# «PROCEEDINGS OF THE 15TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION EDITORS STACY BROWN SEAN LARSEN KAREN MARRONGELLE ...»

Cosine - Inverse Cosine Identities We collected questionnaire data from 45 trigonometry, pre-calculus, and calculus students, We collected questionnaire data from 45 trigonometry, pre-calculus, and calculus students, taught by two instructors, Elizabeth (trigonometry, pre-calculus) and Emmet (calculus). We taught by two instructors, Elizabeth (trigonometry, pre-calculus) and Emmet (calculus). We asked the students to (1) explain what the xxin the intervals meant and (2) why the first statement asked the students to (1) explain what the in the intervals meant and (2) why the first statement had the values -1 and 11and the second the values 00and pi. Because of the complex nature of this had the values -1 and and the second the values and pi. Because of the complex nature of this statement, we anticipated that students would need to coordinate several foundational notions in statement, we anticipated that students would need to coordinate several foundational notions in order to be able to answer the two prompts successfully (Thompson, Carlson, & Silverman, order to be able to answer the two prompts successfully (Thompson, Carlson, & Silverman, 2007) (a detailed concept map illustrating the various concepts involved will be provided in the 2007) (a detailed concept map illustrating the various concepts involved will be provided during the presentation). In follow-up interviews with ten students (4 trigonometry, pre-calculus, 4 4 longer paper).

statement, f-1(f(x)) ==x, and then gave the students the task of explaining the statement in the box.

In an individual interview Elizabeth commented on her own explanation of the meaning of fStudent Responses (f(x)) = x and the connection to the statement in the box. In a joint interview, we asked Emmet The students interviewed were recruited from thethe prompts in our task. Bothbands as defined and Elizabeth to anticipate students’ answers to high- and low-achievement instructors by the two quite complete third week ofinvolving several ideas: the ten participants were in the produced teachers in the explanations class, but only three of (1) this statement is a particular case of f-1(f(x)) = x; (2) Thus, the interviewed student sample includes mostly high-achieving low-achievement range. trigonometric functions are periodic, and not one-to-one; so one must students the functions 21.7, sd =inverses; (3) when dealing students’ interviews function undoes the restrict (mean age = to obtain 6.57). Our analysis of the with inverses, “one using Balacheff’s model of conceptions one obtains an x; (4) the different values2010) revealed that their from the other” which is why (Balacheff, 1998; Balacheff & Gaudin, in the two intervals stem understanding of the statement is basedare composed.conceptionsthe instructors, however, different order in which the functions on particular Neither of about composition, inverse functions, injectiverestrictions infunctions, operated range, and angle measures. In particular, we indicated that the (one to one) each line domain, differently: While the restriction in the first have evidence of students’ difficulties in:function can be calculated, the restriction in the second line is needed to ensure that the inverse line is needed in order to ensure that the equality holds (one does not need to restrict cos(x)). In our sample ofTH students interviewed, only one student had this realization.

15 Annual Conference on Research in Undergraduate Mathematics Education 503

1. Identifying composition as an operation between functions (including interpretations of inverses under composition as multiplicative inverses). (Charlie1, Cathy, Thomas, Tony)

2. Recognizing that the identity for the operation of composition is f(x) = x, and thus that a bijective function composed with its inverse results in that identity. (Tina, Cathy)

3. Interpreting the inverse of trigonometry functions, in particular the need to restrict the function so that it is one-to-one so and can have an inverse. (Carl, Tracy)

4. Recognizing the nature of the statement as a statement of truth and the role of the restrictions for making that statement true. (Peggy, Carl, Corey)

5. Managing multiple representations. (Carl, Peggy, Paul)

6. Choosing examples to justify a statement, without attending to the correctness of the example. (Cathy)

7. Using and interpreting radians, degrees, angles, axis, and periods. (Carl, Peggy, Corey, Paul) The following excerpt illustrates some of these issues, regarding the identification of

**composition, restricting input values, and the selection of examples:**

Cathy: This [line 1] is saying that the domain for the inverse cosine is in between negative one and one and this [line 2] is saying that the domain of the cosine is between zero and pi because this is the one that we are evaluating first here and this is the one that we are evaluating first here. The inverse cosine is giving you like one over cosine where x is the inverse cosine of x [writes 1/cos(x) = cos-1(x)]. So in doing that you end up with like an indeterminate function if you have your value outside of this [the intervals].

Notice that although Cathy states that “this is the one we are evaluating first,” she means to calculate cos(x) first, then take the reciprocal. In this case, restricting x was associated with avoiding a value [1/0] that “does not exist… it gives you error messages because you can’t divide by zero”.

In cases in which the students recognized the composition, they used notions of domain to

**interpret their meaning of the restrictions in the statement:**

I: What happens when x is not between one and negative one?

Tina: Then it’s not a function. I don’t think. I don’t, no, it’s not a function. Doesn’t work.

I: And between zero and pi?

Tina: Same with, yeah.

I: And when you say it’s not a function.

Tina: that, that equation doesn’t work. Mathematically, it’s not provable (pause) Like that number isn’t a possible answer. Whatever number is plugged in.

I: And why is it not possible?

Tina: Because it doesn’t fall between the negative one and one. If it fell, like, if it was two, whatever is plugged in wouldn’t be a possible answer.

Teachers’ Anticipations of Students’ Responses In an individual interview prior to collecting the student interview data, Elizabeth commented on her own explanation of the meaning of f-1(f(x)) = x and the connection to the statement in the box. In a joint interview, conducted after we collected the student interview data, we asked Pseudonyms were chosen so that the first letter would identify the course in which the student was enrolled, Trigonometry, Precalculus, or Calculus.

504 15TH Annual Conference on Research in Undergraduate Mathematics Education Emmet and Elizabeth to anticipate students’ answers to the prompts in our task. Both instructors produced quite complete explanations involving several ideas: (1) this statement is a particular case of f-1(f(x)) = x; (2) trigonometric functions are periodic, and not one-to-one; so one must restrict the functions to obtain inverses; (3) when dealing with inverses, “one function undoes the other” which is why one obtains an x; (4) the different values in the two intervals stem from the different order in which the functions are composed. Neither of the instructors, however, indicated that the restrictions in each line operated differently: While the restriction in the first line is needed to ensure that the inverse function can be calculated, the restriction in the second line is needed in order to ensure that the equality holds (one does not need to restrict cos(x)). In our sample of 10 students interviewed, only one student had this realization.

Teachers’ Reactions and Interpretations of Students’ Understanding of the Task At the time of the data collection Elizabeth had seven years of college teaching experience, while Emmet had 16. We presented the teachers with summaries of data from their students’ written questionnaires and the student interviews. Both teachers were surprised to read the students’ responses and engaged in a search for explanations for why students could have such conceptions. The teachers thought that the results were indicators of larger issues with the curriculum. In particular with when and how functions are introduced and how inverses are taught. The instructors did not deal with the possibility that the curriculum may be set up to obscure that composition can be seen as a binary operation between functions and that inverses are functions that give the identity function, when composed with each other. Both instructors suggested that the problems might lie in the college algebra course, which is a pre-requisite for both trigonometry and calculus but indicated hope that a recent change of textbooks and course organization would better address this in the future. By suggesting these interpretations, the teachers appear to recognize institutional obligations more easily than disciplinary, individual, or interpersonal obligations (Herbst & Chazan, 2011). In other words, instructors did not produce justifications that could be tied to the complexity of the mathematics (for them the mathematics is too simple), differences among their students, or to their shared space during teaching. Instead the externally imposed curriculum is seen both as root of the students’ difficulties and also as solution to address it.

Questions for the audience

1. The analysis of the practical rationality can be complemented with an analysis of teachers’ mathematical knowledge for teaching. What types of analyses could we perform of our teacher data to address that? We have thought about using the scholarship on noticing (Sherin, Jacobs, & Philipp, 2011) and on teachers’ knowledge (Ball, Thames, & Phelps, 2008), and will bring some initial analyses. We have also thought about pursuing a linguistic analysis about the way in which teachers position themselves vis a vis the students and the content (Martin & White, 2005; Mesa & Chang, 2010; Wagner & Herbel-Eisenmann, 2008).

It is still unclear to us what would be the nature of the claims we could do with these analyses.

2. What are possible ways to use this research to inform a faculty development program that would push teachers into thinking about students’ understanding and the role it plays in teaching?

15TH Annual Conference on Research in Undergraduate Mathematics Education 505 References Balacheff, N. (1998). Meaning: A property of the learner-milieu system. Grenoble.

Balacheff, N., & Gaudin, N. (2010). Modeling students' conceptions: The case of function.

Research in Collegiate Mathematics Education, 16, 183-211.

Ball, D. L., Thames, Mark, & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407.

Herbst, P., & Chazan, D. (2011). Research on practical rationality: Studying the justifications of actions in mathematics teaching. The Mathematics Enthusiast, 8(3), 405-462.

Martin, J. R., & White, P. R. R. (2005). The language of evaluation: Appraisal in English. New York: Palgrave-Macmillan.

Mesa, V., & Chang, P. (2010). The language of engagement in two highly interactive undergraduate mathematics classrooms. Linguistics and Education, 21(2), 83-100.

Mesa, V., & Herbst, P. (2011). Designing representations of trigonometry instruction to study the rationality of community college teaching. ZDM The International Journal on Mathematics Education, 43, 41-52.

Sherin, M. G., Jacobs, V. R., & Philipp, R. (2011). Mathematics teacher noticing: Seeing through teachers' eyes. New York: Routledge.

Thompson, P. W., Carlson, M., & Silverman, J. (2007). The design of tasks in support of teachers' developmet of coherent mathematical meanings. Journal of Mathematics Teacher Education, 10, 415-432.

Wagner, D., & Herbel-Eisenmann, B. (2008). "Just don't": The suppression and invitation of dialogue in the mathematics classroom. Educational Studies in Mathematics, 67, 143

Preliminary Research Report We report initial findings of a study that seeks to investigate the changing nature of instructors’ concerns as they learn to teach mathematics courses using inquiry-based learning approaches.

Using year-long data from interviews with faculty and bi-monthly teaching logs, we seek to describe the concerns of instructors teaching with this method. Our initial analysis of pilot data with four instructors new to the method suggests that these concerns are organized into five major themes: Student preparation, motivation, and engagement; Coverage; Rigor; Difficulty of the material; and Student Learning. Additionally, the nature and relative frequency of these concerns seem to suggest that these faculty are more preoccupied with managerial aspects of teaching and less with student learning, consistent with a proposed developmental model of professional expertise in teaching. We seek input on the instrument used to gather the data as current results might be consequential to the organization of the instrument.

**Keywords: inquiry-based learning, teaching expertise development, instruction**