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+ Calculus Students’ Understanding of Area and Volume in Non-Calculus Contexts Allison Dorko December 5, 2011

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+ Calculus Students’ Understanding of Area and Volume in Non- Calculus Contexts Committee: Dr. Natasha Speer (advisor) Dr. Eric Pandiscio Dr. Robert Franzosa

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+ Introduction & Overview 0 Research about the learning of calculus is of critical importance. Calculus students do not do as well as instructors might like like (Anderson & Loftsgarden, 1987, Bressoud 2005, CBMS 2000, College Board 1999, Jencks & Phillips, 2001, Treisman, 1992). In many topics, calculus students are procedurally competent but lack a rich conceptual understanding (Ferrini-Mundy & Gaudard 1992; Ferrini-Mundy & Graham 1994; Milovanovi ć 2011; Orton 1983; Rasslan & Tall 1997; Rosken 2007; Thompson & Silverman 2008).

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+ Introduction & Overview 1 Three overarching topics in calculus: limits, derivatives, integrals These topics are difficult for students (Cornu, 1981 Orton, 1983a; Orton, 1983b; Tall & Vinner, 1981; White & Mitchelmore, 1996; Zandieh, 2000). Researchers have documented specific difficulties students have with integration (e.g., Orton, 1983) but we don’t fully understand why these difficulties exist.

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+ Introduction & Overview 2 Prior understanding may effect learning of new concepts Function (Carlsen, 1998; Monk, 1987) Variable (Trigueros & Ursini, 2003) The premise for my study is that something similar may be occurring with student understanding of other calculus topics. Specifically, students’ understanding of area and volume in non-calculus contexts may interact with their learning of calculus topics which build on area and volume (e.g., related rates, optimization, integration, volumes of solids of revolution).

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+ Overarching Research Questions 1) Do calculus students have difficulties with area and volume? 2) Are the difficulties calculus students experience with area and volume the same as or different from those documented in elementary school students?

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+ Advanced Organizer Literature Review Two stories: 1. Surface Area Students finding Surface Area when directed to find volume 2. Units Sub stories: length units and circle issues

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+ Literature: Elementary School Students’ Understanding of Area and Volume We don’t know much about how calculus students understand area and volume We do know something about how elementary school students understand area and volume.

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+ Elementary School Students’ Understanding of Area We would like students to understand area/volume as arrays of squares/cubes Students tend to use length units for other spatial measure (Battista & Clements, 1996; Lehrer, 2003) Students have trouble with tiling/arrays Unit Square

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+ Elementary School Students’ Understanding of Volume Battista & Clements (1998) Some tasks written, some with manipulatives This is a picture of a unit cube. How many unit cubes will it take to make each building below? The buildings are completely filled with cubes, with no gaps inside. Unit Cube

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+ Battista & Clements (1998) Task TOP FRONT Right Side

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+ Elementary School Students’ Understanding of Volume Battista & Clements (1998) Students counted cubes on the faces Students could not coordinate orthogonal views Area and volume ideas pose difficulties for elementary school students. Specifically, some elementary school students find surface area when directed to find volume.

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+ Area and Volume in Calculus Area/volume in calculus: Riemann sums Integration Optimization Related rates Volumes of solids of revolution Multiple integration These are difficult concepts for students (Orton, 1983)

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+ Area and Volume in Calculus Area/volume in calculus: Riemann sums Integration Optimization Related rates Volumes of solids of revolution - Students see the integral symbol to mean “do something” or as a “rule for antidifferentiation,” and don’t always think beyond that (Orton, 1983; Petterson, 2008)

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+ Area and Volume in Calculus Area/volume in calculus: Riemann sums Integration Optimization Related rates Volumes of solids of revolution -Ferrini-Mundy (1994) noted a reluctance on students’ part to use the geometric notion of integrals for functions like f(x)=|x|

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+ Area and Volume in Calculus Area/volume in calculus: Riemann sums Integration Optimization Related rates Volumes of solids of revolution Integration finds “area under the curve,” but area is not always area: The units associated with integration are difficult for students (Hall, 2011; Rasslan & Tall, 1997; Sealey, 2006)

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+ Area and Volume in Calculus Area/volume in calculus: Riemann sums Integration Optimization Related rates Volumes of solids of revolution You have been asked to design a one-liter can shaped like a right circular cylinder. What are the dimensions (in centimeters) that would minimize the amount of material needed? A spherical snowball is melting. Its radius decreases at a constant rate of 2 cm per minute from an initial value of 70 cm. How fast is the volume decreasing half an hour later?

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+ Area and Volume in Calculus Area/volume in calculus: Riemann sums Integration Optimization Related rates Volumes of solids of revolution In a set of tasks about integration, performance was the lowest on VoR items (Orton, 1983)

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+ Theoretical Perspective Cognitivist framework Byrnes (2001) definition of cognitive: [a focus] on mental processes such as thinking, learning, remembering, and problem-solving… (p. 3) This theoretical perspective has been used widely in mathematics education research (e.g., Battista & Clements 1998; Hall 2011; Orton 1983a; Orton 1983b; White & Mitchelmore 1996).

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+ Research Design Participants: 255 Calc I students; 43 Calc III students Data collection: two phases Written surveys Clinical interviews Data analysis Grounded Theory (Strauss & Corbin, 1990) Made use of other researchers’ findings about area and volume (e.g., Battista & Clements, 2003; Izsák 2005; Lehrer, 1998; Lehrer, 2003;)

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+ Research Instrument Take a couple minutes to do the circle and triangular prism problems. Answers? What difficulties do you think students might have with these problems?

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+ Surface Area Overview There exists anecdotal evidence from calculus I instructors that students learning optimization find SA when directed to find volume I was interested in documenting this phenomenon, if it exists, and finding out why it occurs

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+ Data Collection & Analysis Tasks analyzed: Grounded Theory Glaser and Strauss (1990) Look for patterns in pieces of data Sort based on patterns Each “pile” of data becomes a category The patterns become the properties of the category Continue until there are no new emergent categories and properties Data is “coded” into categories by their properties My modifications

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+ Coding: Triangular Prism Correct volume: 48 ft 3 Correct surface area: 108 ft 2 Did the student write the formulae V=Bh? Did the student write the formula ½lwh? Did the student write 48? Did the student write 96 If so, categorize as “found volume.” If not, proceed to #2. Did the student write 108? Did the student write “area of two triangles plus area of faces?” Did the student do arithmetic that was finding the areas of the three lateral faces? If so, categorize as “found surface area instead of volume.” If not, proceed to #3. Categorize as “other.”

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+ Found Surface Area: Triangular Prism

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+ Results: Raw Numbers Volume of Rectangular Prism Calc I n=198 Calc III n=43 Found Surface Area30 Found Volume19443 Other10 Volume of Cylinder Calc I n=198 Calc III n=43 Found SA100 Found Volume17243 Other160 Volume of Triangular Prism Calc ICalc III Found SA170 Found Volume9543 Other100

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+ Results: Percentages Do calculus students find surface area when directed to find volume? Yes, for some shapes, most notably triangular prisms, trapezoidal prisms, and washers. Rect. Prism CylinderTri. Prism Trap. Prism Washer % of Calc I students who Found SA 1.5 %5.5 %15.2 %71.4 %14.3 %

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+ Coding: Interview Data Written work: same algorithm as survey data Also analyzed student words for strategy E.g., “I dissected the prism into a box and two triangular prisms” E.g., “I added the areas of all of the faces” Interview data was used primarily to explain the thinking that led to certain types of answers Thinking behind finding surface area Thinking behind finding volume

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+ What thinking leads students to find surface area? Finding 1: Some students think that adding the areas of the faces of an object finds the measure of the object’s volume. Finding 2: Some students understand the difference between area and volume, but confuse the formulae or mix the formulae together. I call this mixed formula (e.g., V=2πr 2 h) an amalgam

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+ Finding 1: Students correctly define volume, but find surface area. They think that the sum of the areas accounts for the measure of the object’s three-dimensional space. Meet Geddy

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+ Geddy Understands Volume Geddy: Volume is the amount of units it takes to occupy a space, like a three dimensional space. For this one, say you just have… I don’t know… a cardboard box and for some reason you wanted to put sugar cubes in it, like you have a big area and you want to know how many little individual units it takes to fill that area. So if you think of a box of sugar cubes, like a Domino box, I think when they come packaged they are usually just full of the little sugar cubes and there’s no space in between those cubes. So that’s what volume is. It’s when you have a bunch of little smaller – for this one, 1 cm cubed pieces – combining to fill a space, like the volume of a space, without any gaps in between.

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+ Geddy Thinks She Has Found Volume Interviewer: So if we were going to fill this shape – you talked about filling something- and we want to fill it without leaving gaps, what shape would you used to do that? Geddy: Well, since it’s 108, it’s an even number of cubes. You’d be able to use squares equal to volume 1 ft cubed and you should be able to fit them all in without having any gaps.

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+ Finding 2: Surface Area-Volume Formulae Amalgam Nell: I don’t know the formula for this one. Two pi r squared … times the height. Sure. We’ll go with that one. So you have two circles at the ends, which is two pi r squared [student does calculations] … is that right ? Interviewer: I can’t answer the question, but tell me about this formula. [points to the student’s 2 r 2 h] Nell: Sure. You have the two pi r squared because that’s the area on the top and the bottom so you can just double it, then you have to times it by the height. Interviewer: Why do I have two areas? Nell: You have two circles. V =2πr 2 h 2π (3 in)^2 2π9 in 18π in(8 in) V=144π in 2

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+ Nell cont’d Interviewer: What about this multiplying by the height? Why do we do that? Nell: It gives you the space between the two areas. Volume is all about the space something takes up so you need to know how tall it is.

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+ Surface Area Conclusion (1) Do calculus students find surface area when directed to find volume? Yes. Some calculus students find surface area when directed to find volume. (2) If calculus students find surface area when directed to find volume, what is the thinking that leads them to do so? Finding 1: Some students think that adding the areas of the faces of an object finds the measure of the object’s volume. Finding 2: Some students understand the difference between area and volume, but confuse the formulae or mix the formulae together.

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+ Calc Students vs. Elementary School Student Some students from both populations find surface area when directed to find volume. Difficulty for elementary school students was coordinating the orthogonal views of the object. Elementary school students have trouble with rectangular prisms. Calc students are relatively okay. Difficulty for calc students: They think Surface Area computations = Object volume They have an amalgam formula As shape complexity increases, Surface Area finding increases These are not difficulties experienced by elementary school students.

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+ Instructional Implications and Suggestions for Further Research Suggestion for Further research: Does the Surface Area -Volume Amalgam interact with these students’ understanding of calculus topics that make use of these concepts? Optimization Volumes of Solids of Revolution Instructional Implication: Create opportunities for students to revisit and strengthen their understanding of prerequisite topics in conjunction with the study of new content

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+ Units Elementary school students often misappropriate units of length for other spatial measures (Lehrer 2003) Units are important in chemistry, physics, some math topics, etc. Instructors tend to note that students struggle with units (Saitta, Gittings, & Geiger, 2011)… … but there are few articles about those specific difficulties or why they occur.

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+ Research Questions (1) Do calculus students write the correct or incorrect units associated with various spatial measures? (2) What thinking occurs when calculus students use the wrong unit for area? (3) What thinking occurs when calculus students use the wrong unit for volume?

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+ Data Analysis Grounded Theory Transcribed student work and looked for patterns Modified as in surface area work Based on literature about elementary school students, I was looking for students who used length for other spatial measures Student #Area Rectangle Area Circle Volume Rect. Prism Volume Cylinder Volume Tri. Prism cm 2 25π in cm 3 72π in 2 48 ft cm 2 25π in cm 3 Don’t remember cm 2 25π in cm 3 72π47 ft cm25π in cm 3 72π in 3 48 ft 3

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+ Patterns Length units used for a measure of area Inconsistent unit use for problems of the same type No unit written with answer, even though units were given in the problem Correct units for all four problems One type of unit for all problems

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+ Example 0 This was coded as “2, Inconsistent unit use for problems of the same type.” Student #Area Rectangle Area Circle Volume Rect. Prism Volume Cylinder Volume Tri. Prism cm 2 25π in cm 3 72π in 2 48 ft 3

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+ Example 1 This was coded as “length units used for all tasks.” Student #Area Rectangle Area Circle Volume Rect. Prism Volume Cylinder Volume Tri. Prism cm10π in200 cm150.8 in48 in

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+ Categories All Correct Student has the correct units for all four questions. Magnitudes may be correct or incorrect. No Units Student has no units for any answer. Magnitudes may be correct or incorrect. Length Units Student has linear units for all four questions. Magnitudes may be correct or incorrect. Area Units Student has square units for all four questions. Magnitudes may be correct or incorrect. Other Student has a mix of correct and incorrect units.

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+ Results Spring Calc I (n=110) Fall Calc I (n=68) Spring Calc III (n=43) Total (n=220) % of Total All correct % No units % Length units % Area units % Other %

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+ Two Notes About the “Other” Category (1) Linear Units 34 students of these 91 had at least one linear unit for a problem. (2) Circle Issues Theme for this category was most of the units correct except the units for the area of the circle. 68 of 91 students had units for Area of Circle wrong 6 of 91 had circle and cylinder units wrong In most cases, “wrong” meant “no unit”

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+ Student Reasoning Behind Answers All correct: most of these students understood area/volume as arrays of cubes/squares and/ or understood dimensionality Steven: area is a square – every time we multiply one dimension by the next, we change from a linear to an area – then area to volume. It’s recognizable that volume is a cubic area, as opposed to area which is squared. Linear Units: “that’s the unit the shape was measured in” Interviewer: Why 200 cm? Rae: That’s the unit the shape is measured in.

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+ Student Reasoning Behind Answers: CIRCLES Interviewer: Cary, you wrote units with all problems except the cylinder. Can you tell me why? Cary: I think it’s because I forgot that. Either that or the pi threw me off and then I forgot. Interviewer: Tell me about the pi throwing you off. Cary: Because pi doesn’t have a unit. I think I forgot because of the unitless pi. Interviewer: Do you know what the units would be for that problem? Does it have them? Cary: Inches cubed.

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+ Circles Cont’d Interviewer: You wrote units cubed for all the other volume problems. Why not for the cylinder? Alanis: I probably didn’t even think of it because I was using pi, so I left pi in it and I didn’t think to label it. But I labeled all the rest of them. That’s really weird. Interviewer: Tell me more about “because you were using pi.” Alanis: Well I know pi is an actual value, but I guess I would … I don’t know. It probably just slipped my mind because I was using pi to represent a number rather than saying 3.14 and I probably just forgot to put a label on it. Interviewer: So, you did the same thing on the circle. put any unit. Your other area problems have units. Alanis: I guess the same thing as the other problem. I probably have a tendency to do that with circles because you really only use pi with circles and it kind of doesn’t have a label on it. And I guess it makes sense that I would use it consistently with circles. You can multiply it out [multiply 25 * 3.14], but I tend to leave pi as pi. I don’t know.

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+ Conclusions, Implications, and Suggestions for Further Research Some elementary school students and some calculus students use one-dimensional units for area or volume The units with circles are particularly difficult. SEEMS TO BE: 48 in 2 72π Number letter Further research is needed to see if this is indeed the case. I can say that “something about the π” interacts with the units the student writes for a circle problem.

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+ Further Research & Instructional Implications Further research: find out about unit understanding in calc concepts, such as integration. Instructional implications: Reinforce unit-attribute relations Exponent rules as applied to area/volume E.g., in * in = in 2

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+ Conclusions, Implications, and Suggestions for Further Research Instructional implications: help students learn units and concepts with “Jeopardy” type questions This has been used successfully in physics (Van Heuvelen & Maloney, 1998) N – (60 kg)(9.8 m/s/s) = 0 Example of use in calc: y x z x 2 + y 2 = z 2

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+ Works Cited 0 Artigue, M. (1991) Analysis, Advanced Mathematical Thinking. In D. Tall (ed.) Kluwer Academic Publishers, Dordrecht, 1991, pp. 167–198. Ball, D. L., Lubienski, S., & Mewborn, D. S. (2001). Research on teaching mathematics: The unsolved problem of teachers' mathematical knowledge. In V. Richardson (Ed.), Handbook of Research on Teaching (pp ). Washington, DC: American Educational Research Association. all, D.L., & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics. In J. Boaler (Ed.), Multiple perspectives on the teaching and learning of mathematics (pp. 83– 104). Westport, CT: Ablex. Battista, M.T., & Clements, D.H. (1998). Students’ understandings of three- dimensional cube arrays: Findings from a research and curriculum development project. n R. Lehrer & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (pp ). Mahwah, NJ: Erlbaum. Battista, M.T., & Clements, D.H. (1996). Students’ understanding of three- dimensional rectangular arrays of cubes. Journal for Research in Mathematics Education, 27(3), Bohren, C.F. (2004). Dimensional analysis, falling bodies, and the fine art of not solving differential equations. American Journal of Physics, 72, p Carman, Robert. (1969). Numbers and units for physics. New York : Wiley.

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+ Works Cited 9 Strauss, A. & Corbin, J. (1990). Basics of qualitative research: Grounded theory procedures and techniques. Newbury Park, CA: Sage. Tall, David. (1986) A graphical approach to integration and the Fundamental Theorem. Mathematics Teaching, (113), p Tall, D. O., & Vinner S. (1981). Concept image and concept definition in mathematics, with particular reference to limits and continuity. Educational Studies in Mathematics, 2, Thompson, P., & Silverman, J. (2006). The concept of accumulation in calculus.. In M. Carlson & C. Rasmussen (Eds.). Making the connection: Research and teaching in undergraduate mathematics education. (pp.27-41). MAA Notes #73. Mathematical Association of America: Thompson, P. W. (1994). Images of rate and operational understanding of the fundamental theorem of calculus. Educational Studies in Mathematics, 26(2-3), Wemyess, T.M., Bajracharya, R.A., & Thompson, J.R. (2011).Student understanding of integration in the context and notation of thermodynamics: Concepts, representations, and transfer.” In Proceedings of the 14th Annual Conference on Research in Undergraduate Mathematics Education. eds. S. Brown, S. Larsen, K. Marrongelle, M. Oehrtman (Mathematical Association of America, 2011).

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+ Works Cited 9 3/4 Tall,D., & Blackett, N. (1986). Investigating graphs and the calculus in the sixth form. In N. Burton (Ed.). Exploring mathematics with microcomputers. (pp ). London: Council of Educational Technology. Thompson, P., & Silverman, J. (2006). The concept of accumulation in calculus. In M. Carlson & C. Rasmussen (Eds.). Making the connection: Research and teaching in undergraduate mathematics education. (pp.43-52). MAA Notes #73. Mathematical Association of America: Treisman, Uri. (1992). Studying Students Studying Calculus: A Look at the Lives of Minority Mathematics Students in College. The College Mathematics Journal. Vol. 23, No. 5 (Nov., 1992), pp Vinner, S. (1983). Concept definition, concept image, and the notion of function. International Journal of Mathematics Education in Science and Technology, 14(3), Vinner, S. (1987). Continuous functions – Images and reasoning in college students. In J.C. Bergeron, N. Herscovics, and C. Kieran (Eds.), Proceedings of the Eleventh International Conference of the Psychology on Mathematics Education (pp ). White, P., & Mitchelmore, M. (1996). Conceptual knowledge in introductory calculus. Journal for Research in Mathematics Education, 27(1)

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+ Works Cited 10 Wildi, T. (1991). Units and conversions: A handbook for engineers and scientists. New York: IEEE Press. Woodward, E. & Byrd, F. (1983). Area: Included topic, neglected concept. School Science and Mathematics, 83(4), Zandieh, M. J. (2000). A theoretical framework for analyzing student understanding of the concept of derivative. In E. Dubinsky, A. H. Schoenfeld & J. Kaput (Eds.), CBMS Issues in Mathematics: Research in Collegiate Mathematics Education (Vol. IV(8), pp. 103–127).

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+ TASKSTASKS 3 ft 4 ft 5 ft 8 ft 7.

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+ Tasks By Class and Data Type Calc I Fall Calc I Fall Calc I Spring Calc I Spring Calc I Summer Calc III Fall SurveyInterviewSurveyInterview Survey 1aArea Rectangle XXXXX 1bArea Circle XXXXX 2aVolume Rect. Prism. XXXXXX 2bVolume Cylinder XXXX 3Volume Tri. Prism. XXXXX 6Volume Trap. Prism XXX 7Volume Washer XX

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+ An Example of an “Other” Categorization

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