影響六年級學生立方體計數表現的因素--空間定位與視覺化的角色

Exploring the Factors that Influence Sixth Graders' Cubic Enumeration: The Roles of Spatial Orientation and Visualization

立方體計數作業常被用來測量空間能力，也是數學課程中空間與圖形領域的學習活動，本研究以六年級學生為受試者，探討空間定位與視覺化能力對該作業表現的影響。材料分為高、低規律兩類型，低規律題操弄隱藏立方體個數（分4、5、6、7 個隱藏個數），隱藏立方體分佈的零散程度（分二向度及三向度）。高規律題分為外形完整與不完整兩類。研究採紙筆團體施測收集204 名學生的正確率，以及個別施測測與訪談收集40 名學生的正確率、解題時間、主觀難度的評定，與解題策略。結果，學生解題的正確率、時間，和主觀難度，在高規律題上的表現優於低規律題，隱藏向度二的表現優於向度三，而隱藏個數的效果並不明顯，且立方體外形的完整與否並不影響正確率與解題時間，顯示影響學生立方體計數表現的因素，除了Ben-Haim 等人（1985）與Battista 和Clements（1996， 1998）所主張的，從2D 圖形轉換成3D 的表徵理解能力，以及協調各視點以對隱藏部份立方體產生心像的定位能力之外，對六年級學生而言，群組與挪移等動態心像操弄的視覺化能力更為關鍵。

The task of enumerating the number of 3-D cube arrays is used to evaluate and develop student’s spatial ability. This research investigates the roles of spatial orientation and visualization of sixth graders in cube enumeration. The material included two types of cube arrays – low regularity and high regularity. We tested two independent variables in low regularity arrays – number and distribution of hidden cubes. The former had four levels (i.e.， 4， 5， 6， or 7) of hidden cubes. The latter were divided into two axes and three axes according to the degrees of the hidden cubes. High regularity arrays were divided into outside intact and non-intact types depending on whether the appearance is integral or not. Paper-and-pencil questionnaire was group-administered to collect the hit rate of 204 students. Individual interviews were also conducted to gather data on hit rate， response time， difficulty rating， and strategies from 40 students. Results showed that high regularity arrays yielded better performance than low regularity arrays， and two axes condition yielded better performance than three axes condition. The effects of the number of hidden cubes are not as steady. There are no significantly differences in hit rate and response time between the intact and non-intact groups. Current literature (Battista et al.， 1996， 1998， & 1999; Ben-Chaim et al.， 1985) has claimed that students’ difficulty in cubes enumeration is due to deficiencies of mental imagery or orthogonal coordination. However， we argue that visualization is more influential than spatial orientation.