Interview question 1b

Question 1b of the interview asked students to determine whether g(x) = ±√4−x2 _rep- resents a function. Out of fifteen students, twelve rejected g as a function with correct reasoning. One of these twelve required prompting from the interviewer. Two students re- jected g as a function but gave incorrect reasoning. One student incorrectly affirmed g as a function.

At the beginning of the interview, Ron gave a correct definition of function. He said, “okay so a function being for every input there is only one, only one output.” However, he did not apply this definition in question 1b. As a consequence, he incorrectly determined that g is a function.

R: And then (b) um I saw the same thing because once you solve down, it’s the same thing. You have two different systems of square roots, but once you solve down for x2 you’ll still only get one function for every uh x.

When Ron said,“you have two different systems of square roots,” he seems to describe distributing the radical symbol to each of the terms in 4− x2_{. It should be mentioned}

that Ron was the first student in his interview to state his definition of function and to give a response to question 1a, which was correct. Despite this, Ron’s response to question 1b suggests that he cannot reason about more complicated functions. Therefore, he was considered to have no higher than an action conception of function.

Recall from the previous section that Frank gave an incorrect definition of function, which caused him to incorrectly determine that ρ(θ) = θ in question 1a does not represent a function. During his discussion of question 1b, he said he remembered the definition of function, after hearing the response of another student.

F: Now that I think about it, now that she said it, I actually remember the definition of a function that the, uh, professor gave us.

I: Is part (b) a function? Yes or no? And how does your definition relate? F: Well, my definition does not relate to any of these now. Now that I think

about it, since it’s like, you said a function would give you just one input, uh, output rather, um, this is a plus or minus problem. It would give you two inputs. So it couldn’t be a function.

Frank attempted to answer question 1b based on hismemory of the function definition. He determined thatg is not a function, but his reasoning was not correct. He said, “it would give you two inputs.” Twice in this excerpt, Frank confused the words “input” and “output” which indicates that he is still at the pre-action level for function.

Griffin gave a correct definition of function by stating, “uh, a function is, every input only has one output,” but he appeared to confuse his definition with the definition of one-to- one in his discussion ofg(x) =±√4−x2 _{in question 1b . The following excerpt occurs right} after another student explains how the notation “g(x)” made him think of the rectangular coordinate system instead of the polar coordinate system.

G: I think also because since it was x, it kind of made me think of rectangular instead of . . . I’m not sure what it would look like on a polar graph. With that as being rectangular, then I viewed it as not being a function cause I think since its squared that you would um there would be two . . . You can

get the same input . . .I mean you can get the same output for inputs. I: Mmhm. So is that what makes it a function or not a function, if you can get

the same output for different inputs?

G: I look at it as the vertical line test. So on a rectangular graph . . .So if you put in one um . . . if there is more than one answer for an x, then it’s not a function.

I: Okay. So if like x= 1, what do you get for (b)?

G: For (b), we get . . .plus or minus √3. And then if you put um negative, well, if you put −1 in there too, then you would get the same thing.

In the first sentence of the above excerpt, Griffin explained that the notation “g(x)” made him think in terms of the rectangular coordinate system. The rest of his response indicates that he concludedg is not a function, because of the possibility of getting the same values of g(x) for different values of x. This appears to concern him more than the multiple values of g(x) for one value of x. This is in spite of his accurate function definition and his correct interpretation of the vertical line test. According to Dubinsky and Harel (1992), confusion between the definition of function and definition of one-to-one can only be resolved with a process conception of function (see also Breidenbach et al., 1992).

One student, Whitney, required prompting on question 1b before answering correctly. Her definition of function was, “I think that if you have, um, if you plug in an input, it’ll give you one output.” The following excerpt contains her reasoning aboutg(x) = ±√4−x2_.

I: Did you think it was a function? Yes or no?

W: I think it was . . . I feel like if I plug in any number, it will also give me an answer.

I: Okay. So what are you plugging into? x?

W: Yeah. If a plug a number into x, it would give me an output. I: And how many outputs will you get for each x you plug in? W: So, that would be two. It doesn’t represent a function [laughs].

±√4−x2 _{represents a function because it would “give her an answer” if she plugged in} a number for x. It was not until the interviewer prompted her to consider the number of outputs that she realized thatg is not a function. This is suggestive of an action conception of function.

The remaining eleven students correctly rejected g as a function and gave a response that indicated a process conception of function. For example, Hannah gave her definition of function by stating, “um, that each input of a function has one and only one output. So there should be only one value resulting from each input.” She reasoned that g is not a function by stating:

H: I said (b) was not a function because for every x there would be both a positive and a negative answer, and so that’s two outputs, which means it’s not a function.

Hannah’s definition of function and response to question 1b suggest a process conception of function because she was able to describe the getting “both a positive and a negative answer” for every x without needing to actually perform the calculations.

This section reported on students’ reasoning about real-valued functions given in the form y = f(x). The purpose of this section was to provide at least a rudimentary sense for how the students in this study reason about functions in general. The follow- ing section reports on how students’ reason about parametric functions given in the form p(t) = (f(t), g(t)).