University of Houston-Downtown
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The College Algebra reformers described what they wanted students to "get out of" their course. Here is how one reformer described the goals:
Some of the goals are technology-oriented: the students should be able to use a calculator or computer to plot a table of points. Some of them are strictly by-hand skills: they should be able to solve a quadratic equation, be able to solve a linear equation and a linear inequality. So it's a mixture of hand and technology skills and then understanding concepts. They need to understand the concept of a function-recognize when something is a function, when something's not a function, the domain and range of a function, is this number in the range of this function? So some of our skill goals are very specific, and others are a little bit more broad and general.
(Linda Becerra, Faculty)
What follows are the other major focal points the reformers had, in no particular order of importance:
- provide numerous opportunities to learn
- teach fundamental concepts and skills by using real-world problems
- stimulate student interest by making mathematics relevant to the students' lives
- increase mathematics literacy
- use diverse teaching strategies
- offer a technology-dependent curriculum.
Provide Numerous Opportunities to Learn
An important trademark of UHD is its commitment to help students succeed, regardless of previous performance or failure.
Faculty are aware of the classroom challenges they face, but that only reinforces their resolve. UHD prides itself openly on its ability and commitment to provide many learning opportunities to its students, whether they need remedial activities or are prepared for college-level work. Several instructors speak about UHD's
learning environment (defn),
as an embodiment of the American dream, a place where everyone has an opportunity to learn. One instructor comments on the University's philosophy:
We have our 25th anniversary this year, and the president's motto is "Twenty-five years of excellence, opportunity, and diversity." The key word there for us is opportunity. Even though our students are under-prepared, we want to over and over be providing them more opportunities that are optional that they can take advantage of to be successful students. So we have a tremendously active math lab that offers tutoring by faculty-lecturers and faculty who are not tenure-track. The lab has practice software, drill type software, and it is open during extensive hours all through the week
(Linda Becerra, Faculty)
Teach Fundamental Concepts and Skills By Using Real-World Problems
The tension between having students learn concepts and helping them to be proficient with skills is alive at UHD.
One goal is for students to thoroughly learn certain fundamental algebra concepts, such as functions. To that end, these reformers incorporated a number of activities in the revised course, and course breadth was reduced to accommodate these new activities. On the other hand, they recognized that college algebra is also a skill-building course designed to prepare students for more advanced mathematics courses. The re-designed curriculum focused on mastering a reduced set of skills that faculty felt students must have in order to succeed in future mathematics courses. This strategy did not compromise quality, since most advanced courses often had to subsequently re-teach those skills anyway. An instructor explains:
We wanted to make sure that the fundamental concepts and skills (and we thought a lot about what fundamental means) were emphasized. We were trying to narrow the focus of the course. When our students went on, we wanted to be certain that they could use some small set of skills well. We really wanted to practice those a lot instead of making it a mile wide and an inch deep, which is the way content is traditionally designed in college algebra. You typically spend a little bit of time teaching them all these exotic types of equations that need to be solved and all these different categories of functions. We were trying to get away from that so we would make sure that when they showed up in the applied calculus course, they could solve a quadratic equation confidently, and we wouldn't have to be re-teaching that skill.
(Bill Waller, Faculty)
While this reformed course included the content and skills that students needed to enroll in more advanced courses, it also was useful for students taking their last course in mathematics. The reformers accomplished this by using problems drawn from everyday situations. Thus, this reformed algebra course demonstrated that mathematics could be both valuable and relevant to everyone's life:
We wanted to make sure that they came out with a positive experience from a rounded course, sort of a self-contained course. Even if they didn't go on to another course, it would be justified; they would understand the importance of the material. It wasn't just that we were going to teach them this skill because they're going to need it in calculus: "I'm not going to tell you why it's important here, but you're going to need it later on." It is like a stand-alone course with closure. It is also a skill-building course, though; you can't completely get away from that. Again, [we wanted to] give them the idea that mathematics is practical and meaningful to their lives and that they're capable of doing it a little bit and understand why they're doing certain things. Those are some of the key goals there.
(Bill Waller, Faculty)
Stimulate Student Interest by Making Mathematics Relevant to Students' Lives
Clearly, interest on the students' part in mathematics at UHD was lacking. This population of students, many of whom are commuters and/or returning adults, tend to place work and family obligations ahead of their educational goals and to view courses as a practical means to an end. Thus the University must make a good case for the educational services that it provides and demonstrate to these students the practical value of course content. To that end, the algebra course reformers included real-life problems as a gateway to mathematics to stimulate interest and encourage participation. Said one faculty member:
Well, when we first developed our proposal, we said we wanted our students to see relevant examples from the sciences and engineering. When you open the materials, you get a context to a problem, and then come the mathematics. We try to let our students experience that over and over.
(Bill Waller, Faculty)
Mathematical content in this reformed course is not presented as an end in and of itself but rather as a solution to a problem either from engineering or another science field. A major instructor goal is to provide an everyday context for the content so that students enjoy the course and understand the practical applications of algebra. One reformer told us:
I think we want students to have a positive mathematical experience, if you will, to see that mathematics can be related to their everyday lives. So we present problems that are somewhat whimsical, but if you dig beneath our storylines, they're actually relevant to their lives. It's not just, solve a quadratic equation, factor this, simplify that.
(George Pincus, Faculty)
Increase Literacy in Mathematics
College algebra is a service course taken by both math and non-math majors. The reformers knew that the math majors needed to learn and retain concepts and skills, while the students who were taking algebra as their last mathematics course needed mathematics literacy. To this end, the reformers emphasized algebraic functions and their applications and introduced relevant technology so the students generally would be more informed citizens and also be more aware of current technologies. One of the reformers explains:
For us, functions became the central theme of the college algebra course. We hoped that, by concentrating on functions and then doing variations on a theme, retention of skills and concepts would be improved for the students who do go on to other math courses. For the students who terminate with that course, we hope that they can look back and say, 'Yes I did learn something that I can apply to my life.' We hope they are more knowledgeable, become more informed citizens, and that they don't just play with symbols and numbers that they'll never have to use again. [We also hope that] they can look at technology in new ways of visually illuminating something, or seeing how technology can do a by-hand calculation that they weren't aware of. Symbolic manipulation to them is still completely new; they have no clue that computers now can do some of those things. We hope to just inform them of the possibilities.
(Linda Becerra, Faculty)
Use Diverse Teaching Strategies
With a challenging student population like that at UHD, instructors look for opportunities to diversify their teaching approach. Technology, with its graphing tools, offers one such opportunity, according to a faculty colleague:
The reason I do this is I think that if the students see the concept from different vantage points and different angles, then they will retain the concept; and also once they see it the utility of the concept in different disciplines, they may also be encouraged to learn the concept in the course.
Offer A Technology-Dependent Curriculum
It is interesting to note the reformers' view on technology use per se. According to them, computer technology is fundamental to their reform because the activities they intended to carry out could not be done without computers. A reformer explains how crucial technology is to this project:
We wanted to think about revising the curriculum in such a way that it required the use of technology. We wanted to make sure that the technology wasn't kind of a gismo that got added onto the course, where essentially we would teach the course in a traditional way and then stop and say, "Oh, but look, you can also graph this with a calculator." We wanted to make sure there's no way to teach the curriculum without having the technology.
(Elias Deeba, Faculty)
Given UHD's large minority student population, we expected "access" to figure prominently on the list of goals that faculty and administrators held. It was not. The only concern related to access that faculty had was the capability of the worldwide web to provide students access at home to continually updated software at no cost. The faculty's choice of a software platform was heavily based on providing students free access either through the web or by way of a free player as in the case of Mathwright.
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.