A Cognitive Research Into Cross-Sections Of Cubes Drawn By 9～(th)-11～(th) Grade Students

The research chiefly probes into four hundred and eighty six 9th—11th grade students' cognitive characteristics of various cube sections. The results of the research show: I. With regard to drawing cube arbitrary cross-sections, it's easier for these students to draw triangular and quadrangular sections closely related to daily life than those of pentagon and hexagon. (1) 9th grade students are more inclined to draw quadrangular cross-sections, immediately followed by triangular cross-sections, which is a rather amusing phenomenon. Very few students can correctly draw pentagonal and hexagonal cross-sections. Even a few mistakenly assume that there exist cross-sections about heptagon, octagon and enneagon. (2) 10th grade students have preliminarily acquired the ability to draw arbitrary cross-sections concerning triangle and quadrangle, and they can draw pentagonal arbitrary cross-sections better than those of hexagon. The same is true of 11^th grade students. The chief reason is that they have already considered the midpoints on the corresponding cube edges, and that they are more accustomed to thinking about questions from the perspective of symmetry. (3) Compared with 10th grade students, those ones from 11th grade have no obvious advantage in the perception of triangular and quadrangular cross-sections, but they do better in drawing pentagonal and hexagonal cross-sections. II. Generally speaking, as for drawing cube cross-sections across relatively easier three points on a known edge, the percentage of correctness increases with age. Particularly for 10^th grade students, in drawing cross-sections about quadrangle, pentagon and hexagon they have achieved a great increase in the correctness percentage than 9^th grade ones. With respect to drawing hexagonal cross-sections, the correctness percentage of 11^th grade students doubles that of 10^th grade ones. Regarding the cross-sections of triangle, quadrangle, pentagon and hexagon, students of different grades can always draw triangular ones best; there are the approximate percentages of correctness for 11^th grade students to draw pentagonal and hexagonal cross-sections. III. As for drawing cube cross-sections across more complex three points on a known edge, 9^th -11^th grade students can always draw triangular cross-sections best, followed by drawing those of quadrangle, and they do worst in drawing pentagonal and hexagonal cross-sections; but some students still have some misconceptions of drawn cross—sections, which stems from the difficulty in cognizing diagrams. For students of different grades, their developmental level and developmental speed differ. The students, who can correctly draw quadrangular, pentagonal and hexagonal cross-sections, ground mainly on the proposition: "If two parallel planes simultaneously intersect with the third plane, then their intersecting lines parallel. " Very few 10^th grade students can draw cross-sections according to "the ductibility of the planes", while there is 12% for 11^th grade ones. IV. In the samples investigated in the paper, there exist gender differences between male and female students in the cognition of various cross-sections of cubes, on the whole, male students from different grades have a better understanding of various cube cross-sections than female students; but for different cross-sections, and male and female students of different grades, the degree of difference varies, and generally speaking, gender differences gradually narrow with the increase of grade.Based on the research results, the paper finally puts forward some pertinent suggestions.