摘 要
本研究的目的在探討國中學生學習過方根與π後，對方根和π所具有的本質屬性及特質之概念認知與情意現象、對於方根和π與相關概念聯結時所呈現的思維彈性；以及將方根與π的認知擴充至無理數本質屬性與特質的學習遷移情形。
本研究為描述性研究，主要是透過問卷調查及訪談，收集量與質的資料。此研究以學習過方根和π的國中生為研究對象，樣本共計2077名國中二、三年級的學生。
本研究的主要發現如下：（1）國中學生對方根和π所具有的無理數心像仍有高比例的迷思，例如方根和π是不是大小固定的數？有沒有在數線上？是否等於某有限小數等等。在表達有關方根和π的概念時，只有少數學生能以較完整的敘述表達看法，多數都以片段、不完整的概念來敘述並以簡短的敘述來表達。無理數感知部分，超過半數學生認為無理數存在於身高及時間量中。（2）許多學生對於在圖形幾何、坐標幾何、代數符號等領域的概念中出現無理數並不能接受；至於無理數概念的垂直遷移只有少數學生能夠達到。（3）當學生的思維物件中有無理數時，由於對其本質及特質的不熟悉，多數學生需要將此新數類的思維切換成一具體可掌握的數，而只有少數學生思維彈性的張力可以到達無理數的高階概念；對於π則常以國小經驗的數值憶取作為思維的內容。（4）多數學生對方根和π的存在覺得神秘與驚奇，同時可感受到古人的智慧；也有不少學生面對方根的學習內容覺得無趣，對於生活中出現方根則傾向存疑或盡量避開的態度；但多數學生認同學習方根是為其它數學單元做準備，也覺得應該要學習簡單的根式化簡。雖然只有半數學生滿意自己的學習表現但對自己若用功會有更好的表現有很高的自我信念。學習經驗上，多數學生曾以計算機按出過根號的值；然而雖然約半數教師曾經以計算機檢驗過根式乘法，但仍有不少學生懷疑其等號關係。

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