Jack Koumi, Freelance Educational Media
Consultant, UK
Judith Daniels, OU Mathematics and Computing
Faculty, UK
In this paper the authors describe teaching packages which involve the use of audio-tape together with a personal computer. These are used in the teaching of the third-level UK OU mathematics course M371: Computational Mathematics, and M372: Numerical Methods for Differential Equations. The script design principles were derived from those that have been developed over many years at the OU for the composite point - audio medium. Here, the authors explain the creation of the new packages: the rationale for the composite medium, the structure of the packages, the teaching strategies and the close liaison required between the academics, the programmers and the audio producers in order to implement the ideas successfully. It is argued that these methods could be adapted to other areas of mathematics and indeed to non-mathematical subjects. (This paper is an expanded version of a paper by Daniels and Koumi (1992).)
The Open University third-level mathematics course M371: Computational Mathematics was one of the three courses which pioneered the University's home computing policy in 1988. This requires students to have access to an IBM PC or compatible microcomputer. Software has been developed so that the computer is used as an intrinsic part of the course with the following aims:
An additional advantage is that more advanced and up-to-date numerical methods can be taught than are usually included in an undergraduate course, because students do not need to understand all the details of the algebra required for hand calculations - they can obtain an intuitive understanding of such methods by seeing them work in practice and by having them described graphically through audio-guided 'Teaching packages', as explained below.
The role played by the computer and the audio-cassettes in teaching the numerical methods also means that the other course materials can focus more on the applications and the modeling involved.
The M371 course is an honors level Open University course which teaches those computational methods used in commercial software, as well as giving some experience of modeling real-world problems and of using software packages. The course content is presented in 12 correspondence texts (units), each requiring 12-15 hours of study time. The topics include: systems of equations, linear programming, optimization, queues.
In addition to the correspondence texts, students are supplied with twelve disks containing the course software (application packages and teaching packages, described below), audio-tapes for use with the teaching packages, a computing booklet and a handbook.
Although there is some provision for contact with a tutor, most of the study takes place unaided in the student's home, which means that the course materials, including the software, must be designed so that they can be taken at the student's own pace, without tutor guidance or peer support, and so that students can fit their study into their daily lives, with possible breaks and interruptions.
In 1990, 184 students registered for the course of whom 119 (65%) completed it and sat the examination, with only nine of the examinees failing the course. These statistics indicate that the success rate on the course compares very favorably with other courses at this level. However, it is difficult to make comparisons because M371 is teaching new material as well as using new methods.
The software was developed within the GEM graphics environment, so that dropdown menus, windows, graphics, screen-editing and a mouse provide a simple but powerful teaching medium. (From 1996, students will be required to have access to a machine that will run Windows, and the software will be adapted and improved to run under Windows. The software described here was developed in 1986/87, when GEM was the only available portable graphics package. Another possible future development would be to carry the audio material on CD-ROM. This would enable more flexible control of which audio segments could be listened to.) Students are expected to spend approximately 25 per cent of their study time using the packages. Two types of software are used:
Eleven packages were produced (roughly one per unit). These packages are designed to enable the student to solve a variety of problems using sophisticated methods. The student has a choice of exploring stored problems or of solving their own input problems.
In this paper, we concentrate on the second type of software, described below.
Nine teaching packages were developed to enhance the teaching of the theory in the course. These are regarded as the more innovative development in that they use audiotape in conjunction with the software in a way that has not previously been tried at the Open University, or elsewhere to our knowledge.
The relationship between the words on the audiotape and the images on the computer screen is unusual: the tape is used as a teaching medium in itself, describing the underlying concepts and theory, as well as the more obvious use in prompting the student's progress through the package and in drawing attention to aspects of what is seen on the screen. The software is used to produce animated graphics in response to simple prompts from the user, as well as to perform numerical methods being described. This combination of audio-tape and computer software can be viewed as a development of two successful Open University teaching media:
Figure 1: A 'frame' from an OU audio-vision lesson
While the rich educational potential of video is widely appreciated, the advantages of 'audio vision' are less well recognized. These are summarized below.
Advantages of audio-vision (audio plus printed frames) over print or broadcast radio
The educational benefits of audio-vision are further discussed by Bates (1984) and by Laurillard (1993, ch. 5).
The M371 computer-supported teaching packages retain all the above advantages (except annotation of the frames) plus the additional advantages below:
The student can now concentrate on the theory and the method instead of being distracted by getting involved in detailed calculations. More importantly, without tediously plotting graphs, they can see the graphical implications of what is being taught using the student's own values to experiment with. Values can be keyed in, prompted by the spoken tape, and the mouse or keyboard can be used to control the pace.
Use of the packages
The role of the tape is both to explain the theory behind what the student sees and to prompt the student on how to control the package, via the mouse or keyboard. The use of taped commentary allows us to minimize the amount of text on the screen, so that the student can focus on the graphical and numerical effects.
To demonstrate the ideas, we give examples from the first teaching package, on solving f(x) = 0. There are five 'screens' in this package:
The student would normally work through these in order but can, if required, 'quit' a screen and then return to the main menu, start a screen again or move on to the next screen. The options are made available in a 'drop-down menu' and are controlled by pointing and clicking the mouse. For example, for simple iteration the options are to 'input x0' to start the iterations, or to 'quit'. Once the value for x0 has been provided, the student can choose any of the options at the top of the screen: 'next iteration', 'end iteration' or 'zoom' as shown in Figure 2. The main part of the screen is divided into two areas: the graphics area shown on the left and the text and numerical results on the right.
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Figure 2: Screen 2, once a value of x0 has been keyed in
By clicking the mouse on 'next iteration', an iteration is performed, the calculated value is shown on the right and the next step in the 'cobweb' diagram shown on the left. 'End iteration' will result in a message: 'well done' if the required accuracy has been reached in a minimum number of iterations, how many iterations could have been used if the student has 'overshot', and so on. The complete cobweb diagram is shown when the student chooses to stop. 'Zoom' enlarges the relevant part of the graphics area to show the iterates in more detail. Thus, at each stage, the student is in control and can experiment and work at their own pace.
Note that we have not attempted to synchronize the tape with the progress of the software; instead, the two interact by means of:
The rationale for not linking the software with the audio was as follows:
It was felt that one of the causes of failure in the past of computerized 'programmed learning' was the abrogation of control by the students. Despite the 'branching programmes' in the old style Computer Aided Learning, the agenda is still predetermined by the teacher within a narrow range. Hence, claims that such programs constituted 'individualized learning' were rather exaggerated. In our teaching packages students are at liberty to de-couple the audio presentation from the computer presentation in any way they wish, so that their learning benefits from this greater individual freedom. Moreover, the mere feeling of being in control of their familiar audioplayer is, we believe, psychologically conducive to learning.
We do not deny that there are teaching purposes for which carefully structured synchronization of audio and animated graphics is desirable. However, we believe that these are better served by video than by personal computer (or by personal computer plus CD-ROM carrying video material).
Both the application packages and teaching packages were developed by the Educational Software Group of the Open University Academic Computing Service, based on specifications for the relevant unit authors in the course team. After the initial specifications, considerable liaison and negotiation continued in further development and interactive refinement of the software and teaching requirements.
The two types of package are linked, in that the teaching packages aim to assist the students' understanding of the applications packages. Any numerical output in the teaching packages is therefore designed to simulate output from the corresponding applications package.
For the teaching packages, the audio-tape scripts were developed by BBC OUPC producers and, in working on the scripts with the unit authors, there was additional input from the producers to the software specifications. Thus the role of the BBC producers in the creation of the teaching packages resembled the production of a television programme in that they influenced both the images and the timings on the screen as well as the words spoken on the audio-tape.
The whole production process is labour intensive for programmers, producers and unit authors. A realistic figure for the total time spent by the team per teaching package was one person-year. However, much of this average figure arose from development time during the early packages. The M372 packages being produced currently are taking about half the time.
The point of all the above effort is, first of all, a general one concerning distance teaching.
In the absence of face-to-face contact between teacher and student, it is not possible to provide truly individualized feedback on students' activities. The human teacher has the advantage of being able to start from the student's current understanding and can help to build a bridge between this and the concepts of the course. These concepts are thereby 'plugged in' to the student's existing mental framework and the student can modify that framework appropriately.
In other words, the teacher connects with the student's individual ways of thinking and guides them from there towards a clearer understanding, dealing with difficulties on the way through question and answer.
The problem, in this respect, with distance teaching is that it is impossible to predict all the difficulties that could be encountered by students in achieving an understanding of the concepts. Hence, even the most 'intelligent software' could not explain the concepts in a way that would overcome the particular 'mental blocks' of each and every student.
Nevertheless, one task of the distance teacher is to attempt to predict what will be in the mind of the individual student and to create an environment where idiosyncratic difficulties are likely to be alleviated - an environment which avoids being too prescriptive and includes opportunities for individual exploration.
This is the over-arching guiding principle for the techniques and principles described below. These are illustrated at some length with a total of ten specific extracts.
The first extract is from near the beginning of Teaching Package 1.1 following a prompt on the audio-tape for students to stop the tape and carry out an activity. The activity results in the picture on the computer screen shown in Figure 3.
Figure 3: Screen l, once values of A and B have been keyed in
Students now re-start the audio cassette, which says:
In the graphics area on the left, the computer should have responded to your input values by plotting the points A and B and then enlarging that part of the graph.
Two aspects of this scripting should be noted:
The principle being followed here is: 'Try to get into the student's mind and predict what the student is, thinking/doing/looking at'.
The audio now continues with some comments about the text area on the right and what to expect when the student starts the bisection process. The student is now prompted to carry out the first bisection process:
Incidentally, when you do carry out the bisection, keep your attention on the graphics area, because the picture there will animate automatically. Let's see that now: select the 'next bisection' option at the top of the screen to carry out the first bisection (PAUSE 9 SECS).
This is another example of trying to predict and accommodate what the student is thinking/looking at: the animation happens quite quickly, therefore any student looking at the wrong part of the screen would miss most of it. Hence, the audio alerts students to look at the graphics area and what to expect there.
During the animation, the line at position A moves to a new position. Immediately following this, the information on the right changes so that the screen now looks like Figure 4.
Figure 4: Screen l, once the first bisection has been carried out
The audio then comments as follows:
In the graphics area, one of the end-points stayed where it was ... but the other one, previously on the dotted line, has now moved to the mid-point of the old interval.
So both the text area and the graphics area now display the new, half size interval, on which the function is still negative at the left and positive at the right.
This illustrates a point made earlier, that the screen is kept clear of text as far as possible, because the audio can supply the information. In this case, the audio (but not the screen) tells students that the new interval is half-sized.
This also illustrates one of the roles of the audio; it tells students 'what to make of' the screen. This information is fairly detailed in this case to prepare students for quite a lengthy session of self-study - namely, carrying out a series of bisections.
Incidentally, you may have noticed that the pause on the tape to accommodate the screen changing from Figure 3 to Figure 4 was nine seconds. This duration was determined during rehearsal by timing the student activities (with the producer role-playing the student while the academic read the script).
That pause was built in to the recording because we were confident that it was correct to within plus or minus a second. (Variations in speed of different computers owned by students would give at most half a second difference in the case of this screen-change.) That is, eight or so seconds was predicted as the duration of the keyboard activity, including time for the eyes to settle on the new picture.
On other occasions, instead of building in a pause on the tape, we judge that the student needs to stop the tape to carry out an activity. This was the case, as mentioned above, for the activity leading to Figure 3. This activity needed about 15 seconds when the producer tried it, but involved quite a few keyboard and mouse operations, so the duration could not be predicted with any accuracy for the whole body of students. It was judged that some students might take only ten seconds, while others, especially those with slower computers might take as many as 20 seconds. Hence, a prompt to stop the tape was chosen in preference to a fixed pause built into the tape. However, the most common occasion when students are prompted to stop the tape is when they are supposed to carry out prolonged tasks lasting several minutes rather than seconds.
Incidentally, the prompt for stopping the tape always includes a 'music jingle' (in M371, this is a four-second piece of music), sometimes preceded by a verbal instruction to stop the tape, such as:
Stop the tape now and follow this particular iteration through to its conclusion. (MUSIC JINGLE)
In this case, the jingle punctuated the verbal instruction to stop the tape. However, the jingle obviates the need to say 'stop the tape', so these words are omitted on most occasions.
Returning to where we left off for Teaching Package 1.1, the next item on the audiotape concerning Figure 4 was:
In a moment, I'm going to ask you to continue the bisection process, until you can determine the root to five decimal places. For the time-being, we'll be doing this without the Fudge which you saw in the Unit.
After you have bisected several times, the graphics area will... (SEVERAL SENTENCES OF PREPARATION FOLLOW BEFORE THE STUDENT IS TOLD TO CONTINUE THE BISECTION PROCESS.)
This illustrates another instance of implementing the principle: Try to predict what the student is thinking/looking at. If the first sentence had started Now let's continue the bisection process... instead of In a moment, I'm going to ask you to continue the bisection process. . . then some students may have paused the tape and started the bisection activity prematurely, without the benefit of the preparatory information.
Another principle, illustrated by the second sentence For the time being. . . is: Communicate Assumed External Knowledge. In this case, the knowledge that the teacher assumes the students possess is the 'Fudge' technique which they should have met in the Unit. (The 'Fudge' is an expedient adjustment of the output of the iteration process which speeds up the process once it has neared the result). If the audio had not warned students that the Fudge would be omitted 'for the time being' then, in the subsequent activities, students might have searched in vain (and in panic) for where the Fudge was being used.
In general, communicating assumed external knowledge (eg knowledge gained in pre-work) has two purposes:
Once students have explored the iteration process without the Fudge technique, the package proceeds to show how much faster it is to include the technique. To prepare for this, three keyboard actions are necessary. So, rather than overload the student's memory with a long list of instructions, the prompting is carried out in two stages, as follows:
I'll need to refer to specific n numbers so I'm going to choose the starting values for you. To input these, first select "stop" then choose the INPUT option from the menu. (PAUSE 11 SECS)
This first stage results in the screen shown in Figure 5.
Figure 5: Screen 1, selecting default values for A and B
The second stage of instructions is the following:
Now choose my default values by clicking on OK. (PAUSE 10 SECS)
Apart from being kind to the student's memory load, the above two stages avoided what would have been a very long pause of: 10 plus 11 = 21 seconds.
Such a long pause is inadvisable since it might include a large error in predicting students' durations for their keyboard activities. Another way of avoiding such a long pause on the tape would have been to ask students to stop the tape. But this would not alleviate the memory load.
Incidentally, the decision to supply students with default values was taken close to the end of the production period; the rehearsal revealed that speaking the numbers .739 and .7391 slowly enough for students to type them was far too laborious and unnatural.
A final, important point to derive from this example is that students were not always given a free hand to input whatever values they liked. In order for the teacher to explain the process, it was necessary that all students started with the same numbers, leading to a known sequence of numbers for the teacher to refer to. This relates to a fundamental point about the role of the audio-tape. Most of the above examples have illustrated the 'study guidance' role of the audio-tape (ie prompting keyboard activities and telling students what to look at on the screen). However, this is a minority role: most of the audio actually teaches the subject matter, in a carefully structured narrative, with the computer supplying the visual aids.
There is one final point concerning prompted tape stops. Another reason for stopping, other than for keyboard activity, is for brain activity either before or after accessing a new picture. The following example is where students are advised to familiarize themselves with Screen 3:
Now we're going to look more closely to see when simple iteration works, and when it doesn't. Select the QUIT option now and move forward to Screen 3. Study the screen before I talk about it. (MUSIC JINGLE)
Screen 3 is shown in Figure 6.
Figure 6: Screen 3, to be studied before restarting the tape
After students have studied the screen and restarted the tape, the audio talks them through the right hand side of the screen. But it was considered necessary for them to familiarize themselves with the screen before being talked through it, because of the wide range of reading times for mathematical symbols.
Two final points concerning script-software design concern pedagogic influences on software design and, conversely, the influence of computer drawing speed on script design. These points will be illustrated with a different M371 teaching package, the one for Block III, Unit 3, Constrained, non-linear Optimization. Screen 1 of this package, following some keyboard activity, is shown in Figure 7.
The top halves off the graphics area on the left and the text area on the right were achieved when Screen 1 was first accessed. The text on the right then appears immediately but, unfortunately, the contour diagram takes an inordinately long time to be drawn. It took between 27 seconds on a computer we judged to be as slow as the slowest of students' computers and 13 seconds on a computer we judged to be about average. The script was therefore composed to 'waffle informatively' for 33 seconds in such a way that the Speed of the computer was not a problem, so long as the drawing was completed before about 30 seconds, as follows:
The text area shows the function of two variables which is to be minimized, subject to the equality constraint c(x) =0.
In the graphics area, the curves being drawn are the contours of f. So it's a bird's eye view of the mountains and valleys of the surface f = 0. The contours are symmetrical about the X2 axis.
In fact, so is the curve c(x) = 0. It's a parabola because X2 is a quadratic function of X1.
Note that the first sentence concerns the text area on the right, giving time for some of the contours on the left to be drawn before they are discussed in the second paragraph. The third paragraph described the final curve to be drawn at the top, namely the thick parabolic curve c(x) = 0.
Finally, there follows an illustration of the care taken to compose an educationally effective script and the influence this can exert on the software design so that words and pictures reinforce each other maximally.
Figure 7: Screen 1 from another teaching package
Following some discussion of the top half of Figure 7, the audio prompts keyboard activities which result in the bottom half being added to Figure 7. The audio then explains the new text and explains that the graph at the bottom is the graph of the function g whose minima are the solutions of the original problem, namely, they are the constrained minima of the function f.
The script then continues as follows:
Geometrically, you can think of it like this:
The curve c(x) = 0 can be thought of as an overhead view of a curved fence. This fence, which you're viewing edge-on from above, cuts into the mountains and valleys of f And the curve of their intersection with the fence is illustrated by the graph of g.
This description of the diagram necessitates the scale along the X1 axis to be the same in the two diagrams. This seems an obvious point, but early drafts of the software used arbitrary (and hence unequal) scales for the two diagrams. This was because, during the initial specification of the 'Screens', the unit author was thinking algebraically. (Most real world optimization problems involve many dimensions and cannot be depicted geometrically anyway). Hence, the diagrams on the left were designed to 'look good' independently of each other.
Not all of our scripting for M37l teaching packages managed to be so evocative, although its influence on the design of the software was always significant, and vice versa, in the attempt to culminate in a picture-word whole which was educationally greater than the sum of its parts.
This endeavour owes much to the many years of partnership between the BBC and The Open University in producing 'audio-vision' packages (audio cassettes 'talking through' printed diagrams). This experience has provided much of the expertise and creativity to this new composite medium. To give you an idea of this influence, we attach an Appendix containing the main principles we have developed for scripting our traditional audio-plus-print packages. About half of these principles have been discussed explicitly above.
To summarize, the script and software were carefully developed to accommodate a wide range of student behaviour and to anticipate difficulties where possible while leaving flexibility for individual exploration. In other words, the design was, in the strongest sense, student centred.
Feedback on Open University courses takes several forms. Each course is evaluated in its first year of presentation by the Institute of Educational Technology, using student questionnaires. The feedback on M37l in 1988, when it first appeared, was highly satisfactory and indicated that the use of the software was causing few problems. Compared with the other two non-mathematical home computing courses, which used commercial software, the M371 software was considered the easiest to use, requiring very little time for students to familiarize themselves with the structure and use of the software.
There is also informal feedback from tutors by means of a 'debriefing' after the first year of presentation, and from students and tutors via the regional office, and via telephone calls or letters to the faculty. This form of feedback has consistently indicated the ease of use of the software and the success of the teaching methods in enhancing students' understanding of the course materials.
Finally, because the teaching packages represent a completely new teaching medium, the course team, in conjunction with IET, surveyed students in 1990 on their use by means of a detailed questionnaire. Students were asked to analyze their own learning by classifying their understanding after various stages in their study of the units and the teaching packages.
The survey results showed that the teaching method was particularly successful on methods or concepts which were algebraically complicated and where there was poor or superficial understanding from the printed text alone. The results also indicated the value to students' understanding in leaving them to explore the graphical effects, having listened to them explained for specific examples.
The teaching methods involving the personal computer in the course M371 are widely viewed as a success and are now being used with equal success in its sister course M372: Numerical Methods for Differential Equations, first presented in 1992. The Open University's current home computing policy is expected to change so that in the future a far wider range of courses would have a personal computer requirement (although much work remains to be done to ensure that this does not become a barrier to open access). In particular, plans for the replacement for the Mathematics Foundation Course (for first presentation in 1996) include the use of a personal computer, and it is intended to use the teaching, packages model to enhance students understanding of a wide range of mathematical topics.
M371 materials are being used by other institutions in the UK, Norway, Hong Kong and Australia.
Bates, A.W. (1984) Broadcasting in Education. Constable, London.
Daniels, J. and Koumi, J. (1992) Teaching packages: audio cassette with home computer. Media and Technology for Human Resource Development, Journal of the All India Association for Educational Technology, 5, 1,1-18.
Laurillard, D. (1993) Rethinking University Teaching: A Frame work For The Effective Use of Educational Technology. Routledge, London.
Jack Koumi has degrees in mathematics and psychology, and for six years was a university lecturer and teacher trainer. From 1970 to 1992 he worked for the BBC Open University Production Centre, producing audio and video programmes in mathematics, psychology, biochemistry, genetics, physics and computing. Additionally, since 1981, half his time has gone into training producers and scriptwriters of educational media. He has published many papers on production and management of educational media. He now works full-time in this field, as a freelance trainer/ consultant.
Judith Daniels studied for her BSc in Mathematics in London and obtained her MSc in Numerical Analysis with Functional Analysis in 1975. She has worked as a mathematician/programmer for various organizations. Since 1979 she has worked full time at The Open University as a lecturer in the Mathematics Faculty and has continued as a part time tutor, teaching a wide range of mathematics courses from Foundation Level up to Fourth Level.
Address for correspondence: Jack Koumi, Educational Media Trainer/Consultant, 4 West Street, Weedon, Northants NN7 4QU, UK. Tel: 44(0)327341418. Fax: 44(0)327349655.
Script design for stop-start educational 'audio-vision' (audio with visual material)
* Part of a handout in an audio course at the BBC OU Production Centre.