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PRIMUS: Problems,Resources, and Issues inMathematics UndergraduateStudiesPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/upri20
WRITING CONCEPTESTS FORA MULTIVARIABLE CALCULUSCLASSMark D. Schlatter PhD aa Department of Mathematics , CentenaryCollege of Louisiana , 2911 Centenary Blvd.,Shreveport, LA, 71104, USA E-mail:Published online: 13 Aug 2007.
To cite this article: Mark D. Schlatter PhD (2002) WRITING CONCEPTESTSFOR A MULTIVARIABLE CALCULUS CLASS, PRIMUS: Problems, Resources,and Issues in Mathematics Undergraduate Studies, 12:4, 305-314, DOI:10.1080/10511970208984036
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Schlatter Writing ConcepTests for a Multivariable Calculus Class
WRITING CONCEPTESTSFOR A MULTIVARIABLE
CALCULUS CLASS
Mark D. Schlatter
ADDRESS: Department of Mathematics, Centenary College of Louisiana,2911 Centenary Blvd., Shreveport LA 71104 USA. mschlat@centenary. edu.
ABSTRACT: In a multivariable calculus course, students must master alarge number of concepts in order to successfully learn the material.This paper will discuss one way of addressing this difficulty throughthe use of ConcepTests, that is, multiple choice questions given in thelecture that test understanding as opposed to calculation. In particular, we will look at various types of ConcepTests and the materialthey can cover .
KEYWORDS: Multivariable calculus, ConcepTests, small group work .
INTRODUCTION
As I started preparing for Centenary's multivariable calculus course in thefall of 2000, I was wondering how I could help student understanding. Ihad taught the course the past two years and had incorporated MATLABprograms to help the students visualize surfaces , curves, and vector fields.Even with this help , a significant number of students had difficulties understanding multivariable and vector concepts, particularly in the latter half ofthe course. Unfortunately, due to requirements on coverage of course material (our syllabus covers almost all of the Harvard multivari able calculusbook [2]), I was not able to dramatically slow down the pace. I wanted tofind a way to gauge and improve student understanding and provide a solidfoundation in th e essential concepts.
Let me provide some background. First, our mul tivari able calculus classis offered every fall with between 10 and 20 students (Centenary has 860
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i'~iIilU) December 2002 Volume XII Number 4
undergraduates). These students include mathematics, physics, and engineering majors as well as a few students from other disciplines. The firsttime I taught the class (Fall 1998), we met about 20% of the time in acomputer lab equipped with MATLABj the second year (Fall 1999) we metfull-time in the lab. The class meets five days a week for 50 minutes aday with a typical week consisting of four days of lecture and one day todiscuss homework. Students consistently had problems with the followingconcepts: dot product, interpretations of the gradient, Lagrange multipliers, integrating with cylindrical and spherical coordinates, and line and fluxintegrals. While students could sometimes perform calculations with thismaterial, they often had troubles explaining the concepts or attacking unfamiliar problems .
After attending the Associated Colleges of the South's Teaching andLearning Workshop and being introduced to ConcepTests, I decided to tryusing them in my multivariable calculus course.
HISTORY OF CONCEPTESTS
ConcepTests were developed by Eric Mazur, a Harvard physics professor.He had noticed that his introductory physics students could handle computational problems, but could not solve similar problems if the calculationswere removed and only the underlying concept assessed. In other words,students appeared to be mistaking 'plug and chug' problem solving skillsfor understanding.
To address this problem, Mazur started using ConcepTests - multiplechoice questions that students could answer in their heads if they correctlyunderstood the concepts. In lecture, a test would be presented, studentswould vote on the correct answer and then break into small groups to discusstheir votes, and finally a revote would be taken. Students would thereforeengage the concepts in class through their votes and discussions with theirclassmates. A typical lecture might consist of several tests, with less timespent on examples. A longer discussion of the implementation and effectiveness of ConcepTests can be found in [1] . In addition, Scott Pilzer in [3]shows how these tests can be implemented in a first-year calculus course.
WRITING CONCEPTESTS
In my Fall 2000 multivariable calculus course, I used ConcepTests, withabout 20 minutes of each class period devoted to the tests. I will not bediscussing the specifics of implementing ConcepTests in the classroom - I
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Schlatter Writing ConcepTests for a Multivariable Calculus Class
refer the reader to Scott Pilzer's article [2] for that information. Instead, Iwant to focus on the types of tests I wrote for my class and the materialeach type was best suited for. Roughly speaking, my tests fell into one offive types: 1) visualization, 2) comparison, 3) translation, 4) theorem-using,and 5) theorem-provoking.
Visualization Tests: These tests were designed to develop the students'ability to think in three dimensions. One example is:
Example 1: The set of all points whose distance from the z-axis is 4is the:
a) sphere of radius 4 centered on the z-axis
b) line parallel to the z-axis 4 units away from the origin
c) cylinder of radius 4 centered on the z-axis
d) plane z = 4
As is typical for ConcepTests, note that this problem is answerablewithout symbolic calculation. In this example, I am seeing if studentsclearly understand the freedom of three dimensional space. A commonanswer on the first vote was b) - students saw some of the points thatwere four units away from the z-axis, but not all of them. Anothercommon mistake was a) - students were ready to assume that any setdescribed as an equal distance from something was a sphere. In bothcases, the discussion between votes cleared up the misapprehensionsfor most students.
I used visualization tests early in the semester, when the class focusedon the three-dimensional coordinate system and various cross-sectionsof surfaces. Later on in the semester, I used this type of test to helpstudents visualize vector fields. Here is an example:
Example 2: Which of the following formulas will produce a vectorfield where all vectors move away from the y axis?
a) F(x, y) = (x3)i
b) F(x, y) = (x2 )ic) F(x, y) = (x3)j
d) F(x, y) = (x2 )j
In this case, I wanted students to be able to look at a vector field andcarry out some straightforward visualization. Students were quicklyable to rule out c) and d) based on direction, leaving most of thediscussion on the differences between a) and b).
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December 2002 Volume XII Number 4
Throughout the course, I introduced software to help the studentsvisualize shapes and fields. But before I showed students these tools, Iused visualization tests so that the students developed a solid intuitionand could use software to confirm their suspicions.
Comparison Tests: I used this type of test most often during the semester. These questions involved determining the sign of a quantity or itsrelative magnitude. One example is:
Example 3: In which direction is the directional derivative of z =x 2 + y2 at the point (2,3) most positive? (We are using i and j as theunit vectors in the x and y directions.)
a)b) -i - j
c) -i + j
d) i + j
I would use this test before giving the formula for computing directional derivatives with the dot product. Here I am asking the studentsto combine several concepts: their visual picture of z = x2 + y2 (a surface we had looked at previously), their visual pictures of the fourvectors, and their understanding of the directional derivative. An initial vote on this test produced lots of guessing, but the discussionperiod was very fruitful. Students would quickly review their mentalpictures of the surface and vectors and usually rule out b) and c) .The discussion would then focus on how you might compare the directional derivative in the directions given by a) and d). Even if thesecond vote was split between the two, the class was then primed tolook at a contour plot of the function and argue from there.
What I liked about these type of questions is that they pushed students towards considering the different parameters that affect a scalarquantity in multivariable calculus (e.g., the function and the directionin the above example). Discussions between the votes were then especially helpful since teams of students could usually identify all thenecessary parameters. I used these types of tests extensively whendiscussing vector arithmetic (including velocity and acceleration vectors and dot product), partial and directional derivatives, the gradientvector, and line and flux integrals.
Translation Tests: When I arrived at the chapter on integration, I wasnot sure how to use ConcepTests. Of course, the concept of Riemann sums over rectangles is important, but the bulk of the textbook
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Schlatter Writing ConcepTests for a Multivari abl e Calculus Class
material covers techniques of integrati on and integration in differentcoordinat e syst ems. Since we were using a computer algebra syste min class, I decid ed to focus less on calculation and more on a specific problem student s had in past semesters: translating a particularint egral into a specific coordinate system. One example is:
Example 4: Whi ch of the following is equivalent to
J5 13 J~ x dy dz dx?- 5 0 - .j25-x2
a) Jo" J03J~,2 cos(e) dz dr de
b) J; J~ J; ,2cos(e) dz dr de
c) J02" J; J~ r cos(e) dz dr de
d) J~" J; J~ r2cos(e) dz dr de
Not e that this test does not focus on an underlying concept in thesame way the above test s do. However , student s st ill benefited fromthe voting and discussion phases. One definite adva ntage was that th eclass had to visua lize five different integrals in two different coordina tesystems . Student s quickly picked up the differences between the limitsof th e four answers and what spaces they describ ed. Answers c) andd) were both given to make sure students rememb ered to multiply bya factor of r .
In addition to th e sections on integration, I wrot e ConcepTest s likethe below for parametric curves and surfaces and the parametrizationof line integrals. Here is an example:
Example 5: Whi ch of the following is equivalent to the line integralof F (x ,y) on the line segment from (1, 1) to (3, 4)?
a) J01 F (1 + 2t , 1 + 3t ) dt
b) J01 F (1 + 2t , 1 + 3t ) . (2i + 3j) dt
c) J01 F (3, 4) . (2i + 3j ) dt
d) J01F (1 + t , 1 + t ) . (2i + 3j) dt
Again, the focus is not so much on an underlying concept as it is onstudents recognizin g how to find the line integral of a parametrization.Here, student discussion would focus on the differences between b) andd).
With tr anslation test s I found there to be a higher risk of having alar ge portion of the class making random guesses on the first vote .
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December 2002 Volume XII Number 4
These tests have more of an 'either you get it or you don't' qualitythan the others I developed for the class. I still valued them, however,because I was able to see where students were having problems duringthe discussion time.
Theorem-Using Tests: With these types of tests, I assessed whether thestudents knew how and when to apply a theorem. One example is:
Example 6: Which of the following facts about the vector fieldF(r) = r (where r is a position vector) is implied by Stoke's Theorem?
a) The line integral from (0,0,0) to (1,1,1) is equal to ~ .
b) F(r) = r has positive divergence everywhere.
c) The line integral on any closed curve is zero.
d) The curl of F(r) = r is non-zero.
I did not want the students to carry out a specific calculation usingStoke's Theorem, but to understand its consequences. Here I amlooking for students to combine their mental picture of the vectorfield (we had discussed it earlier in class) with their understandingthat Stoke's Theorem concerns the curl of a vector field. Once theywere able from their picture to see that the curl of F was zero, theywere able to move to c) as a correct answer. (One of the reasons d) isincluded is to prompt students to move in that direction.)
I also used theorem-using tests extensively when we covered optimization and the classification of critical points. In both cases, I was ableto help the students understand the power and limitations of the theorems we used .
Theorem-Provoking Tests: Occasionally throughout the course, I woulduse ConcepTests not as a way of assessing student understanding ofcovered material, but to ready them for new material. One exampleis:
Example 7: The plot in Figure 1 shows the gradient vectors fora (hidden) function f(x, y) and a linear constraint . Which point isclosest to the global min of f(x, y) on this constraint?
a) A
b) Bc) C
d) D
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Figure 1. Plot for Example 7.
My goal here was to help the students see that at a global minimum ona constraint, the gradients of the objective and constraint functionsare parallel. I wanted the students to use their conception of whatthe gradient tells them to prepare them for the method of Lagrangemultipliers.
I also used tests like this when we discussed normals to surfaces inpreparation for parametrizing flux integrals. For those sections whichwere almost purely computational, this type of test gave me the opportunity to engage the students before we got to the symbolic manipulation.
I wrote a total of 89 ConcepTests for my multivariable calculus class,covering almost all of the topics in the Harvard multivariable calculus book.After using these in my Fall 2000 multivariable course, I refined the tests andused them again in my Fall 2001 course. You can download the collectionat http://personal.centenary.edu/-mschlat/conceptests.pdf.
REACTION
When I started using the ConcepTests I discovered how much they enhancedstudent feedback and my understanding of the students' abilities. The useof ConcepTests resulted in an active class that was not afraid to ask questions or make comments. In my experience, the discussion period betweenvotes primes the students for further discussion in class by allowing themto focus on a carefully defined question. I saw a greater range of student
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December 2002 Volume XII Number 4
participat ion (as compared to my ot her classes) when it came t ime for st udent s to explain t heir votes . Indeed , some of the weaker students were themost vocal participants.
I benefited most as a teacher from walking around t he class room duringt he vot e discussions. (Our computer lab has an open area where it is easyto form groups and move between th em. ) I was able to hear st udents'discussion and act as an advocate for different points of view. Quite ofte n ,I would give a group a leading question based on their discussion. When wetook the revote, I frequently discussed the reasoning different groups hadused . All this meant I was better ab le to focus the material following theConcepTest - in fact , it was not uncommon for me to change my lecture ifa ConcepTes t had proven too difficult or controversial. This contact withthe groups also meant that very early in the semester I was ab le to gaugeindiv idu al st udent abilities.
T wo pieces of evidence at the end of the Fall 2000 semester pointed to theeffect iveness of the tests. First, I received some of the best written studentevaluat ions in my career, with several students specifically stat ing how theConcepTests had helped. Second, of the four times I have taught this course,the Fall 2000 semester class was the most successful in keeping st udentinterest. I had fewer students who stopp ed coming to class , stopped turn ingin homework , or had large drops in exam scores than in ot her semesters.When I taught the class in fall of 1998, I had 13 st ude nts initially enrolled,12 who too k the final , and 9 wit h a grade of C or above . In fall of 1999,there were 19 initially enro lled , 18 who too k the final , and 14 wit h a Cor above. In the fall of 2000 when I used the ConcepTests , there were 20init ially enro lled, 20 who too k the final , and 18 with a C or above. Theseclasses are not directly comparable - my exams in fall of 2000 did includeConcepTests while previous classes did not - bu t I did notice fewer studentswho 'gave up ' throughout the semester.
When I repeated the use of ConcepTests in the fall of 2001, I had 12students initially enro lled with 9 who too k the final. All of them earned aC or above. While there was a higher dro p ra te (all three drops came in thefirst half of the semester ), those who staye d in the class fully pa rt icipatedthroughout the semester. There was, however, a problem wit h using ConcepTests with a class that small. T here were fewer viewpoint s expressed andless chance of a correct answer percolating through the discussion period.At the same t ime, the benefits to the instructor were st ill present, and thediscussion t ime was often fruit ful for st udents .
Finally, one of the most int eresti ng pieces of feedback I got was from ast udent who reported that when he started st udyi ng for one of my exams,
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Schlatter Writing ConcepTests for a Multivariable Calculus Class
the first thing he did was to go through all the ConcepTests. His reasonbehind this was not only to prepare for the ConcepTests on the exam,but because reviewing the tests helped him to go back in time to the dayhe learned the material. Apparently, the ConcepTests provided him with'mental landmarks' in the course.
FUTURE PLANS
My first use of ConcepTests in the Fall 2000 semester was primarily intendedas 3, proof of concept - would the tests work in a mathematics classroom andwould student understanding be improved? The answer to the first questionwas a definitive yes. Given my experience with both the Fall 2000 and Fall2001 classes, the answer to the second question is a qualified yes. WhileI have not used a common instrument to compare student understandingbetween those using ConcepTests and those not, the students who havetaken the ConcepTests show a greater comfort in talking about and usingthe material on a conceptual basis. In addition, the use of ConcepTestsappears to prevent students from falling behind or losing interest in theclass. My future plans are to see how ConcepTests can be used in otherclasses, especially our college algebra course.
ACKNOWLEDGEMENTS
I would like to thank the Associated Colleges of the South's Teaching andLearning Workshop, where I was introduced to ConcepTests by Duane Pontius from Birmingham-Southern College and was given the encouragementto experiment with my teaching.
REFERENCES
1. Mazur, Eric. 1996. Peer Instruction: A User's Manual. New Jersey:Prentice-Hall.
2. McCallum, William G., Deborah Hughes-Hallett, Andrew M. Gleasonet al. 1997. Multivariable Calculus. New York: John Wiley & Sons.
3. Pilzer, Scott. 2001. Peer Instruction in Physics and Mathematics.PRIMUS. 11(2): 185-192.
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Decemb er 2002 Volume XII Number 4
BIOGRAPHICAL SKETCH
Mark D. Schlatter received his PhD in mathematics from the University ofCa liforn ia at Berkeley. Originally specializing in mathematical logic with afocus on model theory, he has since branched out to nonnegative matrix theory, the mathematics of art, and curriculum development. After three yearsas a Visiting Assistant P rofessor at Truman State University in KirksvilleMO , he is now an Assist ant Professor at Centenary College.
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