THEMATIC WORKING GROUP 4 ARGUMENTATION AND PROOF
Rudolf vom Hofe
Universität Regensburg
Christine Knipping
Universität Hamburg
Maria Alessandra Mariotti
Università di Pisa
Bettina Pedemonte
CNR Genova
The role and importance assigned to argumentation and proof in the last decade
has led to an enormous variety of approaches in research. Historical and epistemological
issues, related to the nature of mathematical argumentation and proof and
its functions in mathematics, represent one focus of this wide-ranging research.
Focus on mathematical aspects, concerning the didactical transposition of
mathematical proof patterns into classrooms, is another established approach,
which sometimes makes use of empirical research. Most empirical research focuses
on cognitive aspects, concerning students' processes of production of conjectures
and construction of proofs. Other research addresses implications for the
design of curricula, sometimes based on the analysis of students' thinking
in arguing and proving and concerns about didactical transposition. Recent
empirical research is now looking at proof teaching in classroom contexts
and addresses the question of curricular implications based on the results
of these studies. The social-cultural aspects revealed in these studies motivate
a more recent branch of research which is offering new insights. Comparative
studies, trying to come to a better understanding of cultural differences
in student's arguing and in the teaching of proof, especially as they relate
to learning in different cultural contexts, can be seen as part of this new
branch of research.
Papers collected in this section on "Argumentation and Proof" represent
this diversity. Authors raise issues and questions about argumentation and
proof from a wide range of positions and theoretical perspectives. These differences
are reflected in the focus researchers take in their approach, as well in
the methodological choices they make. This leads not only to different perspectives,
but also to different terminology when we are talking about phenomena. Sometimes
differences are not immediately clear, as we use the same words, even though
we assign different meanings to these words. On the other hand, different
categories that we build from empirical research in order to describe students'
processes, understandings and needs are rarely discussed conceptually across
the research field. Conceptual and terminological work is helpful in that
it allows us to progress as a community operating with a wide range of research
approaches. Differences in interests, perspectives and terminology and their
relevance become obvious when looking at the same data together. The experience
we had in our working group, analysing the same video-data from different
research perspectives, turned out to be fruitful and rich. It made the diversity
of our approaches evident and valued this diversity at the same time. In keeping
with this insight, one part of this introduction will give an overview of
the contributions of this group, and another important part will honour the
diversity that became so clear.
The papers deal with issues and questions in four main topics. From different
positions and theoretical perspectives, the authors consider: (A) student's
competence and experiences with proof (including argumentations, concepts
of proof, proving processes etc.) (B) forms and uses of logical and mathematical
reasoning and their relevance for understanding students' reasoning processes,
(C) argumentation and proof in class - comparing different classroom contexts
and (D) the role of mathematical problems and students' epistemological obstacles
in proving.
In the first section (A) four papers discuss cognitive and epistemological
aspects, concerning the processes of production of conjectures and construction
of proofs. Nordstroem reminds us that many students lack the experiences with
proofs that will help them become successful in mathematics in their later
studies. Heinze and Reiss find as well that students at the end of upper secondary
level have deficits in methodological knowledge about proof which are part
of their problems with judging proofs. They describe three aspects of methodological
knowledge about proof: proof scheme, proof structure, and logical chain, that
they consider as important components of proof competence. Their empirical
data support their claim that all three aspects of methodological knowledge
are important when students validate proofs. Küchemann and Hoyles find
in their long-term empirical study that there is a tension for students in
retaining their intuitive sense of the mathematical problem given and producing
deductive explanations fitting social norms. The authors come to the conclusion
that teaching geometrical reasoning and giving students opportunities to progress
in their reasoning requires not only clarifying effective heuristics, but
also finding out ways to revisit and teach proof over time. Here more research
has to be done. Misailidou and Williams point out the important role that
visual elements and cultural contexts can play in argumentations. The authors
conclude from their research that in order to foster students' argumentation
competencies, the teacher requires not only content-specific knowledge, but
a great richness of expertise that is local to the task and context of teaching,
rather than general strategies. This points out the important role of teachers
in ensuring that appropriate cultural tools are made available for students.
Scimone describes the various difficulties students have in trying to prove
a conjecture where it is not clear what approach will be fruitful. He takes
a historical viewpoint in order to investigate mental representations of students
in problem solving.
In the second part (B) logical, historical and epistemological aspects, related
to the nature of mathematical argumentation and proof, and terminological
aspects, based on the differences between explanation, justification, argumentation
and proof in mathematics education are discussed and used to describe students'
activities and products. Durand-Guerrier uses the model-theoretic approach
introduced by Tarski and distinguishes three dimensions: syntax (the linguistic
form), semantic (the reference objects), pragmatic (the context, and the subject's
knowledge in the situation), for a didactic analysis of mathematical reasoning
and proof. She considers these distinctions as important in order to foster
argumentation and the proving processes of students. Durand-Guerrier argues
that the model-theoretic approach calls for continuity between argumentation
and proof, in contrast with the discontinuity seen by researchers working
in a cognitive approach (e.g., Duval). Pedemonte, although taking a cognitive
approach, explicitly takes issue with Duval´s assertion that deductive
reasoning is not like argumentation, that there is a cognitive discontinuity
between proof and argumentation. Pedemonte finds in her research both discontinuities
and continuities in students' argumentations as they come to a conjecture
and in the proofs they produce subsequently. She describes the transition
from abduction to deduction in proving processes and illustrates that a gap
between an abductive argumentation and a deductive proof is possible as well
as a continuity between the two. Reid differentiates between different forms
of abductive reasoning based on Peirce's early and late work. In analysing
students' reasoning within a framework using these distinctions he demonstrates
the value and relevance of these distinctive categories for a better understanding
of students' reasoning processes. Yevdokimov questions the nature of proof
and develops a typology of proofs. In particular he stresses the importance
of intuition in mathematical reasoning processes and puts forward that intuition
and proof are inseparable.
In part (C) different classroom contexts are discussed in which constructions
of proofs and arguments take place. Comparing these different contexts helps
to come to a better understanding of different teaching contexts of proof,
especially as they relate to learning in different social and cultural contexts.
Douek describes early argumentations of young (first grade) students learning
argumentation in the process of writing. She shows how students' involvement
in experimental situations rich in concrete experiences gives them the opportunity
to develop important skills related to mathematical argumentation. The implicit
assumption of Douek's work, that proof and proving processes are strongly
linked to the discourse culture of the class, is made explicit by the research
of Knipping. She identifies various argumentation structures and discourse
cultures in French and German classes. Although the same mathematical topic
is discussed in class, and the proofs seem to be close from a mathematical
point of view, argumentation structures in the classes' discourses differ
substantially, on a local and global level. She argues that these differences
correspond to different functions of proof that are recognized in the class'
culture.
In the last section (D) reality-related thinking, basic ideas and epistemological
obstacles in argumentation and proof are discussed. Here mathematical and
educational aspects, concerning the didactical transposition of mathematical
proof into classroom, and implications for the design of curricula are discussed.
Blum argues that reality-related applications provide interesting contexts
for proof and enable pupils to gain non-formal insights. Vom Hofe offers descriptions
of different argumentations based on conceptual understanding and stresses
the importance of a genetic concept development. The two papers elaborate
the role of reality-related thinking for argumentation and proof. In particular,
the authors are interested in the relation between modelling processes and
proving processes. They explore how basic ideas, individual concept images
and epistemological obstacles are relevant in argumentation and proving processes
and situations. Detailed case studies and long term empirical studies are
envisaged to explore the development of student's concept images and their
role in students' argumentation and proving skills. Analyses of the empirical
data are expected to give theoretical insights into the relation between modelling
and proving processes. The authors expect these studies to be helpful in identifying
levels in reality-related arguing and proving.
Another element in the work of the group, not represented in the papers, was
the video session. As noted above, this experience involved collective analyses
of video data and transcripts from different theoretical perspectives, which
revealed and acknowledged the diversity in our approaches.
The video segment showed three 14 year old students working on the problem
of determining the number of handshakes that occur when n people shake hands.
In the case of 6 people they had found the answer by adding 1+2+3+4+5. Faced
with n = 28 they were attempting to find a formula. The process of hypothesising
formulae and verification of the formulae was the focus of the segment. The
scene can be characterised as an unguided problem solving situation.
The working group participants broke into small groups to discuss the students'
mathematical activity, guided by a set of questions based on theoretical and
methodological elements of the papers presented. Not surprisingly, when the
small groups reported back they had chosen very different foci for their discussions
and raised different questions for further discussion.
The first group discussed the benefit of the three categories syntax, semantic
and pragmatic for analyses of students' argumentations given in the video
data. In particular, the categories were considered for describing differences
in students' argumentations and for a better understanding why students in
the problem solving process did not come to one collective argumentation.
Further, the question arose in how far the use of figures in the problem solving
process could be described in these categories. In particular, the group tried
to find out in how far these categories would be helpful to describe the motivation
for shifts between different figures and the motivation for shifts between
figures and arguments.
The second group discussed the transition from abduction to deduction as described
in the paper by Pedemonte. The process of attempting to analyse the reasoning
of the three students involved in solving a problem was contrasted with the
more structured mathematical activity of the students in Pedemonte's paper,
who were looking for geometry proofs. This allowed the members of the group
to come to a deeper understanding of the use of the category of abductive
reasoning in both Pedemonte's work and in Reid's paper.
The third group discussed the characterisation of proof given by Heinze and
Reiss. The focus of their paper is that three different aspects of methodological
knowledge (considered as an important component of proof competence) may be
distinguished: proof scheme, proof structure, and logical chain. In particular,
these three aspects were used to analyse the students' argumentations given
in the video data. The difficulties met in this analysis allowed the development
of a critical discussion about the terms and definitions referring to proof
in the paper.
The fourth group analysed the video transcripts from the perspective of the
process of mathematical modelling and translating between reality and mathematics.
In particular, the role of 'Grundvorstellungen' (mental models) was considered.
These can be seen as mental links between the real world and mathematics.
From this point of view, reality related proving can be described as the reverse
process of mathematical modelling: While the usual way of modelling takes
a real world situation and transforms it into a mathematical situation, reality
related proving goes the other way, i.e., a mathematical context is transposed
into a corresponding real life situation which forms a new basis for argumentation.
In addition to the different observations reported by the groups, the experience
of examining the same data raised some important methodological questions.
We discussed questions such as:
· To what extent do results of our empirical studies reflect (or not) the teaching and learning experience the students went through?
· How do our research contexts (multiple-choice tests, tests based on open tasks, interviews, classroom observations, etc.) affect students' performances, answers and explanations?
· What are possible implications for our methodology and the applicability of our results in teaching or teacher training?
It is difficult to specify conclusions for a group that spend
considerable time exploring the value of diversity. What is presented here reflects
themes that emerged, rather than points of unanimous agreement.
In our discussions it became clear that there is a need to specify the meaning
of the terminology we are using to describe types of argumentation and proof,
but equally importantly there is a need to understand better their inter-relations
and relevance in the context of learning. Furthermore, the processes by which
types are transformed, from private to public argumentations, in varied contexts,
through teaching, etc. need to be a focus for research.
Our discussions led us to consider several questions for further research:
How can we explain the differences in proving that are observed in different contexts?
How can we deepen our understanding of the relationship between argumentation and proof?
How can we address the methodological and theoretical challenges we face?
What are the implications of this research for school practice and how can the challenges in school practice be addressed by research.
The reader will be confronted with a wide range of positions and
perspectives in the following papers. We hope that this introduction helps both
to establish this diversity and to reveal connections between the papers. The
heterogeneous research foci represented here and the complexity of the outcomes
of this research need further analyses. These analyses cannot be done here,
but questions and issues that emerged from our working group show directions
for future research. Diversity is a richness and challenge at the same time;
therefore, we very much hope that the culture of future conferences continues
to support the diversity we found at CERME 3.