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Calculus Students’ Understanding of Area and Volume in Non-Calculus Contexts

Allison DorkoDecember 5, 2011

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Calculus Students’ Understanding of Area and Volume in Non-Calculus Contexts

Committee: Dr. Natasha Speer (advisor)Dr. Eric PandiscioDr. Robert Franzosa

+Introduction & Overview0

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; Milovanović 2011; Orton 1983; Rasslan & Tall 1997; Rosken 2007; Thompson & Silverman 2008).

+Introduction & Overview1

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.

+Introduction & Overview2

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).

+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?

+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

+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.

+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

+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

+Battista & Clements (1998) Task

TOP

FRONT

Right Side

+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.

+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)

+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)

+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|

+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)

+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?

+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)

+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).

+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;)

+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?

+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

+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

+Coding: Triangular Prism

Correct volume: 48 ft3

Correct surface area: 108 ft2

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.”

+Found Surface Area: Triangular Prism

+Results: Raw Numbers

Volume of Rectangular Prism

Calc In=198

Calc IIIn=43

Found Surface Area

3 0

Found Volume 194 43

Other 1 0

Volume of Cylinder

Calc In=198

Calc IIIn=43

Found SA 10 0

Found Volume 172 43

Other 16 0

Volume of Triangular Prism

Calc I Calc III

Found SA 17 0

Found Volume 95 43

Other 10 0

+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

Cylinder Tri.Prism

Trap. Prism

Washer

% of Calc I students whoFound SA

1.5 % 5.5 % 15.2 % 71.4 % 14.3 %

+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

+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=2πr2h) an amalgam

+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

+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.

+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.

+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 r2h] 

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 =2πr2h2π (3 in)^22π9 in18π in(8 in)V=144π in2

+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.

+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.

+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.

+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

+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.

+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?

+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 #

AreaRectangle

AreaCircle

VolumeRect. Prism

VolumeCylinder

VolumeTri.Prism

114 48 cm2 25π in2 200 cm3 72π in2 48 ft3

115 48 cm2 25π in2 200 cm3 Don’t remember

Don’t remember

116 48 cm2 25π in2 200 cm3 72π 47 ft3

117 48 cm 25π in2 200 cm3 72π in3 48 ft3

+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

+Example0

This was coded as “2, Inconsistent unit use for problems of the same type.”

Student #

Area Rectangle

AreaCircle

VolumeRect. Prism

VolumeCylinder

VolumeTri.Prism

114 48 cm2 25π in2 200 cm3 72π in2 48 ft3

+Example1

This was coded as “length units used for all tasks.”

Student #

Area Rectangle

AreaCircle

VolumeRect. Prism

VolumeCylinder

VolumeTri.Prism

125 48 cm 10π in 200 cm 150.8 in 48 in

+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.

+Results

Spring Calc I(n=110)

Fall Calc I(n=68)

Spring Calc III(n=43)

Total(n=220)

% of Total

All correct

42 42 38 122 55.5 %

No units 1 2 0 3 1.5 %

Length units

2 1 0 3 1.5 %

Area units

1 0 0 1 0.5 %

Other 65 21 5 91 41.0 %

+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”

+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.

+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.

+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.

+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 in2

72π Number letter 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.

+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 = in2

+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

x2 + y2 = z2

+Works Cited0 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. 433-456). 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. 227-248). 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), 258-292.

Bohren, C.F. (2004). Dimensional analysis, falling bodies, and the fine art of not solving differential equations. American Journal of Physics, 72, p. 534-537.

Carman, Robert. (1969). Numbers and units for physics. New York : Wiley.

+Works Cited1 Cavanagh, M. (2007). Year 7 students' understanding of area measurement. In K.

Milton, H. Reeves & T. Spencer (Eds.), Mathematics: Essentials for Learning, Essential for Life (Proceedings of the 21st Biennial Conference of the Australian Association of Mathematics Teachers, pp. 136 - 143. Adelaide: AAMT

CBMS. (2002). Statistical Abstract of Undergraduate Programs in the Mathematical Sciences in the United States: Fall CBMS Survey. Lutzer, D., Maxwell, S. (Eds.). Providence, RI: American Mathematical Society.

Confrey, J. (1980). Conceptual change, number concepts and the introduction to calculus. Unpublished doctoral dissertation, Cornell University, Ithaca, NY.

Cornu, B. (1981). Apprentissage de la notion de limite: Modbles spontanes et modules propres. [Learning the limit concept: Individual and suitable models.]

Actes du Cinquihme Colloque du Groupe Internationale PME (pp. 322-326).Grenoble: Universit6 de Grenoble.

Clements, D. (2003) Teaching and learning geometry. In Kilpatrick, J., Schifter, D. and G. Martin (Eds.), A Research Companion to the Principles and Standards for School Mathematics. pp. 151-178. Reston: NCTM.

De Corte, E., Verschaffel, L., & Van Collie, V. (1988). Influence of number size, problem structure, and response mode on children’s solutions of multiplication word problems. Journal of Mathematical Behavior; 7, 197-216.

Dick, B. (2005). Grounded theory: A thumbnail sketch. Retrieved from http://www.uq.net.au/action_research/arp/grounded.html

+Works Cited2 Dreyfus, T., and Einsberg, T. (1983). The function concept in college students:

Linearity, smoothness, and periodicity. Focus on Learning Problems in Mathematics, 5, 119-132.

Ehrlinger, J., and Dunning, D. (2003). How chronic self-views influence (and potentially mislead) estimates of performance. J. Pers. Soc. Psychol. 84 (2003), pp. 5–17.

Elby, A. (2000). What students’ learning of representations tells us about constructivism. Journal of Mathematical Behavior. Vol. 19, p. 481-502.

Even, R., Lappan, G., and Fitzgerald, W. (1988). Pre-service teachers’ conceptions of the relationship between functions and equations. Proceedings of the Tenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp.238-289). DeKalb, IL: IGPME-NA.

Fay, T.H., and Joubert, S.V. (2002). Dimensional analysis: an elegant technique for facilitating the teaching of mathematical modeling. International Journal of Mathematical Education in Science and Technology. Vol. 33 p. 280-293.

Fennema, E., Carpenter, T., Franke, M., Levi, L., Jacobs, V. R., & Empson, S. B. (1996). A longitudinal study of learning to use children's thinking in mathematics Instruction. Journal for Research in Mathematics Education, 27(4), 403-434.

Ferrini-Mundy, J. (1987). Spatial training for calculus students: sex differences in achievement and in visualization ability. Journal for Research in Mathematics Education (18)2, p.126-140.

+Works Cited3 Ferrini-Mundy, J., & Graham, K. (1994). Research in Calculus Learning:

Understanding of Limits, Derivatives, and Integrals. In J. J. Kaput & E. Dubinsky (Eds.), Research Issues in Undergraduate Mathematics Learning: Preliminary Analysis and Results (Vol. 33, pp. 29-45). Washington, DC: The Mathematical Association of America. 

Ferrini-Mundy, J., and Guadard, M. (1992). Secondary school calculus: preparation or pitfall in the study of college calculus? Journal for Research in Mathematics Education, (23)1, p.56-71.

Fischbein, E. (1987). Intuition in science and mathematics: An educational approach. Dordrecht: Kluwer.

Fuys, D., Geddes, D., & Tischler, R. (1988). The van Hiele model of thinking in geometry among adolescents. Journal for Research in Mathematics Education Monograph 3. Reston: NCTM 1988.

Glaser, B. (1992). Basics of grounded theory analysis: emergence vs. forcing. Mill Valley, CA: Sociology Press.

Glaser, B., & Strauss, A. (1967). The discovery of grounded theory. Chicago: Aldine.

Graham, K.G., and Ferrini-Mundy, J. (1989). An exploration of student understanding of central concepts in calculus. Paper presented at the Annual Meeting of the American Educational Research Association, San Francisco, CA.

+Works Cited4 Hall, Jr., W.L. (2010). Student misconceptions of the language of calculus:

Definite and indefinite integrals. In Proceedings of the 13th Annual Conference on Research in Undergraduate Mathematics Education (Mathematical Association of America).

Hirstein, J., Lamb, C., & Osborne, A. (1978). Student misconceptions about area measure. The Arithmetic Teacher, 25(6), 10-16.

Hirstein, J.J. (1981). The second national assessment in mathematics: Area and volume. Mathematics Teacher, 74, 704-708.

Izsák, A. (2004). Students’ coordination of knowledge when learning to model physical situations. Cognition and Instruction, 22(1), 81-128.

Izsák, A. (2005). You have to count the squares: Applying knowledge in pieces to learning rectangular area. Journal of Learning Sciences, 14, 361-403.

Isbell, Douglas and Savage, Don. (1999). Mars Climate Orbiter Failure Board Releases Report, Numerous NASA Actions Underway in Response. http://mars.jpl.nasa.gov/msp98/news/mco991110.html 10 Nov 1999. Accessed 9 October 2011.

Jencks, C., & Phillips, M. (Eds). (2001). The Black-White test score gap. Washington, DC: Brookings Institution.

+Works Cited5 Kutner, M. H., Nachtsheim, C.J., Neter, J., &Li, W. (2005). Applied Linear Statistical Models.

New York, NY: McGraw-Hill Irwin.

Lehrer, R. (2003). Developing understanding of measurement. In J. Kilpatrick, W. G. Martin, & D. E. Schifter (Eds.), A Research Companion to Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.

Lehrer, R., Jacobson, C., Thoyre, G., Kemeny, V., Strom, D., Horvath, J., Gance, S., & Koehler, M. (1998a). Developing understanding of space and geometry in the primary grades. In R. Lehrer & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (pp. 169-200). Mahwah, NJ: Erlbaum.

Lehrer, R, Jenkins, M., & Osana, H. (1998b). Longitudinal study of children’s reasoning about space and geometry. In R. Lehrer & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (pp. 137-167). Mahwah, NJ: Erlbaum.

Logan, J.D. (1987). Applied Mathematics: A Contemporary Approach. New York: John Wiley and Sons.

Mack, (2001). Building on informal knowledge through instruction in a complex content domain: Partitioning, units, and understanding multiplication of fractions. Journal for Research in Mathematics Education, Vol. 32(3), 267-295.

Mars Climate Orbiter Mishap Investigation Board. (1999). Mars Climate Orbiter Mishap Investigation Board Phase I Report. ftp://ftp.hq.nasa.gov/pub/pao/reports/1999/MCO_report.pdf 10 November 1999. Accessed 9 October 2011.

+Works Cited5 McDermott, L. C, Rosenquist, M. L., & van Zee, E. H. (1987). Student difficulties in

connecting graphs and physics: Examples from kinematics. American Journal of Physics, 55,503-513.

Milovanović, M., Takači, Đ., and Milajić, A.(2011). Multimedia approach in teaching mathematics - example of lesson about the definite integral application for determining an area. International Journal of Mathematical Education in Science and Technology, 42: 2, 175-187, First published on: 17 January 2011 (iFirst).  

Monk, G.S. (1989). A framework for describing student understanding of functions. Paper presented at the Annual Meeting of the American Educational Research Association, San Francisco, CA.

National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Carpenter, J., Gorg, S., and Martin, W.G. (Eds.). Reston, VA: The National Council of Teachers of Mathematics, Inc.

National Governors Association Center for Best Practices and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Author. Retrieved from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf. Accessed 10/27/11.

Nesher, P. (1992). Solving multiplication word problems. In G. Leinhardt, R. Putnam, & R.A. Hattrup (Eds.), Analysis of arithmetic for mathematics teaching (pp. 189-219). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.

Nurrenbern, Susan and Pickering, Miles. "Concept Learning vs. Problem Solving: Is There a Difference?" Journal of Chemical Education 64(6):508-510, June, 1987.

+Works Cited6

Oehertman, M., Carlson, M., & Thompson, P. W. (2008). Foundational Reasoning Abilities that Promote Coherence in Students’ Function Understanding. 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: 2008.

Orton, A. (1983a). Students’ understanding of differentiation. Educational Studies in Mathematics, 15, 235-250.

Orton, A. (1983b). Students’ understanding of integration. Educational Studies in Mathematics, 14, 1-18.

Outhred, L.N., & Mitchelmore, M.C. (2000). Young children’s intuitive understanding of rectangular area measurement. Journal for Research in Mathematics Education, 2, 144-167.

Peled, I., & Nesher, P. (1988). What children tell us about multiplication word problems. Journal of Mathematical Behavior, 7 , 239-262.

Piaget, J., & Inhelder, B. (1956). The child’s conception of space. London: Routledge and Kegan Paul. (Original work published 1948).

+Works Cited7

Piaget, J. Inhelder, B., & Szeminska, A. (1960). The Child’s Conception of Geometry. New York: Basic Books.

Pettersson, K., & Scheja, M. (2008). Algorithmic contexts and learning potentiality: A case study of students’ understanding of calculus. International Journal of Mathematical Education in Science and Technology, 39(6), 767-784.

Price, J.F. (2003). Dimensional analysis of models and data sets. American Journal of Physics. Vol. 71. p. 437-447.

Raghavan, K., Sartoris, M., Glaser, R. (1998). Interconnecting science and mathematics concepts. . In R. Lehrer & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (pp.267-294). Mahwah, NJ: Erlbaum.

Rasslan, S., and Tall, David. (2002). Definitions and images for the definite integral concept. in: A. D. Cockburn & E. Nardi (eds.) Proceedings of the 26th Conference PME, Norwich, 4, 89-96.

Redish, E.F. (1997). Student difficulties with math in the context of physics. Available online at: http://www.physics.umd.edu/perg/papers/redish/talks/math

+Works Cited8

Rosken, B., and Rolka, K. (2007). Integrating intuition: the role of concept image and concept definition for students’ learning of integral calculus. The Montana Mathematics Enthusiast, Monograph 3, p181-204.

Rozema, Edward. Estimating the error in the trapezoidal rule. The American Mathematical Monthly, Vol. 87, No.2 (Feb.,1980), p.124-128.

Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14.

Sealey, V. (2006). Definite integrals, Riemann sums, and area under a curve: what is necessary and sufficient? PME-NA 2006 Proceedings. Vol. 2 pp.46-53. Alatorre, S., Cortina, J.L., Sáiz, M., and Méndez, A.(Eds) (2006). Proceedings of the 28 annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Mérida, México: Universidad Pedagógica Nacional.

Siegler, R. (2003). Implications of cognitive science research for mathematics education. In Kilpatrick, K., Marting, G., and Schifter, D. (Eds.) A Research Companion to Principles and Standards for School Mathematics. P. 289-303. Reston, VA: National Council of Teachers of Mathematics.

Simon, M., & Blume, G. (1994). Building and understanding multiplicative relationships: A study of prospective elementary school teachers. Journal for Research in Mathematics Education, 25, 472-494.

+Works Cited9

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.48-51.

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, 151-169.

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: 2008.

Thompson, P. W. (1994). Images of rate and operational understanding of the fundamental theorem of calculus. Educational Studies in Mathematics, 26(2-3), 229-274.

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).

+Works Cited9 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.156-175). 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: 2008.

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. 362-372

Vinner, S. (1983). Concept definition, concept image, and the notion of function. International Journal of Mathematics Education in Science and Technology, 14(3), 293-305.

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. 177-183).

White, P., & Mitchelmore, M. (1996). Conceptual knowledge in introductory calculus. Journal for Research in Mathematics Education, 27(1) 79-95.

+Works Cited10

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), 343-341.

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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TASKS

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5 ft

8 ft

7.

+Tasks By Class and Data TypeCalc IFall

Calc IFall

Calc ISpring

Calc ISpring

Calc ISummer

Calc IIIFall

Survey Interview Survey Interview Interview Survey

1a AreaRectangle

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1b AreaCircle

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2a VolumeRect. Prism.

X X X X X X

2b VolumeCylinder

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3 VolumeTri. Prism.

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6 VolumeTrap. Prism

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7 VolumeWasher

X X

+An Example of an “Other” Categorization

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