探討高一生對數學敘述與代數式證明之邏輯理解
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2015
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Abstract
本研究的目的是了解高一學生「對數學敘述有效性的判斷」、「理解代數式推理與證明的表現」是否顯著地受到「對數學敘述真實性的認知」的影響，並採量的研究方法。
本研究對象為台南地區某一高中的240位高一學生，學生的國中成績PR值約在86至96之間。研究結果顯示：
1.學生「認為兩敘述是否同時為真」和「判斷兩敘述的有效性」間的關聯性並不顯著，也就是說以高一學生對真實性的認知，並未能有效預測其對有效性的判斷。
2.對兩敘述有效性判斷的表現，「在幾何知識上」與「在代數知識上」沒有顯著差異。
3.真實性p1→p2,p2→p3的敘述，而認為p1→p2,p2→p3敘述為真的高一學生中，在多項方程式敘述上和在不等式敘述上，分別有4%與53%的學生將推理有效性「p1→p2,p2→p3」誤以為是「p1←→p3」，兩者有顯著差異。
4.在判斷代數式證明過程的有效性上，平均約有84%的學生誤以為一個「若p則q」的證明可以同時證了「p→q」和「q→p」。
5.約有95%的學生無法正確判斷循環論證是無效的。
從測驗結果顯示，高一學生對敘述真實性的認知雖然和其判斷敘述的有效性沒有顯著關聯，但仍會影響其判斷某一證明的有效性。並且，學生對敘述有效性的判斷優於對證明結果有效性的判斷，又優於對循環論證是無效證明的判斷。
This research aims to realize if the judgement of the validity of mathematical conditional statements (on logical implications) and the performance of understanding algebraic reasoning and proving for tenth-graders are strongly influenced by cognition of the truth of mathematical conditional statements. The quantitative research method is adopted in this study. The subjects in this research are two hundreds and forty tenth-graders of a senior high school in Tainan, whose knowledge levels in junior high school were between PR 86 and PR 96 or so. The main results are: 1.The tenth-graders' understanding of the truth of two mathematical conditional statements does not correlate strongly to their judgement of the validity of two mathematical conditional statements, that is, we can not predict students' judgement of the validity of mathematical conditional statements by their cognition of the truth of mathematical conditional statements. 2.About the tenth-graders' performance of judgement of the validity of two mathematical conditional statements, there is no significant difference between which of the knowledge geometrically and which of the knowledge algebraically. 3.Among the tenth-graders who think that the statements of p1→p2,p2→p3 are true, the ratio of regarding valid reasonings p1→p2,p2→p3 together as p1←→p3 (false), through the Kruskai-Wallis' test, is more in the conditional statement of polynomial equality (53%) than in the conditional statement of polynomial inequality (4%), and there is significant difference between them. 4.In the judgement of the validity of proving about proof process of algebraic statements, there are 84% of the tenth-graders answering wrongly that the process-proving content of the question which p→q is concerned proves not only p→q, but also q→p . 5.95% of the tenth-graders or so are not able to judge correctly that the circular argument is invalid proving. The result of research tests says that although tenth-graders' cognition of the truth of mathematical conditional statements does not correlate strongly to their judgement of the validity of two mathematical conditional statements, it still influences their judgement of the validity of proving. Moreover, students' judgement of the validity of two mathematical conditional statements is superior to the judgement of the validity of proving, and both of the two kinds of judgement are superior to the judgement that the circular argument is invalid proving.
This research aims to realize if the judgement of the validity of mathematical conditional statements (on logical implications) and the performance of understanding algebraic reasoning and proving for tenth-graders are strongly influenced by cognition of the truth of mathematical conditional statements. The quantitative research method is adopted in this study. The subjects in this research are two hundreds and forty tenth-graders of a senior high school in Tainan, whose knowledge levels in junior high school were between PR 86 and PR 96 or so. The main results are: 1.The tenth-graders' understanding of the truth of two mathematical conditional statements does not correlate strongly to their judgement of the validity of two mathematical conditional statements, that is, we can not predict students' judgement of the validity of mathematical conditional statements by their cognition of the truth of mathematical conditional statements. 2.About the tenth-graders' performance of judgement of the validity of two mathematical conditional statements, there is no significant difference between which of the knowledge geometrically and which of the knowledge algebraically. 3.Among the tenth-graders who think that the statements of p1→p2,p2→p3 are true, the ratio of regarding valid reasonings p1→p2,p2→p3 together as p1←→p3 (false), through the Kruskai-Wallis' test, is more in the conditional statement of polynomial equality (53%) than in the conditional statement of polynomial inequality (4%), and there is significant difference between them. 4.In the judgement of the validity of proving about proof process of algebraic statements, there are 84% of the tenth-graders answering wrongly that the process-proving content of the question which p→q is concerned proves not only p→q, but also q→p . 5.95% of the tenth-graders or so are not able to judge correctly that the circular argument is invalid proving. The result of research tests says that although tenth-graders' cognition of the truth of mathematical conditional statements does not correlate strongly to their judgement of the validity of two mathematical conditional statements, it still influences their judgement of the validity of proving. Moreover, students' judgement of the validity of two mathematical conditional statements is superior to the judgement of the validity of proving, and both of the two kinds of judgement are superior to the judgement that the circular argument is invalid proving.
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敘述與逆敘述, 演繹推理, 證明的有效性, statement and its converse, deductive reasoning, validity of proving