Overcoming Barriers to Student Understanding

Threshold Concepts and Troublesome Knowledge

Jan Meyer, Ray Land

• Publisher: Routledge
• Year: 2006
• ISBN:
  9781134189946
• Edition: 1
• Topic: Education
• Subtopic: Education General

It has long been a matter of concern to teachers in higher education why certain students 'get stuck' at particular points in the curriculum whilst others grasp concepts with comparative ease. What accounts for this variation in student performance and, more importantly, how can teachers change their teaching and courses to help students overcome such barriers?

This book examines the difficulties of student learning and offers advice on how to overcome them through course design, assessment practice and teaching methods. It also provides innovative case material from a wide range of institutions and disciplines, including the social sciences, the humanities, the sciences and economics.

Part I
Towards a theoretical framework

Chapter 1
Threshold concepts and troublesome knowledge

An introduction

Jan H. F. Meyer and Ray Land

Introduction

A threshold concept can be considered as akin to a portal, opening up a new and previously inaccessible way of thinking about something. It represents a transformed way of understanding, or interpreting, or viewing something without which the learner cannot progress. As a consequence of comprehending a threshold concept there may thus be a transformed internal view of subject matter, subject landscape, or even world view. This transformation may be sudden or it may be protracted over a considerable period, with the transition to understanding proving troublesome. Such a transformed view or landscape may represent how people ‘think’ in a particular discipline, or how they perceive, apprehend, or experience particular phenomena within that discipline (or more generally). It might, of course, be argued, in a critical sense, that such transformed understanding leads to a privileged or dominant view and therefore a contestable way of understanding something. This would give rise to discussion of how threshold concepts come to be identified and prioritised in the first instance. However, first we require examples.

A simple illustrative example can be taken from the kitchen. Cooking is fundamentally a process of using heat (in various degrees and sources) to effect desired outcomes. In physics one encounters the concept of heat transfer and its mathematical formalisation (as an equation) that represents heat transfer as a function of something called the temperature gradient. It is not necessary to have a sophisticated understanding of physics to have this principle quite simply illustrated. Imagine that you have just poured two identical hot cups of tea (i.e. they are at the same temperature) and you have milk to add. You want to cool down one cup of tea as quickly as possible because you are in a hurry to drink it. You add the milk to the first cup immediately, wait a few minutes and then add an equal quantity of milk to the second cup. At this point which cup of tea will be cooler, and why? (Answer is the second cup because in the initial stages of cooling it is hotter than the first cup with the milk in it and it therefore loses more heat because of the steeper temperature gradient.) When the physics of heat transfer is thus basically grasped by people in terms of things specific to what goes on the kitchen, it will fundamentally alter how they perceive this aspect of cooking, and they might consequently even filter out what to look for (the signified!) when they watch the better class of television cookery programmes; for example, a focus on the pots and pans that are selected by the chef in context (the heat source in relation to the cooking process to be applied as a function of time and its regulation to the ingredients) rather than simply on the ingredients and, superficially, the ‘method’. So it could be said that, as a stand alone example, heat transfer or, more precisely, controlling the rate of heat transfer, is a threshold concept in cookery because it alters the way in which you think about cooking. And, in the special case where barbecuing is the method of cooking (where heat transfer is via radiation) you also have to take into account the inverse square law, which explains why so many people find barbecuing a ‘troublesome’ notion. We shall return to the notion of troublesomeness later.

Threshold concepts and troublesome knowledge within subject disciplines

Our interviews and wider discussions with practitioners in a range of disciplines and institutions have led us to conclude that a threshold concept can of itself inherently represent what Perkins (1999) refers to as troublesome knowledge — knowledge that is ‘alien’, or counter-intuitive or even intellectually absurd at face value. It increasingly appears that a threshold concept may on its own constitute, or in its application lead to, such troublesome knowledge.

From a student perspective let us consider some examples from Pure Mathematics: first that of a complex number – a number that is formally defined as consisting of a ‘real’ and an ‘imaginary’ component and which is simply expressed in symbolic (abstract) terms as x + iy, where x and y are real numbers (simply put, the numbers we all deal with in the ‘real’ world; for example numbers we can count on our fingers), and i is the square root of minus 1 (√−1). In other words i is a number which when squared (multiplied by itself) equals minus one (−1). So a complex number consists of a real part (x), and a purely imaginary part (iy). The idea of the imaginary part in this case is, in fact, absurd to many people and beyond their intellectual grasp as an abstract entity. But although complex numbers are apparently absurd intellectual artefacts they are the gateway to the conceptualisation and solution of problems in the pure and applied sciences that could not otherwise be considered.

Second, in Pure Mathematics the concept of a limit is a threshold concept; it is the gateway to mathematical analysis and constitutes a fundamental basis for understanding some of the foundations and application of other branches of mathematics such as differential and integral calculus. Limits, although not inherently troublesome in the same immediate sense as complex numbers, lead in their application to examples of troublesome knowledge. The limit as x tends to zero of the function f(x)=(sine x)/x is in fact one (1), which is counterintuitive. In the simple (say, geometric), imagining of this limit is the ratio of two entities (the sine of x, and x) both of which independently tend to zero as x tends to zero and which are also (an irrelevant point, but a conceptual red herring if the threshold concept of a limit is not understood) respectively equal to zero when x equals zero. So the troublesome knowledge here then (based on mathematical proof) is that something which is getting infinitesimally small divided by something else doing the same thing is somehow approaching one in the limiting case.

That mathematicians themselves are aware of issues that surround threshold concepts is evident from the work of Artigue (2001, p. 211) who refers to a ‘theory of epistemological obstacles’ and, by way of summary, gives as a first example of such obstacles: ‘the everyday meaning of the word “limit”, which induces resistant conceptions of the limit as a barrier or as the last term of a process, or tends to restrict convergence to monotonic convergence.’ The idea is then developed by way of more complex examples that, as forms of knowledge, ‘epistemological obstacles’ constitute ‘resistant difficulties’ for students.

Within Literary and Cultural Studies the concept of signification can prove problematic, even ‘subversive’ in that it undermines previous beliefs, and leads to troublesome knowledge insofar as the non-referentiality of language is seen to uncover the limits of truth claims. For example, the recognition (through grasping the notion of signification) that all systems of meaning function like signifiers within a language (that is that terms derive meaning from their relationship to each other, rather than in any direct empirical relationship with a ‘reality’) leads on to an understanding that there are no positive terms. Hence the basis of many systems of meaning, including positivist science and the basis of many religious and moral systems, falls into question. This can be a personally disturbing and disorienting notion leading to hesitancy or even resistance in learners. Other aspects of post-structuralist practice such as techniques of deconstruction for analysing literary texts (with a strong emphasis on the ironic, the contradictory, the ludic) often appear counter-intuitive, looking for absences, or what is not there, in order to gain insights into how the text is currently structured by a prevailing set of (occluded or tacit) values or priorities.

One final illustrative example from Economics will suffice, again from the student perspective. The concept of opportunity cost has been put forward as one of many examples of a threshold concept in the study of economics. Martin Shanahan (as quoted in Meyer and Land 2003, pp. 414–15) assesses the transformative effect of this concept as follows:

‘Opportunity cost is the evaluation placed on the most highly valued of the rejected alternatives or opportunities’ (Eatwell et al. 1998, Vol. 3, p. 719). Fundamental to the discipline of economics is the issue of choice: choosing between scarce resources or alternatives. Economists are interested in how individuals, groups, organisations, and societies make choices, particularly when faced with the reality that resources and alternatives are limited. No-one can have everything, and in most cases the ‘constraints’ faced by the chooser can be quite severe and binding. People choose, for example, how to allocate their time, their work or leisure; firms choose between different methods of production and combinations of inputs; societies choose between different legal regimes, levels of exports or imports etc. Fundamental to the economic way of approaching the issue of choice is how to compare choices. Thus ‘The concept of opportunity cost (or alternative cost) expresses the basic relationship between scarcity and choice’ (Eatwell et al., ibid.); for this reason it is a fundamental (or threshold) concept in Economics.

Thus opportunity cost captures the idea that choices can be compared, and that every choice (including not choosing) means rejecting alternatives. A student who has a good grasp of this concept has moved a long way toward breaking out of a framework of thinking that sees choices as predetermined, or unchangeable. They have also moved toward seeing ‘two sides’ of every choice, and in looking beyond immediate consequences, and even just monetary ‘costs’ towards a more abstract way of thinking.

Thus to quote Eatwell et al. for a final time (ibid.), ‘Opportunity cost, the value placed on the rejected option by the chooser, is the obstacle to choice; it is that which must be considered, evaluated and ultimately rejected before the preferred option is chosen. Opportunity cost in any particular choice is, of course, influenced by prior choices that have been made, but with respect to this choice itself, opportunity cost is choice-influencing rather than choice-influenced’ (emphasis in original). Thus, if ‘accepted’ by the individual student as a valid way of interpreting the world, it fundamentally changes their way of thinking about their own choices, as well as serving as a tool to interpret the choices made by others.

Characteristics of a threshold concept

A threshold concept is thus seen as something distinct within what university teachers would typically describe as ‘core concepts’. A core concept is a conceptual ‘building block’ that progresses understanding of the subject; it has to be understood but it does not necessarily lead to a qualitatively different view of subject matter. So, for example, the concept of gravity — the idea that any two bodies attract one another with a force that is proportional to the product of their masses and inversely proportional to the distance between them — represents a threshold concept, whereas the concept of a centre of gravity does not, although the latter is a core concept in many of the applied sciences.

Our discussions with practitioners in a range of disciplinary areas have led us to conclude that a threshold concept, across a range of subject contexts, is likely to be:

1. Transformative, in that, once understood, its potential effect on student learning and behaviour is to occasion a significant shift in the perception of a subject, or part thereof. In certain powerful instances, such as the comprehension of specific politico-philosophical insights (for example, aspects of Marxist, feminist or post-structuralist analysis) the shift in perspective may lead to a transformation of personal identity, a reconstruction of subjectivity. In such instances transformed perspective is likely to involve an affective component — a shift in values, feeling or attitude. In this regard there are correspondences with Mezirow’s (1978) work on ‘perspective transformation’. A threshold concept may also involve a performative element. Sproull (2002) points out how the gaining of aquatic confidence in Sports Science students leads to a dramatically enhanced appreciation of water as a sporting and exploratory environment. This would be an interesting example of an enactive concept in Bruner’s sense (Bruner 1966).
2. Probably irreversible, in that the change of perspective occasioned by acquisition of a threshold concept is unlikely to be forgotten, or will be unlearned only by considerable effort. As a conveniently graphical metaphor, the post-lapsarian state of Adam and Eve after their expulsion from Eden in the Book of Genesis illustrates how new (and in this case troublesome) knowledge, symbolised by the cunning (i.e. ‘conynge’, knowing) serpent, radically transforms their landscape as they pass through the threshold from innocence to experience (new understanding). They gain freedom, responsibility and autonomy, though this is not a comfortable transition. As they look back to the Gate at the East of Eden their return across the threshold is barred by Cherubim ‘and a flaming sword which turned every way’ (Genesis 3: 24) to prevent return to the tree of knowledge. Though they have learned, and g...

Table of contents

• Cover Page
• Half-title Page
• Title Page
• Copyright Page
• Table Of Contents
• Notes on contributors
• Foreword
• Editors' preface
• Acknowledgements
• PART 1 Towards a theoretical framework
• PART II Threshold concepts in practice
• Index