POLYMINOS: A WAY TO TEACH
THE MATHEMATICAL CONCEPT OF IMPLICATION
Virginie Deloustal-Jorrand
Laboratoire
Leibniz-Université Joseph Fourier-Grenoble-France
virginie.deloustal@imag.fr
Abstract:
In this paper, we present a didactic analysis of the implication under three
points of view: sets, formal logic, deductive reasoning. For this study, our
hypothesis is that most of the difficulties on implication are due to the lack
of links in education between those three points of view. Then, we show, thanks
to the analysis of a problem from our experimentations, how polyminos’ paving
can imply the sets point of view.
Introduction
The existence of the implication as an object
of our everyday life leads to confuse it with the mathematical object. As a
result, the implication seems to be a clear object. Yet, students have
difficulties related to this concept until the end of university, especially
with regard to necessary and sufficient conditions.
The study we present here is a part of our
thesis on the mathematical concept of implication. Our theorical framework is
placed in the theory of french didactics, in particular, we use the tools of
Vergnaud's conceptuals fields theory and those of Brousseau's didactical
situations theory. Our study is based on the work of V. Durand-Guerrier
[Durand-Guerrier, 1999] on the one hand and of J. Rolland [Rolland, 1998] on
the other hand. V. Durand-Guerrier shows, in particular, the importance of the
contingent statements for the comprehension of the implication. J. Rolland is
interested in the distinction between sufficient and necessary condition. Our
researches are also linked to those of J. and M. Rogalski. They try to define
types of structuring of the use of logic when evaluating the truth of an
implication with a false premise.
We present three points of view on the
implication, then the analysis of a problem of paving that allows a work on
implication, we conclude with some results.
Three points of view on the
implication
The mathematical implication seems to be a
model of the natural logic implication. Like any model, this mathematical
concept is faithful from certain angles to that of natural logic but not from
others. This distance between the mathematical concept and the object of our
everyday life leads to obstacles in the use of the mathematical concept. An
epistemological analysis [Deloustal, 2000] enabled us to distinguish three
points of view on the implication: formal logic point of view, deductive
reasoning point of view, sets point of view.
Of course, these three points of view are
linked and their intersections are not empty. We will not develop here the
formal logic point of view (for example truth tables or formal writing of the
implication).
We call "deductive reasoning" the
structure of an inference step: "A is true; A implies B is true; Thus B is
true". Its ternary structure includes a premise "A is true", the
reference to an established knowledge "A ÞB" (theorem, property or
definition...) and a conclusion "B is true" [Duval, 1993, p 44]. In
the deductive reasoning, the implication is used only as a tool. However, in
French secondary education, where this point of view is the only one, it often
acts as a definition for the implication.
Generally speaking, having a sets point of view, means
to consider that properties define sets of objects: to each property
corresponds a set, the set of the objects which satisfy this property.
The sets point of view on the implication can
then be expressed as follows: in the set E, if A (resp. B) is the
set of objects satisfying the property A (resp. B), then, the implication A Þ B is satisfied by all the objects of the set E
excluded those which are in A without being in B,
i.e. by all the objects located in the shaded area below.
Research hypothesis
The experiments carried out for three years,
within the framework of our research, have shown that the implication was not a
clear object even for beginner teachers. Moreover, they showed that, contrary
to a widespread idea, a logic lecture is not enough to get rid of these
mistakes and difficulties.
Following these comments, we formulate the
research hypothesis: it is necessary to know and establish links between these
three points of view on the implication for a good apprehension and a correct
use of it.
In the following paragraph we show that a
problem of paving, using only easy properties, may question the reasoning in a
non obvious way and allow a work on the implication under the sets point of
view.
The Experimentation
The problems we present here result from an
experimentation carried out in 2001 with beginner teachers of mathematics. This
experimentation includes two sessions on the proof and, in particular, on the
implication. The first one contained two problems (one in
geometry, the other one on pavings) the second one proposed a work on written
proofs. For each
meeting, a work by small groups was following an individual work to allow
questionings and discussions. We present here two problems concerning pavings
of polyminos.
first problem
Presentation
Polymino:
collection of square boxes connected by ridges (in the plane)
Domino :
two-boxes polymino 
Size of a polymino: number
of boxes
Even: a
polymino is even if its size is even
Paving a polymino with
dominos: to entirely cover, without overlap, a polymino with dominos.
In this case, we say that a polymino is pavable with
dominos.
pavable 
non pavable 
Balanced: If one
colours the polymino like a chessboard, we say that it is balanced
when the number of white boxes is equal to the number
of black boxes.
Question
Let P1, P2 and P3 be the properties of a polymino:
(P1)
: pavable (by dominos)
(P2)
: balanced
(P3)
: even
Which are the relations between these properties?
Write a proof satisfying all
the group.
Mathematical Analysis
The
implication P1ÞP3 is true because a pavable
polymino can be paved with dominos, and dominos include two boxes. So the
number of boxes included in a pavable polymino, that is to say its size, is an
even number.
The
implication P3ÞP1 is wrong, see counter-example 1:
The
implication P1ÞP2 is true. Indeed, let us suppose
the polymino coloured as a chessboard, then a domino cover exactly one white
box and one black box. Moreover, a pavable polymino can be covered by dominos.
Conséquently, a pavable polymino includes the same number of white or black
boxes, i.e. it is balanced.
The
implication P2ÞP1 is wrong, see counter-example 2:
The
implication P2ÞP3 is right. Since the polymino is
balanced it contains the same number of black boxes and white boxes. Let n be
this number. Then, the polymino’s size is 2n and the polymino is even.
The
implication P3ÞP2 is
wrong, as shows the counter-example 1:
second problem
Presentation
1 – We examine the implication P1ÞP2.
Here is a proof suggested by some and refused by
others:
A domino covers a
white box and a black one. If a polymino is
pavable, it is covered
by k dominos and these k dominos cover
k white boxes and k
black boxes, therefore it is balanced.
Give your opinion on this proof.
Would you have a different answer as a mathematician
and as a teacher?
2- We examine the implication P2ÞP1.
Here is a suggested proof:
In a 2n-boxes balanced
polymino, there are two neighbour boxes: the colours of these two boxes are
different and the two boxes form a domino. Let us remove it. We obtain, then, a
2(n-1)-boxes balanced polymino. When repeating this process, we obtain a
two-boxes balanced polymino. This shows that the polymino was pavable.
Give your opinion on this proof. Justify.
Mathematical Analysis
The
implication P1ÞP2 (pavable Þ balanced) is true and the proposed
proof is valid.
The
implication P2ÞP1 (balanced Þ pavable) is false as shows the counter-example
2. Moreover, the proof is false since in a polymino boxes must be connected
by a ridge. However, when one removes a domino taken randomly, this connexity
is not assured and one obtains not inevitably a
polymino. In addition, it is not always possible to choose the polymino that
one removes so that the connexity is assured as shows the counter-example 2. It
is balanced but whatever the removed domino, one cannot obtain a polymino.
Didactical Analysis
General choices
Mathematical framework for the problem
First of
all, we choose, for our experimentations, very easily accessible mathematical
concepts. Indeed, our hypothesis is that to see a work on the reasoning and to
be able to distinguish difficulties due to the concept of implication, there
must not be difficulties linked to a mathematical concept. Polyminos’
properties are easy to understand and to use. Everyone, even pupils at primary
school, as we have already tested, can start a reasoning on polyminos.
Practical organization of the situation
The
situation contains two different sessions. The
second session allows a work on written proofs that caused difficulties the
week before.
Practical organization of the session
Our
hypothesis is that a research in groups is necessary for our problems. That
allows a confrontation between the various points of view. Furthermore, it
stimulates discussions.
Choices for the first problem on polyminos
The tool of coloration
We chose to
give the students the property “balanced”. Previous experimentations showed
that the tool of coloration appears after a very long time. As we had not
enough time, we decided to remove the difficulties linked to the modelisation
in order to focus our problem on the concept of implication.
The work on implication under the sets point of view
At first sight, one does not know which are the
true implications. To answer the problem, one must express conjectures. This is
different from a problem of geometry where, most of the time, the result is
known before being proved.
Moreover, these implications are true only in
one direction whereas in a geometry problem most of implications are
equivalences. This compels to distinguish sufficient and necessary conditions.
Our activity on polyminos is focused on the
search of counter-examples to refute false implications. We will point out the
counter-examples in the transcripts as characteristic of the use of the sets
point of view.
Furthermore, the properties of polyminos are
not known as are the properties of parallelograms for example. Thus, students
can not try to build a proof by writing properties they know, one after
another. This prevents the students from using the deductive point of view
while it favours the sets one.
Choices for the second problem on polyminos
Second proof
We chose a
proof that could seem to be valid whereas the students had proved the
implication was false. We will distinguish three types of answers :
-
this
proof is false since we showed that the implication is false.
-
this
proof is false since the connexity is not assured.
-
this
proof is valid but the result is false.
Teacher’s opinion
We chose to ask them to answer as a teacher.
That allows a discussion on the status of the proof. How to recognize that a
speech is a proof? Which are the criteria which enable them to validate a
proof?
conclusion
The analysis of the students' answers is still
in progress. However, we can already say that the first problem fulfiled its
role, as for the work on the implication since all the groups have worked on
necessary and sufficient conditions. Most of the groups studied another
implication: Pavable È Non Pavable Þ Non pavable? A group discussed on what is a
false implication. Some implications, for example “even Þ pavable”, are false because there is a
counter-example, that is to say they are sometimes false, whereas some
implications are always false, for example “odd Þ pavable”.
Davy: A
counter-example, that says that in some cases it is false but it isn’t always
false whereas here my proof, it is always false!
The sets point of view appears many times.
Since they do not know any link between the properties they must work with the
different classes of polyminos. For example, when they search a counter-example
to “balanced Þ pavable”, they begin with a
balanced polymino and they grow it while keeping the number of white boxes
equal to the number of black boxes. That it to say they search a non pavable
polymino inside the class of balanced polyminos.
These results are to be placed among others.
Indeed, this problem forms part of a six hour experimentation on implication
and reasoning. Moreover, this experimentation takes sense when one knows that
it was preceded by two others, carried out in 1999 and 2000.This problem is,
thus, to consider as part of a broader context.
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