Project title:
Transforming theory into practice:
A guide for teaching Proof to mathematics undergraduates
grounded on the co-ordinated perspectives and recommendations of
mathematicians and researchers in mathematics education
Final Report
Elena Nardi & Paola Iannone
School of Education and Lifelong Learning
University of East Anglia
Norwich, UK
Address:
Telephone:
Fax:
E-Mail:
Website:
UEA-EDU, Norwich NR4 7TJ, UK
+44 1603 592631 (Nardi), +44 1603 591007 (Iannone)
+44 1603 593446
[email protected], [email protected]
http://www.uea.ac.uk/~m011/, http://www.uea.ac.uk/edu/mathsed.html
On March 20th 2006 we were commissioned by the HEA-MSOR Network to
produce a guide that will provide ‘support to those teaching Proof to Pure
Mathematicians’. The production of the Guide would be based on: findings
from our previous studies in the area; relevant literature; and, an interim
evaluation of a draft sample by groups of mathematicians and researchers in
mathematics education based at UEA and other institutions. The project was
expected to last from April 1st to September 30th. In what follows we report on
the Guide’s production process and outline the dissemination plan regarding
the promotion of the Guide that is currently in progress.
Elena Nardi and Paola Iannone, September 30th 2006
HEA-MSOR Project Final Report – Nardi & Iannone 1
1. Production of the Guide
Production of the Guide was carried out in three phases



Work towards the selection of the themes the Guide touches on and preparation of
a draft sample of the Guide to be used as the basis of the discussion in an Interim
Evaluation Event
Interim Evaluation Event
Work towards finalising the structure and content of the Guide in the light of
recommendations made at the Interim Evaluation Event and further reading and
scrutiny of our previous data.
Below we describe each of the above phases.
Phase I: Towards a draft structure and sample of the Guide (April – May)
Work towards the selection of the themes the Guide touches on consisted largely of
drawing on relevant literature and on our previous studies in this area.
With regard to the former we searched the mathematics education research literature in
order to identify tried-and-tested recommendations regarding the teaching of Proof to
mathematics undergraduates.
With regard to the latter these include the recently completed study ‘Engaging
mathematicians as educational co-researchers’ funded by the Learning and Teaching
Support Network as well as five other studies that Nardi has been involved with since 1992
(funded by: ESRC (2), Wingate Foundation, Nuffield Foundation (2 - Iannone also
involved)). Mathematical Reasoning, and in particular Proof, has been an essential theme
running across the data collection and analyses in these studies. We mostly revisited the
material from the LTSN-funded study in which we had conducted a series of themed
Focus Group interviews with mathematicians from six UK universities. In these interviews
pre-distributed samples of mathematical problems, typical written student responses,
observation protocols, interview transcripts and outlines of relevant bibliography had been
used to trigger an exploration of issues on the teaching and learning of university
mathematics. Two of the six themes explored had been: Students’ perceptions of Proof
and its necessity and Students’ enactment of proving techniques and construction of
mathematical arguments. Eleven half-day interviews had generated about 250,000 words
of transcript sharply focused on:


how the participating mathematicians perceive their students’ difficulties; and,
the pedagogical practices they employ – or wish they could employ more
extensively and more systematically – to support their students’ overcoming of
these difficulties.
An additional resource has been our own teaching experience, in particular Iannone’s fiveyear experience as an undergraduate mathematics tutor at UEA.
Overall the work was underpinned by the principles and theoretical perspectives on the
teaching and learning of mathematics that have been underlying our previous work. In the
following table we outline these principles and demonstrate how they have helped shape
the contents of the Guide.
HEA-MSOR Project Final Report – Nardi & Iannone 2
Theoretical Framework
Relevant Dimension
Selection Criterion
Constructivism
Learning as an individual
sense-making process,
teaching as facilitating this
process
Select activities that foster student
participation, learning from peers and
student – teacher interaction (e.g. project
work, group-work, student presentations
etc.)
Socio-cultural theory
Enculturation into the
language and practices of
university mathematics
Select activities that foster an
understanding of the language and
practices of university mathematics (e.g.
student / lecturer led Scientific Debates on
acceptable forms of argumentation and
Proof; Mathematical Writing Workshops on
acceptable forms of mathematical writing,
etc.)
Enactivism
Teaching and learning as
processes of constant mutual
specification and
codetermination, as “paths
laid while walking”; embrace
and draw on the power in
concreteness and in
specificity of context
Present selected activities through lively
and illustrative examples from the data (e.g.
short stories and vignettes from teaching
events recorded in the previous studies)
Participatory / Partnership
Research
Integration of innovation is
manageable, and sustainable,
if driven and owned by the
mathematicians who are
expected to implement it
Select activities that, either in our data or in
the literature, have been highlighted as
potent by mathematicians themselves,
respond to specific student and curricular
needs and are manageable within the
systemic constraints of university courses
By early June the preliminary structure and content of the Guide was as follows:




Introduction
Five sections each on a theme regarding the teaching of Proof to first-year
undergraduates. These themes were:
o conceptualising formal mathematical reasoning and the necessity for Proof
o the role of examples in Proof: the tension between the general and the
particular and ‘proof’-by-example
o the role of examples in Proof: Proof by Counterexample
o Proof by Mathematical Induction
o Proof by Contradiction
Epilogue
References
Each of the five sections was envisaged as follows:

Example from the data collected in the course of the above mentioned LTSN study
that illustrates the theme. This usually starts with examples of student written
responses to a mathematical problem that required the use of a particular aspect of
Proof that the theme of the section aims to explore. It continues with a brief
reference to the comments made on the student responses by the mathematicians
interviewed in that study and concludes with a listing of issues that the reader is
invited to consider in the light of these examples.
HEA-MSOR Project Final Report – Nardi & Iannone 3


Brief review of some relevant literature with regard to the theme the section aims to
explore. Throughout amongst our priorities has been that this scrutiny of the
relevant literature is presented in a language that is accessible to users with a
limited background in the prose and terminology of educational research.
Exposition on recommended practices as found in the literature and the interviews
with the mathematicians with a particular emphasis on strengths and cautionary
points.
A sample of the Guide entitled ‘Proof’-by-example: Syndrome or Starting Point?’ (see
Appendix) was prepared for use in an Interim Evaluation Event which aimed to collect the
comments and recommendations of practitioners (teachers of mathematics at the
undergraduate level) whose teaching the Guide ultimately purports to support.
Phase II: Evaluation of the draft sample (June)
The event took place in the course of two two-hour lunchtime meetings on June 22nd and
23rd  one with colleagues from UEA and the other with colleagues from elsewhere in the
UK and abroad (Italy and Brazil). The structure of the meetings replicated that of the
numerous focused group interviews we have conducted in the context of the above
mentioned LTSN study. The meetings were audio recorded.
Overall the reception of the sample was very positive and its presence became the starting
point of fervent discussion both about the theme it touched on and the potential uses of the
Guide in general. More or less the discussion followed the structure we proposed in the
concluding page of the sample:
‘In the discussion of this sample from the Guide we would like you to consider the following
questions:
Regarding the particular issue we raise in the sample (‘Proof’-by-example):

Is this an issue that you believe your students face?

If yes, to what extent? And how do you as their teacher usually help them towards a
more comprehensive understanding of the role of examples can play in constructing
mathematical arguments?

Do you find the discussion of generic examples as a means to facilitate such
understanding useful?

Do you use generic examples in your teaching? Could you suggest other occasions,
perhaps more in tune with university-level mathematics than the ones we offer in this
sample, where resorting to a generic example and transparent pseudo-proofs can be
helpful?

Do you find John Mason’s tactics relevant to what you do, or could do, in your own
teaching? Would you like to add any more suggestions to those?
Regarding the production of the Guide in general:
 Content: what other issues regarding the teaching of Proof would you like to see addressed
in the Guide?
 Format:
o do you find the format of the sample … Appropriate? Clear? Engaging?
o do you find the length of discussion devoted to each part of the sample (data,
literature, pedagogical suggestions) appropriate? Would you like to see the balance
altered in any particular direction?
Please remember that the Guide is currently in its infancy. It is our serious intention that your
comments – as thoughtful and frank as we can get them to be! – shape subsequent versions of the
Guide as much as possible.’
In sum the points raised in the interviews were as follows:
Regarding the sample:
HEA-MSOR Project Final Report – Nardi & Iannone 4




Example from the data – there is value in starting each section with an example
from the written work of the students and the reactions from the mathematicians.
The example however needs to be chosen carefully. As it is impossible to find an
example that exemplifies only the particular issue treated in the particular section,
the attention of the reader needs to be directed to which aspect of the example we
want them to concentrate upon. One suggestion was to use one example of student
data to highlight more than one issue, pointing the reader each time to which issue
we want them to concentrate upon.
Literature review – introduce examples for each of the phenomena described in the
educational literature section (for example, in the case of the phenomena described
in the sample, introduce examples for each one of the Proof Schemas). Pay
attention to the language in which the literature review is written. It must be
accurate but also avoid excessive use of jargon. For example: in the literature
review of the sample precede the terms ‘affective’, ‘cognitive’ etc. with the questions
that raise the corresponding issues (affective, cognitive etc.). Unpack the
educational jargon even further: for example, stress the difference between what
constitutes a taxonomy in mathematics and a taxonomy in mathematics education.
Transforming theory into practice – the level of the discussion in the sample was
accepted as suitable and the mathematics in there as fine, even if it treats
apparently elementary statements.
The practical suggestions to the lecturer – these might be better placed at the end
of the “Transforming theory into practice” bit rather than at the beginning, and there
should be plenty of those. There was little consensus amongst interviewees on this
however: some suggested that these might be redundant altogether. One
suggestion was to include some direction on how to use a blackboard in a lecture
(seen to be important).
Regarding the Guide generally:







The title of the Guide must be a shorter and snappier version of the title project.
The five themes / headings for the Guide’s sections are appropriate and relevant.
The length of the sample is only not-too-long but, according to some, even affording
a bit more detail. As the Guide is not intended for reading from cover to cover in one
go, we need not shy away from making our point elaborately even if this implies a
longer text for each section.
Avoid jargon.
Do not be afraid of using (mathematically easy) examples. They can be very
illustrative of the issues in question.
Include an introduction to explain how the Guide works.
Some participants expressed a preference for presenting the Guide in LaTex1.
It is our intention that the final version of the Guide complies with as many of the above
recommendations as possible. Please note that we have not taken all the
recommendations on board: for example, we have not taken up the suggestion to use just
one example of student data that highlights several aspects of the issues touched upon in
the various sections and revisit it in each section from a different perspective. One reason
is a concern for variety: we believe more, and more diverse, examples make the Guide
more engaging. Another reason is that the diversity of examples we draw on reflects the
1
At the time of writing this Report we were at the final stages of writing/formatting and had not decided yet
on whether we will comply with this preference.
HEA-MSOR Project Final Report – Nardi & Iannone 5
diversity of data we collected in our previous studies as well as the breadth and wealth of
pedagogical discourse the participating mathematicians engaged in during the interviews
for these previous studies.
Another suggestion we have not taken up is to include more generalist pedagogical advice
(e.g. on using blackboards). Even though we recognise that there are mathematicsspecific aspects to learning how, for example, to use a blackboard, given the limitations of
this very small-scale project, we have chosen to stay focused on the issues strictly related
to the teaching of Proof. We do however consider this request for more general, yet
mathematics-related, pedagogical advice as one that merits further and systematic work
and we would be happy to consider it for future projects.
Phase III: Towards the final version of the Guide (July – September)
In the light of the above evaluation we finalised the Guide which will be available in the
coming weeks in hard copy (as a booklet) and electronically (as a PDF file available for
downloading from our site – a link to the HEA-MSOR can also be easily established). The
Guide is approximately 12,000 words long and consists of the sections we described in the
Phase I section. Below we outline the plan for the promotion of the Guide.
2. Promotion of the Guide
At present activities towards the promotion of the Guide are planned as follows:






We have referred to the work towards the Guide in a presentation at the Third
International Conference on the Teaching of Mathematics at the Undergraduate Level
(Istanbul, June 2006) entitled ‘You look at these students, you look at their faces, you
know they are lost…’: the pedagogical role of the mathematician as de-mystifier and
en-culturator.
We have obtained an ISBN number for the Guide so that it becomes accessible
through bibliographical and library databases
We are making the Guide available at our site (and HEA-MSOR’s site if requested) as
well as other relevant sites such as http://www.lettredelapreuve.it The International
Newsletter on the Teaching and Learning of Mathematical Proof.
We have been invited to offer a presentation, or working session, at a mathematics
education conference in which the mathematical community is known to participate.
This is the Undergraduate Mathematics Teaching Symposium in Galway – Ireland
(December 15th-16th, 2006). Another relevant conference we will try to present at is the
annual UMTC (Undergraduate Mathematics Education Conference) held in
Birmingham annually (UMTC 2006 is planned for December 18th-19th 2006) or the
annual conference of the Mathematical Association (April 2007).
As promised to the colleagues from UEA and elsewhere who participated in the Interim
Evaluation Event we will present and discuss the final version of the Guide in a
seminar/event within the current academic year.
We will be announcing the completion of the Guide in a short article in a professional
journal. In this we will outline the contents of the Guide and invite readers to access it
in full. Potential venues of publication are The Mathematical Gazette, The MSOR
Newsletter and The Bulletin of the London Mathematical Society.
HEA-MSOR Project Final Report – Nardi & Iannone 6
3. Further work
At the time of completing this report the Leverhulme Trust, following our submission of an
outline proposal in July 2006, has invited a full application of our proposal entitled
‘Transforming theory into pedagogical practice for university mathematics’. We will submit
this by December 1st 2006 and will be notified of the results by end of April 2007. The
proposed work employs the Guide as a starting point for a Programme of Innovation, a
programme of innovative practices that will be implemented and evaluated in the context
of a Year 1 course in Pure Mathematics at UEA and elsewhere. Moreover, and given the
exact match between our research plans and the objectives set by the HEA National
Teaching Fellow Scheme, we are also inquiring our eligibility for this scheme and may
submit an outline proposal by the end of October 2006.
4. References
The Guide includes with a complete set of references to the works mentioned in each
section – for example each section starts with an excerpt from
Nardi, E. (in press 2007) Amongst Mathematicians: Teaching and Learning Mathematics
at University Level. Due by Springer in the Spring/Summer 2007.
However we would also like to acknowledge the following works for their substantial
influence on the theoretical underpinnings of the work towards the Guide.
Dawson, S. (1999). The Enactive perspective on teacher development: “A Path Laid While
Walking”. In B. Jaworski, T. Wood, & S. Dawson (Eds.), Mathematics teacher
education: International critical perspectives (pp. 148–162). London: Falmer.
Holton, D. A. (Ed.). (2001). The teaching and learning of mathematics at university level:
An ICMI study. Dordrecht, The Netherlands: Kluwer.
Iannone, P. & Nardi E. (2005) On the Pedagogical Insight of Mathematicians: ‘Interaction’
and ‘Transition from the Concrete to the Abstract’, Journal of Mathematical
Behaviour, 24(2), 191-215
Mason, J. (2002). Mathematics teaching practice: A guide for university and college
lecturers. Chichester, UK: Horwood Publishing.
Nardi, E., Jaworski, B. & Hegedus, S. (2005) A spectrum of pedagogical awareness for
undergraduate mathematics: from ‘tricks’ to ‘techniques’, Journal for Research in
Mathematics Education, 36(4), 284-316
von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning.
London: The Falmer Press.
Wagner, J. (1997) The Unavoidable Intervention of Educational Research: A Framework
for Reconsidering Researcher-Practitioner Cooperation. Educational Researcher.
Vol. 26, No. 7. p. 13-21
Wenger, E. (1998). Communities of practice: Learning, meaning and identity. Cambridge,
UK: Cambridge University Press.
HEA-MSOR Project Final Report – Nardi & Iannone 7