Introduction
In collaboration with A. Robert,
J.Robinet, and M. Rogalski, we have developed a research program
on the learning and teaching of linear algebra in the first year
of French science universities. This work, which started some
ten years ago, includes the elaboration and evaluation of experimental
teaching based on a substantial historical study and a theoretical
approach within the French context of "didactique des mathmeatiques"(the
French name for a field of research which would more or less correspond
to mathematical education research in the USA). In this paper,
we will try to summarize the main issues of our research, focusing
the concepts of linear dependence and independence.
If a system of linear equations
has as many equations as unknowns (n), dependence between
the equations of the system may be understood in different ways.
If one is not familiar with the notion of linear dependence, but
more concerned with solving the system, the dependence reflects
an indetermination on the solutions of the system. Practically,
it means that, in the process of resolution, one (or more) unknown(s)
will be left undetermined. Therefore n dependent equations
in n unknowns will be characterized by the fact that they
determine unknowns less than n and thus act as if they
were less than n. With regard to the solving of equations,
dependence is therefore revealed by an accident in the solving
these results in the vanishing of at least on equation and the
indetermination of at least one of the unknowns. It is an accident
because n equations usually determine n unknowns
exactly. If the method for solving the system uses linear combinations,
this accident may be connected to the fact that a linear combination
of the equations is zero. If the dependence is "obvious",
one may even see directly that one equation is a linear combination
of the others, although this will not be the central characteristic
of the dependence.
Although it might be difficult
to admit for a modern mathematician so familiar with the vocabulary
and basic notions of linear algebra, the above way of considering
dependence between equations may be found (with the same words)
in a text by Euler dating from 1750. It still prevailed in most
of the texts about linear equations up to the end of the 19th
century (see Dorier 1993, 1995a, 1996b).Euler's text is the first
in which the question of dependence was discussed. The general
idea that n equations determine n unknowns was so strong that
nobody had taken the pain to discuss the exceptional case, until
Euler was confronted with Cramer's paradox and pointed out this
particularity.
He starts by an example with
two equations:
Let us just look at these
two equations
3x-2y = 5 and 4y = 6x -10, one will see immediately that
it is not possible to determine the two unknowns x and
y, as if one eliminates x , then the other unknown
y disappears by itself and one gets an identical equation,
from which it is not possible to determine anything. The reason
for this accident is quite obvious as the second equation can
be changed into 6x -4y = 10, which being simply the first
one doubled, is thus not different.
It is clear -especially by
reading the end of this quotation- that Euler does not intend
to fool his reader, even though he artificially hides the similarity
of the two equations. Yet, it is also clear that it is not the
fact that the two equations are similar that determines the dependence
of the equations, but the fact that something unusual -an accident-
happens in the final step of the solving process. This accident
reveals the dependence of the equations, because, although there
are two of them, these equations do not determine two unknowns.
Mathematically speaking, the two statements are logically connected,
a linear dependence between n equations in n unknowns
is equivalent to the fact that the system will not have a unique
solution; However the two properties correspond to two different
conceptions of dependence. To be able to distinguish these two
conceptions, I will call Euler's conception, inclusive dependence.
I wish to insist on the fact that this conception is natural in
the context in which Euler and all the mathematicians of his time
were working, that is to say with regard to the solving of linear
equations, and not the study of equations as objects on their
own.
Let us now see what Euler
says for three equations:
[…]The
first one, being not different from the third one, does not contribute
at all in the determination of the three unknowns.
But there is also the case
when one of the three equations is contained in the two others.
[…]
So when it is said that to determine three unknowns, it is sufficient
to have three equations, it is necessary to add the restriction
that these three equations are so different that none of them
is already comprised in the others.
It is important to notice
that, for three equations, Euler separates the case when two equations
are equal from the case when the three equations are globally
dependent. This points out the intrinsic difficulty of the concept
of dependence which has to take all the equations in a whole system
into account, and not only the relations in pairs. We will see
that students have real difficulties with this point. On the other
hand, Euler's use of terms such as comprised or contained,
refers to the conception of inclusive dependence as we explained
above. It does not mean that Euler was not aware of the logical
equivalence with linear dependence, but, within his practice with
linear equations, the conception of inclusive dependence is more
consistent and efficient. Yet, there is a difficulty for further
development; indeed, the conception of inclusive dependence is
limited to the context of equations and cannot be applied to other
objects like n-tuples for instance. Therefore inclusive
dependence is context-dependent although linear dependence is
a general concept that applies to any object of a linear structure.
Yet, in his text Euler was able to bring out issues that can be
considered in many aspects as the first consistent ideas on the
concept of rank. Indeed, he discusses, although in a very intuitive
and vague manner, the relation between the size of the set of
solutions and the number of relations of dependence between equations.
We will see now that it took more than a century for the concept
of rank to come to maturity.
1750 is also the year Cramer
published the treatise that introduced the use of determinants
which was to dominate the study of linear equations until the
first quarter of the 20th century. In this context, dependence
was characterized by the vanishing of the determinant. The notion
of linear dependence, now basic in modern linear algebra, did
not appear in its modern form until 1875. Frobenius introduced
it, pointing out the similarity with the same notion for n-tuples.
He was therefore able to consider linear equations and n-tuples
as identical objects with regard to linearity. This simple fact
may not seem very relevant but it happened to be one of the main
steps toward a complete understanding of the concept of rank.
Indeed in the same text, Frobenius was able not only to define
what we would call a basis of solutions but he also associated
a system of equations to such a basis (each n-tuple is
transformed into an equation). Then he showed that any basis of
solutions of this associated system has an associated system with
the same set of solutions as the initial system. This first result
on duality infinite-dimensional vector spaces showed the double
level of invariance connected to rank both for the system and
for the set of solutions. Moreover, Frobenius' approach allowed
a system to be seen as an element of a class of equivalent systems
having the same set of solutions: a fundamental step toward the
representation of sub-spaces by equations.
This brief summary of over
a century of history shows how adopting a formal definition (here
of linear dependence and independence) may be a fundamental step
in the construction of a theory, and is therefore an essential
intrinsic constituent of this theory.
Anyone who has taught a basic
course in linear algebra knows how difficult it may be for a student
to understand the formal definition of linear independence, and
to apply it to various contexts. Moreover, once students have
proven their ability to check whether a set of n-tuples, equations,
polynomials or functions are independent, they still may not be
able to use the concept of linear independence in more formal
contexts.
A. Robert and J. Robinet (1989)
have tested beginners on the following questions:
1. Let U, V and W be three
vectors in IR3.
If any pair of them is non-collinear, are they independent?
2.1. Let U, V and W be
three vectors in IR3,
and f a linear operator in IR3.
If U, V and Ware independent, are f(U), f(V) and f(W) independent
?
2.2. Let U, V and W be
three vectors in IR3,
and f a linear operator in IR3.
If f(U), f(V)and f(W) are independent, are U, V and W independent
?
Beginners generally failed
these questions. In the three cases, they used the formal definition
of linear independence and tried different combinations with the
hypotheses and the conclusions leading to apparently erratic proofs,
that teachers usually reject without further comment.
For instance, to the first
question a majority of students answered "yes" giving
proofs that are examples of the difficulty they have in treating
linear (in)dependence globally. Indeed, many students are inclined
to treat the question of linear (in)dependence by successive steps,
starting with two vectors, and then introducing the others one
by one much like Euler did. We will say that they have a local
approach to a global question. Indeed, in many cases, at least
if it is well controlled, this approach may be correct and actually
quite efficient, yet, it is a source of mistakes in several situations.
The students have built themselves what G. Vergnaud (1990) calls
theoremes-en-acte (i.e. rules of action or theorems that
are valid in some restricted situations but create mistakes when
abusively generalized to more general cases). Here is a non-exhaustive
list of theoremes-en-acte connected with the local approach
of linear (in)dependence, that we have noticed in students' activities:
- if U and V are independent
of W, then U, V and W are globally independent
- if U1 is not a linear combination
of U2,U3,…, Uk , then U1,U2,…, Uk are independent
- if U1, V1 and V2 are independent
and if U2, V1 and V2 are independent, U1,U2, V1 and V2 are
independent.
The historical analysis confirms
the fact that there is a difficulty in treating the concept of
linear (in)dependence as a global property(remember the distinction
made by Euler for three equations). It follows that special care
must be taken in the teaching regarding this point. For instance
the exercise above can be discussed with the students. Moreover,
the teacher, knowing the type of th_or_mes-en-acte, that
students may have built, must help them in understanding their
mistakes and thereby correct them more efficiently.
To questions 2.1 and 2.2 above,
many students answered respectively "yes" and "no",
despite coming close to writing the correct proof for the correct
answers. Here is a reconstructed proof that reflects the difficulties
of the students:
If aU + bV +gW = 0 then
f(aU + bV + gW) = 0
so f being a linear operator:
af(U) + bf(V) +gf(W) = 0 ,
now as U, V and W are independent,
then a =b=g =0,
so f(U), f(V) and f(W)
are independent.
In their initial analysis,
A. Robert and J. Robinet concluded that this type of answer was
revealing a bad use of mathematical implication as characterized
by the confusion between hypothesis and conclusion. This is indeed
a serious difficulty in the use of the formal definition of linear
independence. In the following year, we tested the validity of
this hypothesis with different students. Before the course, we
set up a test to evaluate the students' ability in elementary
logic and particularly in the use of the mathematical implication
(Dorier 1990a and b), and after the course, we gave the same exercise
as above to the students. The results showed that the correlation
was insignificant, in some cases it was even negative. Yet, on
the whole (both tests included many questions), there was quite
a good correlation between the two tests. This shows that if a
certain level of ability in logic is necessary to understand the
formalism of the theory of vector spaces, general knowledge, rather
than specific competence is needed. Furthermore, if some difficulties
in linear algebra are due to formalism, they are specific to linear
algebra and have to be overcome essentially in this context.
On the other hand, some teachers
may argue that, in general, students have many difficulties with
proof and rigor. Several experiments that we have made with students
showed that if they have connected the formal concepts with more
intuitive concepts, then they are in fact able to build very rigorous
proofs. In the case of the preceding exercise for instance, after
the test, if you ask the students to illustrate the result with
an example in geometry, they usually realize very quickly that
there is something wrong. It does not mean that they are able
to correct their wrong statement, but they know it is not correct.
Therefore one main issue in the teaching of linear algebra is
to give our students better ways of connecting the formal objects
of the theory with their previous conceptions, in order to have
a better intuitively based learning. This implies not only giving
examples but also to show how all these examples are connected
and what the role of the formal concepts is with regard to the
mathematical activity involved.
For instance, R. Ousman (1996)
gave a test to students in their final year of the lycee
(just before entering university). Through this test, he wanted
to analyze the conception of students on dependence in the context
of linear equations and in geometry before the teaching of the
theory of vector spaces. He gave several examples of systems of
linear equations and asked the students whether the equations
were independent or not. Of course he noticed mistakes due to
a local approach but the answers showed also that the students
justify their answer through the solving of the system. Therefore
they very rarely give a justification in terms of linear combinations
but most of the time in terms of equations vanishing or unknowns
remaining undetermined. Their concept of (in)dependence is, like
Euler's, that of inclusive dependence and not linear dependence.
Yet, this is not surprising, as these students, like Euler and
the mathematicians of his time, are only concerned with solving
a linear system, therefore inclusive dependence is more natural
and more relevant for them.
However, the formal concept
is the only means to comprehend all the different types of "vectors"
in the same uniform manner, as subject to linear combinations.
In other words, students must be aware of the unifying and generalizing
nature of the formal concept. In our research, we used what we
called the meta lever. Therefore we built teaching situations
leading students to reflect on the nature of the concepts with
explicit reference to their previous knowledge (Dorier 1991,1992,
1995b and 1997 and Dorier et al. 1994a and b). In this approach,
the historical analysis is a source of inspiration as well as
a means of control. Nevertheless, these activities must not only
involve a lecture by the teacher, nor a reconstruction of the
historical development, but take into account the specific constraints
of the teaching context, to reconstruct an evolution of the concepts
with consistent meaning.
For instance, with regard
to linear (in)dependence, French students entering university
normally have a good practice of Gaussian elimination for solving
systems of linear equations. It is therefore possible to begin
teaching linear algebra by making them reflect on this technique
not only as a tool but also as a means to investigate the properties
of the systems of linear equations. This does not conform to the
historical development, as the study of linear equations was historically
mostly held within the theory of determinants. Yet, Gaussian elimination
is a much less technical tool and a better way for showing the
connection between inclusive dependence and linear dependence
as identical equations (in the case when the equations are dependent)
are obtained by successive linear combinations of the initial
equations. Moreover, this is a context in which such question
as "what is the relation between the size of the set of solutions
of a homogeneous system and the number of relations of dependence
between the equations?" can be investigated with the students
as a first intuitive approach for the concept of rank. M. Rogalski
has experimented with teaching sequences illustrating these ideas
(Rogalski 1991, Dorier et al. 1994a and b and Dorier 1992 and
1997).
Finally we give the scheme
of a teaching experiment that we have set up for the final step
in the teaching when introducing the formal theory after having
made as many connections as possible with previous knowledge and
conceptions in order to build better intuitive foundations.
After the definitions of vector
space, sub-space and linear combination, the notion of generator
is defined. A set of generators gathers all the information we
have on the sub-space, it is therefore interesting to reduce it
to the minimum. The question thus is to know when it is possible
to take away one generator, with the remaining vectors still being
generators for the whole sub-space. The students easily find that
the necessary and sufficient condition is that the vector that
can be taken away must be a linear combination of the others.
This provides the definition of linear dependence: "a vector
is linearly dependent on others if and only if it is a linear
combination of them". This definition is very intuitive,
yet it is not completely formal, and it needs to be specified
for sets of one vector. Without difficulty it induces the definition
of a set of independent vectors as a set in which no vector is
a linear combination of the others. To feel the need for a more
formal definition, one just has to reach the application of this
definition. Indeed, students must answer the question: "are
these vectors independent or not?". With the definition above,
they need to check that each vector, one after the other, is a
linear combination of the others. After a few examples, with at
least three vectors, it is easy to explain to the students that
it would be better to have a definition in which all the vectors
play the same role. One is now ready to transform the definition
of linear dependence into: "vectors are linearly independent
if and only if there exists a zero linear combination of them,
whose coefficients are not all zero." The definition of linear
independence being the negation of this, it is therefore a pure
problem of logic to reach the formal definition of linear independence.
A pure problem of logic, but in a precise context, where the concepts
make sense to the students from their intuitive background.
The evaluation of our research
proved that students having followed an experimental teaching
based on this approach are more efficient in the use of the definitions
of linear dependence and independence, even in formal contexts.
Their scores for exercises such as the three questions quoted
above are much higher than the scores of students having followed
a more classical teaching (Dorier 1997).
Moreover, it is quite a discovery
for the student to realize that a formal definition may be more
practical than an "intuitive" one. Indeed, most of them
keep seeing the fact that a vector is a linear combination of
the others as a consequence of the definition of linear dependence.
Therefore they believe that this consequence is the practical
way of proving that vectors are or are not independent, even if
that goes contrary to their use of these definitions.
This example is relevant with
regard to the question about the role of formalism in linear algebra.
Formalism is what students themselves confess to fear most in
the theory of vector spaces. One solution is to avoid formalism
as far as possible, or at least to make it appear as a final stage
gradually. Because, from our historical analysis, we have pointed
out evidence that formalism is essential in this theory, we give
a different answer: formalism must be put forward in relation
to intuitive approaches as the means of understanding the fundamental
role of unification and generalization of the theory. This has
to be an explicit goal of teaching. This is not incompatible with
a gradual approach toward formalism, but it induces a different
way of thinking out the previous stages. Formalism is not only
the final stage in a gradual process in which objects become more
and more general, it must appear as the only means of comprehending
different aspects within the same language. The difficulty here
is to give a functional aspect to formalism while approaching
it more intuitively.
Linear dependence is a formal
notion that unifies different types of dependencies, which interact
with various previous intuitive conceptions. It has been shown
above how in the historical development of linear algebra the
understanding of this fact was essential for the construction
of the concept of rank and partly of duality. In teaching, this
questioning has to be made explicit, if we do not want misunderstandings
to persist. Therefore even at the lowest levels of the theory
the question of formalism has to be raised in interaction with
various contexts where the students have built previous intuitive
conceptions. The construction of a formal approach right from
the beginning is a necessary condition for understanding the profound
nature of the theory of vector spaces. In this sense, formalism
has to be introduced as the answer to a problem that students
are able to understand and to make their own, in relation to their
previous knowledge infields where linear algebra is relevant.
These include at least geometry and linear equations but may also
include polynomials or functions, although in those fields one
may encounter more difficulties.
The theory of vector spaces
is a unifying and generalizing theory, in the sense that, historically,
not only did it allow solving new problems in mathematics, but
it essentially unified tools, methods and results from various
backgrounds in a very general approach. Thus its formalism is
a constituent of its nature. Yet all the problems our students
may solve with this theory could be solved with less sophisticated
tools which they have already learnt (or at least are supposed
to have learnt). Therefore the gains of this unification and generalization
have to be understood by them, if we want them to accept this
formalism and to use the theory correctly. General talk on the
subject will not improve the situation, instead we have to build
teaching sequences in which this idea can be understood in relation
to a mathematical activity so that it becomes a personal reflection
of the students. The students must be able to see the relationship
between their knowledge and intuition in concrete contexts and
the formal language of the theory of vector spaces. In our research
work, we state that one must use this "meta-lever" to
bring students to a personal understanding of the unifying and
generalizing nature of the theory of vector spaces (see Dorier
et al. 1995a and b; Dorier 1992, 1995b and 1997).
Cramer, G. (1750): Introduction
a l'analyse des courbes algebriques, Geneve: Crameret Philibert.
Dorier,J.-L. (1990a): Contribution
a l'eude de l'enseignement a l'universite des premiers concepts
d'algebre lineaire. approcheshistorique et didactique, These
de Doctorat de l'Universite J. Fourier - Grenoble 1.
Dorier, J.-L. (1990b): Analyse
dans le suivi de productions d'etudiants de DEUG A en algebrelineaire,
Cahier DIDIREM no6,
IREM de Paris VII.
Dorier, J.-L. (1991): Surl'enseignement
des concepts elementaires d'algebre lineaire a l'universite, Recherches
en Didactique des Mathematiques11(2/3), 325-364.
Dorier, J.-L. (1992): Illustrer
l'Aspect Unificateur etGeneralisateur de l'Algebre Lineaire,
Cahier DIDIREM no14,IREM
de Paris VII.
Dorier J.-L. (1993):L'emergence
du concept de rang dans l'etude des systemes d'equations lineaires,
Cahiers du seminaire d'histoire desmathematiques (2e serie)
3 (1993), 159-190.
Dorier, J.-L., RobertA., Robinet
J. and Rogalski M. (1994a): L'enseignement de l'algebre lineaire
en DEUG premiere annee, essai d'evaluation d'une ingrnierie longue
et questions, In Vingt ans de Didactique des Mathematiques
en France,Artigue M. et al. (eds), Grenoble
: La Pensee Sauvage,328-342.
Dorier, J.-L, Robert, A.,Robinet,
J. and Rogalski, M. (1994b): 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, Lisbonne, 4 vol., 4:137- 144.
Dorier, J.-L. (1995a): Ageneral
outline of the genesis of vector space theory, HistoriaMathematica
22(3), 227-261.
Dorier, J-L. (1995b): Metalevel
in the teaching of unifying and generalizing concepts in mathematics,Educational
Studies in Mathematics 29(2), 175-197.
Dorier J.-L. (ed.) (1997):L'enseignement
de l'algebre lineaire en question, Grenoble: LaPensee Sauvage.
Euler, L. (1750): Sur une
contradictionapparente dans la doctrine des lignes courbes, Memoires
del'Academie des Sciences de Berlin 4, 219-223, or,
Opera omnia, 3 series - 57 vols., Lausanne: Teubner - Orell
Fussli - Turicini,1911-76, 26:33-45.
Frobenius, G. F. (1875): Uber
das Pfaffsche Problem, Journal fur reine und angewandte Mathematik
82, 230-315.
Frobenius, G. F. (1879): Uber
homogene totale Differentialgleichungen, Journal fur reine
und angewandte Mathematik 86, 1-19.
Ousman, R. (1996): Contribution
a l'enseignement de l'algebre lineaire en premiere annee d'universite,
These de doctoratde l'universite de Rennes I.
Robert, A. and Robinet, J.(1989):
Quelques resultats sur l'apprentissage de l'algebre lineaire
en premiere annee de DEUG, Cahier de Didactique desMath_matiques
53, IREM de Paris VII.
Rogalski, M. (1991): Unenseignement
de l'algebre lineaire en DEUG A premiere annee,Cahier de Didactique
des Mathematiques 53, IREM de Paris VII.
Rogalski, M. (1994):L'enseignement
de l'algebre lineaire en premiere annee de DEUGA, La Gazette
des Mathematiciens 60, 39-62.
Vergnaud, G. (1990): La theorie
des champs conceptuels, Recherchesen Didactiques des Mathmatiques
10(2/3), 133-170.
Historical background
Didactical issues
Conclusion
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