# QED-Tutrix: intelligent tutorial system for student support in problem solving demonstration in flat geometry

In the past years, Quebec’s school system imposes a growing amount of pressure on its teaching staff. They must juggle classes with more and more students while assuring the quality of their teaching to each of them. In this context, the use of intelligent tutoring systems which can assist the teacher in his or her work could allow the teacher to dedicate more energy to each student when it’s needed. However, the offer for tutoring systems for the learning of proof is limited. Moreover, the available systems force the student to work according to a determined order and they don’t provide help in the context of a free exploration of the problem. They also force the student to write formal proofs when high school teachers rarely demand. With this assessment in mind, we established the principal objective for our project which aims at offering an intelligent tutoring system that assists the student in an exploratory approach when solving geometry proofs instead of a formal proof writing approach.

The system we offer is QED-Tutrix which was designed taking into account actual teacher interventions observed in a classroom environment. QED-Tutrix allows the teacher or didactician to construct a number of admissible proofs for a given problem according to the learning goals. The student can then try to solve the problem chosen by the teacher by using the different tools QED-Tutrix puts at his disposal. The student has access to a dynamic geometric figure he can work with in order to make conjectures, as well as to a repertoire of statements to create their proof and an interactive written proof they can use to complete their proof. All the statements he provides are analyzed by the system which then generates feedback through a chat window in order to guide the student during the exploration and solving of the problem. QED-Tutrix’s elaboration was carried out by a multidisciplinary team comprised of experts in didactics of mathematics and in computer science. The system is built in an iterative manner by adopting a design in use approach which consists of a series of many cycles of research and development. Each cycle is ended with an experimentation which aims at validating the work accomplished and at collecting information to be reinvested in the following cycle. A first version of the system (GeoGebraTUTOR) was created to study, among other things, real teacher interventions which inspired the implementation of the second version (QED-Tutrix). This last version is described and analyzed in the following thesis. We do not claim that the present QED-Tutrix version has measurable effects on academic results, since our aim at the moment is to make sure it allows the student to work in a fashion put forward by known didactic principles. Indeed, QED-Tutrix’s conception is rooted mainly in the didactical situation’s theory which represents a didactical context by a student-milieu interaction. However, we use an extended version of this theory in which a tutorial system playing a virtual teaching role may influence this student-milieu interaction. Moreover, we aim at offering a tutoring system that is a true geometrical workspace, meaning that it allows the student to solve problems by engaging in three mathematical processes described in the geometrical workspace model. The didactic theories and conclusions drawn from our observations were implemented in QED-Tutrix. This resulted in a system made of four main software layers. The first of these layers is for the organization of the different proof solutions for a given problem. For each solution, the teacher or didactician registers all the inferences or proof steps which are admissible according to a specific learning context. Each inference includes a justification that is used to produce a result stemming from a group of premises. It is possible to combine the different inferences in order to generate a graph of all the different admissible solutions, since the results of one inference can be recycled as a premise for another. The different pathways of this graph which is the output of the first software layer allow for the account of the different solutions to each implemented problem. In order to offer help that takes into account the cognitive state of the student exploring the problem, it is essential to keep track of the chronology of his or her actions. This memory was implemented in the second layer of the system. It contains information that is dynamic and evolves as problem solving occurs. It also overlays the solution graph which is static. Therefore, we indicate for each of the graphs nodes, the most recent activation time associated with the writing of the statement attached to it. This approach stands out from the way other systems operate and in which chronology is usually not taken into account since a solving sequence is imposed. In order to be able to suggest alternative paths to a student who is stuck in his or her solving process, we chose to treat inferences according to an order of priority. This ranking is carried out by QED-Tutrix’s third layer which uses the data from the first two layers. In order to achieve this ordering, we look for the most advanced solution with the help of an original heuristic, elaborated for this project, which spares the system from running through all the admissible solutions. We then assign the highest priorities to the inferences which are part of the identified solution and that has been worked on recently by the student keeping his or her cognitive state in mind. The ability to suggest other solution option distances us from traditional tutorial systems which offer help only to complete an optimal solution. This list of ranked inferences is used by the fourth and last layer of the system, meaning the layer which generates various feedback. Firstly, QED-Tutrix replies to the writing of each statement with instant feedback in the shape of emojis or short messages. It also allows the programming of particular messages associated with known common mistakes. Lastly, this layer of the system offers help with the next step which is inspired by actual teacher interventions. This last form of help was modelled by a finite state machine that sequentially treats the ranked inferences from the list and produces a series of hints to help the student complete them. Messages must be created for each of the inferences, but mechanisms are implemented in order to reuse messages according to the inferences content, limiting the number of entries. The feedback offered by these messages is similar in form to the feedback offered by other tutorial systems. Approximately 450 inferences were produced and close to 900 messages were created in order to implement the five problems currently available in QED-Tutrix. It’s operating has been verified by an independent expert, which confirmed that the output of messages is true to the identified structure, but the evaluation of the most advanced solution is sometimes problematic. QED-Tutrix was then put through a second trial in a class of 4th year of high school. The students generally found the system to be useful and appreciated the experience. The analysis of the session recordings revealed that the generated messages help some students. Also, we observed different mathematical processes which confirmed QED- Tutrix’s geometrical workspace status. However, the efficiency of QED-Tutrix is limited when helping students with less mathematical abilities since the message structure is built with the average student as a reference. Problems with the identification of the most advanced solution also lead to incoherence between messages and student strategies. In order to enhance the system’s efficiency in helping the student solve problems, we contemplate, among other things, to differentiate tutorial profiles according to students solving profiles. In spite of witnessing shortcomings, QED-Tutrix is an innovative tutorial system. Indeed, in the field of geometry proofs, it is the only automated tutoring system to use emojis and to suggest alternative solution paths. Moreover, the iterative and multidisciplinary approach adopted for its design and development stands out from a traditional approach which aims at reproducing expert reasoning. The next design steps aims at including feedback in the form of related problems and to provide help with building the geometrical figure. Our system could easily be adapted to handle first order logical proofs. An adjustment to process non- formal argumentation could also be considered. Finally, the suggestion of alternative solution paths could be implemented in other tutorial systems.

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