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Doing other activities and coming back to an unfinished problem might be considered an example of incubation, which is the process by which the mind goes about solving a problem subconsciously and automatically, and which happens best when one takes a break from creative work (Krashen, 2001). While there are many reports of experiments on incubation in the psychology literature (Sio & Ormerod, 2009), they typically allow only a short time for incubation. However, both mathematicians stated that when they received the notes, they immediately glanced at them to estimate how long the proofs might take, but both started proving the next day. It is difficult to know whether there was an incubation effect due to actually commencing their proving the next day. How can we gain information on when and how 15TH Annual Conference on Research in Undergraduate Mathematics Education 533 incubation is used in mathematics? Is it important to let students know about incubation? How can we collect all actions that mathematicians use to recover from impasses? Also, can we encourage students to take some of these actions to recover from impasses?

References Burton, L. (1999). The practices of mathematicians: What do they tell us about coming to know mathematics? Educational Studies in Mathematics, 37, 121-143.

Conradie, J., & Frith, J. (2000). Comprehension tests in mathematics. Educational Studies in Mathematics, 42, 225-235.

Krashen, S. (2001). Incubation: A neglected aspect of the composing process? ESL Journal, 4, 10-11.

Meier, A., & Melis, E. (2006). Impasse-driven reasoning in proof planning. In M. Kohlhase (Ed.), Mathematical Knowledge Management: 4th International Conference MKM 2005 (pp. 143-158). Berlin: Springer-Verlag.

Mejia-Ramos, J. P., Weber, K., Fuller, E., Samkoff, A., Search, R., & Rhoads, K. (2010).

Modeling the comprehension of proof in undergraduate mathematics. Proceedings of the 13th Annual Conference on Research in Undergraduate Mathematics Education (pp. 1Raleigh, NC.: Available online.

Moore, R. (1994). Making the transition to formal proof. Educational Studies in Mathematics 27, 249-266.

Samkoff, A., Lai, Y., & Weber, K. (2011). How mathematicians use diagrams to construct proofs. Proceedings of the 14th Annual Conference on Research in Undergraduate Mathematics Education (pp. 430-444). Portland, OR.: Available Online.

Selden, A., & Selden, J. (2003). Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem? Journal for Research in Mathematics Education, 34, 4-36.

Sio, U. N., & Ormerod, T. C. (2009). Does incubation enhance problem solving? A metaanalytic review. Psychological Bulletin, 35, 94-120.

Weber, K. (2008). How mathematicians determine if an argument is a valid proof. Journal for Research in Mathematics Education, 30, 431-459.

Weber, K., & Alcock, L. (2004). Semantic and syntactic proof productions. Educational Studies in Mathematics, 56, 209-234.

534 15TH Annual Conference on Research in Undergraduate Mathematics Education Wilkerson-Jerde, M. H., & Wilensky, U. J. (2011). How do mathematicians learn math?

Resources and acts for constructing and understanding mathematics. Educational Studies in Mathematics, 78, 21-43.

15TH Annual Conference on Research in Undergraduate Mathematics Education 535 First Semester Calculus Students’ Understanding of the Intermediate Value Theorem

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In our calculus courses, we often see students perform poorly on problems involving the Intermediate Value Theorem (IVT), despite being a fairly basic concept. Thus, we designed a study to analyze students' conceptual understanding of the IVT and their ability to express the theorem in their own words. Two groups of students were video-taped while working on an activity designed to guide their construction of an initial understanding of the IVT, and fifty-four students were later asked to state the IVT in their own words. Both video data and student responses on the written work were analyzed to identify common themes. It was found that even though students were able to understand the concepts behind the Intermediate Value Theorem, they were unable to correctly describe the IVT in their own words, largely due to confusing the independent and dependent variables and issues with the if/then structure in a theorem.

Keywords: Calculus, Intermediate Value Theorem, mathematical language

The Intermediate Value Theorem (IVT) is typically the first theorem introduced in a firstsemester calculus course, and quite possibly the first formal mathematical theorem that many students encounter. In Stewart’s Essential Calculus, this theorem is introduced in Section 1.5, which discusses an informal notion of continuity. Recall that the IVT states that if a function f is continuous on the closed interval [a,b] and N is any number between f(a) and f(b), where f(a) ≠ f(b), then there exists a number c in (a,b) such that f(c) = N. (See Figure 1.) y

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While much research has been conducted on student understanding of some fundamental mathematical concepts and theorems, very little work has been done to investigate student

understanding of the IVT. For this study, we have identified 3 aspects of understanding the IVT:

a conceptual understanding, the ability to state the hypotheses and conclusion of the theorem correctly (written language), and the ability to apply the theorem to a problem (finding zeroes, etc.). This preliminary report focuses on the first two aspects of IVT understanding. From prior teaching experiences, we have seen that students may appear to have a conceptual understanding of the Intermediate Value Theorem but are still unable to apply or express this idea when necessary. In this study we specifically investigate whether or not students are able to 536 15TH Annual Conference on Research in Undergraduate Mathematics Education conceptualize the meaning of the IVT and whether or not they are able to express the IVT in written form.

Literature Review In an exploratory study, Monk (1992) found that students conceptualized functional situations in two distinct ways, termed point-wise and across-time, and that if the manner in which they conceptualized function did not meet the demands of a given task, student difficulties arose. We found this to be very similar to the ways in which our students initially interpreted the Intermediate Value Theorem. Instead of determining if a y-value of N existed on the entire function, the students focused on one particular x-value, and evaluated if that input produced an output of N. Carlson (1998) found that even mathematically talented students still have misconceptions about functions, specifically with respect to the language of functions.

Another fundamental mathematical concept that relates to the Intermediate Value Theorem is that of limits and the struggles students have in understanding them. Cottrill et al.

(1996) provide a genetic decomposition of the limit concept and posit that a more complete development of a dynamic view of this concept will promote a better understanding in students.

Oehrtman (2009) describes various metaphors students use when understanding and describing the concept of limit, and he advocates for promoting an approximation metaphor when teaching students, since it is easy for students to understand and also closely aligned with formal mathematics. On the other hand, Williams (1991) found that students held fast to their models for understanding limit and were "extremely resistant to change" (Williams, 1991, p. 219). This emphasizes the need to be careful and deliberate about the ways in which we first introduce these ideas to our students.

Researchers have also investigated how students understand theorems such as the Extreme Value Theorem and Rolle’s Theorem. Abramovitz et al.(2007, 2009) developed a process for learning theorems (the self-learning method) to help students better understand the hypotheses and conclusions of the Mean Value Theorem and Rolle’s Theorem. Much work has also been done on students' ability to prove theorems, but our work does not address proving the IVT, only understanding the statement.

Theoretical Perspective Piaget’s structuralism (1970, 1975) is used as the theoretical perspective throughout this study. Structuralism is a type of constructivism wherein it is believed that students construct an understanding of mathematical concepts not at free will, but within certain constraints. In this particular study, students worked in groups on an activity that guided them to construct an understanding of the hypothesis and conclusion of the Intermediate Value Theorem.

Methods/Subjects Participants in the study were first-semester calculus students at a large, public, research university. Two sections of students participated, both of which were taught by one author. In each class, a group of four students was videotaped while working on the activity mentioned above. This activity was given before the instructor formally introduced the IVT to the class.

Students were asked to draw a series of functions which satisfied some of the conditions given in the IVT. Two class periods after completing the activity, all students (n = 54) were given a pop quiz which asked students to state the Intermediate Value Theorem in their own words. Written responses were collected and analyzed using Corbin and Strauss’ (2008) open and axial coding.

15TH Annual Conference on Research in Undergraduate Mathematics Education 537 Results

As mentioned earlier, we examined two aspects of student understanding of the IVT:

conceptual understanding and the ability to state the hypotheses and conclusion correctly (written, language). At this point, we have not, yet, studied students' ability to apply the theorem to a problem (finding zeroes, etc.), but will do so in a future study. Analysis of the video tapes shows that students in this study do, in fact, understand the concept of the IVT, although they have significant difficulty with formal mathematical language.

In the first question on the in-class activity, students were asked to sketch the graph of a function such that f (c)  13 does not exist. Seven of the eight students sketched a graph where the x value of 13 produced no y-value, instead of avoiding a y-value of 13. Both groups of students needed assistance to recognize their mistake, but all students were easily convinced of their mistake. One student said, "Oh yeah, f of x" with a strong emphasis on x. Some students, who happened to initially graph a function that was one-to-one realized that they could simply "rotate" their graph so that a y-value of 13 would not exist, instead of avoiding an x-value of 13.

However, students did not seem to be aware that functions that were not initially one-to-one would not produce a function when rotated. (See Figure 2.) The instructor or a teaching assistant eventually pointed out the problem to the students, and they were able to fix their graphs to produce appropriate functions.

y y Figure 2: Illustration of a rotation that does not produce a function Another mistake that was prevalent in the video data is the tendency for students to avoid one specific y-value of 13, namely the point (0, 13). Students seemed to be attending only to the place on their graph where they had labeled y = 13, instead of attending globally to any y-value of 13. This aligns with Monk's (1992) classifications of point-wise versus across-time reasoning.

The graphs that our students drew did not cross through the point (0, 13), but sometimes had a yvalue of 13 elsewhere (See Figure 3). Other students were able to draw a graph that never had a y-value of 13, but it was unclear in the video if that was by chance or if it was a purposeful decision. Additional data is needed to more fully understand students’ beliefs.

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Even though the students had difficulty with basic function notation, they were able to understand the ideas behind the Intermediate Value Theorem. Students had no trouble believing that a continuous function must pass through a y-value of 13 if there were a y-value less than 13 and a y-value greater than 13 somewhere in the function. Throughout the semester, students told the instructor that they “get the idea” but have difficulty expressing it. Their verbal descriptions of the theorem often included gestures, which made it easier for them to express. In the written work, the students greatly struggled with function notation and the overall structure of an “ifthen” statement.

Fifty-four responses to the pop-quiz question were collected, and mistakes were categorized according to common themes. One noticeable error was in the students’ attempts to use the standard if/then wording of the theorem. Common errors in this category included the presence of a hypothesis with no conclusion statement or switching the ‘if’ and ‘then’ statements (yielding in an incorrect assumption that the IVT proves that a function is continuous). Often, students would omit one or more parts of the theorem (e.g. not stating that the function must be continuous), resulting in a statement of a theorem that was not always true.

Another common problem in the student responses on the quiz dealt with issues in the x and y-values. A few students used a non-standard notation, but still produced a mathematically correct statement. He wrote, "The Intermediate Value Theorem states that if a function is continuous and there is a point a with y-value x and a point b with y-value z...". As mathematicians and teachers, we would never consider labeling a y-value with x, but it is not mathematically incorrect. Other students were less clear about whether the variables they used referred to x or y values, making it unclear whether or not their statements were correct. Still, other students were clearly wrong in their labeling. For example, one student stated, "The Intermediate Value Theorem is proving that N (y-value) exists by finding an a and b (x-values) on a continuous graph both greater than and less than N.” In this example, the student clearly labeled N as a y-value and said that this y-value should between two x-values.

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