Por otra parte debe considerarse el hecho de que el corredor Tijuana-Ensenada sea frontera con California, uno de los estados más ricos de Estados Unidos y que sea una zona

The data reported in this study were from a wider investigation into the use of

collaborative metacognition during problem solving in primary aged children. Twelve students were observed during their collaborative problem solving sessions in

mathematics. Students in a primary 5 classroom (aged 9-10) were asked to work together in groups of four each week for one full school term lasting 15 weeks.

Research suggests that it is difficult to isolate specific issues which may influence group dynamics on a global basis (Dillenbourg, Baker, Blaye & O’Malley 1995). Issues such as the most effective group size, number of girls versus boys and levels of knowledge will all impact on collaboration to some extent. However, that extent is different for each task and indeed may be different from day to day. It is therefore important to acknowledge that such issues exist but are not issues which can be eradicated in a normal classroom setting. In light of this, and as one of the main purposes of the study was to understand

metacognition in a natural group setting in a classroom, it was left to the teacher to choose how to group students based on what would normally occur.

Students were grouped in four with two boys and two girls in each and the groups reflected the mixed ability of the class. Those students not participating in the study were grouped and worked in the same way as those in the study.

The problem solving sessions lasted approximately 90 minutes. The lessons began with a short whole class discussion led by the teacher, the primary purpose of which was to provide an overview of the task at hand and remind students that they were to work together to jointly solve the problem. Students were then asked to join their groups and work on the problem. During the problem solving sessions, the teacher and researcher joined the groups at various points to monitor progress. Occasionally, if the teacher felt

that all groups were struggling with some aspect of the problem, she would call them together for a whole-class discussion on that particular aspect.

This design differs to the existing research which considers metacognition in group work. Previous research has focussed on students who are high-achieving (Goos et al 2002, Iiskala 2004 & 2010, Hurme et al 2006) show a preference for group work (e.g. Goos et al 2002) and students working in dyads (e.g. Iiskala 2010). To my knowledge there are no published studies which consider collaborative metacognition during mathematical problem solving within a completely natural classroom setting which has not been modified in order to accommodate potential findings.

Students worked together each week and were videotaped over three sessions. One session near the beginning of the study, one in the middle and one near the end of the study. Further information and justification for the design of the study can be found in the methods section.

3.11.1 Data and Coding

Data were in the form of verbal utterances. Data were taken from three groups of students over the three sessions, providing a total of 9 videotaped sessions. In order to understand the extent to which students displayed both metacognition and transactive talk, two coding schemes were developed. The first was to understand the learning processes and type of talk that students displayed, specifically focussing on metacognition. The second coding scheme was required to understand the collaborative nature of the interactions. Further details of the development of the coding schemes can be found in the methods chapter and the coding schemes can be found in Appendix A.

3.11.2 Data Analysis

Utterances were coded according to the coding schemes, firstly to understand the types of talk displayed. The transactive coding was then applied to understand the extent to which students interacted with one another. This double coding of the data meant that a

metacognitive statement could also be coded as transactive and a transactive statement could also be metacognitive. Totals were provided for each type of talk. In order to investigate the relationship between transactive (collaborative) talk and other types of talk, it was necessary to understand the interactive nature of the analysis. Totals of each type of talk provided a static account of the quantities of talk displayed. However, I was interested

in how often a transactive statement would lead to a metacognitive one and how often a metacognitive statement would lead to a transactive one. This interaction provided the operationalisation of collaborative metacognition.

Once the coding schemes had been applied, analysis was performed to produce an

understanding of the type of statements which followed from each. Figures were produced for the amount of transactive statements which led to other statement types and also for the amounts of each statement type which led to a transactive statement. However, these figures did not provide a realistic account of the data since there were far more social statements than cognitive and metacognitive. A more appropriate method was to use proportions of each statement type which were preceded by a transactive statement and also which led to a transactive statement. It is these figures which are reported on in the results section.

As previously mentioned, the notion of collaboration is a theoretical one and the operationalisation of this term is through the use of transactive coding schemes. The research question has therefore been divided into two distinct questions which reflect the operationalisation of the term:

Is there a relationship between the transactive quality of a statement and the type of statement which followed it?

Is there a relationship between the type of statement made and the transactive quality of the statement which followed it?

3.11.3 Results

Is there a relationship between the transactive quality of a statement and the type of statement which followed it?

Proportion of each statement type preceded by a transactive statement 82 157 90 23 22 1615 751 214 129 98 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Social (1697) Cognitive (908) Metacognitive (304) Teacher Teaching (152) Other (120) U tt e ra nc e t y pe Proportion Preceded by Transactive Preceded by Other

Figure 3- 2 Proportion of each statement type preceded by a transactive statement

Is there a relationship between the type of statement made and the transactive quality of the statement which follows it?

What proportion of each utterance type was followed by a transactive utterance?

Proportion of each statement type followed by a transactive statement

73 187 88 14 5 1624 721 216 138 115 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Social (1697) Cognitive (908) Metacognitive (304) Teacher Teaching (152) Other (120) S ta te m e nt Ty pe Proportion Followed by transactive Followed by non-transactive

Figure 3- 3 Proportion of each statement type followed by a transactive statement

Figures (3-2) and (3-3) suggest a relationship between transactive statements and

metacognitive statements. Although there were higher quantities of cognitive statements which both led to and were followed by a transactive statement, a higher proportion of

metacognitive statements fell into this category. Overall, a higher proportion of metacognitive statements were followed by a transactive statement than were other statement types. Similarly when a metacognitive statement was made, a higher proportion of these were preceded by a transactive statement, than all other statement types.

Chi square tests of association were performed to ascertain if the differences were

statistically significant. Test were performed for metacognitive and cognitive statements preceded by a transactive statement and also for the difference between metacognitive and cognitive followed by a transactive statement as these represented the largest proportions.

The purpose of a chi square test is explained differently by researchers. Coolican (2004) suggests a chi square is used to search for an association between data. Cohen, Manion and Morrison (2007) on the other hand describe a chi square test as a test for difference. The different descriptions can be better classified as what the research question wants to ascertain versus what the statistical analysis does. The chi square tests for a difference between expected frequencies and actual observed frequencies. In determining the difference between expected and observed frequencies, inferences can be made regarding the presence of an association between the categories.

The type of data appropriate for use with this test are categorical data where each item must appear in only one of the cells. Furthermore, some statisticians (e.g. Cochran 1954) suggest that no more than 20% of the cells should contain less than 5 counts. If such a case arises, a Fischer’s Exact test is more appropriate than the Pearson’s (Cochran 1954).

The data from this study were categorical and each utterance could only be coded as one type (e.g. metacognitive). Therefore a chi square was an appropriate test for ascertaining if there was difference between the observed and expected frequencies of each utterance type.

Utterances preceded by a transactive statement

Around 30% of all metacognitive utterances were preceded by a transactive statement compared to around 17% of all cognitive utterances. A χ2

analysis of the difference was significant, (1, N = 1212), χ2

Utterances followed by a transactive statement

Around 29% of all metacognitive utterances were then followed by a transactive statement compared to around 21% of all cognitive utterances. A χ2

analysis of the difference was significant, (1, N = 1212), χ2

= 9.05, p=0.003.

Therefore, there was a significant difference between the proportion of metacognitive utterances which were preceded by a transactive utterance compared to cognitive

utterances preceded by a transactive utterance. Similarly, there was a significant difference between the proportion of metacognitive utterances which were followed by a transactive utterance compared to the proportion of cognitive utterances followed by a transactive utterance.