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)