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Programming as mathematical narrative
Yishay Mor, Richard Noss
To cite this version:
Yishay Mor, Richard Noss. Programming as mathematical narrative. International Journal of Con-
tinuing Engineering Education and Life-Long Learning, Inderscience, 2008, 18 (2), pp.214-233. �hal-
00591784�
214 Int. J. Cont. Engineering Education and Life-Long Learning, Vol. 18, No. 2, 2008
Copyright © 2008 Inderscience Enterprises Ltd.
Programming as mathematical narrative
Yishay Mor* and Richard Noss
London Knowledge Lab,
Institute of Education,
University of London,
23-29 Emerald Street,
London WC1N 3QS, UK
E-mail: yishaym@gmail.com E-mail: r.noss@ioe.ac.uk
*Corresponding author
Abstract: This paper describes a narrative-oriented approach to the design and
the analysis of a computational system and a set of activities for mathematical
learning. Our central contention is that programming can offer a key to
resolving the tension between the different representational structures of
narrative and mathematical formalism. In the course of describing our
approach, we make a distinction between the epistemic-cognitive elements of
narrative and the social, cultural and affective elements. We then elaborate the
theoretical grounds of the individual epistemic facets of narrative. We propose
a link between narrative theories of learning and constructionist traditions,
specifically the notion of situated abstraction. This link suggests the possibility
of further dialogue between the two academic communities.
Keywords: collaborative learning; constructionism; Computer Supported
Collaborative Learning; mathematical learning; narrative; programming;
situated abstraction.
Reference to this paper should be made as follows: Mor, Y. and Noss, R.
(2008) ‘Programming as mathematical narrative’, Int. J. Continuing
Engineering Education and Life-Long Learning, Vol. 18, No. 2, pp.214–233.
Biographical notes: Yishay Mor holds an MSc in Computer Science from the
Hebrew University, Jerusalem. After a career in the software industry, Yishay
turned to educational research. He was a Senior Researcher with the WebLabs
project (http://www.weblabs.eu.com) and the Learning Pattern project
(http://lp.noe-kaliedoscope.org). His main research interests are
constructionism, mathematics education, educational programming, and
computer supported collaborative learning.
Richard Noss is a Professor of Mathematics Education, and co-director of the
London Knowledge Lab, an interdisciplinary facility built to explore the future
of learning with digital technologies. He has a Masters Degree in pure
mathematics, a PhD in Mathematical Education and has taught Mathematics at
all levels of the educational system. He was, until recently, the Pro-director
Information and Communication Technologies (ICT) of the Institute of
Education. He has directed some 20 projects, including the EU-funded
Playground Project (1998–2001) and WebLabs Project
(2002–2005). He has edited and authored five books, most recently Windows
on Mathematical Meanings: Learning Cultures and Computers co-authored
with Celia Hoyles in 1996. He was, until 2005, editor-in-chief of the
International Journal of Computers for Mathematical Learning.
Programming as mathematical narrative 215
1 Introduction
“Actually, it is half the art of storytelling to keep a story free from explanation
as one reproduces it (...). The most extraordinary things, marvellous things, are
related with the greatest accuracy, but the psychological connection of the
events is not forced on the reader. It is left up to him to interpret things the way
he understands them, and thus the narrative achieves amplitude that the
information lacks (Benjamin, 1968)”.
This paper describes a narrative-oriented approach to the design and the analysis of a
computational system and a set of activities for mathematical learning. The language of
mathematics is often perceived as propositional; a formalism which defines terms, states
axioms and rules, then derives theorems and proves them. Its structures are static, devoid
of time and person. This view was demonstrated lucidly by Wittgenstein (1989).
“In mathematics, we have propositions which contain the same symbols as, for
example, “write down the integral of…”, etc. with the difference that when we
have a mathematical proposition, time does not enter into it and in the other it
does (p.34)”.
This would appear to be antithetical to narrative form, which is always personal,
contextual and time-bound. By contrast, Bruner (1986, 1990) shows that narrative is a
powerful cognitive and epistemological construct. The main question we explore is: how
can the epistemic power of narrative be harnessed in the construction of mathematical
meaning?
We approach this question from a design perspective. We are concerned with the
design of platforms, tools, and activities for mathematical learning, focusing on the
notion of situated abstraction (Noss and Hoyles, 1996). The idea (since developed in, for
example, Noss, Healy and Hoyles, 1997) highlights the dynamics of constructing
knowledge from activity, by inserting or populating an abstraction with meaning – in the
shape of special cases, particular values, or familiar contexts (or, in the special case of the
mathematical situation, with mathematical objects and relationships). The questions we
ask include: what are the possible contributions of narrative that might facilitate such a
trajectory? What is required from such narrative, and what is required from the learning
activity encompassing it? In brief, we aim at elaborating the role that narrative could play
in the construction of mathematical abstraction.
Our central contention is that programming can offer a key to resolving the tension
between the different representational structures of narrative and the mathematical
formalism. We see programming as an expressive activity, a form of writing or
composing, contingent on context and used purposefully to carry out actions. We claim
that programming can afford a narrative form for representing mathematical meanings.
The issues we address have strong social and cultural dimensions, and occasionally we
refer to these. However, our goal is to highlight the often neglected aspects of individual
knowledge construction within a social environment.
The structure of this paper is as follows. We begin by presenting a review of the use
of narrative in educational theory in general and in the teaching of mathematics in
particular. Inter alia, we present our own perspective on the relationship between
narrative, learning and technology. We then briefly describe the WebLabs project and the
tools developed for it, as an example of a narrative-aware learning environment.
Following this, we present three illustrative episodes from our observations, and
comment on the role of narrative in students’ learning and in the design of technology to
216 Y. Mor and R. Noss
support it. Our concluding discussion highlights the potential of constructionist
programming to provide students with a medium for mathematical narrative.
2 Narrative and education
The concept of narrative has been investigated extensively within a wide range of
disciplines over the last few decades. To name but a few: in literary theory, Genette
(1980) establishes narrative as a fundamental tool; in the social sciences, Gergen (1998)
refers to it as a tenet of social construction; Carr (1986) positions it as a central concept in
the philosophy of history. Since the 1980s, narrative approaches have also become
popular in counselling, where the term refers to a patient’s personal account of her
condition (White and Epston, 1990; Roberts, 2000).
Our own interests centre on the epistemic role of narrative, in the tradition of Bruner.
We focus on storytelling as a means of meaning-making, with an emphasis on the
structural and the semantic components of narrative. In reality, it is hard to separate the
individual epistemic aspects of narrative from the social, affective and cultural aspects.
Nevertheless, the emphasis in this paper is on the former. In his theory of learning and
education, Bruner (1986, 1990, 1991, 1996) and Bruner and Lucariello (1989) identified
narrative as the predominant vernacular form of representing and communicating
meaning. Humans use narrative as a means of organising their experiences and making
sense of them. Parents use narrative as a means of sharing knowledge with their children.
Schank and Abelson (1995) argue that stories about one’s experiences, and the
experiences of others, are the fundamental constituents of human memory, knowledge,
and social communication. They call for a shift towards a functional view of knowledge,
as Schank (1995) explains: “intelligence is really about understanding what has happened
well enough to be able to predict when it may happen again” (p.1). Such knowledge is
constructed by indexing narratives of self and others’ experiences, and mapping them to
structures already in memory. While Schank and Abelson (1995) come from an AI
perspective, their theory is supported by recent psychological studies. Atance and O’Neill
(2005) define episodic future thinking as the ability to project oneself into the future to
pre-experience an event. This, they claim, is a uniquely human phenomenon which
precedes semantic future thinking (Atance and Meltzoff, 2005), and provides the
developmental basis for skills such as planning and causal reasoning. They found that
episodic future thinking emerges around the age of four, and is related to children’s
abilities to construct and comprehend verbal accounts of experiences. Recent
developments suggest a neural basis for the role of narrative in the abstraction of daily
experience to knowledge (Mar, 2004). Narrative comprehension engages a widely
distributed network of brain regions, and is clearly distinct from basic language
comprehension (Nichelli et al., 1995; Ferstl, Rinck and von Cramon, 2005; Xu et al.,
2005).
Following Bruner, we define narrative as a progression of statements describing
something happening to someone in some circumstances. This view entails a form of
language which includes a context (setting) and a plot: a sequence of events bound by
temporal – and implicitly causal – relationships. Likewise, Mar (2004) identifies the
presence of a causal-temporal event structure as imperative, and notes:
Programming as mathematical narrative 217
“The most basic elements of a story include a setting, and an agent who holds a
certain goal […] and whose progress towards that goal is impeded […] or
facilitated by certain events” (p.1415).
In this paper, we explore three constituents of narrative: context, plot and moral. The
context includes the background information assumed or conveyed explicitly with a
narrative. The plot denotes its temporal and causal structure. We use the term moral to
refer to the implicit endpoint of a narrative, the purpose for which it is told.
A narrative is always contextualised. An important contextual element is the
exposition, which lays out the context: time, location, props and characters. Such
an exposition is not limited to imaginative narrative: it also appears in scientific texts
(Bruner, 1986). One particular element of context we focus on is the idea of voice, which
relates to the presence of the speaker. Even in allegedly ‘de-humanised’ arenas, such as
scientific or legal writing, great significance is attached to the voice of a document’s
author. When approaching a scientific paper, one draws on knowledge of the author: past
publications, close collaborators, institution, etc. Likewise, when writing a paper, one is
advised to imagine its readers and engage in a dialogue with them. Familiarity with the
writer’s personal style makes the writing much easier to interpret and understand. A clear
sense of authorship promotes responsibility for the text.
A well-formed narrative must maintain coherence of temporality and causality
(Gergen, 1998). Temporality refers to the chronological ordering of events. In the light of
narrative intelligence theory (Mateas and Sengers, 1999), it is clear that maintaining the
temporal structure is crucial to the reader’s ability to comprehend a story. The
identification of temporal affinity of events also plays a strong role in learners’ inferences
of causality, an important component in the construction of meanings. The sequencing of
events is referred to as the plot. Gergen (1998) adds that events are carefully selected to
support an endpoint.
Yet perhaps the most important part of a narrative is typically left unstated: its moral.
We use this term with an expanded meaning, referring to the narrative’s implicit
endpoint. A story is told for a purpose – establishing norms, conveying knowledge, or
raising a question. It is the implicit layer that holds the narrative together – the causal
relationships along the way and the climactic moral at the end. Without them, all we have
is an arbitrary list of events. As Mar (2004) asserts,
“If a well-crafted story contains mention of an event or a character, it is
assumed that this element is in some way relevant to the goals of the
protagonist” (p.1416).
Recent advances in neural psychology ground these observations in new understandings
of the brain’s inner working (Holyoak and Krogen, 1995; Young and Saver, 2001; Addis
et al., 2004; Mar, 2004; Mason, 2004; Mar et al., in press). Xu et al. (2005) link context
to brain regions responsible for global semantic processes such as inference, coherence,
conceptual association and text integration. Other findings point to a strong link between
narrative comprehension and theory-of-mind processing (Mar, 2004), suggesting that the
cognitive modelling of the storyteller and the protagonists is a critical constituent in
understanding a story. A detailed discussion of the relations between neural and cultural
theories is called far, but is beyond the scope of this paper.
Closer to home, the mechanism described above is consistent with the ideas of
situated abstraction and webbing (Noss and Hoyles, 1996; Noss et al., 1997). The concept
of situated abstraction focuses attention on the process of making meanings through
