Pernille B Sunde

Peter Sunde

Development and variance components in single-digit addition strategies in year one

Keywords: Mental computation, addition, primary education, individual differences

In this paper, we aim to quantify students' development of mental strategies in single-digit addition over a five months period in six Danish year one classes dependent on their initial stage based on assessment interviews of 83 year one students' strategy use when answering 36 single-digit addition tasks. Students were interviewed twice (autumn and spring). Students using direct retrieval the least at interview 1 had the highest increase in proportional use at interview 2, whereas students using it the most showed no mean increase. For counting and derived fact, the change from interview 1 to 2 was constant irrespective of initial use. Analysis of how much of the variation in change in proportional use of different strategies was explained by different factors showed that 60-83% of the total variation was explained by the students' initial strategy use. Factors attributable to the specific learning environment and instructional practice did not explain any significant variation.

Background

Whole number arithmetic is a foundational subject in early years' mathematics. The first arithmetic operation students encounter in school is addition and simple arithmetic, which is related to number sense according to [START_REF] Gersten | Early identification and interventions for students with mathematics difficulties[END_REF]. Thus, knowledge on students' early use and development of strategies in addition contributes to the understanding of students' development of arithmetic proficiency in general (e.g., [START_REF] Laski | Early use of decomposition for addition and its relation to base-10 knowledge[END_REF]. Furthermore, students' early qualitative use of strategies in mental arithmetic has been shown to influence development of mathematics competence and to be a valid predictor of later mathematical achievement and difficulties [START_REF] Carr | Fluency, accuracy, and gender predict developmental trajectories of arithmetic strategies[END_REF][START_REF] Dowker | Young children's use of derived fact strategies for addition and subtraction[END_REF][START_REF] Gersten | Early identification and interventions for students with mathematics difficulties[END_REF][START_REF] Ostad | Developmental differences in addition strategies: A comparison of mathematically disabled and mathematically normal children[END_REF].

Strategies for single-digit mental addition can be divided into three categories as described by [START_REF] Carpenter | The acquisition of addition and subtraction: Concepts in grades one through three[END_REF]: (1) Direct modelling strategies with fingers or physical objects, where counting sequences are used to count all concrete objects. (2) Counting strategies, where mental counting is used in counting-all or counting-on from either the first or, later, the largest addend. (3) Number fact strategies, which involve either a) direct retrieval, which is the direct recalling of the sum, or b) derived fact strategies, where known facts are used to derive the answer. Interesting to note that derived fact strategies has been the focus for many recent studies (e.g. [START_REF] Dowker | Young children's use of derived fact strategies for addition and subtraction[END_REF][START_REF] Gaidoschik | First-graders' development of calculation strategies: How deriving facts helps automatize facts[END_REF][START_REF] Laski | Early use of decomposition for addition and its relation to base-10 knowledge[END_REF].

Students strategy use in arithmetic is influenced by many factors related to the individual student, for example working memory (e.g., [START_REF] Imbo | The development of strategy use in elementary school children: Working memory and individual differences[END_REF] and gender (e.g., [START_REF] Carr | Fluency, accuracy, and gender predict developmental trajectories of arithmetic strategies[END_REF], as well as the learning environment, such as instructional practice (e.g., [START_REF] Timmermans | Gender-related effects of contemporary math instruction for low performers on problem-solving behavior[END_REF], and parents and teachers' strategy preferences [START_REF] Carr | Gender differences in first-grade mathematics strategy use: Parent and teacher contributions[END_REF]. Thus, [START_REF] Carr | Gender differences in first-grade mathematics strategy use: Parent and teacher contributions[END_REF] found that boys would use the strategies their parents use, which include the use of retrieval strategies. These strategies were also those their teachers were more likely to teach them. Furthermore, students tend to use the strategies they are confident with in a feedback loop as described by [START_REF] Bailey | The codevelopment of skill at and preference for use of retrieval-based processes for solving addition problems: Individual and sex differences from first to sixth grades[END_REF], where early preference for a strategy predicts later use, which in turn predicts later preference. Students, who primarily use counting in early arithmetic at the expense of derived fact strategies, will get less practise in working flexibly with numbers. As a consequence, they will have fewer options to develop the more complex comprehension required in advanced arithmetic [START_REF] Gersten | Early identification and interventions for students with mathematics difficulties[END_REF]. Thus, students change in strategy use are found to be dependent on individual factors, such as gender and initial strategy use, as well as factors related to the learning environment and learning opportunities, such as instructional practice.

The study of [START_REF] Gaidoschik | First-graders' development of calculation strategies: How deriving facts helps automatize facts[END_REF] is an example of a study where students' development of strategies for single-digit addition in year one are investigated in relation to both, students' prior knowledge and teachers' instructional practice. [START_REF] Gaidoschik | First-graders' development of calculation strategies: How deriving facts helps automatize facts[END_REF] found that strategy development was significantly related to students' prior number knowledge. Furthermore, he found in a qualitative analysis that although teachers' instructional practice focussed on counting strategies in year one, many students developed derived facts and automatized sums. This finding indicates instructional practice had minor influence on these students development of strategies. Although several studies have documented effects of students' prior knowledge and instructional practice, there is less evidence on the relative contribution of individual differences and instructional practice for students' development of strategies.

The current study

The data presented here are part of a larger project on the relationship between students' development of mental strategies in single-digit addition, teachers' perceptions of the learning of number and arithmetic in year one, and classroom instruction. In this paper, we aim to quantify students' development of mental strategies in single-digit addition over a five month period in six Danish year one classes dependent on their initial strategy use. Furthermore, we investigate how much different variance components contribute to the overall variation in year one students' development of strategies. We are specifically interested in quantifying how much of the total variation in the students' strategy use at time 2 that could be explained by their strategy use at time 1, instruction time and possible differences between classes in general as well as in interaction with time as the latter would include all variation due to possible effects of instruction.

Methods and analyses

The study and participants

The study consisted of 155 assessment interviews on strategy use for single-digit addition conducted on 83 year one students (46 girls; age 7) from six Danish year one classes. In each class, between 11 and 18 students were selected randomly though balanced with respect to gender. 72 students were interviewed in November 2015 and April 2016. 11 students were interviewed only once due to absence. To evaluate differences and similarities in the learning environment of the six classes, the six teachers (three female and three male) were interviewed on the teaching and learning of arithmetic and number. Furthermore, their instructional practice was documented through classroom video observations. These data will be analysed and discussed in detail elsewhere, but initial analysis indicated the teachers differed with respect to emphasis on teaching mental strategies for arithmetic and focus on procedural as opposed to conceptual knowledge. For example, two teachers emphasised rote learning of facts and procedural learning of counting strategies, two teachers focussed on students' skilled use of derived fact strategies and taught these explicitly, and two teachers were conscious of both rote learning of facts and derived fact strategies but did not emphasise one over the other. Furthermore, a previous study showed that the teachers' expectations for their year one students' development in number and arithmetic differed substantially (with one to two years of learning) [START_REF] Sunde | Danish teachers' expectations of year one pupils' additive competence[END_REF]. Thus, with respect to differences in teachers approach to teaching and learning arithmetic the six classes represents a continuum from high level of focus on explicit instruction of strategies with conceptual understanding to an instructional practice focussing on automatization of facts and procedural understanding.

Assessment of students' strategy use

Students' use of mental strategies for single-digit addition was assessed in one-to-one assessment interviews where students were presented with flashcards of the 36 addition tasks with numerals 2-9, including doubles. Manipulatives, paper and pencil were not accessible for the student. The flashcards were presented with tasks of increasing difficulty to avoid less confident students to give up or only use counting if presented with difficult questions at the beginning of the interview. Interviews took place in a quiet room at the school and each session lasted 10-30 minutes. We asked the student to explain how he or she found the answer. Based on students' explanations and the interviewer's observation, answers were categorised in four categories. 'Error' was used if the student gave up or miscalculated. For correct answers 'Counting' included all variations of counting procedures on fingers, verbal or self-report as mental counting, 'Direct retrieval' was used for automatized sum, and 'Derived fact' included all variations of decomposing addends and using automatized sums to calculate the answer (e.g. 4+5 = 4+4+1 or 5+5-1) (for more detail see [START_REF] Sunde | Sex differences in mental strategies for single-digit addition in the first years of school[END_REF].

For each assessment interview, we calculated a strategy use profile on the basis of the proportion of answers (denoted 'proportional strategy use' hereafter) to the 36 addition tasks scored as 'error', 'counting', 'direct retrieval', and 'derived fact'. Hence, if a student used 'counting' to solve 6 of the 36 tasks, the proportion for counting, p counting , was 6/36 = 0.1667. Original values were logit transformed prior to the analysis (logit p i = ln[p i /{1-p i }]). Before transformation, we substituted zero values (not defined on the logit scale) by 0.01, which is 2.8 times lower than the minimum achievable value larger than 0 (1 of 36 items = 0.028).

Analysis of change in proportional strategy use

Development in mean use of each of the four strategies over half a school year was analysed by modelling proportional strategy use in the second assessment interview (p i2 ) as a linear regression function of the students' proportional strategy use in the first assessment interview (p i1 ) five months earlier. From this regression function ( i2 = β 0 + β 1 p i1 ) the mean change in (logit-transformed) p i from a given level in autumn to the next spring could be derived as the vertical distance between the regression line and the reference line y=x that indicated no change in mean strategy use between the two test sessions. Hence, the more students increases the proportional use of a given strategy from autumn to spring, the higher above the y=x line will the data points and the regression line be located and vice versa. From the slope of the regression line, it is furthermore possible to test whether the change in use of a given strategy is conditional on how much it was used in the first interview. If the mean change in proportional use (on logit scale) of a given strategy from interview one to interview two is equal for all students irrespective of how much they used the strategy on the first assessment interview, the slope of the regression line would not be significantly different from 1. On the contrary, a slope < 1 indicate a greater change in proportional use of a strategy by those students that used the strategy least in the first interview whereas a slope >1 would indicate the opposite, i.e. that those students that used a strategy least in interview one are also those that have developed relatively less from interview one to interview two.

Before the linear regression functions were established, we tested for possible quadratic and cubic relationships between p i2 and p i1 , as well as for possible effects of class (for the case development in strategy use varied between classes between interview one and two) and sex (for the case boys and girls differed in change of strategy use between interview one and two) without finding any significant effects of those predictors.

Analysis of variance components

We quantified how much of the variance in proportional use of each strategy was explained by different factors. The factors were time (difference between interview one and two), student identity nested within class (the systematic difference between students over the two interviews within a given class), class (the systematic difference between students in different classes) and the interaction between class and time (systematic differences in how much students from different classes changed from interview one to two). We calculated variance components as each of the aforementioned factors' sum-of-square compared to the total sum-of-squares in General linear models with all 155 interview results as observation units and the aforementioned variables and interaction terms as fixed effects. In this respect, the interaction term class*time encompasses all variation that could be explained by differential instructional practice because it represents the time a student has received instruction in a specific classroom. We derived statistical significances of each factor from F-tests.

Results

Change in proportional strategy use: Conditional on initial strategy use

As expected, mean use of all four strategies significantly changed from autumn to spring (all pvalues ≤ 0.001) as the proportions of errors and counting decreased and direct retrieval and decomposition increased. The coefficients of determination (R 2 ) when modelling results from assessment interview two as functions of the results from interview one showed that the changes in the students' proportional use of a strategy was highly conditional on initial strategy use (Figure 1: R 2 -values: 0.26-0.69). The slopes of the regression lines were significantly < 1 for 'error' (t 71 = 6.52, P < 0.0001) and 'direct retrieval' (t 71 = 2.18, P = 0.03), but did not differ from 1 for 'derived facts' (t 71 = 1.75, P = 0.08) and 'counting' (t 71 = 0.6, P = 0.4).

For 'error', the slope lower than 1 indicates that students with the highest error rates in interview one displayed the highest reduction in error rates between assessment interview one and two, probably as the mere result of the vast majority of the students solving all items correctly in general and especially in interview two. For 'counting' and 'derived fact' slopes not significantly different from 1 indicated that the mean change in use of these two strategies from interview one to two (decrease in counting, increase in derived fact) were constant no matter how much the strategies were used in the first assessment interview situation. For 'direct retrieval' a slope lower than 1 and inspection of the confidence zones of the regression line suggested that students using direct retrieval least in the first interview expressed the highest increase in use of this strategy in interview two. Whereas, students with the highest use of this strategy in interview one did not show any mean increase in use of this strategy in interview two. From inspection of Figure 1, one can see that girls generally used counting substantially more than boys, and boys significantly used derived fact more than girls. These sex differences that were already present at interview one are dealt with elsewhere [START_REF] Sunde | Sex differences in mental strategies for single-digit addition in the first years of school[END_REF]. As regards changes from interview one to interview two, boys and girls did not differ for any of the strategies (additive effect of sex if included in regression models: all p-values > 0.2).

Variance components: Effect of individual differences, time and class

The half school year represented by the mean difference in strategy use between assessment interview one and two explained 3-6 % of the total variation in use of the four strategies, as compared to 60-83% explained by student identity nested within class (all p-values ≤ 0.001) and 3-12 % by class (statistically significant for 'error', 'counting' and 'derived facts': Figure 2). The time-by-class interactions (representing specific instructional practice) explained 0-1 % indicating no measurable class differences in mean progression from interview one to two (Figure 2). use at interview two as function of strategy use at interview one demonstrates that in at least two of four strategies the mean change in strategy use from interview one to two differed significantly between those students who used the strategy most and least. In the present case those students with highest proportion of errors in interview one expressed the largest reduction in error frequency in interview two five months' later. Those who used direct retrieval least in the first interview expressed the highest increase in use of this strategy to interview two. For counting, the mean change in strategy use did not differ significantly between students with high and low initial use. This corresponds with Bailey et al.'s (2012) feedback loop of relationship between skills and preference for specific strategies.

Considering that use of derived fact strategies are related to high arithmetic ability [START_REF] Dowker | Young children's use of derived fact strategies for addition and subtraction[END_REF] and the relationship between excessive use of counting and mathematical difficulties [START_REF] Ostad | Developmental differences in addition strategies: A comparison of mathematically disabled and mathematically normal children[END_REF], it is concerning that the students with high initial proportional use of counting do not decrease the use of this strategy more than students with a low initial use. These findings underline the importance of analysing and reporting changes in strategy use (or any other achievement measure) as function of a given time period or intervention type differentially for student groups with different start level. In that respect, the here used graphical and statistical method based on testing the nil-expectation of a slope = 1, may provide an easily used and comprehendible method. One implication for practice of these results is that instruction should support students in developing a diverse strategy use.

Second, albeit mean use of all four strategies significantly changed over the five months between interview one and two (as expected), these changes, reflecting the fundamental 'effect' of learning and maturation, were modest compared to the variation in strategy use between students (evident as the high correlation coefficients of the scatter plots in Figure 1, as well as the amount of variance components explained by student ID in Figure 2). This illustrates the importance of acknowledging the long known but sometimes implicitly forgotten notion that students within a class may vary in strategy use patterns equalling several years' education time.

Third and finally, our analysis revealed that for Danish primary school children little variation (1-7%) appeared to exist between classes in general (main effect of class), and even less (0-1%) between classes and time (class-by-time interaction). Given that the five months' time span represents a big part of the students' first formal mathematics education, we find it interesting that the different classes in effect appeared to develop similarly between interview one and two despite considerable differences in teaching practice.

We want to stress that from this, one cannot conclude that differential teaching practices do not influence learning outcome, just that such differences were barely measurable over a time scale of five month's education time. A delayed effect of different instructional practices in year one on later arithmetic ability is thus possible. It is also important to notice that effects of differential teaching practices may have occurred before November, when interview one took place. Significant main effects of class for counting and especially derived fact indicate that this might have been the case.

In conclusion, Danish year one students' developmental change of strategy use for single-digit addition is primarily determined by their strategy use when entering year one. Development patterns were similar between classes, suggesting no influence of instructional practice, but mean change in error rate and use of direct retrieval differed for students with high or low initial use. The relatively small influence of five months teaching on strategy use patterns and the lack of difference between classes with different instructional practice is stunning. The findings call for qualitative research on what role different instructional practice play in year one students' development of strategy use.

Figure 1 :

 1 Figure 1: Proportional use (logit-scale: zero-values replaced by 0.01 before transformation) of strategies of 72 year one students in spring (April) plotted against their results achieved in the same assessment interview the preceding autumn (November)

Figure 2 :

 2 Figure 2: Variance components in terms of Type 3 sum-of-squares of strategy use assessed twice in year one (autumn and spring), derived from general linear modelsDiscussionThis statistical investigation of strategy use in single-digit addition amongst year one students revealed the following three results worth noting. First, the slopes of the regression lines of strategy


Xenia Vamvakoussi

Maria Bempeni
email: mbempeni@cc.uoi.gr

Stavroula Poulopoulou

Andioanna Tsiplaki
email: jtsiplaki@yahoo.com

Reflecting on a series of studies on conceptual and procedural knowledge of fractions: Theoretical, methodological and educational considerations

Keywords: Fractions, conceptual/procedural knowledge, individual differences, instruction

We present an overview of four studies investigating secondary students' conceptual and procedural knowledge of fractions qualitatively as well as quantitatively. We draw on these studies and their results to discuss the problem of measuring conceptual and procedural knowledge; the issue of individual differences in conceptual and procedural fraction knowledge; and related educational implications.

Conceptual and procedural knowledge in mathematics learning

The distinction between procedural and conceptual knowledge has elicited considerable research and discussion among researchers in the fields of cognitive-developmental psychology and mathematics education. Procedural knowledge is typically defined as the ability to execute action sequences to solve problems and is usually tied to specific problem types, whereas conceptual knowledge is defined as knowledge of concepts pertaining to a domain and related principles [START_REF] Rittle-Johnson | Developing conceptual and procedural knowledge of mathematics[END_REF]. This definition has been contested on the grounds that it depicts a very narrow picture for procedural knowledge as something that one either has, or does not have, ignoring different qualities that such knowledge may have [START_REF] Star | Reconceptualizing procedural knowledge[END_REF]. Still, this definition captures the distinction between carrying out a procedure that pertains to a domain, and understanding of the entities (e.g., mathematical objects and mathematical relations) that are involved in the procedure. This distinction is valuable, particularly from the point of view of mathematics education that has been tantalized by the phenomenon of procedural skill without understanding before acknowledging that conceptual and procedural knowledge are equally important for students ' mathematical development (e.g., De Corte, 2004).

The relation between the two types of knowledge has been difficult to establish empirically, in particular with respect to the order of acquisition (i.e., which type of knowledge develops first).The current prevailing model is the iterative model that assumes that there is a bi-directional relation between conceptual and procedural knowledge and accounts for many empirical findings, notably the well-documented finding that the two types of knowledge are typically highly correlated [START_REF] Rittle-Johnson | Developing conceptual and procedural knowledge of mathematics[END_REF]. However, such correlations found at group level do not accurately reflect what happens at the individual level. Indeed, research shows that there are individual differences in the way students combine the two kinds of knowledge putting a challenge to the iterative model (e.g., [START_REF] Canobi | Individual differences in children's addition and subtraction knowledge[END_REF][START_REF] Hallett | Individual differences in conceptual and procedural knowledge when learning fractions[END_REF]Hallett, Nunes, Bryant, & Thrope, 2012). In particular, Hallett andcolleagues (2010, 2012) were the first to systematically study such individual differences in the area of fraction learning. They assessed conceptual and procedural knowledge of students at Grade 4 and 5 (2010) as well as at Grade 6 and 8 (2012) and they identified groups of students who were either strong or weak in both types of knowledge. However, they also consistently traced two substantial groups of students for whom there was discrepancy between the two types of knowledge (i.e., students in one group exhibited stronger procedural fraction knowledge, compared to their conceptual knowledge, and vice versa for students in the other group). Hallett and colleagues found that such discrepancies became less salient with age.

Before more progress can be made in understanding the relations between conceptual and procedural knowledge, more attention should be paid to the validity of measures of conceptual and procedural knowledge [START_REF] Rittle-Johnson | Developing conceptual and procedural knowledge of mathematics[END_REF]. This is a challenging endeavor, particularly with respect to conceptual knowledge, which is considered a multi-dimensional construct; and it becomes even more challenging when it comes to the concept of fraction, itself a multifaceted construct pertaining to a vast variety of situations [START_REF] Moss | Pipes, tubs, and beakers: New approaches to teaching the rational-number system[END_REF].

In this paper we present an overview of four studies investigating secondary students' conceptual and procedural knowledge of fractions qualitatively as well as quantitatively. We reflect on the challenges we met and we draw on the findings to discuss the problem of measuring conceptual and procedural knowledge; the issue of individual differences in conceptual and procedural fraction knowledge; and related educational implications.

Conceptual and procedural fraction knowledge of Greek secondary students Study 1: General trends

In this study (Bempeni & Vamvakoussi, 2014) we looked at the development of conceptual knowledge of fractions in the first three years of secondary level (7 th to 9 th grade in Greece).

We administered an open-ended questionnaire to 80 seventh and ninth graders. The questionnaire consisted of 24 items targeting many aspects of fraction knowledge. The tasks included interpreting and constructing fraction representations using the area model and the number line; fraction magnitude estimation and estimation of outcomes of arithmetical operations; fraction comparison tasks that could be tackled with conceptual strategies (e.g., comparing 3/2 and 13/27, for which 1 can be used as benchmark); tasks targeting the inappropriate transfer of natural number knowledge to fractions (e.g., "how many numbers are there between 2/5 and 3/5?"); and contextualized problems where physical quantities were involved, requiring understanding of the unit of reference(see Task 1 in Figure 1 for an example of the latter, albeit in multiple-choice format). The difficulty of items varied, from very simple ones (e.g., representing a proper fraction using the area model) to quite challenging ones (i.e., items targeting the dense ordering of fractions). In addition, the questionnaire included 4procedural items on standard school-taught procedures, namely on fraction operations (e.g., "compute the sum 10/63+8/9"). The students were asked to solve the conceptual tasks and explain their solutions; and to carry out the fraction operations.

The results showed no significant difference between the two age groups' overall performance, and no significant difference in the overall performance for conceptual tasks, norfor procedural tasks. However, we found that older students relied significantly more on procedural strategies (i.e., school-taught procedures). Looking at each task separately, we found that ninth graders performed significantly better in 3 conceptual tasks, albeit relying on transformation strategies. For example, they converted fractions into decimals in order to place them on the number line; and they converted 3/2 and 13/27 into similar ones in order to compare them. Performance in procedural tasks was fairly good (at least 70% succeeded in each task), whereas performance in conceptual tasks varied widely (as expected). An interesting pattern emerged: Although the great majority of students were apt to carry out the four fraction operations, about 60% failed in tasks targeting conceptual understanding of the operations (e.g., in estimating the sum 12/13+7/8). In addition, although practically all students were able to construct an area model for a proper fraction, about 25% asserted that a shaded part of a shape that was partitioned in 3 unequal parts was "one-third"; and more than half of the students failed to represent an improper fraction, or explicitly stated that "it is not possible to take five parts out of three". Further, about half asserted that "eating 3/5 of a pie" necessarily means "eating 3 pieces of a pie" (see Task1, Figure 1). Finally, we also traced some students who were flawless in the procedural tasks, but failed in even the simplest conceptual tasks, indicating an asymmetry between the two types of knowledge. We focused on this issue in Study 2.

Study 2: Individual differences in conceptual and procedural knowledge of fractions

In Study 2 [START_REF] Bempeni | Individual differences in students' knowing and learning about fractions: Evidence from an in-depth qualitative study[END_REF] we recruited 7 ninth graders to participate in in-depth interviews. The selection of the participants was not random. First, based on their school grades, all participants could be characterized as medium to high level students in mathematics. Second, they shared the same mathematics tutor. Based on information provided by their tutor, we had reasons to expect some variation in their conceptual and procedural knowledge of fractions. We used an instrument with 30 fraction open-ended tasks, adjusting and extending the collection of conceptual tasks used in Study 1, and adding few procedural tasks (e.g., operations with mixed numbers). We added one more task focusing on the role of the unit of reference, in contexts where physical quantities were involved (see Task 2 in Table 1, albeit in multiple choice format). We also explicitly asked students not to apply typical school taught procedures in certain fraction comparison tasks that could be tackled with conceptual strategies (e.g., one proper and one improper fraction). The participants were asked to solve the tasks thinking aloud and explaining their answers.

Based on the analysis of their responses in terms of accuracy and strategy used (procedural/conceptual), the participants were distributed in three profiles. The Conceptual-Procedural profile consisted of three students who showed advanced conceptual knowledge of fractions, combined with procedural fluency. The Procedural profile consisted of three students who succeeded in practically all tasks that could be solved via the use of procedures taught at school (e.g., fraction and mixed numbers operations), but failed systematically in conceptual tasks (e.g., in representing an improper fraction; or in comparing dissimilar fractions when they were not allowed to apply the typical procedure). All three failed in both tasks targeting the role of the unit of reference. Finally, the Conceptual profile was represented by one student that failed in all tasks requiring procedural knowledge but managed to deal successfully with most of the conceptual tasks, including the ones targeting the role of the unit of reference.

Thus the findings of Study 2 are consistent with the findings of [START_REF] Hallett | Individual differences in conceptual and procedural knowledge when learning fractions[END_REF][START_REF] Hallett | Individual differences in conceptual and procedural fraction understanding: The role of abilities and school experience[END_REF], indicating that there are individual differences in the way students combine the two types of knowledge. Moreover, they illustrated the possibility that these differences can be extreme, even at grade 9. In following studies, we attempted to further investigate this issue, shifting from qualitative to quantitative methods.

Study 3: Developing and evaluating an instrument to measure for conceptual and procedural knowledge

Bearing in mind Rittle-Johnson and [START_REF] Rittle-Johnson | Developing conceptual and procedural knowledge of mathematics[END_REF] plea for valid measures of procedural and conceptual knowledge, we developed a new instrument measuring conceptual and procedural knowledge of fractions [START_REF] Bempeni | Individual differences in fractions' conceptual and procedural knowledge: what about older students?[END_REF]. In its initial form, the instrument consisted of 39 items, 12 procedural items and 27 conceptual items, a total of 39 items. The procedural ones were paper-and-pencil tasks requiring knowledge of procedures taught at school (e.g., to carry out fractions operations and operations with mixed numbers, to find an equivalent fraction, to cross-multiply, to simplify complex fractions, and to compare dissimilar fractions using the standard procedure).The conceptual tasks were based on extensive literature review (e.g., Baroody & Hume, 1991;[START_REF] Van Hoof | Inappropriately applying natural number properties in rational number tasks: Characterizing the development of the natural number bias through primary and secondary education[END_REF] and on our materials from Study 1 and Study 2 which were adjusted and enriched when necessary, based on our experience with testing these tasks with students. First, we opted for multiple-choice items, instead of open ones (see also [START_REF] Van Hoof | Inappropriately applying natural number properties in rational number tasks: Characterizing the development of the natural number bias through primary and secondary education[END_REF], with a view to discourage the use of procedural strategies in conceptual tasks (e.g., in comparison tasks), an issue that is highlighted in the literature (e.g., [START_REF] Rittle-Johnson | Developing conceptual and procedural knowledge of mathematics[END_REF] and was noticeable in Study 1. Second, we added more tasks on fraction representations that proved good indicators of conceptual knowledge (or the lack thereof) in our previous studies (see, for example, Task 4 in Figure 1).

We conducted a clinical pilot study with 61 students and asked 6 mathematics education experts to assist in the evaluation of the instrument. The instrument was assessed with respect to a) face validity and content validity, through the experts' feedback on clarity, accuracy, and relevance of the instrument; b) convergent and divergent validity, via multitrait analysis; c) internal consistency, calculating Cronbach's alpha; and d) external consistency with the test-retest method, calculating the intra-class correlation coefficient.

The instrument showed strong face validity given that tall items were assessed as clear and accurate by the experts, who were also highly consistent with each other in rating the relevance of each item to the aim of the instrument (Content Validity Index =1>.83). All items of the procedural scale showed convergent validity and divergent validity by demonstrating high correlation with the procedural scale and low correlation with the conceptual scale, respectively. However, eight items of the conceptual scale showed low correlation with the conceptual scale or higher than expected correlation with the procedural scale. Further, the value of intra-class correlation coefficient was high (above 0.8) for all procedural items, but below 0.5 for five conceptual items.

The problematic items were removed, resulting in an instrument consisting of 14 conceptual and 12 procedural tasks, with good indicators of validity, reliability, and objectivity, which we used in Study 4. However, removing conceptual items from the questionnaire is not an insignificant matter, and we will come back to this issue in the discussion. In Study 4 [START_REF] Bempeni | Individual differences in fractions' conceptual and procedural knowledge: what about older students?[END_REF] we administered the aforementioned instrument to 126 ninth graders and we analyzed the data using cluster analysis, testing for individual differences. Following Hallett andcolleagues (2010, 2012), we used in the analysis the residualized scores of the procedural and the conceptual scales, the raw scores being the percentages of correct answers out of the total of answered questions. This is because the two scales are expected to be correlated [START_REF] Rittle-Johnson | Developing conceptual and procedural knowledge of mathematics[END_REF], and this method provides a way to exclude the common part of variation from both scales [START_REF] Hallett | Individual differences in conceptual and procedural knowledge when learning fractions[END_REF][START_REF] Hallett | Individual differences in conceptual and procedural fraction understanding: The role of abilities and school experience[END_REF]. It should be noted that these scores do not represent absolute magnitudes of each type of knowledge; rather, they indicate discrepancies between the two types of knowledge. For example, a positive residual with respect to procedural knowledge means that a person's procedural knowledge is stronger than expected given their conceptual knowledge, and vice versa for negative residuals.

The cluster analysis used the k-means method and Euclidean distance as a distance measure and the optimal number of clusters was determined via statistical methods to be four. Two of these clusters comprised students who did not show great discrepancies between the two types of knowledge and their performance was either rather high in both type of tasks (N=22, 17.5%) or rather low in both types of tasks (N=31, 24.6%). On the contrary, the two remaining clusters comprised students who either performed better than expected in conceptual tasks given their performance in procedural tasks(Conceptual Profile, N=21, 16.7%); or performed better in procedural tasks given their performance in conceptual tasks(Procedural Profile, N=52, 41.3%).

These results supported the hypothesis that there are individual differences in the way that students combine the two type of knowledge and were consistent with the findings of our previous studies, and also with the results of [START_REF] Hallett | Individual differences in conceptual and procedural knowledge when learning fractions[END_REF][START_REF] Hallett | Individual differences in conceptual and procedural fraction understanding: The role of abilities and school experience[END_REF]. Moreover, our findings provided evidence that these differences may persist and remain salient for older students.

We note that within the Procedural Profile, we traced few extreme cases of students who excelled in the procedural items, but showed a severe lack of conceptual understanding, similar to students in Study 2. For example, one student achieved 100% score in the procedural tasks but only 14.29% in conceptual tasks, failing even in some of the simplest ones. However, we did not trace extreme cases within the Conceptual Profile.

Conclusions -Discussion

In a series of studies we examined secondary students' conceptual and procedural knowledge of fractions, qualitatively as well as quantitatively. The findings of these studies converge on the conclusion that there are individual differences in the way that students combine these types of knowledge (i.e., there are students who exhibit stronger procedural than conceptual knowledge, and vice versa); that these differences remain salient up to ninth grade; and that they may even be extreme (Study 2, Study 4). These findings put a challenge to the assumption that conceptual and procedural knowledge develop in a hand-over-hand manner [START_REF] Rittle-Johnson | Developing conceptual and procedural knowledge of mathematics[END_REF].

The findings are strengthened by the fact that we measured conceptual and procedural fraction knowledge reliably and validly via a new instrument (Study 4). However, there were several theoretical and methodological issues that we had to tackle in the process of developing the instrument, and some difficult decisions to make. First, measures of procedural knowledge were easy to construct and validate. However, this is due to the fact that we adopted a simple, perhaps over-simplified, definition of procedural knowledge [START_REF] Star | Reconceptualizing procedural knowledge[END_REF], as discussed in the introduction.

Second, measures of conceptual knowledge were particularly challenging. On the one hand, it is important to address various aspects of conceptual knowledge [START_REF] Rittle-Johnson | Developing conceptual and procedural knowledge of mathematics[END_REF], which requires the use of a great variety of tasks. On the other hand, this variety makes it difficult to construct a measure with good indicators of convergent validity, resulting to the exclusion of tasks (Study 3). Further, the use of multiple-choice items does not exclude the possibility that students in fact use procedural strategies when dealing with conceptual tasks, which was observable in Studies 1 and 2, but not in Studies 3 and 4. This might be an explanation for the fact that some conceptual tasks showed higher correlation with the procedural scale than expected (Study 3), and they also needed to be excluded. For example, locating fractions on the number line were among the problematic conceptual tasks in Study 4, presumably because the students, similarly to the students in Study 1, used transformation strategies to deal with the task. Excluding items may result in a more robust instrument; however, a lot of useful insights in students' understandings are lost. Consider, for example, the three first tasks in Figure 1. These tasks address students' understanding of the role of the unit of reference, in a context where it is necessary to take it into account (Baroody & Hume, 1991). All three had to be removed from the instrument, because the great majority of students, even the ones with overall good performance, failed. Thus, these tasks were not useful in discriminating between "conceptual" and "procedural" students but removing them meant ignoring an important aspect of conceptual knowledge that students appear to lack. This brings us to a consistent finding across all four studies: Students fail in elementary, yet fundamental conceptual tasks. Consider Task 1 (Figure 1) which was used in all studies. This task was challenging for half the students in Study 1; for three out of seven middle-to-high level students (based on their school grades) in Study 2; and for the great majority of students who participated in Study 3. This raises certain issues for instruction. First, Study 1 indicates that conceptual knowledge does not improve from seventh to ninth grade, despite the fact that during these years Greek students recapitulate content regarding fractions and are introduced to rational numbers. However, what seems to change is that students tend to rely more on procedural strategies. This may conceal their lack of conceptual fraction understanding is some cases (e.g., when placing fractions of the number line), but not in tasks that cannot be tackled with school-taught procedures (e.g., Tasks 1-3 in Figure 1). Second, it appears that students with very poor understanding of fractions still manage to get good grades at school (Study2). Third, the majority of the students in Study 4 was placed in the Procedural profile (i.e., they did better in the procedural tasks, than one would expect based on their performance in the conceptual tasks). These are quite strong indications that students' school experiences are more favorable to the development of procedural knowledge. In other words, instruction appears to still over-emphasize procedural knowledge, neglecting students' conceptual difficulties with fractions [START_REF] Moss | Developing children's understanding of the rational numbers: A new model and an experimental curriculum[END_REF].

This assumption cannot, however, explain the existence of (fewer) students who have stronger conceptual than procedural knowledge. In a more general fashion, the source of individual differences in conceptual and procedural knowledge is still an open question, despite the fact that several hypotheses have been formulated. Factors such as prior knowledge in the domain [START_REF] Schneider | Relations among conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge[END_REF], cognitive profile [START_REF] Gilmore | Can children construct inverse relations in arithmetic? Evidence for individual differences in the development of conceptual understanding and computational skill[END_REF][START_REF] Hallett | Individual differences in conceptual and procedural fraction understanding: The role of abilities and school experience[END_REF], general conceptual of procedural ability as well as school attendance at different schools have been tested [START_REF] Hallett | Individual differences in conceptual and procedural fraction understanding: The role of abilities and school experience[END_REF], with unsatisfactory results. Further theorizing and research is needed in this respect.

Figure 1 :

 1 Figure 1: Examples of conceptual tasks Study 4: Individual differences in conceptual and procedural knowledge of fractions: a quantitative study

Acknowledgement

References Baroody, A.J., & Hume, J. (1991). Meaningful mathematics instruction: The case of fractions.

Remedial & Special Education, 12(3), 54-68. Bempeni, M., & Vamvakoussi, X. (2014). Seventh and ninth graders' conceptual and procedural knowledge of fractions: what changes with school experience? Paper presented at the 9 th International Conference on Conceptual Change, Bologna, Italy.


Aikaterini Vissariou
email: vissario@eled.auth.gr

Despina Desli
email: ddesli@eled.auth.gr

Metacognition in non-routine problem solving process of year 6 children

Keywords: Metacognition, Non-routine problem, Problem solving, HISP inventory

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.

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' [START_REF] Anderson | The role of metacognition in second language teaching and learning[END_REF]. It is often presented as comprising three phenomena: metacognitive knowledge, metacognitive experiences and metacognitive skills [START_REF] Efklides | Measurement of metacognitive knowledge of self, task, and strategies in mathematics[END_REF]. 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 [START_REF] Desoete | Metacognitive prediction and evaluation skills and mathematical learning in third-grade students[END_REF]. Fostering metacognition, students avoid "blind calculations" or a superficial "number crunching" approach in mathematics [START_REF] Verschaffel | Realistic mathematical modelling and problem solving in the upper elementary school: Analysis and improvement[END_REF] 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 [START_REF] Dunning | Why people fail to recognize their own incompetence[END_REF]. 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 [START_REF] Zimmerman | Investigating self-regulation and motivation: Historical background, methodological developments, and future prospects[END_REF].

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 [START_REF] Gascoine | The assessment of metacognition in children aged 4-16 years: a systematic review[END_REF] 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, [START_REF] Desoete | Multi-method assessment of metacognitive skills in elementary school children: how you test is what you get[END_REF] Mathematical metacognition is often examined in terms of monitoring skills with respect to solving problems [START_REF] Desoete | Metacognitive prediction and evaluation skills and mathematical learning in third-grade students[END_REF][START_REF] Desoete | Multi-method assessment of metacognitive skills in elementary school children: how you test is what you get[END_REF][START_REF] Panaoura | Young pupils' metacognitive ability in mathematics[END_REF]. 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 [START_REF] Magno | Assessing grade school students' metacognition in solving mathematical problem[END_REF]. 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 [START_REF] Pappas Schattman | Fostering kindergartener's metacognition[END_REF].

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 [START_REF] Elia | Exploring strategy use and strategy flexibility in non-routine problem solving by primary school high achievers in mathematics[END_REF]. 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 [START_REF] Van Streun | Representations in applying functions[END_REF].

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 [START_REF] Schneider | Metacognition and mathematics education[END_REF], 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, [START_REF] Kolovou | Non-routine problem solving tasks in primary school mathematics textbooks -A needle in a haystack[END_REF].

The questionnaire was based on the 'How I Solve Problems' measurement tool of metacognition developed by [START_REF] Fortunato | Metacognition and problem solving[END_REF], after taking into consideration relevant instruments for measuring metacognition compiled from the literature (e.g., Jr MAI developed by [START_REF] Sperling | Measures of children's knowledge and regulation of cognition[END_REF]. The particular instrument was chosen as it is considered valid for research and useful for assessment and intervention in classrooms [START_REF] Gascoine | The assessment of metacognition in children aged 4-16 years: a systematic review[END_REF][START_REF] Sperling | Measures of children's knowledge and regulation of cognition[END_REF]. 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 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 ttest 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 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 Sections

Problem Solving Success Total

Successful Unsuccessful

A. 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' 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. 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., [START_REF] Wilson | Towards the modeling of mathematical metacognition[END_REF]; 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.

Overall Metacognition

Section

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 [START_REF] Desoete | Multi-method assessment of metacognitive skills in elementary school children: how you test is what you get[END_REF], 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 [START_REF] Magno | Assessing grade school students' metacognition in solving mathematical problem[END_REF]. 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.

Figure 1 :

 1 Figure 1: The problem used in the present study

  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.

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

 1

Table 2 : Correlations between overall metacognition and problem solving success

 2

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).


Christophe Finot
email: christophe.finot@u-bourgogne.fr

Optical pulse doublet resulting from the nonlinear splitting of a super-Gaussian pulse

Keywords: pulse shaping, nonlinear propagation, optical fibers

We experimentally demonstrate the generation of a temporal pulse doublet from the propagation of an initial super-Gaussian waveform in a nonlinear focusing medium. The picosecond structures are characterized both in amplitude and phase and their close-to-Gaussian Fourier-transform limited shape is found in excellent agreement with numerical simulations. This nonlinear fiber-based single-stage reshaping scheme is energy efficient, can sustain GHz repetition rates and temporal compression factors around 10 are demonstrated.

I. Introduction

Generation of picosecond optical pulse doublets at high repetition rates remains a technological challenge. Indeed, the optoelectronics bandwidth required to generate these double pulses is much higher than the one usually required to synthesize an isolated waveform. Therefore, all-optical approaches are fully relevant to overcome the electronics bottleneck and to provide optical doublets with a duration below 10 ps. In this context, several linear solutions have relied on photonic first-order derivation using long-period fiber gratings [START_REF] Slavík | Ultrafast all-optical differentiators[END_REF], fiber Bragg gratings [START_REF] Li | Arbitrary-order all-fiber temporal differentiator based on a fiber Bragg grating: design and experimental demonstration[END_REF] or on-chip microdisk resonators [START_REF] Yang | Experimental observation of optical differentiation and optical Hilbert transformation using a single SOI microdisk chip[END_REF]. Nonlinear approaches in optical fibers can also be advantageous thanks to the combination of the dispersive and Kerr effects. Indeed, numerous examples of doublets have been reported in fiber lasers [START_REF] Salin | Observation of High-Order Solitons Directly Produced by a Femtosecond Ring Laser[END_REF][START_REF] Grapinet | Vibrating soliton pairs in a mode-locked laser cavity[END_REF][START_REF] Lapre | Real-time characterization of spectral instabilities in a mode-locked fibre laser exhibiting soliton-similariton dynamics[END_REF]. However, their repetition rates are typically a few tens of MHz. Cavity-free solutions that do not require mode-locking processes also offer attractive solutions and several strategies have been explored such as chirp-induced splitting of pulses [START_REF] Helczynski | Chirp-induced splitting of pulses in optical fibers[END_REF], pulse copying thanks to an external triangular waveform [START_REF] Latkin | Doubling of optical signals using triangular pulses[END_REF][START_REF] Verscheure | Pulse doubling and wavelength conversion through triangular nonlinear pulse reshaping Electron[END_REF], parametric processes in dispersion oscillating fibers [START_REF] Finot | Influence of the pump shape on the modulation instability process induced in a dispersion-oscillating fiber[END_REF] or the compression of higher-order soliton pulse [START_REF] Mollenauer | Experimental observation of picosecond pulse narrowing and solitons in optical fibers[END_REF][START_REF] Stolen | Observation of pulse restoration at the soliton period in optical fibers[END_REF][START_REF] Agrawal | Nonlinear Fiber Optics[END_REF].

In this contribution, we introduce a new solution based on the nonlinear propagation of a super-Gaussian pulse in a fiber with anomalous dispersion. This problem, also known as the boxproblem or the dam-break problem, has recently stimulated a renewed interest in optics, mainly driven by the theoretical study of the nonlinear dynamics of the temporal [START_REF] Audo | Experimental observation of the emergence of Peregrine-like events in focusing dam break flows[END_REF][START_REF] El | Dam break problem for the focusing nonlinear Schrödinger equation and the generation of rogue waves[END_REF][START_REF] Kraych | Nonlinear Evolution of the Locally Induced Modulational Instability in Fiber Optics[END_REF] or spatial [START_REF] Wan | Diffraction from an edge in a selffocusing medium[END_REF][START_REF] Marcucci | Topological control of extreme waves[END_REF] coherent structures that emerge upon propagation. We show here that it can find practical applications. In the first part of this article, we discuss numerical simulations that demonstrate that a pair of Gaussian-like pulses emerge during the evolution of an initial flattop pulse in a singlemode waveguide with a focusing nonlinearity. We then present the experiment that validates our numerical expectations. A detailed study of the phase and intensity profiles, in the temporal and spectral domains, fully confirms our predictions and demonstrate the generation of close-to-Gaussian Fourier transform-limited doublets with a duration of 4.3 ps spaced by 15.2 ps at a repetition rate of 2 GHz.

II. Principle and numerical simulations

We first consider the evolution of a super-Gaussian pulse propagating in an optical fiber with a Kerr nonlinear coefficient  and a second order anomalous dispersion 2 (2 < 0). The longitudinal evolution of the temporal complex field envelope (t,z) in the slowly-varying envelope approximation is analytically described by the standard nonlinear Schrödinger equation (NLSE) [START_REF] Agrawal | Nonlinear Fiber Optics[END_REF]:

2 2 2 2 | | 0 2 i zt           , (1)
z being the propagation distance and t the temporal coordinate. The NLSE can be solved numerically using the split-step Fourier algorithm [START_REF] Agrawal | Nonlinear Fiber Optics[END_REF]. In order to illustrate our approach, we consider parameters typical of the standard single-mode SMF-28 fiber [19] involved in the proofof-principle experiment described in section III. The input pulse is a third-order super-Gaussian pulse   Intensities are normalized at each propagation distance by the peak intensity.

The longitudinal evolutions of the temporal intensity and phase profiles are provided in Fig. 1, panels a and b, and are compared with the dynamics of a Gaussian pulse having the same fwhm duration and peak power,

  0 2 '2 0 exp '( , 0) / P t t T    with   ' 0 1/2
2 ln 2 TT  . The evolutions of the temporal properties appear significantly different. On the one hand, the Gaussian pulse is first temporally narrowed and gets compressed to a fraction of its original width. It then splits into two substructures similarly to a higher-order soliton dynamics (the soliton number defined as N= ( P0 T0 2 / 2) 1/2 being here equal to 7.5) [START_REF] Stolen | Observation of pulse restoration at the soliton period in optical fibers[END_REF][START_REF] Agrawal | Nonlinear Fiber Optics[END_REF]. On the other hand, the super-Gaussian pulse first experiences the emergence of two in-phase ultrashort structures in its temporal wings that then collide at the pulse center. Such a scenario should be preferred to generate a double pulse as it reduces the required fiber length and avoids the instability that affects bell-shaped pulses at their point of maximum temporal focusing [START_REF] Agrawal | Nonlinear Fiber Optics[END_REF][START_REF] Dudley | Supercontinuum generation in photonic crystal fiber[END_REF]. Details of the temporal intensity profile obtained after 3.6 km of propagation are provided in Fig. 2(a). For this distance, the super-Gaussian waveform has reshaped into a pair of well-separated and identical pulses that can be adjusted by a

Gaussian profile having a fwhm duration of 2.9 ps and a temporal spacing of 15.3 ps. This stresses the significant temporal compression occuring during the nonlinear process (here a 14.5 fold compression). The energy remaining in the residual central lobe is minimal. The spectral bandwidth of the doublet (Fig. 2(b)) is significantly larger than the input pulse and is maximum after 3.6 km (see Fig. 1(c2)). The optical spectrum exhibits a modulation typical of the interference of two in-phase pulses. It can be qualitatively adjusted by a modulated Gaussian waveform with a fwhm width of 150 GHz. This leads to a time-bandwidth product of 0.44, suggesting that the Gaussian-like structures under study are rather close to the Fourier transform limit. This is in agreement with the temporal phase profile that is flat over the central part of the pulses (see Fig.

1(b1))

. Note that, in the present letter, our goal is not to provide a theoretical framework of the nonlinear dynamics that is observed and we are only motivated in the qualitative trends linked to the application we demonstrate next section. Therefore, we do not explain the details of the dynamics of the box-problem and we do explicitly interpret the present results in terms of finite background solutions of the NLSE such as second-order Peregrine solitons, soliton gas or dispersive wave structures [START_REF] Marcucci | Topological control of extreme waves[END_REF]. We do not carry spectral analysis in terms of nonlinear Fourier transform [START_REF] Randoux | Inverse scattering transform analysis of rogue waves using local periodization procedure[END_REF]. Let us, however, mention that the duration of the initial pulse has to be carefully chosen and that longer durations may lead to a different and more complex picture as we have experimentally described in [START_REF] Audo | Experimental observation of the emergence of Peregrine-like events in focusing dam break flows[END_REF] carried in a rather similar experimental configuration. Convincing explanations of the influence of the initial width can be proposed in terms of topological control [START_REF] Marcucci | Topological control of extreme waves[END_REF].

III. Experimental setup

In order to validate our numerical study, we have implemented an experiment based on devices that are commercially available and typical of the telecommunication industry. The experimental setup, close to the one we previously developed for the demonstration of the temporal Arago spot [START_REF] Finot | Arago spot formation in the time domain[END_REF], is sketched in Fig. 3. A laser at 1550 nm emits a highly-coherent continuous wave. The initial temporal profile is obtained thanks to a Lithium Niobate intensity modulator operated at its point of null transmission and driven by an electrical pulse generator that delivers super-Gaussian pulses at a repetition rate of 2 GHz. The resulting signal is amplified using an erbium-doped fiber amplifier delivering up to 23 dBm of average power. An optical attenuator is used to control the power sent into the fiber. Particular care has been devoted so that the resulting intensity profile presents a highly symmetric profile, with a strong platitude of the high level. The corresponding optical intensity profile is plotted in Fig. 3(b) (blue solid line) and approximates the sharp edges of the ideal box waveform (red solid line). The experimental intensity profile is well fitted by a third-order super-Gaussian pulse with a fwhm duration of 42 ps. The initial optical spectrum recorded on a high-resolution optical spectrum analyzer (Fig. 3(c)) confirms the high level of symmetry of the pulse which is close to the Fourier-transform limit and the high degree of coherence of the signal.

The nonlinear propagation experiment takes place in a set of four identical 900m long optical single-mode fibers with nonlinear and dispersion parameters provided in the previous section. The high level of dispersion enables us to neglect the impact of third-order dispersion. In order to suppress the deleterious stimulated Brillouin backscattering, we have inserted three optical isolators [START_REF] Takushima | Suppression of stimulated Brillouin scattering using optical isolators[END_REF]. Compared to the widely used phase modulation approach, this mitigation scheme presents the advantage of not impairing the phase of the signal under study, which is crucial for our detection stage. Indeed, the characterization of the output pulse properties is ensured by a complex optical spectrum analyzer that enables us to get access both to the spectral intensity and phase profiles. It becomes therefore possible to measure the temporal phase and intensity profiles with a temporal resolution below 1 ps.

IV. Experimental results

Instead of carrying a study in terms of propagation distance and in order to avoid cuttingback the fiber arrangement, we have measured the nonlinear dynamics as a function of the average input power. The results recorded on the optical spectrum analyzer are summarized in Fig. 4 and compared with numerical simulations of the NLSE that also take into account the distributed losses of the fiber (0.2 dB/km) as well the lumped losses induced by the isolator dB per isolator).

The evolution of the spectrum exhibits a behavior that clearly differs from the propagation in a normally dispersive fiber where the wave-breaking evolution leads to a broadened spectrum that is not affected by any modulation and has a spectral extent that saturates [START_REF] Parriaux | Spectral broadening of picosecond pulses forming dispersive shock waves in optical fibers[END_REF]. The numerical predictions of the spectral profiles are found in excellent agreement with the experimental results. A more detailed view of the temporal profiles obtained for an average power of 80 mW (corresponding to an input peak power of 1 W and a pulse energy of 40 pJ) is proposed in Fig. 6.

For this power, two identical picosecond pulses are generated and spaced by 15.2 ps. The experimental results are in remarkable agreement with numerical simulations. They confirm that the temporal phase profile over the two pulses is very flat and that the two pulsed parts are inphase. This contrasts with other methods such as optical derivation where the leading and trailing pulses are -phase shifted [START_REF] Slavík | Ultrafast all-optical differentiators[END_REF]. Compared to the loss-free case described in section II, the presence of losses leads to a broadening of the output pulse structures that have a fwhm duration of 4.3 ps (i.e a compression factor of 9.8 with respect to the input waveform). A beneficial impact can however be observed with the disappearance of the small bump that was predicted at the center in the ideal case. Finally, the temporal intensity profiles recorded for three different input powers (50, 80 and 125 mW of average power, respectively) are compared in Fig. 7. The splitting of the pulse is already apparent for pulse energies of 25 pJ. In the three cases, the intensity profiles can be adjusted with a good accuracy by the temporal superposition of two or three Gaussian structures. The delay between the two main structures is not heavily affected by the input power level.

V. Conclusions

In conclusion, we have numerically and experimentally demonstrated the generation of a doublet of well-separated pulses resulting from the nonlinear propagation of a super-Gaussian pulse in a focusing medium. The pulses resulting from this single-stage nonlinear reshaping process are significantly shorter than the input waveform. Contrary to optical differentiation, the proposed approach is energy efficient and the losses are limited to the propagation losses. The typical powers that are required (around 1 W of input peak power) are moderate and can be further lowered using a fiber with higher nonlinearity. The approach can fully sustain GHz repetition rates.

We have focused this letter on the qualitative aspects by highlighting that the structures are in-phase and can be well described at their point of maximum focusing by a Gaussian-like profile.

However, a better understanding of the nonlinear dynamics will stimulate further theoretical research with more advanced tools. Using topological perspective will enable to better determine the theoretical guidelines to apply and the balance between the duration, peak power, dispersion/diffraction and nonlinearity that has to be targeted [START_REF] Audo | Experimental observation of the emergence of Peregrine-like events in focusing dam break flows[END_REF][START_REF] Wan | Diffraction from an edge in a selffocusing medium[END_REF].

We carried this proof-of-principle demonstration using a 3.6 km long SMF-28 fiber, but nonlinear compact optical waveguides with anomalous dispersion [START_REF] Choi | Soliton-effect optical pulse compression in CMOS-compatible ultra-silicon-rich nitride waveguides[END_REF] are also suitable, paving the way for chip-scale solutions of ultrashort pulse generation.

  and an input peak power P0 of 0.9 W. They propagate in a fiber with a dispersion 2 of -20 ps 2 /km and a nonlinearity  of 1.1 /W/m.

Figure 1 :

 1 Figure 1 : Comparison of the longitudinal evolutions of a super-Gaussian pulse and a Gaussian waveform (panels 1 and 2 respectively). The temporal intensity and phase profiles are provided in panels a and b respectively whereas the spectral intensity profiles are given in panels c. The horizontal white dotted line represents the distance of 3.6 km.

Figure 2 :

 2 Figure 2 : Temporal and spectral intensity profiles (panels a and b, respectively) of the waveform after propagation in 3.6 km of fiber. Numerical simulations (solid black lines) are compared with a fit assuming a Gaussian shape of the two pulses (blue full circles). The input intensity profile is plotted with a dashed grey line.

Figure 3 :

 3 Figure 3 : (a) Experimental setup. CW : continuous wave, IM : intensity modulator, EPG : Electrical Pattern Generator, EDFA : Erbium Doped Fiber Amplifier, OVA : Optical Variable Attenuator, ISO : optical isolator, OSA : Optical Spectrum Analyzer. (b) Temporal intensity profile obtained after modulation. The experimental results recorded with the complex spectrum analyzer (blue circles) are compared with a fit by a third-order super-Gaussian waveform (blue solid line). The red line represents an ideal rectangular function. (c) Optical spectrum of the initial signal.

Figure 4 :

 4 Figure 4 : Evolution of the optical spectrum according to the average initial power. Numerical simulations (a) are compared with experimental results (b). For clarity, the frequency comb structure of the spectrum has been removed.

Figure 5 :

 5 Figure 5 : Evolution of the output temporal intensity and phase properties (panels a and b, respectively) of the waveform as a function of the initial average power for a fixed propagation length of 3.6 km. Numerical simulations (panels 1) are compared with the experimental results (panels 2). Intensity profiles are normalized by the peak-power.

Figure 6 :

 6 Figure 6 : Details of the temporal intensity and phase profiles (subplots a and b, respectively) obtained for an input average power of 80 mW. Experimental results (solid black lines) are compared with numerical simulations (blue full circles). The input intensity profile is plotted with a dashed grey line.

Figure 7 :

 7 Figure 7 : Details of the temporal intensity profiles obtained for an input average power of 50, 80 and 125 mW (subplots a, b and c, respectively). Experimental results (solid black lines) are compared with Gaussian fits of the substructures (blue full circles). The input intensity profile is plotted with a dashed grey line.

Acknowledgements

We acknowledge the support of the Institut Universitaire de France (IUF), the Bourgogne-Franche Comté Region. CF thanks its colleagues, Frederic Audo, Bertrand Kibler, Alexandre Parriaux, Julien Fatome for fruitful discussions and technical assistance.