Managing epistemic imbalances in peer interaction during mathematics lessons

Abstract

In this study, we investigated how students manage their lack of/insufficient understanding of the content of a mathematical task with the aim of reaching shared understanding and epistemic balance in peer interaction. The data consist of recordings collected during a mathematics project (6 × 75 minutes) in a Finnish lower secondary school. The findings, drawing on conversation analysis, showed two markedly different sequence trajectories: (1) how interaction between a K+ and a K− (more/less knowledgeable) student proceeded relatively smoothly when these positions were accepted by both participants, and consequently the K+ led epistemic work by designing turns that resembled teachers’ practices; and (2) how the K+/K− interaction became extended when a K− challenged the K+’s knowledge claims, and furthermore, how a K− steered the epistemic work using polar and wh-interrogatives. The findings contribute to a better understanding of the ways the management of epistemic imbalance can progress during peer group work.

Keywords

Conversation analysis epistemic imbalance epistemics mathematics peer interaction

Introduction

The classroom is an institutional environment and the primary goals of its practices are expanding students’ understanding and knowledge (e.g. Macbeth, 2004, 2011). When students work in peer groups, they are oriented to co-constructing knowledge in order to achieve shared understanding of a curriculum content (e.g. Zahner and Moschkovich, 2010; Zemel and Koschmann, 2013). In this study, by drawing on conversation analysis (CA), we investigated how students manage their lack of/insufficient understanding of the content in peer interaction within mathematics lessons.

Recent CA studies, focusing on how (lack of) understanding and knowledge are registered, asserted, contested and defended through turns-at-talk in interaction, have generally adopted the concept of epistemics (e.g. Heritage, 2012b, 2013b; Stivers et al., 2011). There has been an increasing body of CA research investigating how epistemic issues are locally managed in the classroom, mainly within teacher-led interaction. Some of these studies have observed an initiative role of students, such as their claims of insufficient knowledge (Sert and Walsh, 2013), or of current knowledge (Solem, 2016). Previous studies on epistemics in mathematics lessons have explored how teachers call for the students to produce evidence of either understanding or knowing, depending on the sequential environment (Koole, 2010), and how they tend to assume the students’ problems and invite them to align with these assumptions, rather than inviting the students to explain what their problems really are (Koole, 2012).

By contrast, there is a scarce body of research on normative orientations to understanding and knowledge in peer interaction in the classroom context (e.g. Jakonen, 2014). Those few studies that exist have demonstrated, for example, how students manage knowledge gaps by requesting information or help (Jakonen and Morton, 2015) or by asking known-answer questions from co-participants (Rusk et al., 2017), how students express uncertainty towards their own displayed knowledge in order to maintain epistemic balance (Hauser, 2018). This study contributes to the small but growing body of research on epistemics in peer interaction within the classroom. More precisely, the aim of the study is to provide a better understanding of peer interaction practices in a context that has received little attention: mathematics lessons.

Epistemics in everyday and educational settings

The first studies concerning the epistemic relationship and its influence on sequence organization in talk-in-interaction were focused on ordinary conversations in everyday settings. In these cases, the participants’ epistemic status related to the epistemic domain defines who has the relative access and rights to know and claim knowledge (Heritage, 2013b; Heritage and Raymond, 2005; Stivers et al., 2011). The speakers display their epistemic status as more or less knowledgeable (hereafter K+/K−) by displaying a ‘knowing’ or ‘unknowing’ stance related to epistemic domain through lexical and grammatical forms, such as declaratives, negative interrogatives or interrogatives (Heritage, 2012a, 2012b, 2013a), or other linguistic resources such as boosters (‘certainly’) or hedges (‘maybe’) (Back, 2016: 511).

The epistemic phenomena are argued to be ubiquitous in social interaction, and to drive the sequence organization (e.g. Drew, 2018; Heritage, 2012b). The participants are oriented towards each other’s epistemic status and treatment of occurring epistemic imbalances, as well as towards epistemic discrepancies caused by diverging versions of reality (e.g. Heritage and Raymond, 2005). In other words, the epistemic positions are dynamically negotiated in interaction by displaying, claiming, contesting and defending (Heritage, 2013b; Stivers et al., 2011). The aim is to achieve mutually adequate epistemic balance or ‘epistemic equilibrium’ (Drew, 2012: 62).

The studies on epistemics in interaction have been criticized in a Special Issue of Discourse Studies (2016) entitled ‘The Epistemic of Epistemics’. Criticism has been directed, among other things, at the assertion that information exchange and treatment of epistemic imbalances are pervasive concerns in conversation (Lynch and Macbeth, 2016; Lynch and Wong, 2016). However, in more recent articles, also published in Discourse Studies, the authors have presented reasoned arguments with systematic analysis of how participants are oriented towards each other’s states of knowledge and how these are embedded, in turn, in design, correction and contesting practices (Drew, 2018; Raymond, 2018).

The pervasiveness of such social organization of knowledge is not limited only to everyday interaction, but is an essential part of educational settings in which the institutional goal is to develop the learners’ content knowledge. In classroom interaction, displays of epistemic stances (K+/K−) can be used as a resource to indicate an individual’s (lack of) understanding or knowledge, and thus what someone has learnt or needs to learn (Jakonen, 2014; Rusk et al., 2016). In teacher–student interaction, the teachers, who generally have assumed K+ status based on their higher content knowledge, are perceived to treat students’ epistemic problems through various verbal, embodied and material resources (e.g. Kääntä, 2010; Sert, 2013). For example, a teacher’s ‘designedly incomplete utterance’ prompts the student to produce the answer by completing the turn which the teacher leaves open (Koshik, 2002; Sert and Walsh, 2013).

In peer interaction, the participants can also have different states of knowledge on the subject content or task (e.g. Melander, 2012), or even contrasting understandings, which may cause epistemic discrepancy (e.g. Rusk et al., 2016). The students are seen to manage epistemic imbalances/discrepancies in various ways, depending on their epistemic positions. Rusk et al. (2017) demonstrated how K+ students use incongruent interrogative (or known-answer question) as part of second-language (L2) learning in order to prompt K− participants (L2 learners) to reach and display knowledge. By contrast, Jakonen and Morton (2015) showed how K− students manage knowledge gaps by displaying an information request to ‘a possible knower’ (K+) with respect to some content knowledge or state of affairs. In addition, Rusk et al. (2016) showed how participants manage emerged epistemic imbalances/discrepancies in L2 lessons, and also that the use of L2 risks expands these discrepancies.

The contribution of this study to the existing literature is in exploring how peers manage their lack of/insufficient understanding or knowledge of the content of a mathematical task, that is, perform epistemic work (e.g. Drew, 2018). In the analysis we explicate how these sequences proceed, and what practices and linguistic resources the participants use either from their K+ or K− epistemic positions in order to reach epistemic balance and shared understanding of the correct answer of the task.

Method

The research data consist of video and audio recordings collected during a mathematics project involving 14- and 15-year-old students in one classroom in a Finnish lower secondary school during autumn 2015. This project was based on student-centred learning (e.g. Neumann, 2013) and minimalist instruction philosophy (e.g. Carroll, 1990) within a quasi-systematic model for mathematical concept-building (Eronen, 2014). The project lasted for six lessons, and each lesson was 75 minutes long. The content to be learnt was a linear function, in which the geometric representation is a straight line and the algebraic representation y = mx + c, where the constant m represents the slope of the straight line and the constant c the line’s intersection with the y-axis. The linear function is part of the core content of ninth-grade mathematics in the Finnish national curriculum. The content was new for all the students, and thus everyone had similar relative access to knowledge and rights to know.

At the beginning of the project, the 18 students formed groups of three participants. The tasks were offered in the form of a ‘problem buffet’, in which the students had the freedom to choose the tasks they wanted to solve at that moment. During the lessons, the peer groups had autonomy to direct their learning processes – no instruction on the topic, no controlled lesson tasks or homework assignments were given. The students used the dynamic Geogebra software that handles tabular, graphical and algebraic linear representations, among other things (see also Granberg and Ohlsson, 2015). The teacher guided and facilitated their learning processes when requested.

In this study, the data consisted of recordings of the third and fourth lessons, by which time the project had got off to a good start. The data of two out of six groups were analysed (6 hours). These groups were selected to represent the variation of the dynamics of epistemic relationships in peer interaction. The first group consisted of one girl (Anna) and two boys (Leevi, Tomi), and the other group included three boys (Otto, Juho, Sami). All the names are pseudonyms. In the examples of this study, however, only two students of the group participate in interaction.

In the analysis, the methodological framework of CA was used as a central tool (Gardner, 2013; Schegloff, 2007), since it enabled an exploration of how students’ epistemic work was constructed turn by turn within a situated activity (e.g. Drew, 2018). The interaction during group work resembled mainly non-formal/informal institutional interaction: the students did not involve any particular pre-allocated turn types or sequence organization, although they used certain resources in conversation (e.g. specific question forms) in order to reach pre-established institutional goals (e.g. Arminen, 2005; Hauser, 2008; Kasper and Kim, 2015). However, on some occasions the conversations included some features of formal institutional interaction (Arminen, 2005; Drew and Heritage, 1992), such as a K+ student’s questions that initiated sequences resembling IRE sequence format: teacher initiation (e.g. known-answer question)–student response (answer)–teacher evaluation (Ingram et al., 2015; Macbeth, 2004; Mehan, 1979).

The sequences were generally composed of question (first pair-part (FPP))–answer (second pair-part (SPP)) adjacency pairs with various insert-expansions (e.g. call for previously learnt information) and post-expansions (e.g. evaluative turn, also called a sequence closing third) (e.g. Schegloff, 2007). The sequences also included correction trajectories following a speaker’s error/incorrect answer. In CA research, ‘correction’ is generally used to illustrate the replacement of an error/mistake by what is correct, whereas ‘repair’ is a generic term that describes the practices for resolving breakdowns, such as problems in hearing or understanding a prior turn (Macbeth, 2004; Schegloff et al., 1977: 363).

The interaction was transcribed utilizing the notation system developed by Jefferson (2004; see Appendix 1). The original talk in Finnish language was idiomatically translated into English using bold typeface. In addition to the talk, the most essential embodied actions (facial expressions, gestures, body postures; e.g. Goodwin, 2012) were transcribed and described by denoting the beginning of the action by a curly bracket ({) within the ongoing conversation (e.g. Kääntä, 2010). This study offers a detailed analysis of two markedly different sequence trajectories on how epistemic work is carried out in peer interaction: a relatively simple and straightforward sequence trajectory initiated and led by the more knowledgeable (K+) student, and a complex, disputed and extended one initiated and (mainly) steered by the less knowledgeable (K−) student.

Performing epistemic work to manage epistemic imbalance in peer interaction

In this section, we present the analysis of two situations during which participants manage their lack of/insufficient understanding or knowledge of the content of the task with the aim of reaching epistemic balance and shared understanding of the correct answer of the task. During the sequences, the participants also treat emerging problems in understanding the steps of task solving (procedural problem) or a particular mathematical concept/term (conceptual problem; Koole, 2012).

K+ student leads the epistemic work

In this situation, the participants solve a mathematical task dealing with the concept of ‘slope of a straight line’ (hereafter ‘the slope’). Before the following sequence begins, Anna announced that next they are supposed to study this concept, and Leevi displayed a K− position by asking What is it? Consequently, Anna took a K+ stance by explaining how to solve ‘the slope’ by using the formula m = (y₂ – y₁) ÷ (x₂ – x₁) and Leevi wrote this explanation in his notebook. Through these actions the participants indicated and accepted their epistemic statuses as K+ and K− related to knowledge of the formula. The following excerpt illustrates how Anna (K+) initiates the instructional sequence in order to resolve an occurred knowledge gap:

Excerpt 1. The slope

1 A: Eli siis nyt niinku tässä (0.9) lasketaan ensin kaks miinuslaskuu ja

So that now like here (0.9) first one calculates two subtractions and

2 sitte niitten tulokset ↓jaetaan. (.)=Okei, .hh elikkä yy koordinaattien erotus

then their remainders ↓are divided. (.)=Okay, .hh so the remainder

3 mikä se ois_

of y coordinates what could it be_

4 L: Öö (0.5) yy koord- ↑yks

Erm (0.5) y coord- ↑one

5 (1.1)

6 A: Siis mitkä tässä n- kat- näytä nyt eka ne yy koordinaatit mitä tässä on_

I mean what here n- lo- first show those y coordinates that are here_

7 L: <Elikkä nelonen ja: miinus kakonen?>

8 A: Mm?

9 L: <Niitten erotus:>

10 A: [Sitte jos ne miinustetaan toisistaan,

[Then if they are subtracted from each other,

11 L: Niin tullee ↓kaks.

Then it is ↓two.

12 A: E:i_

N:o_

13 (1.4) L LOOKS AT THE NOTEBOOK

14 L: E:i[ku ¹

15 A: [Muistatko miinus miinus.

[Do you remember minus minus.

16 L: ↑Aa elikkä sitte (.) miinus kuus.

↑Oh so then (.) minus six.

17 A: E:i_

N:o_

18 L: Nii?

So? GAZES AT ANNA

19 A: Ei vaan siis ku kato se on silleen=

No but I mean look it is like such that=

20 (2.6) A TURNS THE NOTEBOOK TOWARDS HERSELF

21 A: =<n:eljä miinus miinus ↓kaks> muistatko ku on kaks miinusta

=< f:our minus minus ↓two> do you remember when there are two

{A WRITES DOWN SUBTRACTION 4–2 AND RETURNS THE NOTEBOOK TO L

22 ni mitä sillon tulee.

minuses so what do you get then.

23 L: Plus

Plus

24 A: Niieli se on=

Yes so it is=

25 L: =Kutonen NODS HIS HEAD

=Six

26 A: #Joo se on kuus.#

#Yeah it is six.#

27 (0.6)

28 A: #Nii# (.) Laita siihen vastaukseks kuus (.) ↓tonne_

#Right# (.) Add there the result six (.) ↓there_

(2.0)

29 A: ↑Sitte kolme↓ja yks elikkä kolme miinus yks.(.)=Se on helppo.

↑Then three and ↓one so three minus one.(.)=It is easy.

30 M: @Kolme miinus y-@

@Three [minus o-@

31 L: [Öö miinus ↓kaks

[Erm minus ↓two

32 A: Kolme miinus yks on kak- miinus kaks.

Three minus one is t- minus [two. ]

33 L: [Kaks ] kaks

[Two ] two

34 A: Joo se on kaks.

Yeah it is two.

35 L: ↓Kaks eli

↓Two so

36 A: Joo? (.) ja sitten ku ne jaetaan toisillaan kuus jaetaan kahella,

Yea? (.) and then when they are divided by each other six is divided by two,

(2.4)

37 L: Tuo on kolme miinus yks (.) nooin,

That is three minus one (.) like that,

38 A: Sitte ku ne jaetaan toisillaan niinku <kuus jaetaan kahella_>

When they are divided by each other like <six is divided by[two_> ]

39 L: [kolome] kolme

[three ] three

40 A: Niin se on kolme.

Yes it is three.

41 A: .hh Elikkä siis kulmakerroin on <kolme.>

.hh So the slope is <three.>

This sequence begins with Anna’s (K+) turn: first she explains how to solve the slope by using the formula, and then produces a question (BFPP). The question is an incongruent interrogative, also called a known-answer question (e.g. Macbeth, 2004; Rusk et al., 2017), which in this case the K+ produces in order to prompt the K− to produce the correct answer of the remainder of the y coordinates (y₂ – y₁). As a response, Leevi starts to repeat the question, then hesitates and produces an incorrect answer (SPP). This is followed by a 1.1 second silence, indicating a possible trouble source.

In line 6, Anna initiates an insert expansion FPP (to show the’y’s in the task) in order to establish some required knowledge which Leevi needs in order to be able to answer the initial question. In other words, Anna appears to have noticed Leevi’s difficulty, and thus performs epistemic work from her K+ position. Next, Leevi produces an insert expansion SPP to Anna’s instruction. This is done as try-marked, through which Leevi seeks confirmation for his answer. Anna produces a minimal post-expansion in line 8 by accepting Leevi’s candidate answer to the insert expansion FPP.

In lines 10 and 11, both Leevi and Anna repeat the original instruction (BFPP in lines 2–3). Anna’s turn resembles the form of a designedly incomplete utterance (e.g. Koshik, 2002) that prompts Leevi to complete the turn. Leevi responds by completing it, but this is again an incorrect answer. In line 12, Anna produces a minimal post-expansion FPP that evaluates Leevi’s answer explicitly as incorrect. Leevi responds by initiating the self-correction, but Anna interrupts him by initiating another insert expansion FPP (line 15), which is a guiding question (Sahin and Kulm, 2008) do you remember minus minus that recalls previously learnt information (how to operate with two minus signs). Leevi’s response in line 16 is still an incorrect answer to the original question and it is followed by Anna’s post-expansion FPP, a blunt negative evaluation and Leevi’s SPP so?, with rising intonation, indicating that he has not understood his prior error and urges Anna to explain.

As a consequence, Anna continues epistemic work by rephrasing her prior turn (in line 15) and producing an even more explicit insert expansion FPP (lines 19–22). She demonstrates this matter by writing down the subtraction (4 − − 2), by stressing the words minus minus, and ends her turn by posing a guiding question. Now Leevi’s insert expansion SPP is the correct answer plus (line 23) and Anna produces a minimal post-expansion, the positive evaluative turn yes. As this epistemic work succeeded, Leevi returns to the original question (in line 2–3). This turn design resembles again a designedly incomplete utterance form. Anna responds immediately by completing it with a correct answer (line 26). Anna again produces a positive evaluative post-expansion FPP, and consequently in line 28 confirms the correct answer and instructs Leevi to add the result to the notebook. By now, with the help of Anna’s guiding actions, Leevi has succeeded in solving the original instruction (lines 2–3), the subtraction of the y coordinates.

The example continues as Anna moves on to the next part of the original instruction (lines 1–2) by initiating the next BFPP (line 29) concerning the calculation of the remainder of the x coordinates (x₂ – x₁). In line 31, Leevi responds with an incorrect answer. At this point, instead of addressing the error, Anna produces a post-expansion FPP (line 32), during which she repeats the subtraction and Leevi’s incorrect answer. In overlap, Leevi produces a self-correction, and Anna evaluates this answer as correct.

Finally, in line 36, Anna moves to the last part of the original instruction. Here she again produces a designedly incomplete utterance (BFPP), prompting Leevi to solve the last phase of the task, to divide the remainder of the y coordinates (answered in line 26) by the remainder of the x coordinates (answered in line 34). Leevi does not respond to this initiation because he is about to add the previous result into the notebook. Anna repeats her BFPP (line 38) and in overlap, Leevi completes the turn with the correct answer. Anna produces a positive evaluative post-expansion FPP that accepts the answer as correct. However, in line 41 she confirms the definitive answer, that the slope (constant m) is three. As a result, the participants achieve a shared understanding on the correct answer that dealt with the topic of how to solve ‘the slope’.

In sum, this example illustrated how one participant adopted the K+ epistemic status and the right to determine the sequence organization and her co-participant’s actions. The epistemic work included series of sequences that resembled the IRE sequence form typical in teacher-fronted lessons (Ingram et al., 2015). The sequences began with a K+ initiation, which was a known-answer question (Macbeth, 2004; Rusk et al., 2017), a guiding question (Sahin and Kulm, 2008) or a designedly incomplete utterance (Koshik, 2002). All these turns indicated the knowledge of the K+ student, and were followed by a K− answer (SPP) that led to post-expansion FPP (or a third turn), which was the K+’s evaluative turn. The excerpt also included several insert expansions during which the K+ established required knowledge for the K− (line 6) or prompted the K− to remember knowledge they had already learnt (line 15, lines 21–22).

This example shows how the students’ different states of knowledge and accepted relative epistemic statuses as K+/K− can produce interaction that resembles formal institutional interaction (e.g. Hauser, 2008). However, some of the K+ actions differed from teachers’ practices: the K+ responded to the K− ’s incorrect answers generally by addressing them explicitly in post-expansions, with a negative evaluative word no (lines 12 and 17). The literature shows that rather than such blunt negative evaluations, trained teachers typically use other strategies, such as additional questions, that urge the instructed student to produce self-correction (Ingram et al., 2015; Kääntä, 2010; Kapellidi, 2013; Macbeth, 2004).

Overall, in this example the epistemic work proceeded relatively smoothly. By contrast, the following example illustrates how epistemic work can become extended and complex in practice.

K− student steers the epistemic work

In this example, the students deal with a mathematical task in which they are requested to formulate a particular equation of a straight line (the algebraic representation is y = mx + c). Immediately before the excerpt, one student, Juho, drew two points on the coordinate system and started to process this task independently. The following excerpt begins with Juho’s turn that includes a candidate answer designed as a confirmation request directed to a co-participant, Otto. In this example, the epistemic imbalance is realized and epistemic positions (K+ and K−) are negotiated during the extended sequence trajectory:

Excerpt 2a. Formulating the equation of a straight line

1 J: Ni jos tuolla on meillä toi yks ↑piste ja meillä toinen on tuolla?

So if there we have that one ↑point and we have another one there?

2 (1.1)

3 O: mm

4 J: Nii sillon ku x on yks ja että tuo suora- tuon täytyy olla silleen suorassa ni

So when x is one and that this line- this needs to be like straight so

5 sillon ku x on yks ni se on kaks piste (0.7) viis?

when x is one then it is two and a (0.7) half?

6 O: mm

7 J: Ni eiks se oo sillon, (0.8) toi y↑ ni eiks se oo s- <sillon x on (0.7) yhtä kun>

So isn’t it then, (0.8) that y↑ so isn’t it- <then x (0.7) equals>

8 (1.1) eiku °vittu° y on yhtä ku (0.9)(x ker-)

(1.1) eiku¹ °fuck° [y equals (0.9) [(x ti-)

9 O: [y: [kaks puol x=

[y [two and a half x=

10 J: =Nii.

=Yeah.

11 (1.0)

12 O: Plus plus ö kaks puol koska tuo (.) se leikkaa ton (.) öö y akselin tossa

Plus plus erm two and a half because that (.) it crosses that (.) erm y-axis

13 öö kaks puol kohalla,

here erm at the point of two and a half,

14 J: Miks siihen pitää plus laittaa.

Why does plus need to be added there.

15 O: Kyllä pitää koska se (.) leikkaa ton tossa_

Yes it is needed because [it (.) crosses that here_

16 J: [Tos ei oo mitään jär-

[That doesn’t make any sen-

17 O: Ei koska muuten se- muuten se menis tosta.=Muuten se menis tosta (.)=

No because otherwise it-[otherwise it would go there.=Otherwise it=

18 J: [Sit sen arvo on viis.=Sit sen arvo on-

[Then its value is five.=Then its value is-

19 O: =tosta (.) niinku (.) origon kautta.

=would go there (.) there (.) like (.) through the origin.

20 (2.6)

21 J: °täh°

°what°

22 (1.2)

23 O: Niin sit siihen laitetaan lop- perään se plus kaks pilkku

So then an end- is added after that plus two and a

24 viis ni sit se-

[half so then it-

25 J: [↑Niin mut sillon jos se menee origon kautta ni x on ↓nolla siinä kohti.

[↑Yes but if it passes through the origin then x is ↓zero at that point.

26 (2.2)

27 O: E:iku se (0.7) ääh (.) x on yhtä suuri kun eiku y on yhtä suuri kun (.) kaks

E:iku¹ it (0.7) erm (.) x equals eiku y equals (.) two and a half x which

28 puol x ni määrittää tämän (0.4) miinus (0.6) kaks puol x määrittää tämän

defines that this (0.4) minus (0.6) two and a half x defines that

29 (0.7) öö (.) kulman.

(0.7) erm [(.) angle.

30 J: [Ei se mikään miinus kaks puol x oo.

[It is not minus two and a half x.

31 O: On koska se onlaskevasuora.

Yes it is because it is a downward linear.

32 (3.0)

33 J: Tuossa ei oo yhtään mitään järkee jos se tuolleen on.

That doesn’t make any sense if it is like that.

34 O: No niin se vaan miusta on.

Well that’s how it is in my opinion.

35 (0.4)

36 O: On se ehkä vähän (.) hämmentävää mutta,

It is maybe a bit (.) confusing but,

37 J: Onks meil sittemuka aikasemmin ollu pelkästään nousevia suoria ja mie oon

Have we earlier only had upward linears and I have only

38 ymmärtäny vaan ne.

understood them.

39 (0.8)

40 J: Koska ne (-) tehään järkevästi.

[Because those (-) are done in a rational manner.

41 O: [En: tiiä_

[I don’t know_

42 J: Tai meil on ollu siis nousevia suoria jotka menee origon kautta ja jotka-

Or I mean we’ve had upward linears that go through the origin and that-

43 aineit- ainoita jotka lasketaan järkevästi niinkö.

on- only that are calculated in a rational manner right.

44 O: Ehkä,

Maybe,

45 J: Täs ei oo mitään järkee sitte.

Then this doesn’t make sense at all.

This excerpt begins with Juho’s pre-expansion, during which he demonstrates the mathematics task (lines 1, 4, 5). This lays the foundation for the subsequent BFPP. In lines 7–9, he starts to formulate a candidate answer designed in a polar negative interrogative (e.g. Heritage, 2012a). This turn is a BFPP that requests confirmation for his candidate answer and positions the recipient, Otto, as ‘a possible knower’ (K+) (Jakonen and Morton, 2015). The turn includes an error (line 7) and a self-initiated correction (line 8). In overlap, Otto produces other-correction and then continues Juho’s candidate answer (line 9), thus displaying a K+ stance related to the content of the task. Juho accepts this continuation. Next, Otto completes the equation and gives an explanation for this completion (lines 12–13). On this occasion, Juho responds with a wh-interrogative that seeks reasoning (line 14). Otto responds with a knowledge claim while demonstrating it on paper. In overlap, Juho produces a negative assessment that challenges Otto’s knowledge claim (line 16), and thus does not accept the response for his BFPP.

The prior turns reveal that the students have diverging understandings of the correct answer. Thus, the situation develops into a disagreement that lasts several turns (lines 17–36). It consists of Otto’s explanations and knowledge claims concerning his candidate answer (lines 17, 19, 23–24, 27–29, 31), and Juho’s several incursions into Otto’s ongoing turn-constructional units (TCUs) (lines 18, 25, 30), through which he challenges Otto’s knowledge claims and produces counterclaims. In line 33, Juho expresses the disagreeing assessment that doesn’t make any sense if it is like that, indicating that he still disagrees with Otto’s candidate answer. Otto responds by defending his own understanding (line 34) and yet displays empathy concerning Juho’s confusion (line 36).

Next, Juho (K−) initiates the epistemic work (line 37), aiming to resolve the emerged epistemic discrepancy. During the following sequence, he produces polar interrogative FPPs to request information (e.g. Heritage, 2012b) concerning their previous learning experiences (lines 37–40, 42–43). Through these turns he seeks information that would explain his possible insufficient understanding, and thus the occurred epistemic discrepancy. These turns indicate that Juho still orients to Otto as a K+. However, the Finnish particle muka (Hakulinen et al., 2004,§1495) (line 37) indicates that he also questions Otto’s knowledge claims (see also Steensig and Drew, 2008). Otto produces the SPP I don’t know (line 41) and later a ‘hedge’ maybe (line 44), which express his uncertainty concerning Juho’s questions (e.g. Tsui, 1991). These minimal responses also appear to indicate Otto’s reluctance to continue epistemic work. In post-expansion, Juho produces a disagreeing assessment (line 45), indicating that he still has not understood Otto’s candidate answer.

During this excerpt, Otto first attempted to do epistemic work from his K+ position by explaining and demonstrating his knowledge concerning the current task, but subsequently abandoned this work (from line 36). Then, Juho started to steer the epistemic work from his K− position. However, by the end of this excerpt they could not resolve the occurred epistemic discrepancy and reach a shared understanding of the correct answer. The epistemic work continues after 8 seconds of silence and is illustrated in the next excerpt. During the excerpt, Juho (K−) attempts to reach a general understanding of the equation of the straight line and its terms (y = mx + c). These actions reveal that he has to some extent realized his insufficient understanding:

Excerpt 2b

49 J: Eiku jos se ois (1.6) nouseva suora? (1.2) näin?

Eiku¹ if it would be (1.6) upward linear (1.2) like this?

50 O: Niin

Yeah

51 J: Tuo (.) origo on niinku tuolla?

This (.) the origin is like there?

52 O: Nii

Yea

53 J: Ni sillon ku x on yks ni sillon täällä yyki on yks (.) ni sillon se x on yhtä ku

So when x is one then this y is also one (.) so then x equals

54 y (1.0) jos se on tollanen suora, mut jos se on tuolleen että se on niinku

y (1.0) if it is that kind of straight line, but if it is like that so that it is like

55 tuplasti tuolleen kaks pitää olla yks (.) ni se on (0.6) y on yhtä ku ↑kaks

two times like that two have to be one (.) so it is (0.6) y equals ↑two

56 kertaa (2.8) y (0.8) eiku x on yhtä ku kaks kertaa y.

times (2.8) y (0.8) eiku ¹ x equals two times y.

57 O: Nii (.)=Ja sit jos se ei mee origon kautta ni sit siihen laitetaan perään plus tai=

Right (.)=And if it doesn’t run through the origin then plus or minus [is put=

58 J: [eiku¹

59 O: =miinus (.) se määrittää sen kohan missä se leikkaa sen akselin.

= in the end (.) it defines the point where it crosses that axis.

60 J: Tossa ei muuten- (0.8) mä tajusin just et miks toi mun (0.5) logiikka ei toimi.

This doesn’t by the way- (0.8) I just realised why my (0.5) logic doesn’t work.

61 O: Tsh

62 (6.6)

63 J: Eli siis pitääks mun nyt sanoo että (.) y on yhtä ku (2.1) kaks piste viis kertaa

So do I now have to say that (.) y equals (2.1) two and a half times

64 miinus äx.

minus x.

65 (1.3)

66 Sil ei oo välii- sil ei oo välii kumpaan pistää miinuksen.

[It doesn’t matter- it doesn’t matter to which one[you add minus.

67 O: [Miinus kaks puol x [Joo joo mut kuitenki_

[minus two and half x [Yea yea but anyway_

68 (0.6)

69 O: Helpompi on jos miinus kaks puol x plus-

It is easier if it is minus two and a half x[plus-

70 J: [(Totta) se on järkevämpi jos äxässä

[(Right) it is better if there is a

71 on miinus sori.

minus sign in front of x sorry.

72 O: Joo

Yea

73 (0.8)

74 Öö (2.0) Sit loppuun plus (0.9) plus kaks puoli jos se menee siinä.

Erm (2.0) then in the end plus (0.9) [plus two and a half if it goes there.

75 J: [Eli (.) eli tää on niinku y on

[So (.) so this is like y equals

76 yhtä kuin miinus <kaks viis x.>

minus <two and a half x.>

77 O: Plus kaks puol.

Plus two and a half.

78 J: Miks plus kaks puol.

Why plus two and a half.

79 O: Koska se leikkaa y akselin tossako- (.) plus kaks viis kohalla.

Because it crosses the y-axis there (.) at the point of two and a half.

80 J: Mitä välii sil on.=Pitäskö tonne laittaa sitte niinku (.) öö (3.0) näin.

So what.=Should it be put there then like (.) erm (3.0) like this.

{J WRITES

81O: x on (1.8) ei sinne tarvi laittaa ku se menee origon kautta.

x is (1.8) it isn’t needed because it goes through the origin.

{READS J’S CANDIDATE ANSWER

82 J: No niin mut oisko se sitte muka teoriassa tuolleen.

Well but would it be like that in theory.

83 O: Kai.

Maybe.

84 (2.2)

85 O: En mie tiiä kysy Timolta mut näin miulle on opetettu miun mielestä.

[I don’t know, ask Timo but this is how [it has been taught to me in my opinion.

86 J: [(paska) [(no mä haen Geogebran)

[(shit) [(well I’ll go and get Geogebra)

87 koska miun aivot derbbas (.) eilen ja mun aivot ei (toimi).

because my brain collapsed (.) yesterday and my brain doesn’t (work).

In the beginning, Juho produces a declarative that demonstrates his current understanding of the topic (lines 49, 51, 53–56). Otto responds first by displaying the Finnish continuers nii (lines 50, 52), and then by assessing Juho’s turn and producing a knowledge claim concerning the constant term (constant c) (lines 57, 59). Through these turns he maintains his K+ status. In overlap, Juho produces the Finnish repair particle eiku (line 58), and his next turn (line 60) indicates that at this moment he has not oriented to Otto’s turn, but has realized his prior lack of understanding.

Next, Juho returns to the original task (presented in excerpt 2a) by producing a new candidate answer. This turn is again designed in interrogative form and requests confirmation from the co-participant (lines 63–64, 66). In overlap, Otto responds to this BFPP by reformulating the candidate answer with reasoning (lines 67–69), and thus maintains his K+ position. Juho accepts this reformulation (lines 70–71), and repeats it in lines 75–76, which confirms that they have now reached shared understanding concerning ‘the slope’ of that function (constant m). That is, Otto’s candidate answer is now partly accepted by both students.

However, Juho’s turn in line 78 reveals that they still need to continue the epistemic work. The following section consists of multiple question–answer adjacency pairs. In his turn, Juho responds to Otto’s prior turn (line 77) with a wh-interrogative that asks for reasoning why they need to add plus two and a half (constant term) at the end of the equation. This interrogative indicates that he still has not reached an understanding of what this other term represents. Otto answers by producing a knowledge claim (line 79).

This leads to another disagreement, during which Juho first challenges Otto’s knowledge claim by asking so what (line 80), and then designs two polar interrogatives to request information aiming to achieve understanding (lines 80 and 82). At the same time, these interrogatives again question Otto’s knowledge. Otto first responds by producing another knowledge claim (line 81). After a second interrogative he displays a ‘hedge’ maybe (line 83), followed by 2.2 seconds of silence and a claim of insufficient knowledge (I don’t know), thus mitigating his K+ epistemic status (line85) (e.g. Heritage, 2012a). Consequently, Otto suggests that Juho should ask for help from the teacher, the epistemic authority. In the end, he defends his candidate answer and knowledge claims by explaining that these are based on how he has been taught, and then withdraws himself from interaction. Juho expresses frustration at the current state of affairs (line 86). However, instead of accepting Otto’s suggestion, he announces that he will continue with the Geogebra software. This turn indicates that Juho is willing to resolve the emerged epistemic imbalance and reach understanding on his own.

After this sequence, the conversation shifts to the organization of group work, during which the students negotiate who will bring the laptop from the storeroom and choose their next task. Later, the laptop causes difficulties as it does not work properly. It takes some time to solve this technical problem, which lengthens Juho’s incompleted activity to resolve the epistemic imbalance. However, while they are working out the problem with the laptop, Juho goes back to this activity every now and then by expressing his prior difficulties in understanding Otto’s candidate answer. This is one example of Juho’s turns: Otto, now my brain works and I thought it correctly. heh. My brain can think again so that it makes sense. The turn is followed by Otto’s minimal response oh he he.

Finally, 27.22 minutes after the 2b excerpt, Juho has got the laptop and its Geogebra software working. The next excerpt illustrates the sequence during which the participants finally reach shared understanding of the correct answer:

Excerpt 2c

1 J: Joo tos on ihan oikeesti järkee,

Yeah that really make sense now,

2 (2.0)

3 J: Mun aivot vaan pieras (–)

My brain just [farted (–)

4 O: [Pistä siihen se plus kaks (.) puol.

[Put there that plus two (.) and a half.

5 J: Joo mä just (.) tutkin sitä asiaa,

Yea I’m just (.) exploring that,

6 O: Joo ni sit se hyppää tonne.

Yes so then it jumps there.

7 J: Mun aivot vaan pieras.

My brain just farted.

8 O: Joo.

Yeah.

9 J: (–) on järkee.

(–) make sense.

This sequence illustrates how Juho finally displays understanding and acceptance of Otto’s candidate answer to the original task (presented in excerpt 2a). The excerpt begins with Juho’s claim of understanding and is followed by a turn that explains his prior lack of understanding (line 3). Next, Otto urges Juho to complete the equation by adding plus two and a half (constant term). Juho does not accept this directive straightforwardly, but expresses that he is exploring this term with Geogebra (line 5). However, in line 9, (after having checked the correctness with Geogebra), Juho produces another claim of understanding that confirms Otto’s candidate answer. Overall, this very long series of sequences finally results in shared understanding of the correct answer.

In sum, this example illustrated the extended epistemic work practice steered by a K− student. The interaction included epistemic discrepancy as the participants displayed diverging knowledge claims concerning the correct answer. After having realized his insufficient understanding, the K− did persevering work to resolve the epistemic imbalance by designing wh- and polar interrogatives that requested information or confirmation and positioned the recipient as K+ (e.g. Heritage, 2012b). Some of these interrogatives also challenged the K+’s knowledge claims indicating disaffiliation (e.g. Steensig and Drew, 2008). That is, the K+ and K− positions were not simply accepted, but also challenged, contested and defended in interaction (Heritage, 2013b). Overall, the K− could maintain epistemic work, even despite the difficulties with the laptop that interrupted his working, and could eventually reach shared understanding of the correct answer formulated by the K+ participant.

Discussion

This study aimed to identify the ways in which students manage their lack of/insufficient understanding in peer interaction aiming to achieve epistemic balance and shared understanding of the correct answer of a mathematical task. The findings contribute to prior studies of epistemics in interaction by demonstrating how the participants oriented to each other’s state of knowledge and resolved an emerged epistemic imbalance (K+/K−) in peer interaction in the classroom context (see e.g. Drew, 2018; Heritage, 2012a, 2012b). The findings illustrated two markedly different sequence trajectories in which epistemic work was steered either by a more (K+) or less (K−) knowledgeable student, and revealed how epistemic work was constructed primarily through various interrogatives.

The findings illustrated that when a K+ student performed epistemic work (excerpt 1), the sequences resembled the IRE sequence format prevalent in traditional teacher-led lessons (Ingram et al., 2015; Mehan, 1979). The K+ initiated sequences by designing turns similar to teacher’s practices: known-answer questions (incongruent interrogatives) (Macbeth, 2004), guiding questions (Sahin and Kulm, 2008) and designedly incomplete utterances (Koshik, 2002; Sert and Walsh, 2013). Consequently, the less knowledgeable (K−) student oriented to the sequences by answering and requesting the K+ to evaluate his answers.

These results are in line with some prior studies on how K+ students can use known-answer questions and other resources similar to formal institutional teacher–student interaction in order to instruct and guide the learning of K− co-participants (e.g. Melander, 2012; Rusk et al., 2017). These practices and the familiar IRE-sequence format may offer students a practical and well-known routine to resolve epistemic imbalances and co-construct shared understanding. However, if epistemic imbalance becomes established and a K+ participant consistently determines the sequence organization, it can be anticipated that the K− co-participants’ opportunities to talk, and to try out and claim their understanding and knowledge, may become limited (e.g. Back, 2016; Lehtimaja, 2012).

Interestingly, the findings revealed that a K− student can also steer epistemic work (example 2) through designing polar and wh-interrogatives that request information or confirmation from a K+ student (e.g. Heritage, 2012b; Jakonen, 2014). In addition to seeking support, some of these interrogatives also challenged the K+’s knowledge claims (e.g. Steensig and Drew, 2008). Moreover, the interaction included dispute sequences during which K+ knowledge claims were contested and counterclaims displayed. In other words, the epistemic positions were also challenged, contested and defended in interaction. Nevertheless, the K− accepted his co-participant as K+ and did persevering work to reach shared understanding with K+.

It should also be noted that on this occasion the epistemic work was partly extended due to emerged epistemic discrepancy when the students displayed diverging knowledge claims. One factor that has been perceived to complete epistemic work more straightforwardly is that K− and K+ participants not only display understanding/knowledge, but explicitly demonstrate it while orienting to realize others’ demonstrations (e.g. Koole, 2010; Raymond, 2018). Overall, the outcome of extended epistemic work can be fruitful when it is based on a K− student’s desire and possibility to process his insufficient understanding of the content. The findings reveal that mathematical task-solving processes within peer interaction may take time, and suggest that this should be taken into account when organizing student-centred learning activities.

To conclude, the main purpose of this study was to demonstrate how managing epistemic imbalance in peer interaction can be either a rather straightforward or an extended process. In the straightforward process, the sequence proceeded purposefully as determined by the K+ participant. Consequently, the K− co-participant as an interaction partner became subsidiary. By contrast, when the epistemic work was led by the K− participant, the sequence was prolonged but interaction became more reciprocal and explorative as the epistemic imbalance led the K− to explore the difference between their understandings (see also Wood, 2016). This study was limited to focusing on situations in which peers were positively oriented towards achieving shared understanding of the correct answer of the current task. Further studies examining diverse practices and how (lack of) understanding and knowledge are displayed, demonstrated and co-constructed in peer interaction within various contexts should also be conducted.

Footnotes

Appendix 1 Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

Notes

Author biographies

Anniina Kämäräinen is a doctoral student in Special Education at the University of Eastern Finland. She is currently writing her doctoral thesis on organization of social interaction in mathematics lessons based on student-centred learning. Her research interests include classroom interactions in inclusive settings, practices in peer interaction and epistemics in interaction.

Piia Björn is Professor of Special Education at the University of Eastern Finland, who specializes in the field of studies in learning mathematics, as well as intervention studies among students in need of support in learning. She is also interested in teachers’ work and professional development.

Lasse Eronen works as a Senior Researcher at the University of Eastern Finland, School of Applied Educational Science and Teacher Education. He has worked as a mathematics and science teacher for 15 years and as a mathematics teacher educator since 2000. His current research interests concern the student-centred learning environments in mathematics education and integrative approaches to teaching and learning.

Eija Kärnä works as a Professor of Special Education at the Faculty of Philosophy, University of Eastern Finland. Professor Kärnä has been involved as a principal investigator in many national and international development and research projects. The projects have been funded by organizations such as the Ministry of Education and Culture, the Academy of Finland and the Regional Council of North Karelia. All development and research projects have been multidisciplinary. Her research interests are inclusive learning environments, technology for individuals with special needs, communication and interaction of individuals with severe developmental disabilities and autism spectrum disorder, and digital literacy for various age groups.

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