Metacognition in non-routine problem solving process of year 6 children Aikaterini Vissariou, Despina Desli

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Aikaterini Vissariou, Despina Desli. Metacognition in non-routine problem solving process of year 6 children. Eleventh Congress of the European Society for Research in Mathematics Education, Utrecht University, Feb 2019, Utrecht, Netherlands. ￿hal-02401125￿

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Metacognition in non-routine problem solving process of year 6 children Aikaterini Vissariou1 and Despina Desli2 1Aristotle University of Thessaloniki, Faculty of Education, Thessaloniki, Greece; [email protected] 2Aristotle University of Thessaloniki, Faculty of Education, Thessaloniki, Greece; [email protected]

This paper describes a study that investigated the metacognitive strategies used by fifteen Year 6 children while working individually on a non-routine mathematics problem. After attempting the problem, the students completed a questionnaire that elicited their retrospective reports on the metacognitive strategies they had employed while working on the problem. The analysis of the participating students’ written work and questionnaire responses revealed that independently of how many metacognitive strategies were observed; children’s problem solving success was not guaranteed. Interestingly, children’s use of metacognitive strategies was found particularly low. Educational implications for metacognition in both mathematical problem solving and in the learning of mathematics are discussed. Keywords: Metacognition, Non-routine problem, Problem solving, HISP inventory.

Introduction In the last decades, metacognition and its development have been identified as having an important role in education and thus underlay in the center of primary and secondary school curricula. Metacognition is popularly defined as ‘knowledge, awareness and deeper understanding of one’s own cognitive processes and products’ (Desoete, 2008, p. 190) and as ‘thinking about thinking’ (Anderson, 2002). It is often presented as comprising three phenomena: metacognitive knowledge, metacognitive experiences and metacognitive skills (Efklides, & Vlachopoulos, 2012). Moreover, it is commonly divided into two components: knowledge of cognition and regulation of cognition. Knowledge of cognition refers to the individuals’ awareness of their own knowledge and includes their declarative, procedural and conditional knowledge. Regulation of cognition, on the other hand, pertains to the procedural aspect of knowledge that enables the effective linking of actions needed to perform a given task. It includes planning, monitoring and regulation, and evaluation. Of central concern for researchers and educators is the crucial role that metacognition plays in children’s academic performance in general as well as in mathematical achievement in particular (Desoete, 2009). Fostering metacognition, students avoid “blind calculations” or a superficial “number crunching” approach in mathematics (Verschaffel, 1999) and learn how to use the acquired knowledge in a flexible way. Through metacognition, students become able to effectively discern information they know from those they do not know and retain new information (Dunning, Johnson, Ehrlinger, & Kruger, 2003). Metacognitive learners have an increased understanding of

the control over their thinking, they employ this control and, thus, they are able to thrive across the curriculum (Zimmerman, 2008). Different procedures and measures have been used to assess mathematical metacognition. A systematic review of the metacognition assessment in children aged 4–16 years over a 20-year period made by Gascoine, Higgins, and Wall (2017) revealed that the off-line methods, namely the self-reports measures, are the more commonly used, mainly because these measures are perceived as valid, reliable, easy to use and their application doesn’t require much time. However, Desoete (2008) asserts the importance of distinguishing the use of different tools or methods when examining different facets of metacognition in relation to other factors such as children’s age range among others. Her statement ‘how you test is what you get’ (p. 204) highlights this importance. Mathematical metacognition is often examined in terms of monitoring skills with respect to solving problems (Desoete, 2009, 2008; Panaoura, Philippou, & Christou, 2004). Children demonstrating metacognitive functions are usually aware of their own learning and are able to control their learning process. Associated learning skills like these are essential to the development of effective problem-solving ability as well as to academic success in mathematics (Magno, 2009). When working metacognitively students are able to correctly represent and solve mathematics problems, evaluate the effectiveness of strategies and recognize mistakes. In addition, they learn to clarify goals, understand concepts, monitor their understanding, predict outcomes and choose appropriate actions (Pappas Schattman, 2005). Wilson and Clarke (2004) view children’s problem solving as the purposeful alternation between cognitive and metacognitive activities. In particular, the completion of a mathematical problem is a cognitive process that requires the use of cognitive strategies (e.g., adding up). The selection and use of these cognitive strategies, however, shows the solver’s reflection on her existing knowledge (e.g., what do I know about the problem to help me work it out?). This is when metacognitive behavior is coming to the fore. It is possible that school practice could inhibit the development of metacognition during problem solving, since the mathematics problems used within school context are often solved by means of a known method or formula. It is of great interest to study children’s engagement in metacognitive activity when solving problems that are unfamiliar to them, like ‘non-routine’ problems. These problems ‘make cognitive demands over and above those needed for solution of routine problems, even when the knowledge and the skills required for their solution have been learned’ (Mullis et al., 2003, p. 32). Because non-routine problems require a flexible and strategic way of thinking, children often have difficulties and appear low success (Elia, Van den Heuvel-Panhuizen, & Kolovou, 2009). Interestingly, even good calculus students are not able to solve non-routine problems successfully. Such a failure does not necessarily result from restricted mathematical knowledge, but may be linked to ineffective use of that knowledge (Van Streun, 2000). Based on the hypothesis that metacognitive functions are more likely to employ during challenging tasks and knowing that metacognition and problem solving is usually studied with older students and adults (Schneider, & Artelt, 2010), the present study aimed to investigate the relationship between metacognition and non-routine problem solving in primary school children. Investigating

young children - apart from providing us with information concerning their ability to reflect upon their problem solving - will make us more sensitive about how metacognition might be further fostered. Research questions are as follows: a) to what extend do children use metacognitive strategies when solving a non-routine mathematics problem?, b) is there an association between children’s success in problem solving and their deployment of metacognition?

Method Participants Fifteen Year 6 children (8 male, 7 female), coming from the same class of a state primary school in a Greek island, participated in the study. They were randomly selected as they covered from low to high socioeconomic and academic statuses. The average age of the students was 11 years and 8 months (range: 11 years and 4 months - 12 years). The participating students had not received any training on metacognition. Design of the study - Instrument All participants were presented with a non-routine mathematics problem that was designed for the purpose of the study in order to examine children’s mathematical achievement as well as to establish a situation for revealing their problem solving strategies. The problem was also used as a basis for participating students’ reflections on their metacognitive strategies. After attempting the problem, the students completed a questionnaire that asked them to report retrospectively on the metacognitive strategies they had used while working on the particular problem. More specifically, the non-routine problem used in the present study (see Figure 1) did not require a particular algorithm to be applied neither was connected to a specialized mathematical concept. Because it is recognized that routine mathematics problems often require little student reflection, it was tried to follow the criteria that define the non-routine problems and reveal solvers’ critical thinking (e.g, Kolovou, Van den Heuvel-Panhuizen, & Bakker, 2009). The questionnaire was based on the ‘How I Solve Problems’ measurement tool of metacognition developed by Fortunato, Hecht, Tittle, and Alvarez (1991), after taking into consideration relevant instruments for measuring metacognition compiled from the literature (e.g., Jr MAI developed by Sperling et al., 2002). The particular instrument was chosen as it is considered valid for research and useful for assessment and intervention in classrooms (Gascoine et al., 2017; Sperling et al., 2002). A few modifications were done to the original version of the tool, mainly concerning rewording a few items in order to make the questionnaire more appropriate for younger children. In the present study, the questionnaire consisted of twenty-one action statements, as many as in the original version. The statements covered a range of likely metacognitive activities during problem solving process. Students were asked to respond to these statements using a five-point Likert scale (ranging from ‘1=never to 5=always”) based on their way of working when solving the mathematics problem.

‘The cost of the theater ‘Avlea’ with 250 seats for a two-hour play is 750€. Last Sunday 30% of the seats were empty. Which will be the profit of the theater if the cost of the ticket for the audience is 9 € each?’

Figure 1: The problem used in the present study

The questionnaire was subdivided into four sections. He first, named ‘Before you started to solve the problem’ section referred to reading and understanding the problem as well as planning of a solution method (6 statements). The second, the ‘As you worked on the problem’ section involved employment of the solution method when working on the problem (5 statements). The third, the ‘After you finished working on the problem’ section presented with ways of confirming and checking for the solution (4 statements) And the ‘Ways of working on the problem’ section listed particular heuristics as strategies that a student might have used when working on the problem (6 statements).

Procedure All participants were asked to solve the problem and answer the questionnaire individually in their classroom during school hours. Their anonymity was ensured. They were initially required to read carefully the non-routine mathematics problem. The first section of the questionnaire was then given and a 30 minutes period was allocated for solving the mathematics problem. The students were provided with a specific area on a paper in which they were asked to write their working and solution. Immediately after solving the problem, the rest of the questionnaire was administered: the students were required to respond to the statements that were included in the last three sections of the questionnaire regarding specific metacognitive strategies they might have or might have not used.

Results Problem solving success Of the fifteen students who attempted the problem, only eight solved it correctly and, for the purposes of the data analysis, they are called ‘successful students’, whereas those who failed in solving the problem (seven students) were called ‘unsuccessful students’. Although children’s problem solving strategies were recorded, their analysis will not be described as it goes beyond the purpose of the present paper. In order to examine whether the successful and unsuccessful students provided similar or different responses to the questionnaire regarding their metacognitive strategies, an independent samples t- test was conducted. The analysis revealed that, although the overall metacognitive level mean score of the successful students (3,46) was higher than that of the unsuccessful students (3,09), this difference was not found statistically significant (t=-1,159, df=13, p=.267). When the same analysis was repeated for each questionnaire section separately, the original results were confirmed (t=-

1,182, df=13, p=.258, t=-1,100, df=13, p=.291, t=-,383, df=13, p=.708 and t=-,926, df=13, p=.371 for Sections A, B, C and D, respectively). Thus, no statistical significant differences were observed in children’s responses to the total of the statements in each section of the questionnaire among successful and unsuccessful students. Table 1 shows these findings. The mean level of students’ metacognitive strategy for the ‘Ways of working’ section (Section D) was the lowest mean level of all the sections for both successful and unsuccessful students (3,02 and 2,76, respectively). This finding shows that the heuristics listed for children to indicate whether they had used when working on the problem were rather under-recognized. In particular, more than 80% of the students reported that they never/ rarely draw a picture or a diagram for the better understanding of the problem (55, 3% and 26, 7%, respectively for never/rarely).

Questionnaire Problem Solving Success Total Sections Successful Unsuccessful A. Before you started 3,62 (0,93) 3,07 (0,87) 3,34 (0,91) B. As you worked 3,5 (0,93) 3,03 (0,69) 3,26 (0,81) C. After you finished 3,85 (0,58) 3,71 (0,73) 3,78 (0,65) D. Ways of working 3,02 (0,69) 2,76 (0,29) 2,89 (0,49) Overall 3,46 (0,66) 3,09 (0,54) 3,27 (0,61)

Table 1: Mean scores1 (and standard deviations) for children’s metacognitive level in the questionnaire by their success in problem solving

On the contrary, children’s response rates for the ‘After you finished’ section (Section C) that might imply their use of strategies for verifying their solutions were quite high (3,85 and 3,71 for successful and unsuccessful students, respectively). For example, 53% of the students said that they always go back and check if their calculations are correct. However, no participant was found to say that she always thinks about a different way to solve the problem. Correlation between metacognitive strategies and problem solving success Aiming to investigate the existence of correlations between children’s responses to the statements in the four sections of the questionnaire (metacognitive strategies) as well as between children’s questionnaire responses and their written working of the problem (problem solving success), correlation analysis was conducted, as presented in Table 2. Interestingly, no correlations were found either between problem solving success and overall metacognition (Pearson’s r=.306, p=.267) or between problem solving success and metacognitive strategies at each questionnaire section separately (Pearson’s r=.312, p=.258, Pearson’s r=.292, p=.291, Pearson’s r=.106, p=.708 and Pearson’s r=.249, p=.371, for sections A, B, C and D, respectively). However, participants’

1 Responses were measured on a scale from 1 (Never) to 5 (Always)

overall metacognition was positively correlated to their responses to each questionnaire section (Pearson’s r=.953, p<.01, Pearson’s r=.817, p<.01, Pearson’s r=.869, p<.01 and Pearson’s r=.691, p<.01, for sections A, B, C and D, respectively) This result shows that participants who responded highly to statements at one section tended to respond highly to statements at any other section. General Discussion The present study has attempted to investigate the metacognitive strategies used by sixth graders while working on a non-routine mathematics problem. The analysis of students’ written work and questionnaire responses revealed two main findings. First, students reported that they are using metacognition. However, their demonstration of metacognitive strategies is quite low both in general and at different stages of the solution process. In particular, a few reported that they greatly made connections to their previous experiences, reviewed their progress towards the problem solution, checked their calculations while they worked, attempted to verify the accuracy and sense of their answer to the problem. Additionally, an unequal use of metacognitive strategies at different stages of the solution process was also observed. Overall Section Α Section B Section C Section D Metacognition Problem .306 .312 .292 .106 .249 Solving Success Section A .953** .684** .854** .603* Section B .817** .657** .338 Section C .869** .430 Section D .691** *Significant correlation at the p<.05 level **Significant correlation at the p<.01 level Table 2: Correlations between overall metacognition and problem solving success

Second, a successful solution to the problem was found not necessarily connected to the employment of metacognitive strategies. Although a challenging problem was chosen for use in this study –mainly because it would raise the opportunity for metacognitive strategies to be called into play- so few students succeeded in obtaining a complete solution. This finding is consistent with previous studies in which the success in a mathematical problem is considered a cognitive process that requires the use of cognitive strategies (e.g., Wilson & Clarke, 2004); thus, low metacognitive strategies during problem solving do not undoubtedly lead to failure. This issue further reveals that even if students behave metacognitively (e.g., they are able to recognize when they are stuck), their difficulty with solving the problem remains (e.g., they fail to solve the problem). It is possible that their worthy efforts for metacognitive strategies will be restricted if they are unable to identify alternative successful strategies in the problem solving process. More needs to be known about the impact of the school practice on the development of problem solving and metacognitive strategies in children and implications for teaching should be considered.

From a methodological perspective, it is important to note that a self-report questionnaire as the one used in the present study might provide only a little about children’s metacognitive strategy use. Further research with larger samples is required in order to get detailed information on both children’s actual problem solving behavior and children’s actual use of metacognitive strategies. Multi-method approaches (Desoete, 2008), such as on-line procedures, might work towards this aim. Interestingly, metacognitive instruction leads not only to the empowerment of students’ metacognition but also to the development of the practices from the part of the teachers concerning metacognition in both mathematical problem solving and in the learning of mathematics (Magno, 2009). As many factors as possible need to be further considered in order that any teacher intervention on metacognitive engagement can be informed and effective.

About the authors Aikaterini Vissariou is a Ph.D student at the Aristotle University of Thessaloniki in the Faculty of Education, in the field of Didactics of Mathematics. She is also doing her second Bachelor degree at the School of Philosophy and Pedagogy at the Aristotle University. She works as a primary school teacher at the Ministry of Education. Despina Desli is associate professor of mathematics teaching and learning at the Aristotle University of Thessaloniki in the Faculty of Education. Her main research interests concentrate on children’s mathematical reasoning, teacher education and mathematics curricula. She has worked on research projects in both primary and secondary schools concerned with mathematics education and in-service teacher training. Acknowledgment This research has been financially supported by General Secretariat for Research and Technology (GSRT) and the Hellenic Foundation for Research and Innovation (HFRI) (Scholarship Code: 384).

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