Publication Type: Conference Paper/Unpublished Manuscript
Review Method: Peer Reviewed
Abstract: The author looks at post-World War II development theory and practice to delineate a set of myths that appear – and reappear in different guises – in the dominating post-World War II paradigms of how richer nations can help catalyze “development” in poore

Publication Type: Session Paper
Abstract: In late October 2008, politico.com reported that the Republican National Committee had spent $150,000 of donations given in support of the McCain presidential campaign on clothes and make-up for the Republican Vice Presidential nominee, Sarah Palin. Primary expenditures were made at Saks Fifth Avenue and Neiman Marcus, high-end department stores far outside the price range of the "working-class hockey mom" that Palin billed herself as being. Though supporters argued that she needed a new wardrobe to compete on a national stage and that gender bias made it critical for women to cut a stylish figure, neither Palin nor the McCain strategists could answer why her new wardrobe required such financial largesse. More broadly, we might argue that Palin was merely participating in the form of image transformation we see celebrated on reality TV, where style and self-hood coalesce to make one worthy of citizenship in Makeover Nation. These narratives depart, however, in who pays for such changes (a television show or donor dollars) and the degree to which the makeover is a means to an end or the end in itself.

Publication Type: Conference Paper/Unpublished Manuscript
Review Method: Peer Reviewed
Abstract: Related rates problems require students to have a strong understanding of differentiation, function, and variable, and their application beyond strictly computational problems. It has been suggested that students do not fully understand the concepts of differentiation or variable (Orton, 1983; White & Mitchelmore, 1996). The purpose of this study is to gain a better understanding of the process involved in completing related rate problems, including knowledge of the obstacles that first semester calculus students commonly encounter when completing these tasks.

Martin�s (1996, 2000) results indicate that students have particular difficulty with the conceptual aspects of solving related rates problems, especially those requiring the solving of an auxiliary problem. For example, students may have difficulty identifying the need to use similar triangles to relate specific variables prior to expressing the volume of a cone in terms of radius. In a similar study, White & Mitchelmore (1996) found that students tended to have a �manipulation focus� when solving these types of problems, that is, they merely searched for symbols they could push around with known acceptable operations. Frequently, the symbols carried no meaning for students.

Simon (1994) claims that transformational reasoning is a critical component of mathematics learning and understanding. The idea behind transformational reasoning is that students can create a mental model that can be manipulated to visualize and understand relationships. As Simon states, �Central to transformational reasoning is the ability to consider, not a static state, but a dynamic process by which a new state or a continuum of states are generated.� (pg 5) It seems plausible that this applies to solving related rates problems, as students must understand how particular quantities are changing in the given problem.

To solve a related rates problem, the student must be able to understand the concepts of variable and function. It is also important that students recognize geometric relationships in the situation, and are able to understand and apply implicit differentiation. The notion of variable is key to being able to appropriately label a diagram and construct the necessary relationships. From here, the concept of function must be utilized to create a function in terms of the desired variable. It is my hypothesis that these first two steps rely heavily on the students� transformational reasoning skills. After a desirable function has been established, the student may then apply the tools of implicit differentiation and solve the problem.

After the unit on related rates was completed in a first semester calculus class, student interviews were conducted. Subjects were asked to solve two or three related rates problems they had not seen before while being video taped. The interviews were transcribed and analyzed for student understandings and misconceptions using the open and axial coding techniques of Strauss and Corbin (1990). The initial one or two questions were in the familiar form of a plane flying over a tower (application of the Pythagorean Theorem to a right triangle) and a balloon (volume of a sphere) being deflated. It was the third question, a coffee cup being filled (see interview excerpt below), which was not posed in a standard form. The researcher asked questions and prompted for further understanding throughout the interviews.

My results revealed that students focused on the three procedural steps that Martin (2000) outlined. They generally drew a diagram and labeled the constants, chose a formula and differentiated it, then plugged in values. This abbreviated procedure worked well on the standard problems, particularly ones that do not require an auxiliary problem to be solved. Problems arose for students when a nonstandard question was posed as is evidenced in the interview excerpt below.

Student: Ok. (Reads question 3 � Cup problem: Coffee is poured at a uniform rate of 20 cm3/sec into a cup whose inside is shaped like a truncated cone. If the upper and lower radii of the cup are 4 cm and 2 cm, respectively, and the height of the cup is 6 cm, how fast will the coffee level be rising when the coffee is halfway up the cup?) Ok, so the first thing we have to do is draw the picture. Truncated cone, I�m not an artist if you couldn�t tell.
Interviewer: Neither am I.
Student: So, then we have 2 cm for the lower one and up here it�s 4 cm, and then the height between them is 6 cm. So, um, the volume of a cone, can you tell me that? Can I look it up? I don�t have it memorized.
Interviewer: It�s 1/3 pi r squared h.
Student: Ok, um, so then, how fast will the coffee level be rising when the coffee is halfway up the cup? Um, so that would make the height 3 cm. So, first ting we have to do is take the derivative. Um, we want to find the change it the height. So, um, cause we know the volume, the derivative of the volume. So, first we�ll solve for the height cause that�s what we�re looking for.

Rather than attending to the dynamic nature of the problem, she proceeded to create a formula that subtracted the smaller cone, leaving her with a formula in two variables. Later in the interview, she attempted to compute the derivative of her formula, and appeared to experience cognitive dissonance when her attempt to use the chain rule resulted in the realization one cannot have two variables in the formula. Another student did not draw diagrams without prompting; he immediately searched for a formula to differentiate and plug values into. These behaviors were not uncommon among interviewees.

The geometry of the situations in the problems also caused difficulties for the students. One student was compelled to express her negative thoughts about geometry with statements like �Dude, I so suck at geometry� throughout her interview. Others demonstrated a desire to verify every geometric relationship with either the interviewer or the book.

Three major student difficulties that emerged in the data were:
� Students were focused on the procedural steps
� General relationships, such as the radius and height in the cup problem, were not taken into consideration
� Students had algebraic and/or geometric deficiencies
Based on my analysis of the data, I conjecture that their inability to engage in transformational reasoning is a major obstacle to their successful completion of the problem.

It is my hypothesis that one of the critical components of solving related rates problems is being able to diagram and visualize the situation. Even though most students appeared to be proficient at drawing an appropriate diagram, difficulties arose when they began their labeling phase. Although students labeled their diagrams with the constants given in the problem, they did not attend to which quantities were considered to be changing. This data supports a lack of transformational reasoning in the students� solution approach. Instead, after drawing their diagram, students immediately looked for a geometric formula that would fit the situation, differentiated it, and plugged in values. Their focus was on the procedural steps. What became evident, particularly in the cup problem (noted in the excerpt above), was that students did not account for the general relationship between the radius and height of the cone. They did not appear to have a model that indicated these two quantities were continually changing, again demonstrating a lack of transformational reasoning.

As Martin (1996, 2000) found, when students were required to solve an auxiliary problem, they struggled more. Solving an auxiliary problem requires the student to investigate and understand the general relationship between changing quantities. This suggests that the students are not using transformational reasoning in solving the problems. Students� inability/reluctance to form an image of the dynamically changing event appears to foster students� dependence on the procedural steps, which in turn also revealed students� deficiencies in basic algebraic and geometric knowledge.

The results of this study provide valuable insights for the development of activities to facilitate students' ability to understand and complete related rate type problems. Activities that may facilitate improved understanding of these problems are also suggested. First, I have been developing worksheets to remind students of geometric relationships and functional knowledge. Second, I am working to develop worksheets that will ultimately elicit students� transformational reasoning skills in solving related rate type problems. Ongoing studies will test the efficacy of these worksheets.

References

Martin, T. (2000). Calculus Students� Ability to Solve Geometric Related-Rates Problems, Mathematics Education Research Journal, Vol. 12, No. 2, 74-91.
Martin, T. (1996). First-Year Calculus Students� Procedural and Conceptual understandings of Geometric Related Rates Problems. In E. Jakubowski, D. Watkins, & H. Biske (Eds.) Proceedings of the 18th Annual Meeting of the North American Chapter of the International Group for Psychology of Mathematics Education, Vol. 1, 47-52.
Orton, A. (1983). Students� Understanding of Differentiation, Educational Studies in Mathematics, 14, 235-250.
Simon, M. (1994). Beyond Inductive and Deductive Reasoning: the Search for a Sense of Knowing, Presented at Annual Meeting of the American Educational Research Association.
Strauss, A. & Corbin, J. (1990). Basics of Qualitative Research: Grounded Theory Procedures and Techniques. Sage Publications, Newsbury Park, California.
White, P. & Mitchelmore, M. (1996). Conceptual Knowledge in Introductory Calculus, Journal for Research in Mathematics Education, Vol. 27, No. 1, 79-95.

Publication Type: Conference Paper/Unpublished Manuscript
Review Method: Peer Reviewed
Abstract: Italian foreign policy specialists stress the importance of the country’s post-World War II identity as a force for world peace that works through the United Nations. How, then, can we explain Italy’s decision to contribute fifty-eight planes to the 1999

Publication Type: Conference Paper/Unpublished Manuscript
Review Method: Peer Reviewed
Abstract: The relation between �territoriality� and foreign/international politics has always interested students of both Geopolitics and Foreign Policy Analysis (FPA). These fields of study are much more interrelated than traditionally is acknowledged. In this paper, we study the important work of Harold and Margaret Sprout, who played a key role in introducing their insights on �geopolitics� into what later would become known as �foreign policy analysis�. First, we will try to gain a better insight into the ideas of the Sprouts. What did their suggested �ecological triad� entail? What are the consequences of the distinction between the �operational� and the �psychological� milieu for the study of foreign policy? How is �cognitive behavioralism� related to other possible epistemological appreciations regarding the relationship between �territoriality� and �foreign policy�? What does the distinction between �foreign policy analyses� and �capacity analyses� mean? Second, we will identify three ways in which Harold and Margaret Sprout influenced IR and FPA. Third, we will reconstruct the ontological, epistemological and methodological assumptions and traits of �Cognitive Geopolitics�. Finally, we will formulate some critiques on �Cognitive Geopolitics�, while at the same time developing some ideas on the continued relevance today of the work of Harold and Margaret Sprout.