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五年級“確定位置”三思
作者:智慧數學研究所 陳士文 點擊數:1916 更新時間:2016年10月7日 文章錄入:智慧數學
   

五年級“確定位置”三思

智慧數學研究所   陳士文

關于五年級“確定位置”的教學,我仔細看過同行的教學視頻,認真讀過同行的教學實錄。感受是:嘆服!嘆服!還是嘆服!

嘆服:“行”“列”數學概念建立的巧妙;嘆服:“數對”創寫中對學生的激發;嘆服:“數”“形”結合思想的滲透感悟。

作為一名同行,不能除了嘆服還是嘆服,我的思考呢?

一思、除了建立“行”“列”數學概念,還可以借助“橫”“豎”的生活經驗嗎?

“行”“列”確實是學習“數對”時的輔助概念。首先是表達上的需要,因為要說清楚位置在第幾行第幾列。其次是建構“數對”概念的需要,數對的文字通式是(列,行)。

但實踐中也遇到困擾:

一是生活語言表述順序是“行”“列”,“行”在前,“列”在后,而數對中卻是“列”在前,“行”在后。學生一時不習慣。

二是“列”沿y軸豎著往右數,“行”沿x軸橫著往上數,這與將來在坐標系中找 “點”的方式(先橫軸后豎軸)不一致。

三是現在二維平面有“行”“列”的輔助概念,將來學習到z軸的時候,并沒有與之相隨的名詞。“行”“列”只是曇花一現。(行列式又當別論)

面對困擾,可以借用“橫”“豎”的概念。好處有三:不出現“行”“列”,概念系統簡化;“橫”“豎”學生熟悉,且習慣上“橫”就是從左往右;“橫”“豎”與將來的x軸、y軸相對應,即數對中的(橫,豎)對應(xy)。

鑒于以上思考,在教學“確定位置”時,可以借助“橫”“豎”經驗,規避“行”“列”困擾,直至“數對”在坐標系中的形態。

二思、除了著力“數對”的創寫,還可以有意“數軸”的建構嗎?

關于數對創寫,我發現學生的答案不外乎(橫4,豎3)、(→4,↑3)、(43)幾種,學生探索創寫數對的意義和空間非常有限。對于同一個點,學生可能出現(43)、(34)兩個不同的答案,這是因為生活中位置的確定就有先豎后橫的。例如,住戶編號302,就是指樓房的第三層第二戶。

面對上述狀況,可以有意數軸的建構。一維空間中,依據“從左到右的方向規定及數數”確定位置,這其中“從左到右的方向規定及數數”就是數軸。同樣的道理,依據“從下往上的方向規定及數數”確定位置,這其中“從下往上的方向規定及數數”也是數軸。學生在學習數對之前,已經認識數軸,在此可以有意建構橫豎兩條數軸(x軸、y軸),并將其整合。

當出現(43)、(34)兩個不同的答案時,需要引出一個規定,即數對中兩個數的順序是“先橫后豎”。這是在橫豎兩條數軸整合的基礎上,建構坐標系的規則。

至此,課堂上激發的不僅僅是學生創寫的激情,更有從生活中的位置到數軸的逐步抽象與理性提升。

三思、除了感悟“數”與“形”的結合,還可以探索“二維”到“三維”的空間嗎?

在教學用數對確定位置時,教師們常常講述笛卡爾的故事:

有一天,笛卡爾生病了,躺在床上還在思考一個問題:用什么辦法才能把“點”和“數”聯系起來呢?

笛卡爾凝神望著天花板,他看見屋頂角上有一只蜘蛛,蜘蛛一會兒拉著絲垂下來,一會兒又順著絲爬上去……上、下、左、右地吐絲。如果把蜘蛛看作一個點,它在網上來回移動,能不能把蜘蛛的每個位置用一組數確定下來呢?

蜘蛛的“表演”使笛卡爾豁然開朗:平面上的一個點可以用一組數來表示。

笛卡爾是個勤于思考的人,他創立了坐標系。

故事呈現后,教師們往往有這樣的話語:笛卡爾的思考,使數與形結合了。今天學習的數對,是用“數”來表示“點”。

我覺得此處感悟數形結合有點牽強,因為“點”還不是小學生心中的“形”。對于確定位置,可以找到另一個智慧的增長點,即引領學生從二維空間到三維空間的探索。示例如下:

 

公園大門外的報亭可以用數對表示嗎?



2.怎樣用數對表示正方體頂點A



此時,不在意問題的解決,而有心留給學生探索的接口,打開思維的天窗。我覺得這種探索比數形結合感悟更實在,更讓人深思。

“確定位置”三思是在嘆服同行教育藝術的基礎上而作,欣賞別人,創造自我。最后再贅一句自勵并想與您共勉的話:

借鑒別人的路,走出自己

   
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