Teaching with Duke's CCP Materials

Author(s): 
Stephanie Fitchett

For nearly five years now, I have incorporated Duke's Connected Curriculum Project (CCP) materials into various lower division mathematics courses, including calculus and linear algebra. I have used the materials with different technologies, and at institutions with different student populations. There are some common threads in my experiences, but each class has also taught me something new about using the materials effectively, and I would like to share some of what I learned. My hope is that this discussion will complement John Hannah’s recent article (Hannah, 2001 ) on using Duke’s CCP materials in differential equations, and that perhaps my (our) experiences will encourage others to review this collection of materials and consider adopting modules for use in their own courses.

Stephanie Fitchett is Assistant Professor in the Honors College of Florida Atlantic University.

In the next section, I describe the CCP materials and discuss how and where they might be incorporated into courses. Sections 3 and 4 describe my experiences using the CCP materials, respectively, with junior-level students who had experience with a computer algebra system, and with first-year students unfamiliar with such technology. In Section 5 I discuss some of the logistics of using the materials, particularly handling and grading electronic files. Section 6 contains my observations on some effects the use of these materials may have on students. The final two sections include a brief summary and a list of references.

Published November, 2002
© 2002 by Stephanie Fitchett

Teaching with Duke's CCP Materials - A User's Perspective

Author(s): 
Stephanie Fitchett

For nearly five years now, I have incorporated Duke's Connected Curriculum Project (CCP) materials into various lower division mathematics courses, including calculus and linear algebra. I have used the materials with different technologies, and at institutions with different student populations. There are some common threads in my experiences, but each class has also taught me something new about using the materials effectively, and I would like to share some of what I learned. My hope is that this discussion will complement John Hannah’s recent article (Hannah, 2001 ) on using Duke’s CCP materials in differential equations, and that perhaps my (our) experiences will encourage others to review this collection of materials and consider adopting modules for use in their own courses.

Stephanie Fitchett is Assistant Professor in the Honors College of Florida Atlantic University.

In the next section, I describe the CCP materials and discuss how and where they might be incorporated into courses. Sections 3 and 4 describe my experiences using the CCP materials, respectively, with junior-level students who had experience with a computer algebra system, and with first-year students unfamiliar with such technology. In Section 5 I discuss some of the logistics of using the materials, particularly handling and grading electronic files. Section 6 contains my observations on some effects the use of these materials may have on students. The final two sections include a brief summary and a list of references.

Published November, 2002
© 2002 by Stephanie Fitchett

Teaching with Duke's CCP Materials - The CCP Materials

Author(s): 
Stephanie Fitchett

The CCP materials are discovery-based learning materials that incorporate widely available technology in a student-centered learning environment. The modules are described in greater detail in (Smith and Moore, 2001), so the description here is somewhat brief. The learning materials on the Duke CCP site are mostly web-based modules that can be integrated into undergraduate mathematics courses, including precalculus, calculus, ordinary differential equations, linear algebra, and engineering mathematics. The modules are single-topic units that students complete in approximately two hours, using a web browser and a computer algebra system – or, in a few cases, using just a Java-enabled browser. Nearly all modules are currently available for Maple and Mathematica, many are available for Matlab, and some are available for Mathcad. The modules can also be used with graphing calculators.

Each module consists of several HTML pages containing discussion and guidance for exploring a topic, which may be theoretical or application-oriented. The discussion pages break the material into reasonably-sized chunks, and they include questions with a wonderful knack for enhancing students' understanding of the material, rather than just their computational skills. The Summary section of each module ties together the activities, observations, generalizations and theorems of the unit, forcing students to reflect on what they have learned. The modules follow a standardized format – for example, the same buttons are found in the same locations on each page, and do the same things in all the modules – making the site easy to navigate.

The modules have been developed with enormous thoughtfulness and attention to detail. They are easy to use, and you do not need to be an expert with Maple (or any other computer algebra system) to adopt these materials successfully. In fact, if you would like to learn more about a computer algebra system, the CCP materials offer an excellent way to proceed. The computer algebra system tutorial modules provide a nice introduction for both students and instructors.

One advantage of these materials over other web-based supplementary activities is that many of the CCP modules are not just a supplement to a standard lesson in a course, but can actually replace a lesson. This allows instructors to incorporate the modules without cutting large portions of material. For example, I have used Taylor Polynomials I, a module on Taylor polynomials about x = 0, to introduce Taylor polynomials in second semester calculus. We spent one class period in the lab, with students working in pairs on the module, which they then finished outside of class. In the next class period, we reviewed the module material and discussed polynomial approximation about a general point x = a. (The module handles only the x = 0 case.) Since I usually spend from one and a half to two class periods on Taylor polynomials, using the module takes no more time, and in fact, the students seem to have a stronger understanding of what they are doing, since they discover the formula for the Taylor polynomial coefficients themselves. The sample syllabi available on the Resources for Teachers page are helpful for those who would like more examples of how the CCP modules can be used in different courses.

Many of the modules focus on mathematical concepts, asking students to use the computer algebra system to experiment, look for patterns, and then explain why the patterns occur. For example, in the Systems of Linear Equations module, students are asked to explore various systems involving 2 or 3 variables, write down systems that would give lines or planes that intersect in a given way, then summarize what they learned by answering summary questions:

  • When we try to solve a system of m linear equations in n unknowns, there are only three possible outcomes for the number of solutions. What are those possibilities?

     

  • If the system has 3 unknowns, then it can be described in terms what geometric objects? What are the possible configurations of these objects? Which configurations represent consistent systems? Which represent inconsistent systems?

     

  • If the system is consistent, then there is a relationship between the number of free variables in the solution, the number of nonzero rows in the reduced row echelon form of the augmented matrix, and the number of unknowns. What is that relationship?

These questions allude to the process: experiment, analyze, justify, and generalize, a theme repeated throughout the CCP modules. The process encourages students to discover mathematics through exploration, and write mathematics to explain their observations and make predictions. The computer laboratory provides a discovery-based learning environment that mimics the experiences many students have in their science laboratories, providing a view of mathematics as a science .

In another parallel with the discovery-based learning common in the sciences, many CCP modules use real data to explore interesting applications. These applications, and the mathematics that accompanies them, range from elementary to quite sophisticated. For example, the purposes of the Marine Pollution module are

“To carry out a short study of the relationship between concentration of a marine pollutant and shell thickness of mussels; to practice writing about the results of a mathematical study,”

and the mathematical content is a linear model. At the other end of the spectrum, Harvesting an Age-Distributed Population uses eigenvalues and eigenvectors to determine sustainable harvesting policies in an age-structured population of sheep.

Some modules provide hyperlinks to additional information related to the module topic, such as biographical sketches of mathematicians who contributed to the material discussed in the module. Students may explore these at leisure, and the most curious students often do. On the other hand, my experience suggests that many students are often motivated to finish the assignment in an efficient manner, and thus tend to avoid what they perceive as optional distractions. Instructors, though, will certainly appreciate the supplemental information.

While I have generally used roughly equal numbers of conceptual and application-oriented modules in a course, as a whole, the collection is very diverse, allowing instructors to choose modules that fit their course and pedagogical preferences.

One last point before discussing specific courses: Every time I have used the materials, students have worked in pairs in the computer lab, with one pair of students at each computer. This arrangement forces students to discuss the mathematics of the modules and eases the learning curve for the software (Bookman and Malone, to appear; Hannah, 2001). The collaborative atmosphere turns the lab into a student-centered learning environment rather than a teacher-centered one. Students discuss the material among themselves, primarily with their lab partners, but often with other pairs of students as well. In fact, when students ask me a question while working on the modules, I will often refer them to another pair of students who have encountered and overcome the same difficulty.

Teaching with Duke's CCP Materials - Linear Algebra with Maple-experienced Students

Author(s): 
Stephanie Fitchett

The first time I used the CCP linear algebra materials, I was teaching at Duke University, and I was fortunate to have the mentorship of the primary authors. Both Lang Moore and David Smith had taught linear algebra with the CCP materials they had largely created, and I was able to modify their syllabi for my course. This sharing of syllabi worked well in part because the student audience for our courses was essentially the same, though I readily admit that making their syllabus work at another institution, with a different student audience, would take more effort. Aside from the advantage of a tried and true syllabus, the fact that Moore and Smith were colleagues was perhaps more of a psychological crutch than a real one. I would have been more tentative about diving in and using the materials for such a large portion of the course had I not known that I could always consult with them if I encountered difficulties. But we rarely discussed either the contents of the modules or the logistics of using them, while the course was in progress. (On the other hand, we discussed the modules at great length the summer after the course.) The modules are so well written, the accompanying worksheets so clear, and the questions so perfectly tailored for student exploration, that I rarely had any question about the intent of the modules or their authors. We used David Lay’s Linear Algebra and its Applications (Lay, 1997) for the course, and the modules fit well with the text, in the sense that when we were ready to use a particular module, the students already had the necessary linear algebra prerequisites, whether or not they had completed “earlier” CCP modules.

My linear algebra class met three times each week, twice in a traditional classroom and once in a computer laboratory. Over the course of the semester, we spent twelve class sessions in the computer lab and completed eleven CCP modules. Most of the students, having taken a lab calculus course that used Maple, were already familiar with the computer algebra system, but even those who were not caught on quickly with the Maple Tutor for Linear Algebra, the first CCP module we completed, which provides a great introduction to the software.

On lab day each week, the students would come to lab, launch a web browser that opened with the Materials for Linear Algebra page as the default homepage, and set up the day's assigned module by downloading the Maple file and arranging the browser and Maple windows.  After the first two days in the lab, students never waited for me to tell them what to do – they just got started on the day's module, the title of which was on the chalkboard when they arrived.  Their self-motivation was wonderful, and I was elated to see that they were taking responsibility for their own learning.  Part of their motivation stemmed from wanting to make good use of their time.  Our lab sessions were only 50 minutes long, but the modules generally took between 90 and 120 minutes to complete. Most students strongly preferred to be able to consult with one another, and with me, as they worked through the modules, so they worked hard during class time.  Students who did not have another scheduled after ours often stayed and finished their modules immediately, rather than going back to them later.

The careful integration of a computer algebra system with the web-based materials has great advantages to students.  In many cases, using Maple for computations allowed students to focus on the bigger picture, rather than becoming lost and frustrated during the course of a long computation, such as finding the inverse of a 4 by 4 matrix.  Because they could do many computations quickly, they were also able to look for patterns in ways that would have been horribly inefficient were they doing the computations by hand. 

Students made strong gains in their conceptual understanding of linear algebra as they used Maple to complete the modules.  One of my favorite examples is in the Inverses and Elementary Matrices module, where students are prompted, “In general, for any two row equivalent matrices A and B, describe how to find a matrix P such that PA = B.” This requires considerable thought, effort, scratch work and verification (usually with Maple), on the part of students, but working together, they are almost all able to give a general construction of P and an explanation of how and why it works.

Using the CCP modules influenced other aspects of the course as well.  I was fortunate to have Maple demonstration capabilities in the regular classroom, so I often used Maple to illustrate points or do computations even when we were not in the lab.  The materials also affected how I assessed student learning.  The Maple worksheets were submitted by pairs of students after the completion of each module and graded as homework.  Moreover, because students tend to equate the importance of a classroom activity with its effect on their final grades, and I wanted to communicate my belief in the importance of the laboratory activities and the utility of Maple as a tool, I followed Moore’s lead, and gave exams in two parts:  a traditional in-class paper and pencil exam, and a take-home portion that required (or was made easier by) the use of Maple.  This arrangement worked well and allowed me to evaluate hand computation and proof writing, while still allowing for longer, application-oriented or computationally intensive questions on the take-home portion of the test.

Teaching with Duke's CCP Materials - Calculus at Florida Atlantic University

Author(s): 
Stephanie Fitchett

For the last three years, I have been using the CCP materials at Florida Atlantic University’s Honors College, primarily in second semester calculus. My first attempt at using the materials in this new setting was not particularly successful. Unlike the more advanced students in my linear algebra class at Duke, my first year calculus students had no experience using Maple.

That first semester, we completed three modules from the PostCalc Project collection, a set of longer modules intended for high school students who have finished calculus. The modules were fantastic – the difficulties arose from the way I structured the course and the fact that I needed to provide more support for learning and using Maple. I did not set aside enough class time for the CCP projects. I did not force the students to work in pairs– some did, but some did not, and those that chose to work individually often struggled. And the length of time between projects allowed students to forget a lot about Maple in the interim.

While I was unhappy with my arrangement, the students were not overly bothered. On their course evaluations, where I specifically asked them to comment on the laboratory portion of the course, they said they generally enjoyed completing the modules and felt that they learned a lot of mathematics in doing so. Their biggest frustration was in having to complete the modules outside of class, when I was not around to quickly answer questions, especially questions about Maple.

Properly chastised, and recognizing at least some of my errors in structuring the first course, the next time I taught the course we completed seven modules, meeting in the lab roughly once every two weeks. I used shorter modules from the Integral Calculus and Differential Equations sections of the main CCP site, with just two excursions to the longer PostCalc modules, and in those two cases, we did only portions of the available modules. All of the students worked in pairs, and I felt the course progressed much more smoothly. As others have noted -- see (Bookman and Malone, to appear) and (Hannah, 2001), for instance -- students do struggle with Maple at times, but having them work in pairs alleviates this difficulty somewhat. The pairing also helps facilitate discussion of mathematical concepts and seems to encourage students to experiment – even struggle – with material more before becoming frustrated.

In recent semesters, partly because of limited lab availability, I have been using the CCP modules with Texas Instruments graphing calculators instead of Maple. Moreover, our college emphasis on environmental issues has led me to choose calculus modules that emphasize applications such as Accumulation (on air pollution), World Population Growth, the Logistic Growth Model, and Predator-Prey Models. Using calculators has the advantage of portability (I gave students printed versions of the modules), but does not allow for the same sort of interaction as pairs work through the materials. For instance, in the lab, students almost invariably divide control of Maple: One student runs the keyboard and the other runs the mouse -- Bookman and Malone (to appear) mention this, and it seems to be a common observation of instructors teaching in similar settings -- whereas with a graphing calculator, this sort of sharing is impossible. Students must either work individually with the technology, or one works while the other watches, and neither option is optimal. Pedagogically, I far prefer to teach in a computer lab with Maple, but the modules are flexible enough to work outside of a lab.

Teaching with Duke's CCP Materials - Handling and Grading Student Work

Author(s): 
Stephanie Fitchett

For an instructor integrating computer algebra assignments for the first time, the mechanics of file submission and grading can be somewhat daunting, so I would like to address these briefly. At Duke University, students were given accounts on the department's UNIX system, so they saved their files in their personal accounts and e-mailed them to me as attachments. At FAU Honors College, students use a WindowsNT lab, where they have no assigned disk space and must save their files to floppies. Either system works, though there are definite advantages to not having to deal with floppy disks.

I generally have students submit their Maple worksheets by e-mail, and I filter them into a separate mail folder so I can easily keep track of them. I grade them electronically, opening the files, adding comments and a grade in color, then returning the graded file to both contributors, again by e-mail. A few students at the Honors College submit their files on a disk, in which case I save the graded copy of the file to the same disk and return it. While handling the files takes time, it has not been overly taxing. For example, my linear algebra class had 28 students, and each week I got 14 files that I could grade and manage in about two hours. My strategy has been to skim the body of the worksheet, then carefully read the responses to the summary questions, all of which can be done quickly. The more time-consuming aspects are managing the files (saving, sorting, sending) and carefully reviewing material in the body of the worksheet when a summary response is incorrect.

One strategy that helps the grading go more quickly is to require students to write in complete sentences. The modules lead students through explorations and computations with Maple, and ask students to make observations and respond to specific questions along the way. I tell students that I will not refer to the questions in the module when grading their reports, so their responses must be complete sentences that provide context as well as answers. This forces them to articulate a complete thought, which facilitates the grading process for me. A satisfying side effect of the complete-sentence policy has been that students' ability to communicate their observations and explanations steadily improves as the semester continues, provided I give them feedback and suggestions, especially on the early assignments.

Teaching with Duke's CCP Materials - Effects of Using CCP Materials

Author(s): 
Stephanie Fitchett

I have not designed and implemented a careful study of the effects of the CCP materials on students – my college is small enough that there is no feasible way to maintain a control group – so I can only offer the conclusions that I have drawn from my own observations and experiences as a teacher. While readers may legitimately question the value of such anecdotes, perhaps a few might be intrigued enough to give the materials a try, or, even better, design a study to assess the validity of the observations.  (Some such studies are under way, I believe, including ongoing assessment work at Duke University.) What follows are student reactions that I perceive to be due to the use of the CCP modules, either directly or indirectly.

  • Students retain the material covered in the modules longer and better than many topics covered in a more traditional format. Rarely will students miss questions on a quiz or an exam that are closely related to the material covered in one of the CCP modules they have completed. I believe the fact that they discovered what they have learned aids retention.

  • Students tend to be willing to work longer and harder on a mathematical problem when they are working in pairs. They also seem to be more willing to experiment and explore with Maple than with their graphing calculators. I am not sure if the latter is because Maple is faster than the calculators are, if it is because a collaborator can more easily see the results of an experiment, or if there is some other reason entirely. In any case, when students are thinking longer and harder about the obstacles they encounter, they more often overcome those obstacles, and they generally learn something in their struggles.

  • Students who regularly use the CCP materials, and more generally a computer algebra system, become more comfortable using technology. Certainly, their ability with the particular technology improves, helping them in later mathematics courses, and often in other disciplines. For instance, students have mentioned to me that they have used Maple to plot data from a science lab, and to compute integrals for physics homework problems. Perhaps more importantly, confidence with technology seems to be cumulative, even across applications. Students familiar with one technology, like Maple, seem more comfortable picking up another, whether it is another software package such as SPSS or the computer interface to a gas chromatograph mass-spectrometer in the chemistry lab.

  • Students come to view Maple as another tool in their mathematical toolbox. The questions in the modules consistently encourage students to think about whether or not their answers make sense, and, if not, to investigate what may have gone wrong. Students respond to this re-examination of their work much more readily than might be anticipated. In addition to using Maple, they will consult their text, use their graphing calculators (odd though that sometimes seem to me!), and work with pencil and paper to facilitate their understanding.

  • In a broad educational sense, I believe we will see (through student surveys being administered each semester---we will eventually have data for or against this assertion) that students who are regularly exposed to the CCP modules, or are consistently engaged in similar active and application-rich learning environments in mathematics, will exhibit an increasingly strong awareness of mathematics as a science. Even students in their first semester with the CCP materials seem to begin to perceive that there is a continuous scientific process in mathematics, just as in biology or chemistry. I expect that these similarities between mathematics and the sciences will become more apparent to students as they continue in discovery-based mathematics and science courses.

Finally, I want to mention an effect of the materials not on students, but on my colleagues. My institution allows individual faculty members a great deal of autonomy in the content and conduct of courses, but the combination of my enthusiasm, and that of my students, has convinced my mathematics colleagues to incorporate the CCP materials in their courses.

Teaching with Duke's CCP Materials - Summary

Author(s): 
Stephanie Fitchett

Students learn by exploring, by recognizing patterns, by generalizing, and by writing about what they've learned. The CCP materials thoughtfully encourage all of these, and my students have consistently responded to that encouragement, meeting and exceeding my expectations in problem-solving, writing mathematics, and working together. If you are looking for interesting, application-rich materials that can be tailored to almost any lower division undergraduate mathematics class; if you want your students to experience mathematics as a science of discovery, practicing the process of experiment, analyze, justify, and generalize; and if you want materials that incorporate pedagogy that has proved successful in facilitating student learning; then the CCP modules are exactly what you are looking for!

Teaching with Duke's CCP Materials - References

Author(s): 
Stephanie Fitchett

Bookman, Jack, and David Malone (to appear), "The Nature of Learning in Interactive Technological Environments: A Proposal for a Research Agenda Based on Grounded Theory", Research in Collegiate Mathematics Education V.

Duke Development Group of the Connected Curriculum Project, The Connected Curriculum Project, http://www.math.duke.edu/education/ccp/index.html.

Hannah, John (2001), Using Connected Curriculum Project Modules in a Differential Equations Course, Journal of Online Mathematics and its Applications.

Lay, David C. (1997), Linear Algebra and Its Applications, 2nd edition, Addison-Wesley.

Smith, David, and Lang Moore (2001), The Connected Curriculum Project, Journal of Online Mathematics and its Applications.