臆測取向教學能建構學生在面臨高層次數學問題時的臆測認知展現。本研究採個案研究法，選擇就讀新竹市實驗國小六年級學生且三位皆是通過國小一般智能資賦優異鑑定的資優生為研究對象，分析三位在臆測認知元素下做共同與差異性的綜合性比較以及在教學前、臆測取向教學中、教學後臆測認知的臆測認知元素與歷程的展現情形。
研究結果發現，在有特殊化的認知元素下，因其簡化題目的複雜度，能幫助解題者啟發對題目的熟悉感，但需要的是進一步的觀察、辨別數量關係與結合所學習的數學相關知識連結，方能繼續往前探索。學生在探索問題階段，運用系統化認知的元素，對於解題的認知有其正向助益。在類比向度中，發現學生類比認知元素與特殊系統化會成為一連動的關係，亦即藉由特殊系統化的例子，察覺對應相似概念與結構的相關數學知識結合。在教學後的驗證階段中，學生有意識對初步的猜想，會想要主動的驗證更一般化的可能性或有缺漏的不確定性，試圖使自己可以有信心的相信猜想正確性。小六生雖然不會用數學式的演繹證明，但其證明一般化的概念是可以建立學習認知的。對於未來數學學習的證明有其助益。在臆測認知元素的思維導向中，能使得學生對題目的條件分析與資訊的探索顯得更完整，其能做到一般化證明的認知可能性相對提高許多。
因此本研究認為，臆測取向教學對高年級資優生的數學臆測認知學習有其助益，使學生能在面臨高層次思考的數學問題展現解題的思維歷程，對學習數學能影響學生的思維模式。

Conjecture-oriented teaching can construct students' cognitive performance in conjecture when facing high-level mathematics. This study adopted the case study method and chose three gifted students, who were sixth-graders at Hsinchu Experimental Elementary School and had been identified as exceptionally gifted children, as the study subjects. The three were comprehensively compared with each other to see their commonality and difference, as well as the display of cognition and experience before, during and after the conjecture-oriented teaching.
The study results showed that under the specialized cognitive elements, simplifying the complexity of problems can inspire solvers' familiarity with problems. However, further observation and identifying quantitative relations and the connection with the math knowledge learned are required for further exploration. Adopting elements of systemized cognition for the exploration of problems has positive benefits in students' cognition of problem-solving. With respect to the analogy dimension, it is found that students' analogy and specialized and systemized can become a linked relation, that is, the examples of specialized and systemized can integrate related mathematical knowledge similar in concepts and structures. In the post-teaching justifying the conjecture phase, students are conscious of preliminary conjectures and will want to proactively justify more generalized possibilities or uncertainties, trying to convince themselves to confidently believe in the correctness of conjectures. Sixth-graders may not adopt mathematical dedction, but it proves that generalized concepts can be learned, and it can be helpful to the proof of future mathematical learning. In a conjecture-oriented way of thinking, students can show more completeness of their analysis of problems and exploration of information, further improving the cognitive possibilities of generalized proof.
Therefore, this study believes that, conjecture-oriented teaching can help gifted students in their senior year in terms of their conjecture learning of mathematics. Students can demonstrate their problem-solving thinking process when facing high-level math problems, and the way of thinking can impact how they learn mathematics.

壹、緒論 1
第一節 研究背景與動機 1
第二節 研究目的與待答問題 3
第三節 名詞釋義 3
第四節 研究範圍與限制 3
貳、文獻探討 4
第一節 數學臆測認知歷程 4
第二節 數學臆測教學模式實徵性研究 13
第三節 臆測模式取向 15
第四節 補習班的數學學習與教學相關研究 18
參、研究方法 20
第一節 個案研究法 20
第二節 研究對象與場域 20
第三節 研究架構 21
第四節 研究流程 23
第五節 研究設計 25
第六節 資料來源與分析 40
肆、研究結果 44
第一節 三位個案認知元素的綜合性比較 44
第二節 探索階段、形成猜想階段 46
第三節 驗證猜想階段 78
第四節 證明猜想階段 88
伍、結論與建議 98
第一節 研究結果與討論 98
第二節 未來研究的教學和建議 101
參考文獻 102
第一節 中文文獻 102
第二節 英文文獻 103
附錄 106

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