以學前診測與自由擬題探討九年級學生的自發性機率概念

Investigation of Ninth Graders'Spontaneous Probabilistic Concepts in Pop Pretests and Problem Posing

臺灣從92課綱起，機率即首次出現於九年級，但是八年級學生參與TIMSS國際評量時，在機率統計主題的表現卻已然很好，似乎透露著學生在未經學校正式的數學科課程之前，已經擁有某種程度的機率知識。本研究因而好奇：未經機率教學的九年級學生，到底具備何種自發性的機率概念？換言之，本研究意欲探討臺灣九年級學生對機率知識之自發性潛力。本研究設計機率學前診測（解題）與自由擬題兩種活動，將機率概念分為三種類型：主觀機率、古典機率、以及頻率機率；在前述的類型上，又辨識出四種概念層次：單一事件、複合事件、獨立性、條件機率，用於區分學生在解題與擬題各自具備的機率知識。結果顯示，未經機率教學的九年級學生，七成以上有能力解出古典機率的單一事件問題，顯示他們具有自發性的機率知識；再從學生所擬題目來分析，發現他們能展現機率概念的每一個層次。本文因而建議機率課程可提早於八年級開始，並宜注意各種機率類型的均衡發展，亦可在九年級導入獨立性與餘事件。 Many students exhibit some knowledge of probability before they have formally been taught.For example, in Taiwan, probability is not taught until the ninth grade, but Taiwan's eighth graders perform above average in the probability category of the Trends in International Mathematics and Science Study. This fact inspired the authors to investigate ninth graders'spontaneous conceptions of probability. In the present study, two activities were designed for data collection: a pop pretest and free problem posing. Two-dimensional analysis was employed to identity the spontaneous concepts demonstrated in the activities. Analysis was based on three probability types: subjective, classic, and frequentist. These three types encompassed four conceptual levels: single event, compounded events, independency, and conditionality. The results revealed that more than 70% of the students could solve single-event classic probability problems before they were formally taught relevant methods. Additionally, the students' spontaneous concepts extended to all conceptual levels. We therefore suggested that the introductory probability curriculum can be implemented in grade eight, and independency and complementary-events could be taught in grade nine.