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Using the Computer AlgebraSystem Maple to GenerateResearch Questions for Pre-Service Teachers in a CapstoneCourseRosemary Carroll FarleyAccepted author version posted online: 12 Mar2013.Published online: 10 May 2013.

To cite this article: Rosemary Carroll Farley (2013) Using the Computer AlgebraSystem Maple to Generate Research Questions for Pre-Service Teachers in a CapstoneCourse , PRIMUS: Problems, Resources, and Issues in Mathematics UndergraduateStudies, 23:4, 367-376, DOI: 10.1080/10511970.2012.736452

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PRIMUS, 23(4): 367–376, 2013Copyright © Taylor & Francis Group, LLCISSN: 1051-1970 print / 1935-4053 onlineDOI: 10.1080/10511970.2012.736452

Using the Computer Algebra System Maple toGenerate Research Questions for Pre-Service

Teachers in a Capstone Course

Rosemary Carroll Farley

Abstract: At Manhattan College, secondary mathematics education students take acapstone course designed specifically for them. In this course, students revisit impor-tant topics in the high school curriculum from a mathematically advanced perspective;incorporating the mathematical knowledge they have attained in their college mathe-matics classes to an analysis of high school topics. Various technological tools are usedwhere appropriate. This paper will provide specific examples of what has worked in thiscourse, including information about student research projects that evolved from coursematerials.

Keywords: Mathematics education, technology in mathematics, undergraduateresearch.

1. INTRODUCTION

At Manhattan College, mathematics education students who intend to becomesecondary school teachers take a capstone course designed specifically forthem. In this course, students reconsider important topics in the high schoolcurriculum from a mathematically advanced perspective. The students areexpected to incorporate the mathematical knowledge they have attained in theircollege mathematics classes to analyze high school topics in a mathematicallymature way. While selected topics may change depending on the interests andconcerns of particular students, the level of mathematical maturity demandedof the students remains constant.

This is a senior level mathematics course and the topics chosen for thiscourse reflect this fact. The course content is offered in an environment that

Address correspondence to Rosemary Carroll Farley, Department of Mathematicsand Computer Science, Manhattan College Parkway, Manhattan College, Bronx,New York, NY 10471, USA. E-mail: rosemary.farley@manhattan.edu

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encourages pre-service education students to explore, discover, and understandmathematics using the types of technological tools they will be expected to usein their own teaching classrooms. This paper will provide specific examples ofwhat has worked in this course, including information about several researchprojects that developed from the selected topics.

2. OUR MOTIVATION

Prior to a recent redesign of the course, our mathematics education studentswere experiencing a “disconnect” between what they were learning in thehigher mathematics classroom and what they thought they would be teach-ing. Many students reported that they felt that the most important thing thatthey could do in this course was to review the mathematical concepts thatthey would be teaching. We also felt the need to make more explicit the con-nections between the content in their college mathematics courses and themathematics they would be teaching. There were specific topics that theywere concerned about teaching. At the top of the list was trigonometry. Newgraduates, informally polled, tended to agree with this assessment.

Many of our students were also experiencing a technological disconnect.The use of technology in middle school and secondary mathematics classroomshas been evolving very quickly. More powerful calculators that include a com-puter algebra system are becoming commonplace. The simple fact is that atour college, some of our mathematics graduates will have rarely used even agraphing calculator in their mathematics classrooms. Some faculty use techno-logical tools regularly and some do not use them at all. There is no departmentalpolicy about the use of technology in the mathematics classroom. That couldchange in the near future, but at the present it is a fact. As a result, some ofour mathematics education students were at a disadvantage because they couldnot use available technology in an appropriate manner. The use of these toolsis mandated in their school districts and some of our students simply were notprepared. This need and other needs were addressed in a redesigned seminarfor mathematics education course.

3. THE COURSE

The seminar we describe here meets for three 55-minute periods per week.Lectures are rare and the classes tend to have between 4 and 12 students.A good portion of class time is spent with students solving problems, manyof which originate from Mathematics for High School Teachers: An AdvancedPerspective by Usiskin et al. [1] Many times, as certain questions are answered,others arise quite naturally. One class per week is held in a computer lab. In thissetting, students are expected to solve problems using any one of the tools they

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Using the Computer Algebra System Maple 369

have at hand. This includes paper and pencil, scientific calculators, graphingcalculators, and the computer algebra system Maple. Since each of these stu-dents takes geometry and the appropriate technological tools are used in thatcourse, we have not used them in our course. The students tend to work withpartners.

3.1. Conversation flows

People make hypotheses, not only about how to approach a given problem, butwhat tool should be used to accomplish the task. They have debates about themerits of one mathematical model over a different one. They construct viablearguments and critique the reasoning of others, while simultaneously learninghow to use appropriate tools strategically. The syllabus lists the topics includedin this course as the real numbers, the complex numbers, functions, curve fit-ting, angle measure, trigonometric ratios, trigonometric functions, and theirconnections. The reality is that each semester reveals different needs for dif-ferent students and that it is very important to remain flexible during class.There are certain topics that are important in each semester, not only for theirmathematical content, but also for the questions they raise about topics in themiddle school and high school curriculum. Here, we consider work we do infour areas: the complex number system, best fit curves, rational functions, andprobability. Examples of the kinds of interesting, and sometimes unexpected,directions this work takes the class will be provided in this article.

4. COMPLEX NUMBERS

Students are very comfortable with manipulating complex numbers when theyare in the rectangular form a + bi. However, when polar coordinates are intro-duced their comfort level changes dramatically. Students’ lack of confidenceabout trigonometric functions becomes evident very quickly. When complexnumbers are written in their polar form r (cos θ + i sin θ ) and in their expo-nential form reiθ , interesting things happen in the class. For example, at thebeginning it is stressed that the exponential form is just notation. It is fascinat-ing to watch very bright students use the rules of exponentiation with absolutelyno proof. Inevitably, these students realize—and this is a powerful moment—that these rules probably will hold because if they do not, the notation is notterribly good. They hypothesize that the notation exists because the rules ofexponents work. However, they also realize that they do not know if these ruleswork for the complex numbers. They understand the need for the mathematicalproof. This is simply a perfect place to be in a mathematics class. The stu-dents want something to be true, and they strongly believe that it is true, butthey have to prove it. With some coaxing, students consider the polar forms of

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the given complex numbers. They realize that some trigonometric identities,perhaps forgotten, or perhaps never learned, will be important in any proof.It is even more exciting when they realize that a proof by induction might beneeded. The connection between the mathematics they are learning and themathematics they will be teaching is obvious.

The complex number section of the course has led to some very acces-sible research questions for students. It has also provided an environment inwhich collaboration among students thrives. Two examples are provided in thefollowing sections.

4.1. The Orbit of a Complex Number

The orbit of a complex number is defined as the set O(z) = {z, z2, z3 . . . }. In labwe discussed some conditions under which there would be a finite number ofpoints in O(z). Using Maple, the students experimented with different valuesof z, and calculated zn for increasing values of n. They were very quick to stateand prove that if z = reiθ and either r > 1 or r < 1, then there will be an infinitenumber of points in the orbit of z. The case when r =1 was harder to handle, forexperimentation with many different values of z showed that sometimes O(z)was a finite set. Two students decided to continue to research this question andultimately proved that the complex number z = reiθ has a finite orbit if r =1 andarg(z) = pπ/q, where p and q are relatively prime integers and q is nonzero.Experimenting with Maple with many z values of this type, they noticed thatsometimes O(z) had q points while at other times O(z) had 2q points. Theirability to experiment so easily allowed these students to form a hypothesisbased on the results of that experimentation. They ultimately proved that if p isan even number, then the number of points in the orbit of z is q but if p is anodd number then the number of points in the orbit of z is 2q.

4.2. The Orbit of a Linear Transformation

On another day, the discussion about the orbit of a complex number led to avery brief discussion about other orbits—including the orbit of a linear trans-formation. We let T be a linear transformation from R

2 to R2 and let A be the

2 x 2 matrix associated with the transformation. We chose an arbitrary pointin R2 and called it x0. We let x1=Ax0, x2=Ax1, . . . , xn=Axn-1. We definedthe set {x0,x1, x2, ... } to be the orbit of the transformation T for the pointx0. This can also be written {x0, Ax0, A2x0, ..}. The question that arose quitenaturally was whether one could find the conditions under which there wouldbe a finite number of points in the orbit of a transformation. Two studentsdecided to research this question. They created several graphs of orbits andfound that the points in the orbit of certain linear transformations all lie on an

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Using the Computer Algebra System Maple 371

ellipse. Ultimately they realized that the existence of a finite orbit depended onwhether the matrix associated with the transformation was diagonalizable andon the matrix’s complex eigenvalues. These students were working at the sametime as the students working on the orbit of a complex number problem. Therewas great excitement in both groups when they realized that the orbit of a lineartransformation depended on the orbit of any existing complex eigenvalues. Thesecond group needed to use the results of the first group! It was a wonderfulexample of the need for collaboration in mathematics. One group was happy toshare, while another group was happy that their work was done.

5. BEST FIT CURVES

One day in lab a problem was posed. Given n points (x1, y1), (x2, y2), . . . .,(xn, yn) in the xy plane, find formulas for m and b in the line y = mx + b suchthat the sum of the squares of the vertical distances from the points to the lineis a minimum. Most students began this question by drawing points on a pieceof paper, drawing the best line they could, and talking about the line of bestfit. Some students decided to look up the formulas and try to work backwards.With some coaxing, most eventually realized that this was a minimization prob-lem and that they first had to create a formula for the “sum of the squares of thevertical distances from the points to the line.” The first partial derivatives wouldhave to be set equal to zero and solved for m and b. Reluctantly, students startedto use summation notation. A few students decided to use Maple immediately.They had to be helped with some of the syntax. The rule in lab is simple. If youexplain in precise mathematical terms what you want Maple to do, the syntaxwill be provided. Typically, students need little help with syntax once they areable to explain in precise mathematical terms what they are trying to do. Thesestudents were rather surprised to get the desired results immediately.

The Maple commands

f := (m, b) →∑n

j=1(m · x(j) + b − y(j))2

and solve

({∂

∂mf (m, b) = 0,

∂

∂bf (m, b) = 0

}, {m, b}

)

yield the solution

{b = −

(∑nj=1 x(j)

)(∑nj=1 x(j)y(j)

) − (∑nj=1 y(j)

)(∑nj=1 x(j)2

)n

(∑nj=1 x(j)2

) − (∑nj=1 x(j)

)2 , m

= n(∑n

j=1 x(j)y(j)) − (∑n

j=1 x(j)) (∑n

j=1 y(j))

n(∑n

j=1 x(j)2) − (∑n

j=1 x(j))2

}.

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This began a discussion of whether these students should go back and do thecalculations using paper and pencil. Then a deeper mathematical discussionensued. We began to discuss what we could do if we wanted to find a plane ofbest fit or even a hyperbolic paraboloid of best fit. Students saw rather quicklythat modifications could be made to the work above to create the appropriatefunction and simultaneously solve the appropriate number of partial deriva-tives. The fact that the calculus would be done by Maple made the idea evenmore appealing.

6. INTEGRATING PARTICULAR RATIONAL FUNCTIONS

Research topics can arise very unexpectedly. In class one day, the studentswere asked what the quadratic formula was. As expected, everyone knew it.Then they were asked to prove it. This proved to be challenging for manystudents. After some thought, one student suggested that the method ofcompleting the square would work. Using this hint, every student was able tocomplete the proof.

The students were then asked if they remembered using the method ofcompleting the square in any of their college classes. They agreed that theyused it somewhere in integration, but no one remembered where. They wereasked how they would find the value of an integral of type

� 1

(ax2 + bx + c)dx.

A few examples were given. They included

� 1

(x2 + 6x + 8)dx,

� 1

(x2 + 6x + 9)dx, and

� 1

(x2 + 6x + 10)dx.

There was a general discussion about whether or not one method could be usedto evaluate all three integrals. In fact, many students speculated that since thedenominators differed only by the constant term, there was a good chance thatone method would work for each of the three integrals. Apparently Calculus2 was a distant memory!

The students decided that they should try to factor the denominator. Witha great deal of prompting, the students recalled the algebraic method of partialfractions. We spoke at length about the verb “simplify” as it is used in thehigh school mathematics class. For example, high school students are regularlyasked to simplify an expression of the type

1

2 (x + 2)− 1

2 (x + 4).

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Using the Computer Algebra System Maple 373

The act of finding a common denominator and rewriting the expression as onefraction is well known. But when the students were asked if the equivalentexpression 1

/(x + 2)(x + 4) was simpler than the original, there was some dis-

agreement. However, when the students tried to integrate this rational function,everyone agreed that the original form was simpler. This led to a discussionabout whether the topic of partial fractions should be in the high school curricu-lum. One interesting part of the conversation was that most students thought ofthe phrase “partial fractions” as a method of integration rather than an algebraicmethod to separate one fraction into the sum of two or more fractions.

We returned to the original three integrals given above. Using Maple, theintegration was completed quickly. Then the students were asked to explainthe different integration methods used and to explain the algebra involved.Factoring did work on two of the three integrals. One of the denominatorsresulted in a perfect square and u substitution worked. Another required themethod of partial fractions. The third did not factor, and students struggledwith what to do. They knew that arctangent was involved because Maple toldthem that it was. They had to be coaxed to use the algebraic method of com-pleting the square. A very interesting discussion ensued about whether one ofthe three methods used above had to work for every integral of this type. As onestudent said:

In our calculus classes we have been taught several different methods to findintegrals of this type. We students try all the different methods until one of themworks. But since our teachers make up the questions, one of the methods theytaught us has to work for these selected integrals.

This led to the research project “Integrating Certain Rational Functions.”Two students decided to question whether they could evaluate all integrals ofthe type

� 1

(ax2 + bx + c)dx.

These students proved that the integral has to be one of the three types, depend-ing on the roots of the quadratic in the denominator, and that they did knowhow to integrate each type. Once again, these students were convinced tobegin this project when they integrated dozens of integrals of this type usingMaple and they always got one of the three types of integrals. These studentswere extremely excited by their ability to prove something without simplyregurgitating steps found in a textbook.

7. AN INTERESTING EXPECTED VALUE

One day in lab, a question about random numbers was raised. Although prob-ability is not in the syllabus, everyone in the class had previously taken a

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semester of probability, so we decided to experiment. We used both the TexasInstruments 84-Plus and Maple to generate random numbers. We then tried thisexperiment. Randomly select real numbers between zero and one. Stop select-ing numbers when the sum of the randomly generated numbers is greater thanor equal to one. Let X equal the number of values selected. We ran this exper-iment 25 times, writing down the different values of X on the board. We thenfound the average of the 25 numbers. It was approximately 2.7. We ran theexperiment another 25 times. Again, the average was about 2.7. Then one stu-dent wrote a Maple program to perform this experiment 10,000 times. Theaverage was 2.7185. The program was changed to perform the experiment100,000 times. The average was 2.72168. The students recognized that theseaverages were approximately equal to the irrational number e.

One student realized that the average we were finding was giving us anapproximation for the expected value of our random variable X. She hypothe-sized that the expected value of X was e and set out to prove her claim. Thisstudent began her research by dividing the interval from zero to one into n equalsubintervals and analyzed the question as if there were only the n-1 possibili-ties 1/n, 2/n, . . . , (n–1)/n when a number is selected from the interval (0, 1).She calculated that, in this case, the probability that X = 2 is

1

n× 1

n+ 1

n× 2

n+ . . . + 1

n× n − 1

n·

.Recognizing this as a Riemann sum and letting n approach infinity, she foundthat the probability that X = 2 is equal to

� 10 x dx. She continued with this pro-

cess, and used Maple when the integration became more complicated. Shecalculated

P(X = 3) =1�

0

1�y

x dx dy = 1

3;

P(X = 4) =1�

0

1�z

1�y

x dx dy dz = 1

8;

P(X = 5) =1�

0

1�w

1�z

1�y

x dx dy dz dw = 1

30;

P(X = 6) =1�

0

1�a

1�w

1�z

1�y

x dx dy dz dw da = 1

144.

Then the student found an approximation for the expected value by calcu-lating 2 · P( X = 2 ) + 3 · P( X = 3 ) + 4 · P( X = 4 ) + 5 · P( X = 5 ) +6 · P( X = 6 ) = 1 + 1 + 1/2 + 1/6 + 1/24.

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Using the Computer Algebra System Maple 375

The student realized that these are the first five terms of the Maclaurinseries expansion for ex evaluated at x = 1. Using Maple, she calculated severalmore probabilities and added to the above sum. She found that she was addingmore terms in the Maclaurin series expansion for ex evaluated at x = 1. Sheconcluded that the expected value was e.

All of the students mentioned here have presented their findings at theSpuyten Duyvil Undergraduate Mathematics Conference, a regional under-graduate mathematics conference partially funded through the MAA RegionalUndergraduate Mathematics Conferences program. Several of these confer-ences have been held at our college and the most recent was in April, 2013.Students are typically apprehensive when they present at a conference for thefirst time. They return exhilarated by the experience. We are trying to intro-duce younger students to research opportunities so that they have more thanone experience presenting in such a supportive and nurturing environment.

8. THE ROLE OF THE COMPUTER ALGEBRA SYSTEM

In each of the examples presented here, it is important to emphasize the placethat the computer algebra system Maple had on students’ decisions to begintheir research. In each instance, students used this computer algebra system toexperiment with ideas, to visualize results, and ultimately to form hypothesesthat they proved. Maple made the experimentation easy. Ideas were formedand Maple was used for the calculations. Ideas were quickly discarded toowhen results were not as expected. Without the power of Maple, it is difficultto imagine these students forming the hypotheses they did. There was simplytoo much calculation to do. The graphing capabilities of Maple help studentsto visualize the problem under consideration and hypotheses often are made asa result. Once again, the ability to graph hundreds of points rather than a fewdozen points affects a student’s ability to hypothesize. Mathematically, some-thing wonderful happens when a student actually creates a graph consistingof hundreds or thousands of points and those points all lie on a figure like anellipse. Even though it might look like the entire ellipse is graphed, the studentknows there are “more” points on the ellipse. In the two cases involving orbits,each student involved in the research realized that the orbit could never consistof all the points on the resulting ellipse because the number of points in theorbit is at most countably infinite while there is an uncountably infinite numberof points on the ellipse. This is a powerful idea. The use of Maple fosters thedevelopment of such ideas.

9. SUMMARY

The capstone course designed specifically for pre-service secondary educa-tion mathematics students creates an environment in which mathematically

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mature students reconsider important topics in the high school curriculumfrom a mathematically advanced perspective. These students make connec-tions between what they will be teaching and what they are learning in theircollege level mathematics courses. In this environment, students learn to makeappropriate decisions about when and how to use the technological tools attheir disposal. They are also encouraged to use the computer algebra systemMaple as a tool to experiment and to use that experimentation to create math-ematical hypotheses which they ultimately prove or reject. These students arealso learning how to teach using technology by modeling the atmosphere in thecapstone course classroom. Here no classroom topic is driven by the technol-ogy. Rather, technological tools are chosen by students and used at appropriatetimes so that problems are solved. Students learn that they must be acquaintedwith the capabilities of the technology in order to make informed decisions, butit is the subject that must shine through. Students attain a deeper understand-ing and appreciation for secondary mathematics when the link between thesetopics and those they learned in their college courses is made explicit.

REFERENCE

1. Usiskin, Z., A. Peressini, E. Marchisotto, and D. Stanley. 2003.Mathematics for High School Teachers an Advanced Perspective, NewJersey: Prentice Hall.

BIOGRAPHICAL SKETCH

Rosemary Carroll Farley is an Associate Professor of Mathematics andComputer Science at Manhattan College. Her current interests include the useof technology as an experimental tool in undergraduate mathematics researchand the use of visualizations in the linear algebra curriculum.

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