Operationalising the ZPD as a Semantic Space in the 'I Spy' Trope

Contemporary Approaches to Research in Mathematics, Science, Health and Environmental Science 2013 1

Operationalising the ZPD as a Semantic Space in the ‘I Spy’ Trope

John Kusznirczuk

Melbourne Graduate School of Education

[email protected] This paper reports on a methodology that operationalises ‘objects of pedagogical knowledge’ in natural examples of mathematics classroom interaction. Using Vygotskii’s zone of proximal development (ZPD) as an example of a ‘knowledge object’, it presents a generic episode of mathematics classroom interaction, identified here as the ‘I spy’ trope, which indicates the possibility of the ZPD’s operation. The presentation of this trope is a practical demonstration of a methodology that reconciles theory and practice by providing educators with a critical vocabulary and recognisable examples of interaction that indicate the operation of ‘objects of pedagogical knowledge’ so that they can talk about their work in terms that are both empirically grounded and theoretically coherent.

The Importance of Being Able to Describe Classroom Interaction In 2007, the Melbourne Graduate School of Education (MGSE) introduced a

Master of Teaching programme that, in common with most teacher-training programmes, places a premium on graduating professionals that exhibit “a deep understanding of how students learn” (Mclean Davies 2012). What is, perhaps, unique to this programme is its emphasis on the application of this deep understanding by clinical professionals that “use theory and evidence to inform interventionist practice” (op. cit.). According to Mclean Davies it is not enough that teachers know “what works”, they need to be able to explain why it works, and they need to be able to do so on the basis of empirical evidence. The “key thing” driving this Masters’ programme, according to Maclean Davies, is that “theory and practice are not divided”. She summarises what master teachers need, and what the MGSE requires of them, with the deceptively simple observation that “teachers need to be able to talk about their work.” (Mclean-Davies, op. cit.).

My aim is to contribute to teachers’ and researchers’ abilities to talk about their work by reconciling theory and practice with respect to the interaction that happens in the middle-school mathematics classroom and the explanatory principles that account for the possibility of learning school-mathematics there. This paper demonstrates how I do so using a methodology that provides educators with a critical vocabulary and recognisable examples with which to continuously test and develop their own understanding of objects of pedagogical knowledge in a shared discourse of practice.

Kusznirczuk Operationalising the ZPD


Systematically Operationalising the ZPD The methodology demonstrated here is designed to systematically operationalise

objects of pedagogical knowledge by describing actual pedagogical situations. The mathematics period, an object of social order, is its unit of analysis, where ‘periods’ are conceptualised as being constituted by ‘episodes’ of interaction (Clarke 2001) and, in turn, as constituting topical ‘units of work’.

Using explanatory principles taken from psychology, sociology, linguistics and poetics, the following steps have been applied to the analysis of video-records of Australian middle-school mathematics periods to arrive at the description of a familiar episode of interaction, identified here as the ‘I spy’ trope, that indicates the possibility of the ZPD’s operation.

1. Draw on the literature to synthesise an interdisciplinary organising framework for the objects used to describe classroom interaction. [Figure 1a shows this framework’s three ‘planes’ of analysis; expression, content and context, divided into six analytical strata, each with qualitatively distinct units of analysis and analytical tools.]

2. Draw on records of natural examples of mathematics classroom interaction to identify recognisable episodes, referred to here as tropes, and analyse their structural and functional features at the scale of ‘rhetorical units’ (Cloran 1994). [The ‘text semantics’ stratum in Figure 2 shows how rhetorical units are conceptualised in relation to other units of analysis in the organising framework. Figure 3 shows Cloran’s rhetorical units on a cline of deictic distance (from nearest, on the left, to most distant, on the right). Because my analysis is concerned with semantic objects, and following Halliday’s invocation of ‘trinocular vision’ (Halliday 1978), the three analytical strata of interest here are the social order, text semantics and lexicogrammatical strata, shown in Figures 1b & 2. Figure 2 is a representation of these three strata that includes each stratum’s units of analysis. Appendix 1 shows a coded transcript of an episode of middle-school mathematics classroom interaction.]

3. Use the literature to unpack characteristic features of the ZPD (an object of pedagogical knowledge). [See Appendix 2 for my reading of the ZPD.]

4. 'Translate' these characterisations of the ZPD into the semantic features of actual classroom interactions that indicate the possibility of the ZPD's operation. [Table 1 shows this step carried out on the episode analysed here.]

The Trope as a Recognisable Episode In the social order stratum, I treat the episode as my radical (irreducible) unit of

analysis and take it that certain episodes of interaction are so familiar that they can be classified as tropes. In common use, the term ‘trope’ refers to ‘a figurative use of a word or expression’. I extend the use of the term to include the figurative labelling of familiar episodes of interaction. According to this use, tropes are realised by generic forms of symbolic influencing behaviour that, historically, have been shown to ‘get things done’ in coordinated interaction. ‘Show and tell’ is a well-known example of what I would refer to as a classroom trope. A specific instance of interaction can be described as a trope when it is of a recognised generic kind. I can recognise a trope



because I belong to a community of practice. Through my interaction in a community, I incidentally cultivate, and I am implicitly trained in, a ‘native literacy’. This implicit symbolic knowledge can be made explicit, systematically, by describing the material features that characterise recognisable forms of symbolic interaction. By describing material interaction using explanatory principles that account for the resources that make that interaction possible, I am able to make a plausible case for the explicit description of the material conditions that identify a recognisable trope.

Figure 1a: A stratified view of language as a social semiotic

Figure 1b: The trinocular view of text semantics



Figure 2: Units of analysis, organised according to their analytical strata

Features that Indicate the ZPD’s Operation in the ‘I Spy’ Game We play ‘I spy’ with our children because it is fun, and we play it, I suggest,

because it has proven itself to be a useful pedagogical activity. To successfully take part in the game the child must recall abstract features of concrete things, the first letter in a thing’s name, for example. The game works as a learning activity when the participants already share the ability to practice recalling objects’ features. The game works as a developmental activity because it establishes a recognised context of situation in which the child is ‘pulled through’ to a future stage of development, to a stage of inner speech, where they must silently recall the abstracted features of conventionally named objects. With reference to the ZPD’s ‘unpacked’ features (see Appendix 2), it can be said that it is possible that the ZPD emerges in the ‘semantic space’ between players in the game of ‘I spy’.

The ‘I spy’ game can be characterised semantically, according to its rhetorical units, as a generalisation embedded in a commentary (“I spy with my little eye…”) (Figure 3 lists Cloran’s rhetorical units). The game entails an expectation that players are concerned with identifying this co-present concrete object based on the commentary’s dependant clause (“something beginning with…”).

Figure 3: Rhetorical units on a cline of deictic distance (Cloran, 2010, p. 59)



Features that Indicate the ZPD’s Operation in the ‘I Spy’ Trope The episode of mathematics classroom interaction presented in Appendix 1 can

be treated as a trope because it is recognisable (teachers will ‘know’ it). My warrant for calling it the ‘I spy’ trope is that it shares structural and functional features of the game after which it is named. This trope is described according to its rhetorical units as a generalisation embedded in a conjecture.

The difference between this trope and the game that it emulates is a developmental one; it concerns the different natures of the two activities’ objects of interest. In the ‘I spy’ trope, the principle rhetorical unit is a conjecture concerning two abstract objects in a class relationship. In the children’s game, the principle rhetorical unit is a commentary based on the correspondence between a co-present concrete object and its generalised description.

In the tropical episode of ‘I spy’ analysed here, the teacher wants to know if her pupils can ‘see’ what she ‘sees’ with respect to the abstract relationship between squares and rectangles. The evidence shows that her pupils have learned to generalise about the facile features of rectangles and squares (a plane shape with four sides and angles of such and such disposition and measure). The evidence also shows their teacher ‘pulling them through’ to a higher order of thinking with respect to a consequence of the relationship between those generalised features, that is, that a square is a special kind of rectangle.

The analysis performed here is possible because, from the outset, my native literacy acts as a bridge to a critical literacy whereby I can explicitly describe the features of a form of interaction that I implicitly recognise. Along with my analysis of this episode as an instance of the ‘I spy’ trope, Table 1 summarises my analysis of the empirical evidence of the ZPD’s operation in it.

Conclusion The reflexivity that connects knowledge objects to the material substance of

social interaction is both central and common to any attempt to reconcile theory and practice. In the methodology used here, knowledge objects assist in the description of aspects of social interaction, and these same aspects of social interaction constitute the empirical evidence for visible occurrence of the knowledge objects.

The value of this approach to operationalising an object of pedagogical knowledge is not in the certitude of its descriptions, it is in its systematic and transparent analysis of empirical evidence – always open to criticism and improvement in a shared discourse of practice. I am happy that my analysis be held accountable on the basis of; (i) its coherence to the theories that provide the warrants for its analytical tools, and (ii) the plausible connections made between its characterisations of knowledge objects and the empirical features of actual maths classroom interaction.

The demonstrated feasibility of an evidential base for objects of pedagogical knowledge makes a valuable contribution to theorising and practice by creating the expectation that the use of such knowledge objects be expressed in terms both empirically grounded and theoretically coherent, and by providing the means to do so in a shared discourse of practice.



Table 1 Translating figurative characterisations of the ZPD into literal indicators

Figurative description:

More ‘literal’ description:

Catch each others’ activity

Thematic cohesion is used here in lieu of the ‘coherence’ suggested by Clarke (op. cit.,) which is taken to be an object of the individual psyche and, as such, as being beyond direct observation. Cohesion, on the other hand, can be directly observed in the thematic choices made in interaction. In the episode here, it manifests itself in a sustained sequence of interaction on the theme of ‘rectangles’.

Psychological distance

Deictic distance is used here as a semantic indicator of psychological distance. Psychological distance is concerned with the construal of objects and the formation of a subject’s actions. It is measured here according to four dimensions; spatial, temporal and social distance, and an expectancy that is measured according to hypotheticality and accountability. The ZPD’s emergence depends on interaction concerning proximal objects of concern to participants. The episode analysed here indicates activity with respect to co-present objects (representations of rectangles and squares). The ZPD’s operation also depends on there being psychological distance between what is understood and what is yet to be understood. The conceptualisation of shapes like the square and the rectangle begins with the description of their superficial features as discrete ‘things’. This level of understanding is demonstrated in the episode analysed here by pupils’ abilities to recall generalised features of these shapes. Conceptualisation of the square as a rectangle represents a higher order of understanding that, apparently, pupils in this episode have not yet attained. In this episode the teacher attempts to ‘pull these pupils through’ to this higher order understanding, creating the conditions for psychological distance by asking them “what can we say about a square (given what we know about rectangles)?”

Teaching leads learning

Evidence of a learner being ‘pulled through’ to a socially sanctioned future can be found in the pupil (Leon) who performs a facile imitation of the teacher’s mode of action when he conjectures “couldn’t a rectangle be a kind of square”, in a challenge to his teacher’s generalisation, “a square is a kind of rectangle”.

Learning leads development

Evidence of ‘learning leading development’ can be found in the behaviour of pupils (Joel, Earl, Antonis, and Leon) who demonstrate a facile understanding of rectangles’ and squares’ features (without yet recognising the special relationship between these objects).



References Bernstein, B. B. (1990). Class, codes and control: Vol.4, The structuring of

pedagogic discourse. London, Routledge. Bernstein, B. B. (2000). Pedagogy, symbolic control and identity: Theory, research,

critique (often refered to as 'class, codes and control, v.5'). Lanham, Md., Rowman & Littlefield Publishers.

Clarke, D. (2001). Perspectives on practice and meaning in mathematics and science classrooms. Dordrecht, Kluwer Academic Publishers.

Cloran, C. (1994). Rhetorical units and decontextualisation: an enquiry into some relations of context, meaning and grammar Thesis (Ph.D.) School of English and Linguistics, Macquarie University, 1993., University of Nottingham.

Cloran, C. (1999). Contexts for learning. Pedagogy and the shaping of consciousness : linguistic and social processes. F. Christie. London ; New York, Cassell: 296 p.

Cloran, C. (2010). "Rhetorical unit analysis and Bakhtin's chronotype." Functions of Language 17(1): 29-70.

Halliday, M. A. K. (1978). Language as social semiotic: the social interpretation of language and meaning. London, Edward Arnold.

Maturana, H. R. (1980). Biology of cognition (1970). Autopoiesis and cognition: the realization of the living. R. S. Cohen and M. W. Wartofsky. Dordrecht, D. Reidel Publishing Company: 1 - 58.

Maturana, H. R. and F. J. Varela (1980). Autopoiesis and cognition: the realization of the living. Dordrecht, D. Reidel Publishing Company.

Mclean Davies, L. (2012). "Master of teaching information session." Retrieved Oct. 10, 2013, from http://content.lecture.unimelb.edu.au:8080/ess/echo/presentation/1e4d5e9a-2ee3-492f-923e-17dc6e01a911/media.m4v.

Meira, L. and S. Lerman (2001). The zone of proximal development as a symbolic space. Social Science Research Papers No. 13. London, South Bank University, Faculty of Humanities and Social Science.

Trope, Y. and N. Liberman (2010). "Construal level theory of psychological distance." Psychological Review 117(2): 440-463.

Valsiner, J. and R. van der Veer (1993). The encoding of distance: the concept of the zone of proximal development and its interpretations. The development and meaning of psychological distance. R. R. Cocking and K. A. Renninger. Hillsdale, N.J., L. Erlbaum: 35-62.

Vygotskii, L. S. (1978). Mind in society: the development of higher psychological processes. Cambridge, Harvard University Press.

Vygotskii, L. S. (1981). The genesis of higher mental functions. The Concept of activity in Soviet psychology. J. V. Wertsch. Armonk, N.Y., M.E. Sharpe: 144-188.

Vygotskii, L. S. (1994). Tool and symbol in child development. The Vygotsky reader. J. Valsiner and R. v. d. Veer. Oxford, UK ; Cambridge, Mass., USA, Blackwell: 99-174.



Appendix 1: A Coded Transcript of an Episode of Interaction

A1#L12lesson#clause131114.xlsx,3egIspy

13of32

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time

cls#

subj

Vctr

Clau

se/M

essage

RUsubject

finite

CEEO

17:34:28

601

TTC

O.K.3[to3all]33That,3if,

604

TTC

if3this3were3the3de

finition3[[that3we3were3using3for3a3rectangle]],3

conjecture

we

can3say

cop

hypo

thetical

605

TTC

and3it:is3((the3de

finition3of3a3rectangle)),3

observation

itis

cop

habitual

606

TTC

then

3what3can

:we:say3abou

t3a3squ

are?33

(see3#604)

17:38:08

607

TTC

This3is3th

e3last3th

ing:I'll3say,3

commen

tary

Iwill3say

cop

non#habitual

608

TTC

and3then

:we'll3move3on

.33plan

we

will3m

ove

int

volitional

17:55:19

609

TTC

For3what3can

:we:say3abou

t3a3squ

are?33

conjecture

we

can3say

int

hypo

thetical

610

TTC

What3d

oes3a3square3look3like3at3the

3mom

ent?

gene

ralisation

a3square

does

gen

habitual

17:59:06

611

TTP

[to3Joel]3Joe

l?18:00:29

612

Joel

PTall3the

3sides3are3equ

al3

613

Joel

PTand3[...]

18:03:08

614

TTP

Yeah.

18:05:09

615

TTP

[to3Earl]3E

arl?3

616

TTP

Earl,3if3a3rectangle3has3opp

osite3side

s3eq

ual,3op

posite3sides,3sorry,

620

TTP

parallel,3eq

ual,3and3four3right3angles,3and

3so3do

es3a3squ

are,3

624

TTP

what3can3we3say3abou

t3a3squ

are?

18:18:04

625

Earl

PT[...]

18:22:16

626

TTP

[Smiles]3A3squ

are?

628

TTP

It3seems3to3m

e3that3a3squ

are3might3be...3A3squ

are3might3be3a…

18:32:20

631

TTP

[to3Anton

is]3A

nton

is?

18:33:04

632

Anton

isPT

((a3square3m

ight3be))3E

quilateral

18:34:04

633

TTP

A3what?

18:35:21

634

Anton

isPT

((a3square3m

ight3be))3E

quilateral

18:36:12

635

TTP

A:squa

re:is3an3eq

uilateral.33

gene

ralisation

a3square

isgen

habitual

636

TTP

An3eq

uilateral3w

hat?33

637

TTP

No,3sorry,3

639

TTP

a3square3is...33

640

TTP

I3think3you

're:trying:to

:say3parallelogram

3there,3are3you

?33

commen

tary

you

say

int

non#habitual

643

TTP

No3((I3m

isun

derstood

?)),3((you))3go3on

.33645

TTP

Sorry,3I'm3con

fusing3th

e,3th

e3issue.3

18:36:12

648

Leon

PP[to3self]3[...]333333

18:48:15

649

TTC

[to3all]33All3I3tho

ught

650

TTC

if3that's3all3we3ne

ed3to

3get3a3rectangle,3if3that's3our3rectangle,

652

TTC

doesn't3it3loo

k3like,3our3squ

are3might3be3a3rectangle?33



A1#L12lesson#clause131114.xlsx,3egIspy

23of32

16/11/2013

time

cls#

subj

Vctr

Clau

se/M

essage

RUsubject

finite

CEEO

18:48:15

654

Pepe

PP

[to3Leon]3[...]3

18:57:29

655

S2

PP

[...]

19:03:07

656

TTC

[to3all]33It3seems3to3m

e,3that3a3square3could3be3a3rectangle.33

658

TTC

The3only3difference3about:the

:squa

re3is

gene

ralisation

the3square

isgen

habitual

659

TTC

that3it's3a3special3type3of3rectangle.33

660

TTC

And3what's3special3about3this,3what's3special3about3the3square?3

.33

662

TTC

(there3is))3Nothing,3((special3about3a3square))3apparently

19:22:25

663

Earl

PT

all3sides3equal,3((and))3all3sides3parallel.

19:24:03

665

TTP

[to3Earl]33Yep,3yep.

19:26:06

667

TTC

[to3all]33A3square3is3it's3special,3

668

TTC

if3that's3the3definition3we3use3for3a3rectangle,3

669

TTC

then3a3square,3yes,3is3a3special3type3of3rectangle,3

672

TTC

it's3special3case3of3a3rectangle,3

gene

ralisation

(a3square)

isgen

habitual

673

TTC

and3in3fact3all3the3sides,3

674

TTC

it's3a3rectangle3where3all3the3sides3are3equal.33

19:42:27

675

TTC

And:I3know3I3haven

't3convinced3m

any3of3you.

commen

tary

Ihaven't

int

non#habitual

19:45:17

677

TTP

[to3Leon]3Leon.

19:47:02

678

Leon

PT

By3that3((generalisation3about3sides3of3recatangles3&3squares))

conjecture

a3rectangle

could3be

gen

hypothetical

679

Leon

PT

couldn

't:a:rectan

gle:be

3a3special3kind3of3square?

19:50:00

680

TTP

Um,3do

es3a:sq

uare:have3sides3that3are3not3equal,3then?

gene

ralisation

a3square

does

gen

habitual

19:55:14

682

Leon

PT

No,3that's3why3I'm

3saying3a3rectangle3would3be3a3special3square.33

686

Leon

PT

It's3different3to3the3rest3of3them.33

687

Leon

PT

[laughs]3

688

Leon

PT

What's:everyon

e3laughing3at3me3for?

commen

tary

everyone

iscop

non#habitual

20:04:12

689

TTP

�I'll3talk3to3you3about3this3later,3

plan

Iwill

int

volitional

690

TTP

'cause3I,3I:w

ant3p

eople3to,3

commen

tary

Iwant

int

non#habitual

692

TTP

I3think3you've3asked3m

e3a3very3important3question,3

694

TTP

but3I3can't3spend3too3m

uch3tim

e3on3it3

695

TTP

'cause3some3people3aren't3really3interested3in3that,3

696

TTP

but3it's3a3good3question.3333

20:05:04

697

Pepe

PPWhere!did!you

,(where(did(you

(pull(tha

t(shit(from?

commen

tary

you

did

int

non7ha

bitual

20:08:00

699

Leon

PP[Lau

ghs]



Appendix 2: The ZPD Unpacked as a Semantic Space Vygotskii’s zone of proximal development (ZPD) is at risk of losing its explanatory

power if it is not understood in relation to the rest of his work. To mitigate this risk, complementary theoretical accounts of meaning-making and learning and development such as those of Vygotskii (1978; 1981; 1994), Valsiner and Van der Veer (1993), Trope and Liberman (2010), Maturana (1980), Halliday (1978) and Bernstein (1990; 2000), have been drawn on to arrive at the following characterisation of the ZPD’s operation. The ZPD is treated here as a rhetorical device, like gravity. The following characterisations are figurative descriptions of a complex explanatory principle. 1. Following Meira and Lerman (2001), the ZPD can be treated as a quality of the

‘symbolic space’ between subjects in interaction. a. By analogy with Maturana’s (1980) biological account of the autopoiesis of

living things, symbolic interaction can be characterised as ‘mutual symbolic orienting behaviour’ – the mutual influencing, or rhetorical behaviour that emerges between biologically closed and autonomous subjects in symbolic interaction. Considering communication1 as the material substance of symbolic interaction, and considering the semantic content of communication as information, then, as a consequence of the subject’s physiologically closed and autonomous nature, there is no direct transfer of information between subjects in interaction, there are only the consequent effects of subjects’ mutual orienting behaviour on their respective making of meaning, which, for any given object, may, or may not, approach some kind of congruence.2

b. If the ZPD emerges when subjects “catch each other’s activity”, as described by Meira and Lerman (2001), then such a state must be indicated by one or more kinds of congruence in their interaction. That is, there must be manifestations of some degree of alignment between subjects’ orienting behaviours, such as co-location, co-operation, co-ordination, consent, cohesion, similarity, imitation, and so on. In the physiological domain, these kinds of conditions are consistent with what Maturana describes as organisms’ ‘structural coupling’ (Maturana and Varela 1980) – their ‘mutual orienting behaviour’ in material interaction in their ‘ecological niches’. In the psychic domain, ‘mutual orienting behaviour’ can, by analogy, be considered as the ‘abductive3 coupling’ of subjects in symbolic interaction in their ‘cultural niches’. Considering speech as one semiotic mode of interaction, if there is to be any alignment between subjects’ semantic orientations, these kinds of conditions must necessarily manifest themselves with respect to some lexical, grammatical, semantic and contextual features of that interaction, as suggested by Halliday’s (1978) notion of ‘trinocular vision’4.

1 Communication, considered in its broadest sense, includes all forms of undifferentiated material interaction, from reciting a poem to punching someone in the nose. 2 Paradoxically, while information belongs to the individual subject there is a sense in which it does not. While objects and their meanings belong to the subject in his or present, they are, historically, only made possible by others. 3 After Peirce, a kind of inferential logic or ‘guessing’ where a subject abduces an hypothesis from observed phenomena on the basis that the explanation ‘seems’ to fit the observation (is sufficient, but not necessary). 4 Halliday shows us that an account of a particular class of object requires an account of other objects; (i) within the same analytical stratum, (ii) in the stratum ‘below’ that object, and (iii) in the stratum ‘above’ it. This is his ‘trinocular vision’.



2. The ZPD has no past or future (Valsiner and van der Veer 1993; Meira and Lerman 2001);

a. The ZPD exists only in the instantaneous present as a quality of the ‘symbolic space’ between subjects in interaction, as an emergent effect of their interaction (Meira and Lerman, op. cit.).

b. The likelihood of the ZPD emerging at any given moment increases with the proximity of an object of interaction to the subjects in interaction. The value that a subject attaches to an object in an activity is always referentially centred on the subject’s own present time and place (after Trope and Liberman 2010). A subject’s valuation5 of an object varies with a subject’s construal of it and its distance from the subject in time and space. The subject’s self-centred valuations of objects form his or her choices.

3. The ZPD, as an emergent effect of interaction in the instantaneous present, generates a developmental potential that is socially directed with respect to the subject’s past and future.

a. Learning leads development (after Vygotskii). A subject learns on the basis of what he or she already knows. A subject’s learned behaviour conditions the possibility of the ZPD’s emergence in any given interaction.

b. Teaching leads learning through subjects’ mutual orienting behaviours. Two of the rhetorical functions of language that feature in the classroom (in fact, in any institutional/pedagogical setting) are instruction and regulation in the sense of Bernstein’s principles of instructional and regulative discourses (ID and RD) (Bernstein 1990). Instruction and regulation serve the purpose of ‘pulling the learner through’ to a recognised future. The institution thus acts to ensure that the subject’s developmental potential is directed towards socially sanctioned ends. With respect to the role of the subject’s play and self-help in their own learning and development, and in keeping with the Vygotskian notion of ‘experience’ as a proper unit of analysis in psychological theorising (described here by means of rhetorical unit analysis), social-teaching and self-teaching become equivalent, so that the notion of the ZPD comes to emphasise “the process of construction of the future structure of the functions on the basis of the present experience6 by the child” (Valsiner and van der Veer 1993, p. 44).

Drawing on the rhetorical unit (RU) analysis of the substance of spoken interaction (Cloran 1994; 1999; 2010), which is used here to describe the movement of the ‘semantic space’ between subjects in interaction, these figurative descriptions of the ZPD are used to systematically identify the semantic features that more literally indicate the possibility of the ZPD’s operation in recognisable episodes of mathematics classroom interaction, referred to here as tropes.

5 A subject’s valuation of an object can be described (after Trope and Liberman 2010) according to a subject’s ‘construal level’ with respect to a given object (the degree of abstraction that a subject’s construal of an object takes) and ‘psychological distance’ (an object’s distance in space-‐time from the construing subject). 6 Valsiner and van der Veer (1993) suggest that ‘personal experience’, from the original Russian perezhivanie, is better translated as “the process of experiencing” and “the state of ‘living-‐through’” (p. 44).

Operationalising the ZPD as a Semantic Space in the 'I Spy' Trope

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