科學(xué)院院士陳佳洱、王大中聯(lián)袂誠(chéng)意推薦，適合高中生及教學(xué)工作者使用。
2016
年10 月8 日，教育部考試中心下發(fā)《關(guān)于2017 年普通高考考試大綱修訂內(nèi)容的通知》（教試中心函〔2016〕179 號(hào)），增加了數(shù)學(xué)文化的要求。這一文件的公布，引發(fā)了社會(huì)對(duì)數(shù)學(xué)文化的極大關(guān)注，成為教育界的熱門(mén)話題。
事實(shí)上，當(dāng)人們不滿足于對(duì)數(shù)學(xué)本身的研究，開(kāi)始從外部審視數(shù)學(xué)的價(jià)值時(shí)，對(duì)數(shù)學(xué)文化的研究就開(kāi)始了。人們開(kāi)始思考數(shù)學(xué)是什么與數(shù)學(xué)的價(jià)值是什么，并逐漸認(rèn)識(shí)到數(shù)學(xué)已經(jīng)成為人類文化的重要組成部分，在過(guò)去已經(jīng)對(duì)人類文明做出了巨大的貢獻(xiàn)，在未來(lái)也必將做出更大的貢獻(xiàn)。
把數(shù)學(xué)作為文化來(lái)進(jìn)行研究，國(guó)外要比我們?cè)绾芏�，比較有代表性的是美國(guó)的數(shù)學(xué)家懷爾德與數(shù)學(xué)史專家莫里斯·克萊因。懷爾德認(rèn)為數(shù)學(xué)是一個(gè)由內(nèi)在力量與外在力量共同作用，而處于不斷發(fā)展和變化之中的文化系統(tǒng)。數(shù)學(xué)文化是由文化傳統(tǒng)和數(shù)學(xué)本身所組成。這些觀點(diǎn)在他的兩部著作《數(shù)學(xué)概念的演化》和《作為一種文化系統(tǒng)的數(shù)學(xué)》中得以充分體現(xiàn)。莫里斯·克萊因在他的著作《西方文化中的數(shù)學(xué)》《數(shù)學(xué)：一種文化探索》《數(shù)學(xué)與知識(shí)的探求》中則系統(tǒng)論述了數(shù)學(xué)對(duì)西方文化、理性精神、現(xiàn)代人類思想發(fā)展所產(chǎn)生的影響。他側(cè)重于數(shù)學(xué)與各種文化以及社會(huì)因素之間相互作用的分析。我國(guó)開(kāi)展數(shù)學(xué)文化研究是近二三十年的事情，時(shí)間雖短，但是相關(guān)研究也取得了極大進(jìn)展。徐利治探討數(shù)學(xué)一般意義上存在的某些相同或不同的方法，開(kāi)創(chuàng)了數(shù)學(xué)方法論的研究與教學(xué)；鄭毓信教授在他的《數(shù)學(xué)文化學(xué)》中基于數(shù)學(xué)哲學(xué)觀點(diǎn)構(gòu)建數(shù)學(xué)文化學(xué)的理論體系，嘗試從理論層面論述數(shù)學(xué)作為文化的內(nèi)涵；齊民友在《數(shù)學(xué)與文化》一書(shū)中則論述了中西數(shù)學(xué)文化的差異。
我國(guó)教育主管部門(mén)對(duì)數(shù)學(xué)文化也很重視。早在2003 年，教育部頒布的《普通高中數(shù)學(xué)課程標(biāo)準(zhǔn)（實(shí)驗(yàn)）》中就突出強(qiáng)調(diào)了數(shù)學(xué)的文化價(jià)值數(shù)學(xué)是人類文化的重要組成部分，對(duì)數(shù)學(xué)文化給予了特別重視，要求數(shù)學(xué)文化貫穿整個(gè)高中數(shù)學(xué)課程并融入教學(xué)中。這次教育部考試中心又再次以發(fā)布正式文件的方式對(duì)數(shù)學(xué)文化重點(diǎn)提及，明確提出要在高考試題中考查數(shù)學(xué)文化，足以看出國(guó)家對(duì)數(shù)學(xué)文化的重視。
以往提及數(shù)學(xué)文化，大多停留在理論的層面，而且很多研究?jī)H僅把史實(shí)、案例、故事等稍作整理，將數(shù)學(xué)史等同于數(shù)學(xué)文化。這種做法不過(guò)是給數(shù)學(xué)文化穿了件數(shù)學(xué)史外衣而已。類似的研究總給人以不接地氣之嫌，沒(méi)有引起人們的觸動(dòng)。但這次之所以引發(fā)對(duì)數(shù)學(xué)文化思考與研究的熱潮，主要原因是考試主管部門(mén)從考試命題的角度提出了要求，第一次正式地明確要求把數(shù)學(xué)文化滲入數(shù)學(xué)試題，所以未來(lái)高考數(shù)學(xué)命題肯定會(huì)遵照?qǐng)?zhí)行。雖然我們提了很久的數(shù)學(xué)文化，但如何在數(shù)學(xué)試題中體現(xiàn)出來(lái)，對(duì)很多人來(lái)說(shuō)還是比較陌生的。數(shù)學(xué)文化本來(lái)是個(gè)籠統(tǒng)的概念，傳統(tǒng)意義上文化在文科中出現(xiàn)得較為普遍，在文科類的命題中是很容易做到的。而數(shù)學(xué)是邏輯的科學(xué)、思維的科學(xué)，如何把文化滲透其中，是一個(gè)擺在廣大數(shù)學(xué)教育工作者面前的新課題。正是在這樣的背景下，我根據(jù)多年積累的豐富資料和大量的研究工作，捉筆成書(shū)，對(duì)相關(guān)問(wèn)題作出了系統(tǒng)的回答。
《高考中的數(shù)學(xué)文化》一書(shū)共分八章。第一章為數(shù)學(xué)文化總論，主要闡述了數(shù)學(xué)文化的含義、內(nèi)容和基本特性等問(wèn)題；后面七章分別為數(shù)學(xué)歷史、數(shù)學(xué)精神、數(shù)學(xué)應(yīng)用、數(shù)學(xué)之美、數(shù)學(xué)語(yǔ)言、交匯拓展、數(shù)學(xué)游戲，遴選近十五年高考數(shù)學(xué)中涉及數(shù)學(xué)文化的試題進(jìn)行分類賞析，系統(tǒng)揭示命制背景，展示解題過(guò)程，凸顯數(shù)學(xué)文化價(jià)值。全書(shū)盡可能把數(shù)學(xué)文化出題的方向一網(wǎng)打盡，讓讀者明確數(shù)學(xué)文化是什么怎么考怎么練，為備考數(shù)學(xué)文化提供清晰的復(fù)習(xí)方向。同時(shí)每個(gè)主題后面提供與主題相關(guān)的數(shù)學(xué)文化練習(xí)題，方便讀者練習(xí)鞏
固使用，從而更有針對(duì)性地復(fù)習(xí)。本書(shū)的編寫(xiě)具有實(shí)用性、針對(duì)性和時(shí)效性三個(gè)鮮明特點(diǎn)，密切結(jié)合當(dāng)下學(xué)生、教師對(duì)數(shù)學(xué)文化試題認(rèn)識(shí)相對(duì)陌生的客觀實(shí)際，對(duì)高考中曾經(jīng)出現(xiàn)的數(shù)學(xué)文化試題進(jìn)行分類賞析，系統(tǒng)闡述最新考綱考查數(shù)學(xué)文化的內(nèi)涵，同時(shí)為學(xué)生備考提供科學(xué)性的建議。
我國(guó)對(duì)數(shù)學(xué)文化的系統(tǒng)研究時(shí)間不長(zhǎng)，尤其是數(shù)學(xué)文化與試題的結(jié)合研究更是不成熟。此書(shū)盡管對(duì)數(shù)學(xué)文化的理論探討與應(yīng)用研究取得了較大突破，然而也不免出現(xiàn)不完善或者紕漏之處。但我相信，此書(shū)的問(wèn)世將會(huì)受到廣大同行的重視和學(xué)生的歡迎。在此，我也懇請(qǐng)讀者批評(píng)雅正。
齊龍新，北京市數(shù)學(xué)特級(jí)教師，山東省教學(xué)能手，山東省新課程研修團(tuán)隊(duì)專家，山東省基礎(chǔ)教育教師培訓(xùn)專家，山東省優(yōu)質(zhì)課一等獎(jiǎng)、教育部一師一優(yōu)課一等獎(jiǎng)獲得者，現(xiàn)為北京市育英學(xué)校經(jīng)濟(jì)學(xué)實(shí)驗(yàn)班班主任。作者致力于新數(shù)學(xué)自學(xué)輔導(dǎo)教學(xué)法的實(shí)踐與研究，所編自學(xué)教材曾獲得山東省優(yōu)秀課程資源一等獎(jiǎng)，在日常教學(xué)實(shí)踐中也取得了非常顯著的教學(xué)效果。其微信個(gè)人公眾號(hào)龍新數(shù)學(xué)(zxfd66)曾經(jīng)獲得2016年數(shù)學(xué)文化雜志社主辦的攜手北大數(shù)學(xué)文化節(jié)最紅公眾號(hào)評(píng)選全國(guó)第二名。
把數(shù)學(xué)教得通俗易懂，讓學(xué)生學(xué)得幸福有趣是作者一以貫之的精致追求，其教學(xué)風(fēng)格獨(dú)特，教法細(xì)膩、技巧純熟，教學(xué)成績(jī)突出，與學(xué)生之間亦師亦友，深得學(xué)生的喜愛(ài)。作者教學(xué)之余寫(xiě)出了數(shù)十篇教育教學(xué)論文，撰寫(xiě)了《初高中數(shù)學(xué)銜接教材》等多部教研論著。
第一章 數(shù)學(xué)文化總論 ··················································································· 1
第一節(jié) 什么是數(shù)學(xué)文化 ········································································ 1
第二節(jié) 數(shù)學(xué)文化的四個(gè)層面 ································································· 3
一、知識(shí)系統(tǒng) ······················································································· 4
二、工具系統(tǒng) ······················································································· 4
三、價(jià)值系統(tǒng) ······················································································· 5
四、史實(shí)系統(tǒng) ······················································································· 7
第三節(jié) 數(shù)學(xué)文化的人本特性 ································································· 7
一、精神特性 ······················································································· 8
二、物質(zhì)特性 ······················································································· 8
三、行為特性 ······················································································· 9
四、審美特性 ······················································································· 9
第二章 數(shù)學(xué)歷史 ························································································ 11
第一節(jié) 數(shù)學(xué)名著 ················································································ 11
一、《九章算術(shù)》 ················································································ 12
二、《數(shù)書(shū)九章》 ················································································ 19
三、《算數(shù)書(shū)》 ··················································································· 24
第二節(jié) 數(shù)學(xué)故事 ················································································ 27
一、畢達(dá)哥拉斯的故事 ········································································· 27
二、高斯的故事 ·················································································· 36
三、哥德巴赫的故事 ············································································ 41
第三節(jié) 數(shù)學(xué)名題 ················································································ 44
一、米勒問(wèn)題 ····················································································· 45
二、斐波那契數(shù)列 ··············································································· 57
三、阿基米德窮竭法 ············································································ 66
四、柯西不等式 ·················································································· 73
五、角谷猜想 ····················································································· 80
六、阿波羅尼奧斯圓 ············································································ 85
七、回文數(shù) ························································································ 93
八、勾股數(shù) ························································································ 98
九、平均數(shù) ······················································································· 101
十、皮克定理 ···················································································· 108
十一、數(shù)字黑洞 ················································································· 114
十二、伯努利不等式 ··········································································· 119
十三、正整數(shù)方冪和 ··········································································· 124
十四、切比雪夫多項(xiàng)式 ········································································ 130
十五、馬爾科夫定理 ··········································································· 135
十六、四色問(wèn)題 ················································································· 143
十七、萊布尼茨三角形 ········································································ 146
十八、黃金分割比 ·············································································· 151
第三章 數(shù)學(xué)精神 ······················································································· 155
第一節(jié) 理性求真 ··············································································· 155
第二節(jié) 創(chuàng)新意識(shí) ··············································································· 167
第四章 數(shù)學(xué)應(yīng)用 ·······················
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