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Discussion  The proposed paper addresses a general question of what tools can facilitators use to address the challenges of managing public reflections on instruction in research and teacher education settings. We use the context of study group sessions with community college trigonometry instructors as a setting for exploring the work involved in this type of facilitation. These sessions 15TH Annual Conference on Research in Undergraduate Mathematics Education 337 are unique because of their participants, learning goals, and research agenda, but the issues of sharing reflections for the purpose of education and research are shared across contexts.

We situated the work of asking questions to facilitate productive discussions in the “facilitation triangle” (Figure 1). In this triangle the vertices are the facilitator [F], the participants [P], and the representation of teaching [R]. We also recognize that these discussions take place in environments. Important features of the session that promote productive conversations can be located in this triangle, but we are focusing on the arrow between the facilitator and the connection between the participants and the representation of teaching. We see the proposed paper as contributing to understanding these interactions and therefore improving our capabilities to design and enact productive conversations among participants.

Questions for the audience

1. What are other challenges to facilitating sessions where public reflections are managed? 2.

Could different research or learning goals lead to a different type of facilitation? 3.What do you think participants could learn from a session like this? 4. How can the nature of the artifacts used (video, animations of teaching) shape these conversations?

338 15TH Annual Conference on Research in Undergraduate Mathematics Education


Borko H. (2004). Professional Development and Teacher Learning: Mapping the Terrain, Educational Researcher, 33(8), 3-15.

Cohen, D. K., Raudenbush, S., and Ball, D. L. (2003). Resources, instruction, and research.

Educational Evaluation and Policy Analysis. 25, 119-142.

Elliott, R., Kazemi, E., Lesseig, K., Mumme, J., Carroll, C., Kelley-Petersen, M. (2009).

Conceptualizing the Work of Leading Mathematical Tasks in Professional Development, Journal of Teacher Education, 60(4), 364-379.

Hatton, N. & Smith, D. (1995). Reflection in teacher education: towards definition and implementation, Teaching & Teacher Education, 11(1), 33-49.

Herbst, P. and Chazan, D. (2003). Exploring the practical rationality of mathematics teaching through conversations about videotaped episodes: The case of engaging students in proving.

For the Learning of Mathematics, 23(1), 2-14.

Koellner, K., Schneider, C., Roberts, S., Jacobs, J., & Borko, H. (2008). Using the ProblemSolving Cycle model of professional development to support novice mathematics instructional leaders. In F. Arbaugh & P. M. Taylor (Eds.), Inquiry into Mathematics Teacher Education. Association of Mathematics Teacher Educators (AMTE) Monograph Series, Volume 5, 59-70.

Mesa, V. (2010). Student participation in mathematics lessons taught by seven successful community college instructors. Adults Learning Mathematics, 5, 64-88.

Mesa, V. (Accepted). Achievement goal orientation of community college mathematics students and the misalignment of instructors' perceptions. Community College Review.

Mesa, V., Celis, S., & Lande, E. (2011). Teaching approaches of community college mathematics faculty: Do they relate to classroom practices? (Manuscript under review). Ann Arbor: University of Michigan.

Mesa, V., Suh, H., Blake, T., & Whittemore, T. (2011). Examples in college algebra textbooks:

Opportunities for students' learning (Manuscript under review). Ann Arbor, MI: University of Michigan.

Nachlieli, T. & Herbst, P. (2010). Facilitating encounters among teachers with representations of teaching: Two registers. Retrieved on January 12, 2010 from Deep Blue at the University of Michigan, http://hdl.handle.net/2027.42/64852.

Suzuka, K., Sleep, L., Ball, D. L., Bass, H., Lewis, J. M., & Thames, M. (2009). Designing and using tasks to teach mathematical knowledge for teaching. In D. S. Mewborn & H. S. Lee (Eds.), Scholarly practices and inquiry in the preparation of mathematics teachers (pp. 7-23).

San Diego: AMTE.

–  –  –

,'-/1/&'&*!+ 4$"+*,+,'-/1/&')*%+ 54"$)*%$+ !"#!"$"%&'( #'!)-/#'%&$+ )*%+*,+ &"'-./%0+ "%2/!*%3"%&+ Figure 1: The facilitation triangle (Adapted from Cohen, Raudenbush, & Ball, 2003)

–  –  –

Abstract In this report we examine linear algebra students’ conceptions of inverse and invertibility. In the course of examining data from semi-structured clinical interviews with 10 undergraduate students in a linear algebra class, we noted a proclivity for students to identify 1 as the result of the composition of a function and its inverse. We propose that this may stem from the several meanings of the word “inverse” or the influence of notation from linear algebra. In addition, we examined how students attempted to reconcile their initial incorrect predictions with their later computational results, and found that students who succeeded in this reconciliation made heavy use of what we termed “do-nothing function” ideas. The implications of this work for classroom practice include a possible method to help students develop object conceptions of function, as well as the need to pay more explicit attention to often-backgrounded notational issues.

Keywordslinear algebra, function, linear transformation, process/object pairs

15TH Annual Conference on Research in Undergraduate Mathematics Education 341 Background The nature of students’ conceptions of function has been well-studied (e.g., Sfard, 1991, 1992; Dubinsky & McDonald, 2001; Carlson, Jacobs, Coe, Larsen, & Hsu, 2002). Sfard (1992) suggests that there are both structural (object-like) and operational (process-like) facets to the function concept, that structural conceptions are the result of reification of operational conceptions, and that processes are reified into objects that are then operated on by yet other processes. Sfard observes further that students’ conceptions are usually closer to operational than structural, and that many students develop pseudostructural conceptions – that is, object-like conceptions that they cannot unpack to obtain the underlying process.

Much work has also been done examining students’ understanding in the field of linear algebra in general (Dorier, Robert, Robinet, & Rogalski, 2000; Hillel, 2000; Sierpinska, 2000) and of the concept of transformation in particular (Dreyfus, Hillel, & Sierpinska, 1998; Portnoy, Grundmeier, & Graham, 2006). This study contributes to these bodies of research by examining the link, or lack thereof, made by students between the closely-related concepts of function and transformation, and the influence that knowledge from the one context has on the other.

Methods Data for this analysis comes from semi-structured clinical interviews with 10 undergraduate students in a linear algebra class at a large public university in the southwestern United States. Interviews were videorecorded and transcribed. In addition, students’ written work was retained. Grounded analysis (Strauss & Corbin, 1994) was employed to analyze the data.

The interview covered a wide range of topics relating to students’ understanding of the relationship between two mathematical contexts, functions in high-school algebra and transformations in linear algebra. As we began examining the data, we became particularly interested in how students reasoned about inverse and invertibility. In particular, we noted that all ten students predicted that the composition of a function with its inverse would yield 1. This surprising result informed our research questions: How can we account for these predictions?

What reasons do students give that the composition of a function or transformation with its inverse should be 1? Also, how do students reconcile their incorrect predictions with the correct answer they later obtain? Accordingly, this analysis focuses on students’ responses to the last

few questions of the interview:

• Find the inverse of f(x) = 3x – 9.

• Find the inverse of T(x) = x.

1 −2

• What will you get when you compose f(x) with its inverse that you found earlier?

o Perform the composition. Does the result match your prediction? If not, is there some reason your result makes sense?

• What will you get when you compose T(x) with its inverse?

o Perform the composition. Does the result match your prediction? If not, is there some reason your result makes sense?

Results When asked to predict the result of composition of f(x) with its inverse, every one of the ten students answered 1 rather than the correct x. For several of the students, this is likely linked to conflating algebraic inverses and functional inverses. For example, when asked to give an example of an invertible function, Nicholas confused algebraic and functional inverses: “So say you have x, the inverse is x to the negative 1, or 1 over x.” Nicholas’s mistake appears to stem 342 15TH Annual Conference on Research in Undergraduate Mathematics Education from the confusion between two concepts with very similar names that use the same notation, a superscript -1. Similarly, Naheem attributed properties of the algebraic inverse to the functional inverse. When asked to predict the result of composition of a function with its inverse, she stated that “if you take this one [f(x)] and multiply it by this one [the inverse], it's supposed to give you 1.” This statement, while entirely incorrect in the realm of functions, is entirely accurate in the context of algebraic inverses.

Other students did not conflate algebraic and functional inverses, but symbolized their answers incorrectly. For instance, Grant described the identity function fairly clearly, but chose 1

to represent the result of the composition:

Int: If I do f of f inverse of x, what do you expect it to come out with?

Grant: Input, the input that you put in there. It shouldn't modify it.

Int: If I haven't put in any input though, I'm just doing a calculation?

Grant: [writes] It's just 1.

Int: It would be 1?

Grant: It's not going to change what you put in there, because if you do something and then you undo it, has it really changed?

This is attributable to backward transfer (Hohensee, 2011): the influence of the notation of linear algebra, where the notation representing a linear transformation, T(x) = Ax, is often abbreviated to the matrix A alone. In particular, students may think that the identity matrix represents the identity transformations and overextend analogies. After all, as Grant reasoned, “this [circles 1] means this [circles identity matrix] when you’re dealing with matrices,” so since the identity matrix represents the identity transformation, 1 must represent the identity function.

Of the ten students, six were able to resolve the discrepancy between their prediction and their result. These six were exactly the six who expressed what we called “do-nothing function” (DNF) ideas, describing the result of the composition of a function (or transformation) and its inverse as being the function (transformation) that does nothing to the input. Joseph, for example, explained that he originally saw the function and its inverse as canceling to yield 1.

Then, however, he decided that x is a more reasonable answer, because “whatever you put in there, is whatever you’re getting out.” Joseph then used these DNF ideas to inform his correct prediction that the result of the composition of a transformation and its inverse should be x, because “you’re pretty much transforming it into something else, and … transforming it back to what it originally was.” We conclude that DNF ideas provide students with a helpful lever to reason about functions, their inverses, and the composition of the two. In addition, we theorize that DNF ideas may indicate a robust process conception of function, as well as providing a bridge to object conceptions of function.

In our talk, we will present a case study of one student who was able to resolve the discrepancy between their prediction and their result, and a case study of one student who was not; these students have been chosen to be more or less typical of their respective categories. In addition, we will present as a third case that of Liam, who appeared to transition from a less sophisticated to a more sophisticated understanding with the help of the interviewers. We will also discuss several implications for classroom practice, warn of possible consequences of common notational abuses for students’ conceptions of function, and outline what these data suggest teachers may be able to do to help their students develop object views of function.

15TH Annual Conference on Research in Undergraduate Mathematics Education 343 Discussion Questions

1. How can students effectively distinguish between functional and algebraic inverses? At what point in students’ education should we expect this not to be an issue?

2. What other instances of “backward” transfer might there be in students’ undergraduate mathematical studies?

3. How might DNF thinking relate to process and object conceptions of function?

References Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 33 (5), 352-378.

Dorier, J.-L., Robert, A., Robinet, J., & Rogalski, M. (2000). The obstacles of formalism in linear algebra. In J.-L. Dorier (Ed.), On the teaching of linear algebra (pp. 85-124).

Dordrecht: Kluwer.

Dreyfus, T., Hillel, J., & Sierpinska, A. (August, 1998). Cabri-based linear algebra:

Transformations. Paper presented at the First Conference on European Research in Mathematics Education, Osnabrück, Germany.

Dubinsky, E., & McDonald, M. (2001). APOS: A constructivist theory of learning in undergraduate mathematics education research. In D. Holton, M. Artigue, U.

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