This paper investigates the discourse structure, mathematicising and the participatory approach in an early education classroom community whose pedagogy in the learning of mathematics draws on the sociocultural perspective. The social interactions of the classroom community, as well as the role of pre-symbolic sign vehicles as semiotic tools in supporting interaction building, mathematicising, and strategy selection, were subjected to a qualitative micro-level analysis based on applied discourse and semiotic procedures. The results of the study suggest that the young children's mathematical ability develops during the two-year observation period from nonverbal participation to verbal participation through the following core elements: attention to numeracy, the learning of number words, object counting and mathematical story telling. The iconic and indexical pre-symbolic sign vehicles and semiotic chaining applied in the learning situations provide the learning community with the means to structure their experiences of their everyday practise and to relate them to the culture of mathematicising.

Earlier, a semiotic perspective has been applied in school contexts [

This study offers an example of emerging mathematical activity, called here

The theoretical and pedagogical background of this paper holds a conception of the child as a cultural and historical subject, embedded within and constituted by a network of social relationships and interaction within the culture in question [

Central to this process is the creation of collective zones of proximal development [

Segregated modern and postmodern social practise is founded on different kinds of discourses like the discourse of politics, the discourse of religion, the discourse of business, and, for example, the discourse of everyday small-talk. We call the discourse of politics a certain kind of discourse with its own language, concepts and rules, or discourse principles. Some discourses are highly specialised, such as the discourse of mathematics containing abstract symbolic signs, language, rules, and principles. Using appropriate discourse moves is one of the rules to promote communication and mutual understanding [

We use theory of sign to clarify semiotic tools and semiotic chaining used in promoting and supporting interaction building, children’s mathematicising, and strategy selection. Peirce’s theory of sign [

In this paper, the collaborative practices of the mathematicising community in the early education classroom are highlighted with case-based descriptions derived from five activities, one from each of five consecutive terms during the observation years 2007–2009.

The specific research questions posed for the study are as follows.

What is the nature of the discourse moves or the conversational exchanges between the members of the early education classroom?

How are semiotic tools applied as meditational means in the participation of the mathematicising community?

What are the core elements of mathematicising in the early education classroom?

In this study the interplay between discourse structures, mathematicising, and the participatory approach will be discussed.

The day care centre that participated in the study is a private day care centre located in the northern part of Finland. The pedagogical philosophy of the day care centre consists of partnership—which means close collaboration with parents in child care—and action evaluation and participation, which involves listening to the children’s voices in the design and implementation of the curriculum. The usual starting age for the children is two years, and they continue in the day care centre until they start their primary school at the age of six to seven.

Altogether 11 children, their kindergarten teacher, and a university researcher participated in the study. The children, three male and eight female, were two to three years old at the beginning of the 29-month observation period. The kindergarten teacher and the university researcher shared a joint responsibility for the mathematical activity in the classroom community. During the first 12-month observation period, the mathematical activity was taught weekly, and during the second 17-month period it was taught five times per term. In the project, a research-based instructional emphasis was placed on the role of visual models as strongly iconic or strongly indexical sign vehicles (such as Cuisenaire rods, diverse geometrical models) in supporting the children’s mathematical skills. An example of visual models for the number four is presented in Figure

Example of visual models for the number four.

The adult members of the preschool learning community studied in this research share joint pedagogical principles (cf., page 7 [

This study draws on videotaped and transcribed data gathered from five joint mathematicising sessions. The classroom interactions are subjected to a qualitative discourse and semiotic analysis in order to investigate, evaluate, and disseminate the pedagogical ideas behind the classrooms. In the first three activities, the applied mathematical big book was

The discourse analysis procedure [

In order to develop the analysis to investigate the collaborative inquiry and the activity construction in the early education classroom, the data were analysed in several phases. Firstly, the video material capturing the social activity of the classroom in question was closely examined and reflected upon. Next the discourse occurring in the classroom during the course was transcribed. The analytic categories were constructed from the interaction data of the study, on the basis of the transcriptions, supported by the video data. Although the code names of the analytic categories represent context-free codes, the meaning behind the code names has been contextually defined to represent the interaction data of this study. The application of the coding scheme is realised through the microanalysis of the evolving classroom interaction by focusing on each conversational turn, using mutually exclusive and exhaustive categories.

Discourse moves identified in the data are

The method of analysis for mathematicising activity.

Discourse moves | Mathematicising | Participatory approach | |

Mathematicising strategy | Interaction building | ||

(i) accepting | (i) attention to quantity | (i) evaluation | (i) suggesting a participatory turn |

(ii) answering | (ii) number sequencing | (ii) touching ducks one by one | (ii) giving a participatory turn |

(iii) commenting | (iii) one-to-one correspondence between a number and the quantity | (iii) pointing at each duck one by one and counting orally | (iii) taking a participatory turn |

(iv) continuing | (iv) recognising a written symbol to a quantity | (iv) counting the first five dips in a ten-egg carton and then the second five dips | |

(v) counting | (v) addition task | (v) story telling | |

(vi) questioning | (vi) subtraction task | (vi) mathematical story telling | |

(vii) extending | (vii) subitising | ||

(viii) initiating | (viii) counting mentally | ||

(ix) tutoring | |||

(x) nonverbal participation |

The results will next be discussed with case-based descriptions. In the first activity; the applied mathematical big book was

Joint big book reading and mathematicising (videotaped in the autumn term 2007).

No. | name | Social interaction | Discourse moves | Mathematicising | Participatory approach |
---|---|---|---|---|---|

1 | teacher | how many ducks do you see there | questioning | attention to quantities | |

4 | Elli | yes I know how to count | initiating | evaluation | |

5 | Kalle | nonverbal participation | continuing | touches ducks one by one | |

6 | Inka | nonverbal participation | |||

7 | Elli | you do not count like that | commenting | evaluation | |

8 | Antti | can I count | questioning | ||

9 | teacher | now it is Kalle’s turn to count | tutoring | ||

10 | Elli | one, two, three, four, five | counting | one-to-one correspondence between a number and the quantity | points to each duck one by one and counts orally |

11 | teacher | okay | accepting | ||

12 | Inka | one | continuing |

In this activity, the teacher directs the children’s attention to quantities by asking:

This example demonstrates that the signs with strong indexical and iconic dimensions used in the teaching material can be applied to guide children’s attention to quantities; the children’s response can be either non-verbal, counting by pointing at each one of the objects, or verbal counting where the number names are produced orally and the mathematical content of the participation is the one-to-one correspondence between a number and the quantity.

In this phase of instruction, the everyday activity can be interpreted as the school life of ducks, and visual signs with strong indexical and iconic dimensions for the number five can be interpreted as a manipulative to represent the specific practice. The developmental path of the child in mathematicising can be sketched as a progression from the narrative function of a gesture to its grounding function, as the cases of Inka and Elli demonstrate.

In the second activity, represented in Table

Counting the book characters and the quantity of objects.

No. | Name | Social interaction | Discourse moves | Mathematicising | Participatory approach |
---|---|---|---|---|---|

1 | teacher | I have a great puzzle for you, look how many ducks there are in the picture | questioning | attention to quantities | |

2 | Inka | one, two, three, four, five, | counting | number sequencing | points to each one, one by one, and counts orally |

3 | Antti | first Selma | tutoring | giving participatory turn | |

4 | Selma | one, two, three, four | counting | one-to-one correspondence between a number and the quantity | points to each one, one by one, and counts orally |

5 | Antti | then me | tutoring | giving participatory turn | |

6 | Inka | I should, one | counting | suggesting participatory turn | |

7 | Antti | I have to, one, two, three, four | one-to-one correspondence between a number and quantity | taking a participatory turn |

In this activity, the teacher again starts by directing the children’s attention to quantities by saying,

This activity shows that Inka had developed her nonverbal participation to number sequencing, as she now uses her skill in order to count the number of objects.

In the next activity, presented in Table

Story telling and mathematicising.

No. | Name | Social interaction | Discourse moves | Mathematicising | Participatory approach |
---|---|---|---|---|---|

1 | Selma | a bus is travelling with those ducks | initiating | story telling | |

2 | Kalle | and there is a pig who is driving | continuing | ||

3 | Elli | me me me me | initiating | ||

4 | Veli | handing the book to Elli | non-verbal participation | giving a turn | |

5 | Elli | okay here is one duck and there are three ducks skating | extending | attention to quantities | subitising |

6 | Selma | on the road for cars | continuing | story telling | |

7 | Elli | Alma will start | tutoring | giving a turn |

Here Selma initiates by storytelling, saying,

In this activity, the children collaborate by themselves. This activity shows that the children do not necessarily pay attention to quantities in their social interactions without the adult guidance, although the learning material provides signs in visual modelling. In Elli’s turn, there is mathematical content and she is able to subitise in the situation correctly.

Elli’s developmental path shows that her mathematical ability has developed from object counting to that of subitising in the number area of four.

In the next activity, presented in Table

Mathematicising during the Mayday holiday period, videotaped in spring term 2009.

No. | Name | Social interaction | Discourse moves | Mathematicising | Participatory approach |
---|---|---|---|---|---|

6 | visiting-teacher | okay so how many go here | questioning | attention to quantity | |

7 | Elli | six | answering | subitising | |

8 | Inka | one, two, three, four, five, six | counting | One-to-one correspondence between a number and quantity | |

9 | visiting-teacher | if four go here and six go here, so how many altogether | questioning | attention to quantity | |

10 | Inka | one, two, three, four, five, six, seven, eight, nine, ten | counting | One-to-one correspondence between a number and quantity | counting the first five dips in a ten-egg carton and then the second five dips |

11 | visiting-teacher | okay good, what about, there are balloons there, so how many are there, it has been marked with a (written) number, can you find it | questioning | ||

12 | Elli | eight | answering | recognising a written symbol for quantity | |

13 | visiting-teacher | okay | accepting |

This activity starts by the teacher questioning,

This activity shows that Inka’s developmental path has moved from non-verbal participation through number sequencing to object counting. Elli, at her current stage, is able to subitise up till six and make a connection between the number eight and its written symbol.

In the next activity, presented in Table

Mathematical story for the visual model.

No. | Name | Social interaction | Discourse move | Mathematicising | Participatory approach |
---|---|---|---|---|---|

6 | Kalle | I have four | initiating | one to one correspondence between a digit and quantity | mathematical story telling |

7 | visiting-teacher | yes, and what about that, then | commenting | ||

8 | Kalle | if I added six it would make I do not know | continuing | addition task | |

9 | visiting-teacher | yes you said that you have four, and if you add six it will make ten | extending | ||

10 | visiting-teacher | okay good Inka | initiating | ||

11 | Inka | I have here six | continuing | subitising | |

12 | visiting-teacher | yes, will you tell us something else | questioning | ||

13 | Inka | noo | extending | ||

14 | visiting-teacher | it is a good story with six, you can tell a little bit more next time, okay Selma | commenting | ||

15 | Selma | I have ten, if I subtracted those I would have four and if I subtracted those I would have six | continuing | one-to-one correspondence between a digit and quantity | subtraction task |

16 | visiting-teacher | okay Elli | initiating | ||

17 | Elli | eight | answering | one-to-one correspondence between a digit and quantity | subitising |

18 | visiting-teacher | hmm | commenting | ||

19 | Elli | if I subtracted these two I would have six | continuing | subtraction task | counting mentally |

20 | visiting-teacher | oh-hoh what about alma | commenting | ||

21 | Alma | I have now seven if I subtracted | initiating | subtraction task | |

22 | visiting-teacher | yes | commenting | subtraction task |

Here Kalle starts by saying,

This example demonstrates that Inka has developed from nonverbal participation through number sequencing and object counting to a subitising level in her mathematical thinking. Elli is able to construct a subtraction task on the basis of signs that are less indexical and iconic and closer to symbolic signs and count it mentally.

The results of the study suggest that these children are eager and able to participate in mathematicising activity that is grounded on the instructional usage of semiotic chaining and indexical and iconic signs closer to their way of thinking and mental capacity. From the social and mental perspectives, the teacher role in guiding the participation becomes crucial. The children’s eagerness in participation can raise social conflicts that need to be resolved within the learning community. It should also be noted that children do not necessarily pay attention to the quantifying property of the model without teacher guidance.

The results of the study suggest that the usage of proper signs as semiotic tools in guiding participation supported diverse social roles such as initiator, extender, executor, and evaluator. In this study, the teacher had an active role as an initiator in guiding child attention to quantities. A child in participation can be viewed as an executor of the activity in question when continuing the teacher initiation. However, the children participated also as evaluators, tutors, initiators, and extenders.

The mathematical activity, of the children varied across situations. In mathematicising, the children’s participation moved from non-verbal participation to that of verbal participation. In verbal participation, they paid attention to quantities, used number sequencing in counting the number of objects, and found one-to-one correspondences between a number and the quantity. In the story telling activity, they constructed addition and subtraction tasks and counted the solution for them mentally.

This study in the participatory approach deals with content and with the mental and social strategies which the children chose. The signs applied in this study-mediated mathematical story telling, subitising, and connection building between a quantity and a written symbol. The children used metacognition in evaluating the nature of participation, and they used cognitive benchmarks such as pointing and touching the object when counting. Furthermore, they applied the mathematical model of grouping in fives when counting the number of ten objects. The social strategies that the children selected were giving participatory turns, suggesting participatory turns and taking participatory turns.

On the whole, this study suggests that educational interaction grounded on semiotic chaining in mathematics and the choosing of appropriate signs in mediating mathematicising activity offered an educationally rich environment in the early education classroom described in this study. Multidimensionality was seen in diverse social roles, making visible the core elements of the young children’s mathematical ability development, and diverse social and mental strategies emerged in the interaction. To conclude, signs with indexical and iconic dimensions applied in the learning situations provided the children with the means to structure their experiences of their living environment and to relate them to the culture of mathematicising. The role of the teacher then is to help children to see the culture of mathematics in the world around them and to support the children’s personal growth in the area of mathematics.

This study makes visible the nature of educational interaction as regards orientation towards mathematics which is possible in the early education classroom. From the educator’s point of view, the educational interaction investigated in this study requires expertise in mathematics pedagogy, knowledge of the child, and application of semiotics. These are not the questions typically resolved in instructional methods for teaching mathematics in early education classrooms. In this study, these questions were approached in close collaboration between an early childhood educator and an expert in mathematics pedagogy.

The present study challenges early childhood education programs to take into account the nature and role of mathematics in the pedagogy of young children. This study brings in the dual nature of learning in early mathematics: on the one hand, the mediational role of semiotic vehicles in raising the numerical awareness of young children and on the other hand, the teacher’s ability to assist the child in building up the meanings for mathematics in everyday life.