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Some comments on "The role of proof in comprehending and teaching elementary linear algebra" by F. Uhlig
DORIER, Jean-Luc, ROBERT, Aline, ROGALSKI, Marc
Abstract
Frank D. Uhlig published in a recent issue of this journal (ESM 50.3) a very interesting article about the question of proof in linear algebra. We have been doing research in the field of mathematics education about the teaching of linear algebra since the 1980s. In this paper, we want to underline the common points in Uhlig's approach and some of our work. We also want to bring a new light of some of his ideas and give perspective for a further didactical development of Uhlig's first experiments.
DORIER, Jean-Luc, ROBERT, Aline, ROGALSKI, Marc. Some comments on "The role of proof in comprehending and teaching elementary linear algebra" by F. Uhlig.
Educational Studies in Mathematics, 2002, vol. 51, no. 3, p. 185-191
DOI : 10.1023/A:1023624601930
Available at:
http://archive-ouverte.unige.ch/unige:16636
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Dorier J.-L., Robert A. & Rogalski M. (2002) Some comments on “ Proof to comprehending and teaching elementary linear algebra” by F. Uhlig. Educational Studies in Mathematics 51(3), 185- 191.
JEAN-LUC DORIER,ALINE ROBERT AND MARC ROGALSKI
SOME COMMENTS ON “ PROOF TO COMPREHENDING AND TEACHING ELEMENTARY LINEAR ALGEBRA” BY F. UHLIG
1ABSTRACT: Frank D. Uhlig published in a recent issue of this journal (ESM 50.3) a very interesting article about the question of proof in linear algebra. We have been doing research in the field of mathematics education about the teaching of linear algebra since the 1980s. In this paper, we want to underline the common points in Uhlig’s approach and some of our work. We also want to bring a new light of some of his ideas and give perspective for a further didactical development of Uhlig’s first experiments.
KEY WORDS: Linear Algebra – Proof – Linear Dependence – Meta-Lever – Reflective Abstraction – Higher Education – History of Mathematics
The authors of this paper are researchers in mathematics education, who have been involved in several research projects on the teaching of linear algebra at the beginning of French university since the late 1980s. A book (Dorier 2000)2 published by Kluwer in the series Mathematics Education Library gives an overview of their work as well as of other authors from various countries.
We met Frank Uhlig during the annual meeting of the International Linear Algebra Society, in 1996. We have exchanged ideas since then and we have been very interested by his textbook (Uhlig 2002a) which gives a very original approach to the teaching of linear algebra.
The paper published recently in Educational Studies in Mathematics (Uhlig 2002b) gives interesting keys to better appreciate the value of Uhlig’s approach. Our aim in this paper is to reflect on this approach with the experience from research in mathematics education and a long practice of teaching linear algebra and analyzing this teaching.
We totally agree with Uhlig on the fact that the “classical” teaching of linear algebra, which he characterises as the “sophisticated Definition-Lemma-Proof-Theorem-Proof-Corollary (DLPTPC) approach”, leads to important difficulties in the students, which we have characterised as the “obstacle of formalism”. Not only students have difficulties in understanding proofs, but they are also overwhelmed by the number of new definitions and feel like they are landing on a new planet (an expression first used by Hillel). Uhlig sees a general problem there, stemming from the lack of practice in proving in mathematics. In the French educational system, students may be better prepared for proof in mathematical teaching. Indeed, they encounter proof at the beginning of secondary school
1 Uhlig, F.: 2002, 'The role of proof in comprehending and teaching elementary linear algebra', ESM 50.3, 335-346.
(age 13-14) in the context of elementary geometry. Using different characterisations of a parallelogram, the Pythagorean and Thales Theorems are an important area in the curriculum of geometry in French secondary schools. Moreover, the teaching of Calculus (which is more the beginning of real analysis) tends to be more formal and proof oriented at the end of secondary school and at the beginning of French science universities. For three years now, the teaching of elementary number theory is also part of the curriculum for students specialising in mathematics at the end of secondary schools. So, it seems that, indeed, French students are more accustomed to proving in mathematics than in the USA. Nevertheless, they encounter similar problems when faced with their first classes in linear algebra. This means that the difficulties with proof and formalism in understanding linear algebra are content-specific. We have worked on this question for a long time.
An extensive work in the history of mathematics (Dorier 1995a and 2000, part I) supported an epistemological reflection that led to the idea of unifying and generalizing concept, first presented in (Robert and Robinet 19963). A unifying and generalyzing concept (or theory) is characterised by the fact that it did not emerge essentially to solve a new type of problems in mathematics (like the derivative or the integral for instance). Its creation and its use by mathematicians were more motivated by the necessity to unify and generalise methods, objects and tools, which had been independently developed in various fields. Therefore the formalism attached to a unifying and generalizing concept is constitutive of its existence and creation. This means that the formalism is not just a pure convenience of language or communication, but is an unavoidable part of the nature of the concept itself. In other words, formalism cannot be avoided when learning linear algebra; furthermore, learning linear algebra includes appreciating the value of formalism. This does not mean that unifying and generalizing concepts have no intuitive background. In fact they have several such backgrounds which result from an abstraction of the common characteristics of various objects from a less formal nature. Therefore, learning linear algebra requires that students look back to different previously learned fields of mathematics and step aside in order to have a reflective attitude towards what they already know. For instance, we share with Uhlig the idea that reflecting upon the solvability of systems of linear equations is an important starting point in order to access to more formal ideas in linear algebra. A detailed description of a teaching project can be found in (Rogalski 1996) (see also Dorier, Robert, Robinet and Rogalski 1994 and 2000). We have especially worked on the concepts of linear dependence and rank. The main results of this work, including an interactive epistemological approach between the historical and the didactical contexts, can be found in (Dorier 1998). We will try to summarise the results now.
It is important that students have acquired a good technical level in solving systems of linear equations before we start to teach linear algebra. Several solving methods can be used. The determinants, which historically dominated the subject from 1750 up to the beginning of the twentieth
2 See also Dorier and Sierpinska (2001) and Dorier (2002).
3 We only give here the most accessible reference but the ideas presented in this paper were first published around 1993.
century, are to be avoided since their technicality tends to mask basic ideas. Algorithmic approaches are to be favoured for their systematic nature. For cultural reason, more than anything else, we use the Gaussian elimination, rather that Row Echelon Form (REF), favoured by Uhlig, yet, we totally agree with him on this point. An historical analysis of several texts showed that the basic idea of linear dependence was not so easy to formalise, even by great mathematicians like Euler, who were first faced with the question of non-determination of n unknowns by any n equations when trying to solve what is known as the Cramer’s paradox (Euler 1750). Indeed, the idea of dependence between equations is not immediately attached to the idea of linear dependence. An equation is dependent on the other ones when it gives no new restriction on the unknowns; this is what Dorier named the concept of “inclusive dependence”. This view on dependence of equations is totally consistent with the fact that the main goal was to solve equations, not to reflect on them as objects. Of course, the inclusive dependence is logically equivalent to the concept of linear dependence, but it is very different from a cognitive point of view. Historically, it prevented mathematicians from using the same concept of dependence on equations and on n-tuples and, therefore, of having a clear idea of duality, which is an essential step in order to totally build the concept of rank. It took nearly a century and a half before Frobenius reached this stage. A didactic study showed that sophomore students, before any teaching of linear algebra, have, in their great majority, very similar pre-conceptions about dependence of equations. Indeed, as mathematicians of Euler’s time, they are uniquely concerned by solving equations. Therefore, an essential stage to prepare students to the formal concept of linear dependence is to make them step aside and reflect on the solving process and their actions when solving a system of linear equations. This can be done by reflecting on the reasons why a pivot is zero at the end of the Gaussian elimination. Here, we come to another similarity with Uhlig’s approach, even if we use different contexts. Indeed, to achieve the goal of having students reflect on their actions, we think that it is essential to ask them questions like:
What happens if…?
Why does it happen?
How does it occur?
What is true here?
This is what we call using the meta lever (Dorier 1995b, Robert & Robinet 19964). Meta means that there is a reflection upon previously acquired knowledge. Lever means that this is something to be used at an appropriate time, in order to reach a higher level of knowledge. It is essential that the activities designed with use of meta lever do not reduce to a discourse from the teacher followed by an activity by the students. The students have to be engaged in a mathematical task, which becomes problematic; the meta lever is then to be used at a stage where the students have started a reflection based on actions. It can be a discourse from the teacher, an historical text with questions, or just an
4 In spite of the dates of these papers, the ideas are originally Robert’s.
unusual question, that is chosen in order to make students step aside and change their point of view.
We have designed several such activities, for instance about the introduction of the axioms of vector space, or the use of a general method in problems of interpolation (Dorier 1995b).
In the teaching project that we experimented and improved several times, the progression of new material is deliberately slow, to allow for the necessary maturation of ideas in students. It also starts from several investigations and “meta lever activities” in various contexts (linear equations, geometry, magic squares, Cartesian and parametric representations of geometrical objects, …). A first stage of formalism in Rn is attained, before formal vector spaces are introduced. However an essential step is the use of a unique letter to name the vectors in Rn and the differentiation between a matrix and the transformation associated via the original basis. This is an important difference with the usual North American curriculum, which allows to avoid the problem related to the notion of coordinate vectors analysed by Hillel and Sierpinska (1994) (see also Dorier 2000, pp. 191-207).
As can be seen, we share many ideas with Uhlig. Yet his approach is totally original.
However, it raises new questions. Indeed, the extensive use made of the REF has the advantage of providing the students with a technical ground that is essential. The danger is that this becomes a
“magic tool” that hides some essential ideas.
Let us examine for instance the third exercise that he proposes in section 4 of his text :
For which vectors u ∈ R4 are the three vectors u, u+e2, and u+e3 ∈ R4 linearly independent, where ei denotes the ith unit vector in R4?
Uhlig proposes a solution using his intuitive definition of linear dependence and challenges the reader to solve this question with the formal definition. Let us take up the challenge. The easiest is to start from the fact that the three vectors are linearly dependent. This is equivalent to the fact that there exist three scalars a, b, c not all equal to zero, such that au + b(u + e2) + c(u + e3) = 0. This is equivalent to (a+b+c)u = - be2 – ce3 (1). If a+b+c = 0 then be2 + ce3 = 0, which implies that b = c = 0 since e2 and e3
are unit vectors. This, together with a+b+c=0 implies that a = 0, and therefore the three scalars should be zero, which contradicts the assumption that the vectors are dependent. Therefore a+b+c≠0 and one can divide (1) by a+b+c. This implies that the three vectors are linearly dependent iff there exists two scalars m and n such that u = me2+ ne3, which means that u belongs to span(e2,e3).
We totally agree that this proof is complicated and requires quite a lot of sophisticated skills in logic, as well as in linear algebra. Yet, it gives a very different light on the result compared to the computational proof proposed by Uhlig. Moreover, it is very doubtful that the computational proof will prepare students for such a formal proof. They will be certain of the result, but they will still lack a certain level of comprehension that it encloses. The question is to know whether we want our students to attain this level of comprehension? If yes, is it possible and how?
Uhlig seems to assume that his method leads “naturally” to a more formal stage. What formal stage does he have in mind? Is it possible to attain the level of the preceding formal proof? He claims
that his approach provides an intuitive basis for linear algebra concepts. We would rather say that it gives a technical basis that allows to attain a large number of conceptual results. The question is to know how students can get free from this technique that governs most of their actions. It would be interesting to have more information on this point.
Uhlig’s paper presents an original teaching proposal, which shares many ideas with our own research5. A didactical challenge would be now to confront this proposal with an experimental didactical analysis in order to probe the effectiveness of its goals. Moreover, it seems important to evaluate the epistemological background vehiculated by the proposal.
REFERENCES
Dorier, J-L. (2002) Teaching Linear Algebra at University, in Li Tatsien (ed.) Proceedings of the International Congress of Mathematician, Beijing 2002, August 20-28, Vol III (Invited Lectures) pp.
875-884.
Dorier, J.-L. & Sierpinska, A. (2001) Research into the teaching and learning of linear algebra, in D.
Holton et al. (eds) The teaching and learning at university level – An ICMI Study, Dordrecht : Kluwer Academic Publisher, pp. 253-271.
Dorier J.-L (ed.) (2000) On the teaching of linear algebra, Dordrecht : Kluwer Academic Publishers, 288 + xxii p.
Dorier J.-L., Robert A., Robinet J. and Rogalski M. (2000) On a research program about the teaching and learning of linear algebra in first year of French science university, International Journal of Mathematical Education in Sciences and Technology 31(1), 27-35.
Dorier J.-L. (1998) The role of formalism in the teaching of the theory of vector spaces, Linear Algebra and its Applications , 275-276, 141-160.
Dorier J.-L. (1995a) A General Outline of the Genesis of Vector Space Theory, Historia Mathematica 22(3), 227-261.
Dorier J.-L. (1995b) Meta Level in the Teaching of Unifying and Generalizing Concepts in Mathematics, Educational Studies in Mathematics 29(2), 175-197.
Dorier, J.-L, Robert, A., Robinet, J. and Rogalski, M. (1994). The teaching of linear algebra in first year of French science university, in the Proceedings of the 18th conference of the international group for the Psychology of Mathematics Education, Lisbon, 4 vol., 4: 137- 144.
Euler, L. (1750) Sur une contradiction apparente dans la doctrine des lignes courbes, Mémoires de l'Académie des Sciences de Berlin 4, 219-223, or in Opera omnia, (3 series - 57 vols.) Lausanne:
Teubner - Orell Füssli - Turicini, 1911-76., 26:33-45.
5 An issue that we have not discussed here concern the relation with geometry. For recent ideas on this matter see (Gueuet-Chartier , in press).
Gueudet-Chartier G.(in press) : Using geometry to teach and learn Linear Algebra, Research in collegiate mathematical education, AMS, Providence, Rhode Island.
Hillel, J. & Sierpinska, A. (1994): On one persistent mistake in linear algebra. Proceedings of the 18th International Conference on the Psychology of Mathematics Education, Lisbon, August 1994., Vol.III 65-72.
Robert, A, and Robinet, J. (1996). Prise en compte du méta en didactique des mathématiques, Recherches en Didactique des Mathématiques 16(2), 145-176.
Rogalski, M. (1996). Teaching linear algebra : role and nature of knowledge in logic and set theory which deal with some linear problems, in L. Puig et A. Guitierrez (eds), Proceedings of the XX°
International Conference for the Psychology of Mathematics Education, 4 vol., Valencia : Universidad, 4: 211-218.
Uhlig, F. (2002b) The role of proof in comprehending and teaching linear algebra, Educational Studies in Mathematics 50.3, 335-346.
Uhlig, F. (2002a) Transform Linear Algebra, Upper Saddle River: Prentice-Hall, 504+xx p.
Jean-Luc DORIER [email protected] Equipe DDM – Laboratoire Leibniz
46, Ave F. Viallet 38 031 Grenoble FRANCE
Aline ROBERT [email protected] Marc ROGALSKI [email protected] Equipe DIDIREM – IREM - Case 7018 Université Paris 7
2, Place Jussieu 75252 Paris Cedex 05 FRANCE
