Camila Peres Nogues
email: camilapnogues@gmail.com

Beatriz Vargas Dorneles
email: beatriz.dorneles@ufrgs.br

Number line estimation and quantitative reasoning: two important skills for mathematical achievement 1

Keywords: Number estimation, quantitative reasoning, mathematical learning

Number estimation and quantitative reasoning have been indicated as significant skills for mathematical achievement. In this sense, this paper aims to analyse the students' performance in these two skills and verify possible relations between both. For that, 143 students from 3 rd and 4 th grades of elementary school were evaluated using two number line estimation tasks. Results showed that there is a direct relation between children's performance in both abilities. From this, it is suggested that both number estimation and quantitative reasoning tasks can be used in daily classroom activities to promote the learning of initial mathematics concepts considering the importance of estimation in daily activities.

Introduction

Mathematics is an important area of knowledge for children's cognitive development, as it involves several domain-specific skills related to numerical knowledge. Learning mathematics requires the understanding of initial concepts about numbers, quantities and relations, and the ability to make connections between these concepts [START_REF] Nunes | Development of maths capabilities and confidence in primary school[END_REF]. Therefore, it is important to provide different teaching and learning situations with the intention of promoting children's mathematical development. In this sense, our study highlights two skills indicated in the literature as significant for mathematical learning: number estimation and quantitative reasoning. Number estimation is a cognitive ability that has been studied and pointed out as important for mathematical performance [START_REF] Geary | Cognitive predictors of achievement growth in mathematics: A 5-year longitudinal study[END_REF][START_REF] Moore | Children's mathematical performance: Five cognitive tasks across five grades[END_REF][START_REF] Schneider | Associations of Number Line Estimation With Mathematical Competence: A Metaanalysis[END_REF][START_REF] Siegler | Development of numerical estimation in young children[END_REF][START_REF] Siegler | The logarithmic-to-linear shift: One learning sequence, many tasks, many time scales[END_REF]. The evidence is that the ability to estimate is related to arithmetic, number categorization and numerical magnitudes [START_REF] Laski | Is 27 a big number? correlational and causal connections among numerical categorization, number line estimation, and numerical magnitude comparison[END_REF][START_REF] Link | On the relation between the mental number line and arithmetic competencies[END_REF][START_REF] Schneider | Associations of Number Line Estimation With Mathematical Competence: A Metaanalysis[END_REF][START_REF] Siegler | The logarithmic-to-linear shift: One learning sequence, many tasks, many time scales[END_REF]. In number estimation tasks, students identify numerical magnitudes and perform number decompositions. More than that, studies have shown that children's ability to accurately locate numbers on a number line is predictive of later mathematical achievement [START_REF] Geary | Cognitive predictors of achievement growth in mathematics: A 5-year longitudinal study[END_REF][START_REF] Siegler | Development of numerical estimation in young children[END_REF]). The number line estimation task, in the version number-to-position (NP), is the task most commonly used to evaluate the ability to estimate numerical magnitudes. In this task, participants are given a number, in the form of Arabic numerals, and are asked to locate that number on a line usually bounded in a numerical range. This task has been used to trace the development of the pupil's comprehension of numerical magnitude throughout the school grades [START_REF] Booth | Developmental and individual differences in pure numerical estimation[END_REF][START_REF] Schneider | Associations of Number Line Estimation With Mathematical Competence: A Metaanalysis[END_REF][START_REF] Siegler | The development of numerical estimation: Evidence for multiple representations of numerical quantity[END_REF].

An alternative version of the number line estimation task has also been proposed. The intention of this version is to investigate estimation accuracy differences between tasks and to gain a broader understanding of the development of the ability to estimate. In this version, called position-tonumber (PN), participants are asked to estimate a number corresponding to a location on a number line also bounded by a numerical range. Figure 1 shows examples of NP and PN tasks. These two versions of the number line estimation task are complementary since they embody the central property of the ability to estimate, which is the translation between numerical and spatial representations [START_REF] Siegler | The development of numerical estimation: Evidence for multiple representations of numerical quantity[END_REF].

Regarding quantitative reasoning, we can say that it is essential for the construction of initial mathematical concepts, since it is based on the understanding of the relations between quantities, for example, if they are increasing or decreasing [START_REF] Nunes | Teaching and Learning About Whole Numbers in Primary School[END_REF]. Evidence indicates that quantitative reasoning developed from the beginning of elementary school contributes significantly to mathematical achievement and that understanding the relation between quantities is essential for learning numerical and arithmetic representations [START_REF] Nunes | The contribution of logical reasoning to the learning of mathematics in primary school[END_REF][START_REF] Nunes | The relative importance of two different mathematical abilities to mathematical achievement[END_REF]. Quantitative reasoning is divided into two categories, additive and multiplicative reasoning, which are classified by situations according to the relation established between quantities [START_REF] Nunes | Teaching and Learning About Whole Numbers in Primary School[END_REF]. Additive reasoning is characterized by the ability to join or separate quantities from a total quantity (quantity composition problems), to recognize the inverse relation between addition and subtraction (transformation problems), and to understand comparative situations (comparison problems). Multiplicative reasoning involves the ability to perform one-to-many correspondence and to perform distributions equally. It is classified in situations of direct or inverse relations between quantities and product of measures, in which a third quantity is formed by two other quantities [START_REF] Nunes | Teaching and Learning About Whole Numbers in Primary School[END_REF]. Examples or these quantitative reasoning tasks that were used in this study are presented in Figure 2. Thus, we perceive that different quantitative relations are necessary in order to understand mathematical concepts and operations. When students understand the relation between quantities and operations, it becomes easier to correctly decide the strategies and schemes to be used to solve the problems. Consequently, this understanding will have a direct influence on mathematical achievement.

Concluding this review, some studies on this topic have been presented in a previous edition of CERME. About number estimation, we found a comparison study between two estimation tasks in 2 nd and 3 rd graders´ performance [START_REF] Dorneles | Number estimation in children[END_REF], and another comparing preschool children's performance in a number line estimation task with initial math skills [START_REF] Elofsson | 5-and 6-Year-Old Children´s Number Line Estimation Patterns[END_REF]. On quantitative reasoning, we highlight a study carried out with 2 nd graders checking their solution strategies when solving transformation situations to support the development of their mathematical thinking [START_REF] Serrazina | The role of representations in promoting the quantitative reasoning[END_REF].

In this sense, what was intended in this study was to analyse the performance of 3 rd and 4 th grade students on number estimation and quantitative reasoning and to investigate the relation between their performance in these abilities. This relation, as far as we know, has not yet been the focus of studies. Therefore, this research fills a gap in the literature and provides more evidence of cognitive abilities important to mathematical performance.

Method

The sample comprised a total of 143 children (62 girls and 81 boys) from two public elementary schools in Porto Alegre/Brazil. Schools were chosen by criteria of convenience, number of students and attending to communities of similar socioeconomic classes. The students were 3 rd and 4 th graders (76 from 3 rd and 67 from 4 th grade), and ages between 8 to 11 years old (M=9.8, SD=.74). They were evaluated using two number line estimation tasks (NP and PN) and one quantitative reasoning task. The 3 rd and 4 th school grades were chosen because in these grades the students are in similar moments of learning considering the curricular contents, especially in relation to numerical knowledge. And the tasks used require some numeric knowledge established to be solved, like the understanding of the numerical system and of the four mathematical operations. It is important to say that estimation and problem solving are part of the curriculum in Brazil, but these tasks specifically of number estimation and quantitative reasoning are not usual in classrooms. Even though, the children were able to solve the tasks after the researcher's explanation.

To evaluate number estimation ability, two number line tasks were used adapted from [START_REF] Siegler | The development of numerical estimation: Evidence for multiple representations of numerical quantity[END_REF]. The first one was number-to-position (NP) task, which consists in locating the position of a number on a number line bounded by 0 on the left and 100 on the right. The second one was position-to-number (PN) task, which consists of estimating a value that corresponds to a mark indicated on a number line also bounded by 0 and 100. There were 22 numbers to be estimated (2,3,5,8,12,17,21,26,34,39,42,46,54,58,61,67,73,78,82,89,92,97), taken from the study by [START_REF] Laski | Is 27 a big number? correlational and causal connections among numerical categorization, number line estimation, and numerical magnitude comparison[END_REF]. These numbers were presented in a random order and children wrote down their estimates in a notebook containing a number line and a number to be estimated on each page. The task was applied in groups of 10 students and lasted about 30 minutes per group.

The quantitative reasoning task was adapted from [START_REF] Nunes | Teacher notes[END_REF] and aimed to evaluate children's arithmetic reasoning in problem solving. The quantitative relations involved were additive reasoning (including composition of quantities, situations of transformation and comparison) and multiplicative reasoning (containing situations of direct and inverse relations between quantities and product of measures). The task was composed of 18 situations, nine of each kind of reasoning. Task application occurred collectively in groups of 10 students. Each participant received a notebook containing one situation per page, in which only illustrations were presented without written information. The instructions were given orally by the assessor, so the children did not depend on their reading ability.

The analysis of performance in number estimation was determined by calculating the accuracy with which the students estimated each number requested, that is, by calculating the percentage of absolute error of each child´s estimate. This computation is adapted from [START_REF] Siegler | Development of numerical estimation in young children[END_REF] and can be explained by dividing the difference between the child's estimate and the number to be estimated by the number line scale (which in this task is 100). For example, consider that the number to be estimated was 40 and the child made a mark on the number line corresponding to number 30, then the computation will be done as follows: |30-40|/100, obtaining an accuracy of 0.1. Therefore, the lower is the value found, the more accurate is the estimation made by the participant.

For the quantitative reasoning task, the total score of correct answers in the task and the hits per reasoning category (additive and multiplicative reasoning) were considered.

Initially, the quantitative analyses were conducted to identify any differences in performance on number estimation and quantitative reasoning tasks according to gender, age and school grade. Then, an association analysis between the performance in the two types of number line estimation tasks was conducted. After which, the children's performance in quantitative reasoning and number line estimation tasks was compared.

Results

Firstly, the results are presented by task type, describing the students' performance in number line estimation tasks and in the quantitative reasoning task. Finally, the results for the correlation between both math abilities evaluated are shown.

The ability to estimate was evaluated using two number line estimation tasks (NP and PN tasks). The performance in both tasks was organized according to the mean accuracy of each child by task.

In a first analysis, the performance in each of the two number line estimation tasks was related with gender, age and school grade, and a statistically significant difference was only found (NP: =1827, <.05; PN: =1767.5, <.05) with school grade. This indicates that 4 th graders accuracy (mean NP=.086, SD=.038; mean PN=.074, SD=.037) was better than 3 rd graders (mean NP=.11, SD=.055, mean PN=.099, SD=.056) in both tasks. After this, from a correlation analysis, accuracy of NP (M=.10; SD=.05) and PN (M=.09, SD=.06) showed a moderate but significant correlation (rs=.66, <.01), indicating that, the better is the accuracy in NP task, the better it is in PN task. A descriptive analysis was also conducted to identify which proposed numbers the participants estimated accurately. From this, we can observe (Figure 3) that children were more accurate in numbers at the ends of the number line (numbers 2, 3, 5 and 97).

Quantitative reasoning

The quantitative reasoning task assessed the children's ability to solve situations involving arithmetic reasoning using the four fundamental mathematical operations. The results were computed by the total of correct answers in the task and by the number of correct answers in each reasoning category (additive reasoning and multiplicative reasoning). In general, students presented a mean of 10.55 hits (SD=3.22). When scores were verified according to the reasoning category, the students obtained better scores in additive reasoning (M=6.03, SD=1.92) than in multiplicative reasoning (M=4.51, SD=1.81). When correlating performance in the quantitative reasoning task with the variables gender, age and school grade, there was a statistically significant difference with school grade ( =3256, <.05) and a direct but weak correlation with age (rs=.18, <.05). These results indicate that there was an increase in total score from 3 rd to 4 th grades (3 rd grade: M=9.74, SD=3.32; 4 th grade: M=11.46, SD=2.87). However, this increase was significant ( =3486.5, <.05) only for additive reasoning (3 rd grade: M=5.42, SD=2.06; 4 th grade: M=6.73, SD=1.47).

When analysing the situations involved in both additive and multiplicative reasoning (Figure 4), there is a higher percentage of correct answers in situations of additive reasoning involving quantities composition and transformation. Concerning situations of multiplicative reasoning, students obtained higher scores in situations involving direct and inverse relations between quantities.

Number line estimation and quantitative reasoning

A correlation analysis was conducted to verify associations between the measures of number line estimation and quantitative reasoning. Thereby, moderate and significant correlations were found between quantitative reasoning total score and both the NP task (rs=-.39, <.01) and PN task (rs=--.42, <.01). In the same way, there were also moderate and significant associations between score by reasoning category and accuracy in both number line estimation tasks. Correlations index can be seen in Table 1.

Discussion and conclusions

The aim of this research was to verify the relation between number estimation and quantitative reasoning performance of a third and fourth graders group, as well as to analyse their performance in each of these abilities. For this, two number line estimation tasks (NP and PN tasks) and a quantitative reasoning task were used.

Among the research results, we can highlight the significant difference found between the school grades, with 4 th graders performing significantly better than 3 rd graders in both abilities. These findings indicate that experience and increased schooling improve accuracy in number line tasks, as found in previous studies [START_REF] Booth | Developmental and individual differences in pure numerical estimation[END_REF][START_REF] Laski | Is 27 a big number? correlational and causal connections among numerical categorization, number line estimation, and numerical magnitude comparison[END_REF][START_REF] Siegler | The development of numerical estimation: Evidence for multiple representations of numerical quantity[END_REF]. This is a consequence of children becoming more aware of numbers and their relations, learning that larger numbers are located to the right on a number line, and that the distance between 10 and 20 is the same as that between 80 and 90, for example [START_REF] Friso-Van Den Bos | Longitudinal development of number line estimation and mathematics performance in primary school children[END_REF][START_REF] Laski | Is 27 a big number? correlational and causal connections among numerical categorization, number line estimation, and numerical magnitude comparison[END_REF].

An association was also found between children's performance in both number line estimation tasks, that is, participants who performed well in the NP task also performed well in the PN task. When analysing the numbers in which children obtained better performance, it was found that in the NP and PN tasks the numbers with the best precision (2, 3, 5 and 97) are close to the extremities of the number line. The worst performances were found in different numbers for each task. Nevertheless, in both cases, the majority of the numbers are located between 25 and 50 and between 50 and 75. This result is similar to previous studies that indicate that children up to the 4 th grade are not very accurate in numbers far from the extremities of the number line, [START_REF] Laski | Is 27 a big number? correlational and causal connections among numerical categorization, number line estimation, and numerical magnitude comparison[END_REF][START_REF] Siegler | The development of numerical estimation: Evidence for multiple representations of numerical quantity[END_REF].

In quantitative reasoning task, 4 th graders scored better than 3 rd graders as expected. When analysing the reasoning categories separately, 4 th graders presented better scores in both additive and multiplicative reasoning, but this difference was only significant for additive reasoning. This indicates that due to the similarity of contents proposed in the Brazilian school curriculum in these grades, students are learning multiplicative reasoning, that is, the relations between multiplication and division operations, which may explain the fact that the difference between 3 rd and 4 th graders was not significant for these types of reasoning.

A direct association between number line estimation and quantitative reasoning was found. Students' performance in the quantitative reasoning task showed a correlation with both number line estimation tasks, specifically additive reasoning, which had the highest correlation index in both NP and PN tasks.

However, one limitation of the research must be considered: the collective application of the tasks proposed. Nevertheless, the findings of this study contribute to the planning of tasks that may improve abilities that are not currently approached in classrooms, such as number estimation. As previously mentioned, the development of the ability to estimate can improve students' skills in arithmetic computations. There are several daily situations when an estimation is enough, without the need to get an exact answer, for example, to choose the smallest cashier row in a supermarket, or if it is necessary to run to get to school on time. Therefore, in mathematics teaching it is important to highlight the use of alternative strategies to solve problems, allowing students to increase their set of solution methods. In this way, the use of number estimation in mathematics classes is an option to develop ability in arithmetic computations and may help to explain and to check the results of computations, while allowing students to establish their own solution strategies and appropriate different ways to solve mathematical tasks.

These results add new elements to the discussion of the role of number estimation in math achievement and provide evidence regarding estimation ability. However, future research on the subject is needed, such as longitudinal studies that describe the development of the ability to estimate and studies that verify the strategies used by students in each school grade.

Figure 1 :

 1 Figure 1: Examples of the number estimation tasks NP and PN

Figure 2 :

 2 Figure 2: Examples of quantitative reasoning problems

Figure 3 :

 3 Figure 3: Accuracy mean on NP and PN tasks

Figure 4 :

 4 Figure 4: Scores Organized by Category of Problem Types

Table 1 : Correlation between number estimation and quantitative reasoning

 1

		Number Estimation	Quantitative Reasoning
	Measures	1	2	3	4	5
	1. NP	1
	2. PN	0,660**	1
	3. AR	-0,378**	-0,394**	1
	4. MR	-0,292**	-0,328**	0,502**	1
	5. QR -Total	-0,389**	-0,420**	0,857**	0,864**	1
	*p<0,05; **p<0,01

1 This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -Brasil (CAPES) -Finance Code 001


Ioannis Papadopoulos

Styliani Panagiotopoulou
email: stella.panagiotopoulou@gmail.com

Michail Karakostas

Mental calculations with rational numbers across educational levels

Keywords: Mental calculations, rational numbers, written algorithm

In this study primary, secondary and tertiary education students try to calculate mentally the very same tasks involving operations with rational numbers (mainly fractions and mixed numbers). The aim was to record the range of strategies used by the students and at the same time to examine how these strategies are distributed across these three educational levels. The findings give evidence that no matter the educational level, the dominant strategy is the use of mental form of the written algorithm. All the other strategies are underrepresented. Moreover, it seems that the use of rational numbers in mental calculation is in itself an obstacle for the students since a great percentage of them either did not give any answer or the answers they gave were incorrect or non-codable.

Introduction

Mental calculations in arithmetic refers to the process of calculating the exact result of an arithmetical expression given that the solvers use neither a calculating device nor any kind of writing and which usually demands an application of mental strategies. In 1924, the well-known mathematician Carathéodory, during a lecture he delivered for the Hellenic Mathematical Society highlighted the vital importance of the students' ability "to calculate mentally instead of merely writing down, for example, simple subtractions or multiplications" (p. 87). The main interest of the majority of the relevant research studies lies particularly in identifying and organizing the strategies the solvers use. A considerable body of these studies examines mental calculation in the realm of whole numbers and their four basic operations [START_REF] Rezat | Mental calculation strategies for addition and subtraction in the set of rational numbers[END_REF]. Lately, researchers express a growth interest on the mental calculations in the set of rational numbers concerning either comparison of rational numbers [START_REF] Yang | Number sense strategies used by pre-service teachers in Taiwan[END_REF] or operations with rational numbers [START_REF] Caney | Mental computation strategies for part-whole Numbers[END_REF].

In this study our focus lies on the strategies used by students across all the educational levels when they mentally make calculations that involve rational numbers. More precisely, our research questions are:

 What is the range of the strategies used by the solvers while they mentally make operations that involve rational numbers?  How are these strategies distributed across the three different educational levels and what similarities or/and differences are identified?

Research literature

Cathy [START_REF] Seeley | Do the Math in Your Head!" President's Message[END_REF], NCTM president for 2004-2006, in her "Do the Math in Your Head!" article is making a call to the educational community to realize the significance of the ability to perform mental calculations. She associates mental math with both the students' ability to perform calculations quickly, and their conceptual understanding and problem solving. Furthermore, mental calculations constitute a useful means for calculations, given that they are used by almost everyone on a daily basis, as a direct and quick way of calculating. According to relevant studies [START_REF] Mcintosh | Mental computation of school-aged students: Assessment, performance levels and common errors[END_REF] over three quarters of the mathematical operations performed by individuals in their daily lives are based on mental calculations. Moreover, they promote creative and independent thinking, develop sound number sense, constitute the basis for developing estimation skills, and finally they facilitate a natural progression through informal written methods to standard algorithms [START_REF] Thompson | Issues in Teaching Numeracy in Primary Schools[END_REF].

As it has already been mentioned, the majority of the research studies focus on mental calculations involving whole numbers, and a wide variety of strategies associated to the four operations has been recorded. However, the corresponding body of research concerning mental calculations with rational numbers is rather limited. A systematic documentation of the strategies used by students in primary and secondary education in operations with fractions, decimals and percentages have been realized by [START_REF] Caney | Mental computation strategies for part-whole Numbers[END_REF]. One main finding of their study was that many of the strategies used in operations with whole numbers are also used for operations with rational numbers. Furthermore, they identified two kinds of strategies for mental calculations with rational numbers: instrumental and conceptual strategies. The instrumental work refers to the use of procedural strategies learned by rote. Moreover, this work is not accompanied by an explanation displaying conceptual understanding of the involved processes. On the contrary, conceptual work takes place when students make use of their knowledge on the specific set of numbers and operations to calculate mentally. Finally, [START_REF] Caney | Mental computation strategies for part-whole Numbers[END_REF] added the 'mixed' strategy in case the students combine a rote procedure with a conceptual explanation. In their study they identified a series of strategies the students used to solve 12 problems. More specifically, the students 'changed operation' (division to multiplication, subtraction to addition), 'changed representation' (fractions to decimals and vice versa, percent to fractions), 'used equivalents' (use of equivalent fractions), 'used known facts' (times table ), 'repeated addition/multiplication/division' (by doubling, halving), 'used bridging' (bridged to one/whole), 'worked with parts of a second number' (split by place value or by parts), 'worked from the left/right', 'used mental picture', 'used mental form of written algorithm' and 'used memorized rules'. [START_REF] Carvalho | Students strategies and errors in mental computation with rational numbers in open number sentences[END_REF] examined grade-6 students' mental calculations involving fractions, decimals and percentages. They found that students mainly changed representations (fractions to decimals) or used the relationship between numerical representations when possible. Moreover, data analysis showed that pupils utilized depictive representations such as models or images and descriptive representations like the propositional representations in the sense that these mental models arise as a representation of the real world (i.e., the students used the context of money, clocks, pizzas, etc.). [START_REF] Callingham | A developmental scale of mental computation with partwhole numbers[END_REF] working with students in grades 3 to 10 found that fractions items were easier than decimals and percents and that multiplication and division operations with fractions were more difficult for students than addition and subtraction, thus mirroring similar findings for whole numbers. In the same spirit [START_REF] Lemonidis | In-service teachers' Content and Pedagogical Content Knowledge in mental calculations with rational numbers[END_REF] found that teachers exhibit a lack of flexibility in the use of strategies in mental calculations with rational numbers. [START_REF] Mcintosh | Mental computation of school-aged students: Assessment, performance levels and common errors[END_REF] examined the errors students make during mental calculations with rational numbers and identified two different kinds: procedural and conceptual errors. Procedural errors are connected with simple operations between the terms of fractions. They are mostly careless errors or errors that appear while carrying a step of a strategy even though the steps are known to the student. For example, the student knows that multiplication of fractions means to multiply separately the numerators and denominators of the participating fractions but calculates mistakenly the product of these terms. So, if the numerators of the fractions are 6 and 7 the student might find that 6x7=48. Conceptual errors are connected with a limited understanding of the rational numbers or the operations they have to execute.

In this context our study focuses mainly on the strategies (rather than on the errors, although they have been recorded) used by primary, secondary and tertiary education students given that all of them had to calculate mentally the very same operations involving rational numbers.

Study design and methodology

The participants were 295 students on a voluntary basis from all the three educational levels (primary, secondary and tertiary education). More specifically, there were 78 grade-6 students [SP], 97 grade 10-12 students [SS] and 120 University students (63 from the Department of Primary Education [SU1] and 57 from the Mathematics Department and the Faculty of Engineering [SU2]). These last ones were chosen due to their strong mathematical background that would provide flexibility in their strategies. The tasks and the process were identical for all the three different groups.

Eleven tasks (Figure 1) were posed and the participants were asked to calculate mentally their results. The tasks included all the four operations and their design incorporated two main principles. First, the answers can be calculated in more than one way thus giving freedom to the participants to work in the way they feel more familiar. Second, the tasks are amenable to a fast and easy calculation if the correct mental strategy is chosen. For example, in task-a the answer is obvious if the solver can see that it is about the sum of two halves. Similarly, in task-f the wanted operation is 2 x 3.5. Aiming to cover a variety of different combinations within the spectrum of the operations with rational numbers we included operations between fractions, whole numbers and fractions, mixed numbers and fractions, mixed and whole numbers. The criterion for each task was always to meet the standards for primary school students. Individual interviews where used and each participant was asked to solve one-by-one the whole collection of tasks in the order presented in Figure 1. The students were asked to vocalize their thoughts while solving the tasks and were prohibited from taking notes or making written calculations. There was not time limitations and the duration of the interviews varied from 4 to 9 minutes since the solvers could skip tasks they couldn't solve. Neither help nor any feedback were provided to them. The only clarification that was given concerned mixed numbers since they are used mainly in primary education and the older students confuse mixed numbers with the case of multiplication in algebra. Given that 2x means the multiplication 2 times x, in a similar way the was mistakenly translated as . All interviews were recorded and transcribed. These transcribed protocols constituted our data. The students' efforts were coded according to the following two levels: (i) whether the answer was correct, incorrect, left unanswered, or non-codable, and (ii) whether the strategy used by the student fits to a certain category of the [START_REF] Caney | Mental computation strategies for part-whole Numbers[END_REF] model.

The data were coded independently by the authors and validity and reliability were established by comparing sets of independent results, clarifying codes and re-coding data until agreement.

Results and Discussion

Table 1 summarizes the distribution of the total number of answers (3274 answers in total) across the various categories of strategies according to the categories of [START_REF] Caney | Mental computation strategies for part-whole Numbers[END_REF]. It is worth mentioning that not all their strategies were identified in our sample and also a new one has been added. We've been able to identify the 'changed representation', 'used equivalents', 'used known facts', 'repeated addition/ multiplication', 'worked with parts of a number', 'used a mental picture', and 'used mental form of written algorithm' strategies. The new one is named 'used reduction'. Finally, the incorrect, non-answered and non-codable strategies have also been recorded. The 'changed representation' strategy refers to the conversion of a fraction or a mixed number to decimal. Some students considered (task-e) as . In task-b some prefered to use also decimal numbers to calculate . In a similar way some students made use of this strategy to reach faster the answer in task-f converting the product to . We identified the 'used equivalents' strategy when a fraction is replaced by an equivalent with an explicit reference to this. Some students' mental solution for task-a was , but this was not because they made an immediate reference to the fact that there are two halves but to the fact that ½ is equivalent of 4/8. The 'used known fact' strategy was mostly used in task-h ( ) since the solvers made use of their knowledge of the times table. They started reciting the multiples of 5 and they noticed that and which made them to say that the answer is 15. All the three strategies presented until now had a very low frequency (less than 2%). This frequency becomes even lower for the next three strategies. 'repeated addition/multiplication' was only used by two participants in three different tasks. The first example is from task-b. The solver's reaction to the task was to calculate the product as follows:

Strategy

. The next example is drawn from another student's effort to cope with task-g ( ). The solver considered more convenient to substitute 3 with . This substitution facilitated the addition that follows since all the fractions have now the same denominator. SS14.7.1: I will substitute 3 with . SS14.7.2: Then, .

SS14.7.3: And then,

The 'used parts of the numbers' strategy was recorded only in task-e and task-f. In the first case the solvers decomposed the mixed number to its parts, (i.e., the whole number and the fraction) and then made use of the equality : . In the second case the students decomposed again the mixed number and applied the distributive property of multiplication over addition: . The 'mental picture' strategy is connected to the solvers' effort to visualize the activity as a means to mentally calculate the result [START_REF] Carvalho | Students strategies and errors in mental computation with rational numbers in open number sentences[END_REF]. So, some students while trying to solve task-h preferred to consider 20 as a whole unit, 'like a circular pie' which was divided mentally in 4 equal pieces and 3 of them were taken. SS17.8.1: I am thinking of a circular pie and draw its .

However, the strategy that dominated with extremely high percentage (42,95%) was the 'used mental form of written algorithm' strategy. Solvers worked mentally in a way that appeared to reflect the written algorithm. For task-d a relevant reaction was" "I will make sure that denominators are the same… " (SP7.4.1). For task-f the students converted the mixed number to fraction and then they followed the rule for multiplying a whole number with a fraction: "3x8=24, plus 4…28, times…56, 56/8…therefore, 7" (SU1.55.6.1). The 'used reduction' strategy was the one added by the research team. Instances of this strategy were identified in three tasks. The students simplified the whole number (in tasks b, c and h) with the denominator of the fraction. So, in task-b, they simplified the 4s:

. In task-c, the simplification (4 and 8 were divided by 4) resulted in a fraction equal to one: . In task-h, the word 'of' was translated as 'times' and then the whole number was again simplified with the denominator: , which means (20 and 4 were divided by 4).

Sometimes it seems to exist an overlapping between the 'used reduction' strategy and strategies such as the 'used equivalence' or the 'written algorithm'. But it was the solver's aim as this was expressed verbally that made us to decide if the solution is associated with one strategy or another. For example, in task-a, the solution might be considered as the 'use of equivalence'. However, the student's wording reveals pure use of fraction reduction: "I will simplify the fraction 4/8 which…I divide by 2… is 2/4 which in turn is ½. Therefore, ½ + ½" (SU1.45.1.1). In the same spirit, for the same task, the answer sometimes was considered as 'use of equivalence' and others as 'written algorithm' (see below extracts SU1.9.1.1-2 and SU1.58.1.1-3 respectively) SU1.9.1.1: I will make two equivalent fractions. I will multiply both the numerator and denominator by 4. Errors were surprisingly the second most popular result [START_REF] Mcintosh | Mental computation of school-aged students: Assessment, performance levels and common errors[END_REF]. Both procedural and conceptual errors were detected. Procedural errors were mainly due to miscalculations, for example: . Conceptual errors were mirroring students' limited understanding and misconceptions relevant to fractions. We could mention the flawed idea that when we add fractions we just add separately numerator and denominators, (task-d) or (task-a). We could also add the misconception that when we multiply a whole number with a fraction then the whole number must be multiplied by both the nominator and the denominator: . Moreover, some the students hesitated or were not able to give an answer. Almost 16 out of 100 instances were actually non-answered. Finally, we sorted as non-codable answers that did not reveal any kind of thought. In this category we also included efforts that were based in a strategy that was not suitable for the specific task. In task-k some solvers tried to calculate the results as .

This last multiplication was quite demanding, and the students decided to abandon the task. It is more than evident the dominance of the written algorithm approach over the remaining strategies. This might raise questions about the effectiveness of the way we teach. This becomes more interesting if we take account of the incorrect, non-answered and non-codable answers. The total percentage of these three categories is 44.86% which actually is more than the percentage of the algorithm. If we consider together these two broad groups of participants (those who used the written algorithm and those who had difficulties in answering) then it can be said that almost 9 out of 10 participants either used the written algorithm or were unable to solve the tasks (42.95% + 44.86% = 87.81%).

Table 2 presents in detail how students' answers are distributed across the groups in the different educational levels. The percentages in each column correspond to the specific group of students and not to the total sample and confirm the strong influence of the written algorithm in each group. A closer look reveals also some interesting issues. On the one hand it appears that both groups of university students rely more than primary and secondary students to the written algorithm (which by far dominates the landscape). Forty five out of 100 preservice teachers and 6 out of 10 Mathematics and Engineering students resorted back to the written algorithm to respond to the tasks. Moreover, it is seeming paradox that the highest percentage for algorithmic use (Table 2) is got by students from the Mathematics Department and the Faculty of Engineering [SU2] who are the ones with the strongest mathematical background.

On the other hand, it appears that as we move from primary to tertiary education there is an increased variety of the correct strategies used (though very small number of instances compared to the total number of answers). We refer to all but the 'written algorithm' correct strategies. In total 3.96% of the primary school students' answers used these strategies. This percentage is more or less the same for secondary education students (5.04%). However, for the tertiary education students this becomes almost three times bigger (13.07%). This is not because the use of the 'written algorithm' strategy has been reduced. On the contrary, the relevant percentages of the algorithmic use for SU1 and SU2 students are extremely high and greater than the corresponding ones for younger students. It is the number of the incorrect answers that became smaller.

Having said that, it remains impressive that the main finding of this study is the dominance of the algorithmic use across all the educational levels [START_REF] Rezat | Mental calculation strategies for addition and subtraction in the set of rational numbers[END_REF]. This students' preference becomes more interesting because the tasks were designed to be amenable to a variety of strategies for their solution. For example, task-f ( ) can be solved directly if one notices that the wanted is the double of 3 and half. But there are other possibilities also. According to the 'changed representation' this can be seen as . According to 'written algorithm' it is . There is also the case of the 'worked with parts' strategy: or its modified version . In this last version the fraction might be the result of the 'equivalence' or the 'reduction' strategies. However, for this specific task, according to our data, 3 out of 4 who answered correctly preferred the 'written algorithm' which obviously is not a straightforward calculation. On the contrary it is rather demanding and time consuming. The situation is more or less similar for the total set of the tasks. This highlights the students' obsession with the use of the written algorithm which disqualifies the core idea of mental calculations.

Conclusions

Given the importance of the ability to execute mental calculations involving rational numbers it became evident that although the students exhibit a broad range of different strategies the majority of these strategies is underrepresented. The students mainly follow the steps of the written algorithm. Moreover, it seems that this is not an age-dependent reaction. Very interestingly, the 'written algorithm' strategy was the most frequent choice for primary, secondary, and tertiary education students. We do not doubt that possibly the solvers were aware of alternative strategies and able to use them for mentally calculating the results. What can be said on the basis of the data is that their first choice was the application of the written algorithm. Moreover, there was a great number of participants (44.86%) who either did not give any answer or the answer they gave was incorrect or non-codable. This makes evident the difficulty the students have with rational numbers especially when they are invited to handle them mentally. Additionally, it seems that university students exhibit flexibility in their choice of strategies but in an extremely small percentage. All these deepen our understanding of the topic but at the same time challenge us for a future study that will try to give answers to questions such as: Why do they choose these strategies? What alternative strategies do they know? Why do they not use those alternative strategies? Would the results be the same if the students were asked to solve written the same tasks?

Figure 1 :

 1 Figure 1: Tasks posed to the students

SU1

  .1.1: I will make fractions with the same denominator and I will add them. SU1.58.1.2: I need the Least Common Multiple which is 8. SU1.58.1.3: It becomes , therefore 1.

Table 1 : Distribution of answers across strategies

 1

		Changed	Equivalents Known	Repeated	Parts of a	Mental
		representation		facts	add/ multip	number	picture
	Frequency 56 (1.71%)	59 (1.80%) 43 (1.31%) 3 (0.09%)	31	30 (0.92%)
						(0.95%)
	Strategy	Written	Reduction	Incorrect	Non-	Non-
		algorithm			answered	codable
	Frequency 1406	177	907	519	43
		(42.95%)	(5.41%)	(27.7%)	(15.85%)	(1.31%)

Table 2 : Number of total responses per strategy per group

 2


Pernille Ladegaard Pedersen

Peter Sunde

Students' ability to compare fractions related to proficiency in the four operations

Keywords: Arithmetic, whole number division, fractions

In this paper, we investigate the relationship between 99 fourth-grade students' ability to compare fractions ( and ) and solve tasks in each of the four operations (68 + 753, 547 -64, 12 72 78 3) in a Danish primary school. The students faced more challenges answering the question where the fractions were equal compared to the non-equal fraction items (32% vs 49% correct answers) although the probability of solving the two types of questions were significantly positively associated (odds ratio = 11.0). Relative to the four operations items, solving the nonequal fraction item associated significantly positively with solving the multiplication (odds ratio = 4.5) and division item (odds ratio = 3.9) while the correct solving of the equal fraction associated significantly positively with the solving of the division item (odds ratio = 2.6). These results support the notion that the understanding of fractions is closely connected to multiplicative reasoning.

Introduction

Research has shown that students' knowledge of fractions in middle school greatly affects their overall progression in general mathematics (e.g., [START_REF] Bailey | Early predictors of middle school fraction knowledge[END_REF]. There is also evidence that students need to have mathematical proficiency in the four operations of whole numbers to enable them to effectively comprehend rational numbers [START_REF] Behr | Teaching rational number and decimal concepts[END_REF]. Additionally, multiplicative reasoning seems especially connected to the development of students' understanding of fractions [START_REF] Thompson | Fractions and multiplicative reasoning[END_REF]. Therefore, it is relevant to further investigate the variety of associations between the students' proficiency in the four operations and in fractions. This paper aims to uncover the fourth grade students' abilities to answer equations that require an understanding of fractions as well as the relationship between their abilities to answer tasks and their understanding of the four operations. The primary research question for this paper is: How do students' abilities to solve arithmetic tasks in the four operations (e.g., division) relate to their abilities to answer items that require them to compare fractions?

We expect that the students' proficiency in division and multiplication, but not their capability related to addition and subtraction, will show a strong association to their ability to answer fraction equations because we assume that an understanding of fractions is closely connected to multiplicative reasoning. This paper primarily focuses on students' difficulties with comparing fractions and how this is associated with the four operations. Therefore, the theoretical framework is based on theories of fractions and how these are connected to, for example, multiplicative reasoning.

Theoretical framework

Over the last three decades, several researchers, such as [START_REF] Bailey | Early predictors of middle school fraction knowledge[END_REF], Behr and[START_REF] Behr | Teaching rational number and decimal concepts[END_REF][START_REF] Ni | Semantic domains of rational numbers and the acquisition of fraction equivalence[END_REF], have studied aspects of students' understanding of fractions and the complexities related to teaching and learning these concepts. For example, [START_REF] Kieren | On the mathematical, cognitive, and instructional foundations of rational numbers[END_REF] described how fractions can be seen as a multifaceted construct. He characterized the understanding and knowledge of rational numbers as sets of sub-constructs, which have since been further developed by [START_REF] Behr | Rational Number Concepts[END_REF]. They defined these sub-constructs as: Part-whole, decimal, ratio, quotient, operator, and measure. They recommended that the sub-construct part-whole should be considered to be distinct from the others as a fundamental scheme. Although we are aware of the limitation, in this paper we choose to focus on the sub-construct of quotient even though rational numbers are a more complex system of sub-constructs, and we are aware of for example the importance of ratio in the proportional understanding of fraction [START_REF] Behr | Rational Number Concepts[END_REF][START_REF] Behr | Teaching rational number and decimal concepts[END_REF].

Quotient

For the sub-construct quotient, the denominator designates the amount of recipients, and the numerator represents the amount of quantities that have to be shared [START_REF] Behr | Teaching rational number and decimal concepts[END_REF][START_REF] Toluk | The development of children's understanding of the quotient: A teaching experiment[END_REF]. Thus, the sub-construct of quotient has a connection to partitive division, also known as sharing division, where an object or a group of objects are divided into a number of equal parts [START_REF] Fischbein | The role of implicit models in solving verbal problems in multiplication and division[END_REF]. Moreover, division is the only one of the four operations where a rational number can be the outcome. Therefore, division can be regarded as an integral part of an understanding of rational numbers; hence students must develop an understanding of the connections between division and fractions [START_REF] Behr | Teaching rational number and decimal concepts[END_REF][START_REF] Toluk | The development of children's understanding of the quotient: A teaching experiment[END_REF]. [START_REF] Toluk | The development of children's understanding of the quotient: A teaching experiment[END_REF] conducted a case study of students' development of fraction schemes and the importance of division in the progression from the part-whole subconstruct to the conceptualisation of the quotient sub-construct. They illustrated the developmental process from Fraction-as-Part-Whole and Whole Number Quotient Division-as-Operation to the final stage, which was Division-as-Number (a ÷ b = a/b, a/b) [START_REF] Toluk | The development of children's understanding of the quotient: A teaching experiment[END_REF]. When comparing fractions, it is necessary to understand that the magnitude depends on the relation between the two quantities; the denominator and the nominator [START_REF] Ni | Teaching and learning fraction and rational numbers: The origins and implications of whole number bias[END_REF], rather than simply viewing the fraction notation as two independent cardinal numbers above and below a bar [START_REF] Stafylidou | The development of students' understanding of the numerical value of fractions[END_REF].

Equivalent fraction

Overall, there is substantial support that students' whole number multiplicative understanding is an important resource for developing fraction knowledge [START_REF] Hackenberg | Students' whole number multiplicative concepts: A critical constructive resource for fraction composition schemes[END_REF]. In addition, multiplicative reasoning appears to be essential for the effective understanding and accurate representation of equivalent fractions. One basic fraction perception is to understand fraction equivalence [START_REF] Ni | Semantic domains of rational numbers and the acquisition of fraction equivalence[END_REF]. Fraction equivalence can be explained as the constancy of a quotient and as the constancy of the multiplicative relationship between the numerator and the denominator [START_REF] Behr | Rational number, ratio, and proportion[END_REF]. Here, the quotient is involved in an understanding of equivalence; for example, can be viewed as a division, which represents the equivalence between and 0,5 [START_REF] Behr | Rational Number Concepts[END_REF]. Each fraction belongs to a unique equivalence class represented by a multiplicative equation. Each class denotes a distinct rational number; for example = .

Difficulties related to the understanding of equivalent fractions have been connected to the students' lack of multiplicative reasoning [START_REF] Ni | Semantic domains of rational numbers and the acquisition of fraction equivalence[END_REF]. If the students do not develop this understanding of equivalence, they will view for example tasks that involve adding two fractions with different denominators as a purely technical algorithm [START_REF] Arnon | Where do fractions encounter their equivalents? Can this encounter take place in elementary-school[END_REF]. Therefore, both multiplication and division of whole numbers might be more closely connected to understanding fractions than additional reasoning. Hence, we expect to find an association between the fraction equations and the multiplication and division items, meaning that if the students can solve items involving the two operations correctly, they will more likely be able to solve the fraction equations. However, if the students have not developed a proficiency with multiplication and division, we expect that those students will experience difficulties answering the fraction equations correctly, especially the equivalent fraction equations. We will analyse how the two fraction items differ in proportion of correct answers as well as whether and how they differ in association with the four operations.

Methods and analysis

Data consisted of answers provided by fourth-grade students (ages 10-11) on a computerised test.

The test consisted of 110 items, including problems on the four operations, fractions, and algebra as well as word problems. The test was time restricted (45 minutes), so not all students finished all of the items. The questions on fractions included in the analysis were items 50 and 52, and only the students who answered these items within the timeframe were included in the analysis. However, we are aware of the constraints of interpreting the students' answers on a computerised test in which the student just enters the answer to the item. This only allows us to review a student's correct or incorrect answers and not the working process itself. In addition, we are aware that neither a whole number multiplication nor a division item can be seen as a strong indicator of student's multiplicative reasoning. A quick review of these answers showed that the students who answered this section were generally quick at calculating and, on average, had the highest scores in the first section of the test (whole number arithmetic operation tasks). In other words, the students who reached the fraction items (items 50 and 52) scored above average on the test (N = 99). The items we chose to analyse were items where the students were asked to compare to , to , and to

The students could choose between the following three symbols: <, >, and = (Figure 1). Earlier in the test, the students were asked to solve a whole number equation using the same symbols. There were not any significant differences found in the items containing the equal sign.

The first item was created to see the students' answers on a unit fraction equation. Often, students have whole number biases in their learning processes of fractions such that they look at the biggest digit and conclude that this is the largest number; for example 5 is larger than 4 [START_REF] Ni | Teaching and learning fraction and rational numbers: The origins and implications of whole number bias[END_REF].

The second item, compared to , was likewise designed to examine the students' answers on an item where there is a whole number bias since both 5 and 11 are bigger than 3 and 5. In the solving process of this item, the student might use a different strategy, such as the benchmark strategy where is smaller than a half and is bigger than a half [START_REF] Clarke | Students' fraction comparison strategies as a window into robust understanding and possible pointers for instruction[END_REF]. We considered this item to be the most difficult item for the students to solve because of the different numerator and denominator as well as the fact that the item did not contain any unit fraction. The last item was the equal fraction item where is equal to . This item included the unit fraction , which we considered to be a commonly used fraction in the fourth-grade curriculum. Therefore, we assumed that the students might have a god mental representation of this fraction. As for the analysis of the relationship between the four operations, we have selected one whole-number problem from each of the operations (68 + 753, 547 -64, 12 72, 78 3) based on the following criteria: The item where the students had the fewest correct answers, the most wrong answers, and the most times it was not answered, in that order. Each of the arithmetic items was designed to evoke solution strategies based on number sense approaches. For example, in the case of item 12 72, it involves breaking up one of the factors 12 into a 10 and a 2 and then multiply it in parts (10 72) + (2 72).

Analysis

The analysis focused on how the students' answers to one fraction equation associated to the other fraction equation and to answers on the four operations items. The association between the probability of returning a correct answer as opposed to an incorrect one or omitting an answer for one item in relation to whether the student answered another item correctly was analysed as a binary logistic regression (Logistic procedure in SAS 9.5) function. Logistic regression models yield similar p-values for statistical significance of associations as conventional 2 x 2 contingency 2tests as well as additionally producing predictions of the conditional probabilities of returning a correct answer for one item in relation to whether the student had answered the other item correctly. The difference between these two predicted probabilities on a logit scale (coefficient of the difference between the two levels in the regression equation) furthermore serves as a coefficient of association in the response patterns between the two items.

The anti-log of this coefficient (the odds ratio 'OR') equals how many more times a student is likely to answer item Y (e.g.,

) correctly if he/she has answered item X (e.g., 78 3 =) correctly as compared to if he/she has not answered X correctly. Hence, an association coefficient of 1.0 indicates that the students answering item X correctly are exp. (1.0) = 2.71 times more likely to answer item Y correctly as compared to the students who did not answer item X correctly. Note that the ORs derived from the logistic regression equation exceeds the ORs calculated from the arithmetic values of the probabilities because of the infinite state space on the logit scale, whereas values are on the arithmetic scale  0;1. The conditional probabilities of getting correct answers as a function of correct vs. incorrect or missing answers in other item types were modelled and illustrated graphically for the item as a function of the item and The same was modelled and illustrated for the item as a function of item . A matrix table of association coefficients between items that represent the different concepts of fractions, addition, subtraction, multiplication, and division was calculated for each of the items.

Results

When analysing data from the test, it was found that the students showed difficulties solving items involving equivalent fractions. Only 32% of the students answered correctly while 49% answered correctly (Figure 2). Overall, the two non-equal fraction equations were similar in their answer patterns since both had about 50% correct, about 30% incorrect, and 20% with no answer. When comparing, the probability of answering the three items representing equation problems correctly were all strongly and highly statistically significantly associated (Figure 3, association coefficient values: 2.4-4.0, i.e. ORs: 11-54). The two non-equal fraction items showed the same pattern when compared to the equal fraction.

Hence, in all comparisons, the probability of answering correctly on the item was about 90%

if the student had answered the predictor item correctly compared to about 50% if they had answered it incorrectly (Figure3). When comparing the two non-equal fraction items, we found that if the students had answered the predictor item correctly, only about 10% could then not solve < . Therefore, we chose to focus on the equal fraction item and the non-equal fraction item

. When comparing all problem types (Table 1), correct solutions were most strongly associated with the two equation problems (OR = 11.0, p < 0.0001), followed by the association between the addition and subtraction (OR = 6.8, p < 0.0001), multiplication and division (OR = 5.4, p < 0.0001), and subtraction and multiplication (OR = 5.0, p = 0.001) items, respectively. The results of the equation item were highly significantly positively associated with the results of the multiplication (OR = 4.5, p = 0.0009) and division items (OR = 3.9, p = 0.003). The results of the equation item only associated positively with the division item (OR = 2.6 p = 0.02) (Table 1). The lowest associations (all non-significant) were found between the results of two equation items and the results of addition and subtraction items (all four ORs 1.7-2.4), between equation item , and the multiplication item (OR = 2.1) (Table 1).

Discussion

This statistical investigation of the two fraction items' differences and their associations to the four operations revealed the following three results: Firstly, when comparing the three items with each other, the students' answers showed that the equal fraction item (30% correct answers) was more difficult than the two non-equal fraction items (50% correct). This was unexpected since one of the fractions in the equal item was a commonly referenced unit fraction of . It is a fraction which is common in the introduction of fractions in the fourth grade, whereas is a rare fraction notation in school curriculums. As referred to earlier, we therefore expected the item to be easier than the other one. The students' difficulties with the equal item could indicate that the students have difficulties understanding fractions as an equivalence class made by a multiplicative equation [START_REF] Ni | Semantic domains of rational numbers and the acquisition of fraction equivalence[END_REF]. This is emphasized by the results in Figure 3, which show that the probability of answering correctly on the item = was still only 50% if the student had answered incorrectly. However, a probability of only 10% was found for answering correctly if they had answered incorrectly, which indicates that the comparison of equal fraction items differed. This could be due to the fact that when solving the equivalent fraction items, the students have a whole number bias [START_REF] Ni | Teaching and learning fraction and rational numbers: The origins and implications of whole number bias[END_REF], the equivalence can be difficult for the students to comprehend because it is the first time they have experienced numbers that can be labelled differently and still refer to the same numerical quantity. Secondly, we found that the result of the non-equal fraction was highly associated with the result for both the multiplication (OR = 4.3) and the division (OR = 3.9) items, whereas there was no significant association with the results of the addition and subtraction items. This result coheres well with the theory that fractions have a stronger connection to multiplicative reasoning than to additive reasoning [START_REF] Thompson | Fractions and multiplicative reasoning[END_REF]. According to this theory, one should have expected a closer association with division than with multiplication [START_REF] Toluk | The development of children's understanding of the quotient: A teaching experiment[END_REF]. A possible explanation might be that the result patterns of the two operations were also closely related (OR = 5.4), suggesting that the basis of the understanding necessary for solving division and multiplication problems is so closely related that they could not be discriminated in the analysis.

Thirdly, we found that the results of the non-equal item correlated more strongly and significantly with the results of the division and multiplication items than was the case for the result of the equal fraction item (that only associated modestly with the result of the division item). Thus, there is a component of the equal fraction item making this more difficult for the students to answer. An explanation could be that students have not yet developed the conceptualisation of the quotient as Division-as-Number [START_REF] Toluk | The development of children's understanding of the quotient: A teaching experiment[END_REF]. Hence, when students look at equal fractions, they can see a quotient as a division, which represents the equivalence between both and 0.25 [START_REF] Behr | Rational Number Concepts[END_REF]. Overall, these results support the notion that the understanding of fractions is closely connected to multiplicative reasoning; however, it is essential to pursue further investigation since there seems to be a different pattern in the students' proficiency in the equivalence concept. Otherwise, they will experience difficulties in understanding other concepts, like common denominators [START_REF] Arnon | Where do fractions encounter their equivalents? Can this encounter take place in elementary-school[END_REF]. Further research should collect qualitative data to examine the students' working process to overcome the strong limitations of a computer test, which can only offer limited insights. We need to investigate further ways in which the students experience difficulties in the equal fraction task. It may be that this is strongly connected to their understanding of the equal sign or that there is a connection to a whole number bias. On the one hand, the students' whole number multiplicative understanding might support their understanding in some fraction contexts, and on the other hand, the students' understanding of whole numbers can distract them in other fraction contexts.

Figure 1 :

 1 Figure 1: The items used in the analysis. Translated to: A. Choose the right symbol: >, < or =. (No symbol was preselected). B. Calculate the task

Figure 2 :

 2 Figure 2: Percent of correct, incorrect or no answer to the three fraction items (N = 99)

Figure 3 :

 3 Figure 3: Percent of correct answers on one item type conditional to whether the student answered another item type. Error bars indicate 95% confidence intervals. All differences were highly significant (p < 0.0001)

Table 1 : Coefficients of association (log odds ratios) of correct vs. incorrect/missing answers of different problem items: The higher the coefficient value, the more likely a correct answer of one item type associated with a correct answer of the other item type. Significance levels: * p < 0.05, ** p < 0.01

 1

		1/4 = 2/8	5/11<3/5	68+753		547-64	12 72		78 3
	5/11<3/5	2.40	****
	68+753	0.85		0.53
	547-64	0.75		0.88		1.91	****
	12 72	0.72		1.51	***	0.83		1.60	***
	78 3	0.96	*	1.36	**	1.02	*	0.68		1.68	****
	N items:	99		99		142		142		142		142
	N correct items:	50	51%	67	68%	32	23%	49	35%	98	69%	87	61%
													,
	*** p < 0.001, **** p < 0.0001. (A log odds ratio of 1 equals a difference of exp (1) = 2.7)


Farah Gabsi

Frédéric Hamelin

Nathalie Sauer

Dominique Sauter

Fault-tolerant economic model predictive control for building temperature regulation using ℓ_ϵ-regularization

come   L'archive ouverte pluridisciplinaire

Fault-Tolerant Economic Model Predictive Control for Building

Temperature Regulation using ε -Regularization* Farah Gabsi 1,2 , Frédéric Hamelin 1 , Nathalie Sauer 2 and Dominique Sauter 1

Abstract-This paper presents an active Fault-Tolerant Control (FTC) scheme for the temperature regulation of buildings subject to actuator faults. An optimal control strategy based on a regularized predictive economic criterion is proposed to determine the remedial actions to be taken when actuator faults are diagnosed. These corrective actions can be used either to generate a less "aggressive" control scenario by minimizing the number of on/off cycles of the faulty actuator or to reduce the number of actuators used at any time. One of the interests of this strategy is to prevent simple faults from developing into serious failures for actuators particularly vulnerable to repeated on/off cycles (in particular heat pump compressors). To account for the "aggressiveness" of a control scenario, the penalty term associated with the regularized economic criterion is defined using the parsimony property of ε -norm (ε small).

I. INTRODUCTION

In the context of smart buildings, Building Energy Management Systems (BEMS) improve energy efficiency while providing the comfort levels expected by users. They monitor and control the building's energy needs. Through the use of advanced control techniques, BEMS can ensure that the building operates at the highest level of energy efficiency. They can be optimized to take full advantage of the flexibility of the building environment or to take into account aging and faulty equipment.

In recent years, the use of control techniques in building management has been a very active research area. To effectively control the temperature inside a building, many approaches have been proposed [START_REF] Salsbury | A survey of control technologies in the building automation industry[END_REF], [START_REF] Dounis | Advanced control systems engineering for energy and comfort management in a building environment-a review[END_REF]. Model Predictive Control (MPC) is considered the most appropriate method to ensure user comfort while minimizing energy consumption [START_REF] Gabsi | Hygrothermal modelling and MPC-based control for energy and comfort management in buildings[END_REF], [START_REF] Godina | Model predictive control home energy management and optimization strategy with demand response[END_REF], [START_REF] Serale | Model predictive control (MPC) for enhancing building and HVAC system energy efficiency: Problem formulation, applications and opportunities[END_REF]. Indeed, this technique makes it possible to take into account predictions of disturbances such as the occupation of the zones, the outside temperature or the occupant behavior [START_REF] Mirakhorli | Occupancy behavior based model predictive control for building indoor climate-a critical review[END_REF]. Economic Model Predictive Control (EMPC) is becoming increasingly popular [START_REF] Zong | Challenges of implementing economic model predictive control strategy for buildings interacting with smart energy systems[END_REF] because of the value of economic cost functions that are more realistic than traditional quadratic cost functions.

Failure of some components of the Heating, Ventilation and Air Conditioning (HVAC) system or its sensors can have *This work has financial support from the Contrat de Plan État-Région (CPER) Lorraine 2015-2020, project "Matériaux, Énergie, Procédés". 1,2 Farah Gabsi is with CRAN and LGIPM, Université de Lorraine/CNRS, F-54000 Nancy (CRAN), F-57000 Metz (LGIPM), France Farah.Gabsi@univ-lorraine.fr FirstName.LastName@univ-lorraine.fr

2 Nathalie Sauer is with LGIPM, Université de Lorraine, F-57000 Metz, France Nathalie.Sauer@univ-lorraine.fr a significant impact on the expected energy performance of the building and on users comfort (high energy consumption, low thermal comfort and unacceptable air quality) [START_REF] Zhang | Modeling of HVAC operational faults in building performance simulation[END_REF]. Faulttolerant control systems aim to recover performance close to nominal value after a fault has occurred. Several studies have been conducted as part of the fault-tolerant control applied to building temperature regulation [START_REF] Bengea | Fault-tolerant optimal control of a building HVAC system[END_REF], [START_REF] Sauter | Fault tolerant control of HVAC systems for energy efficient buildings[END_REF]. Some of them allow to accommodate the sensor faults [START_REF] Papadopoulos | Distributed adaptive sensor fault tolerant control for smart buildings[END_REF], [START_REF] Du | Tolerant control for multiple faults of sensors in VAV systems[END_REF] while others allow to reconfigurate the control law in the presence of HVAC failures [START_REF] Darure | Fault-adaptive control of VAV damper stuck in a multizone building[END_REF], [START_REF] Subramaniam | Fault tolerant economic model predictive control for energy efficiency in a multi-zone building[END_REF] (stuck faults in VAV dampers).

The main contribution of this work is the design of an active FTC scheme to best accommodate actuator faults affecting HVAC systems. The novelty of the proposed FTC strategy is that it is based on an EMPC, whose cost function is regularized by ε -norm penalties in the case of an actuator failure. By a judicious choice of the regularization terms, this approach makes it possible to smooth the control input of the faulty actuator, for which too frequent starts/stops would be critical. It also makes it possible to reduce the number of active control inputs at any time and to take into account stuck actuators. Regularization terms are defined on the basis of the LASSO (Least Absolute Selection and Shrinkage Operator) theory, particularly used in signal processing, which has led to the emergence of new predictive control strategies called " asso MPC" [START_REF] Gallieri | l asso MPC: Smart regulation of over-actuated systems[END_REF], [START_REF] Rao | Sparsity of linear discrete-time optimal control problems with l 1 objectives[END_REF]. Considering the 0 penalties [START_REF] Aguilera | Quadratic model predictive control including input cardinality constraints[END_REF], [START_REF] Aguilera | Quadratic MPC with l 0 -input constraint[END_REF] allows for more parsimonious capacities but makes the optimization problem non-convex.

The paper is organized as follows. Section 2 presents the general objectives of the proposed fault-tolerant economic model predictive control. It specifies the control objectives not only in the fault-free case by means of an economic criterion, but also under actuator fault occurrence in order to better accommodate it. Section 3 specifies these different objectives in the form of a non-convex ε -regularized economic criterion. The resulting NP-Hard minimization problem is solved using an iterative algorithm recently derived from [START_REF] Gabsi | Hygrothermal modelling and MPC-based control for energy and comfort management in buildings[END_REF]. Finally, conclusions and perspectives are presented in Section 4.

II. PROBLEM STATEMENT A. Dynamic Model

The general structure of the dynamic model considered in this paper is chosen in order to take into account the thermal modeling of a multizone building integrating the different equipment as well as all influential disturbances [START_REF] Gabsi | Building hygrothermal modeling by nodal method[END_REF]. For a N-zone building characterized by n u independent control scenarios (no action, Heat Pump (HP, on/off), Continuous Mandatory Ventilation (CMV, on/off), automatic tilt-turn windows (open/close), venetian blinds (open/close),...), a set of n u state-space models can be defined. The following bilinear state-space representation allows us to consider this set of linear models:

     x(k + 1) = n u -1 ∑ i=0 u i (k) (A i x(k) + B i q(k)) T OP (k) = C i x(k) (1)
with:

• T OP (k) = T OP i (k) 1 ≤ i ≤ N ∈ R N : the output vector de- fined from the operational temperature T OP i (k) in each zone i ( 1 ≤ i ≤ N ); • u(k)= u i (k) 0≤i≤n u -1 ∈ R n u : the control vector defined
from the control inputs of all zones i . Each element u i (k) is equal to 1 or 0, these two values corresponding respectively to the switching on or off of the ith control input; • x(k) ∈ R n : the state vector;

• q(k) ∈ R n q : the disturbance vector that groups uncontrollable exogenous inputs and all heat flows in and out of the N zones i ; • A i , B i and C i : matrices of appropriate dimensions that may differ depending on the control inputs u i (k).

B. Economic Model Predictive Control (EMPC)

The principle of predictive control [START_REF] Rockett | Model-predictive control for nondomestic buildings: a critical review and prospects[END_REF] is to optimize a cost function to describe the control objectives over a time horizon of length N P . The economic objective function J MPC (u, x) that we propose in the context of the temperature regulation of a multizone building is as follows:

J MPC (u, x) = min u N P ∑ j=1 T OP (k + j) -T COMF (k + j) 2 Ψ j + ∆u(k + j) 2 e ∆u j + u(k + j) 2 e u j (2)
where:

• T COMF (k + j) = T COMF i (k + j) 1 ≤ i ≤ N ∈ R N
and T OP (k + j) represent the estimated comfort temperature and operational temperature in each zone ( 1 . . . N ) according to [START_REF] Mccartney | Developing an adaptive control algorithm for europe[END_REF] and model [START_REF] Salsbury | A survey of control technologies in the building automation industry[END_REF]. As for ψ i , j , it reflects the importance attached to T OP i (k + j) being close to

T COMF i (k + j). • T OP (k + j) -T COMF (k + j) 2 Ψ j = ∑ i = 1 ,..., N ψ i , j |T OP i (k + j) -T COMF i (k + j)| 2 represents
a discomfort criterion at the time (future) k + j.

• ∆u(k + j) 2 e ∆u j = n u -1 ∑ i=0 e i, j |u i (k + j) -u i (k + j -1)| 2 and u(k + j) 2 e u j = n u -1 ∑ i=0 e i, j |u i (k + j)| 2 reflect the
economic cost of the control law implemented at the time k + j. The weightings e i, j and e i, j allow to specify the cost in euros of each possible action on the system. The e-terms reflect the energy cost of starting a HP or VMC for a given period of time while the e -terms reflect the opening or closing of tilt-turn windows or venetian blinds.

At any given time k, an optimal control sequence {u * (k + j)} 1≤ j≤N P is calculated to minimize J MPC (u, x) and only the first element u * (k + 1) is applied to the system.

C. Fault Tolerant Economic Model Predictive Control (FTEMPC)

In the presence of incipient actuator faults, it may be useful to continue to use these actuators but with less stress to allow the system to operate in slightly degraded mode. In the context of building temperature control, several scenarios may arise. If, for example, the compressor of a heat pump does not work well, it would be interesting to continue to use the heat pump but to reduce the number of start-up cycles. In this case, the optimal control scenario {u * (k + j)} 1≤ j≤N P must be modified so that the control input associated with this actuator has few fluctuations over the horizon N P . As a second example, if we consider a central Air-Handling Unit (AHU) supplying air to a large room by several Variable Air Volume (VAV) boxes, a failure of the supply fan may result in a decrease in the air flow. One solution to this problem could be to close some modulating dampers in these VAV boxes. In this case, {u * (k + j)} 1≤ j≤N P mut be modified so that the control inputs associated with these dampers are 0 on the N P horizon.

The objective of the FTEMPC approach proposed in this paper is to adapt the optimal control scenario {u * (k + j)} 1≤ j≤N P in the presence of an actuator fault so that the control input is smooth over the entire N P horizon of the faulty actuator (case of the first example) or the number of actuators used is reduced (case of the second example). To do this, the economic objective function J MPC (u, x) (2) is regularized.

III. REGULARIZED EMPC

A. Regularized Criterion

It is proposed to consider a regularized criterion such as:

J λ ,Ω (u) = (1 -λ )J MPC (u, x) + λ Ω (u) (3)
with J MPC (u, x) defined by ( 2) and where 0 ≤ λ ≤ 1 is a regularization parameter whose value depends on the fault diagnosed on the system. The additional term λ Ω(u) in the criterion amounts to regularizing the solution. In the context of the problem presented above, λ Ω (u) should be a penalty that increases the parsimony of u or the first derivative of u. Based on the Tikhonov regularization method [START_REF] Engl | Regularization of inverse problems[END_REF], one possible technique is to include a linear operator R in the regularization term Ω (u), and solve the following problem :

u * λ ,p = arg min u∈R nu (1 -λ )J MPC (u, x) + λ Ru min(1,p) p ( 4
)
where

• p is the p -norm z p := n z ∑ i=1 |z i | p 1 p
for a vector

z ∈ R n z .
The linear transformation R can take different forms:

• 0 th -order regularization:

R = R 0 = I (5)
Penalty R 0 u min (1,p) p gives priority to solutions with a small norm.

• 1 st -order regularization:

R = R 1 =       -1 1 0 • • • 0 0 -1 1 . . . . . . . . . . . . . . . . . . 0 0 • • • 0 -1 1       (6)
The penalty term R 1 u min(1,p) p penalizes control increments between two consecutive time samples. Minimizing the regularized criterion (3) smoothes the control signal, which is useful when it is necessary to avoid sudden changes in control. All control inputs must be stored according to the vectors u i and u defined as follows:

u T i = u i (k + 1) . . . u i (k + N P ) u T = u T 0 . . . u T n u -1 (7)

B. p Penalization

There are several types of penalty functions [START_REF] Hastie | The elements of statistical learning: data mining, inference and prediction[END_REF]. The Ridge regression is a penalty of type 2 -norm. The LASSO regression, introduced by Tibshirani [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF], is a linear regression with a 1 -norm penalization, which increases the parsimony of the solution when R = I. An even more natural method to contract the value of the different elements of Ru but also to force some of these elements to be strictly zero for λ large enough is to take into account a constraint Ru ε ε (with 0 ≤ ε 1). To illustrate the advantages and disadvantages of each possible penalty, consider the following criterion:

J λ ,p (u) = (1 -λ )J 1 (u) + λ u min(1,p) p (8) with J 1 (u) = (u 1 -2 + u 2 ) 2 + (u 2 -0.5) 2 .
First, we observe that ∀p ≥ 0, we have:

   u * 0,p = 1.5 0.5 T = arg min u J 1 (u) u * ∞,p = 0 0 T = arg min u u min(1,p)
p Between these two extreme values, the trajectory Γ λ ,p followed by the optimal point u * λ ,p in the u 1u 2 plane as a function of λ is represented in red in Fig. 1. The ellipsoids and the filled contour plot in the background of these figures are isolines of J 1 (u) and u min(1,p) p respectively. We can see on each of the subfigures (Fig. 1a-1h) that the shape of the trajectories Γ λ ,p is very different depending on the p values. In particular, the LASSO selection (Fig. 1c) gives a more parsimonious solution than the Ridge selection (Fig. 1b), which tends to make the coefficients very small without canceling them. More generally, for p > 1, the trajectory Γ λ ,p tends towards u * ∞,p for λ increasing but without being "attracted" by the axes u 1 = 0 and u 2 = 0. On the other hand, as soon as p ≤ 1, we observe that this convergence towards u * ∞,p takes place along one of the axes u 1 = 0 or u 2 = 0. Another phenomenon appears for p < 1; the trajectory Γ λ ,p followed by u * λ ,p becomes discontinuous as the value of p decreases. This is due to the non-convexity of the term u min(1,p) p which increases for small values of p and makes the criterion J λ ,p (u) non-convex.

In the sequel, an explanation is given on the effect of p on the trajectory Γ λ ,p followed by u * λ ,p . It is based on the shape of the isolines u min (1,p) p shown in Fig. 2 for different p-values. First, the minimization of the regularized criterion J λ ,Ω (u) (4) is reformulated into a problem of minimizing the function J MPC (u, x) subject to an inequality constraint, It can therefore be observed that (4) corresponds to the Lagrangian form of the optimization problem [START_REF] Bengea | Fault-tolerant optimal control of a building HVAC system[END_REF]. As a result, if the cost function J MPC (u, x) is convex and if the regularization term Ru min (1,p) p is a p -norm with p ≥ 1 then the solutions to the problems ( 4) and ( 9) are identical. The exact relationship between β and λ is then data dependent.

Fig. 3 gives a geometric interpretation of the solution to the optimization problem [START_REF] Bengea | Fault-tolerant optimal control of a building HVAC system[END_REF] in the case of the example considered above [START_REF] Zhang | Modeling of HVAC operational faults in building performance simulation[END_REF], characterized by a strongly convex cost function J 1 (u). It helps to understand the interest of ε -norm compared to p -norm with p > 1. Depending on the p-values and their associated contour lines (Fig. 2), the reddashed borderlines in Fig. 3 plane for which the constraint u min (1,p) p ≤ β is fulfilled for different values of β . For a given β -value, the optimal point u * * β ,p , represented by a red dot on each subfigure corresponds to the intersection of the isoline u min(1,p) p = β with the lower-valued isoline J 1 (u) represented in blue. The β -evolution of u * * β ,p for β ∈ 0, u * 0,p min(1,p) p generates the black trajectory Γ β ,p . A geometric interpretation can be given to explain the "attractiveness" of the axes u 1 = 0 and u 2 = 0 for the trajectory Γ β ,p associated with the regularization term u min(1,p) p with p ≤ 1. This is due to the fact that the isolines associated with these norms have a sharp corner on the axes u 1 = 0 and u 2 = 0 that they do not cut orthogonally.

An analysis of Fig. 1 and3 shows that the trajectories Γ λ ,p and Γ β ,p are identical in the case of p = 1, 2 and +∞. We therefore observe that the solution of the problems ( 4) and ( 9) is the same for p ≥ 1. On the other hand, for the particular value p = 0.2, it appears that the trajectory Γ β ,p (Fig. 3d) differs from Γ λ ,p (Fig. 1g). Nevertheless, it can be noted that:

Γ λ ,p ⊆ Γ β ,p (10)
this result having been demonstrated in [START_REF] Soussen | Homotopy based algorithms for 0 -regularized least-squares[END_REF].

C. Fault-Tolerant Regularized Criterion

In the context of a fault on the actuator u i , the penalty term Ω (u) (3) can be used to:

• reduce the number of on/off cycles of the faulty actuator. In view of the previous paragraph, the minimization of J λ ,Ω (u) (3) reflects this goal with Ω (u) =

R 1 u i (k) u T i T ε ε
and u i defined by ( 7); • use a reduced number of actuators at each prediction time k + j, which results in the minimization of J λ ,Ω (u) with Ω (u) = R 0 u(k + j) ε ε and u(k + j) = u i (k + j) 0≤i≤n u -1 ;

• have the control input u i (k + j) constant and equal to u i, f ∀ j when the actuator u i is stuck at u i, f . Minimizing

J λ ,Ω (u) with Ω (u) = R 0 u i -u i, f 1 ε ε satisfies this constraint.
Remark 1: It is interesting to note that the latter type of penalty can be used to assign a 1 or 0 value to the control inputs (on/off or open/closed). Minimizing J λ ,Ω (u) with Ω (u) = R 0 (u (u -1)) ε ε achieves this goal if u (u -1) refers to the element-wise product of the two vectors u and (u -1).

Remark 2: In the fault-free case, it may be worthwhile to reduce the number of actuators used or to smooth the control input over the entire horizon N P . It is therefore interesting to take into account the regularization term Ω (u) not only in the presence of an actuator fault, but also when the system is healthy.

Ω (u) is then defined as:

Ω (u) = γ 1 R 0 NB l=1 (u -u l 1) ε ε + γ 2 R 0 u i -u i, f 1 ε ε +α 1 R 1 u i (k) u T i T ε ε + α 2 N P ∑ j=1 R 0 u(k + j) ε ε (11)
with:

• (u l ) 1≤l≤NB : constant values that can be taken by at least one of the control inputs. For on/off controls, we have NB = 2, u 1 = 1 (On) and u 2 = 0 (Off).

• NB l=1

(uu l 1) = (uu 1 1) . . . (uu NB 1);

• α 1 , α 2 , γ 1 , γ 2 : weighting coefficients. γ 1 is very large because the norm it weights must be zero. The values of α 1 , α 2 , γ 2 depend on whether or not an actuator has failed. In the fault-free case, a strategy based on Bryson's rule allows to choose α 1 and α 2 , which are then equal to the inverse of the maximum value of the weighted terms. In the presence of a fault on the actuator u i , α 1 and α 2 can be considerably increased depending on the magnitude of the fault and its criticality. As for γ 2 , it is non-zero only if u i (k + j) is stuck at u i, f ∀ j.

The parameter λ (3) allows to find a compromise between the cost function J MPC (u, x) and the penalty term Ω (u). The problem of the "best" choice of λ , which corresponds to the optimal compromise between the minimization criterion J MPC (u, x) and the respect of the "choice criterion" (Ω (u) "not too large"), is difficult and remains, from a theoretical point of view, largely open. As is often the case in methods for estimating the regularization parameter λ , it is proposed to evaluate the criterion for different values of λ defined on a grid Λ. The optimal regularization parameter λ * is determined using a classical technique widely used in this type of problem, the L-curve method [START_REF] Hansen | Analysis of discrete ill-posed problems by means of the l-curve[END_REF]. It consists in plotting the value of the two parts of the criterion for multiple values of λ . As an illustration, Fig. 4 shows these L-curves associated with the regularized criterion J λ ,p (u) (8) for different small values p. It should be noted that in the general case, the values of λ * are multiple and depend on the desired degree of parsimony (number of "0") in the penalty Ru. The determination of λ * is quite simple if the penalty is in ε -norm. Indeed, as can be seen in Fig. 4, the appearance of a "0" in the penalty Ru results in the presence of a discontinuity (negative step) in the L-curve. Thus, if we accept that the penalty term Ru has only one "0" then the optimal solution u * λ ,p is associated with the first discontinuity of the L-curve examined from left to right (J MPC (u, x) increasing). In Figure 4, this corresponds to the leftmost red dots. The following discontinuities in the curve reflect the presence of additional "0" in the penalty Ru. The value of λ * therefore depends on the desired degree of parsimony in the penalty Ru.

Schematically, the active actuator fault-tolerant control design we propose in this paper is shown in Fig. 5. Fig. 5: FTEMPC scheme.

D. Minimization of the regularized criterion

The above optimization problem is NP-Hard due to the non-convexity of the ε -norm. To solve this problem, we use a recently developed iterative algorithm [START_REF] Gabsi | Hygrothermal modelling and MPC-based control for energy and comfort management in buildings[END_REF], whose objective is to estimate the optimal control scenario with a lower computation load. The idea is to keep at each time of prediction k + j only a reduced number of scenarios among all those that are possible. For this purpose, a principal component analysis is performed on a limited number of points properly selected in the variables space (T OP 1 (k + j), . . . , T OP N (k + j)) in order to determine a suitable basis for the representation of all possible realizations of T OP i (k + j) with i = 1 , . . . , N . By prioritizing the information, this makes it possible to replace the set of all these realizations by a smaller subset S j whose cardinality is defined a priori. This procedure is repeated N p times iteratively for j ranging from 1 to N p . Of course, the larger the number |S j |, the better the approximation of the optimal control scenario. This prioritization technique allows to consider not only a non-convex regularized criterion J λ ,Ω (u) (3) but also large values for the forecast horizon N P .

IV. CONCLUSION

This paper presents an innovative Fault-Tolerant Economic Model Predictive Control (FTEMPC) based on the definition of a regularized economic criterion. In the fault-free case, this approach generates an optimal control scenario resulting from a compromise between thermal comfort in occupied zones and energy consumption. In the presence of actuator faults, the control scenario is optimally adjusted according to the type of fault assumed to be known. In this case, the regularization terms are used to allow both to smooth the control input of the faulty actuator and to reduce the number of operating actuators at any time. Their role is also to constrain the control inputs in the presence of stuck actuators. The proposed approach is particularly interesting in the context of building temperature regulation where some actuators are particularly sensitive to repeated on/off cycles (e.g. heat pump compressors). A detailed analysis showed the importance of choosing a ε -norm (ε small) to define the regularization terms. The proposed fault-tolerant control strategy will be implemented on the "Eco-Safe" platform at CRAN Nancy [START_REF] Gabsi | Energy efficiency of a multizone office building: MPC-based control and simscape modelling[END_REF], and will be reported in a follow-on paper.

Fig. 1 :

 1 Fig. 1: Trajectory Γ λ ,p (in red) followed by u * λ ,p according to λ in the u 1u 2 plane.

1 Fig. 2 :

 12 Fig. 2: Mesh and contour plot of u min(1,p) p in the u 1u 2 plane.

2 Fig. 3 :

 23 Fig. 3: Trajectory Γ β ,p (in black) followed by u * * β ,p according to β in the u 1u 2 plane.

  (a) p = 0.8. (b) p = 0.6. (c) p = 0.4. (d) p = 0.2.

Fig. 4 :

 4 Fig. 4: L-curve Ru min(1,p) p versus J 1 (u) associated with the optimization problem (4).