Software to support the learning and teaching of elementary algebra
 

The Lingot project : overview

The Lingot project is a multidisciplinary project involving computer scientists, educational researchers in mathematics, cognitive psychologists, mathematic teachers trainers and mathematics teachers.

1.1 Objective :

    The objective of the Lingot project is to design situations including ITS for learning algebra in secondary schools. On the one hand, the aim is to allow teachers to take into account their students’ cognitive diversity in order to manage the classroom and regulate individual learning. On the other hand the aim is to provide researchers with tools for studying systematically and on the long term, the effects of teaching strategies on learning.

1.2 Key idea

    The basic idea is to start from a multidimensional model of algebraic competencies in secondary schools to study the instruction given in schools. The individual competencies (accurate or not) a student has built are situated in relation with competencies the educational institution is expecting. From these analyses the research team build learning situations in order to introduce new concepts or to make students’ conceptions evolve to the expected skills. The key point is to allow the teacher to manage different learning processes which is a crucial issue, especially for students with big difficulties.

1.3 Metaphor

    The metaphor is to look for the nuggets (in French “les pépites”) from which we can build ingots (in French “les lingots”).

1.4 Main issues

Three main issues are then considered in the Lingot project:

The Pépite project

    This is the diagnosis part of the Lingot project. The basic idea is that students’ answers to a set of well chosen problems show coherences in their reasoning and computing. Understanding these coherences will help teachers to tune the learning situations better.

2.1 Didactical foundations

    The foundation of the project is a research work in mathematical education from Brigitte Gruegeon a resercher from DIDREM (Université Paris 7). From a review of theoretical and empirical work published and from a long term study of student algebraic activity, Brigitte Grugeon built a multidimensional model of algebraic competencies expected from students in French secondary schools (15-16 years old).

A multidimensional model of algebraic competencies

The main dimensions considered are:

A paper and pencil diagnosis tool

In order to situate students in comparison with this model, she proposed a paper and pencil diagnosis tool.

Students’ algebraic competencies and difficulties are analysed with three entries:

    She proposed a test : a set of about twenty exercises with closed questions and open questions.

    Then an analysis grid derived from the model of competencies is used by a human (a teacher or more probably a researcher) to code the students’ answers to each exercises of the test. Then carrying out a global analysis of the coding results, the teacher (or researcher) builds a cognitive profile of the student in algebra. Figure 3, 4 and 5 show the Lauren’s cognitive profile built by a teacher with the software that supports this diagnosis method.

Student’s cognitive profils in algebra

It is a three levels description of the student’s competencies:

This paper and pencil diagnosis tool is too complex to be used by teachers in every day classrooms. So the first research work of the Pépite project was to explore the idea of the automation of the paper and pencil tool. This work [9] intended to demonstrate:

The first Pépite software

    Pépite is a free software available in french in the web. The first version was implemented in Delphi by S. Jean. Then a second version improving the automatic diagnosis and including users’ suggestions for improving usability and effectiveness has been implemented by D. Prévit. Following the diagnostic strategy proposed by Grugeon, Pépite software is made of three modules.

3.1 PepiTest : the students’ software.

    It proposes 22 exercises derived from the paper and pencil tasks and it gathers students’ answers to problems. It  contains

    For educational researchers it is important for the diagnostic that the students can provide explanations by themselves.

    A great attention has been paid to HCI design issues because it is crucial for the diagnostic that the data collected be indicators of students’ competencies and not indicators of interface manipulation problems. In particular, the mathematical educational researchers in the project team were very suspicious at the beginning about the modifications in the mathematical tasks due to the difficulties to enter algebraic expressions with a keyboard and a mouse. Figure 1 shows a student’s answer to an exercise.

3.2 PepiDiag : the analysis module

    It “interprets” students’ answers to every exercise of PepiTest. Like in the paper and pencil tool, it matches every student’s answer with an item of diagnostic. We call that operation coding student’s answers. PepiDiag automatically fills a “diagnostic matrix” of 55 questions and 35 items derived from the multidimensional model of algebraic competencies. In fact it partially fills the matrix. The closed answers and algebraic expressions are analysed. Answers in natural language and mixed answers are very partially analysed by key words analysis. So 75 % of the students’ answers are automatically analysed.

3.3 PepiProf : the teacher’s software

    It establishes the student’s profile from the filled matrix by transversal analysis and presents it to the teacher. It also provides an interface to modify the coding of student’ answers (i.e. to modify the diagnosis matrix without showing it to the teacher) in order to allow the teacher to control the software coding and to correct or complete it when necessary.

    Figure 2 shows this interface. In Lauren’s answer shown in figure 1, PepiDiag was successful when diagnosing: incorrect treatment, correct uses of letters, unsuitable use of brackets leading to correct result, correct translation from natural language to algebraic expression, explanation through algebra. The teacher can modify the coding if necessary. Figure 3, 4 and 5 shows a student’s profile with the three descriptions instantiated with Lauren’s profiles.

Figure 1 : Lauren’s answer to an exercise in PepiTest

Figure 2 : PepiProf teachers' coding interface with the six dimensions of the diagnosis

Figure 3 : Lauren's cognitive profile, quantitative description

Figure 4 : Laurens’ cognitive profile, the qualitative description

Figure 5 : Lauren’s cognitive profile, the diagram of flexibility between representations

Uses of the first Pepite software

We have observed uses of Pépite in different contexts.

Table 1 synthesize these contexts.

Table 1 : different contexts of use for Pepite 1

Current work

Publications (in English)

Contributors

Acknowlegment

This research has been funded by

Teachers trainers from IUFM of Creteil and mathematiques teachers are acknowledged for testing Pépite with their classes.

References

[1] I. Arroyo, A. Schapira, B. P. Woolf, Authoring and sharing word problems with AWE, in J.D. Moore et al. (Eds.), Artificial Intelligence in Education, IOS Press, 2001, 527-529.

[2] I. Arroyo, R. Conejo, E. Guzman, B. P. Woolf, An Adaptive Web-Based Component for Cognitive Ability Estimation, in J.D. Moore et al. (Eds.), Artificial Intelligence in Education, IOS Press, 2001, 456-466.

[3] M. Artigue, T. Assude, B. Grugeon, A. Lenfant, Teaching and Learning Algebra : approaching complexity trough complementary perspectives, In Helen Chick, Kaye Stacey, Jill Vincent, John Vincent (Eds), The future of the Teaching and Learning of Algebra, Proceedings of 12 th ICMI Study Conference, The University of Melbourne, Australia, December 9-14, 2001, 21-32.

[4] J. M. Carroll, G. Chin, M. B. Rosson, D. C. Neale, The development of Cooperation: Five Years of Participatory Design in the Virtual School,, in John M. Carroll (ed.), Human-Computer Interaction in the New Millenium, Addison Wesley, 2001, 373-418.

[5] H. Chick, K. Stacey, J. Vincent, J. Vincent (Eds), The future of the Teaching and Learning of Algebra, Proceedings of 12 th ICMI Study Conference, The University of Melbourne, Australia, December 9-14, 2001.

[6] É. Delozanne, B. Grugeon, P. Jacoboni, "Analyses de l’activité et IHM pour l’éducation ", In Proceedings of IHM’2002, International Conferences Proceedings Series, ACM, 2002, Poitiers, France 25-32

[7] É. Delozanne, P. Jacoboni, S. Jean, B. Grugeon, Assessing Students' Competence in Algebra, Workshop "Learning Algebra with the Computer, a transdisciplinary Workshop”, ITS 2000, Montreal, June 2000.

[8] B. Grugeon, Etude des rapports institutionnels et des rapports personnels des élèves à l’algèbre élémentaire dans la transition entre deux cycles d’enseignement : BEP et Première G, thèse de doctorat, Université Paris VII, décembre 1995.

[9] S. Jean, E. Delozanne, P. Jacoboni, B. Grugeon, A diagnostic based on a qualitative model of competence in elemantary algebra, in S. Lajoie, M. Vivet, AI&ED’99, IOS Press, Amsterdam, , Le Mans (1999) 491-498

[10] C. Kieran, The Learning and Teaching of School Algebra, Handbook of research on Mathematics Teaching and Learning, Douglas Grouws Ed. Macmillan publishing company, 1992.

[11]K. R. Koedinger, J. R. Anderson, Intelligent Tutoring Goes to School in the Big City, IJAIED (8), 30-43, 1997.

[12] S. M. Land, M. J. Hannafin, Student-Centred Learning Environments, in D.H. Jonassen, S. Land (Eds.), Theoritical Foudations of Learning Enviroments, Lawrence, Erlbaum Associates, 2000, 1-19

[13] P. Rabardel Instrument Mediated Activity in Situations, in Blandford A., Vanderdonckt J., Gray Phil (eds.), Joint Proceedings of HCI’2001 and IHM’2001, 17-30

[14] D. Rasseneur, E. Delozanne, P. Jacoboni, B. Grugeon, Learning with virtual agents: Competition and Cooperation in AMICO, Proceedings of ITS’2002, Biarritz (France), 5-8 juin 2002. Cerri S., Gouardéres G., Paraguaçu F. (eds.), Springer-Verlag, p. 61-70.

[15] M.E. Verona, D. Curtis, D. Shaffer, Supporting Teacher development in Enacting the RiverWeb Water Quality Simulator, in J.D. Moore et al. (Eds.), Artificial Intelligence in Education, IOS Press, 2001, 87-98.

[16] http://pepite.univ-lemans.fr