Teaching with Duke's CCP Materials - The CCP Materials

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.