Journal of Online Mathematics and its Applications
Using Connected Curriculum Project Modules
This article is about my experiences, and those of my students, the first time we used modules from the Connected Curriculum Project (CCP). The CCP modules are part of an integrated approach to learning mathematics, taking in not just the use of technology, but also problem solving, cooperative learning and communication skills. The modules aim to combine the interactivity and accessibility of the Web with the power of a computer algebra system like Maple. They are quite adaptable, and could be used either as an integral part of a course, or as supplements to classroom discussion, or even for independent study by individuals. Each of the modules I used was a single-topic unit designed to be completed in one to two hours with students working in pairs in a computer lab environment. For more about the philosophy behind CCP see Smith, Bookman and Malone, or the For Teachers link on the CCP home page.
The rest of the article is laid out as follows:
The next four sections deal with issues raised by what happened during the lab sessions.
The article concludes with
Copyright 2001 by John Hannah
Published December, 2001
The CCP modules were used in the second half of a one semester second year calculus course, the first half being about multivariate calculus, and the second being about differential equations. There were 107 students in the class, all of above average ability (getting at least B+ in their first year mathematics course). About 60% of the class were engineering students, with most of the remainder being science majors.
The class met for 4 one-hour lectures and 1 one-hour tutorial each week. For the first half of the course, the students were divided into six groups which met for traditional tutorials: discussing previously assigned problems from the text book (Anton), working through additional problems set during the tutorial itself, and getting help from one another or the tutor if they got stuck. Homework problems were not graded, the main incentive to do them being that the exam would be based on them. (This was insufficient incentive for some students, who came to the tutorials unprepared and tended to take little part in the discussions.)
The same six groups were retained in the second half of the course, and the tutorials became lab sessions in which the CCP materials were used. In this more active environment, it was much harder for students to avoid participating. Tutors too needed to adapt to the new environment. For an account of the dilemmas facing tutors in a lab environment, see Winter et al.
Homework problems were still being set, this time from Boyce and DiPrima. To compensate the students for the loss of the traditional tutorials, three extra consultancy hours were offered each week to help students with any problems they might have with their homework or with the CCP modules.
The differential equations part of this course deals with second order linear equations, series solutions, the Laplace Transform and Fourier series -- essentially chapters 3, 5, 6, and the first half of chapter 10 from Boyce and DiPrima. I wanted the students to try each module about a week after I had lectured and set homework on the relevant subject matter. This would give them enough time to prepare for each module. To fit in with this schedule, I postponed the series solutions topic until the end of the course and decided to use the following five CCP modules.
The first of these modules reviews what it means to solve an differential equation. The next two offer different views of the solutions to second order linear equations, the first one analytic and the second one physical. These complemented the lectures, where I mainly talked about ways of finding these solutions. The fourth module lets the student explore the Laplace Transform, and how it can be used to solve a differential equation, without getting bogged down in improper integrals and partial fraction calculations. Finally, the last module offers a visual interpretation of the Fourier series, complementing the analytic approach taken in the lectures and homework.
Ideally, these modules would be graded each week as they were being done but, because of other commitments, I decided to collect and grade the modules all together at the end of the course. To keep the students working steadily at the modules, I had weekly reporting stages in which the students had to show their tutor some crucial graphic or summary statement from the previous week's module.
Sources of data
Data on students' attitudes were collected in two open-ended surveys conducted during normal lecture time, one at the end of the first half of the course -- before any of the CCP materials had been used -- and the second just before the end of the course. Copies of the survey questions can be seen in the Appendix. Such surveys provide quite limited information, of course, compared with interviews. Furthermore these surveys sought a picture of the course as a whole -- as a part of my own course development strategy -- so only a fraction of the responses deal with the CCP modules. However the students' responses provide a first glimpse of how they viewed the CCP learning experience.
Both surveys were completed by 96 of the 107 students taking part in the course. To get a reliable picture of the students' responses I coded and categorised them following the recommendations in Miles and Huberman. This is similar in spirit to the grounded theory approach in Bookman and Malone, in that the data generate the categories to be discussed. All student quotes in the following sections come from the second survey, the numbers indicating which survey form the comments come from.
My own observations come from grading the students' final submissions, along with recall of incidents that occurred while I was tutoring in the laboratory sessions, and which were recorded after comparing experiences with the other two tutors at our weekly debriefing session. (I was the tutor for two of the six weekly sessions, and helped in the other four for the first two modules.)
Do CCP modules encourage greater understanding?
It is difficult to get reliable data on this question when dealing with a large number of students. I was worried that the time students spent on the CCP modules would mean less time being spent on the more traditional elements of the course, leading to a poorer performance in the exam on those skills we all hold so near and dear -- e.g., doing a double integral, solving a DE, finding a Fourier series, all by hand. But in fact they did really well in the exam. There were some confounding influences -- e.g., one student explained to me that they had done DEs in three separate courses that semester, so they were very familiar with Des by exam time. But it would appear the traditional skills were still being learnt -- the "no harm done" conclusion mentioned by Heid et al.
On the other hand, the students were learning things in the labs that I would never have tried to teach in a more traditional setting, such as the detailed behaviour of solutions to initial value problems, and how this behaviour responds to parameter changes. So my impression is that the students acquired a deeper understanding by using the modules, combining graphical and physical interpretations with the mechanical methods learned in more traditional courses.
What did the students themselves think about this question? In my second survey, 62 students commented about whether using the CCP materials had helped them to a better understanding of the mathematics. Of these students, 37 made generally positive comments, 14 made generally negative comments, and the rest voiced mixed feelings.
A few students offered their own explanations for why the lab sessions had helped their learning. Some of their explanations could apply to any active learning experience. Thus, in one case, the prospect of a session where they knew they would have to do something encouraged the student to do some preparatory study:
#45: It does help me to understand more because I have to read the lecture notes and reference book to do the Maple assignment.
Another student observed the converse:
#4: I know that we are suppose to do H.W. [homework] problems before Maple sessions so we can understand but, most of the time, I don't have time to do H.W. problems before the session and I end up understanding nothing.
The CCP modules are designed to be completed in about two hours. As I had only one hour of class time available each week, I expected the students to finish off each module in their own time (although they were always welcome to take advantage of spare computers at one of the other five scheduled sessions each week). One student saw this as a useful way of encouraging deeper learning:
#54: . . . having labs which you couldn't quite finish meant you had to come back and learn the stuff again [giving] greater understanding.
On the other hand, some of the students' explanations of the efficacy of the sessions refer to specific features of the technology. Thus, for one student, the Maple laboratory sessions
#95: Allows you to explore enough possibilities to see the patterns.
while for another the sessions were
Technology: help or hindrance?
Ideally the browser and the helper application will be transparent to the student, as they are primarily a means of getting the student to understand and engage with the mathematics. Despite an initial hostility to an unfamiliar operating system (Unix instead of Windows), and perhaps an unfamiliar browser (Netscape rather than Internet Explorer), most students settled quickly into using the browser pages of the CCP materials.
The Maple worksheets that accompany each CCP module have most of the necessary commands already entered, and most of the exercises can be answered by copying and pasting these commands to new command lines, and making minor modifications to them. One student had a colorful way of expressing the advantages of this approach:
#43: Not too much emphasis on knowing the exact commands to input for Maple which means time is well spent on the actual mathematical concepts not pissing around trying to get it to work.
Some sensed a temptation inherent in this approach:
#30: Useful tool for learning concepts. Perhaps it was too easy just to follow the instructions without actually understanding it, but still an effective method of learning.
to which other students succumbed:
#20: . . . I feel I didn't learn a lot as we seemed to be just pressing enter, so when it came to explain what we learned, it was difficult.
but others felt the approach worked in their case:
#96: Often with computer learning it is easy to race through the exercises just pressing the enter key, but these labs often made me stop and think.
Just pressing the enter key led a few students astray in the fifth module, Experiments with Fourier Series, where about 10% of pairs commented on the convergence of the Fourier series approximations (Part 2, Step 5) using just the plot with n = 1, the value supplied by the original worksheet, as their evidence. (By this stage many students had discovered the Execute Worksheet command in Maple's Edit menu, and they didn't always look very carefully at the resulting output.)
Where more than copying and pasting was involved, students needed to be more or less familiar with the basic commands and syntax of Maple. There are hints of the problems some people experienced in the responses above (see #43 and #85 on page 5). Some students seemed almost philosophical about it:
#46: Good. Although you can get a bit distracted from the task at hand due to trying to figure out how to work Maple, ie syntax etc . . Can get a bit annoying.
But others became quite angry about the experience:
#93: BAD. Maple is both boring and aggravating, and doesn't seem particularly useful either. The assignment took ages, and most of it was trying to get Maple to execute commands which we'd left a bracket out of or something, who really knows where. All the difficult bits we just worked out on paper anyway.
Bookman and Malone also offer several examples of how exasperating the Computer Algebra System can be and, as they found, the problems with Maple can lead to problems with time management too:
#76: Useful for understanding concepts but the interface is so user-unfriendly, unpredictable and unreliable that on several occasions we have spent the entire hour in tutorials fighting with computers crashing or just taking 10 minutes to load a large file. This leaves us no time to ask the tutor questions and as I cannot attend any other tutorials this means I cannot ask for help! Don't get me wrong, this is a useful tool for visualising complex ideas but maybe we could shift across the hall to a (slightly) more reliable computer lab.
Some of the students' problems can be put down to unfamiliarity with Maple. Although most of the students would have met Maple in their first year courses, only a few had used it intensively. I offered an introductory Maple session at the start of my course, but only a third of the students attended. In the first survey (before the CCP modules), 36 of the 96 responding students said that they had either never used Maple or hadn't had time to use it during this course. A further 18 students voiced negative feelings about Maple (either from experience in a previous course, or from trying to use it in the first part of my course). As Bookman and Malone report (see their Vignette 6), sometimes even the tutor can't figure out why Maple isn't doing "the right thing," and the only option is to start all over again. So it is not too surprising that even after the CCP modules were completed there were still 18 students in the second survey who either disliked Maple or felt that syntax problems had hindered their progress.
Having to communicate with fellow students encourages students to articulate, and perhaps clarify and deepen, their mathematical thinking. However a computer can stifle such communication. In my own institution, Boustead [Section 8.3, page 8-12] reported on four tutorial groups of first year students, two groups working with the CD-ROM version of a calculus text and two working in more traditional, computer-free, tutorials. The former students
opted to work alone with their CD-text. After a few weeks there was little if any interaction between students as each worked at their own pace on different aspects. During one week in which the students were forced to share a CD between pairs, there was increased student discussion, the students took longer to go through the work and there was considerable frustration. Some students wanted to concentrate on the concepts, others wanted to do just the exercises. All the students opted to return to one CD ROM per student.
Experience at Duke University also suggests that more communication takes place when the CCP modules are tackled by students working in pairs Bookman and Malone. Furthermore (David Smith, personal communication), the interaction is often better if the two students are relative strangers, perhaps because in that case their thought processes have to be made more explicit for communication to take place. Because of this, I paired my students off more or less at random, picking neighbours off an alphabetic listing of the particular tutorial group the students belonged to (these groups having been assigned at the start of the course to fit in with timetable constraints).
In the first half of the course, I had offered the students the option of doing their first assignment either individually or in teams of two or three students. About half the students chose the team approach. With this background, and with Boustead's experience, I might have expected some student resistance to enforced team work, but as an observer I felt that it worked well. Most pairs worked as teams both physically and verbally. Thus, it was very common for one student to control the mouse while the other typed -- a significant feat in communication and cooperation. Furthermore some students, who had seemed to be struggling in the lectures and ordinary tutorials, now became confident investigators.
In the end-of-course survey only 15 of the 96 students thought this aspect was worthy of comment, and they were marginally in favour of the idea of working in pairs. Their comments reinforce the issues raised above. One student voiced the ideal of cooperative learning:
#57: group work was good as we learned off each other well
while another felt that it helped to pull them up to a higher level:
#79: I found it useful to work in pairs as some of the work was too difficult for me to do by myself . It was also good motivation to work in pairs.
but a third student felt this arrangement was unfair:
#4: I take long time to understand what the questions mean while my partner is "too brainy" and understands quickly so, he does most of the work during lab so it is not fair for him. But I go back to lab and go over the questions in my own time. So it is not fair for me because I spend a lot of time.
although, as we have already seen, this same student admitted not preparing for the labs.
Some of the difficulties reported by students may be the consequence of having a non-homogeneous class. Thus some reported difficulty in finding times when both partners were available to finish off the work started in the lab sessions. Language differences also caused difficulties for (at least) two of the pairs:
Some of these difficulties were apparent before I conducted the second survey, and I asked a student from one of the dysfunctional pairs whether it would have been better to let everyone choose their own partner. To my surprise he said No, and went on to explain that all the other pairs he knew were working well together, and it was to be expected that there would be one or two unsuccessful pairings. This, together with the small number of students who felt moved to comment, suggests that cooperative learning is an accepted (and perhaps unremarkable) feature of the educational landscape for most of these students.
Getting students to check their answers is a feature of most CCP modules. This encourages students to engage in self-monitoring, to become more independent learners. It can also have an indirect benefit, encouraging deeper understanding or developing connections between ideas which might otherwise remain compartmentalised in the student's mind. For example, a solution to an equation can be checked by substituting it into the original equation (thus clarifying what it means to be a solution), or again, an integral can be checked by differentiating the answer to get the original integrand (thus illustrating the fundamental theorem of calculus).
How students reacted to this idea was best seen in the lab sessions themselves. Sometimes checking occurred without being asked for. For example, in Part 1 of the Spring Motion module, several students copied and pasted the data for the spring system incorrectly, missing the initial decimal point of the first reading. This made the first reading much larger than the others and the resulting plot of the data appeared to represent a constant zero function. A typical student query was "What's gone wrong here?" Here an expectation (presumably of oscillatory motion) has been thwarted, and the students have gone into checking mode, looking for the source of the problem.
A similar attitude was apparent in Part 1 of the Second-Order Linear Homogeneous Differential Equations with Constant Coefficients. Step 3 of this Part looks at the differential equation y" + y = 0 with varying initial conditions:
Reset y(0) = 1, and change y'(0) first to 2, then to 3. How do the solutions change as y'(0) varies through positive values? Use the symbolic solution to explain what you see in the solution graphs.
Some students tried relatively large values for y'(0), and then the Maple command given in the accompanying worksheet:
DEplot(DE, y(t), t= 0..15,[[y(0)=y0,D(y)(0)=y1]], y=-4..4, stepsize=0.1, linecolor=blue);
produces a plot where y appears to grow indefinitely. Again the self-monitoring attitude showed itself in the resulting query: What has gone wrong here? (Incidentally, modifying the y-range overcomes this problem.)
Sometimes, however, students were unsure about what checking actually meant. Part 11 of the Helper Application Tutorial module provides a case in point. The introduction explains that checking is part of the forthcoming task:
In this part we explore Maple's ability to solve the logistic equation
Step 1 uses Maple to find the general solution to this differential equation. In Step 2 the student is led through the calculations needed to check that the general solution does satisfy the original equation (although, perhaps significantly, the word check is not used):
Differentiate your solution expression with respect to t to get an explicit expression for dy/dt Then use your solution expression to find an explicit formula in t for y(1 - y). Is this formula the same as the one for dy/dt? You may want to simplify the output before you try to answer this.
In Step 3 an initial value is added to the problem, and Maple is again used to find the solution. Finally in Step 4 the student is asked:
What do you have to do to check the answer from the preceding step? Have you done it already? If not, can you get the checking technique from what you did in Step 2?
Although all students eventually got correct answers to Step 2 (with more or less help from the tutors), most of them checked the solution to the initial value problem in Step 4 by just substituting the proposed solution into the original differential equation, ignoring the initial value part of the problem.
Failing to check the initial condition here probably indicates some uncertainty about what checking actually means. On the other hand, substituting a particular solution into a differential equation after they have already checked that the general solution works may mean that they haven't understood what is meant by a general (or particular) solution. Another possibility is that Step 2 was not perceived as a check. Apart from the absence of the word check in the instructions for Step 2, the actual calculation of y(1 - y) was complicated by the fact that Maple did not store the solution in a variable called y. The purpose of Step 2 may have got lost in the intricacies of the calculations.
A similar uncertainty about checking surfaced in the Experiments With the Laplace Transform module. In Part 1 the student uses Maple to calculate the Laplace transforms of some standard functions, including exp(-at), cos(at), and sin(at). Then in Step 2 they are asked to verify that the Laplace transform of df/dt is s F(s) - f(0) for the functions exp(-at) and cos(at). Many students asked what they were supposed to do here, almost as if they did not know what was meant by the word verify.
As an aside here, it is interesting that most students ignored the accompanying advice:
If you use the results of Step 1, you should not need additional computer algebra calculations for this step.
and used Maple to do the calculations anyway. Bookman and Malone noted student uncertainty about which tool to use for a particular calculation (see their Vignette 3). In this case it seemed that most students preferred to use Maple, although as a tutor I felt that they would learn more from a hand calculation -- and the advice quoted above suggests that the authors of the module felt this way too.
None of the students mentioned checking in the surveys, but I did get some positive feedback from individual students who had experienced the warm glow of knowing their answer was correct in the end-of-course exam.
Summary and discussion
Were the CCP modules a worthwhile experience? In the second survey, most students seemed to think so: 53 students made only positive comments while 19 made only negative comments (another 21 students found both good and bad points in the experience). The key question for me is whether using the CCP modules has helped the students learn, and both my students and I feel (on balance) that it has.
Bookman and Malone address the question of how students learn in an environment like a CCP laboratory, and we have seen some of the factors that helped or hindered learning for my students. Some of the things which my students felt helped them were
To this list I (as the teacher) would like to add CCP's insistence that students check their answers although, as we have seen, this did not always work well in this instance. To get round this problem, I hope next time to give more regular feedback, rather than postponing all the grading until the end of the course.
On the other hand, the main hindrances that my students noticed were
Next time I hope to reduce the latter difficulties by being a little more flexible in the way I assign students to teams.
Difficulties with the computer algebra system may be a little harder to cure. These difficulties were also common in Bookman and Malone's study (see their Vignettes 1, 2, 3, 4 and 7). For my students the difficulties might be partly explained by lack of experience with Maple. However, the fact that even experienced tutors cannot always resolve the difficulties suggests that we may never eradicate this problem, and should rather content ourselves with trying to minimise it. Next time I plan to give my students a more thorough grounding in Maple at the start of the course, but it is clear from some of my students' comments (and from the work of Bookman and Malone) that students also need advice on self-monitoring skills (knowing when to quit and start again, if necessary).
It is easy to get the impression, when reading accounts of student difficulties, that students are not coping very well. Despite the problems mentioned above, and those reported by Bookman and Malone, it needs to be remembered that most of my students did not think the problems were serious enough to merit mention on their survey forms. Indeed many felt that the lab sessions
Acknowledgment and References
These CCP lab sessions were conducted while David A. Smith was visiting the University of Canterbury on an Erskine Fellowship. I would like to thank David for his advice and help in running the sessions.
Anton, H. (1999), Calculus, John Wiley, New York.
Bookman, J. and D. 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.
Boustead, T. M. (1999), Undergraduate algebraic skills and mathematical comprehension, Ph.D. thesis, University of Canterbury (NZ).
Boyce, W. E. and R. C. DiPrima (2001), Elementary Differential Equations and Boundary Value Problems , John Wiley.
Heid, M. K., G. Blume, K. Flanagan, L. Iseri, W. Deckert, W. and C. Piez (1998), "Research on mathematics learning in CAS environments." In G. Goodell (Ed.), Proceedings of the Eleventh Annual International Conference on Technology in Collegiate Mathematics (pp. 156-160), Addison-Wesley.
Miles, M. B. and A. M. Huberman (1994), Qualitative Data Analysis (2nd ed.), Sage Publications.
Winter, D., P. Lemons, J. Bookman, and W. Hoese (2001), "Novice Instructors and Student-Centered Instruction: Identifying and Addressing Obstacles to Learning in the College Science Laboratory," Journal of Scholarship of Teaching and Learning, Volume 2, Number 1, July 2001.