Computer Enabled Learning Activities
Computer-enabled activities are learning activities (defn), that require the use of computers or are significantly enhanced by computers. At UHD, the reformed college algebra course embraces visualization and real-world examples while taking advantage of computer speed to present images, graphs, and relevant problems and data. An overarching point that faculty made is that students also need to use computers to carry out rote calculations, because doing so allows them time to focus on conceptual learning:

Any instructor who has taught this course to beginning students knows that if students experience difficulty, it's typically in carrying out the procedure. "Where did I make my arithmetic mistake?" It's like looking for a needle in a haystack. It's not that the student doesn't understand the concept, it's that he or she has made a mistake in carrying it out, in the manipulation. So a high percentage of the time when you're teaching students these procedures, these algorithms to solve systems of equations, you're spending a large amount of your time saying, "Look, you didn't multiply two times all of the left-hand side of the equation." We just said, for the sake of time, let's turn that over to the computer, because the manipulations are only part of the problem. We really want to answer these conceptual questions at the end. (Linda Becerra, Faculty)

Visualization
Instructors in College Algebra use visual aids in a way similar to the way they might use a graphing calculator. The visual aids provide students a means to graph and manipulate functions with a friendly user interface. However, a computer is a lot more powerful than a graphing calculator, one reformer indicates, because it shows the graph in color on a 15" monitor or on a larger area through a projection system. Graphs offer instructors an avenue to interact with students. Discussion can begin with a function's graph, so students can attempt to determine the properties of the function, or it can begin with the mathematical function, so students can determine the graph and its properties. One instructor explains the importance of graphing as a teaching strategy:

Those are certainly the high points: changing representation, beginning a problem with function, exploring properties by looking at tables more closely or by looking at a graph, or starting with a graph and trying to develop the function formula. We constantly go back and forth between the representation and the mathematical function, showing how you can illuminate certain properties depending on the form that you choose to work with, and we try to cut down on the tedious calculations. (Linda Becerra, Faculty)

Course discussion focuses on the course's central theme, functions, and it emphasizes interpretation of functions, not rote calculations. This fundamental strategy is based on the connections between mathematical functions and their graph and table representations:

The whole course revolves around functions. Almost everything is done in that context. We do lots of lab activities where we just explore some functions that have meaning to the students and are practical. We do a lot of graphing and a lot of creating tables of functions, so we take a formula and make a table out of it or we graph it. (Bill Waller, Faculty)

Student interaction seems to be an important learning strategy, and when computer-enabled visualizations are utilized, student interaction increases. Students feel engaged when instructors use graphing tools to help them understand difficult concepts. One student explained:

I like the graphs. The instructor will ask, "What does this mean?" And the class is like, "Oh, I don't know," and he'll say, "It means all the points in this plane." And then we're like, "what plane?" And then he'll show us on Maple, and it has a 3-D graph that shows the colored plane. And then suddenly we all understand what he's talking about. Or two intersecting planes, and it'll be a line between the two planes. It has a very good visual interface. (Student)

Connecting to Real World Data
In this college algebra course, instructors attempt to stimulate students' interest and engage them in the learning process by giving them mathematical problems to solve from everyday situations. An instructor comments on this approach:

For example, in lab activities we give them a postage rate table and ask them to calculate the postage for a letter weighing so many ounces. Another example is a single taxpayer with this much income. They are asked to calculate what the taxes will be. Or we give them a tuition table and ask what the tuition cost will be for a student enrolling in twelve credit hours. So all these examples come from their everyday life experiences. It's true that some of our students have never worked with a tax table before or gone to the post office and paid anything different than regular first-class postage. Nevertheless, these activities, because of their relevance or connection to real life, keep them interested. (Linda Becerra, Faculty)

The connection to real-world problems lets students discover mathematics, while the interaction between students and instructor creates a sense of apprenticeship. Here's a student's viewpoint on this:

He would give us projects where he'd say, "we have three companies and they produce these products." He would ask us a real-life question that would require that we use a matrix approach, but he wouldn't tell us how to set up the matrix or what variables to put where. We'd have to set it up on our own. I think that's a good way because it made us talk to each other. We got to know each other, and we learned to rely on each other. You know, if some of us had problems, we'd show each other how to do it. (Student)

There are additional advantages in using computers in a curriculum. One is that they enable instructors to include, not just canonical problems that illustrate concepts, but also more complicated problems that have a real-world basis. Said one faculty member:

With the computer, you get an opportunity to give more complicated examples. Then there is the question: What's the meaning of the example, what's going on here? You might be doing an example from finance or from science, so you get a chance to discuss the context of the problem. That helps them, I think, and gives me a lot more opportunity to interact with the students. (Bill Waller, Faculty)

These sentiments, namely that computers have an ability to expose students to more real-world problems, are echoed by a student. He explains:

A lot of the faculty members like the technology because you can actually do some real world problems. You don't have to restrict it to one- or two-digit numbers and only five data points when you are dealing with calculations. You can really do some high-powered nice problems.

Finally, one stereotype of the academic world is that teaching sociology or communications is more enjoyable than teaching mathematics because instructors can draw their students into conversations about content. With contextual examples and computer-enabled visualizations to illustrate concepts, mathematics instructors are able to create similarly enjoyable classroom environments. In addition, they are finding it easier to make seamless transitions between one topic and another:

Using graphs and providing a context for problems also help me as an instructor. They provide me a good opportunity to make connections between topics. How am I going to move from this topic to another topic? In math classes, when you're looking at a graph, you almost always stop, and almost always something that you're leading to is showing up in that graph. Then you get to at least ask the question that introduces the next topic. (Bill Waller, Faculty)

Speed
Earlier, when we discussed faculty and institutional goals and computer-based technology, it was clear that cost efficiency was one important goal of the institution, while faculty place student learning as their priority. Given the large number of under-prepared students at UHD, faculty are especially concerned with the efficient utilization of classroom time. We asked instructors if they attempted to maximize student learning by using computers. An instructor answered:

You asked if technology helps you make more efficient use of class time. Yes, it does, because without it you might have spent a lot of time graphing functions by hand. That's a time-consuming process. Your graphs are usually not very good, and the way that we teach students to graph functions by hand is not the way we usually do it in practice. We'll either use a tool or we'll just roughly think about what the function looks like. Often we won't bother to draw a very accurate representation, or we don't need to draw a very accurate picture of the function in order to answer the sorts of questions that we want to answer. So, yeah, technology is great for helping you spend a lot more time looking at graphs and thinking about what they mean and the information that they convey rather than going through the tedious process of drawing. That's what is really important about doing graphs with the computer. (Bill Waller, Faculty)

While technology provides tools to produce accurate visualizations quickly and hence increase the time available for student learning, this "extra" time is not used at UHD to pack more content into a given time period. Rather, it is used to explore topics more thoroughly.

Using the computer's speed to graph functions not only allows the students to focus on concepts as opposed to arithmetic, it also gives them the opportunity to perform additional tasks and to consider more examples in a relatively short time period. This ability to work through more examples aids both the students' the learning process and the instructor's teaching process.

Enhancement of Student Collaboration
Collaborative learning is an essential part of the reform of this college algebra course. Students are assigned to groups and asked to work together on problems with a computer and also to interact with an instructor. However, the success of collaborative methods in the Department of Computer and Mathematical Sciences is mixed from the points of view of both instructors and students. On one hand, instructors see the benefits of group work, and they experiment with ways to engage students in collaborative activities and come up with a working model. An instructor explains:

The collaborative learning I've experimented with takes many different forms, but I've yet to find one that really works for me because our student body is strictly a commuter one. It's hard to get groups that can continue and build relationships beyond the classroom. How many students should make up a group, two versus three versus four? I don't know. And should it be set groups or rotating groups? Should you let them self-select, or should you appoint them? I've tried different variations, and I still haven't found one that works every time. So I think those are typical problems that most people have in trying to implement a collaborative learning method. (Linda Becerra, Faculty)

On the other hand, students do not appear to appreciate the social benefits of learning via group work. They are still operating in a competitive individual mode, where everyone needs to outperform their classmates in order to progress. One student explains:

As a student, I think there's advantage and disadvantage in doing group work. There's an advantage if you are a little slower than your peers, because you will have ample time to benefit from your peer's knowledge. But it is a disadvantage for the one who learns quicker. He gets bored working with the group. (Student)

In general, the people we talked to at UHD realized that their work on College Algebra has not yet yielded the results they had hoped for, but preliminary results are promising. Moreover, the faculty are determined to continue their reform efforts and seem willing to consider other options, particularly those relying on computer technology.

In order to get colleagues to accept (not necessarily adopt) reform ideas in a lower-division, required course like college algebra, it is important to have evidence that the reforms work. At UHD, this evidence takes the form of both anecdotes and student performance data. The college algebra course was reformed at a time when a sizeable fraction of students were failing traditional algebra sections. Passing rates in the reformed course sections significantly increased in comparison to passing rates in the traditional sections:

We went from a 38% passing rate in 1996 in our traditional section to a 46% passing rate in our "unified" technology section. (We use the term "unified" to describe our approach.) During the spring 1998 semester, the gap was even bigger. The passing rates were 35% and 51% for the traditional section and the unified technology section, respectively. That's quite an accomplishment. (Bill Waller, Faculty)

The table below provides comparisons between reform and traditional sections over a two-year period. No only does this table contain passing rates in college algebra, it also shows passing rates for students in the subsequent math courses.

Semester	Type of Section	# sections /students	Grade C or better	Grade C or better in next course
Fall 1996	Traditional	6/202	38%	35%
	Reformed	6/185	46%	36%
Spring 1997	Traditional	3/91	38%	32%
	Reformed	3/75	39%	26%
Fall 1997	Traditional	3/74	45%	20%
	Reformed	3/51	48%	29%
Spring 1998	Traditional	3/82	35%	19%
	Reformed	3/69	51%	28%

Passing rates are not the only evidence of success, according to the reformers. Instructors report that there is anecdotal evidence of an improvement in students' experiences--students seem to enjoy the course more:

I think that we may not be able to quantify success with passing rates alone. But in discussing with students, asking them to consider their overall experience, and reading some of the general comments, I do find a number of them who say it was a worthwhile course. Maybe they actually enjoyed it. We don't really have data that we could use to compare [comments about the reformed course] with those of the traditional course. (Linda Becerra, Faculty)

If students enjoy being in this course, so do instructors. Before reform, the classroom atmosphere was less than enticing for an instructor. Attendance was low, as were interest and participation. An instructor explains:

I can only give you anecdotal evidence, but the ambiance in the classroom is totally different compared to before. The biggest thing that I've enjoyed about it is when you go into the classroom, you're not dreading going. You're not thinking, How many are going to show up today? How many are going to be paying attention? Are they going to ask any questions? Am I going to get people to ask any questions? It's just a much different classroom atmosphere than before. (Bill Waller, Faculty)

Nevertheless, not all the faculty members in the department share the reformers' enthusiasm about the success of this reformed algebra course. Many question the adequacy of course content and the students' ability to use their algebra skills proficiently, a charge that is answered by one reformer as follows:

We did reduce the content in designing our unified course. But in tracking the students in later courses, we haven't found any noticeable difference in performance between those who attended the traditional course and those who went through our unified section. A lot of the skills that you teach in traditional college algebra, you treat them so superficially that you wind up re-teaching them anyway in later courses. So the students in the traditional sections really didn't have much of an advantage because they didn't remember that one specialized skill they had been taught for one lecture. (Bill Waller, Faculty)

Some faculty also imply that there is a certain amount of grade inflation in the reformed course, that students are graded on the basis of activities that have nothing to do with college algebra. One reformer responds to this allegation by bringing attention to the data shown in the table above:

When you're teaching the college algebra course here, you can expect that the average number of times that a student in your class has taken the course is more than one, and that's on average. I've had a lot of students who have been able to pass my reformed class, but weren't able to pass the traditional course. A lot of people will say, "Well, there you go; it's too easy. They shouldn't be passing." But in tracking these students, we find that they seem to do okay, not any worse than other students in later courses. I think to at least be successful once in a college math course is important for them. (Bill Waller, Faculty)

There is also skepticism on the part of some faculty regarding the validity of the performance data (see table above), given that students enrolling in both the reformed and traditional sections are not selected at random (the sections that rely heavily on computers are so noted in the sign-up schedule). However, the reformers point out that at UHD the dominant factors in class selection are course availability and section time, not the methods used to teach content. Said one reformer:

People always question the validity of those statistics because students self-select the section they want to enroll in. The sections are identified in the schedule, so they know which ones are going to be using computers. How much does that affect our outcomes? I don't know. But I strongly believe most students are selecting classes based on availability and their schedules. I don't think that we're having a lot of people who are gravitating toward the tech sections just because they want to be in that section. (Bill Waller, Faculty)

While some questions about college algebra reform exist within the department, we should note that in many respects the objections of faculty are tempered by the fact that at UHD they do not have to adopt these reforms. At UHD, because there is no institutional push to adopt reforms department- or campus-wide, being a reformer is not very threatening to non-reforming colleagues.

Getting Started

Even when a faculty member is faced with serious challenges in his/her learning environment (defn), and thus is eager for reform, there is still a fairly high activation barrier in starting to develop a reform agenda. Potential examples of barriers at UHD are the high teaching load, the university's reward system, and student demographics. In this section, we consider the internal and external factors that propelled this group of three algebra instructors to begin the reform process.

Instructor Traits and Motivations
When queried about their motivation to start this reform effort, the faculty gave several reasons. There was, of course, the high failure rate in College Algebra that they wanted to reduce. Also, as already mentioned, they wanted to change the learning environment (defn), to make the classroom atmosphere more enjoyable both for them and their students so they didn't have to think about "how many [students] are going to show up today; how many are going to be paying attention; are they going to ask any questions?" Another motivation came to light when one reformer was asked if faculty used computers at UHD to optimize classroom learning and make efficient use of students' classroom time since most students are commuters working part- or full-time jobs.

I wouldn't say we use computers to optimize the learning time, but to accommodate more learning styles. Some students learn better, of course, in a certain mode-visual, auditory, hands, skills-and so the computer somehow seems to tell students or inform them, illuminate things for them, in ways sometimes that we can't do using the board. It seems to be another dimension for them as a learning tool; so by presenting with it, you sometimes reach a student that nothing else could have reached. (Linda Becerra, Faculty)

Computers seem to provide instructors this new dimension, or teaching medium, that might reach a student with a different learning style.

One issue that was not a motivation for these reformers was to pack more content into the course, and into the students' brains when they are physically present, by using a more "efficient" teaching mode. Instead, instructors were more interested in improving the quality of learning experiences.

External Factors and Support
There were also external factors, some of them serendipitous, providing an initial environment that allowed this algebra project to incubate. One external factor, the Interactive Math Text Project (IMTP), funded by IBM and the National Science Foundation, began early in 1990. The Interactive Mathematics Text Project had as its goal the improvement of mathematics learning through the use of computer-based interactive texts. To achieve this goal, IMTP held summer workshops on the writing and use of interactive texts and supported selected individuals as text developers. UHD was one of six national sites selected for these workshops:

Well, actually the history [of our reform] goes back to the IMTP. That's what got us started here. This was a program that was funded by IBM and NSF for MAA, and was directed by Jerry Porter from the University of Pennsylvania. We were one of six sites selected across the country. I was the project director for our campus site, and that's what got us involved in using some form of technology in the classroom here. (Elias Deeba, Faculty)

The IMTP workshops gave UHD's math faculty access to then state-of-the-art computer equipment and software. It also introduced them to national reformers and reform ideas in mathematics:

In 1992, there were some computers put here because of the IMTP, whose results, I think, were not all that great. But people were coming here. Realize that the IMTP infrastructure was put here, not for the faculty, but for the workshops. Of course, we got to use it when it wasn't otherwise being used. That was an initial catalyst that provided us equipment access. (Bill Waller, Faculty)

In order to validate a reform plan in their colleague's eyes, the College Algebra faculty needed to align themselves with a reform movement that had a national reputation. For UHD's group of soon-to-be reformers, the Harvard calculus reform movement provided this anchor.

We began this work because of the mother of all reform programs, Harvard Calculus. What got us started were the problems [high failure rates] we had before. We knew about reform in part because of the IMTP, which was itself influenced by Harvard. There were these two things that were sort of converging. Technology was more accessible because of the IMTP. Then these reform texts [from Harvard Calculus] were being published at about the same time. (Bill Waller, Faculty)

While IMTP provided computer access for faculty, much more was needed in order to begin implementation of reform in this college algebra course. Real reform would provide wide access to computers, not only faculty, but also for students. In addition, the faculty needed time to develop the new curriculum. UHD'S group of three found an invaluable resource in the NSF's ILI grant program. It provided them with the initial funds that externally validated their plan in their colleagues' and administrators' eyes.

I don't know what NSF's view on equipment is anymore, but ILI grants used to be a really terrific resource. I don't know if the NSF thinks now that all universities can afford their own computers, but that's a place to start looking. We also had the Interactive Mathematics Text Project (IMTP) workshop in the early nineties, which gave us the initial hardware and some software for college algebra. Then there was a curriculum development grant, which we used to buy release time for developing curriculum. (Linda Becerra, Faculty)

Gradually, therefore, the UHD reformers had computer equipment, software, reform-oriented texts, and time to develop a new curriculum.

Implementation Issues: Institutional and Cultural Issues

Organizational issues arise from the institutional context provided earlier. However, the focus here is on specific institutional factors that have affected reform in college algebra.

Rewards
UHD has a number of incentives for faculty who excel in teaching. One is the University-wide teaching award. While one administrator was quick to point out that this award has no substantial financial benefit, it is clear that it is considered prestigious among faculty colleagues and administrators. It is interesting to note that teaching reform that incorporates technology is considered important enough at UHD that some technology savvy reformers have received the award several times in row. In fact, the principal worker in college algebra reform was given the teaching award six times in the last seven years:

The teaching awards are institutional here. In the last seven years, Professor Waller has won the teaching award six times. This college, the College of Science and Technology, and I remind everyone of that, has excelled in teaching. (George Pincus)

The University also offers financial support through its Teaching with Technology Learning Center. The TTLC offers small grants often used for release time to help faculty initiate courseware development with technology. A faculty collaborator explains:

The University was very supportive. There was equipment support and small grants from the TTLC, Teaching with Technology Learning Center. You can apply for a faculty development grant there; you can get money to develop courses; so there is support from the administration to do things like that. (Elias Deeba)

Status
Use of technology in one's curriculum reform seems to bring some status to UHD faculty. There are two avenues for being recognized when doing such work: First is the teaching award mentioned above. It is important to note that although the college algebra reformers have not yet been given the teaching award, their colleague who uses technology in his teaching of linear algebra has. This award winner comments on the campus culture.

I think my use of technology had an impact on them selecting me for the award. (Elias Deeba)

Second, UHD enjoys the exposure and marketing benefits from the technology work done by its faculty. As a result, faculty involved in these activities enjoy a small status advantage.

Cultural Shift
Faculty who have implemented reform at UHD have faced challenges from both students and faculty colleagues who are hesitant to accept it. The faculty's central objection was the issue of content reduction in the college algebra course, but there was also opposition to new technology, as explained by one of the reformers:

I think the biggest challenge is trying to get your peers to accept the idea, the change. That still hasn't happened. The main issue has to do with content. They feel like content is being taken out. In addition, they are not familiar with the technology, and they don't want to learn anything new. They don't want to change the way they teach. But fortunately no one has told us we can't change. (Ongard Sirisaengtaksin, Faculty)

The use of technology in the reformed college algebra course also provided some peer pressure for colleagues to use modern techniques at least to present, if not reform, course content. At this time, faculty colleagues feel comfortable because they don't have to engage in these activities, but some are wondering, if in the future, they will be asked to. An administrator gives his assessment of faculty and student views of the math department's use of technology:

In the math group, you'll find a lot of faculty who are very resistant to using much technology in the classroom. In courses like college algebra and calculus, many of the instructors still want to use the old methods. They are finding that some students are reluctant to accept the technology as well. When we can, we'll put in two sections at the same time, one being traditional and another using some technology. When we do that, we find a lot of switching around as we move along. I've had students come in here and say, "I don't want to buy a calculator," or "the course I had, I'm repeating it, I don't want to learn something new. I want to do it the old way."

This comment quickly reminds one of the UHD culture where reform, even if it is successful, cannot be imposed on other colleagues. The departments at the University appear to be very compartmentalized. Each group managing one or more courses is fairly free to implement new approaches, as long as others are not required to use them. Hence use of new methods on the part of some faculty does not seem to have created department conflicts or affected tenure and promotion decisions.

I don't think that there have been any problems or conflicts. The faculty seem to be happy with the fact that it's ok to use various methods, as long they aren't forced on the rest of us. That is sort of the attitude. So I don't think there's any problem at all. People are, I think, working together very well in the department in terms of the program committees. Whether they use technology or don't use technology has not been an issue in terms of getting tenure.

Like some faculty, some students also have not been easily convinced of the efficacy of using technology. Given that most of them can spend only a limited amount of time on coursework, any activities they perceive as superfluous are not welcomed. A collaborator comments:

Another challenge is to convince the students that these computer-based activities are good for them. That still continues to be a problem because they don't see the need for these methods. They want to do it fast and go. My job is to convince them how very important this type of activity is for them. (Elias Deeba)

Time is not the only factor in the students' resistance. Some are still not entirely comfortable using computers and have completely internalized the value of traditional methods for teaching mathematics. One reformer explains:

The computer is still something new to our students. Talk about student resistance, I've had this comment more than once, and this is really funny. They'll be complaining about the way the course is taught, and they'll say, "You know I've had this course before," meaning that they've taken it and either dropped it or failed it. "I've had this course before, and this is not the way it's supposed to be done." So even though they may be successful in the technology-based course, they are still telling me that I've got to get with the program. "You're not supposed to be teaching it this way," they'll say. "This is a math course; this isn't a writing course. Why are you asking me to write a sentence?" (Linda Becerra, Faculty)

Organizational Savvy
Getting reform implemented in the college algebra course at UHD required little organizational know-how for the faculty. Classroom and laboratory modifications needed for the reform to take hold were and are being done by the College for reasons not directly connected to college algebra. However, one important change worth noting was the reformers' ability to get UHD to agree to commit one technical support person to the math department, so they would not need to rely on the less responsive college-wide technical staff:

In terms of having our own technical support in the department, we've been able to convince the administration that it's a necessity. The reason is that we got external grants to build the lab, and in the grants they promised that we would have technical support. So they didn't have much choice, if they signed off on it, but to go along with it. (Bill Waller, Faculty)

Implementation Issues: Technical Resources

Technical resources needed for the reforms at UHD fall into two categories: equipment/hardware and technical staff support.

Hardware
Nearly all the Computer Science and Mathematical Science Department instructors at UHD who use technology in their classrooms pointed out problems with computer hardware reliability. Sometimes the problem was operating system robustness; other times it had to do with software contaminated before class and laboratory activities. A collaborator reflects on his frustration with hardware:

It was hard for me at the beginning. Really, I mean I was completely frustrated. The machine will break down on me-my God, I'm going to have a fit. This is the way it was: it was lousy. You expect everything is going to be perfect, and it's not perfect, by no means. There will be bugs here and there; the machine is going to break down. (Elias Deeba)

Technical Staff
Adding a dedicated technical support person to the Math Department has significantly reduced the frustration with technical problems experienced at the beginning of the reform efforts. A faculty collaborator explains:

In the past, getting the lab ready, in the sense that students can use the equipment and there won't be any problem during class time, was an issue. Before, we had problems like crashing, for example, or the computer wouldn't start. Now, we have much better technical support than we had in the past. Our support person is now entirely devoted to the department. That is a big difference. (Ongard Sirisaengtaksin, Faculty)

This improvement in technical support is primarily due to the fact that a staff person is now located right in the general Math Department office area. It makes it easy for instructors needing technical support to simply drop by the next-door office, rather than seek help from the distant Teaching with Technology Learning Center, which provides University-wide technical support responsibility. The technical staff person comments on this:

I think the Math faculty like that I am near them, and they feel comfortable working with me. I think they prefer to come to me, because the TTLC not only assists our department, but everybody in the university; so it takes a longer time to get help from them. If the faculty come to me, I usually have less to do, so I can get to them a lot quicker than the TTLC would. (Veronica Martinez)

Implementation Issues: Time and Workload

Many issues can stymie reform. The previous section focused on two obvious, tangible issues: hardware and technical support personnel. Somewhat more elusive is finding the time to implement reform programs. Specifically, is the reform effort placed on top of an already heavy workload? In addressing the workload issue at UHD, an administrator is quick to point out the TTLC's small assigned time grant program:

At this institution, we have the learning center in the technology area. Faculty submit a short proposal to the center, and it offers small grants to professors competitively. It's a positive approach. I've never found that the punishment approach works. (George Pincus)

However, even with an assigned time grant, a major impediment to the kind of reform implemented in college algebra at UHD is lack of time for instructors to design new curriculum methods. With a four-course teaching load per semester, committee assignments, and extensive student contact hours, finding blocks of time to work individually and collaborate with others on curriculum matters is nothing short of a miracle. Moreover, because of the additional learning activities (defn), associated with reform, such as group work, incorporating technology into teaching further increases the workload. At UHD, this added workload is handled without graduate student help. An instructor comments:

Using technology, my own workload, of course, has increased tremendously. I've never done a course where I had to grade exercises, labs, and projects. It does increase my teaching load, not the number of teaching hours, but the kind of work I do. I want to collect homework, and I want to collect the labs and the projects and grade them myself. We don't have graduate assistants to help us in this process. (Elias Deeba)

There is some concern about the possibility that these reformers may burn out, particularly if the department continues to assign the same workload to all faculty, regardless of the kind of activities offered and the methods used

Implementation Issues: Attitudes and Peer Support

Attitudes and Teaching Philosophy
At the root of this ten-year commitment to implementing reform in the college algebra course at UHD are the reformers' attitudes and teaching philosophy (defn). What values made this reform happen? Playing the skeptic, we asked if the purpose of reform was to make the course more appealing to students-a somewhat superficial objective for using technology. One of the reformers who answered provided some interesting insights into his classroom teaching principles (defn):

I don't think we were thinking of using technology to make the course sexier; that was in the back our mind. They [the students] come in with weak skills, so we were using the technology as kind of a skill replacement. We wanted to do more relevant problems. We wanted to present the material in a different way. We didn't want them to be repeating their high school algebra class. Technology really gives you an opportunity to sort of remake the course. We call it reinventing the course, so it gives you something to structure your syllabus around. (Bill Waller, Faculty)

We also inquired about using technology to interact and communicate with students, in light of the fact that many students are working long hours outside UHD. Another reformer answered:

We don't use technology as a communications tool; we use it mainly to teach content. One great thing about the technology is that it really gives you a good context for interaction. So if the students are doing something in class, for example making a graph, it gives you a good opportunity to talk with them about that graph or have them ask you questions about it. It really helps bring the instructor and the student closer together. (Bill Waller, Faculty)

The road to reform has had its bumps, in the form of criticisms from faculty outside UHD who use the curriculum materials developed for the course and also from faculty within the UHD Department of Computer and Mathematical Sciences who have not adapted the materials. Both groups have expressed concern about the scope of the new course. As a result, the reformers have had to make a case for reducing some of the content and skills that are taught in the traditional course. They questioned the old paradigm of teaching students every detailed step in the mathematical solutions to canonical examples, and, in line with the national reform movement, they included in their curriculum problems with real-world relevance. Here is how they made their case:

In many cases, people say, "you didn't teach students the step-by-step way of solving a three-by-three system of equations, which is a standard part of the curriculum." People who have used our material off campus have complained about that, but it's a decision that you make, and here is why you might come to that sort of conclusion. When you're teaching students to solve a three-by-three system by hand, what are you really trying to teach? There is something useful there--the whole idea of substitutions. What else is there about solving a system of equations that's truly useful? What happens often in algebra courses is that it's never pointed out to students that algebra is a problem-solving method that's useful in other areas as well. It merely becomes a blue box in a book: "Follow these steps to solve a system of equations." When it becomes that rote, all the value of the method is missing, but all the pitfalls are still there, the arithmetic mistakes, all the things that frustrate students. So we thought that what students learn by us teaching them elimination is not worth it at this point. We'll just let them solve the system with the computer, which is the way they're going to do it in practice anyway. They're probably not often going to do it by hand with a realistic system of equations, maybe only for small systems like a 2 x 2. So those are the kinds of decisions that we were facing throughout the curriculum. We asked, "What's really important about this problem?" And then, "How important are the manipulative skills that you must have to carry out to the next level?" (Bill Waller, Faculty)

A faculty collaborator reinforces the case for reform by commenting on his experience teaching a reformed linear algebra course at UHD:

My experience so far teaching this class has been relatively successful. In a course like linear algebra, from my point of view, if I were to do it the traditional way--the theorem proof type of thing--it wouldn't be of value to our students and they wouldn't appreciate it. But in the interactive setting that they have, they're creating something of their own; they have to create their own examples. The examples are there, but they can change the examples and try to conjecture from them what's going on. They can derive some of the concepts, but proofs are very minimal in this course. So that's basically what I do in this linear algebra course. (Elias Deeba, Faculty)

When this same collaborator was asked why he continued down the reform path even though he faced a series of daunting challenges (increased workload, technical difficulties, etc.), he replied:

Yes, I could have quit, but I felt a value in it for me, because I saw the utility of these [software] packages--in particular, in my case, Maple. I saw how important it was for me to make a presentation and create a dynamic environment in the classroom using these software packages. (Elias Deeba, Faculty)

In reflecting on the last few years of reform, one instructor considers the many benefits of including technology-based activities in the college algebra curriculum:

People who are thinking about using technology may just be wondering what it offers. It really allows you to open up in the classroom and gives you a lot of avenues for discussion. Sometimes when you're lecturing and you're so focused on finishing up five examples, or whatever you had planned for that day, you're really just reciting the material. You ask for questions, but it's kind of perfunctory, and of course the students don't ask anything. They're willing just to sit there and let you regurgitate the material. But when you're trying to do examples with technology, when you're making graphs in a math class or showing graphs, it always raises points of discussion. There's always something that you see in the graph that suggests a point that maybe you hadn't thought about. When you're doing more complicated examples than before, it brings up the whole question of "What's the meaning of the example? What's going on here?" (Bill Waller, Faculty)

Peer Support
At educational institutions, even those that are teaching-focused, there is limited financial gain for faculty who engage in curricular reform. Often the institutions provide financial support or other types of resources (e.g., additional space) only after external foundations, such as NSF, have approved a project. Hence peer support is a key element that usually keeps reformers engaged and motivated in the initial stages of their work. A variety of factors (such as the Interactive Math Text Project [IMTP] workshops in the early 90's) resulted in the UHD college algebra reformers receiving external support early on. Thus peer support was important, not to get the project started, but to continue its implementation for almost a decade. One should remember that this reform project relied upon a close working relationship among three faculty members for nearly ten years. UHD's environment encouraged this collaboration and allowed the relationship to flourish. Nevertheless, one reformer spoke about the commitment required to continually build support from departmental colleagues. Here is some advice for future adapters:

You have to constantly sell yourself. Give it a chance. There will be times you'll be lost, there will be times you'll be confused, but if you're patient and keep trying, then things will get better. It's not always smooth sailing; it's constantly trying to convince them that it will work in the long run. (Linda Becerra, Faculty)

The environment at UHD is unusual in that faculty are free to experiment with small curriculum segments, as long as they do not attempt to impose these changes on their colleagues. While departmental support per se was not strong, the changes implemented in college algebra were also not opposed by the department.

Well, the department doesn't require that we do things in a certain way; so it's been supportive in the sense that we can pretty much try anything that we want. We've streamlined the curriculum, we've brought in technology, we've tried collaborative learning, and we've never been told to stop doing something. (Linda Becerra, Faculty)

Implementation Issues: New Instructor Role

UHD's implementation of college algebra reform has resulted in an enormous transformation for the faculty in their role as teachers. Simply put, the faculty involved in this reform moved from the model of teacher-as-expert to the model of teacher-as-facilitator who coaches and helps students who are engaged in an active learning process. In that new role, faculty have faced frustration from some students who expect a learning environment (defn), where the teacher presents content while the students sit passively and listen. One reformer describes the challenges inherent in this transformation:

I think we're all sharing the same experiences. The old-fashioned way of the lecture method is very efficient. You have a syllabus and, if the students don't ask any questions, you can tell yourself, "Well, even if they don't understand it, I've covered every topic on this syllabus." But with the new way, you try to stop being the "Sage on the Stage" and become a facilitator. You try to become someone who initiates discussion. Of course, there goes your time, so you don't cover everything that's listed on the syllabus. Sometimes instead of trying to give answers directly when they ask a question, you try and give hints to get them to see what the answer is, what points they're missing. As a result, you sense a little frustration. They say, "Why don't you just tell me the answer?" You try and get them to be more independent learners, to learn to teach themselves, to be lifelong learners. How do you accomplish that? I don't know, but it's not by just giving answers. I know that much. So this has influenced me in every one of my courses. I try not to be simply an answer machine but instead someone who somehow engages the students to break down problems so that they can become better learners. (Linda Becerra, Faculty)

The change in roles resulting from the reform process has other benefits for faculty as well, namely richer interactions with students and an opportunity to reflect upon the new curriculum and teaching methods. Said one reformer:

I feel like it has made me a lot closer to the students. I talk with them more, so I understand their point of view better. Overall, it made me connect to the students a lot better. Just thinking about changing the curriculum makes you a more effective teacher all the way around, in all of your courses. In your other courses, you naturally start asking yourself, "Why am I teaching this course this way? Am I just teaching it because this is the way I was taught, or just because this is the way the textbook presents the material? What's a better way to teach the class?" And for someone who is thinking about technology, curriculum reform is a side effect, a real benefit. If they start to think about how they're going to incorporate technology into their classes, naturally they're going to start thinking about the curriculum. They're going to benefit from that, being reflective about the curriculum. (Bill Waller, Faculty)

Faculty role transformation does not appear to be limited to changes in this college algebra course. These reformers are bringing newfound knowledge and practice to other courses as well. One of them comments:

It changed the way I teach all my courses. I haven't been teaching math courses now for five years-I've been teaching exclusively computer science courses. But in those CS courses, I use that [reform] experience in the way I present the material. I always give an example first that's similar to something in engineering. We have the example and tie it into the concepts. That's the way I now teach all my courses. (Ongard Sirisaengtaksin, Faculty)

In teaching courses where reform has not yet been integrated, these reformers are finding it difficult to revert to the old ways, and they are identifying numerous opportunities for incorporating reformed-minded activities. Said one instructor:

I'm teaching a remedial math course, something I haven't done for a long time. It's intermediate algebra, so it's really high school algebra in one semester. I haven't had time to work on developing anything new, so there's no software. It's just a blackboard lecture-style class with no cooperative learning in it, and it's very difficult for me. I'm always thinking; I see all these opportunities where we could use technology, or all these places where I wish they had a group activity to work on because it's just fruitless for me to stand there and talk to them. So what I'm saying is, once you've engaged in this kind of reform, you realize that it's powerful. It has something to offer. (Bill Waller, Faculty)

It's clear that having experienced the efficacy of active learning techniques in one course, instructors know that the new teaching methods offer powerful opportunities for students to learn, so they want to incorporate them throughout the curriculum.

Implementation Issues: Student Issues

The one audience not yet addressed is the students. We wondered how the change from a traditional course to a computer-enabled curriculum affected them. How easy was it for the students to use the new technology? Did it increase or decrease their workload significantly? Did they have easy access to the computers they needed to complete their work? To what degree did computer use in this course improve or diminish the learning environment (defn), in their eyes?

Ease of Using Technology
Faculty who implemented computer-based activities in this college algebra course initially encountered resistance from students, not because of typical computer access problems, but because of the software interface used in designing the activities. The syntax and programming skills required for proficient use of programs like Mathematica or Maple were too demanding for lower-level students. An instructor explains:

Obviously in College Algebra we couldn't teach them anything about Mathematica. It's difficult to begin with, so students are going to resent and resist that. They'll say, "This is not a class about programming. This is algebra." So we created a visual basic interface. It was one of the first technology things we worked on, because we wanted to use Mathematica or Maple (we just started with Mathematica). We wanted to see if we could use it in the lower-level courses and have an interface that would allow students to carry out certain operations in Mathematica, without any of the programming. It's a fill-in-the-box kind of interface. (Bill Waller, Faculty)

The solution to this proficiency problem was to develop a visual basic interface that would provide access to all the computational and graphing tools of Mathematica, but provide students with computer interactions that were as easy as using a scientific calculator. With this interface, students could focus on learning college algebra concepts rather than programming.

Figure 3. The MILTON computer interface using Mathwright®.
Click on the image to see a larger image.

Change in Course Workload
If students at UHD thought that using computers in College Algebra would reduce their workload, they were mistaken, since the workload has apparently increased. Students are now required to engage in more activities than they did before. Moreover, simply completing these activities does not necessarily lead to a course grade of "A." Students have discovered that active learning is hard work. A faculty collaborator comments on the experience of students who took his reformed linear algebra course:

The students are finding that, especially in linear algebra, the use of technology is taxing them a lot. They have to do all this work and eventually they may end up getting C's and B's. At the beginning, using the technology caused grade inflation. Then, after I learned my lesson in a few semesters, things changed. Students think that if they do all these activities, then automatically they are securing a B or an A in the course, so many of them are disappointed after they end up getting C's or B's. They feel that it's too much work doing all these things, given that they are transit students; it's not like that's the only responsibility they have. (Elias Deeba, Faculty)

This is an important observation, since it is often incorrectly assumed that computer use in a classroom reduces the time required from the student. Although students may initially accept the increased workload, if every course they enroll in requires a significantly increased time investment, problems will undoubtedly surface.

Ease of Access to Technology
The problems typically associated with computer access have not surfaced in UHD's college algebra course. This is surprising, given the fact that UHD provides services to a large population of busy commuter students who have full-time jobs, families, and tight schedules. There have been issues of access, but these focus more on the students' financial inability to obtain the latest versions of software, not their ability to gain access to campus computers. In choosing a software platform, the reformers' top priority was to select one with effective Internet access. That way, only the instructors had to have the most current software, and the students were no longer financially responsible for purchasing it.

The software is continuously improving, but you can't expect a student to keep investing, investing, and investing as upgrades and improvements occur. So that's the fabulous thing about the Internet-it's becoming available everywhere at essentially no charge to the student, because schools are providing computer labs and sometimes even giving students computers. That way, instructors can keep updating the software and the interface without any expense to the students, who can take advantage of the most current technology simply by being connected to the Internet. This is especially important for our student body. Textbooks alone are a major expense for them, and to in any way burden them with something beyond that-they don't like it. They'll choose other options if they're available rather than spend extra money. (Linda Becerra, Faculty)

Increasingly, however, students who do own their own computers are purchasing student software versions of the platform used in class. They find it convenient and helpful to be able to have access to the software at home. One explains:

I think I retained more of the material than I would have if I hadn't done it with the computers. This is so because I have the Maple software at home, and I still use it for other things. It's still pretty fresh in my mind. (Student)

Improvements or Detriments From Use of Technology
Whether technology use has a positive or negative effect on student learning in College Algebra depends upon whom one asks. In general, though, students feel that technology provides them with the means to work on more complex problems and to spend time focusing on interpretation of concepts and learning through parameter variations instead of working out tedious calculations. One explains:

You don't get all caught up in doing the calculations part of the work, so you can actually focus on the concepts. For example, if you're going to see how a transform is going to affect a larger matrix, you don't have to sit there and do every single entry, which would take forever by hand. But you can do, like, a six-by-six matrix on the computer. It'll go through it real quick and it'll show you the results. You can change the values and see the effects, you know? If you did it by hand, you'd have to wait ten minutes just to see the result, and you get really tired of just doing the work itself. (Student)

The respite from tedious calculations notwithstanding, some students expressed frustration to the faculty because of the increase in workload. In addition, one collaborator is not convinced that all students are benefiting from these computer-enabled activities:

The more you involve them in the technology and demand that they use the technology, the more frustrated they are. They don't like it that much, not all of them. Last semester, in my linear algebra class, I would say out of twenty students, probably seven to eight were doing really well. The others were carried over by those, somehow. (Elias Deeba, Faculty)

Roles
Just as computer-enhanced curricular reform has changed the role of the teacher, it has also changed the role of the student. In the past, students typically came to the UHD campus only to attend classes. Now, however, they are engaged in community and group activities. Moreover, they are talking about mathematics, as one faculty collaborator describes:

In general, the students come in, take classes, and go; but this type of activity [cooperative learning] is getting them to know each other, be with each other, and talk about mathematics. In no other class do they get to talk about mathematics. But now they are saying words that have meaning in mathematics and discussing math problems. I think their math literacy has improved. (Elias Deeba, Faculty)

However, this change in role has not always been a smooth transformation. The reformers feel a continuous, strong pull from the students who want to revert back to the traditional teacher-student paradigm. Students feel comfortable in the traditional setting, even if they have not been successful in it. One of the reformers comments:

Students resist the change to a technology-based curriculum. Even if they're failing in the traditional classroom, they're failing comfortably. They're used to teachers coming in, lecturing, writing on the board, assigning homework, and doing practice problems. Maybe the homework is worked on, maybe not. But when they come in to test day, they are going to see problems just like the ones they've been practicing, and they're very comfortable with that. That's essentially the entire experience that they've ever known. (Linda Becerra, Faculty)

Conclusion

In order to sum things up, we asked these college algebra reformers what efforts, if any, they had made to try to disseminate their teaching methods. Intrinsic to that question was our knowledge of the measure of success they had in convincing their peers outside UHD to adopt their methods. One reformer answered:

We went to conferences and talked to people about our method. We were traveling salesmen for a while. When we talked to people and they got interested, they wanted to see more, wanted to know more about it, and asked us to give them a workshop. First we gave one workshop here, just to introduce them to our method. Then we made presentations at conferences and workshops. A few other instructors are using our text, not many, but it has been used off campus--at several schools, actually.

In the early 90s, because UHD was selected as one of six national sites to host IMTP summer workshops, these three math department faculty members were often on the workshop circuit, promoting mathematics reform. Later, they became authors of a college algebra text, and much of their dissemination efforts were made in the context attempting to market that text. Currently the college algebra text is being used at a few other institutions.

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1. Wilson, B. G. (1995). "Metaphors for instruction: Why we talk about learning environments. Educational Technology", 35 (5), 25-30, available at http://www.cudenver.edu/~bwilson/metaphor.html.