Publication Type: Conference Paper/Unpublished Manuscript
Abstract: How can students learn Lagrange multipliers in under 20 minutes, understand divergence and Stokes’ theorem, and use the gradient of a function correctly even as they are just beginning to learn to calculate mixed partial derivatives, double integrals, and the mathematics of three-dimensional space? Although vector calculus and multivariable calculus share many concepts, many procedures in the former rely upon the mastery of skills learned in the latter. How can the two courses be combined into one in such a way that students understand how to use the material in later science courses?

Corinne Manogue and Tevian Dray, designers of the Vector Calculus Bridge Project (http://www.physics.oregonstate.edu/bridge), have found that using geometry, and emphasizing differentials and geometric visualization over algebraic manipulations, can help students “bridge the gap” between the presentation of vector calculus in mathematics and the application of those concepts to physics and engineering courses. Not surprisingly, this approach extends nicely to a multivariable calculus course. This talk will give an overview of the Vector Calculus Bridge Project and highlight the changes required to bring this approach to a typical multivariable calculus course. It will point out the unexpected traps, surprises, and nice consequences for students which occur as a result of this approach, and show how language can help students correctly use skills which they have not yet mastered.

Publication Type: Conference Paper/Unpublished Manuscript
Review Method: Peer Reviewed
Abstract: Comparative analysis of conceptual understanding of Calculus ideas among teachers with different Calculus experience

Abstract
The purpose of this study is to examine the differences in conceptual understanding of calculus ideas among in-service teachers with various calculus backgrounds. One of the main components of the research was to explore how effectively could big calculus ideas be introduced to students with different mathematical experiences.
The research sample included 24 participants –in-service teachers that had various backgrounds, ranging from middle school teachers to community college level instructors in various disciplines (mathematics, science, language arts, music, etc.). The mathematical level of proficiency varied: 9 participants had never taken calculus classes, 8 participants had taken Calculus-1 and/or 2, and 7 participants had taken up to Calculus-3.
The participants were pre-assessed on current level of content knowledge by testing their understanding of calculus concepts of graphing, derivative, integration, and optimization. Teaching intervention consisted of a set of conceptual activities to support students’ understanding of calculus ideas. Finally, a concept post-assessment was given to observe if the activities helped to improve the participants’ conceptual comprehension of big calculus ideas.
Theoretical Framework
During the last two decades Calculus is at the forefront of research and curriculum reforms in mathematics education. The majority of research in Calculus learning has been done at the level of undergraduate education and some at the high school level. Researchers observed that students enter calculus courses with a primitive understanding of concepts of function, change, continuity, etc. (Tall, D., 1996, Ferrini-Mundy, J., & Lauten, D., 1993). They also noted that students have cognitive difficulties in coordinating function concept in algebraic and graphical representations, which is critical in constructing a foundation for fundamental calculus ideas (Schnepp, M., & Nemirovsky, R., 2001). Other researchers concentrate on different approaches to teaching calculus principles: comparison study on technique-oriented approach vs. conceptual and infinitesimal approaches of learning calculus shows that different approaches have different impact on students’ language use and sources of conviction (Frid, S., 1994).
Researches have also determined that cognitive obstacles to the learning of calculus arise in at least two different ways – one related to linguistic/representational aspects and the other related to intuitions. Given that so many of our algebra and calculus courses are immersed in symbolic manipulation, often at the expense of understanding, it is not surprising that linguistic/ representational factors give rise to cognitive obstacles. Since learners basically want to understand and make sense of what they are being asked to learn, the intuitions that students bring to bear on the concept of calculus often play a crucial role in the appropriate construction of those concepts. Researches “propose that a potentially useful framework in which to embed considerations of cognitive obstacles lies in the framework of Krutetskian cognitive processes of reversibility, flexibility, and generalization” (Norman, A., & Prichard, M., 1994, p. 76).
There is an emerging importance of making connections between different representations (concrete, visio-spatial, numeric, graphical, algebraic, etc.) in helping students’ to learn calculus concepts. Visualization may be a major tool to develop this understanding. One of the guiding principles of Harvard Consortium Calculus text is the multiple representations, “which says that wherever possible topics should be taught graphically and numerically, as well as analytically. The aim is to produce a course where the three points of view are balanced and where students see each major idea from several angles” (Hughes-Hallett, D., 1990, p. 121).
One of the most significant points that come from the analyses of research in Calculus learning is that there should be more emphasis on conceptual learning using multiple representations and connections before students immerse into symbolic manipulations. In order to build a rich conceptual foundation for successful learning of Calculus at the high school and college level there should be a lot of preparatory work done at the early years of schooling. “Calculus needs to be studied across many years of school, from early grades onward, much as a subject like geometry should be studied” (Kaput, J., 1994, p. 132). While this approach is common among upper class and private schools, it is lacking in high minority lower socioeconomic school programs.
The project is based on the following key assumptions about learning and teaching:
• Conceptual learning leads development of cognitive acquisition of formal procedural operations. Lev Vygotsky claims that development of formal operations depends on the creation of a successful learning environment, which includes the use of conceptual tools (Vygotsky, 1987). The development of students’ procedural calculus skills is a derivative of students’ conceptual understanding of big calculus ideas.
• An application of the Davydov’s accelerated pedagogy of teaching and learning mathematics using method of ascending from big, general ideas to specific procedures is a key methodological tool for designing a rigorous conceptual mathematics currículum (Davydov, 1990). Thus, development of students’ conceptual understanding of calculus principles should be achieved by ascending from multivariable calculus concepts to single-variable principles.
• Cognitive-visual conceptualization (CVC) through the use of modeling and technology plays a critical role in learning of calculus principles.

Research Design
The 24 participants were given a concept pre-test consisting of four open-ended questions. The first question assessed the participants’ ability to read a graph that showed the distance traveled by a car and to analyze it. The second question evaluated the participant’s competence to create a distance vs. time graph and a speed vs. time graph for a given context. The third question was related to finding the distance a car travels based on two given graphs: a distance vs. time graph and a speed vs. time graph. The last question was linked to understanding the concept of slope by using contour diagrams.
Teaching intervention consisted of a set of conceptual activities. These activities mainly dealt with calculus concepts of derivative and slope, integration, gradient, and optimization. In contrast to previous remarkable attempts in introduction of advanced Calculus concepts (e.g., SimCalc project, CoVis project), which basically considered development of single-variable Calculus concepts, this study focuses on teaching and learning Calculus principles from general multi-variable to single-variable concepts: from generic 3-D surface to arbitrary 2-D curves and then to specific elementary curves (linear, quadratic, exponential, etc.), from tangent plane to tangent line (including concept of gradient), from general infinitesimal methods to procedural calculations of derivative and integral, etc. In teaching multi-variable Calculus concepts we used one of the advantages of local Greater El Paso landscape – mountains (a natural model of generic arbitrary 3-D surface). In parallel with this students were introduced to basic 3-D Geometry concepts (3-D coordinate system, projections of 3-D objects, sections of arbitrary 3-D surface, etc.). 3-D Geometry is a mathematically natural way to introduce/ illustrate multi-variable Calculus concepts.
Piaget’s idea that development of geometric concepts follows an anti-historical order is probably familiar to most readers. The idea is that, whereas historically the earliest geometrical operations were developed to deal with specific problems of terrestrial mensuration and hence had a Euclidean character, the child only arrives at the specific concepts of similarity, congruence, and proportion after a long process of developing these refined concepts from more global, or general, ideas about spatial relations. “Historically, the development has been from the particular, measurement-bound, practical “real-world” geometry to the more general, abstract, and non-metrical relationship found in projective geometry and ultimately in topology. For the child, according to Piaget, the earliest and easiest spatial relations to grasp (in a very intuitive way) are those concerned with general futures such as contiguity, neighborhood, closed contour, and so on – that is, topological futures” (Dodwell, 1971, p. 179). These developments are held to occur through the agency of the child’s own active exploration of, and interaction with, its environment (Piaget, J., & Inhelder, B., 1956). Current research on designing learning environment for developing students’ understanding of geometry and space seems to support the Piagetarian idea (Lehrer, R., & Chazan, D., 1998, Balomenos, R., Ferrini-Mundy, J., Dick, T., 1987, Yakimanskaya, I., 1991).
Following the activities, a concept post-test was administered to the participants to evaluate if there was a significant change in calculus reasoning after the activities took place. A concept post-test consisted of four open-ended questions and was similar to the pre-test in terms of the main Calculus ideas taught through the intervention.

Results and Recommendations
The research showed that the participants that had taken no calculus classes gained the most from the activities (28.20 percent). Where as the group that had taken Calculus-1 and/or 2 gained the least (3 percent). The highest post-test result came from the group of people that had taken up to Calculus-3 (77.00 percent). The following table (Table 1) consists of the research results.
Table 1. Results of the study

Groups Pre-Test ( percent) Post-Test ( percent) Gain ( percent) Pre-Test
(max-25) Post-Test
(max-25) Gain
1 summer semester (excluding Action Research) 38.50 percent 48.70 percent 10.20 percent 9.6 12.2 2.6
4 semesters (including Action Research) 50 percent 70.50 percent 20.50 percent 12.5 17.6 5.1

"Non-Takers" 32.80 percent 61.00 percent 28.20 percent 8.2 15.25 7.05
Calculus 1-2 53.00 percent 56.00 percent 3.00 percent 13.25 14.00 0.75
Calculus 3 52.00 percent 77.00 percent 25.00 percent 13 19.25 6.25

Total 45.90 percent 64.70 percent 18.80 percent 11.48 16.17 4.69

Recommendations:
1. Long-term sustainable Professional Development is a critical component in strengthening teachers’ conceptual understanding of big calculus ideas. Classroom research activities support teachers’ further improvement of content and content pedagogical knowledge of Calculus ideas.
2. In-service teachers with traditional Calculus 1-2 coursework are reluctant to re-learn Calculus concepts. Once student has formalized a procedure, it is difficult to re-visit the underlying concept for deeper understanding (Hiebert & Carpenter, 1992, Lee & Wheeler, 1989, Skemp, 1978, Wilson & Goldenberg, 1998).
3. In-service teachers with no Calculus experience are more open to learn Calculus concepts than teachers with traditional Calculus 1-2 coursework. It sounds like “no knowledge is better than a little knowledge”.
4. Traditional Calculus-3 coursework seems to help students to overcome “the zone of conceptual resistance” and to be able to generalize calculus procedures.
5. Teachers’ content knowledge of calculus concepts should be vertically aligned and flexible enough to make transitions from multi-variable to single-variable Calculus concepts and reverse.
6. Method of ascending from general to specific helps teachers to focus on conceptual generalization of calculus ideas.
7. Connections between different representations (concrete, visual-spatial, numeric, graphical, algebraic, etc.) are important tools in improving teachers’ understanding of calculus concepts.

References
Balomenos, R., Ferrini-Mundy, J., Dick, T. (1987). Geometry for Calculus readiness. In: Learning and Teaching Geometry, K-12. – Reston, VA: NCTM. Pp. 195-209.
Davydov, V. (1990). Types of Generalization in Instruction: Logical and Psychological Problems in the Structuring of School Curricular. Reston, VA: NCTM.
Dodwell, P. (1971). Children’s perception and their understanding of geometrical ideas. In Piagetarian cognitive-development research and mathematics education. Eds. M. Rosskopf, L. Steffe, S. Taback. – Washington, DC, NCTM. Pp. 178-188.
Ferrini-Mundy, J., & Lauten, D. (1993). Teaching and learning calculus. In Research ideas for the classroom. High school mathematics. Eds. P. Wilson, & S. Wagner. Macmillan: NY. Pp. 155-176.
Frid, S. (1994). Three approaches to undergraduate calculus instruction: Their nature and potential impact on students’ language use and sources of conviction. In Research in collegiate mathematics education. Vol. 1. Eds. Dubinsky, E., Kaput, J. Washington, D.C. AMS and MAA. Pp. 69-100.
Hughes-Hallett, D. (1990). Visualization and calculus reform. In Zimmerman, W., & S. Cunningham (Eds.) Visualization in Teaching and Learning Mathematics. MAA Notes # 19. Washington, D.C.: The MAA Inc.
Kaput, J. (1994). Democratizing access to calculus: New routes to old roots. In Mathematical thinking and problem solving. Ed. A. Schoenfeld. – Hillsdale, NJ: Lawrence Erlbaum. – Pp.77-156.
Lehrer, R., & Chazan, D. (Eds.) (1998). Designing learning environments for developing understanding of geometry and space. Mahwah, N.J.: Lawrence Erlbaum.
Norman, A., & Prichard, M. (1994). Cognitive obstacles to the learning of Calculus: A Krutetskian perspective. In Research issues in undergraduate mathematics learning: Preliminary analyses and results. Eds. J. Kaput, E. Dubinsky. Washington, D.C.: MAA Notes, #33. Pp. 65-77.
Piaget, J., & Inhelder, B. (1956). The child’s conception of space. London: Routledge & Kegan Paul.
Skemp, R. (1987). The Psychology of Learning Mathematics. Hillsdale, NJ: Lawrence Erlbaum Associates.
Schnepp, M., & Nemirovsky, R. (2001). Constructing a foundation for the fundamental theorem of calculus. In The role of representation in school mathematics. Eds. A. Cuoco, F. Curcio. Reston, VA: NCTM. Pp. 90-102.
Scientific Visualization Project, http://www.covis.nwu.edu.
SimCalc Project: Democratizing Access to the Mathematics of Change, http://www.simcalc.umassd.edu/.
Tall, D. (1996). Functions and calculus. In International handbook of mathematics education. Part 1. Eds. Bishop, A., Clements, K., Keitel, C., Kilpatrick, J., Laborde, C. – Dordrecht, The Netherlands: Kluwer. Pp. 289-325.
Vygotsky, L. (1987). Thinking and Speech. In R. Rieber, & A. Carton (Eds.). The Collected Works of L.S. Vygotsky. Vol. 1. NY: Plenum Press. Pp. 38-285.
Yakimanskaya, I. (Ed.) (1991). The Development of Spatial Thinking in Schoolchildren. Reston, VA: NCTM.
Zimmerman, W. (1990). Visual Thinking in Calculus. In Visualization in Teaching and Learning Mathematics. Zimmerman, W. & S. Cunningham (Eds.). Washington, D.C.: The MAA Inc.

Publication Type: Conference Paper/Unpublished Manuscript
Abstract: Developed at Purdue University Calumet, CaluMath is open-source mathematics web page construction software. It is designed to be compatible with all web browsers and is freely available. Its goal is to allow instructors to create interactive mathematics web pages for use with their students or to tailor existing pages for their particular classrooms. This session will demonstrate how the software can be used to develop web pages that enable students to create and manipulate mathematical objects and receive feedback concerning their interactions with the page. We will illustrate how to add graphs that students can manipulate, boxes that will accept student input, and buttons that enable students to perform various actions. We will also illustrate how “replies” to students’ actions may be incorporated into the web pages and how the software can randomly generate new versions of a given problem. No programming is necessary since the software provides the web page developer with an array of menu-driven options and boxes used to construct the interactive web pages.

Publication Type: Conference Paper/Unpublished Manuscript
Abstract: Bioaccumulation is the increase in concentration of a substance in organisms, as they take in contaminated air, water, or food more rapidly than can be eliminated by metabolization and excretion. As pollutants move from one link in the food chain to another they concentrate through the process of biomagnification. Through a module designed for use in an introductory calculus course, students explore these concepts using data derived from research on mercury levels in the environment and aquatic food chain. Students begin by tracking mercury across the food chain-- from algae and bacteria, to insects, small fish, larger fish, and eventually to birds, mammals, and humans. Concentrations and bioacculumation factors are calculated and literature is reviewed for transfer ratios to fetuses and eggs. Students are asked to use their typical weekly consumption of fish to calculate their average daily dosage of mercury per kilogram of body weight. Based on models of absorption of mercury from food, a separable differential equation for mercury elimination in humans and the fraction of total body mercury in blood, a time dependent equation for blood mercury concentration is derived. Students then find their steady-state level of blood mercury concentration and compare this to the U.S. Environmental Protection Agency’s “safe benchmark blood level.”

Publication Type: Conference Paper/Unpublished Manuscript
Abstract: In a typical pre-calculus class, students work with linear, polynomial, exponential, logarithmic, and piece-wise defined functions. A student is asked, in a project assignment, to obtain paired data on a topic of interest to him/her and examine the use of various types of algebraic functions to model that data. The student must assess each model and select the best overall model. A final part of the project report (worth 20% of the project grade) requires the student to describe why it makes sense that the chosen model truly describes the relationship between the two variables. For this reason, it is important that the student’s data are drawn form an area of interest to the student. Two recent examples include a quadratic model to describe the relationship between hits and runs scored by Major League Baseball players and the exponential decay model describing the relationship between a player’s height and the average number of digs per game for NCAA Division I volleyball players.