Question 2B: All rhombuses are parallelograms

This statement is also a true statement since a rhombus can be defined as a parallelogram with congruent sides. Twenty-two students said that the statement was true. Of these twenty-two

students, six of them drew pictures along with their answers. Four students thought that the statement was false. The following are representative examples from each group of responses. I will start with a representative example of a student who gave a satisfactory response.

Anna gave a response that was correct without any drawings. The following is her written work:

Figure 39: Anna's response to All Rhombuses are Parallelograms

Anna succinctly shares the most pertinent characteristic of rhombuses for this problem: “a rhombus has 2 set of parallel sides.” Since all parallelograms have two sets of parallel sides, her logic holds. Without a drawing, Anna has interiorized the characteristics of a rhombus and parallelograms. Her response is indicative of a process conception of the hierarchical nature of rhombuses and parallelograms.

Mary’s answer is an example of a response that uses drawings. Her work is below:

Mary uses a hierarchical definition that “a rhombus is a kite + parallelogram.” This definition means that all the properties of a kite and a parallelogram reside in a rhombus. Therefore, a rhombus must also be a special case of a parallelogram. Mary uses a drawing to illustrate how all four sides must be congruent in this special parallelogram. Mary’s response is indicative of at least an action conception of the hierarchical nature of rhombuses and parallelograms.

Three of the students who answered “false” came from the same interview group and two of them built a similar argument for why they thought the statement was false but eventually changed their minds:

Lydia: I put not because a kite could also be a rhombus, right? So it could be similar to a kite and it doesn’t have parallel sides?

Interviewer: Ok

Julie: I put false because all rhombuses are kites and a rhombus is a special kind of kite. Some rhombus…. rhombuses could be parallelograms but not all of them. I said some but not all.

Amity: I said false. It can be a square or a rectangle. Interviewer: A square?

Amity: Uh huh

Interviewer: And is a square a parallelogram? Amity: I’m not sure

Interviewer: So how did you define a parallelogram? What did you say? Amity: At least on pair…

Interviewer: Parallelogram? Amity: …of parallel sides, yeah. Interviewer: and what about rhombus?

Amity: Rhombus does not have parallel sides. It doesn’t. Interviewer: Does not?

Amity: No Lydia: I… Interviewer: ok

Julie: Parallel means that they don’t meet and rhombuses they tend to meet…ok…no never mind. Hold on.

Lydia: A Kite

Julie: Yeah. Kites meet each other but rhombuses…I mean look at a rhombus a different way it looks like a parallelogram so never mind, they are parallelograms.

Lydia and Julie built their arguments around how rhombuses are also kites. In their responses, they considered kites as separate objects from parallelograms. However, they did not consider that the rhombus was an object that was both a kite and a parallelogram. Similarly, Amity argues that rhombuses do not have parallel sides, so they cannot be parallelograms. Yet when Julie turns her drawing of a rhombus around she realizes that rhombuses also look like parallelograms as well. Julie’s work is below:

Julie’s original picture of a rhombus is a prototypical diamond-shaped figure. Her parallelogram is also a prototypically oriented parallelogram. However, when she turned her rhombus in the same orientation as her parallelogram, she realized that all rhombuses are parallelograms. Her image of a rhombus was rigid to a specific orientation. Consequently, her final correct response indicates at least an action conception of the hierarchical relationship between rhombuses and parallelograms.

Sarah also thought that the statement was false. Like the other students above she thought that rhombuses do not have parallel sides:

Interviewer: ok the next…all rhombuses are parallelograms

Sarah: I said false because rhombuses don’t have… not all the sides are parallel of a rhombus.

Sarah also included a counterexample for a drawing:

Figure 42: Sarah's response for All Rhombuses are Parallelograms

Sarah’s counterexample is a rectangle drawn with tic marks for parallel sides. Perhaps she thought she needed a figure that had parallel sides which was obviously not a rhombus. Her idea that the opposite sides of a rhombus are not parallel may stem from her prior definition of a rhombus: “a special type of kite resembles a diamond all lengths are the same.” Sarah included a picture with her definition:

Figure 43: Sarah's definition of a rhombus

Sarah’s drawing is a prototypical diamond. However, her drawing focuses only on the four congruent sides and is not drawn carefully to show opposite parallel sides. Consequently, Sarah does not consider that the opposite sides need to be parallel. Sarah’s response illustrates a pre- action conception of the hierarchical nature of rhombuses and parallelograms.

The next section reports on the analysis of a question concerning rhombuses and kites.