Ernest A Semiotic Theory of Mathematical Text

Thus, in any given context,a mathematical text or sign utterance, like any utterance, is indissolubly associated with apenumbra of contextual meanings including its purpose, its intended

A SEMIOTIC THEORY OF MATHEMATICAL TEXT

Paul Ernest

University of Exeterp.ernest @ ex.ac.uk

In this paper I consider the content and function of mathematical text from a semioticperspective My enquiry takes me beyond the written mathematical text for I also consider thespoken text and texts presented multi-modally I explore both the reading/listening and thewriting/speaking dimensions of mathematical text in this broader sense In addition to makingthe enquiry more extensive this necessitates the inclusion of a further vital dimension ofwritten mathematical text in use, namely, the social context For texts do not exist in theabstract, but are always and only present via their utterances and instantiations Thus to open

up the mathematical text from a semiotic perspective is to explore its social uses andfunctions, as well as its inner meanings and textures Mathematical text is unlike fiction, for it

is not merely a doorway to a world of the imagination It is not just a told tale rendered intowritten language This insight is readily available to schoolchildren But it is one thatphilosophers, mathematicians and educational researchers have to struggle to attain Why isthis? How do mathematical texts differ from fiction? This brings me back to the mainquestions of my enquiry What do mathematics texts ‘say’? What light do the tools ofsemiotics shed on the content and functions of mathematical text?

These are not simple questions, and they are rendered all the more obscure becausemathematical texts are not viewed or read neutrally Irrespective of the reading subject who isneeded to make sense of the text, mathematics is thickly overlaid with ideologicalpresuppositions that prevent or obscure a neutral or free reading So my first enquiry is intosome of these ideological presuppositions that distort a reading of mathematical texts, that is,into what mathematics texts do not say

WHAT MATHEMATICS TEXTS DO NOT SAY

Rorty (1979) has described the ideology gripping traditional philosophy that sees it, the text,and the human mind as ‘mirroring nature’ In other words, he critiques the traditionalassumption that there is a given, fixed, objective reality and that mind, knowledge and textcapture and describe it, with greater or lesser exactitude This traditional philosophy reachedits apogee in Wittgenstein’s (1922) Tractatus with its picture theory of meaning.Wittgenstein’s doctrine asserts that every true sentence depicts, in some literal sense, thematerial arrangements of reality Language, when used correctly, floats above material reality

as a parallel universe and provides an accurate map or picture of it However, a claim thisstrong is hard to sustain, and even the logical positivists withdrew from this overly literalposition about the relationship between language and reality They adopted instead theverification principle which states that the meaning of a sentence is the means of itsverification (Ayer 1946).1 For without this revised view of meaning the predictive power and

1 In both the cases of Wittgenstein and the Logical Positivists mathematics is treated differently from empirical

or scientific claims or texts In the Tractatus, Wittgenstein argues that mathematics is a by-product of the grammar of world description The logical positivists argue that mathematics is analytic a priori knowledge, derived purely by logic and telling us nothing new Both of these accounts trivialize mathematics, and focus on the tasks of accounting for scientific knowledge of the empirical world and the rejection of metaphysics From the perspective of mathematical philosophy these accounts are fundamentally unenlightening

generality of scientific theories is compromised Ironically, Wittgenstein (1953), in his laterphilosophy, also rejected this position himself having pushed the picture or mirror view to itslimits in the Tractatus.2

This ideology applied to mathematics remains very potent in Western culture Mathematics isseen to describe an objective and timeless superhuman realm of pure ideas, the necessity ofwhich is reflected in the ineluctable patterns and structures observed in our physicalenvironment The doctrine that mathematics describes a timeless and unchanging realm ofpure ideas goes back to Plato, if not earlier, and is usually referred to as Platonism Many ofthe greatest philosophers and mathematicians have subscribed to this doctrine in thesubsequent two plus millennia since Plato’s time In the modern era the view has beenendorsed by many thinkers including Frege (1884, 1892), Gödel (1964), and in some writings

by Russell (1912) and Quine (1953) According to Platonism, a correct mathematical textdescribes the state of affairs that holds in the platonic realm of ideal mathematical objects.Mathematical texts are nothing but descriptions or mirrors of what holds in this inaccessiblerealm 3

Although I shall reject it, quite a lot is gained by this view First of all, mathematicians andphilosophers have a strong belief in the absolute certainty of mathematical truth and in theobjective existence of mathematical objects, and a belief in Platonism validates this It posits aquasi-mystical realm into which only the select few – initiates into the arcane practices ofmathematics – are permitted to gaze Secondly it puts epistemology beyond the reach ofhumanity’s sticky fingers and earth(l)y bodies, locating mathematical knowledge in a safe andinaccessible zone Because of this strategy of removal, any claims that mathematics is socially

constructed are disallowed tout court without having their specific merits or weaknesses

to the realm of meaning (i.e., the interpretation of the sign in the platonic universe of number

where 2+2=4 holds), is not illuminating, per se, in providing the meaning of the expression.

Alfred Tarski (1935) in his famous paper on the semantic interpretation of truth argues the

sentence ‘Snow is white’ is true, if, and only if, Snow is white Per se, this is trivial and

uninformative However, Tarski’s project is fundamentally technical He is concerned with thedefinability of the concept of truth in formalized languages, and as a consequence of hisformal explication of ‘truth’ provided the foundations of model theory and arrived at one ofthe important limitative results of modern mathematics Namely, that truth is indefinable informal languages, on pain of inconsistency For technical purposes he clarified the notion ofsyntactic expressions mirroring states of affairs in the semantic realm, with valuablemathematical and foundational results But this does not lend support to the general

2 Wittgenstein (1953) went on to develop what might be termed one of the first postmodernist epistemologies, with his doctrines that ‘meaning is use’, and that all text and knowledge emerges from language games embedded in human forms of (social) life, with mathematics made up a multiplicity of language games

3 Otte (1997: 50) contrasts Cartesianism, the view that the (mathematical) world is already known in principle, and that in time scientific deduction will fill all the lacunae, with Pascal’s ‘subtle intelligence’ according to which “the world can be appropriated cognitively only by the constructive incorporation of ever new types of objects into thought”.

ideological ‘mirroring’ presupposition ‘as below, so above’ (to invert the fundamentalprinciple of astrology) that I am critiquing

Philosophers have long been concerned about mathematical ontology: the study of theexistence and nature of mathematical objects and abstract entities in general Balaguer (2004)provides a state of the art survey of ‘Platonism in Metaphysics’, and concludes thatmathematical objects can be accounted for in three ways: physical (in the world), mental (inthe mind), and abstract (in Plato’s realm of ideal objects or some equivalent) His account is

of course much more subtle and nuanced than this one sentence summary It is an impressivedisplay of intellectual prowess, although I get the feeling that professional philosophers likeBalaguer turn the virtue of subtle reasoning, through excessive application, into a vice Thesethree possibilities seem limited in that mathematical objects cannot obviously be regarded aspurely physical or mental4, which leaves open only the possibility that mathematical objectsare abstract objects, confirming Platonism, or some variant of it Balaguer himselfacknowledges some concerns about his categorization, perhaps as a gambit to stave off thepossible criticism that mathematical objects could also be seen as social constructions

There are a couple of worries one might have about the exhaustiveness of thephysical-mental-abstract taxonomy First, one might think there is anothercategory that this taxonomy overlooks, in particular, a category of social objects,

or perhaps social constructions It seems, though, that social objects wouldultimately have to reduce to either physical, mental, or abstract objects Balaguer(2004: note 3)

Here he makes the unsubstantiated assumption that any other category, such as that of sociallyconstructed objects, can be reduced to one of his three categories This introduces anotherideological presupposition, namely that of reductionism By reductionism I mean thepositivist doctrine, much beloved of the Logical Empiricists, that that there is an hierarchy ofobjects, theories or disciplines, and it is possible to translate and replace objects, theories ordisciplines further up the hierarchy into the objects, terms, concepts and theories lower downthe hierarchy without loss of generality or scientific significance Thus according to thisscheme, sociology can be translated (reduced) to psychology, psychology to biology, biology

to chemistry, chemistry to physics, and finally physics to mathematics (A further step, thereduction of mathematics to logic, the goal of Logicism in the philosophy of mathematics,was shown to be a failure See, for example, Ernest 1991, 1998, Hersh 1997, Kline 1980).5

One of the most notable critics of reductionism is Feyerabend (1975), who argues that there issemantic instability within and across theories or disciplines In Feyerabend (1962) he claimsthat meanings of the same terms and concepts in different theories are not only different but

4 I put to one side Intuitionism in the philosophy of mathematics, which argues that the concepts and results of mathematics are purely mental constructions Intuitionism is problematic in that it rejects much of classical mathematics (the non-constructive and infinitary parts) and has never managed to explain how different minds all arrive at identical mathematical results, especially as it holds that mathematics is a prelinguistic activity Lakoff and Núñez (2000) also claim that mathematical concepts are mental in origin, but unlike Intuitionism their theory of embodied mind rejects the separation of mind and body

5 There has been strong critical and negative reaction among many mathematicians and philosophers to social philosophies of mathematics such as proposed in Hersh (1997) and Ernest (1998) Part of the common critique strategy is to mis-represent such philosophies as saying that mathematics can be reduced to sociology (i.e., that mathematical matters are decided by ‘mob rule’) If these critics subscribe to reductionism, which is not uncommon, then the threat is that by ‘joining the ends’ mathematics is knocked off its pedestal as the foundation

of knowledge, and the whole chain of disciplines closes on itself in a vicious circle, knowledge eating itself like the worm Ouroboros No wonder social constructivist philosophies of mathematics are viewed as a nightmarish threat!

are also incommensurable, and that ‘theoretical reduction’ is not reduction but the

replacement of one theory and its ontology by another The same year Kuhn (1962) also

published his seminal work on the structure of science revolutions in which he argues that theconcepts of competing theories are incommensurable Although the claims of strongincommensurability did not stand up to criticism, there is a powerful argument from holismthat refutes the reductionist claims

Consider for example, the reduction of psychology to biology We know that human minds areultimately based in a material organ Furthermore, specific areas and functions of the organ(the brain) can be correlated with particular mental activities A range of different chemicalshave profound and sometimes predictable impacts on thinking Nevertheless, there is noforseeable possibility that the full complexity of human thinking and behaviour as a wholecould be reduced to a biological model The kinds of mental elements that can be found tocorrespond to biological processes are so very simple and disconnected that there is noprospect that current biology could explain human thoughts, feelings and behaviour as awhole The earlier attempt by Behaviorism to ignore the mind and to try to explain behaviourscientifically is a well known failure

I have suggested that both epistemological and ontological reductionism are fatally flawed

We cannot simply define away complex objects or bodies of knowledge in terms of simplerones In particular, I strongly believe that social constructions cannot be reduced to eitherphysical, mental, or abstract objects, for they combine elements of all three This tripartiteontology mirrors Popper’s (1979) definition of three distinct worlds, each with its own type ofknowledge

We can call the physical world ‘world 1’, the world of our conscious experiences

‘world 2’, and the world of the logical contents of books, libraries, computer

memories, and suchlike ‘world 3’ (Popper 1979: 74)

Ironically, this also mirrors the tripartite division, much beloved of ‘New Agers’ into Body,Mind and Spirit I draw this parallel out of more than naughty playfulness For it seems to methat the Platonic realm of abstract entities, World 3, the world of the Spirit, and even Heavenand the Kingdom of God, all require irrational belief or faith Positing them does not simplifythe task of understanding the mathematical text Rather it defers a key element of thatunderstanding, removing it to what I regard as an inaccessible realm What is needed instead

is what Restivo (1993) has aptly termed the Promethean task of bringing mathematics to earth

One approach, which has some currency if not widespread acceptance, that I have appliedelsewhere (Ernest 1998), is to redefine ‘objective’ mathematical knowledge (using the term

‘objectivity’ without subscribing to absolutism in the epistemological realm or idealism inontology) as social and cultural knowledge that is publicly shared, both within themathematical community, and more widely as well In mathematics, this includes all thatPopper counts as objective knowledge, including mathematical theories, axioms, problems,conjectures, proofs, both formal and informal However, I also want to include the shared butpossibly implicit conventions and rules of language usage, and a range of tacit understandingsthat are acquired through participation in practices These are shared in that they are deployedand learned in public, for persons to witness But they may remain implicit if only theirinstances and uses are made public, and any underlying general rules or principles are rarely

or never uttered According to such an account, explicit objective knowledge is made up oftexts that have been socially constructed, negotiated and accepted by social groups and

institutions Naturally, such texts have meanings and uses both for individuals and for socialgroups

What I have briefly indicated is a social theory of objectivity that resembles, at least in part,proposals by Bloor (1984), Harding (1986), Fuller (1988) and others In some variant oranother, such a view also underpins much work in the sociology of knowledge and in post-structuralist and postmodernist epistemology.6 By subscribing to an approach that demystifies

‘objectivity’ I am suggesting doing away with the ontological category of abstract objects,which often amounts to Plato’s world of pure forms That is, I have applied a principle ofontological reduction I am in good company here Quine (1969) has argued for ‘ontologicalparsimony’, the principle that we should apply Occam’s razor and not allow entities and theirtypes to multiply beyond what is necessary Ryle (1949) has also argued convincingly that theontological separation of mind and body is also a mistake Although the mental cannot besimply reduced to the physical, this is an epistemological non-reducibility, not an ontologicalone So if we reject separate ontological categories for the mental (mind) and the physical(body), we end up with a unified realm of being

In my view, the universe is made up of not three types but one type of ‘stuff’, namely thematerial basis of physical reality Within this unified world there are among the myriads ofthings and beings, humans with minds and groups of humans with cultures Human minds, theseat of the mental, are not a different kind of ‘stuff’, but are a complex set of functions of selforganising, self aware, feeling moral beings Mathematical knowledge, like other semiotic andtextual matters, is made up of social objects These are simultaneously materially represented,given meaning by individuals and created and validated socially

This discussion may seem like an excursus on the way to opening up the mathematical text,but its function is to show that there is a very deeply entrenched ideology, all the way up tothe highest intellectual levels, that regards mathematical text as a vehicle that describessuperhuman and objective mathematical reality According to this view, when the text iscorrect, and it thus counts as expressing mathematical knowledge, it truly describes thisreality So from this perspective, mathematical knowledge texts are mirrors that reflect a truestate of affairs in a timeless, objective, superhuman Platonic realm

Richard Rorty (1979) argues that the assumption the knowledge or the mind mirror nature is amajor stumbling block in philosophy, and in rejecting this, identifies himself as postmodern,

“in the rather narrow sense defined by Lyotard as 'distrust of metanarratives'.” (Rorty 1991:1) He goes on to argue that mathematical knowledge, for example, the Pythagorean Theorem,

is accepted as certain because humans are persuaded it is true, rather than because it mirrorsstates of affairs in ‘mathematical reality’

If, however, we think of "rational certainty" as a matter of victory in argumentrather than of relation to an object known, we shall look toward our interlocutorsrather than to our faculties for the explanation of the phenomenon If we think ofour certainty about the Pythagorean Theorem as our confidence, based onexperience with arguments on such matters, that nobody will find an objection to

6 Some thinkers in postmodernity, e.g., Nel Noddings (1990) and Richard Rorty (Ramberg 2002) have styled themselves as post-epistemological, because they want to reject the absolutist and foundationalist presuppositions of traditional epistemology However, just as I want to retain but redefine ‘objectivity’ in a non- absolutist and non-idealistic way, so too do I wish to retain the term ‘epistemology’ for the theory of knowledge without any similar presuppositions I don’t believe we need to reject important terms because we don’t agree with the ways they have been used by others Terms are all the while growing, changing in meaning, and in mid- renegotiation, as Lakatos (1976) demonstrates for mathematical concepts

the premises from which we infer it, then we shall not seek to explain it by therelation of reason to triangularity Our certainty will be a matter of conversationbetween persons, rather than an interaction with nonhuman reality Rorty (1979:156-157)

Thus Rorty shares the view expressed above that it is social agreement, albeit in a complexand non-whimsical way, that provides the foundation for mathematical knowledge The truth

or otherwise of a mathematical text lies in its social role and acceptance, not its relation tosome mysterious realm

WHAT IS MATHEMATICAL TEXT?

In keeping with modern semiotics I want to understand a text as a simple or compound signthat can be represented as a selection or combination of spoken words, gestures, objects,inscriptions using paper, chalkboards or computer displays, as well as recorded or movingimages Mathematical texts can vary from, one at extreme, in research mathematics, printeddocuments that utilize a very restricted and formalized symbolic code, to at the other extreme,multimedia and multi modal texts, such are used in kindergarten arithmetic These can include

a selection of verbal sounds and spoken words, repetitive bodily movements, arrays of sweets,pebbles, counters, and other objects, including specially designed structural apparatus, sets ofmarks, icons, pictures, written language numerals and other writing, symbolic numerals, and

so on

The received view is that progression in the teaching and learning of mathematics involves ashift in texts from the informal multi-modal to the restrictive, rigorous symbol-rich writtentext It is true that, for some, access to the heavily abstracted and coded texts of mathematicsgrows through the years of education from kindergarten through primary school, secondaryschool, high school, college, culminating in graduate studies and research mathematics But it

is a myth that informal and multi-modal texts disappear in higher level mathematics Whathappens is that they disappear from the public face of mathematics, whether these be in theform of answers and permitted displays of ‘workings’, or calculations in work handed in tothe school mathematics teacher, or the standard accepted answer styles for examinations, orwritten mathematics papers for publication As Hersh (1988) has pointed out, mathematics(like the restaurant or theatre) has a front and a back.7 What is displayed in the front for publicviewing is tidied up according to strict norms of acceptability, whereas the back where thepreparatory work is done is messy and chaotic

The difference between displayed mathematical texts, at all levels, and private ‘workings’ isthe application of rhetorical norms in mathematics These concern how mathematical textsmust be written, styled, structured and presented in order to serve a social function, namely topersuade the intended audience that they represent the knowledge of the writer Rhetoricalnorms are social conventions that serve a gatekeeper function They work as a filter imposed

by persons or institutions that have power over the acceptance of texts as mathematicalknowledge representations Rhetorical norms and standards are applied locally, and theyusually include idiosyncratic local elements, such as how a particular teacher or anexaminations board likes answers laid out, and how a particular journal requires references toother works to be incorporated Thus one inescapable feature of the mathematical text is itsstyle, reflecting its purpose and most notably, its rhetorical function

7 Hersh draws his analogy from Goffman’s (1971) work on how persons present themselves in everyday life.

Rhetoric is the science or study of persuasion, and its universal presence in mathematical textserves to underscore the fact that mathematical signs or texts always have a human or socialcontext I interpret signs and texts as utterances in human conversation, that is withinlanguage games embedded in forms of life (Wittgenstein 1953) or within discursive practices(Foucault 1972) Texts exist only through their material utterances or representations, andhence via their specific social locations.8 The social context of the utterance of a text producesfurther meanings, positionings and roles for the persons involved Thus, in any given context,

a mathematical text or sign utterance, like any utterance, is indissolubly associated with apenumbra of contextual meanings including its purpose, its intended response, the positioningand power of its speaker/utterer and listener/reader.9 Such meanings are both created andelicited through the social context and are also a function of the meanings and positions madeavailable through the text itself Different types of meanings and intentions are intended, butperhaps the most central and critical function (and hence meaning) of mathematical texts inthe mathematics classroom is to present mathematics learning tasks to students Amathematical learning task:

1 Is an activity that is externally imposed or directed by a person or persons in

power representing and on behalf of a social institution,

2 Is subject to the judgement of the persons in power as to when and whether it is

successfully completed,

3 Is a purposeful and directional activity that requires human actions and work in the

striving to achieve its goal,

4 Requires learner acceptance of the imposed goal, explicitly or tacitly, in order for

the learner to consciously work towards achieving it,

5 Requires and consists of working with texts: both reading and writing texts in

attempting to achieve the task goal.10

A more general concept of mathematical task includes self-imposed tasks that are notexternally imposed and not driven by direct power relationships However, in researchmathematicians’ work, although tasks may not be individually subject to power relations,particular self-selected and self-imposed tasks may be undertaken within a culture ofperformativity that requires measurable outputs So power relations are at play at a levelabove that of individual tasks Even where there is no external pressure to perform, theaccomplishment of self-imposed task requires the internationalization of the concept of task,including the roles of assessor and critic, based on the experience of social power-relations, toprovide the basis for an individual’s own judgement as to when a task is successfullycompleted

8 Especially in the age of mechanical reproduction we speak as if different utterances (tokens) represent the same text (type) In such cases much can be shared between the different utterances, e.g., when the students in a class each has a copy of a set school mathematics textbook, because they are located in a shared social context Even

in this example there can be significant differences in reader reception of the text utterances When the tokens are located in different social contexts it makes less sense to refer to them as the ‘same’ text, even if this is common parlance.

9 Clearly these are attenuated if not lost when a text is taken beyond its intended audience or appears accidentally

in some social context Such appearances are a different utterances with a different social contexts, meanings and intensions For example, a school mathematical text taken from the shelf of a bookshop does not have the specific associated directives and positionings for the customer as it would have for a reader in a particular classroom (although it might elicit some comparable meanings through memories of schooling) A Babylonian clay tablet would be unlikely to have any of the same meaning for a US Army soldier finding it in Iraq as it did for the ancient scribes who created and used it

10 Gerofsky (1996) adds that tasks, especially ‘word problems’, also bring with them a set of assumptions about what to attend to and what to ignore among the available meanings

Mathematical learning tasks are important because they make the bulk of school activity inthe teaching and learning of mathematics During most of their mathematics learning careers,which in Britain continues from 5 to 16 years and beyond, students mostly work on textuallypresented tasks I estimate that an average British child works on 10,000 to 200,000 tasksduring the course of their statutory mathematics education This estimate is based on the notunrealistic assumptions that children each attempt 5 to 50 tasks per day, and that they have amathematics class every day of their school career

A typical school mathematics task concerns the rule-based transformation of text Such tasksconsist of a textual starting point, the task statement These texts can be presentedmultimodally, with the inscribed starting point expressed in written language or symbolicform, possibly with accompanying iconic representations or figures, and often accompanied

by spoken instructions from the teacher, typically a metatext Learners carry out such set tasks

by writing a sequence of texts, including figures, literal and symbolic inscriptions, etc.,ultimately arriving, if successful, at a terminal text the required ‘answer’ Sometimes thissequence involves a sequence of distinct inscriptions, for example, the addition of twofraction numerals with distinct denominators, or the solution of an equation in linear algebra.Sometimes it involves the elaboration or superinscription of a single piece of text, such as thecarrying out of 3 digit column addition or the construction of a geometric figure It can alsocombine both types of inscriptions In each of these cases there is a common structure Thelearner is set a task, central to which is an initial text, the specification or starting point of thetask The learner is then required to apply a series of transformations to this text and itsderived products, thus generating a finite sequence of texts terminating, when successful, in afinal text, the ‘answer’ This answer text represents the goal state of the task, which thetransformation of signs is intended to attain

Formally, a successfully completed mathematical task is a sequential transformation of, say, n

texts or signs ('Si') written or otherwise inscribed by the learner, with each text implicitly

derived by n-1 transformations ('ði').11 This can be shown as the sequence: (S0 ð0) S1 ð1

S2 ð2 S3 ð3 ðn-1 Sn S1 is a representation of the task as initially inscribed or recorded

by the learner This may be the text presented in the original task specification However the

initial given text presenting the task (S0) may have been curtailed, or may be represented insome other mode than that given, such as a figure, when first inscribed by the learner In thiscase an additional initial transformation (ð0) is applied to derive the first element (S1) in the

written sequence Sn is a representation of the final text, intended to satisfy the goalrequirements as interpreted by the learner The rhetorical requirements and other rules at play

within the social context determine which sign representations (Sk) and which steps (Sk ðk

Sk+1, k<n) are acceptable Indeed, the rhetorical features of the transformed texts, together

with the other rules at play and the final goal representation (Sn), are the major focus fornegotiation (or correction) between learner and teacher, both during production and after thecompletion of the transformational sequence This focus will be determined according towhether in the given classroom context the learner is required only to display the terminal text(the answer) or a sequence of transformed texts representing its derivation

11 Normally learners of school mathematics are not expected to specify the transformations used Rather they are implicitly evidenced in the difference between the antecedent and the subsequent text in any adjacent (i.e., transformed) pair of texts in the sequence In some forms of proof, including some versions of Euclidean geometry not generally included in modern school curricula, a proof requires a double sequence The first is a standard deductive proof and the second a parallel sequence providing justifications for each step, that is specifications for each deductive rule application Only in cases like this are the transformations specified explicitly.

The final transformational sequence of texts displayed by the learner, and the actualtransformations derived during the work on the task may not be identical The former may be

a ‘tidied up’ version of the latter, constructed to meet the rhetorical demands of the context,rather than the working sequence actually used to derive the answer (see, e.g., Ernest 1993).This distinction is most clearly apparent during the construction of a proof by a researchmathematician, as in the distinction between the ‘front’ and the ‘back’ of mathematicalactivity (Hersh 1988) Here the proof as first sketched and the final version for publication,both transformational sequences of texts, will almost invariably be very different Lakatos(1976) and others have criticized the pedagogical falsification perpetrated by the standardpractice of presenting advanced learners with the sanitized outcomes of mathematical enquiry.Typically advanced mathematics text books conceal the processes of knowledge construction

by inverting or radically modifying the sequence of transformations used in mathematicalinvention, for presentational purposes The outcome may be elegant texts, but they alsogenerate learning obstacles

Typically there are strict limitations on the modes of representation employed in a researchmathematicians’ published proofs, although these will vary according to the rhetoricalstandards of the particular journal and mathematical subspecialism Nevertheless, there maywell be written linguistic text, mathematical symbolism, arrays of signs, and diagrams, forexample, combined in the final text An even greater number of different modes ofrepresentation may be employed singly or together in a school mathematics text, includingcombinations and selections of symbols, written language, labelled diagrams, tables, sketches,models, and so on (even including arrayed objects and gestures, etc., where the text isspoken) In school mathematics it is common for the transformational sequence of textsproduced by students during problem solution activities to use more modes of representationthan the starting text of the task specification, or the final text, ‘the answer’ Furthermore, thetransformational text sequence produced during task directed activities may be neithermonotonic nor single branched Solution attempts may result in multi-branched sequenceswith multiple dead ends and only one branch terminating in the final ‘answer’ text, and thisonly appears when the activity is completed successfully

My claim is that virtually all mathematical activity, throughout schooling, but also in graduateand research mathematics, can be understood in terms of the production of sequences of textsthrough the application of textual transformations What distinguishes lower level from higherlevel school mathematical activity in this description are the types of texts andtransformations involved Routine mathematical activity typically involves relatively simpleinitial texts and the deployment of restricted (algorithmic) transformation rules in theproduction of sequences of texts Non-routine mathematical activities, such as problemsolving, applications, or investigational work typically involve more complex taskformulations and require some novelty and insight in selecting which transformations to applyand which elements to apply them to, in producing the sequence (Arcavi 2005)

Semiotic Systems

One of the central characteristics of mathematics both in school and in research, is the rulebased production and use of signs However, except in degenerate cases, the use of signs andrules is always underpinned by meaning.12 Even in degenerate cases, such as blind rulefollowing in school mathematics, there are meanings to the signs and processes, it is just that

12 Of course electronic computing systems need no recourse to meanings as all processes/processing is carried out by unambiguous and fully specified rules and functions (in the mathematical sense)

the student involved is not accessing them Thus signs, rules and meanings are the threecomponents of a semiotic system

Semiotic system is the main theoretical tool developed in this paper The theory of semioticsystems provides a structure for describing the signs (texts) and the transformational rulesapplied to them in both school and research mathematics This model includes both thepublicly observable features of mathematical activity and the underlying meanings thatunderpin the activities, especially the rules of textual transformation

A semiotic system is defined in terms of three components:

1 A set of signs;

2 A set of rules for sign use and production;

3 An underlying meaning structure, incorporating a set of relationships between these signsand rules (See also Ernest 2005, 2006)

Signs

The set of signs comprises both elementary signs and compound signs made up of

concatenated sequences of signs These signs constitute abstract types (after Peirce), but their tokens (i.e., instances) can be spoken or uttered via various media: written, drawn, encoded

electronically or represented by any material means

Semiotic systems can have multimodal sets of signs, such as the semiotic systems of nursery

or kindergarten arithmetic These can include a selection of: verbal sounds and spoken words,repetitive bodily movements, arrays of sweets, pebbles, counters, and other objects, includingspecially designed structural apparatus, sets of marks, icons, pictures, written languagenumerals and other text, symbolic numerals; and these inscriptions can be represented on thechalkboard, in printed texts and charts, on computer and other ICT displays, and in children’sown work on paper.13

In contrast, the semiotic system of school algebra at the lower secondary school level has forits signs constants (numerals), variable letters (x, y, z, etc.), a 1-place function sign (-), 2-placefunctions signs (+, -, x, /), a 2-place relation sign (=), and punctuation signs (parentheses,comma, full stop).14 These are typically represented as textual inscriptions on the chalkboard,

in printed texts or worksheets or in student written work In practice, the set of signs changesover the course of schooling Early on, in the introduction to algebraic notions during theelementary school years, a blank space ‘ ’, empty line ‘_’ or empty box ‘□’ may be usedinstead of a variable letter Later, after the introduction of school algebra in secondary schoolincluding the signs listed above, further primitive function signs are introduced, includingsin, cos, tan, 1- and 2-place function signs (xc, xy), etc At all levels in school algebra, theformal signs may also be supplemented with written language (e.g., English, French, etc.)

13 Unlike some researchers, e.g., Duval (1995), Hayfa (2006), I do not see the need to create separate semiotic systems for different types of sign/register/mathematical topic I prefer a simple all-encompassing definition of a semiotic system because it is more flexible and offers more generality than multiple system types Although there are complexities involved in coordinating different registers within one system, especially for learners, the ultimate educational goal is for unity to triumph over difference

14 Technically I should put, e.g., ‘=’ for =, in this account, but instead I am following common usage to allow = to stand ambiguously for both a 2-place relation sign in the object language and for the metalinguistic sign (‘=’) that names it in the metalanguage, where my discussion takes place

Strictly speaking, if we add new primitive signs to a semiotic system we have changed it to anew semiotic system However, in practice we often act as if we are extending a singlesemiotic system, or uncovering further parts of a single semiotic system.15

Rules

The rules for sign use, combination and production in a semiotic system can be analysed into

3 types, syntactic, semantic and pragmatic, after Morris (1945) The syntactic rules are based

only on the signs qua signs, such as rules for producing well formed formulas (WFFs), etc.

Thus ‘2x4=8=’ is not a WFF whereas ‘2x4=8’ is one, because ‘=’ is a 2 place relation sign thatfor syntactic correctness must be combined with 2 well formed term (WFT) signs Thecomplexity property of WFFs and WFTs described below is also syntactically defined.

Semantic rules concern the dimension of sign interpretation and meaning(s) Thus, forexample, deriving ‘2x=4’ from ‘2x+3=7’ in a semiotic system incorporating school algebracan be justified in terms of the meanings of the signs ‘2’, ‘3’, ‘4’, ‘7’, ‘x’, ‘+’ and ‘=’ Interms of significance the dominant sign in these expressions is the binary equality relation ‘=’

and this has an underpinning informal meaning of balance that must be respected to preserve

truth The import of this is that whatever operation is applied to one of the binary relationsign’s arguments (one of its ‘sides’) must also be applied to the other Another feature at play

in this example is an implicit heuristic of simplification This seeks to reduces the complexity

of terms in an equation en route to solution.16

Pragmatic rules include contingent and rhetorical rules and these are determined purely bysocial convention Examples include teacher requirements that students label answers with the

prefix ‘Ans =’, and double underline the answer and this prefix (Ernest 1993) Likewise, in

university and research mathematics the end of a proof is commonly signified with theHalmos bar ‘ ’ (analogous to the classical QED) Such pragmatic rules are socially imposed oragreed conventions which are immaterial to syntactic and semantic correctness

Semiotic systems in school and research mathematics extend from highly informal systemswith rules that are largely implicit or tacit, to at the other extreme, totally formalized systems

in which the rules are fully explicit An example of an informal system is the semiotic systemand practice of arithmetical word problems The symbols are alphanumeric; letters, numbers,words, and elementary arithmetical operations (+, -, =); and most rules are implicit;concerning the translation of written sentences into numerals and operations, and thencomputing the answer For example, “Mary has four sweets Jim has three sweets How many

15 Within limits this practice is justifiable, as we can embed a ‘substructure’ (a structure in which each of the constituents sets of a semiotic system is a subset of the corresponding sets within a greater ‘superstructure’) conservatively, within the said superstructure However, the enlargement of a semiotic system, such as extending the semiotic system of natural numbers to that of integers can lead to new metatheoretical results (e.g., multiplication can produce products less than either multiplicand) contradictory to the state of affairs in the system of natural numbers, leading to epistemological obstacles (Ernest 2006) Arzarello (2006) extends the notion of a semiotic system to that of a semiotic bundle This comprises a collection of semiotic systems and a set of relationships between the systems This notion would allow a natural way of treating families of semiotic systems related by the addition of further signs, rules or meanings, as discussed in the text

16 The complexity C of an expression (term or formula) is defined inductively in terms of its syntactic structure The complexity of an atomic expression t, denoted C(t) = def 1 Given a set of k expressions t 1 , t 2 , …, t k , the maximum complexity of which is n, and a k-place function or relation symbol F, the complexity of the expression Ft 1 t 2 …t k , denoted C(Ft 1 t 2 …t k ) = def n+1 Complexity is used as a measure of the impact of transformational rules on a term constituting one of the arguments (‘sides’) of an equation When a legitimate (i.e., rule following) transformation of the equation achieves a reduction of the complexity of the equation or its terms, normally, the task is closer to completion The simplification heuristic motivates the use of rules to reduce the complexity of terms or expressions, and it is normally implicated throughout the solution of algebraic and arithmetical equations, for task goals are typically based on maximal simplification of texts.

do they have altogether?” This can be translated into “Four and three; Add”, or ‘4+3=_’ Butthis translation is based on heuristics, such as ‘How many altogether’ suggests ‘add’ Thiscannot be turned into an explicit rule or algorithm, as the correctness of this interpretationdepends on the context; the sense of the whole set of statements As has been noted in theliterature (Brown and Küchemann, 1976, 1977, 1981), children’s attempts to algorthmicizethis heuristic leads to incorrect surface rules such as ‘more’ or ‘total’ translate to ‘+’.

An example of a totally formalized system in which the rules are made as fully explicit aspossible is given by the formulation of the propositional calculus in Church (1956) Here therules of deductive inference are specified explicitly and completely.17 Even in totallyformalized semiotic systems, the rules are based on the underlying meaning structure It is justthat the process of distilling these meanings into explicit rules has reached a stage ofcompleteness, and so the rules can be applied in the generation of signs without reference tothe underlying meaning structure The possibility of neglecting meanings in formalizedmathematics was noted three centuries ago by Berkeley (1710: 59): “in Algebra, in which,though a particular quantity be marked by each letter, yet to proceed right it is not requisitethat in every step each letter suggest to your thoughts that particular quantity it was appointed

to stand for.” Thus meaning gives rise to rules with the result that the meaning structure can

be neglected (sometimes only temporarily) during the use of a semiotic system The entireedifice of mechanical calculation upon which digital computing is based depends on thisproperty Computers apply rules to signs in completely formalized semiotic systems in whichthere is no possibility of meaning once the computing structure is finally realized by humans Historically, the development of semiotic systems involves the distillation of the implicitmeanings of sign use, with their constraints and affordances, embodied in meaning structures,into the more explicit system rules But it should be noted that any translation of rules andmeanings from a implicit system (the meaning structure) into a more explicit form (the rules

of a semiotic system) is at best a homomorphism and can never be an isomorphism Intranslating relatively vague meanings into something more explicit, some elements ofmeaning must always be lost and further elements must always be added Refinement andexplication of meanings involves the social construction of new meaning, and considerablehuman creativity and ingenuity may be involved

But except for artificial examples of totally formalized systems, virtually none of the semioticsystems at play in mathematics education or mathematical research have fully explicit sets of

rules This because semiotic systems in use, equivalent to Saussure’s systems of parole, are

typically based on implicit or tacit rules inspired by the underlying meaning structure Insemiotic systems, transformations of signs are typically permitted provided that they preservekey meanings in the underlying meaning structure In school algebra this is the truth (balance)

of equations, and also the relevance of transformations In arithmetical word problems this isthe arithmetical meanings and values of the inherent arithmetic in the task statements Inpropositional calculus it is the truth value of the propositional signs However, for each ofthese three types, the rules are in many cases implicit, and are very often acquired by learners

or practitioners as ‘case law’ from the social use of semiotic systems

17 But note that the rules specifying, for example, which axioms may be inserted into deductive sequences must either be given as metalinguistic schemas, e.g., P(QP), where P and Q are metalinguistic variables ranging over and replaceable consistently by any well formed formulas, or as an infinite set of all possible instances of replacements in this expression within the object language (i.e., the signs of the semiotic system), or where there

is a single instance included as a privileged sign (axiom), e.g., p(qp), where p and q are elementary propositional variables, coupled with a metalinguistic rule of replacement for Q by Q(p/q), whereby all instances

of p occurring in Q are replaced by instances of q in Q(p/q), such that if Q is true or assertable, so is Q(p/q).

In research mathematics, especially in the foundations of mathematics, the ideal semioticsystem aims to dispense with all uses of the underlying meaning structure, so that the rules forsign use and production can be specified fully explicitly in syntactic terms However, this cannever be fully achieved except in trivial cases because the specification of syntactical rulesrequires metamathematical formulation Thus even the specification of the propositionalcalculus in purely syntactical terms (as discussed above) requires metamathematicalconstants, variables and expressions, and their deployment depends on an underlying meaningstructure Hilbert (1925), in his contributions to the philosophy of mathematics, freelyacknowledged that a meaningful but finitistic metamathematics is ineliminable whenmathematical theories are presented fully formally, i.e., with all their rules presented explicitly

in syntactical form (Neumann 1931)

Meaning Structure

The underlying meaning structure of a semiotic system is the most elusive and mysteriouspart, like the hidden bulk of an iceberg It is the repository of meanings and intuitionsconcerning the semiotic system which support its creation, development, and utilitization(Arcavi 2005) For individuals it can range from a collection of tenuous ideas and fleetingimages (Burton 1999), to something more well defined, akin to an informal mathematicaltheory

The meaning structure of a semiotic system can be described in three (equivalent) ways, as:

1 A set of mathematical contents;

2 An informal mathematical theory;

3 A previously constructed semiotic system

A meaning structure is a loosely associated set of mathematical contents that can include:signs, concepts, objects, properties, functions, relationships, rules, procedures, methods,heuristics, classifications, problems, examples, ideas, images, metaphors, models, structures,representations, propositions, theorems, arguments, proofs, theories, etc It is a reservoir ofmeanings that can be drawn upon in formulating, developing and operating a semiotic system,such as the metaphor “equality is balance” for simple algebra (Sfard 1994)

An informal mathematical theory can serve as the meaning structure for a semiotic system inwhich the sets of signs and rules constitute a formal mathematical theory All formalmathematical theories, I claim, have an underlying informal theory serving as its meaningstructure Lakatos (1978) argues that such an informal theory provides the touchstone forevaluating a formal mathematical theory The theorems of the informal theory are potential

‘heuristic falsifiers’ through which the success of the formal theory can be judged, according

to whether it captures or contradicts them

Two different formal theories are generally regarded as equivalent if their signs and rules areinter-translatable, and they both share the same informal mathematical theory as theirmeaning structure Thus various formulations of Peano arithmetic, which may vary in theirsets of signs or rules, e.g., the choice of 0 or 1 as the first numeral sign, are regarded asequivalent in this way, even though they are distinct semiotic systems

Additionally, a previously constructed semiotic system can serve as a meaning structure for anew semiotic system Since the meaning structure of a semiotic system can include signs andrules, this possibility is already inherent the first of the descriptions given above It is alsopossible for more than one existing semiotic system to be drawn upon or combined to make

up a meaning structure, and not all of these need to be mathematical systems The

incorporation of entire or elements of non-mathematical semiotic systems into the meaningstructure provides the potential for links between mathematics and other human ways ofrepresenting our experiences and environment

This third case sounds circular, but it is not Rather it is a case of what Peirce terms ‘unlimited

semiosis’, in which the interpretant of a sign is itself a new sign, and so on, ad infinitum For a

semiotic system as a whole can be regarded as a sign, with its underlying meaning structureconstituting its interpretant (from the triadic, i.e., Peircean, perspective of signs) or itssignified (from the dyadic, i.e., neo-Saussurian perspective of signs) Like any other sign this

is linked via its origins, its uses in practice, its meanings and its associations with other signs,

or in this particular case, with other semiotic systems

Often it will be the case that a newly developed or elaborated semiotic system (for anindividual or group of learners) will be more formally and explicitly specified than thepreviously constructed or utilized semiotic system which serves as its meaning structure Inthe teaching and learning of mathematics it is commonly one of the goals of instruction toincrease the abstraction, complexity and formality of the semiotic systems to which learnersare introduced and inducted, over the course of time Hence this gradient of increasedformality and explicitness

Such processes are evident in institutionalized mathematics teaching at all levels In schoolmathematics the study of number properties and manipulations in numerical calculationprecedes and provides the meaning structure for elementary algebra Operations in numbersystems not only grow in complexity in the passage from the Natural Numbers, via Integers,Rationals and Algebraic Numbers to Real Numbers, but also each of these transitions takesthe semiotic system as the meaning structure for the next semiotic system of number that issequentially developed Typically in university mathematics the study of concrete structuressuch as sets and number systems precedes the study of algebraic structures such as group, ringand field theory, and the study of informal or ‘nạve’ set theory (Halmos 1974) precedes thestudy of axiomatic set theory In each these examples the study of an relatively informalsemiotic system precedes the study of its relatively more formal counterpart, and I claim,provides a central part of the meaning structure of the subsequently developed, more formalsemiotic system

The theory of semiotic systems provides a model for describing the teaching and learning ofmathematics in school In learning any school mathematics topic in the form of a semioticsystem, learners are inducted into a discursive practice involving the signs and rules of thatsystem Teachers present tasks in the form of signs, and present rules for working ortransforming the signs for accomplishing the tasks Most commonly the rules will beexhibited implicitly through worked examples, particular instances of rule applications, ratherthan explicit rules stated in their full generality Through observing the examples, working thetasks, and receiving corrective feedback, learners internalize, build and enrich their personalmeaning structures corresponding to the semiotic system

Trying to teach rules explicitly rather than through exemplification can lead to what I term the

‘General-Specific paradox’ (Ernest 2006) If a teacher presents a rule explicitly as a generalstatement, often what is learned is precisely this specific statement, such as a definition ordescriptive sentence, rather than what it is meant to embody: the ability to apply the rule to arange of signs.18 Thus teaching the general leads to learning the specific, and in this form it

18 This also applies to any general item of knowledge that is applicable in multiple and novel situations, such as a mathematical concept, rule, generalised relation, skill or strategy.

does not lead to increased generality and functional power on the part of the learner Whereas

if the rule is embodied in specific and exemplified terms, such as in a sequence of relativelyconcrete examples, the learner can construct and observe the pattern and incorporate it as arule, possibly implicit, as part of their own appropriated meaning structure This is howchildren first acquire the grammatical rules of spoken and written language Thus the paradox

is that general understanding is achieved through concrete particulars, whereas limited andspecific responses may be all that results from learning general statements This resembles theTopaze effect (Brousseau 1997), according to which the more explicitly the teacher stateswhat it is the learner is intended to learn, the less possible that learning becomes For thelearner is not doing the cognitive work (meaning making) that constitutes learning, butfollowing surface social cues to provide the required sign – the desired response or answer

The pattern whereby a learner first learns the use of signs through observation andparticipation in public sign use in discursive practices embodies the well known dictum ofVygotsky 1978: 128) “Every function in the child’s cultural development appears twice, ontwo levels First, on the social and later on the psychological level; first between people as aninterpsychological category, and then inside the child as an intrapsychological category.” ThisVygotskian scheme can be represented as a cyclic pattern for learners’ appropriation of signsand the rules of sign-use through participation in a discursive practice The pattern isillustrated in Figure 1 (after Ernest 2005)

Figure 1 Model of Sign Appropriation and Use in Learning

S OCIAL L OCATION

Public utilization of sign toLearner’s public

express personal meaning (Public & Individual)

Conventionalisation



Social (teacher & others) negotiated and conventionalized (via critical acceptance) sign use (Public & Collective)

M ANIFESTATION Publication   Appropriation

Private Learner’s development of

personal meanings for sign and its use (Private & Individual)



Transformation

Learner’s own unreflective response to and imitative use of new sign utterance (Private & Collective)

The cycle shown in Fig 1 has four phases: Appropriation, Transformation, Publication andConventionalisation, any one of which can be taken as the beginning, for the cycle repeatsendlessly In the figure Vygotsky’s two levels are shown, first, by the top right quadrant,where the socio-cultural is represented as both public and collective, and second, by thebottom left quadrant, where the (intra)psychological is represented as both individual andprivate This latter constitutes the notional space where a learner constructs his or her meaningstructure The other two quadrants are crossing points on the boundary between these twolevels, and these are the locations where the learner semiotic agency is acted out

In the development of a personal meaning structure a learner draws on further resourcesbeyond those indicated in Figure 1 These include existing meanings and the meaningstructures of other semiotic systems already partly mastered, as well as meta-discussions ofsign production and use The latter are partly included in Fig 1 in terms of social negotiationand critical acceptance

The model indicates schematically the route through which learners appropriate the rules ofsign-use, mostly through observing their exemplification in practice The earliest uses of asign or rule can be based on simple imitation, corresponding approximately to Skemp (1976)and Mellin-Olsen’s (1981) notion of instrumentalism, because of the performativity involved.Later, after a sequence of related appropriations, performances and conventionalisations, theuse of the sign is transformed through the development personal meanings including a sense

of where and how the sign is to be used acceptably, and a whole nexus of other associations

The successful appropriation and transformation of a sign, with its nexus of associated

meanings and meta-discourse, parallels Skemp’s (1976) notion of ‘relational understanding’.This involves both being able to use the sign correctly, corresponding to conventionallyaccepted usage within the micro-community of the classroom under the authority of theteacher, and being able to offer a rationale or explanation for the usage

The next phase is that of publication, in which individual learners engage in conversational

acts of sign utterance These can vary from quick, spontaneous verbal, gestural or writtenresponses to a question or other stimulus, through to constructing extended texts elaboratedand revised over a period of time, prior to offering them to others A group of learners canelaborate such texts co-operatively, but such processes subsume several or even many sub-cycles in which individuals utter signs to others in the group in an extended conversationgiving rise to jointly elaborated, negotiated and agreed texts

The cycle is completed through the process of conventionalisation, in which signs are uttered

within the classroom conversation and are subjected to attention, critique, negotiation,reformulation and acceptance or rejection, primarily by the teacher Teacher approval isnormally the final arbiter of acceptance, because of the power and authority relations in theclassroom Typically the conventionalized sign utterance that is accepted will satisfy threecriteria:

1 Relevance The sign or text is perceived to be a relevant response or putative solution (or

an intermediary stage to one) to a recognised (i.e., sanctioned) starting sign which hasthe role of a task, question or exercise This might be teacher imposed or otherwiseshared and authorized

2 Justification The mode and steps in the derivation of the sign from the authorized starting

point will normally be exhibited as a semiotic transformation of signs, employingacceptable rules or sign transformations within the semiotic system, or justified meta-linguistically.19

3 Form Both the signs and their transformations will normally exhibit teacher-acceptable

form, thus conforming to the rhetoric of the semiotic system involved as realized anddefined in that classroom This system could be that of spoken verbal comments, drawnand labelled diagrams, numerical calculations, algebraic derivations, or somecombination of these or other sign types (Ernest 1993)

An overall schematic model of a semiotic system within its social context of use is given inFigure 2 This summarizes the three parts that make up a semiotic system; the signs, rules andmeaning structure The signs can be elementary, or composite, and indeed sequences of signswill often be all that is explicitly exhibited in the operation of the semiotic system Many rules

19 Sign transformations do not always mean the replacement of one or more parts of a compound sign by different parts, with the retention of the unreplaced parts It may involve the construction of a wholly new sign in the sequence For example, axiom use in a logical proof, can involve the insertion of a new sign with no components shared or overlapping with the previous step.

will often be implicit, exhibited only via specific applications as sign transformations, as well

as explicitly presented rules For learners the meaning structure will be developed as thesemiotic system is utilized, although learners will also draw on other, partially mastered,semiotic systems as well as their general repertoire of meanings, including some of the itemslisted, from signs and concepts to the relevant informal mathematical theory

Figure 2 Model of Semiotic System within its Social Context of Use

SOCIAL CONTEXT OF USE

Meta-Discussion of Semiotic System in Use Aims, Goals, Purposes, Concept of TaskRoles, Positions, Power Relations, Relations with Social Institutions

The use of semiotic systems always takes place within a social context Within social settingsthere are persons and their roles, positions, power relations, and relations with socialinstitutions such as schools An important dimension of social understanding as it relates tosemiotic systems is the concept of school learning task and the aims, goals, and purposes ofschool work, that is presupposed by operating semiotic systems in school settings Thetransformation of signs in semiotic systems is directional, and the understanding ofdirectionality in general is socially acquired in a variety of social settings including home andschool.20 Ultimately, directionality in activities results from directions given by a person in

20 Note that the simplification heuristic described above plays a central role in operationalizing directionality in mathematical tasks That is, a significant part of the appropriation of directionality is associated with the implicit