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Baccaglini-Frank, A., Antonini, S., Leung, A., & Mariotti, M. A. (2013). Reasoning by contra-
diction in dynamic geometry. PNA, 7(2), 63-73.
REASONING BY CONTRADICTION IN
DYNAMIC GEOMETRY
Anna Baccaglini-Frank, Samuele Antonini, Allen Leung, and Maria Alessandra
Mariotti
This paper addresses contributions that dynamic geometry systems
(DGSs) may give in reasoning by contradiction in geometry. We present
analyses of three excerpts of students’ work and use the notion of pseudo
object, elaborated from previous research, to show some specificities of
DGS in constructing proof by contradiction. In particular, we support
the claim that a DGS can offer guidance in the solver’s development of
an indirect argument thanks to the potential it offers of both constructing
certain properties robustly, and of helping the solver perceive pseudo
objects.
Keywords: Dynamic geometry; Indirect argument; Proof; Proof by contradiction;
Pseudo object
Razonamiento por contradicción en geometría dinámica
Este artículo aborda las contribuciones que los sistemas de geometría
dinámica (DGSs) pueden dar al razonamiento por contradicción en
geometría. Presentamos un análisis de tres extractos del trabajo de es-
tudiantes y el uso de la noción de pseudo-objeto, elaborado a partir de
investigaciones anteriores, para mostrar algunas especificidades del
DGS en la construcción de pruebas por contradicción. En particular,
afirmamos que un DGS puede orientar en el desarrollo de un argumento
indirecto gracias a las posibilidades que ofrece tanto para construir só-
lidamente algunas propiedades como para ayudar a percibir los pseudo-
objetos.
Términos clave: Argumento indirecto; Geometría dinámica; Prueba; Prueba por
contradicción; Pseudo-objeto
Literature shows that although much research has been conducted on the themes
of proof and argumentation in mathematics education, rarely do the studies focus
on particular proof structures, such as proof by contradiction. The research cen-
A. Baccaglini-Frank, S. Antonini, A. Leung, and M. A. Mariotti
PNA 7(2)
64
tered on proof by contradiction has pointed to various difficulties it presents for
students (see for example, Antonini & Mariotti, 2007, 2008; Leron, 1985; Mar-
iotti & Antonini, 2006; Wu Yu, Lin, & Lee, 2003) especially the difficulties re-
lated to the formulation and interpretation of negation, to the managing of impos-
sible mathematical objects, to the gap between contradiction and the proved
statement.
Some literature takes into consideration contributions that DGSs may give to
students’ production of indirect arguments. Within the very little literature in this
area, there is a study conducted by Leung and Lopez-Real that describes a proof
by contradiction produced by two students working in a DGS. This case study
triggered the development of a framework on theorem acquisition and justifica-
tion in a DGS that the authors used to put together a scheme for “seeing” proof
by contradiction in a DGS (Leung & Lopez-Real, 2002). We will illustrate as-
pects of this framework that we will make use of and develop further in the fol-
lowing section.
With the present paper we intend to contribute to better describe roles that a
DGS can have in reasoning by contradiction. We will further elaborate and make
use of notions from Leung and Lopez-Real’s theoretical framework, in particular
that of pseudo object, to analyze such roles. Moreover, we will provide analyses
of three excerpts of students’ work to show particular construction choices in a
DGS can guide/promote significantly solvers’ development of indirect argu-
ments/reasoning by contradiction.
METHODOLOGY
The data presented was collected during two different studies on the role of a
DGS in processes of conjecture-generation and proof in the context of open prob-
lems in geometry. One study (Leung & Lopez-Real, 2002) was conducted with
Form 4 (Grade 10) students in a band one secondary school in Hong Kong. Hong
Kong’s secondary schools are streamed according to students’ ability. A band
one school is for the most able students. The second study (Baccaglini-Frank,
2010; Baccaglini-Frank & Mariotti, 2010) was conducted with Italian high
school students (16-18 years old) from three different licei scientifici. The partic-
ipants of both studies had been working with dynamic geometry for at least a
year prior to when the studies were carried out. Data was collected in the forms
of audio and video tapes and transcriptions of the introductory lessons, Cabri-
files worked on by the instructor and the students during the classroom activities,
audio and video tapes, screenshots of the students’ explorations, transcriptions of
the task-based interviews, and the students’ work on paper that was produced
during the interviews.
Reasoning by Contradiction in Dynamic Geometry
PNA 7(2)
65
THE NOTION OF PSEUDO OBJECT
When working with paper and pencil and reasoning by contradiction, slight inac-
curacies in the drawing allow the figure to represent properties, which a “proper”
construction would not permit. For example, on paper, with no trouble one can
assume to have drawn a triangle, of which two bisectors intersect at a right angle.
In this case, one may easily be unaware of his/her assumption of contradictory
properties, and it is completely up to him/her to become aware of a contradiction.
In a DGS a similar situation to that described in paper and pencil occurs
when the solver constructs a figure with a robust property (Healy, 2000) while
mentally imposing on it a contradictory property without a robust construction.
By robust construction in a DGS, we mean a construction that can keep the de-
sired properties of a figure invariant under dragging. What happens if, instead,
the solver attempts to construct both properties robustly? However, the solver
may be uncertain whether such a construction is possible or not, or s/he may real-
ize the impossibility when interpreting the DGS’ feedback. Such feedback in-
cludes the making explicit, robustly, of all properties that are derived from the
properties constructed robustly during the construction steps of the figure. This is
the case we find particularly interesting. In this paper we report on ways of rea-
soning that seem to be induced by the feedback provided by the DGS.
As mentioned above, in a DGS no constructible figure can be realized by ro-
bust contradictory properties. So to represent a geometrical object with contradic-
tory properties (at least) one property must not be constructed robustly, but only
conceived (or projected onto the figure) by the solver. Therefore, the solver is
completely in charge of conceiving any contradiction. In this paper, we will pre-
sent three DGS cases of (attempts of) reasoning by contradiction by students that
involve solvers projected non-constructible properties onto geometrical figures.
To analyze these cases, we further elaborated Leung and Lopez-Real’s (2002)
notion of pseudo object in a dynamic geometry environment as follows: A pseu-
do object is a geometrical figure associated to another geometrical figure either
by construction or by projected-perception in such a way that it contains proper-
ties that are contradictory in the Euclidean theory.
We stress that the notion of pseudo object is solver-centered. Thus, the same
dynamic figure can be a pseudo object for one solver, but not for another, de-
pending on whether the solver has projected upon the geometrical objects contra-
dictory properties. In this sense, any dynamic figure defined through a construc-
tion has the potential of becoming a pseudo object for any given solver. In DGS,
this potentiality can be realized through a cognitive process of dragging in which
conceiving a pseudo object is critical in reasoning by contradiction. To facilitate
the analyses of this process, we introduce a notion of proto-pseudo object: A pro-
to-pseudo object is a geometrical object that has the potential of becoming a
pseudo object—such potential is exploited when the solver perceives a property
of such object as being contradictory with respect to another of its properties.
A. Baccaglini-Frank, S. Antonini, A. Leung, and M. A. Mariotti
PNA 7(2)
66
Thus a proto-pseudo object can become an actual pseudo object once (and if)
the solver consciously projects a property upon it that s/he is aware of as contra-
dictory. We will use the notions of pseudo object and proto-pseudo object to
show how DGSs seem to provide cognitive support in (a) offering the potential
of constructing certain properties robustly, (b) generating feedback in the form of
robustness of all properties that are consequences of the constructed ones, and (c)
the possibility of dragging parts of the dynamic figure to explore compatibility
between the robust properties and those the solver has projected upon it. As we
shall see in the following cases, these features seem to guide solvers to conceiv-
ing pseudo objects in processes of reasoning by contradiction.
THE ROLE OF DYNAMIC GEOMETRY IN THREE SOLUTION
PROCESSES
Consider the following task from Leung and Lopez-Real’s (2002) study.
Given a quadrilateral in which the sum of the pairs of opposite angles is
180°, prove that it’s cyclic.
This task was given to the participants in Leung and Lopez-Real’s study. We re-
port on Hilda and Jane’s solution.
Excerpt 1. The Case of Hilda and Jane
Hilda and Jane constructed a quadrilateral ABCD of which the vertices A, B, D
lie on the same circle with center E, while C, D, B lie on a distinct circle with
center F. Then they marked the measures of the angle in A and in C as
a
and
a-°180
, respectively, and proceeded to construct the quadrilateral BEDF (Figure
1).
If the labeling were Euclidean-correct this construction would not be possi-
ble since the circles would coincide, thus ABCD is biased and it leads to the ex-
istence of a non-degenerate quadrilateral BEDF, which Hilda and Jane conceive
as a pseudo object. This can be seen both in Jane and Hilda’s proof, and in an ex-
cerpt of the transcript of a follow-up interview the researchers had with the girls.
Reasoning by Contradiction in Dynamic Geometry
PNA 7(2)
67
Figure 1. Diagram of the proof
Statement in the proof: “From the diagram we see that it has a contradiction as
the sum of the opposite angles of the blue quadrilateral (EBFD) is 360° which is
impossible.” We present an excerpt of the interview.
7
Interviewer:
So before you did that presumably you first of all drew a circle
through three of the points and then you did the same for these three
points.
8
Hilda:
Yes.
9
Interviewer:
So then you marked these two centers. What did you say after that?
10
Hilda:
Because the angle sum of a quadrilateral is 360 and these two [refer-
ring to ∠E and ∠F] already add up to 360 so this is not possible.
Our analysis suggests that the quadrilateral ABCD is initially a proto-pseudo ob-
ject, and it becomes a pseudo object for the solvers once EFBD is perceived as
“not possible” (Line 10). EFBD possesses two contradictory properties which the
solvers perceive simultaneously as (a) a quadrilateral with 4 angles whose sum is
360 degrees (as all convex quadrilaterals), and (b) a quadrilateral in which the
sum of only two angles is 360 degrees —statement in proof and Line 10. This
pseudo object EFBD contains the contradiction necessary for a proof by contra-
diction. The proof is completed by noticing that when EFBD is being dragged to
degenerate (disappear) the two circles, C1 and C2 in Figure 1, coincide. In other
words, the presence of the pseudo object implies the negation of the conclusion
of the statement to prove. By arriving at the proof, the solvers are aware that their
original quadrilateral ABCD also possess contradictory properties; that is, (a) its
four vertices are on different circles and (b) the sum of two opposite angles is
180°. Hence the proto-pseudo object ABCD becomes a pseudo object.
C1
C2
a
180 - a
E
A
D
B
C
F
A. Baccaglini-Frank, S. Antonini, A. Leung, and M. A. Mariotti
PNA 7(2)
68
The support offered by the DGS (in this case, Cabri) consisted in guiding the
solvers’ transition of ABCD from its status of proto-pseudo object to a genuine
pseudo object. In this case the transition occurred through the perception of a
pseudo object (EFBD) associated with ABCD. Finally, it seems quite remarkable
that Hilda and Jane decided to construct two distinct circles (Line 7) through two
sets of thee points of the original quadrilateral, showing the case in which ABCD
is cyclical in terms of coincidence of the two circles via dragging in DGS. In par-
ticular, they transform the problem, which does not contain any impossibility in
its original statement, into a problem of constructing something impossible. This
is the type of problem that Stefano and Giulio, and Tommaso and Simone were
given in Baccaglini-Frank and Mariotti (2010).
Excerpt 2. The Case of Stefano and Giulio
Similarly to what we described above, in the following cases, awareness of the
presence of a pseudo object determines the impossibility of a construction, thus
validating a statement such as “this construction is impossible.” In the following
two cases, we encounter similar solution processes to those described by Mariotti
and Antonini (2009). The solvers conceive a (or various) “new” object(s) that is
(are) used to “show” a contradiction. However, having the DGS at their disposal,
the solvers make use of it in significant ways that we will describe. The task was
as follows.
Answer the following question: Is it possible to construct a triangle with
two perpendicular angle bisectors? If so, provide steps for a construc-
tion. If not, explain why.
Giulio and Stefano immediately advance the hypothesis that the construction is
not possible, but quickly transit to constructing a figure in Cabri to try to explain
their intuition.
1
Stefano:
No, the only way is to have 90 degree angles… [unclear which these
may be, as Ste was not constructing the figure nor looking at the screen.]
2
Giulio:
That for a triangle is a bit difficult! [giggling]… So… they have to be.
3
Stefano:
If triangles have 4 angles…
4
Giulio:
No, I was about to say something silly…
Immediately Giulio starts constructing two perpendicular lines and refers to them
as the bisectors of the triangle (Figure 2).
Reasoning by Contradiction in Dynamic Geometry
PNA 7(2)
69
Figure 2. Bisectors of a triangle drawn by Giulio
5
Stefano:
Yes, these are bisectors, right?
6
Interviewer:
Yes.
7
Giulio:
So, now we need to get… bisectors… how can we have an angle
from the bisector?
…
8
Giulio:
the symmetric image?… It's enough to do the symmetric of this one.
So the solvers have constructed a figure with two robust angle bisec-
tors that intersect perpendicularly [Figure 3].
9
Stefano:
The only thing is that this [Figure 3] isn’t a triangle!
10
Giulio:
Therefore now we could do like this here [drawing the lines through
the symmetric points and the two drawn vertices of the triangle].
11
Interviewer:
Yes.
12
Stefano:
It’s that something atrocious comes out!
13
Giulio:
And here… theoretically the point of intersection should be … .the
points… very small detail… hmmm
…
14
Stefano:
No, we proved that this is equal to this [pointing to angles], and this
is equal to this because they are bisectors… these two are equal so
these are parallel.
…
15
Stefano:
These two [referring to the two parallel lines] have a hole so it is not
a triangle.
A. Baccaglini-Frank, S. Antonini, A. Leung, and M. A. Mariotti
PNA 7(2)
70
Figure 3. Solvers’ figure
We interpret this episode as follows. The solvers use the DGS to construct two
perpendicular lines and the symmetric image to construct the property of them
being bisectors. Once the construction is completed, they discern properties that
are consequences of these two robustly constructed properties, and consequently
notice that “the figure must have two adjacent angles with two parallel sides”
(Lines 9-15). As soon as they recognize “a hole” in the triangle-to-be (Line 15)
the pseudo object exists: that is, a figure that has a “base” and two parallel sides,
and that has the property “triangle” projected onto it. The appearance of this
pseudo object reveals to the solvers the impossibility of accomplishing a correct
robust construction and thus allows them to solve the problem.
Excerpt 3. The Case of Tommaso and Simone
Tommaso and Simone proceeded by constructing a proper triangle and two of its
bisectors. Then they marked an angle formed by the bisectors and start dragging
one vertex of the triangle in the attempt to get the measure to say “90°” (Figure
4).
Figure 4. Tommaso and Simone’ construction
1
Simone:
It’s endless!
2
Simone:
91.2 [reading the measure of the angle between the bisectors.]
91.8°
Reasoning by Contradiction in Dynamic Geometry
PNA 7(2)
71
3
Simone:
Well, yes, in any case it will come out!
4
Tommaso:
How do you know? maybe…
5
Simone:
Well, of course! It's not like it can go on forever! At the end it will
make it to be 90!
…
6
Tommaso:
I don’t think it is possible. The solvers seem unsure about the possi-
bility of constructing such a triangle, but now seems to think it is not
possible. They start reasoning differently.
7
Simone:
Eh, it is impossible to construct it! Because… I only have these two
bisectors.
8
Interviewer:
Hmm.
9
Simone:
How can I…
10
Simone:
Since… the perpendicular bisectors… it means here there is a rhom-
bus… or a square
11
Simone:
If like here… [he draws a segment]… Here… there were… a rhom-
bus… this would be 90, 90… or a square. And therefore… then…
Eh, I mean, if this is like a rhombus, no? here there is 90 and here
there is 90, and these are the bisectors.
12
Simone:
And then… and then I bring these up [pointing to the vertical-
looking sides of the triangle] and I find their point of… of intersec-
tion.
With respect to Stefano and Giulio, here the solvers choose a different pair of
properties to construct robustly: (a) the triangle, and (b) the bisectors. They do
not construct but (we assume) project the property perpendicular bisectors onto
the figure. Nevertheless, they are not able to conceive a contradiction in it or in
the new object they conceive —the rhombus. Hence this rhombus is a proto-
pseudo object, and the solvers do not seem to make the transition to conceiving it
as a pseudo object. It is significant that the solvers say “it has 4 right angles”
(Line 11) pointing to the figure that even has a marked measure of one of the an-
gles, and the measure says “91°!”. No contradiction among the properties of the
“rhombus” is perceived and the solvers are not able to reach a conclusion. We
advance the hypothesis that if they had been able to conceive the rhombus as a
pseudo object, they would have been able to solve the problem geometrically.
Instead, they resort to an algebraic explanation that they cannot coordinate with
what they see on the screen. They seem to keep on believing that the triangle al-
ways has a third vertex “somewhere up high”.
When we compare Excerpt 2 and Excerpt 3, a determining difference, from a
cognitive point of view, is that, in one case, the solvers conceive a pseudo object,
and, in the other, they do not. This can be explained by the solvers’ different
A. Baccaglini-Frank, S. Antonini, A. Leung, and M. A. Mariotti
PNA 7(2)
72
choice of the properties to construct robustly. The choice determines the type of
guidance that the DGS can provide to reasoning by contradiction. In Excerpt 3,
starting from the triangle and trying to obtain perpendicularity of the bisectors
through dragging allows the solvers to use the DGS (only) as a sort of “amplified
paper-and-pencil drawing” in that it allowed the exploration of many cases with-
out having to redraw the figure. On the other hand, in Excerpt 2 the DGS gener-
ates two robust parallel lines as a consequence of the constructed properties, thus
“guiding” the solvers in perceiving “a hole” in the triangle-to-be and thus such
object as a pseudo object.
CONCLUSION
We have introduced the concept of pseudo object and illustrated how it can con-
tribute significantly to reasoning by contradiction in Euclidean geometry. In par-
ticular, in a DGS environment, construction and dragging strategies leading to
degeneration of a pseudo object could guide to ascertainment of a geometric the-
orem or property. The hybrid nature of a pseudo object seems to be conducive to
formulating an exchange of meaning between dynamic visual reasoning in DGS
and theoretical reasoning in the Euclidean axiomatic system (in this case proof
by contradiction). We have shown that there can be a strong subjective element
in the process of producing a geometrical proof (or a convincing argument) via
the solver’s conscious choices of construction and dragging in a DGS. We hope
this paper will open up a window of discussion to view proof in dynamic geome-
try environment in ways that can enrich the formal deductive reasoning ap-
proach.
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This paper was originally published as Baccaglini-Frank, A., Antonini, S.,
Leung, A., & Mariotti, M. A. (2011). Reasoning by contradiction in dynamic ge-
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Anna Baccaglini-Frank
Università degli Studi di Siena
anna.baccaglini@unimore.it
Samuele Antonini
Università degli Studi di Pavia
samuele.antonini@unipv.it
Allen Leung
Hong Kong Baptist University
aylleung@hkbu.edu.hk
Maria Alessandra Mariotti
Università degli Studi di Siena
marialessandra.mariotti@gmail.com
Recibido: enero de 2012. Aceptado: abril de 2012
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