Kirsti Nordström
Department of mathematics
The aim of my study is to explore mathematicians’ pedagogical views,
values and intentions concerning proof. In my earlier study (Nordström, 2003) I found that the university entrants’
attitudes were positive towards learning of proof. In this paper, I will
discuss the problems of access and transparency as they are defined by Lave and
Wenger (1991) and the ways in which the students are drawn to share the
mathematicians’ views on proof.
I have started the study by interviewing mathematicians and, in this
paper, I will analyse five of the interviews as a pilot study in order to find
out methodological and theoretical implications, which might influence and even
change the original research questions, the conceptual framework and the
methodology of the global study.
Lave and Wenger’s theory of legitimate
peripheral participation describes at least partially the conditions of the
teaching- and learning environment in the context of university mathematics and
thus, offers an appropriate frame for my study. In my paper I deal with some
theoretical problems in applying the theory of situated cognition in the
context of university mathematics and how it can be recontextualised
in order to better frame my area of study.
I consider proof, in a very similar way as Jill Adler (1999) in her
study of multilingual mathematics classrooms considers talk, as a resource for mathematical learning. Then
it needs to be both seen (be visible) and to be used and seen through (be
invisible) to provide access to mathematical learning. I argue that Lave and
Wenger’s concept of transparency even captures this dual function of proof as a learning resource in the
practice of university mathematics. Access
to artefacts in the community both through their use and through understanding
their significance is crucial.
But not are only the mathematicians and the students to be seen in the
social, cultural and historical context but the ideas of proof itself as well.
That is the reason for why I have, in my global study, even investigated the
history of proof and the teaching and learning of proof in parallel to the
empirical studies. From these studies and some didactical studies, I have
created a conceptual framework in order to describe and analyse features and
patterns in mathematicians’ utterances and consider them in their social,
historical and cultural context. The aim of the pilot study is to test and
improve this frame. In this summary, I will give a brief description about the
conceptual framework.
Conviction/Explanation
Many mathematicians have emphasised the explanatory aspects of proof
(e.g. Hersh, 1993;
I argue that even conviction can be of importance for the
pupils/students depending on what we mean by proof. If we think that proof
exists in all mathematical activity where we justify every step, then the
conviction is essential. In
Deductive / inductive approaches
The axiomatic deductive approach that is still usual in mathematics
teaching and textbooks, especially in a higher level, has been criticised since
the 70’s by Freudenthal, Hersh,
Human, Kline, Fischbein, Lakatos,
Van Hiele and others (de Villiers,
1986). Lakatos (1976) called this style deductivist.
“…, deductivist style tears the proof-generated definitions off their
‘proof-ancestors”, present them out of the blue, in an artificial and
authoritarian way. It hides the global counterexamples which led to their
discovery.” (p. 144) Lakatos advocates heuristic style that, on the contrary,
highlights these factors and emphasises the problem-situation, the logic which
gave birth to the new concept.
De Villiers (1986) suggests a variability of
approaches. “The axiomatic deductive approach may, in terms of time-saving,
perhaps become more and more essential as students progress into higher
mathematics, provided they had already acquired a sound understanding of
axiomatic structures by their own participation in its construction (or as
re-enacted by the teacher).” (p. 23)
Lerman (2000) describes an approach, which
runs contrary to the tendency of working inductively from, for instance
everyday examples to general principles. Vygotsky
called it the ascent from the abstract to
the concrete. One development of Vygotsky’s
perspective has, according to Lerman, been towards
the teaching of general principles before the applications (p. 65). Lerman gives even account of the results of some studies
that support the argument for a ‘theoretical learning approach’.
Discussions on the different teaching styles are in no way a recent
phenomenon but have been more or less common during a several hundred years (Nykänen, 1945).
Aspects of formality, the level of rigor and
the language
These aspects are overlapping with the former in a sense that working in
an investigative, inductive level is often associated with intuitive and
informal ways of reasoning. Formal/ informal representations have even
connections with the notion of abstraction. Formality and rigour in mathematics
are relative and context dependent concepts (Hersh,
1993). In my paper I will discuss for example the theories of Skemp (1987) and Fischbein (1987)
in order to relate these theories to my data.
The pilot study is based upon data collected by tape-recording and/or
taking detailed notes of five face-to-face semi-structured interviews with
mathematicians teaching undergraduate courses in
After transcribing the interviews I first identified the issues touching
the questions above and organised the data according to them. After that I
analysed the transcripts from my theoretical point of view. Finally, I used a
more open approach without any deductive a
priori theory. The findings in the second part of the analysis would
hopefully give some interesting implications in form of new categories to the
initial theory. The qualitative analysis was made by NVivo
software.
Very clearly, the mathematicians consider
proof as an essential part of mathematics. They talk about proof as a means to
come to grips with the essence of mathematics. “…to understand the essence of mathematics.”, “Gives
insight to the essence of mathematics.” “Proof is a fundamental idea of mathematics…”
Conviction/ explanation
One of the five mathematicians stressed the function of conviction: “In mathematics you cannot be convinced and
continue if you do not have a proof.” Another explicitly denied the
importance of this function: “Not for the
conviction, the students already are convinced…”
Explanation was present in mathematicians’
utterances in different ways which I sum up to mathematical constructions,
structures, relations between different concepts like connections and
hierarchies.
Inductive/
deductive approaches
An interesting result of the pilot study is
that two opposite positions was taken by the mathematicians concerning the
advantages of Euclidian geometry in the teaching and learning of proof. Responses to possible use of
students’ own investigations and conjectures were positive. However, some of
the mathematicians said it was quite difficult to change the contents and
methods of the courses. “They can be time-consuming
but you come to grips what mathematics actually is.”
Aspects of rigour and language
No one stressed the aspects of rigour in the teaching and
learning of mathematics. However, two of them talked about the importance of
the language and the using of symbols like the sign of equivalence. “The students have great difficulties,
especially in precise formulation and language.” One of them considered the learning of proof
as the learning of a special kind of language. “It is also a language you have to learn, maybe by first memorising
some proofs.” The same person also
stressed the importance of understanding the role of definitions when proving
statements. One of the mathematicians
talked about intuition “The aim of my
current course is to give some intuitive understanding and computational skills
in problem solving, so I have hardly proved any statements.”
Some
further aspects
A common aspect when discussing the meaning of
learning proof was that it can help problem solving in other contexts, both
mathematical and outside mathematics.
Indeed, de Villiers (1990) mentioned an aesthetic function of proof and this
very aspect was clearly present in many of the utterances in my data. “Proofs can be beautiful.” “…like a piece of poetry that can be as attractive
as the entire theorem.” One of the mathematicians talked about proof as a
mean of communication. Even
communication is one of the roles de Villiers (1990)
describes in his study on different roles and functions of proof.
Pedagogical aspects
All the mathematicians convey a very careful position
concerning the treatment of proof in the basic courses. Reasons for this
caution can be divided into ‘external’ and ‘internal’ reasons. ‘Internal’
reasons refer to mathematicians’ own pedagogical choices to avoid proving tasks
because they think for example that the students do not need to learn proof
yet. ‘External’ reasons are those circumstances that the mathematicians blame
for not dealing with proof in their lessons. The most of the reasons the mathematicians
gave were external, for example students’ lack of interest, students’
difficulties and fear of doing proofs, lack of time, lack of students’ prior
knowledge.
PRELIMINARY CONCLUSIONS
During the process of analysis of these five
interviews several new aspects emerged. The aspects of aesthetic, communication
and transfer might be valuable to add into the conceptual framework. It might
also be valuable to consider the mathematicians age, teaching experiences and
research area as well as the culture where they come from when analysing their
views. The clusters of internal/ external reasons for dealing/not dealing with
proof will hopefully help me in the analysis of the whole sample.
References
Adler, J. (1999). The Dilemma of
Transparency: Seeing and Seeing Through Talk in the Mathematics Classroom. Journal for Research in Mathematics
education Vol 30, No 1.
De Villiers, M. (1986). The Role of Axiomatisation
in Mathematics and Mathematics Teaching. Research Unit in Mathematics Education.
Stellenbosch.
De Villiers, M. (1990). The Role and Function of
Proof in Mathematics. Pythagoras 24:
17-24.
Fischbein, E. (1987). Intuition in Science and Mathematics, An
Educational Approach. Kluwer Academic Publisher, Holland.
Hanna, G. (1995). Challenges
to the Importance of Proof. For the Learning of Mathematics, 15(3), 42-49.
Hersh, R. (1993). Proving is Convincing
and Explaning. Educational
Studies in Mathematics 24: 389-399.
Lakatos,
Lave, J and Wenger, E. (1991). Situated Learning: Legitimate
Peripheral Participation.
Lerman, S. (2000). Some Problems of
Socio-Cultural Research in Mathematic Teaching and Learning. Nordisk Matematik Didaktik (NOMAD) Vol. 8, No 3.
Nordström K. (2003).
Nykänen, A. (1945). Geometrian Opetus Suomessa Erityisesti Oppikirjojen Kehitystä Silmällä
Pitäen. A doctoral thesis.
Skemp, R. (1987). The Psychology of Learning Mathematics. Expanded American Edition.