本研究之目的在瞭解國三學生對數型線性組合命題的論證表現並探究其進行形式論證的認知障礙。並根據這些認知障礙及相關文獻發展教學策略以提升學生的數型命題論證能力。
學生主動建構的數型命題的論證方式主要可分成經驗論證、敘述說明及形式演繹等三類。本研究首先對兩個來自不同學校的國三班級，進行論證能力及論證偏好問卷的施測，由施測結果選取4位學生進行測後訪談以初步瞭解經驗論證的學生及敘述說明的學生進行形式論證之認知障礙。而後再由兩個班級中分別選取主動建構論證與論證偏好均為經驗論證及敘述說明的學生各一名，及論證偏好為形式演繹且主動建構論證時為敘述說明的學生各一名，組成兩個小組進行教學訪談活動。
針對本研究之研究目的，本研究主要研究結果如下：
(一)國三學生進行演繹證明的認知障礙主要應為：(1)未察覺數字例的侷限性(2)表達理由能力不足(3)無法形式的表徵數型(4)無法對代數表徵進行有意義的推論。
(二)國中學生數型命題論證的學習路徑可為：經驗論證到敘述說明，再到形式演繹。學生論證方式由經驗論證到敘述說明的主要認知需求為能察覺經驗論證的不足、能察覺兩數型間的關係及能以語文表達命題中數型的一般性。由敘述說明到代數演繹證明的認知需求為能形式的表徵數型且能物化表徵，以代數表徵進行運算推理。
(三)批判經驗論證有效性的活動是啟動學生開始考慮進行其他一般性論證方式的必要過程。透過評論及比較各種論證方式的有效性，能促使原本認為經驗論證即足夠驗證命題之真偽的學生，逐漸察覺經驗論證僅對所舉的例子有效，而認知到在證明命題時須採取更一般性的論證方式。
(四)數型樣式的一般化活動能增進學生對數型結構的理解，有助於學生進行利用數型樣式的結構說明相關數型的性質，但對學生使用代數表徵進行推論可能無直接的幫助。
(五)透過區辨及分類不同的代數表徵並討論其意涵，能發展學生對代數表徵式的認知，將有助於其將命題中數型表徵成代數式。但研究結果亦發現，雖然表徵的辨識活動有助於學生寫出數型的代數表徵式，但若要使學生能物化此符號表徵，而能以此表徵進行有意義的操弄及推論，仍有待進一步的教學介入。

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