Volume 8, Number 9: June 27, 2002
The College Algebra E-tutor
by Robert H. Hoar,
The e-tutor is the result of an ongoing project funded by the Learning Technology Development Council. It consists of an integrated collection of dynamic web-based materials designed to help students develop or recover the algebra skills needed to succeed in higher education. The materials directly benefit the group of students who are not fully prepared for general education mathematics courses, as well as students in those courses outside of mathematics that rely upon these skills. The need for remediation among this group of students, and also students for which certain topics have become rusty, has been detected by placement examinations. The growing population of returning students and transfer students also comprise a significant part of this group.
The project, about to begin its third year, has been a collaborative effort involving principle investigators from three System universities and five additional mathematicians from around the System. It has three primary goals:
In addition, the site was designed to track each student individually. A login routine allows the site not only to identify the student, but also to link the student to an instructor and/or course. This has allowed us both to assess student performance and to assess (and modify) the site.
The code necessary to house the dynamic materials is located at http://www.compute.uwlax.edu/expert/. A blueprint has been developed in the form of a "How-To'' Manual that guides developers in the creation of materials appropriate for inclusion onto the site (the manual is available at the website). This manual also guides developers through the process of placing materials on the site and explains how these materials may later be edited and/or extended. The third goal of the project requires the inclusion of other members of the UW System and widespread dissemination of the materials, and was the primary goal of the second year of the project. In the second year, five additional members were added to the team. The five new members from different units within the UW System, together with the original four, were able to increase the amount of content available at the site and helped disseminate the project to more UW campuses. (View a list of principal investigators and additional developers.) The third year of the project will involve the creation of additional content for the site as well as the assessment of the materials.
The need for these materials was highlighted in a report1 presented to the Board of Regents, which noted that, for the past few years, about 1 in 9 (11.9% in 1999) freshmen entering the UW System require mathematics remediation. The report also shows that, of those needing mathematics remediation, only 37.8% completed the requirement in the first year. Further, of those who completed the remedial mathematics requirement during their first semester, 95.3% were retained for the spring semester, as compared with 78.5% of those who did not complete the requirement during their first year.
The courses in each mathematics program in the System (including the colleges) are described in a flowchart contained in University of Wisconsin publication Early Mathematics Placement Testing Program: Mathematics Requirements by Major2. The publication designates five skill levels, ranging from weaknesses in basic mathematics (arithmetic) to the skills required for calculus.
Many students not mentioned in the BOR report will also benefit from the e-tutor materials. Instructors are often faced with students who have strengths and weaknesses in their understanding of prerequisite material. The Placement Exam provides recommendations based on a student's overall performance on a variety of topics. Most students who get placed into courses are, in fact, not completely prepared for the course. These students require "selective'' remediation. The e-tutor provides instructors with a set of materials to which they can direct students, in order to prepare for up-coming topics. Such resources allow students to better prepare for courses and to keep up once in the course. The self-guided materials allow students to visit the site at any time. Properly utilized, these materials will allow instructors and students to concentrate on the primary topics in the course. One of the outcomes to be measured during the assessment phase of the project is the hypothesis that these materials may reduce the frequency of course withdrawals. These benefits are not limited to mathematics courses; the need for selective mathematics remediation is likely to occur in courses that require students to possess a prerequisite set of mathematics skills.
The specific algebra topics covered by the site were selected using the same criteria used by the developers of the UW-System Placement Exam. The primary strength of the e-tutor is its ability to guide students through mathematics topics in a way that is pedagogically sound. These topics are covered at a pace that we believe, from past experience, the students can handle.
Figure 1 (below) shows a snapshot of the start of one of the sections at the site. The screen is split into four frames. The upper left frame contains navigation information, the upper right contains the content for the section, and the lower two frames are used to display examples, problems and guidance.
As a student enters the section, he/she sees a short introduction (B) to the topic covered by the section. The first discussion (D) presents a simple discussion of the topic and a related example is shown (E). As is often the case, a single example rarely answers all of the questions that may arise in a discussion (in fact, it may raise more questions than it answers). The site allows students to ask for as many examples as they like, and these are displayed on the same page as the discussion, so that students can review the discussion in the context of the example. The student may request more examples of this same type by clicking on the word Example. Each example is randomly generated using parameters set by the developer of the section. Each of the boxes in the frame may be opened in a new frame by clicking on the Open in new window link (C).Once students feel they understand the concept, a randomly generated problem is posed (F). Students then answer the problem and submit it for review. The software we have developed to run the site then analyzes the answer and either tells students they are correct, or a message is displayed with a response that is tailored to each student's mistake (G). The website's ability not only to determine right from wrong but to provide feedback tailored to the mistake is perhaps the most valuable educational strength of the site. Once students are provided the feedback, they have the opportunity to correct the mistake and re-submit. The new answer is assessed and the appropriate response is provided. Students can choose to work as many problems of a given type as they wish before moving on to a more complicated problem set. Each section also contains an index (H) which allows the student to return to earlier presented examples and discussions. To view the examples and problems related to multiplying and simplifying products of polynomials, visit http://www.compute.uwlax.edu/expert/2ci, login as guest and click on the words Examples and Show me a problem.
At any time, the user can view his/her performance by selecting the My Statistics button (J). This will bring up a complete set of statistics describing the number of problems, examples, and sections that the student has attempted. If the user is a developer or a faculty member, the statistics shown would include all users or those assigned to the faculty member.
The editing buttons (I) are not made available to the general user, but they provide site developers with an easy to use interface to modify/add/delete content on the site. When the Editing Mode button is pressed, the words Edit (A) appear on each box in the section and the developer needs only to click on the box to gain access to the source code for the box (popped to a new window). This feature allows easy remote modification to the site (the current set of developers are scattered across eight campuses).
In order to implement the dynamic portion of the website, it is necessary to program the computer so that it can interpret the student answers. Multiple-choice questions are passive in nature and, while useful in assessing the student's current level of understanding in a particular area, are not as useful in the learning process. Fill-in-the-blank questions require the student to take a more active role in the development of a solution; they cannot work backwards from proposed solutions. The e-tutor requires the student to input the answers and, since most algebraic expressions have many equivalent forms, it is necessary to create a program that can synthesize the various versions of the correct solutions. The mathematics package MATLAB (by MathWorks) is a powerful package which can be programmed to interpret, compare, and solve algebraic expressions and equations, and can easily create graphs and charts. In order to allow the web materials to utilize (communicate with) the MATLAB package, a web-interface was created. A similar interface allows the website http://www.compute.uwlax.edu to provide computational tools to students in mathematics and science.
There are two reasons for interfacing with MATLAB. The first reason is one of practicality and contributes to the future success of the project. We have produced a framework in which other mathematicians can easily create materials that their students will use. If the computational tools were developed using more technical software, it is unlikely that the average mathematics and/or science instructor would have the time and skills necessary to design the needed material. The second reason for selecting MATLAB is more technical. In order to depart from multiple choice quizzing/evaluation instruments in favor of the more educationally useful open-ended questions, the software needs to be able to compare logically equivalent statements. For example, MATLAB will recognize y = 3x-2, y = -2+3x and y = 3(x-2/3), as equivalent answers automatically, alleviating the need for advanced programming skills.
To help assess the site, the site developers and the instructors of the students who use the site are given access to user specific statistics. Through the use of an easy to use login procedure, the web server will keep an electronic record of how often the site is accessed and how the students are using the site (number of visits to each section, types of problems and examples requested, number of right and wrong answers, etc.). If the site is being regularly accessed by an individual throughout a semester, one can infer that the student feels the e-tutor is useful. The computational design of the site will be enhanced in the third year to include an interface to Access, a powerful database system developed by Microsoft. An interface to the data base will be developed that will allow developers and instructors to sort through the large amount of information gathered about student usage. The instructor will be able to sort the data to investigate how a particular student is performing, or sort the data of all students in a class to determine how they are performing on a particular topic.
As mentioned above, one of the primary strengths of the site is its ability to give students "mistake specific advice." In order to do this, however, it is important that the site contain code relevant to each (predictable) mistake that a student might make. As educators of mathematics, the development team has years of experience that aids them in predicting these mistakes; however, a few possible mistakes can be missed. The current student tracking program that we have developed for the site (which will be further enhanced by Access), has allowed us to view student answers (together with the problems that they were asked). Upon viewing this list, the developers have been able to spot mistakes that they had not envisioned, allowing the site to be edited and improved. Using student mistakes to enhance the site is the sort of feedback loop assessment tool that will continue to help the content of the site improve. While it has not happened yet, it is also conceivable that this feedback loop will prompt a developer to reword a particular discussion within a section of the site in an attempt to guide the users away from a recurring mistake.
The e-tutor project is both a collection of mathematics modules and an experiment in System cooperative efforts. The ongoing project has produced a framework in which mathematicians from around the System can take part in the development of state-of-the-art technology based educational tools. And, participation in the project does not require programming skills, only the logical, systematic reasoning skills that the instructor utilizes when preparing lectures and examinations, along with a history of observing the related outcomes.
The framework developed to house the e-tutor could be used to house many different sites that cover any number of topics, even some non-mathematical topics. MATLAB, the e-tutor's computational core, is not simply a mathematics package, it is widely used in industry (particularly in engineering), and it continues to expand in capabilities. If a new subject area is not within the capabilities of MATLAB, the interface could be replaced by an interface to a more relevant package.
If you are interested in using the e-tutor, contributing to the e-tutor or developing a site utilizing the framework of the e-tutor, please contact me or any of the principle investigators listed below.
1 The 1999-00 Report on Remedial Education the UW-System: Demographics, Remedial Completion, and Retention and Graduation, presented to the Board of Regents (BOR) on March 10, 2000. Agenda Item I.1.b.(1).
2 The UW-System Mathematics Requirements by Major at each campus is available at the site http://wiscinfo.doit.wisc.edu/exams/mathematics requirements by major.htm.