e-Learning in College Mathematics – an Online Course in Algebra with
Auto-matic Knowledge Assessment
Przemysław Kajetanowicz
Institute of Mathematics and Computer Science
Wrocław University of Technology
przemyslaw.kajetanowicz@pwr.wroc.pl
J drzej Wierzejewski
Institute of Mathematics and Computer Science
Wrocław University of Technology
jedrzej.wierzejewski@pwr.wroc.pl
Abstract
A college-level online course “Algebra with Analytic Geometry” is presented. Math concepts and methods are taught through a collection of highly interactive web pages that can be easily turned into an IMS or SCORM package. Nearly 80 types of math problems have been implemented in the form of Java-powered interactive tests that can further be combined into comprehensive exams. The whole process of knowledge assessment can be made automatic. The course was delivered on the blended-learning basis to a group of 55 students in the spring semester 2005, with stunningly good results.
1. Introduction
In the fall 2004, the authors created an experimental Web-based secondary level math lesson with fully automatic knowledge assessment system based on Java technology (see [4]). Encouraged by the obviously promising results of the experiment, we designed, created and taught a complete course in college-level algebra.
Throughout the paper, we describe both the functionality of the e-course and its “real-life” implementation at Wro-claw University of Technology in the spring semester 2005.
2. Organization of math content
The math content of the course is logically divided into five main chapters: Complex Numbers, Polynomials and Rational Functions, Matrices and Linear Systems, Ana-lytical Geometry in Space, and Conic Sections. Each chapter represents a multi-page Shareable Content Object (SCO). The whole content can be easily aggregated into an IMS or a SCORM package (see [1]). Actually, it was turned into an IMS package that was further imported into WebCT as a Content Module.
From the presentation point of view, the content of the course can be categorized as follows.
Lecture notes: those are Web pages where traditional exposition of material is supplemented with concept-illustrating calculators, interactive animations, and power-ful self-assessment tools in the form of interactive Java-powered tests. In that way, a student is given instructional environment in which the study of concepts and methods is supported by a big amount of interaction.
Exercise pages: those are pages where interactive exer-cises (again in the form of Java-powered tests) are gath-ered together to address a group of related notions or methods. Sometimes, exercises are further split into groups of varying difficulty.
Exams: individual single-type-problem tests can be com-bined into comprehensive exams that can be used both for practice (so-called practice exams) and for administrative assessment purposes in the instruction process. Many parameters, such as the number, the type, and the diffi-culty level of exam problems, the time assigned to student for the submission, the scores for individual problems, and the whole exam grading system, can all be controlled at design time, thus offering an instructor a very flexible assessment tool.
3. Functionality of tests and exams
Nearly 80 types of math problems are supported in the form of interactive tests, varying from simple, drill-type exercises to quite sophisticated graphing problems or problems where the user is supposed to demonstrate mas-tery of a given method.
3.1. Functionality of a single test
An individual test addresses a problem of a given type and has these features.
Random generation of data – the user can generate an unlimited collection of problems.
The level of difficulty can be set at design time. More specifically, it is possible to control the test behavior in such a way that problems with desired level of difficulty are generated according to predefined probabilities. Immediate grading is available as soon as the student submits his or her solution in the applet window.
Immediate correct step-by-step solutions are available. Various methods of grading for a given problem can be set at design time, depending on the purpose that the un-derlying test serves (an individual drill-type exercise or a part of an exam).
Time constraint can be imposed on every test, forcing the student to provide the solution within prescribed amount of time.
Fig. 1. Single-concept test
3.2. Functionality of an exam
An exam is a collection of problems gathered in a com-mon window, and serving the purpose of more compre-hensive knowledge assessment. In the actual implementa-tion of the course, the assessment strategy was completely based on automatic exams. This is discussed in the sequel. All the features of single tests that we listed before carry over to an exam. Additionally, exams have the following functionality features.
Random generation of problems – the type of problems that the student will be given can be controlled at design time. The designer can either predefine the exact type of each problem on the exam, or the probability with which problems of given types will appear. In that way, two students get two different sets of problems.
The order in which the problems of individual types appear in the exam window can be fixed or randomized.
The letter-grading system can be defined at design time, so that the exam total score can be transformed into a letter grade.
The correct solution of each problem is available as soon as the student submits the exam. The students greeted this feature with true interest. Many students pointed out that immediate access to the correct solution helped them identify their own errors easily.
Fig. 2. Comprehensive exam
An exam can be used both for practice and as a formal assessment element in the course. In the latter case, addi-tional authentication and security mechanisms are avail-able, and the exam results can be stored in an SQL data-base. In the future, this will be replaced by SCORM mechanisms.
3.3. Special tools
Many mathematical methods are actually algorithms that are difficult to understand without a considerable amount of practice. This, in turn, often involves tedious computa-tions. Typical examples are algorithms used in geodesy, or the Gauss algorithm in algebra. Tools are provided that release a student from the burden of a great deal of hand computation, letting him or her focus on successive steps of the algorithm instead. A student only “prescribes” a specific step (e.g., “add the third row of the matrix to its first row”) by setting appropriate parameters in an applet window. The actual numerical computation is left to the software. Another obvious benefit of having a utility of that kind is that a student is protected from the risk of frustrating computational errors.
Fig. 3. Tools for elementary row/column operations
4. Calculators and interactive animations
In addition to automatic tests and exams, the student is given a collection of interactive calculators that allow him or her to experiment with some of the math concepts and methods. The elementary row/column operations calcula-tor, for example, enables the student to easier grasp the idea of Gauss elimination.
Additionally, interactive animations have been created to illustrate certain geometrical phenomena (e.g., the geo-metrical definitions of conic sections).
5. Implementation
The course was delivered (on a blended-learning basis) to a group of 55 students of Wroclaw University of Tech-nology in the spring semester 2005. Two hours of lectures and one hour of recitation per week were the in-class part of the course. Necessary students’ drill and practice work having been supported by software, more time in class was available for a teacher to discuss advanced topics and applications.
WebCT Learning Management System was used to host the electronic content of the course. In addition to the LMS-independent functions described above, the authors made an extensive use of WebCT functionality, such as online quizzes, discussions, mail, chat, student tracking etc. Online quizzes were a part of the whole assessment system. Two mid-term exams and the final exam were
administered in a computer lab during the semester and were all fully automatic.
It should be emphasized that the students’ final grades turned out evidently better as compared with similar courses taught in the traditional way, even though the local mathematical community definitely judged the prob-lems given on the automated exams as more difficult than those given traditionally.
6. Student evaluations
An anonymous survey was carried at the end of the course, in order to obtain both the quantitative and the descriptive evaluation from the course participants. The students were asked 8 questions. The first 6 questions were intended to gather students’ quantitative judgment of various course elements. Each student was asked to pro-vide his or her evaluation, preference or the degree of agreement in the form of a grade ranging from 1 (very low) to 6 (very high). A question about the amount of time spent on the course was included to verify the claim (expressed by a few math faculty during the course) that technology-supported learning will cause so much time burden for students that they might neglect other courses. Question 8 was the descriptive element of the survey – students were encouraged to provide written remarks or comments that might have occurred to them.
6.1. Survey results
Below we present the survey questions together with the distribution of students’ responses. Survey was released to students by means of the WebCT Survey tool. The stu-dents were confident that the responses were anonymous. Question 1. To what extent did you find the new form of the course (in particular, automatic tests and exams) help-ful in your mastering the course material (as confronted with traditional way of learning)?
Grade 4 29% Grade 5 30% Grade 6 26% Grade 1 6% Grade 3 9%
Question 2. Would you be interested in other math courses, were they offered in a similar form?
Grade 4 6% Grade 5 9% Grade 6 71% Grade 1 3% Grade 3 11%
Question 3. Did you find the Web-based lecture notes complete enough to give up classroom lectures?
Grade 4 9% Grade 5 17% Grade 2 3% Grade 6 62% Grade 3 9%
Question 4. How strongly would you agree with the statement that the new form implies no more need for classroom recitations? Grade 4 29% Grade 5 11% Grade 3 20% Grade 1 20% Grade 6 20%
Note: the majority of students are willing to drop class-room lectures meetings, yet they are not just as willing to drop the recitations.
Question 5. Did the instructors provide sufficient techni-cal support so that you had no problem operating the interactive elements of the course?
Grade 6 63% Grade 3 6% Grade 5 31%
Question 6. How fairly, in your opinion, was your knowledge assessed by the system of automatic exams?
Grade 2 3% Grade 6 26% Grade 1 9% Grade 3 9% Grade 4 25% Grade 5 28%
Note: the exam questions were similar to those given on traditional exams. No such survey has been carried out among students taught traditionally, so no comparison can be made here in terms how students generally judge the fairness of exam questions in algebra courses. The results show that the majority of students judged the system as fair (grades 4 to 6).
Question 7. How many hours weekly did you spend working on this course? Do not include the time spent in class. 37% 53% 5% 5% 2 to 5 hrs less than 1 hr 6 to 10 hrs 16 to 20 hrs
Note: contrary to the concern expressed by some math faculty, the amount of time spent on the course by the majority of students does not indicate that students’ atten-tion was drawn away from other courses.
6.2. Students’ written comments
The students were asked this additional question survey: Question 8. Please give us any remarks or comments regarding the course that may occur to you.
Except for 3 persons that showed strong frustration and were highly skeptical about the advantages of the new form over the traditional way of teaching, the comments on the course were close to enthusiastic. Here are a few excerpts from students’ responses:
What I especially liked was the availability of all the lecture materials and exercises that could be taken re-peatedly, until you mastered a problem completely. Prac-tice exams were of particular value, since they motivated us to work.
I liked the fact that all the materials were available online and thus I could concentrate on listening rather than taking notes during lectures.
At the beginning I was terrified at the perspective of tak-ing a course that was delivered in that way, but now I would like to heartily thank the instructors for the oppor-tunity of being a part of this “experiment”. I can defi-nitely say that thanks to this course I understood things that had been all Greek to me before (...) I truly believe that more courses should be taught that way. I wish to thank again for the opportunity to have been in this course.
7. The present and the future
As of this writing, in the fall semester 2005, the second edition of the course is being taught. A group of over 400 freshmen are enrolled in the course. The course is now nearly one-third into the semester. Due to a very large number of students, the grading procedures have been altered. Each student is supposed to individually take six online tests. Obviously, there is no security check (e.g., no guarantee that a student does all the work without some-one else’s help). Therefore, at the end of the semester, an in-lab comprehensive exam will be administered for all the students. The results of the online tests will contribute to the final grade only if a student obtains a predefined minimum score on the final exam.
One thing can be immediately noted: except for fewer than 20 cases, the whole group “seamlessly jumped” into the system. Self-registration did not posed a problem for students, nor had they any trouble getting used to interac-tive tools. There is a large group of participants who show very active attitude towards the use of the course utilities.
It is the authors’ strong belief that other math courses would be equally successful if offered in similar form. Active exploration of study concepts and automatic knowledge assessment are obviously an element whose role in successful online math teaching cannot be overes-timated (see [2] and [3]).
8. References
[1] ADL Technical Team, “Sharable Content Object Ref-erence Model (SCORM)” Version 1.2, Advanced Distrib-uted Learning, www.adlnet.org,, 2001
[2] E.Cosyn, J-P.Doignon, J-C.Falmagne, N.Thiery, “The Assessment of Knowledge, in Theory and Practice”, www.aleks.com/about/Science_Behind_ALEKS.pdf, 2004
[3] J.Engelbrecht, A.Harding, “Teaching Undergraduate Mathematics On the Internet”, Educational Studies in Mathematics, 58 (2005), p. 235-252.
[4] P.Kajetanowicz, J.Wierzejewski, “E-lesson on Quad-ratic Function. A Step Towards an Online Remedial Math Course”, Proceedings, 5th International Conference Vir-tual University, Bratislava, Slovakia, December 2004
