Parallelograms

Parallelograms have three special cases: rectangles, rhombuses, and squares. The difficulty in this question is increased from other problems because there are three special cases to remember. Also, the hierarchical nature of special quadrilaterals means that the square is also a special case of both the rhombus and the rectangle. Thus, a square could be considered not a direct special case but rather another level down on the hierarchical development of the special quadrilaterals. Consequently, students must consider special cases (e.g. the square) of special cases (e.g. rhombus or rectangle).

Only eight students correctly stated all the special cases for a parallelogram: rectangle, rhombus, and square. Many students (fourteen) gave an incomplete answer for the special case of a parallelogram. Seven students gave a single response instead of listing all three figures: Rectangle (two), Rhombus (four), and Square (one). A few students recognized two of the figures: Rectangle and Square (six) and Rectangle and Rhombus (one). One student said there were no special cases for a parallelogram. Three students did not answer the question.

All the students who responded to this question correctly did not draw any pictures on their papers. Two of the students who were correct in the oral response of the interview did have some errors on their paper; one student added trapezoids and the other student left off rhombuses. Furthermore, all eight students had a correct definition of a parallelogram. Consequently, the responses of these eight students are indicative of a process conception of the special case of a parallelogram. The responses of the remaining eighteen students in the interview who had various errors exemplify a pre-action conception of the special case of a parallelogram.

4.3.1.2 Rectangles

A rectangle has only one special case out of the standard options for special quadrilaterals: square. Since there is only one correct answer to the special case of a rectangle, there were no incomplete answers. Twenty-one students gave the correct response of square. However, there were four wrong answers: Parallelogram or trapezoid (1), Square and some kites (1), None (1), and Rhombus (1). Only one student did not answer the question.

All the students who correctly answered squares did not draw any pictures with their work. The responses of these twenty-one students are indicative of process conceptions of a special case of a rectangle. The incorrect response of “parallelogram or trapezoid” reversed the idea of special case and the student finds a more generic quadrilateral. The response of “square and some kites” accurately lists the square as a special case, but then the student mentions “some kites” which are not hierarchically related to the rectangle. The student who said “rhombus” also tried to connect a figure that is not directly related to a kite. The fourth incorrect response of “none” gives an example of a student who could not discern the characteristics of a square that are inherited from a rectangle. All four of these responses are indicative of a pre-action conception for the special case of a rectangle.

The next section is a report of the analysis of the special case of a rhombus.

4.3.1.3 Rhombuses

Like rectangles, there is only one special case of a rhombus (a square), so there were no incomplete answers given. Twenty-three students gave the correct answer without any drawings. Their responses are indicative of a process conception for the special case of a rhombus. Two students gave the answer of a kite as their special case for a rhombus. Their responses show a reversal of not finding a true special case but rather a more general quadrilateral. These two

incorrect responses exemplify a pre-action conception for the special case of a rhombus. One student chose not to respond.

The next section reports on the analyses for the special cases of a kite.

4.3.1.4 Kites

Kites have two special cases: rhombuses and squares. Like rectangles, students have to think of two hierarchical levels of special cases since squares are the special case of rhombuses. Consequently, several students struggled answering the special case of a kite. Only nine students gave correct responses (Rhombus and Squares). The nine correct responses were indicative of the process conception of the special cases of a kite. Some students (7) did give incomplete responses: Rhombus (5) and Square (2). These students did not consider all the possible special cases that were allowed for kites. Of the remaining six incorrect responses, five of the students replied with “none” and one student said “ASA,” which may refer to the Angle Side Angle Theorem for Triangle Congruence. All thirteen incorrect responses exemplify a pre-action conception of the special cases of a kite. Four of the responses were left unanswered.

The next section reports on the analysis of the special case of a square.

4.3.1.5 Squares

Many students (fifteen) answered that the square had no special case. Their responses exhibit evidence of a process conception of the special case of a square. Of the eight incorrect responses, six students mentioned “rhombus”, one student mentioned “rectangle”, and one student listed “rhombus, kite, rectangle, trapezoid, and parallelogram.” All eight students show a misinterpretation of the meaning of special case and they found more general quadrilaterals. These responses demonstrate a pre-action conception of the special case of a square. Three students did not have a response.

The next section reports on the analysis of the special case of an inclusively defined trapezoid.

4.3.1.6 Trapezoids (Inclusive)

The inclusive definition of a trapezoid includes parallelograms and all figures that are special cases of parallelograms: rhombuses, rectangles, and squares. Students must conceptually consider three hierarchical levels of special cases: squares are special cases of rhombuses and rectangles; rhombuses and rectangles are special cases of parallelograms. Consequently, the inclusive definition of a trapezoid was the most difficult for students to find all correct special cases; only three students answered this problem correct. These three responses indicate a process conception of the special cases of an inclusively defined trapezoid. Yet fourteen students had some various partially correct answers: five students stated “parallelograms”, two students only said “rectangle”, two students included both “parallelogram and rectangle”, one student said “square and rectangle”, one student said “rectangle, rhombus, and square”, one student said “all trapezoids, rectangle, square, and rhombus”, one student said “parallelogram, rhombus, and square”, and one student said “parallelogram, rectangle, and square.” An additional four students had completely incorrect responses: three students said “none” and one student said “isosceles trapezoids.” All eighteen incorrect responses exhibit a pre-action conception of the special cases of an inclusively defined trapezoid. Five students had no response.

4.3.1.7 Trapezoids (Exclusive)

Students were more successful (twelve) with naming the lack of a special case for an exclusive definition of a trapezoid than finding all the special cases of an inclusively defined trapezoid. The responses of “none” of the twelve students that were correct indicated a process conception of the special case for an exclusively defined trapezoid. Of the four incorrect answers,

two students said “parallelograms”, one student said “trapezoid,” and one student said “parallelogram and rectangle.” These four responses exemplify a pre-action conception of the special case of a trapezoid (exclusive). Ten students did not have a response to this question. The unusually high number of non-responses may show the confusion students had toward what is an exclusively defined trapezoid.

The next section is a summary for all the results for special cases of quadrilaterals.