الرئيسية Journal of Educational Technology Systems Computer-Based Multi-Media for Mathematics Learning
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J. EDUCATIONAL TECHNOLOGY SYSTEMS, Vol. 1(3), pp. 225-233, Winter, 1972 Computer-Based Multi-media For Mathematics Learning RUTH IRENE HOFFMAN Professor of Mathematics D i m tor, Mathematics Laboratory University of Denver ANITA S. WEST Research Mathematician Denver Research Institute University of Denver ABSTRACT This article traces the various roles of the computer in mathematics learning. It begins with the computer’s use for curricular enhancement through several modes of classroom usage. I t ends with a description of the computer as a manager of a learning center, where it directs students through a sequence of learning experiences using appropriate media at each step. An article in this journal in June, 1972,contained a description of a laboratory systems approach to individualized learning. The equipment therein described did not include any type of computer or calculator and yet was a total laboratory atmosphere and a very effective system for the individualized learning of mathematics. On the other hand, the computer can be a laboratory by itself for mathematics learning, or can finally become the “manager” of a coordinated collection of multi-media. When used alone, the student will interact with the computer in many different ways. These can be classified into two categories. 0 The student plans and analyzes a problem situation, writes a program, studies the results, and draws conclusions. 225 0 1972, Baywood P u b l i i n g Co. doi: 10.2190/XHVQ-62P0-0PY0-BKE8 http://baywood.com 226 / R. I. HOFFMAN, A. S. WEST The student responds t o preprogrammed material. The common procedure for this first category is to teach students a 1anguagFFORTRAN or BASIC, most commonly-and assign him problems. These are computerized to avoid bulky computation, as in statistics; to print out quantities of data for making conjectures, as in the study of limits; for determining characteristics of functions by examining print-out of graphs, as in f(x) = k sin (cx) + 1 for various values of k, c, and 1. Perhap; s a more creative use is in the assigning of projects for students to compose music, program tests, make ski surveys, and create various simulations and games. A t all times the student enjoys the process, the involvement, the creativity; but sometimes he learns mathematics in a disjointed, unrelated way. The ideal development is t o use the computer to extend the mathematics of an on-going course. Curriculum materials for this use of a computer are scarce. Some computer extension materials have been written and published-two of the best sources are nineteen units in algebra, trigonometry, and calculus published by the CMCP' (Computer and Mathematics Curriculum Project) at the University of Denver, and similar materials published in units by the University of Pittsburgh.2 The second category, preprogrammed materials, may be simple or involved. 0 The student may sit at a teletype and respond t o a complete course or a single lesson, which is written by an author, and is a programmed sequence, just as with a programmed book, except the student has immediate response on answers, reinforcement for learning, and evaluation. This is called CAI (Computer Assisted Instruction) and does not have extensive usage because of the time, cost, and writing talent needed to develop such programs. The student may be given a punched tape for a certain situation, this is particularly adaptable to simulations. For example, the simulation of population growth may be programmed and punched, and students can feed in different values for variables in the growth pattern. The computer prints out sequential years' population based on these figures and the student can analyze the results. An entire class may participate if a teletype is brought into the classroom preprogrammed for a special problem in which COMPUTER-BASED MULTI-MEDIA FOR MATHEMATICS LEARNING I 227 they, as a group, will take part. For example, if the subject is the roots of polynomial equations, the program is written to give the value of a given function at a given point. The class creates a polynomial equation, one student feeds various values of the independent variable to the teletype, gets function values; the class notes the changes in sign, and by successive “guesses” develops a true feeling for the zero points, interval-halving, and related ideas. The computer does the awkward computation while the students, as a group, center their attention on the basic concept, the solution of polynomials. The computer can be preprogrammed to give individualized assignments on a given topic for students to do off-line. For example, if the topic is characteristics and graphs of quadratic equations, a not too involved program can be written by the teacher, using the random number generator to give various characteristics of the graph of the quadratic to the student, and he must write the equation that meets these conditions. This kind of assignment is more demanding on the students’ knowledge than the text book exercises and can be individualized. For example, a student can be asked to write a quadratic equation that opens downward, has two real roots, has an axis parallel to the y-axis, five units to the left, while another student gets an assignment to write a quadratic equation that opens upward, has no real roots, and has an axis on the y-axis, and so on and on with each student receiving a different set of specifications. The student creates his equations off-line, returns to the computer, calls in a program which asks for his specifications and his resulting equation, and tells him whether he has written an equation (from the family of equations) that meets the specifications. Thus, the computer begins to assume a role of helper to the instructor, and can handle individualized assignments. An extension of this is an evaluation program that tests a student, analyzes the results, and recommends follow-up work. A computer can be a component of a learning center when it is one of many media and an organized program of learning directs the student to it for appropriate use when computations are needed just as he is cycled through audio-tape cassettes, film strips, etc., as part of an instructional pattern. The final step-up in this phase of computer usage is computer-managed instruction, and can be used in various ways to manage students and multi-media through a sequence of learning 228 / R. I . HOFFMAN, A. S. WEST experiences. Such a program is being developed in Denver and is in the first phase of a long-range program. For the past several years, the U.S. Air Force (Human Resources Laboratory, Technical Training Division) has been committed to the development of a prototype computer-based instructional system which will combine and integrate the latest techniques in individualized instruction, teaching techniques, media, and computer equipment into one of the “most complex and efficient training programs ever d e v i ~ e d . ”Demonstration ~ of this functioning model is scheduled for 1975. Since 1970, the University of Denver has had a contract with the Human Resources Lab at Lowry Air Force Base in Denver t o provide support in several phases of this overall instructional design system. The scope of the effort has included the evaluation of computers in the training process, fabrication, and evaluation of multi-media training devices, and modification and analysis of student characteristics and performance. A major emphasis during the last year has been the development of a computer managed instructional (CMI) presentation that demonstrates a number of the developing facets of the system-a prototype of a system prototype, if you will. Although the University of Denver system has been designed primarily for demonstration (the final system requiring far more extensive hardware facilities), there are several functions that this system fulfills: 1. it demonstrates a CMI learning environment, the original purpose, 2. it develops a language in which CMI courses and lesson material may be developed, 3. it provides a vehicle for experimentally testing course materials and selected media devices, using small samples of students, and 4. it can be used to gather and correlate data, such as student proficiency levels, course effectiveness, and time and cost estimates. The demonstration system (using a Digital Equipment Corp. PDP-11 as the processor) required that a file be created and maintained for each active student, thus necessitating a non-trivial file management system responsive to key interrupts opening, updating, allocating buffers, and releasing student record files at the appropriate times. In addition, several students progress through a course, each at his own pace through a different pathway; thus, a mechanism is provided to define and direct a course for several students simultaneously. The resulting system is not so much a demonstration simulation, as a “little bit of the real FILE 5 Figure 1. CMI System Diagram. STUDENT FILE 1 TEST DEFINITION FILE FILE 3 - . i6 sD rn I -I D K n 2 230 / R . I . HOFFMAN, A. S. WEST thing.” The system, as diagrammed in Figure 1,has been designed to aptitude 1. consider input and progress variables that range from a characteristics through actual performance parameters, as well as personality and aspiration indices. 2. to provide for prediction of optimal matches of media assignments, level of difficulty, amount of practice, sequence of reviews, etc. 3. to provide for current and predictive progress reports, and update descriptive and predictive efficiency as new data is continually being processed. To illustrate, the following are examples of the types of actions that can be specified by the course file (File 6 in Figure 1).This file is generated by a course specification language card deck and contains complete program logic for the CMI t o interpret. (The CMI interpreter is a reentrant routine which is shared by students at the terminals.) 1. Type out statements. 2. Type out questions and branch on several responses. 3. Expect next key-in interrupt from a specified station by this student. 4. Test key-reader station busy. 5. Request a student proficiency rating and branch on less than, equal to, or greater than. 6. Ask a test question and store results in a test answer array. 7. Call test grading routine to score test, and to update test file and student record file. 8. Write a line to the student report file. 9. Allow a student to write comments on his report file. 10. Calculate various proficiency rates and update student record file. 11. Update student lesson status (e.g., lesson started, test A complete.) 12. Read in next lesson, No. n, block from student record file. 13. Student logging off teletype. SUPPORT HARDWARE AND SOFTWARE STUDIES Several relevant research tasks have been and are being performed concurrently with the development of the computer managed instructional capability. (See Figure 2.) KEY-READERS Basic to the adaptive instructional model was the development of what are called “key-reader” devices. These plastic systems COMPUTER-BASED MULTI-MEDIA FOR MATHEMATICS LEARNING / 231 Figure 2. Support hardware of pilot CMI system. work in conjunction with an on-line PDP-11. The device serves as a monitor for each trainee and each learning station; it outputs 16 bits of data which actuate an information system for scanning a trainee’s records to determine if he is at the appropriate station, for updating his records, and for timing him at the media station. COMPUTER GRAPHICS A graphics system to allow a user to create files of pictures for a Computek Series 400/20 CRT display has been completely designed and will soon be implemented. Pictures will be created via input from a Computek GT50 Graphics Table (to indicate points) and a graphics console key-board (to indicate functions such as drawing lines, curves, rotations, and translations). The system will give the user the capability to dynamically reference and modify components of the picture during its creation and to store pictures in a file or call up previously filed pictures on the display tube, particularly useful for creating course materials. OPTICAL SCANNER AND SLIDE PROJECTOR INTERFACED WITH PDP-11 COMPUTER Interfaces between two GAF ESP2000 Random Access Slide Projectors and two Bell and Howell MDR8000 Optical Mark Readers have been designed and constructed for the PDP-11 Computer. These interfaces allow for selection of slides under computerized control of an instructional program, and the reading 232 / R . I . HOFFMAN, A. S. WEST Figure 3. AV devices interfaced with digital computer. of test forms (answers sheets and punched cards) from optical readers under computer control of a FORTRAN “read” statement. (See Figure 3.) MICROFICHE MEDIUM Some rather interesting small scale demonstrations and evaluation studies have been performed as part of the media development. One of these was an effort to explore the comparative advantages and disadvantages of microforms in training applications. A 30-hour instructional sequence entitled, “Basic Computer Operation” was selected as the experimental class. A two-stage filming procedure was used to convert the training manual used in the instructional sequence to an innovative microform format in both positive and negative film polarities. Students in three experimental classes used the microform presentation as the instructional medium for the course. Use patterns and course performance of these students were determined in three control classes. The major result of this comparative analysis was that Air Force trainees can, and did, use the microform systems effectively and intensively over a one-week period. No significant performance decrements were encountered in the experimental classes. This study also examined a number of important considerations involved in utilizing microforms for training purposes, including the impact of microform use on instructional routine, administrative-logistics considerations, and COMPUTER-BASED MULTI-MEDIA FOR MATHEMATICS LEARNING / 233 student study habits. The significant accomplishments of this study were the demonstration of the feasibility of the microform medium for classroom instruction and the development of an effective, innovative format which utilizes the unique presentation characteristics of microform to facilitate instructional communication. The next research step planned is presentation and diagnostics via CAI with referenced text and illustrations supplied on microfiche. REFERENCES 1. Send t o Dr. Ruth I. Hoffman, CMCP, University of Denver, Denver, Colorado 80210, for copies. 2. Send to Dr. Tom Dwyer, Project Solo, Department of Computer Science, University of Pittsburgh, Pittsburgh, Pennsylvania, for copies. 3. U.S. Air Force, Technical School Briefing on the Advanced Instructional System, December, 1970.