An approach to integrating educational software into the curriculum

Journal of Computer Assisted Learning (1990) 6,190-201

An approach to integrating educational software into the curriculum

N. Zeha vi & M. Bruckheimer The Weizmann Institute of Science,

Rehovot

Abstract A scheme for teacher involvement in exploring the use of mathematical software was designed to meet practical needs of effective implementation. The scheme consists of observation of students at work, followed by didactical and cognitive workshops, in which teachers reflect on the implications of their observations for further classroom activities. The details of the scheme are described via an actual algebra software implementation in Grade 8. Observa- tion of students working on the software and an informal evaluation of their achievement as related to the learning gods, led to professional teacher development toward adaptive implementation of the software for individual students.

Keywords: Algebraic expressions: Learning processes: Mathematical software; Professional development.

Introduction

For the purposes of this paper, we distinguish three types of software: 0 self-contained and 'additional' to the regular curriculum:

designed to be integrated-using an approach made possible by the existence of the micro: a redesigned curriculum with the micro as an essentiaI part.

Software development is rapidly moving through the second type onto the third-but software implementation is lagging far behind.

Much instructional software is available, but evidence of its effective implementation as part af the curriculum is just beginning (Fraser et al., 1987). A part of the problem is undoubtedly purely technical-the micros are not in the right place in sufficient numbers just when they are needed. But there are also

Accepted: 30 December 1989. Correspondence: Prof Maxim Bruckheimer, Department of Science Teaching, The Weizmann Institute of Science, Rehovot 76100, Israel.

190

An approach to integrating educational software into the curriculum 191

more serious problems-in particular teacher preparedness for the strategic shift implied in the use of microcomputers.

To help ensure the applicability of software to classroom practice and its effective implementation. we need to consider the possible changes in the teacher’s role in a ‘micro-classroom’ and provide support to effect this change. Probably the most obvious and significant change is that, whereas previously the teacher was busy delivering the curriculum and thus could not effectively observe the student learn with the computer, a teacher can observe ‘over students’ shoulders’ the way they think and learn. One would expect the teacher thus to become something between a guide and adviser in the learning process, rather than necessarily the driver. According to Shavelson et al., (1985) we should be concerned that teachers are able to ‘orchestrate’ the various modes of computer use in their classes for various students. Therefore, the design of a teacher development programme should involve didactical and cognitive studies at relevant stages.

In the following we describe an approach that connects development, practical evaluation and implementation of educational software. A survey of mathematical software implementation and formative evaluation of software, led to the establishment of a scheme for interactive support. The cognitive work in this scheme is based on Piaget’s notion of reflective abstraction (Beth & Piaget, 1966). In this paper, we shall be concerned with software of the second type, which we believe are a convenient and possibly necessary stage in achieving effective implementation of a computerized curriculum of type three. The major part of the paper is devoted to the description of an application of the scheme using a particular software package in algebra.

Survey of software implementation

The mathematics group in the Science Teaching Department at the Weizmann Institute engages in the development of educational software, as an integral part of the ongoing development of the junior high school mathematics curriculum in Israel. A survey was conducted (in the school year 1986-1987) of a sample of 100 schools that had purchased the software developed by the mathematics group (Coldberg et al., 1987). The picture obtained and the consequent conclusions are similar to those of other surveys reported (Green & Jones, 1986; Dickey & Khirlopian, 1987). The teachers were asked to rank on a scale of 1 to 3, both the desired and current situation of software usage in their schools. Types of usage were grouped into three categories covering software of the first two types above (software of type three does not exist, but is being planned): (1) integrated with the existing curriculum, (2) individual usage, and (3) activities for pleasure and variety.

Results clearly showed that the teachers think that there should be greater use of software in all three categories. As was to be expected, the smallest gap between the desired and current situation was in activities for pleasure and variety, and the largest for items that were intended to be integrated with the

192 N. Zehavi & M. Bruckheimer

existing curriculum. It is a positive sign that teachers are aware of the need to integrate the software with the curriculum: however, for two items, not only is the current situation very poor, but the teachers also ranked the desired state relatively low, One of these items is intended to present material in place of the textbook, the other provides informal preparation for more advanced topics. Both of these types of usage require planning and a high degree of teacher involvement in order to succeed. These latter items reflect our objectives in the design and development of some of the software, and are documented in detail in the teacher manuals. They were also emphasized in in-service courses in which many teachers participated. The survey, therefore, would indicate that both teacher manuals and in-service courses are not sufficient. There is a need for much more basic teacher development.

The teachers were also asked to indicate the importance of various in- service activities. As expected, the activity identified as the most important was the opportunity to try out the software hands-on and gain experience and self-confidence. Most of them (84%) also expressed great interest in workshops to discuss the integration of software with the regular curriculum. About 50% were very interested in receiving help in guiding their students in computer sessions in order to manage effective usage adapted to individual student needs.

A scheme for interactive support

In the traditional repertoire of activities in junior high school algebra, the student is mainly concerned with the manipulation of expressions, word problems and solving equations and inequalities. The introduction of microcomputers enables the design of novel activities, which may help to bridge the arithmetic/formal symbolic divide and thus enhance the basic learning processes. It is reasonable to assume that computer technology can provide flexible representations, which may promote the learning of the algebraic symbol system, by presenting a more coherent view of the mathematical content (Kaput, 1986).

Several of our software packages were designed with the intention of developing an informal understanding of more advanced topics in algebra, such as graphs of quantitative relations and sets of simultaneous (inlequalities. These packages are composed of a variety of tutorials on the basic tasks at different levels, and motivating game environments, in which the strategies are intimately related to the cognitive skills. The question is how can such software be effectively integrated in the curriculum and implemented successfully in classroom practice. Some partial answers were obtained in previous studies (Zehavi, 1988a, b).

From these practical studies we consolidated a scheme which is designed to give teachers confidence in using software and, gradually, to make them independent in managing computer use as an integral part of the curriculum. The scheme is referred to as an interactive support programme and has three levels. It is repeated for each software, with increasing emphasis on the higher levels as the teachers get more experienced.


Level 1: Familiarity

This level consists of three modules.

(11 Observation of student work. Teachers not familiar with a software, observe students’ first exposure and thus get a feeling for its surface structure. They are advised to focus their observation on some behavioristic and affective aspects. Our experience indicates that this is a crucial phase, since teachers sometimes transfer their own anxiety to the students I‘My class will not understand it’). The anxiety decreases after their observation of students using the software.

( 2 ) Hands-on experience. After the first acquaintance ‘over the student’s shoulder’, it is essential that the teachers get extensive hands-on experience. Thus, they get acquainted with its deep structure and acquire the facility to operate the auxiliary interfaces.

(3) Mathematical-didactical workshop. The mathematical background of the topic and the integration of the software are presented at the beginning of the workshop. Then the relevant didactics are discussed, together with practical suggestions how best to integrate the learning package with the regular curriculum, under the constraints of particular schools and student populations.

Level 2: Cognitive workshops

At the second level the teachers deal more with the structure of the software. Researchers discuss the results of their research using the package. The teachers are introduced to student difficulties in the specific area and to common student strategies. Gradually they learn how to adapt teaching approaches to the individual.

Level 3: Creative observation

The third level involves creative observation by the teachers of individual students. They focus on cognitive aspects of the learning process and are encouraged to use similar techniques to those in the research. As a result they initiate, via the software, student activities in an attempt to achieve effective implementation. Thus, the teachers become partners in the process of their own development towards effective classroom use of software.

This scheme reflects our approach to the role of mathematics software in theory and in practice, with emphasis on the latter. In order to give it meaning, it needs to be presented via some specific mathematical content. We, therefore, describe an application of the scheme with a software in algebra, TheExpression Strikes Back (Zehavi & Taizi, 1987). We first describe the software and analyse the nature of the cognitive processes involved.


The software and its didactics

The rationale for the development of The Expression Strikes Back, was to offer activities that combine both skill and reasoning in substitution in algebraic expressions, as part of the presentation of techniques for solving (inlequalities in Grade 8.

The software contains two tutorials and two games. The basic task is to separate an ordered sequence of numbers, according to the sign of the result of their substitution into a given expression. Initially the expressions are of the form

b b ( & x + a ) or -

f x + a '

which involve only one change of sign. A typical situation in the first tutorial is shown in Fig. 1. The numbers (in the upper row) to the left of the bar give negative results when substituted in x - 7, and those to the right positive results.

-16 -11 -7 - 2 1 3 1 8 10 15 19

7 ~ 5 - -23 -18 -14 -9 -6 - 4 1 3 8 12

Fig. 1. Substitution in x-7

The first game, Warring Expressions, for two players, offers a strategic environment which requires mathematical-logical reasoning in addition to skill- drill. Each player is given a sequence of numbers, which remains throughout the game, and an open phrase which changes at each turn (see Fig. 2). The aim of each player is to be the first one to 'turn on' all the numbers in hislher Sequence. To achieve this a player can choose to 'turn on' numbers in hislher sequence that give positive results, or 'turn off numbers that give negative results in the opponent's sequence. To illustrate the skills and. reasoning which are involved, we consider the situation in Fig. 2. It was player B's turn. She chose sequence B and moved the divider to the right and placed it between -10 and - 7, lighting the numbers to the right. (If a player places the divider incorrectly (s)he loses the turn. A common cause for such a mistake is difficulty with negative numbers.) If player B had chosen to cause player A trouble, the divider should have been


Player B Player A Vl

Fig. 2. Warring expressions

placed between -10 and - 5 in sequence A, and then the two numbers to the left would turn off. Note that at each turn a player can (and should) consider the other player's expression. In the example, player B should have stopped player A, who won the game in the following move.

The second tutorial unit deals with expressions which have two changes of sign, of the form

f x + a ( + x + a ) ( + x + b l or ~ + x + b '

One row of increasing numbers is given, and the object is to turn on all the 'lights' in a minimal number of steps. Two expressions are presented (see Fig. 31, of which one is to be chosen. The two dividers are then placed where the signs of the substitution results change. If the dividers are placed correctly, all numbers which give positive results are turned on. If the dividers are not placed correctly, all the numbers that give negative results are turned off. Once an expression has been used, it is replaced by a new one. In the situation in Fig. 3, the choice of

Choose an expression

Fig. 3. Tutorial: expressions with two dividers

198 N. Zehavi 81 M. Bruckheimer

expression 2 can light two numbers on (4 and -91, whereas expression 1 can turn on all the numbers.

The final game, The Expression Strikes Back, from which the software takes its name, requires a higher level combination of skill and reasoning. The game was designed to provide an opportunity to crystallize and generalize the tasks in the first three activities.

There is one list of numbers, two dividers and an expression of the form

where the blank is to be filled by a number of the player's choice. Numbers are turned on if they give a positive outcome when substituted in the current expression, and turned off if the outcome is negative. The first player aims to turn on all the numbers in the list and the second aims to turn them off. In Fig. 4, it is the turn of the second player, who has to choose one divider (rod), move it and then fill in the blank so that the dividers separate numbers that give a positive result from those that give a negative result, In this case, if the player reasons well, (slhe will move divider 1 to the right of -1 and write -13 in the blank. The computer will then turn off all the lights, except for that above 3.

Choose a rod

( ~ 1 . 0 1 )

Fig. 4. The expression strikes back

In the local junior high school curriculum, algebraic techniques for solving linear (in)equalities are presented in the first trimester of Grade 8. The first tutorial and the first game are meant to be integrated at this stage. The second tutorial and the second game are intended to be presented in the third trimester of Grade 8, in preparation for the study of and/or systems of linear (inlequalities, including items of the form (x+ a)(x+ b) > I < 0. The retention of concepts from the first two activities and the preparation provided by the two others, is designed to help creare meaningful understanding of the abstract ideas necessary for and/or systems of in(equa1ities).


The cognitive processes

Theoretically teachers using computers in their classes are expected to be aware of learning processes. In order to be able to apply analysis of the psychological process involved in learning with the software, we worked out simplified interpretations of two methods. The first was used by Davis et al., (19781, who looked at cognitive types of student behaviour and strategies in learning algebra, They classified examples of recognition of instances (pp. 88-89) and dis- tinguished different extents in the search for patterns (pp. 128-1331, The second method is Dubinski's suggested interpretation of Piaget's notion of reff ective abstraction mubinski & Lewin. 1986). The framework we use contains two types of process: internalization of primitive actions into an action, and formalization of many actions into one pattern' described with standard algebraic notation. Both types of process are reversed in tasks that require general coordination of actions.

We now analyse theoretically, the cognitive processes involved in each programme of the software. The empirical observations are described in the next section.

The first tutorial (Fig. 11

The activities presented should lead the student to internalize the fact that for the given expressions there are two sign-regions. A formal pattern is obtained when the student associates the position of the divider with the value of a in b( k x+ a).

The first game (Fig. 2)

At the beginning of the game the coefficients of x in the two expressions have different signs (e.g., 11 - x and 9 + x in Fig. 2). This is intended to lead students first to internalize the fact that +x for the numbers which give negative outcomes are on the left and for -xon the right, and then to be able to explain it formally in terms of increasing and decreasing expressions. However, if one expression contains +x and the other -x it is difficult to win, since one player can turn on the lights and the other can turn them off. Thus, after a while, the expressions are selected randomly to increase motivation and to provide practice and more opportunities for the coordination of the roles of all the elements of an expression.

The second tutorial (Fig. 3)

The student has to coordinate the following actions: application of the sign-of- result pattern of (+x+a) to 9-x and 10+x and then the use of rules for 'multiplying signs'. A more generalized coordination occurs when the learner recognizes instances of the u . v > 1 < 0 principle, and formalizes a pattern that


all numbers between the dividers give positive results and the rest negative results.

The secondgame (Fig. 41

The expression contains a blank, therefore one cannot apply actions without formalization. The game strategies require the reversal of the formal pattern. A higher level of coordination of actions is required if one divider is already in place; the second divider can then be moved to any position (even to an edge), to maximize or minimize the distance between the dividers and thus gain an advantage.

Application of the scheme

As an example of the application of the interactive support scheme we describe activities in one junior high school. The teachers had not used mathematics software in their classes. However, they had participated in a two-day course on algebra-software (not the particular one in this paper) and purchased some packages. They wanted to use them in class but had the usual anxieties. The mathematics coordinator approached us and we decided to apply our interactive support programme using the software described in this paper, with variations in the sequence of observation, hands-on experience, didactics and cognitive work, that are always necessary in any specific application. The cognitive framework was also adapted a s necessary.

One Grade 8 class (n = 28) and six teachers participated in the study. As recommended above, we wanted to implement the software in two parts, in the first and third trimesters, but management constraints forced us to perform the whole application in the second trimester. At that stage the students had already learned to solve linear equations and inequalities, and were working on word problems. A lesson period was devoted to each one of the four programmes, at weekly intervals. A teacher observed 4-6 students, and after each period the teachers participated in a 90 min workshop.

Familiarity

In the first period the teachers observed naively the class working on the first tutorial. In the following workshop they had initial hands-on experience and a didactic discussion. The main issue was the formal mathematical reasons for the relations between the position of the dividers and the expressions.

The observation of students playing the first game was more directed. The teachers occasionally asked students to explain and justify their strategies. The teachers recorded the development of strategies based on mathematical patterns discovered by the students related to the three elements (a, the f sign prefixed to x, and b) of an expression b( _+x+ a).


Cognitive workshop

In the workshop that followed, the teachers were asked to report their observations. The main finding was that students (about one-third of the class) who started the game without internalization (after the first tutorial) of the connection between the ‘free’ number, a, and the place of the divider, did not gain much from the game. They checked by substitution and made mistakes, but at some stage in the game they came to realize the connection (with a hint from a peer or observer). Although they could then place the divider correctly, they did not control the game, and most of the time they were surprised by the lights turning on or off. However, the remaining two thirds of the class, who started the game knowing the role-of-a-pattern, quickly mastered the role of b. The role of f x was internalized by about half the students and a sound formalization was given by a little over a quarter. As a result of these findings, additional activities were suggested, some of which were designed to build reverse processes to internalize the role of a and f x (.e.g., to fill the blank in x+ 0 to correspond to a certain state of the lights). The rest of the workshop was devoted to further experience with the software in preparation for the following programmes.

Creative observation

The observation of the second tutorial can be considered as the beginning of creative observation. Based on the previous work, we asked teachers to identify levels of student performance and to suggest possible guidance activities. We use the expression (1O+x)(9-x) from Fig. 3 to illustrate the levels of student performance which were identified.

I. At the lowest level, a student checked the substitution-result of many numbers, without assuming that there may be three sign-regions.

11. The student’s decision was based on substitution of ‘key numbers’-at ends and adjacent to the dividers.

111. The student substituted key numbers in each of 1O+x and 9-x, deduced their sign-regions, and then combined this with the rules for the sign obtained by the multiplication of the two factors.

IV. The student applied the sign-of-result-pattern of a + x to each of 9 - x and 10 + x, and then used the rules for sign-multiplication.

V. The student explicitly referred to the u . v > / < 0 principle and concluded that all the numbers between the dividers give positive-substitution-results, and the rest give negative results.

It was suggested that students who performed on levels IV and V were ready to benefit from the final game. For students who operated on level I, the teachers concluded that they should play the first game again. For those at levels I1 and I11 (60% of the class), some activities that should lead to coordination were suggested (e.g. to suggest values for missing numbers to correspond to certain states of the lights and certain expressions).


The teachers were now ready and interested to see if, in fact, their conclusions were valid and their various recommendations necessary. In the last computer session the whole class played the game and teachers played individually with some of the students. They were instructed to make good moves and to provide short explanations only if asked. On the other hand, they were told to ask for explanations of students' moves. It was confirmed that students who had been identified as at level IV and V, were motivated by the game and achieved the goals of the software. Level I11 students learned by imitating the teacher but did not achieve mastery in the game. For students at levels I and I1 the session was considered an almost total waste of time.

Conclusion

In the last workshop, a concluding discussion was held in which ideas for appropriate activities and timing of teacher-intervention were presented. The teachers were satisfied that the type of remedial intervention suggested in the previous workshop was generally along the right lines, and expressed reasonable confidence that they could manage the software for individuals and groups of students.

In the following school-year, four of the six teachers integrated this software in their Grade 8 classes, in the first and third trimesters. They implemented the suggested activities to lead students to the general coordination of actions and reported satisfactory results. Two of these teachers joined us in a similar experience in another school with another software.

Implications

Since the application described in this paper, we have used the interactive support scheme again and again. The sequence of activities begins by removing teachers' technical apprehension of the medium itself, and goes on to build didactical confidence. The teachers are offered a clear rationale for the design and use of a package, observe the package in use, and reflect on the implications of their observation for future classroom activities.

In the process, both the developers of the software and the teachers have an opportunity, not previously available to anywhere near the same extent, of studying students' cognitive processes and effective intervention where necessary. Presently, software developers have only partial understanding of student difficulties in algebra and they also lack a good theory of effective didactics. The activities described here gave a lot of useful information, that is being used in updated versions of this software and others. It is interesting to note that the same scheme which serves as a 'formative evaluation' tool for the developers, serves as a medium to lead teachers to effective implementation.

The approach described here is one possible approach to the exploration of complex communications between software developers, students and teachers. The introduction of educational software creates pressure on teachers for a


fundamental change in their direction of the class. The sociology of the classroom is affected: there is a need for new competencies and training for both students and teachers in order to face together new and challenging situations; the monitor is not a copybook: direction of student work is shared by teacher and computer; the teacher is free to focus on what is really happening to the learner.

Involving teachers in the cognitive work may be rewarding to them and to the developers by enriching the understanding of student difficulties and the relevant didactics. This may contribute to future development of computer use, and lead to some professional development toward the changing role of the teacher who wants to integrate software in the classroom.

References

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Davis, R. B., Jockusch, E. & McKnight, C. (1978) Cognitive processes in learning algebra. Journal of Children's Mathematical Behavior, 3,lO-320.

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An approach to integrating educational software into the curriculum