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數學基礎教育:三個饅頭,還是三明治?
作者:智慧數學研究所 陳士文 點擊數:1852 更新時間:2016年8月23日 文章錄入:智慧數學
   

數學基礎教育:三個饅頭,還是三明治?

智慧數學研究所   陳士文

本文緣起一則趣聞。

某年某月某日,某地。

中國數學教育家顧泠沅遇見美國卡耐基基金會主席舒爾曼,他們談起了數學教育。

顧泠沅說,我們中國有個寓言:有一個和尚吃饅頭,吃了第一個饅頭沒有飽,又吃第二個,第二個吃下去還不飽,接著吃第三個,這時吃飽了。于是和尚發表議論,說早知道第三個饅頭能吃飽,何必要吃第一個、第二個饅頭呢。

舒爾曼說,在我們美國,有一個教育家叫加涅,加涅也認為教學好似三個饅頭,第一個是概念;第二個是概念與概念之間的連接,即原理;第三個是解決問題的策略。

舒爾曼還說,美國人吃三明治,三明治是可以夾起來吃的,不同的年齡有不同的夾法,講究其營養。

……

趣聞引發了我關于教育、關于基礎、關于數學的思考:

一、何為教育?

教育有廣義和狹義之分,在教育學界,關于“教育”的定義多種多樣,可謂仁者見仁、智者見智。

“大學之道,在明明德,在親民,在止于至善。”(孔子語)

“教育是幫助被教育的人給他能發展自己的能力,完成他的人格,于人類文化上能盡一分子的責任,不是把被教育的人造成一種特別器具。”(蔡元培語)

“教育是依據生活、為了生活的生活教育,培養有行動能力、思考能力和創造力的人。”(陶行知語)

……

關于教育的話題,除了引經據典,還可以追古溯源,還可以暢想規劃,限于篇幅,我只是直接去想想教育的本質。

教育的本質有五種之說,即:上層建筑說;生產力說;特殊范疇說;多重屬性說;意識替代說。

一時半會,我是說不清楚教育的本質是什么了,那轉個角度看看教育是干什么的?我試著從教育的功能來理解教育。

教育的功能大致可分為個體發展功能與社會發展功能。教育最首要的功能是促進個體發展,包括個體的社會化和個性化。

既如是,那教育乃是奠定“學生發展”與“人格成長”的基礎。

二、何為基礎?

基礎指建筑底部與地基接觸的承重構件,它的作用是把建筑上部的荷載傳給地基。

這可能是基礎的本意,留意工程建設時發現:

“基礎”是慢的,打樁、排水、灌漿……“基礎”施工未曾見過一周一層樓的速度。

“基礎”是寬厚的,“基礎”的長寬遠遠超過地面建筑物的長與寬,而“基礎”的厚度也決定了建筑物的高度。

“基礎”是不裝潢的,它深埋于地下,樸素堅韌,從不追求外在的富麗堂皇。

有俗語:基礎不牢,地動山搖;基礎不平,房倒屋傾。在語言的發展過程中,“基礎”引申意為事物發展的根本或起點。

難道事物發展的根本或起點也需要“慢”、也需要“寬厚”、也是“不裝潢”嗎?

既有“基礎教育”一詞,難道這些和教育有關嗎?

三、何為基礎教育?

基礎教育,狹義講是指九年義務教育,而九年義務教育是國家的法律規定。

顯然,基礎教育是國家法律意志,必須接受,但接受的不僅僅是冷冰冰的條文規定。

基礎教育是一種享受的教育,它是免費的,表面上免去的是經濟支出,實質真正享受的應該是也必須是精神的平等。

基礎教育是人生的起點,它需要“慢的”、“寬厚的”基礎,這基礎對于每一個最年輕的公民(小孩)來講,應該是均衡的。

基礎教育面對的是蒙童,此時受教育的個體差異是不顯著的,不必有精英教育和大眾教育之分。

四、數學基礎教育,是吃饅頭,還是吃三明治?

回到文章開頭的趣聞

數學基礎教育是小學數學教師面臨的現實性話題,教育家顧泠沅和加涅借用寓言和譬喻來表述,把學術性問題通俗化了,顯得輕巧有趣,給人們以啟示。

啟示一:數學基礎教育,要慢慢來,三個饅頭一個一個地吃,只吃第三個饅頭是不行的,基礎教育的過程不可短斤少兩,不可急功近利。

啟示二:數學基礎教育,要講次序,第一個饅頭是概念,第二個饅頭是原理,第三個饅頭是策略,基礎教育的內容不可顛三倒四,不可隨意組合。

啟示三:數學基礎教育,要注重改造,三明治是可以夾起來吃,講究其營養,基礎教育的方式不可照本(學術形態的數學)宣科,不可脫離兒童。

當然,任何比喻都是有缺陷的。

三個饅頭僅僅是數量的疊加嗎?知識概念、關系原理、解決問題三者之間不是割裂的。三個饅頭有其基本的共同的營養,還應有各自的營養素。同樣是饅頭,可以是小麥饅頭、蕎麥饅頭、燕麥饅頭,抑或除了饅頭,還有燒賣、包子、油糕等。

關于東西方的數學教育,我們不能停留在故事譬喻上,不是簡單的三個饅頭的巧合,也不是三明治品種的變化,改造數學絕不是做饅頭、三明治那么簡單。一段趣聞無法承載我們對數學教育的理論思考,應該有更為深刻的闡述,我看到史寧中先生在《關于數學的反思》一文中有這樣一段話:

一個贊美神明,偏愛抽象、辯論及邏輯;一個崇尚自然,依賴經驗、實用及運籌。毫無疑問,這些都深刻地影響了數學的產生和發展。我們已經論及到,無論是古希臘之路,還是古中國之路,都不可能引導數學走得很遠。因為一個缺乏外部世界實踐的活力,一個需要內部世界邏輯的動力。

古代中國人善于講故事、作譬喻,如寓言故事:守株待兔、掩耳盜鈴,但缺少抽象的概念和定義。相應的西方哲學則較早的抽象出教條主義、主觀主義的概念,并在思辨中建立學派理論。盡管中國還有“刻舟求劍”“鄭人買履”“疑鄰竊斧”“罰人吃肉”等類似的故事笑話,但仍停留在樸素的自然階段。

至于此,關于數學基礎教育,絕不是三個饅頭和三明治的故事譬喻所能闡釋的,數學基礎教育呼喚相應的數學教育理論。

學術形態的數學,實用主義的數學,他們都不是數學基礎教育的主體與目的,數學基礎教育需要對學術化的數學進行改造。

教育數學的理論因而滋生了。

   
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