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Projeto ZK -
Informática & Educação |
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Série:
Artigos |
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An algebraic approach to algebra through a
manipulative-computerized puzzle for linear systems
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José Eduardo Ferreira da Silva
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Roberto Ribeiro Baldino |
Introduction
The
literature about mathematics education seems to indicate a discontinuity between
the arithmetic and algebraic domains. Filloy & Rojano [1989] consider “cut
points” separating the two kinds of thinking; Linchevski & Herscovics
[1996] indicate a “cognitive gap” that they endeavor to “cross” and
overcome”; Schmidt&Bednarz [1995:84] refer to a “dichotomy” between
arithmetic and algebra. Kieran [1981] distinguishes between arithmetical and
algebraic uses of the equality sign.
However, the vocabulary used to express the
relation arithmetic/algebra also indicates
continuity. At least in one case the word continuity appears explicitly:
the Argentinean school programs “emphasize continuity with arithmetic” [Panizza,
Sadovski & Sessa, 1996:107]. Continuity is also hinted by signifiers
such as “progress” (from informal to formal level of doing algebra) [Reeuwijk,
1995:1-143],
“transition” (from arithmetic process-oriented thinking to proceptual
algebraic thinking) [Graham & Thomas, 1997:10], “evolution” (from
arithmetic to algebraic language) [Filloy & Rojano, 1989:19], “transition
from arithmetic to algebra” [Bouton-Lewis et al 1997:185]. In spite of
arguments that “algebra cannot be considered as a arithmetical generalization”
[Bodin&Capponi, 1996: 587], expressions
denoting continuity like “generalized arithmetic” are still current [Wong,
1997:285, Graham & Thomas, 1997:9, Sfard & Linchevski, 1994:195,7,
Kutscher & Linchevski, 1997:169].
The
aim of this paper is 1) to argue that the discontinuity between arithmetic and
algebra and, in general, between operational and structural ways
of thinking [Sfard, 1991], is more radical than announced by words such as
“cut”, “gap”, “dichotomy” or “duality”; 2) to argue that
attempts to teach algebra starting from arithmetic [Linchevski & Herscovics,
1996] lead to difficulties, if not to impossibility; 3) to argue in behalf of a
manipulative-computerized puzzle to solve linear systems of two equations
in two unknowns to teach introductory algebra courses.
The dialectical
discontinuity: a difference of essence
According to Sfard
[1991] the literature on epistemology of mathematics teems with allusions to
various dichotomies: abstract/algorithmic, declarative/procedural, process/product,
dialectical/algorithmic, figurative/operative, conceptual/procedural,
instrumental/relational. She proposes another opposition, operational/structural,
that should be considered not as a dichotomy but as a duality. She argues:
“The structural
approach should be regarded as the more advanced stage of concept development.
We have good reasons to expect that in the process of concept formation,
operational conceptions would precede the structural” [10]. “The history
of numbers is “a long chain of transitions from operational to structural
conceptions” [14].
We have some
difficulty in thinking a duality relation in terms of “precedence” and “transition”.
However, if we take the idea of precedence in the sense of ancestry, hence of genesis,
we cannot avoid verging upon the famous Marx’s aphorism: “Human anatomy
contains a key to the anatomy of the ape” [Marx, 1973:105]. Can we infer that
structural thinking contains a key to operational thinking that “precedes”
it? What does this mean?
“Although
it is true that that the categories of structural thinking possess a truth for
all forms of thinking, this is to be taken only with a grain of salt. They can
contain them in a developed or stunted, or caricatured form etc., but always
with an essential difference. The so-called presentation of cognitive
development is founded, as a rule, on the fact that the latest form regards the
previous ones as steps leading up to itself, and since it is only rarely and
only under quite specific conditions able to criticize itself (...) it always
conceives them one-sidedly” (
).
Since
“twentieth-century mathematics seems to be deeply permeated with the
structural outlook” [Sfard, 1991:24] and since the structural way of thinking
is “the more advanced stage of concept development” [ibid. 14], we
infer tat structural thinking, for instance developed algebraic thinking, will
tend to regard all forms of computational thinking preceding it, for instance
arithmetic, as steps leading up to itself. From this point of view, algebraic
thinking contains arithmetic regarded as a step leading up to itself (hence
continuity) but also as a caricatured form essentially different of
itself (hence discontinuity). In order to overcome this one-sided view of
arithmetic as a step leading to itself, the upper form will have to “criticize
itself”. What does this mean? We resort to another author who has thoroughly
discussed the above Marx’s paragraph from Hegel’s
perspective.
“This essential difference
-and
here is the decisive point-
should be considered as crossed by destruction and generation. The movement from
a continuity to a discontinuity perspective corresponds to the movement from a
naive to a critical focus on the upper form” [Fausto, 1987:18].
Therefore, in its
self-criticism structural (algebraic) thinking should recognize itself as being
generated insofar as the previous operational (arithmetical) thinking is being
destroyed as such and, incorporated to it as something essentially different.
The mathematician’s movement back and forth from computational to structural
approaches to a problem is not autonomous because the structural outlook is not
reversible.
According to this argument, no “progress”, “transition”, “evolution”
or “generalization” can lead from arithmetic to algebra. Simultaneous
destruction-genesis (death/birth is just a non-simultaneous approximation) is
the decisive dialectical concept to understand how arithmetic is embodied in
algebra: the tree has to die so that the flower can born from its rotten trunk.
This conclusion will be important for teaching.
Looking
at the continuity/discontinuity ambiguity pointed out at the beginning from the
point of view of the Piagetian theory of equilibration, what our argument
amounts to is that instead of thinking about the relation arithmetic/algebra as
a completive generalization [Piaget and Garcia, 1984, p. 10] we should
think of it as an abstractive reflection [Piaget, 1975, p. 39] which
implies a difference of essence.
Didactical difficulty
of the continuity point of view
The
prevalent teaching strategy in introductory algebra courses is to conduct the
student step by step from procedural to structural thinking following a
supposedly continuous path. Research is then organized to observe, evaluate and
boost progress along this path using teaching experiments, mathematical ability
tests and interviews [Linchevski & Herscovics, 1996]. Instructional
materials are supplied as needed: geometric models [Filloy & Rojano, 1989],
balances [Linchevski & Herscovics, 1996, Aczel, 1998, da Rocha Falcão,
1995] or spreadsheets [Arzarello, Botazzini & Chiappini, 1995].
In
their teaching experiment, Linchevski & Herscovics [1996] adopted a clear
continuity strategy. They accepted the initial use of the inverse operations in
reverse order naturally employed by the students for solving equations with a
single occurrence of the unknown and let them proceed until their method became
“lengthy and tedious”. Then they assumed that the students were “ready to
be exposed to new points of view” [40] and started teaching them a
decomposition-and-cancellation method to solve equations with two occurrences of
the unknown on the same member. Next they introduced equations with the unknown
on both sides. To the students they “pointed out that none of the methods they
knew could effectively solve this type of equation” [53] and introduced the
balance model to help them. However, in spite of all efforts, the students
continued to solve the equations with only one occurrence of the
unknown by performing reverse operations in reverse order. “It should
be pointed out how stable this procedure remained in the seven months since our
initial assessment” [59]. So, the step by step strategy did not lead to the
expected change. Why?
How
much do these students praise the arithmetical undoing strategy that they have
learned for solving one-unknown equations? When they found it “lengthy and
tedious” and claimed that “there must be another way” [62], were they
actually ready to be “exposed” to another point of view? What is the effect
of “pointing out” to a student that his/her method is not “effective”?
What is the nature of the change expected by this strategy?
The
authors do not seem to acknowledge that the students may have a special taste
for wrong answers as long as these are their answers, obtained by their methods and make them big, strong and worth of love for the gaze that
they imagine to be watching them. Will they recognize in the instructors eyes
the referential gaze for which they are willing to play the scene of their lives?
What should be the nature of the demand enjoined by these eyes if the
expected change is to be produced?
Drawing
on previous research the authors attribute certain students’ answers in
ability tests about first degree equations to a “limited view of
algebraic expressions”, “failure to grasp the meaning of operations,
“inability to spontaneously operate with or on the unknown” [39-40,
emphasis added]. Next, inability is identified to an obstacle: “This
inability to spontaneously operate on or with the unknown constituted a
cognitive obstacle” [41]. The teaching experiment was designed to cross this
obstacle.
Thus,
a too restrictive conception of obstacle [Brousseau, 1997:82] considered
to be a mere “failure”, seem to have presided the whole teaching experiment.
The students had to conform to new methods or... fail. Would they be willing to
recognize in the instructors eyes the referential point of their imaginary
identifications? Or did they recognize in these eyes the gaze of the school
system, always ready to classify their answers as “failure” and “inability”?
“The students experience we conjecture is not of a straightforward switch from
arithmetic to algebra, their storying backdrop needs to be extended at the same
time” [Brown & Wilson, 1998:171].
“So
far as obstacles appear as repetitions of failure, they provide a measure
of the persistence of the subject's jouissance organization: the subject
hesitates to abandon what worked well in previous situations and insists on
justifying his statements in terms of the notions that he does not want to give
up. This is why the nature of the demand makes a big difference in the didactic
situation” [Baldino, 1997: 237].
So,
not only Linchevski & Herscovics’ [1996] teaching attempt framed on
continuity assumptions revealed itself difficult but indicated that the
obstacle to pass from arithmetic to algebra is arithmetic itself. Arithmetic
is the knowledge that the students refuse to destroy in order that
structural thinking can be generated. For the students, learning has the
dimensions of death, this is why it is difficult. Insofar as the authors’
teaching method started by recognizing the students’ arithmetical procedures,
they could only reinforce the obstacle and make students more confident
of their arithmetical knowledge, more attached to their past life stories,
instead of developing the courage of reformulating these stories from an
algebraic point of view. Paradoxically, the more we focus on the supposed
“gap” in order to consciously try to bridge it, the wider it becomes. If
there is a “gap”
between arithmetic and algebra, it is not to be crossed, it has to be ignored,
forgotten, dissipated. If we want the students to think algebraically, we
have to start by assigning them typical tasks of the algebraic domain. We have
to seriously take into account that reflective abstraction implies that “every
cognitive system relies on the following one for guiding and the achieving its
regulation” [Piaget, 1975:40]. There is no path to the top of the mountain, we
have to parachute the students up there. This is the objective of the following
puzzle in its three settings, manipulative, computerized and symbolic.
An
algebraic puzzle: the doublequal
The
material consists of a board with two pairs of squares connected by equality
signs, black and white pieces of three shapes, say black and white stones (s, D),
, black and white buttons (l, m),
black and white Montessori cubes (q, n).
To start the activity, four handfuls of randomly chosen pieces are spread in
each of the four squares. This will be called the initial situation.
Examples of initial situations are in figures 1a and 1b.
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Fig. 1: (A) Randomly chosen
initial situation; (B) Simpliefied initial situation; (C) Final situation. |
The
objective is to pass from the initial situation to a final situation through
a series of intermediate situations. In the final situation, square A
must contain only white stones (D),
square C must contain C only white buttons (m)
and squares B and D only Montessori cubes, either white or black (q
or n).
See figure 1c.
The
activity abides by this one single rule: one
passes from a situation to the next by simultaneously adding (or removing) equivalent
handfuls of pieces to (from) squares A and B or to (from) squares C and D.
The
equivalence of handfuls is given by the following rules:
1.
two handfuls of pieces are equivalent if they are made of the same number
o pieces of each shape and color (general equivalence) or if:
2.
they occupy squares A and B or squares C and D in any of the situations
(local equivalence).
3.
A handful of pieces made of equal number of pieces of different colors is
considered equivalent to the empty handful (cancellation).
4.
Dividing or multiplying the number of pieces of two equivalent handfuls
by the same integral number, leads to equivalent handfuls.
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Fig 2: Computerized setting
II, III and IV. |
Besides
the manipulative there are four computerized settings. The first one reproduces
the manipulative setting (fig. 1). The next ones are shown in figure 2. Notice
that in the last setting, predicative signs are turned into operative
signs.
Discussion
of pilot studies
Unlike Dienes and Gategno,
we are not “trying to realize a perfect correspondence between the structure
of the mathematical knowledge involved and the structure of the educational
material” [Szendrei, 1996:420], nor are we assuming that the puzzle hides any
kind of “hidden or frozen mathematics” [Gerdes, 1996:914]. We are assuming
that buttons, stones and cubes are three-dimensional signifiers whose meanings
are given by positions, movements and gestures. The material should be regarded
as an amplifier of language resources, nothing else.
Pilot
studies on the manipulative material were carried out with mathematics teachers,
pedagogy and computer undergraduate students and high school 6th-graders.
The aim of the preliminary studies was to: 1) adjust the written form of the
rules; 2) verify the amount of extra help that should be given for the players
to understand the rules; 3) verify whether the students became engaged in
solving the puzzle.
The single rule of
the doublequal is the pivotal point offered to the students around which
they can start developing a new life story; a story of operational proficiency
instead of one of failure. Further studies may determine what will become of
the arithmetical domain for students who have passed through the doublequal:
will they use algebraic strategies to solve former arithmetic problems? Will
they spontaneously group multiple occurrences of the unknown and operate
simultaneously on both sides?
The
pilot studies
revealed that the nature of the demand makes a big difference. In the first
experiments, we made reference to a balance in order to introduce the
operational rules. Whenever we tried this simplification, most of the groups
started developing trial and error strategies. We tried to assign more difficult
tasks to these groups, forbade them to use pencil and paper, and proposed
situations where the solutions would not be integers. Instead of looking for
another method, they stubbornly went on specializing their method of
systematically investigating solutions with denominators 2, 3, etc. When we
finally showed them the substitution methods, they revolted and complained that
we were “cutting their creativity”. However, even in groups with expert
mathematics teachers, it took quite some time to identify the presence of a
linear system behind the idea of the puzzle.
The
computerized setting is programmed according to the mathematical rules of the
linear systems, not according to the step-by-step manipulation to pass from one
situation to the next, as stipulated in the rules. Therefore, the players may
skip situations by condensing several transformations into one. Mathematically,
this amounts to performing composition of operators in action. Operations
on operators are necessary in order for them to become reified as objects.
“A
person must be quite skillful at performing algorithms in order to attain a good
idea of the ‘objects” involved in these algorithms; on the other hand, to
gain full technical mastery, one must already have these objects, since without
them the process would seem meaningless and thus difficult to perform” [Sfard,
1991:32].
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Some pilot studies were carried out by Rute
Henrique da Silva and Patrícia da Conceição Fantinel, UFRGS, Porto
Alegre, RS, Brazil
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PROJETO
ZK 2007 © Copyright by: José Eduardo Ferreira da Silva |
