MOVING BETWEEN SCHOOL AND COMPANY

Toril Eskeland Rangnes

Bergen University College

The case study to which this paper refers focuses on 8th grade pupils' conversation in a mathematics learning situation related to a building company. The paper illustrates how school and company, having different goals for the use of mathematics, create a field of tension where the pupils meet differing languages and modes of thought. Taking mathematics conversations in and out of school as a starting point, the polyphony of this field of tension is analysed and discussed on the basis of Bakhtin’s ideas of dialogicity and how the encounter with different voices influences the pupils’ positioning and strengthens their participation in making decisions.

INTRODUCTION AND AIM

In the Norwegian curriculum, the development of pupils’ literacy is, inter alia, manifested through principles of education, where cooperation outside of school (e.g. with local companies) is encouraged, in order to learn and be motivated. In addition, oral skills e.g. in mathematics are emphasized, as well as being able to apply problem solving and investigation on the basis of practical, everyday situations (LK06, Norwegian National Curriculum for Knowledge Promotion in Primary and Secondary Education and Training). Mathematics skills are seen as a tool for everyone in their life, and as a necessity for participating in democratic society (Ibid). This mirrors OECD´s (2006) definition of mathematical literacy as something good and necessary for succeeding in society and participating as a democratic citizen. In Norway we also have National tests (written), which aim to measure numeracy with results published on the web. Applying mathematics in practical settings and reflections on applications of mathematics described in the concept of mathemacy (Skovsmose, 2005, p. 46), appear to be difficult to give priority and is not common in Norwegian secondary classrooms.

The intention of this paper is to shed light on conversations where pupils in lower secondary school (8th graders) meet different mathematics practices and how this influences the pupils positioning and language usage. This also entails exploring how dialogues can close or open up possibilities for further discussion and options and give potential for reflection about school mathematics vs. mathematics in company. Johnsen-Høines (2010) describes the pupils´ movement between school and company as a learning loop. This movement is not confined to location – rather, it is about how the moving between is present in conversations both in school and in the building company.

The pupils meet two kinds of practices, the building company, guided by production, efficiency and profitability, and the school, governed by learning goals in mathematics as defined in the curriculum and effectuated by the teacher. This creates

a field of tension where participants are confronted with different goals, practices and language and where they have to cope with the challenges this entails.

There are earlier studies of conversations where everyday discourse and school mathematics discourse meet (e.g. Rønning, 2009). There have also been studies of adults attending courses to develop numeracy associated with work and everyday life (e.g. Wedege, 2010). Essential in this paper is that the conversations are taken from a project where the aim is that secondary school pupils will learn mathematics through meeting with a company they initially know little about. It is not the pupils’ everyday life, and it is different from the school mathematics practice they know. The teacher and the pupils are developing a new practice together.

The pupils (aged 13-14) were told beforehand what mathematical and social goals they were to work towards. The assignment given to them by the teacher and the building company was to construct 3D models of a rorbu, a combined boathouse and seaside cottage, popular as holiday resorts in this island district. Initially, the building company sent the pupils several construction drawings of rorbu of different sizes. These were to be taken as a basis for the pupils’ own construction drawings and suggestions for possible room plans. The group would take their construction drawing to the company, and discuss their drawing with a carpenter. Back in the classroom, the pupils would realize their construction drawing in a 3D model in scale 1:25.

THEORETICAL FRAMEWORK

Wedege (2006) describes differences between working with mathematics at work and mathematics in school on the basis of the experience of her adult students. For instance, she notes that in professional life, one has to find the relevant information oneself, whereas in school, one is given problems cleansed of unnecessary information. Even “reality” has a different function, Wedege claims. In professional life, reality gives opportunities to use mathematical ideas and techniques, and solutions have practical consequences, whereas in school, it serves as pretence for using mathematics, and the results usually have no practical consequences. Furthermore, the tasks in professional life are governed and structured by technology, whereas in school, the mathematical problems structure the teaching (Wedege, 2006, p. 217).

At a workplace one can expect to meet many activities which can be linked to mathematics. What should be counted as mathematical knowledge is a political question, associated with the power of definition. Mathematical activities in a company fall into what Mellin-Olsen calls ”folk mathematic”, and he makes it clear that it is a political choice whether to include it as part of "school mathematics" (Mellin-Olsen, 1987).

The benefit that a meeting between school and company with so different cultures and different aims can produce, is a moot point. To understand and to question one’s

own culture, outsideness can be a most powerful factor, according to Bakhtin (1986, p. 7). The dialogue between cultures, does not result in merging and mixing, he says, “each retains to its own unity and open totality, but they are mutually enriched” (Ibid, p.7.). In Bakhtinian dialogism, the utterance is the unit of analysis. An utterance, which can be a word, a sentence, a drama or a dissertation, can never be studied in isolation, it must be seen in relation to the preceding utterance and its continuation in the utterance that follow. Utterances must, according to Bakhtin, be seen in light of time; present, past and future. They must be seen in relation to context: social, cultural and historical (Bakhtin, 1986).

Bakhtin never explicitly define polyphony, he just provided a great deal about it (Morson & Emerson, 1990). Bakhtin (1981; 1986) describes polyphony where different opinions, understandings and linguistic settings are given room. The voices can be identified through choice of theme, expressivity and purpose. One utterance can contain several voices. For creating understanding it has to be more than different voices, there is also a need of tension and struggle between them (Dysthe, 1999, p. 76). A dialog which opens up for polyphony can be contrasted by monologic talk where one person or a group has the power to decide what to talk about and how to talk. Polyphony opens up for different possible positions where critique and negotiation of power are included. It fits with understanding “power as a relational capacity of social actors to position themselves in different situations and through the use of various resources of power” (Valero, 2004, p. 15). Bakhtinian theory about dialogue, polyphony and positioning is an approach to describing pupils’ movement between different argumentations which can be positioned in a school mathematics discourse or in a company discourse.

The form of the conversation has a link to how power is divided between the participants. An inquiry dialog is seen as a conversation where there are symmetrical relationships between the participants and where the participants investigate each other’s perceptions (Alrø & Skovsmose, 2002). Lindfors (1999) stresses that the object of an inquiring utterance must be an authentic wish to seek others' help to investigate what lies beyond that which one understands. Through an inquiring attitude one also shows what one knows. To ask in order to invite others to investigate is a risky act. One demonstrates one’s need for the other’s good will to listen and interact (Ibid). Dialogue is not only about question/answer; it is about construction of meaning and is included in a social practice (Bakhtin, 1981, p. 121). With Lindfors’ (1999) inquiry dialog and Alrø & Skovsmose’s (2002) inquiry co-operation model (IC-model) as a background, I will in this paper describe dialogues where the participants show an intension to listen and contribute, as an “inquiry dialog”. In such a dialog, awarenesses meet, and the participants’ awarenesses have a potential for change (Lindfors, 1999, p. 150). In this respect, dialogical conversations stand in contrast to monologic ones where an authority holds the truth and tries to persuade or guide other people’s choices.

These perspectives give me a background for studying what are the characteristics of the pupils’ conversations when they encounter in school and workplace. What different positioning do the pupils take in the conversations in movement between different and to some degree contradictory voices? Implicitly, this enlightens how the relational power moves between the participants in conversations.

METHOD

The research project “Learning Conversation in Mathematics Practice” (LCMP) [1] of which this study is a part, cooperated with teachers in a municipality where an object was to carry out the aim about practical learning in mathematics in cooperation with local enterprises. In this project teacher students participated as active actors in developing the teaching. One of these students wanted to try cooperating with an enterprise in her first year as a teacher. One of her arguments for doing this was the diversity in her class, some pupils have preferences to practical work, and most of her teaching was theoretical. On the teacher’s request, I was her conversation partner in the teaching project. The teacher clearly has the main responsibility for the design of the teaching plan and the implementation of the teaching. My role as a researcher was explained to the pupils; it was OK to communicate with me, but the teacher was the one responsible and had the authority.

Teacher and researcher met the carpenter in the building company for an initial clarifying conversation, where it was made clear that it was their practical application of mathematics in the company that was of interest. The carpenter’s role in the company is, among other things, to advise customers and to plan annexes and minor new buildings.

My colleague Gert Hana and I followed two groups with video cameras and sound recorders for seven sessions, one of which was at the company. The groups we followed were put together by the teacher. The grouping was not based on ability to do mathematics; the only criteria were the participants’ ability to participate in conversations and that they had consented to take part.

In order to study both form and content of the conversations, the analysis has been inspired both by conversation analysis (Nielsen & Nielsen, 2005) and pragmatism (Svennevik, Sandvik & Vagle 1995). The analysis is still in progress. In this paper I present three conversation sequences from one group of five pupils. In the conversations presented the active participants are two girls, Anne and Hilde, two boys, Daniel and Jonas, the teacher and the carpenter. The sequences are selected because of the possibilities they offer to find traces of meetings between different voices in the dialogues, based on the contextual dissimilarities represented by school and company, according to Bakhtinian polyphony. A risk to the trustworthiness of interpretations of conversations is that you find what you are looking for. Therefore, possible interpretations have been discussed in the research group that I am part of and with the teacher. Possible alternate interpretations continually have to be tested

against each other; due to limitations of space this cannot be demonstrated in a paper of this format.

CONVERSATIONS ON CHOICES RELATED TO RORBU MODEL

Using external authority

Anne needs information about the construction drawing of the rorbu, since she was not present when the group made the sketch. She asks Hilde about the ground floor and is told that there won’t be two floors; they will just make the top floor.

Anne: Won’t it look like this, then, with only one floor? (Forms a one-floor shack with her hands) and not like this? (Then forms a two-floor shack with a slanting roof.)

Hilde: Teacher said not to make a roof. She said we could just… imagine a shoebox, she said (Forms an imaginary shoebox with her hands.)

Anne: OK. I imagine a shoebox. (Forms a shoebox with her hands.)

The conversation between Hilde and Anne must be seen in light of the upcoming conversation with the carpenter which they are preparing for. Anne’s question reflects her experience with rorbu. Her question, “won’t it look like this with only one floor?” as she demonstrates the shape, can be seen as an argument for discussion. Her realistic model with two floors and a roof drawn from her experience becomes a counterargument to the group’s artificially low model with only one floor without consideration of a slanting roof. Hilde responds by conveying the teacher’s simplification; no roof, imagine a shoebox. I interpret this as a counterargument based in a school context. The choice between the two models has implications related to floorage. The choice has implications for how they move on.

Hilde refers to the teacher’s utterance by saying “teacher said” or “she said”, thereby implicitly giving her argument authority. While Anne positions her utterance in a realistic 3D model, Hilde positions hers in a school context. The utterance with reference to the teacher, the external authoritative voice, halts further discussion. Anne chooses to accept the school context.

Solution oriented, possibilities for choices

Shortly after, the whole group talks with the carpenter about their construction drawing:

Cptr: What have you got?

The carpenter reads out loud the different rooms the pupils have drawn in, among them a room for computer games, which the pupils have had heated arguments about whether such a room should be in a “rorbu”. Both floor use, space limitations, and special interests in computer games have been elements in the discussion. The pupils discuss loudly and show great disagreement as the carpenter listens. The carpenter does not enter this conflict but directs his attention to the kitchen:

Cptr: You have placed the kitchen cabinets mainly on the knee wall. You can have base cabinets there, but not top cabinets.

Fig 1: Carpenter points to base cabinet on the knee wall

Daniel: [It] was a bit like that (points to construction drawing from company)

Anne: Yes, we looked at that (refers to a work sketch lying on the table)

Cptr: The base cabinets will protrude 60 cm (Takes his folding rule and measures 60 cm from the knee wall, makes a mark on the drawing table.) So you can stand there doing the dishes for a while (addressed to Anne, who is standing closest to the knee wall under the slanting roof in the room we are in)… but not for very long. So the kitchen bench should probably be placed a bit more, so if you turned it.

Jonas: Couldn’t we do it more like this, then (points)

Cptr.: Yes, there (takes his pencil). I think I would take, put the kitchen cabinets along here (same place as Jonas suggested).

The carpenter asks what the pupils have to show him. The question is open and focuses on the pupils’ contribution. He then takes a quick overview before he turns to the kitchen. This communication is both oral and written, one might say that the carpenter paraphrases the pupils’ written contribution and in this way shows interest in and reinforces the pupils’ contribution to the dialogue. Rather than following the pupils as they argue about responsibility and guilt for a controversial room they have drawn in, the carpenter zooms in on a problem he notices, the pupils have not considered the slanting roof and knee walls. This reveals his realistic approach to the model; it should be functional as if it were to be built in reality, in full size. This supports Anne’s realistic thinking. As a response to the carpenter’s utterance, Daniel and Anne refer to a work sketch they have been given from the company. I assume they are trying to give the company some of the responsibility for what they have done, through this reference. They defend what they have drawn and are retrospective in their positioning. The carpenter does not use what they say; he just goes on, communicating through a combination of words and action. A kitchen cabinet will protrude 60 cm from a knee wall; he measures with a rule and demonstrates physically how low the ceiling is there. A grown up can’t stand upright. His expression and use of tools to demonstrate consequences of the pupils’ drawing, is polyphonic; he speaks with the authority of a professional, concretizing to someone

KKnee wall

who is learning, a master-swain situation, at the same time he is pointing out consequences of choices, as a consultant would advise a customer planning his cottage. He points ahead to a possible solution, the kitchen bench could be turned. The carpenter’s statements continue to be solution-oriented; he shows what he would do. He uses many words to soften his statements, and he shows tentativeness in the way he enters the dialog (Lindfors 1999, p. 119). The kitchen bench should probably be placed a little more. When he then says they could turn it a little, Jonas is quick to suggest a concrete placement where he sees the kitchen bench turned 90 degrees. A suggestion the carpenter agrees to, saying “yes” and goes on using the personal pronoun “I”: I think I would. His use of I suggests there can be others who would hold a different opinion. The pupils in this sequence choose to position the utterance in different ways; most of them are defensive and preoccupied with distribution of responsibility and giving reasons for choices that have been made and only Jonas chooses, like the carpenter, to look ahead towards possible solutions.

To examine each other’s perspective

This last conversation sequence was recorded about a month after the visit to the company. In the meantime, the pupils have, among other things, practiced constructing a 90° angle with a compass and ruler and measuring and drawing angles using a protractor.

Jonas and Daniel discuss whether to cut the inner walls according to a shoebox model (wall height in real life: 1.4 m) or whether to allow for knee walls and full ceiling height, that is, a realistic model.

Fig. 2 ”If you have a cabinet there, right?” Fig. 3 ”It slants”

Daniel has arguments for both the models. He use a practical argumentation, seen in fig. 2, if they place a cabinet, it will look rather odd if the inner wall is as in a shoebox model. But a shoebox model is easier to make he argued. And they refer to what the teacher has said them to do. Then he goes back to the cabinet argument again (fig. 3).

Daniel: But then they must slant, then we have to make the correct angle on all of them, that's a drag.

Jonas asks me what they should do. I tell them I don’t know what teacher has said. Daniel comments: No, well she says different things. Teacher comes in:

Jonas: How should the inner walls be?

Daniel: Should we make them with slanting ceiling?

Jonas: Should there be a slanting roof here or should it just go straight along?

Daniel: That’ll be a bit silly ’cause if you have a cabinet, right, it should fit with the slanting ceiling.

Teacher: What do you want to do?

Daniel: It will be easier if … but a slanting ceiling is better.

Jonas: No, it’s difficult (Daniel signals his agreement through sounds and gestures)

Teacher: Is it difficult?

Daniel: Yeah, you have to know the angle of…

Teacher: Yes, or the height here and here, and then it will slant automatically

Fig. 4. Teacher points out the heights the pupils need

The boys get going at once. Their concern has been addressed.

In the conversation with the teacher Daniel and Jonas first clarify the question they ask the teacher, they refer to the possible choices they are facing. Thereby, they also show their insight in the problem area (Lindfors, 1998). They take the pupil role, asking the teacher as an authority. At the same time, Daniel starts arguing against the teacher’s former shoebox model. The carpenter’s realistic model, with his problematizing of the location of the kitchen bench and lack of room for top cabinets, is used freely in Daniel’s argumentation – the cabinet will be too tall in relation to the walls. Daniel’s argumentation and the teacher’s open follow-up questions about what the pupils want and what is difficult, give room for polyphony. The teacher empowers the pupils when she asks them what they prefer. Through teacher’s response to the pupils’ problem with the angles, she gives them an alternative way to find out how the walls slant. The teacher's statement is also polyphonous. It can be interpreted from a practical context, where she gives the pupils a practical solution to the problem, but it can also be seen as positioned in a school mathematics context pointing ahead. The statement’s continuation can be application of characteristics of similar triangles and the connection with trigonometry, that is, an extended concept of angle. The result of teacher’s statement is that Daniel and Jonas are given a real choice as to how advanced they are going to make the model. Their voices and choices have been clarified.

CLOSING REMARKS

The conversation between Anne and Hilde appears to be a balanced conversation between two pupils who position themselves differently. Hilde, positioning herself in a school context and referring to an external authority, gives little room for negotiation. In the group’s work they have accepted the teacher’s suggestion. The conversation between the pupils and the carpenter is dominated by the carpenter, who

still shows genuine interest in the drawing the pupils have made. The pupils are on the carpenter’s home ground, showing their drawing to an expert. Even though the carpenter in many ways speaks to the pupils as he would to customers, pointing out possible solutions, the pupils do not have real power like a customer/buyer would have. Therefore, there is little symmetry in the conversation. The pupils do not choose to go into any extent of dialog with the carpenter. I choose to call this conversation pattern a master/apprentice conversation. The analysis of the conversation between Jonas, Daniel and the teacher demonstrates what I would characterize as an inquiry dialog where the participants are investigative in relation to each other’s perspectives. Here, the pupils’ argumentation is given room, they are given the opportunity to explain their arguments and what they consider difficult.

The characteristic polyphonical property I have identified in this paper is how the participants, both the pupils and the teacher, make flexible use of voices both from school practice and from the company practice in their argumentations. Being in a field of tension where the participants are confronted with different goals for the use of mathematics and also different tools (Wedege, 2006), seems in this case to opens up for polyphony (Bakhtin, 1981). Sometimes voices will collide. Differences between enterprises give the participants input relating to argumentation, language usage and choices. Polyphony opens up for the pupils voices to be used and strengthened.

The three conversations have different purposes and qualities. The pupils´ participation in polyphone dialogs give reasons to believe there is potential for developing critical reflectiveness about practice of mathematics in and outside school.

NOTES

1. This is part of my ongoing ph.d.-study in the research project Learning Conversation in Mathematics Practice (LCMP, leader: Marit Johnsen-Høines). The study is financed by the Research Council of Norway (NFR) and Bergen University College. For details on the project see www.hib.no/fou/limp (in Norwegian) and Johnsen-Høines (2010).

REFERENCES

Alrø, H., & Skovsmose, O. (2002). Dialogue and learning in mathematics education: intention, reflection, critique. Dordrecht: Kluwer Academic Publishers.

Bakhtin, M. (1986). Speech genres and other late essays. Austin: University of Minnesota press.

Bakhtin, M. (Ed.). (1981). The dialogic imagination. Austin: University of Texas Press.

Dysthe, O. (1999). Dialogical Perspective and Bakhtin. In O. Dysthe (Ed.), The dialogical Perspective and Bakhtin. PLF Report 2-99. (pp. 73-88). Bergen: Program for læringsforskning, Universitetet i Bergen.

Johnsen-Høines, M. (2010). Interpretative Research as Collaborative Inquiry. In B. Sriraman, C. Bergsten, S. Goodchild, G. Palsdottir, B. D. Søndergaard, & L. Haapasalo (Eds), The First Sourcebook on Nordic Research in Mathematics Education. Charlotte NC: Information Age Publishing.

Lindfors, J. W. (1999). Children´s inquiry. Using language to make sense of the world. New York: Teachers college press.

LK06. (2006). Læreplan for grunnskole og videregående opplæring. Lastet ned 25.08.2010. http://www.udir.no/Store-dokumenter-i-html/Prinsipp-for-opplaringa/Post.aspx?pageNumber=2

Mellin-Olsen, S. (1987). The politics of mathematics education. Dordrecht: Reidel.

Morson, G. S., & Emerson, C. (1990). Mikhail Bakhtin: creation of a prosaics. Stanford, Calif.: Stanford University Press.

Nielsen, M.F., & Nielsen, S. B. (2005). Samtaleanalyse. Roskilde. Samfundslitteratur.

OECD (2006). Assessing scientific, reading and mathemathical literacy: A framework for PISA 2006. Organisation for Economic Co-Operation and Development.

Rønning, F. (2009). Tension between an everyday solution and a school solution to a measuring problem. Paper presented at the CERME 6. Lyon.

Skovsmose, O. (2005). Travelling Through Education; Uncertainty, Mathematics, Responsibility. Rotterdam: Sense.

Svennevig, J., Sandvik, M., & Vagle, V. (1995): Tilnærminger til tekst – Modeller for språklig tekstanalyse. Oslo. Cappelen.

Valero, P. (2009). Socio-political perspectives on mathematical education. In P. Valero, & R. Zevenbergen (Eds), Researching the socio-political dimensions of mathematicseducation: issues of power in theory and methodology. Boston: Kluwer Academic Publishers.

Wedege, T. (2010). Researching workers´mathematics at work. Paper presented at the EIMI 2010. Educational Interfaces between Mathamtics and Industry, Lisbon.

Wedege, T. (2006). Menneskers matematikholdige kompetencer. Kunne det tænkes? – om matematiklæring. In O. Skovsmose & M. Blomhøj (Eds), Albertslund: Malling Beck A/S og forfatterne.