EXPERT SYSTEMS FOR CALCULUS TEACHING

EXPERT SYSTEMS FOR CALCULUS TEACHING EXPERT SYSTEMS FOR CALCULUS TEACHING
Cleide Regina L. Paladini1; Mirian Buss Gonçalves2 e Diva Marília Flemming3 1,2 Mathematics Department - UFSC – Federal University of Santa Catarina and
3 Physical Sciences Department - UNISUL – University of Southern Santa Catarina Brazil.

GEIAAM – Group of Studies on Artificial Intelligence Applied to Mathematics Departamento de Matemática - UFSC - Campus Universitário 88049-900 - Florianópolis/SC - Brazil Key Words: Artificial Intelligence – Educational Expert Systems – Calculus Teaching.
ABSTRACT
This paper shows the results of the research carried out by the Group of Studies on Artificial Intelligence Applied to Mathematics – GEIAAM (Grupo de Estudos de Inteligência Artificial Aplicada à Matemática) of the Mathematics Department of the Federal University of Santa Catarina – UFSC, Brazil, targeted at Calculus Teaching in engineering courses. Concepts of Artificial Intelligence and Expert Systems are shown. These concepts have been applied to the development of small Expert Systems used in specific contents of Calculus. The systems are developed and briefly presented together with the guidelines for their effective use in teaching activities. Experiments previously conducted with some of these systems as well as the relevant results obtained from them are reported.
1. INTRODUCTION
Mathematics teaching has been intensively questioned over the past few years (see Bertoni, 1994, 1994; D'Ambrosio, 1994). Such discussion has been aroused for various reasons. On the one hand are the difficulties naturally ascribed to this subject and its disciplines. On the other hand is the need for redesigning syllabuses, in view of the changes that traditional courses where Mathematics is taught are undergoing. That is the case of the engineering courses, for instance. Furthermore, there is some remarkable development in terms of new tools which can be useful for teaching purposes.
The latter undoubtedly includes Artificial Intelligence (AI). In fact, AI has been associated with a wide range of activities in several areas of knowledge (Rich, 1988; Yazdani, 1986). Recorded uses of AI refer to decision-making processes, information searching and analysis, planning and use of new problem-solving strategies, algorithm development, processing methods and several other mechanisms involving reasoning, demonstration of propositions and theorems, development of automatic translators, control of goods or information flow, etc.
The areas which have stood out within the context of AI involve, with no doubt, devices of Pattern Recognition (Fu, 1982), both in analytical versions and in functional processes, and the so-called Expert Systems (Chorafas, 1988; Levine et al., 1988; Passos, 1989; Waterman, 1985), which consist of special types of software using specific data to propose solutions to problems by using inferential procedures and that require specialized knowledge or qualified information.
This work was moved by the hypothesis that the development and application of Expert Systems to solve problems can be deployed as a support tool in education for those disciplines characterized by their objectiveness. This is the case of Calculus. Such hypothesis, based on practical experience from classroom, has as a starting-point the lack of logical-deductive reasoning as one of the greatest difficulties shown by Calculus students, preventing them from analyzing a given mathematical situation and identifying what is necessary to propose a solution which is adequate to its structure.
Within this context, a group of professors, researchers and students of the Federal University of Santa Catarina – Brazil, set out to study and do some research on AI as an alternative tool to shape knowledge and didactic processes (Balachef and Vivet, 1994). The general objective was to make use of the potentialities of AI in the construction of small Expert Systems aiming at minimizing students' difficulties observed in classroom. These activities were developed with the help of the research project called Artificial Intelligence Applied to Mathematics Teaching, carried out in 1993-94.
The results obtained with this project (Paladini et al., 1995) led to the creation of the Group of Studies on Artificial Intelligence Applied to Mathematics – GEIAAM, of the Mathematics Department of the Federal University of Santa Catarina, Brazil. GEIAAM has since 1994 considering and working on an investigation involving two contexts: the educational and the technological.
From an educational standpoint, the group's motivation derived from the authors' large experience in Mathematics Teaching for courses of the technological area, and also from a continuous concern about the present reality and the need for finding ways of improving the quality of education. From a technological point of view, upon studying and analyzing what is proposed by AI, we realized that it could be an interesting tool to teach Mathematics. It would be a means of bringing together Mathematics Teaching and technological development, in addition to attending to another aspect observed in classroom, i.e., the students' facility and willingness to manipulate computers.
This work done by GEIAAM has yielded relevant results so far. Thus, we now proceed to the following items: general notions of AI and Expert Systems; motivation and survey of data relevant to this work; brief description of the systems so far developed; guidelines for the use of such systems; reports on the use of such systems in classroom; conclusions.

2. GENERAL NOTIONS OF ARTIFICIAL INTELLIGENCE AND EXPERT SYSTEMS

In broad terms, Artificial Intelligence can be defined as 'a set of programs that make computers seem intelligent' (Passos, 1989). Another concept more widely used is that presented by Rich (1988:1): 'it is the study on how to make computers perform tasks which are, up to now, better performed by people.' Barr and Feigebaum (1981) define AI as a part of Computer Science related to the design of intelligent computer systems, i.e., 'systems bearing characteristics which we associate with intelligence in humans, such as: how to understand a language; learning ability; reasoning; problem solving; etc.'. It is worth pointing out that intelligent programs are those capable of using decisions in a way similar to that of human intelligence, in order to find solutions to problems. Intelligent computers, in turn, are those which deploy, process or handle intelligent programs.
For the purpose of this work, the AI characteristics considered relevant are: manipulation of symbols instead of numbers; utilization of inferences and deductions by using information available; use of knowledge for solving problems and use of knowledge in the form of associated rules, in order to limit exponential growth, which happens in complex situations in the real world. Some of these characteristics are typical of the majority of syllabuses of Mathematics.
Expert Systems are computer programs (or software) targeted at solving problems in specific areas of knowledge. Their main feature is a knowledge basis concerning the restrict domain in which the problem is found. This knowledge basis is fed into the computer by experts who, as a matter of fact, make the system richer with their specialized information. The Expert Systems offer suggestions of which decisions to make and are able to justify their own choices.
Basically, six elements are used in the architecture of an Expert System: a knowledge basis containing facts, rules and standards related to certain situations; an inference device capable of making decisions within a given domain; a language through which the rules will be written and the machine-person communication will take place; a 'shell' including the inference device, the knowledge manager and the interfaces with the user; the environment and other parts that can go together with the 'parts kit' and an explanation generator (optional) to explain how the system came to a given conclusion.
An Expert System may be classified according to functioning characteristics. In general, such categories are: Interpretation Systems – make inferences related to descriptions of situations; Diagnosis Systems – detect faults resulting from data interpretation; Monitoring Systems – interpret signal observations; Predicting Systems – by comparing past and present data, they make predictions for the future; Planning Systems – prepare an initiative program in order to achieve an objective; Depuration Systems: provide solutions for malfunctioning caused by data distortion; Repairing Systems: design and execute plans to repair defects detected during the diagnosis phase; Instruction Systems: have mechanisms to check and corrects students' learning behavior; Controlling Systems – control the overall behavior of other systems.
The systems presented in this work have characteristics of some of these categories, however not being classified within a specific category. Hence, we call them Educational Expert Systems (EES).
The basic characteristic supporting the application of Expert Systems within the context of this work is precisely its intensive interactivity with the users, which allows to design a constant feedback process with those students to whom a given subject is being presented. The System, thus, starts acting as a partner in the activities of research or analysis in which the user is involved.
Thus, a structure here is proposed that uses simple stimulus-response procedures. Additionally, it is worth pointing out that hardly any training is required in order to use the systems already developed and that are being currently utilized.
It is important to mention that the system makes possible to have a follow-up of learning progresses. In specific situations, the system make use of devices which offer explanations on how and why a certain conclusion was arrived at. The student has, in this way, the guarantee of a method which enables him/her to know how an answer was obtained (in reality, we aim at showing the student how the system uses its own inference system and, consequently, the logical reasoning applied). In addition, it is possible to separate the facts that served as a basis for the decision-making process, ranking them as relevant and non-considered.
The advantages of using Expert Systems are thus evinced as an adequate didactic resource available to Mathematics Teaching. The computational characteristics of the support in question involve further advantages to users, since they introduce them decision-making in an interactive processing environment. The basic advantage, however, is the reduction of difficulties frequently present in the Mathematics Learning process.

3. MOTIVATION AND EXPECTATIONS

In Brazil, the vast majority of students have their first contact with the notions and concepts of Calculus after their entrance to the university. In the engineering courses, that happens in the first term. In general, at this point, students experience strong impacts caused by the gaps between pre-university and university education.
In pre-university education students are frequently mere passive agents in the educational process, whose role is simply to memorize the information conveyed by the teachers and do the exercises by them proposed. When at the university, students are expected to change their behavior, moving to a position of active agents in the educational process. He/she him/herself must have the initiative to look for books, do exercises, etc., seeking to go deeper into concepts and develop his/her ability to solve problems. Since many of them are not used to it, they let time pass by and when they are submitted to the first examination, the subjects are still not sufficiently matured. As a result of this, the students do poorly in the exam, have difficulties understanding the subsequent topics, lose motivation and, finally, get a failing grade. This justifies, partly, the high failure rate in the discipline of Calculus, in the first term of the courses of the technological area.
In the view of this situation, and starting from the hypothesis that motivation is propeller towards the learning process, we decided to carry out a poll among the engineering students of UFSC, asking them about their expectations in relation to using Information Technology in the disciplines of Mathematics (Flemming and Paladini, 1997). A questionnaire comprising several questions about possible causes of failure, didactic resources used by the professors, etc., was applied to all (frequent) students attending the disciplines of Mathematics, during November 1996.
Through the analysis of this questionnaire we could see that the vast majority of professors still used only chalk and blackboard as didactic resources, in association or not with texts. The vast majority of students, on the other hand, expected computational resources to be employed, with an emphasis to classes in computerized laboratories (83.39%).
That was how motivation to adjust Mathematics Teaching to the students' current reality arose. In fact, we are living in a present context where technology is omnipresent in the routine of every citizen. Several sectors of society have already been computerized, making use of the technological advances and handling the 'possible impacts' thereof. In the educational sector, however, such familiarity has still not been achieved in our country, i.e., there has not been much use of such technology as of yet. The educational process, however, has its own complexity. Thus, not only equipment is necessary in order for the technological resources to be used by the educational system in their completeness; a set of actions involving the institutional sectors, students and professors is also required. Mathematics plays an important part in this process.
According to Grou and Costa (1996), the Mathematics teaching-learning process faces an apparent paradox: 'If, on the one hand, computers save us from calculations and programmable operations, giving us the impression – at first sight – that we could even abolish elementary arithmetic from syllabuses, on the other hand, comparative analysis of a large and intricate volume of data now made possible, requires the mastery of mathematical concepts progressively more sophisticated.' If there is the need for redesigning the whole process, Mathematics teaching can no longer be based on instruction passed on to students by the teacher either. We aim at an educational process where the students construct their own knowledge, develop their competence to learn and search for information and be able to understand it so as to apply it to solve problems.
Within this context, AI techniques and the structures of the Expert Systems provided the conceptual support in order to develop educational software tools. We initiated then the design and development of some expert systems directed to some specific topics in the disciplines of Calculus. The educational theories, within the constructivist context, have accounted for the philosophical assumptions that outline the pedagogical proposals that come together with such systems. The systems developed so far will be briefly described in the section below. Their main feature is the interaction with the user (student), allowing him/her to participate in the construction of his/her own knowledge.

4. A BRIEF DESCRIPTION OF THE SYSTEMS SO FAR DEVELOPED

4.1. Encadin/Functions ( Diva Marília Flemming).

This system aims at revising the basic characteristics and properties of the elementary functions.
From a pedagogical standpoint, in order to understand more advanced concepts such as differentiability, integration, etc., students must master more elementary concepts of functions. Even though these concepts are part of pre-university syllabuses, they are not sufficiently clear for many of the students entering our Universities.
The Expert System as such provides the students with a quick review of the characteristics and properties of the following elementary functions: polynomial up to grade 3, trigonometric, logarithmic and exponential. Upon starting the system, students can view a screen with graphical examples of the four previous types of functions being dealt with. After selection, the system requests that some data be entered and starts operating with specific examples.
Once they have the example, the students can choose which property of characteristic to review, such as: type, domain, image, graphic, roots, increasing or decreasing intervals, minimum and maximum points, continuity, etc. After the students have entered a choice, the system launches its knowledge basis aiming at: encouraging students to solve or think about that question; giving pertinent information; providing answers.
The interface is user-friendly, colored and has barely any repetitions. This, of course, makes it attractive to the users.

4.2. Encadin/Introduction to Derivative (Katiani da Conceição / Mirian Buss Gonçalves).

This system aims at introducing the concept of derivative. Therefore, students are invited to navigate through two practical problems: tangent line of curve and instantaneous speed. At this stage, the system encourages students to think over the situations presented. The definition of derivative is then introduced in a quite natural way.
Next, some examples are dealt with. In this phase the system allows students' active participation and they can choose to solve the problem in a pencil/paper environment and later on refer back to the system in order to verify their answers.

4.3. Fun 97 ( Karin Siqueira, André Meurer e Mirian Buss Gonçalves).

The objective of this system is to offer the students some guidance on how to use concepts and theorems related to the analysis of behavior of functions, in addition to helping settle these topics.
The system has the following options: critical points, maximum and minimum points, increasing and decreasing intervals, concavity, inflection points, etc. Once the choice has been made, the system presents the students with the procedures that can be used in order to analyze the situation in question, as well as the theorems underlying such procedures.
When navigating through the system, the students will no find specific answers to a given function being analyzed, i.e., the system does not perform the calculations. Instead, this has to be done by the students, who need to work simultaneously in the pencil/paper environment.

4.4. Sinde97 ( Dayan Pablo Dose e Diva M. Flemming).

The general objective of this system is to introduce the concept of definite integral, discussing specifically: exhaustion method; Riemann's summation; related historical topics; definition and properties. The system also motivates students to go through the formal study of the fundamental theorem of Calculus.
Sinde 97 has two main subdivisions working as the basic structure of the system: the historical part and the part related to concepts, properties and examples.
Upon clicking on the 'Start' button, the students begin to interact with the system, in its conceptual subdivision, by means of color screens with texts and/or figures. The system guides students' reasoning by taking in their choices and encouraging them to work on important phases by means of messages such as 'Think carefully and double check it', 'Take your time and double check it', etc.
Some screens are able to bring back results from previous phases, thus letting students organize their thoughts and reasoning before going on. Mathematical formalism and notation rigor is gradually introduced as the students advance.
The examples suggested for an analysis are developed step by step, in accordance with the direction taken by the students. Students are encouraged to perform calculations using pencil and paper. Thus, the system creates an interface between computerized environment and pencil/paper environment. All results and calculations can verified by the system.
Upon launching the historical subdivision, the students can find out historical details related to mathematicians of different times who contributed to the evolution of the concept of integral.

4.5. Encadin/Series (First version developed by Maria da Graça Pereira e Mirian Buss Gonçalves; new version implemented by Katiani da Conceição).

The understanding of the notion of convergence of infinite series as well as the verification of convergence of specific series, requires a certain degree of maturity from students. In the analysis of convergence of series, one of the greatest difficulties consists of identifying, among the several existing levels, that one which best suits each situation in particular. To give the student some guidance in order to overcome these difficulties and master the concepts and theorems related to the numerical series is the main goal of this system.

Amidst the convergence tests deployed to analyze the character of a series, i.e., to decide whether a series is convergent or divergent, the following stand out: general term test; comparison test; integral test; ratio test or D'Alebert's test; root test; absolute convergence test; and Leibnitz's test. All of these tests make part of knowledge basis of the system and are presented to the students by means of interactive screens, where his/her participation is fundamental.
We point out that the analyze of convergence of a series is carried out analytically and not numerically. The system presents the theoretical basis and the students are to perform the limit calculations necessary.
5. GUIDELINES TO THE USE OF EESs

The Educational Expert Systems (EESs) described in this work must be used as educational resources and due to their architecture, they induce specific didactic sequences. When designing a didactic sequence, one must have in mind the behavior expected from students and the means (syllabus, procedures and resources) that will be used to systematize teacher-student interaction. In general, one takes into consideration, initially, the decision on the part of the teacher to use an EES in his/her classroom. It is important that such decision has been made after careful thought has been given to it so as to guarantee a good class planning (Flemming et. Al, 1996)
Once the decision is made, the objectives of the class must be established. Three different goals belong here: reviewing topics; make sure that students are familiar with the topics taught; and get them motivated for the introduction of new topics.
Having defined his/her goals, the teacher must plan the didactic sequence of his/her class. According to Sant'Anna (1996), a good didactic sequence must take into account stimuli, be flexible and have a good topic order. Our proposal takes into consideration these three items: it includes stimuli suitable to the students' level, providing teacher – student – computer interaction; it is flexible, since it allows students' participation as the main agent in his/her own learning process; and has a good topic order, with no gaps along the way.
During class planning it is important that the teacher take into consideration small details such as: environment adequacy (computer laboratory); the system must be in perfect conditions for use and support material (class directions, exercise/activity lists, etc.) must suit the level of each group.
After class conclusion, the teacher must list success and failure indicators (if they exist). This generates a new pondering process, which serves as a basis for a new planning (Paladini et al., 1997). It should be noted that the proposal is presented in circles, which makes it dynamic and realistic. In the next section are reported some experiments conducted in classes of engineering courses, in the discipline of Calculus, at two universities, namely UFSC (Federal University of Santa Catarina) and UNISUL (University of Southern Santa Catarina).

6. REPORT ON THE EXPERIMENTS

Experiments conducted with the systems Sinde 97, Encadin/Series and Fun 97 are reported here. All of them were followed up with post-application tests, aiming at assessing the systems.

6.1. Using ENCADIN/SERIES

It was initially tested on students of scientific initiation and subsequently on students of the Mathematics course (licentiate degree) of UFSC.
The class in the computer laboratory was given after the topics on numerical series had been taught by means of explanatory, dialogue-based classes. The objectives were: 'To dwell on the convergence tests of numerical series and develop students' ability to identify the most suitable tests for various particular situations.' A list containing several kinds of series was handed over to the students for analysis of their convergence, by using pencil, paper and computer. It should be noted that the computer was used as an expert giving the students guidance as to what tests to choose.
During the class, students got quickly familiarized with the system. The activities included in the didactic sequence were all covered, thus fully reaching the goals. It is interesting to point out that a mistake which frequently occurs in tests involving this subject, i.e., 'If the limit of the general term of the series is zero, then the series converges,' was absent from all of the evaluations.

6.2. Using SINDE 97

This system was initially applied to two classes of the engineering course at UFSC and was later used with two groups at UNISUL as well.
The objective of this class was: 'To introduce the Definite Integral, to formally present its definition and properties and to encourage students to use the fundamental theorem of calculus'. In this class students had free access to computers, without any directions previously given and took notes in their notebooks. During the class we could observe that in the beginning students were very curious and eager to reach their goals. Later on, students returned and began to direct their learning procedures by navigating through the system. Some students turned to the teacher in order to make sure that their reasoning was correct; some others discussed with their classmates.
The teacher played effectively the part of a mediator and had enough time to observe and guide from closer some students having prior, basic difficulties. In the subsequent class, the teacher evoked was had been previously learnt and could see that the students were ready to proceed to definite integrals and figure out problems related to area calculations.

6.3. Using FUN 97

This system was experimentally applied to two engineering classes at UFSC. The objective of this class was: 'To reinforce the analysis of function behavior.' The students received a list including several types of functions to interact with the system. They were expected to determine: the roots; critical, maximum and minimum points; increasing and decreasing intervals; concavity and inflection point of each one of them. Once having such data available, they were supposed to draw some graphs.
The next step consisted of using another software tool (e.g. DERIVE) to carry out comparative analyses.
In this class we detected a strong concern about 'making calculations', which to a certain extent impaired the realization of the dynamics of FUN 97. This, of course, was taken into consideration by the designing team and they are currently making a few alterations to the system.

7. CONCLUSIONS

This study bears results indicating that AI can fit to Mathematics teaching.
If we take into account that the effective use of computer tools for teaching purposes demands an analysis of several didactic-pedagogical aspects therein involved, then a more in-depth analysis of the systems so far developed is also required. The relation between traditional environment (pencil & paper) and new environment (computer) must be carefully investigated. All hurdles must be tackled, aiming at a better adaptation to the new environment and the new methodological proposals to Mathematics teaching as well.
The expert systems presented are in an experimental and upgrading phase. They are being applied to specific groups of students for analysis of their adequacy to the objectives established and consequent validation. The first evaluation has revealed positive points, which is a motivating factor for the continuity of this work.
An important detail for the educational context is herein emphasized – the architecture of the systems presented in this study allows direct interaction between student-computer and indirect interaction between teacher-student. This feature makes them suitable for applications in the context of distance education.

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