Cascella Clelia
email: clelia.cascella@manchester.ac.uk

Giberti Chiara
email: chiara.giberti@unibz.it

Order of factors in multiplying decimal numbers and gender differences: a comparison of tasks

Keywords: Mathematics achievement test, Misconceptions, Gender gap, Rasch model, Differential item functioning

Differential Item Functioning (DIF) analysis carried out within the framework of the Rasch model has been used to compare scholastic performance in sub-groups of students matched on ability and clustered by some features (such as, for example, gender). In this paper, we use DIF to understand if misconceptions about multiplication with decimal numbers act on boys and girls differently, i.e. if this can lead boys and girls to approach a mathematics item differently and, in some cases, to encounter it not successfully. We present empirical evidences revealed on a sample of 1647 students attending grade 8 of lower intermediate school. Results highlighted that misconception about multiplication of decimal numbers strongly disadvantages girls relative to boys and pose new research questions about mechanisms causing these results.

Introduction.

Italian National standardized assessments, called INVALSI test, are administered every year to all Italian students in different grades from primary to secondary school. In recent years the importance given to these tests has increased and researchers in mathematics education are starting to use the INVALSI results.

Results of national and international surveys have highlighted that in many countries there is a strong gender gap in mathematics in favour of males and Italy is one of the countries in which this gap is more remarkable [START_REF] Mullis | TIMSS 2015 International Results in Mathematics[END_REF]OECD, 2016;[START_REF] Invalsi | Rilevazione nazionale degli apprendimenti 2016-2017[END_REF]. This phenomenon has been deeply studied in mathematics education research. Recent works overcome the idea that the main causes of gender gap are related to biological differences (Hill, Corbett & St Rose, 2010): the main reasons of this gap must be sought considering factors related to the social and cultural context in which the students live, such as the level of gender inequality which is strictly related to the gap in mathematics [START_REF] Guiso | Culture, gender, and math[END_REF]. Furthermore, the beliefs of teachers and parents on students' ability and differences in metacognitive aspects such as self perception, self concept and anxiety related to mathematics are fundamental to explain the reason of the observed gender gap (Fredericks & Eccles, 2002;[START_REF] Cargnelutti | How is anxiety related to math performance in young students? A longitudinal study of Grade 2 to Grade 3 children[END_REF]OECD, 2016;[START_REF] Pajares | Gender differences in mathematics self-efficacy beliefs[END_REF]. Finally, recent research has revealed that factors related to classroom practices, learning strategies and curriculum variables have a strong influence on gender differences in mathematics [START_REF] Leder | Mathematics education: new perspectives on gender[END_REF][START_REF] Giberti | Gender differences and didactic contract: analysis of two INVALSI tasks on powers properties[END_REF]OECD, 2016). Moreover, in previous studies we have shown the influence on gender differences of micro-social factors, strictly related to the milieu habits [START_REF] Brousseau | Le contrat didactique: le milieu[END_REF] and to the interrelation between the student and the teacher. From this perspective we highlighted that misconceptions, as a product of classroom practices, have a stronger impact on females rather than males [START_REF] Bolondi | Highlights on gender gap from Italian standardized assessment in mathematics[END_REF] and our aim is to investigate more in depth this issue.

In this paper we propose the analysis of a specific item, which is part of a wider research program called "Variazioni 2" and financed by INVALSI, aimed at exploring the relationship between variations in item formulation and in its psychometric functionality. Much research in mathematics education has studied the impact of a variation in item formulation with a particular attention to 'word problems'. An updated literature review has been proposed by Daroczy and colleagues (2015). Literature considers the influence of linguistic variations as well as other kinds of minor changes on students' responses and solving strategies (e.g. [START_REF] Nesher | Levels of description in the analysis of addition and subtraction word problems[END_REF][START_REF] De Corte | Influence of number size, problem structure, and response mode on children's solutions of multiplication word problems[END_REF][START_REF] D'amore | Lingua, matematica e didattica[END_REF]. Duval in 1991 defines all these modifications as "redactional variables", and Laborde in 1995 uses this term to include also non-verbal changes, e.g. introduction or modification of pictures. Using a specific statistical strategy, in this paper we analyze data collected in the project "Variazioni 2" to understand if and how a specific variation might affect differently male and female solving strategies to solve a mathematics item. In particular, we analyze an item included into a mathematics achievement tests to explore a specific didactical issue concerning misconceptions in multiplying decimal numbers [START_REF] Fischbein | The role of implicit models in solving verbal problems in multiplication and division[END_REF][START_REF] Sbaragli | Il ruolo delle misconcezioni nella didattica della matematica[END_REF].

Research questions.

The term misconception is widely used in research in education and takes different meanings; in this research, following D'Amore & [START_REF] D'amore | Analisi semantica e didattica dell'idea di "misconcezione[END_REF], we define a misconception as a concept momentarily incorrect, standing by for a more critical and developed cognitive arrangement. A typical misconception (D'Amore [START_REF] D'amore | Analisi semantica e didattica dell'idea di "misconcezione[END_REF][START_REF] Hart | Children's understanding of mathematics: 11-16[END_REF] is due to the premature formation of a conceptual model of multiplication when students operate exclusively with natural numbers. Students learn multiplication with natural numbers and, observing that the product of two numbers is always greater than its factors, they are led to believe that "multiplication always increases". This is true for natural numbers (excluding 0 and 1), but this is not always true for decimal numbers. This misconception, thus, leads students to make mistakes when they operate with decimal numbers, and, for instance, D'Amore and [START_REF] D'amore | Analisi semantica e didattica dell'idea di "misconcezione[END_REF] explain that the answer of many students of different grades to the question "What is the result of 4 × 0.5?" is 8. In order to explore this phenomenon, we compared two versions of the same item (Table 1). The two formulations differ only in factor order. This allows us to understand if the idea that "multiplication always increases" is related to both factors of the multiplication or mostly to one of the two factors. In particular, our hypothesis is that the intuitive model [START_REF] Fischbein | The role of implicit models in solving verbal problems in multiplication and division[END_REF][START_REF] Mulligan | Young children's intuitive models of multiplication and division[END_REF] of multiplication as a repeated sum is easier to apply in the second version (0.5 × 4) and this strategy led students to overcome easier the misconception. This is of course also related to the language used: in Italian we read the multiplication 4 × 0.5 as "4 repeated 0.5 times" while in English and in many other languages it is "4 times 0.5". This only one of several misconception related to the transition between natural numbers and decimals and, in previous studies focused on misconception in comparing decimal numbers [START_REF] Bolondi | Highlights on gender gap from Italian standardized assessment in mathematics[END_REF][START_REF] Giberti | Differenze di genere e misconcezioni nell'operare con le percentuali: evidenze dalle prove INVALSI[END_REF] we observed that females have more difficulties than males. For this reason, we decided to investigate if this misconception and the variation carried out, affects differently boys and girls. Therefore, our research questions are: 1) Does the misconception here analyzed act on boys and girls differently? In other words, does this misconception cause different item functioning in male relative to female subgroup? 2) What is the impact of the inversion of factors into a multiplication (e.g., 4×0.5 in place of 0.5 ×4) on students' answers? What is its impact on item functionality? Does the inversion of factors have different effects on boys and girls?

Participants and measures.

Starting from a mathematics achievement test developed by INVALSI for students attending grade 8 (lower intermediate school), in the project 'Variazioni2', we developed three further mathematics achievement tests, complete in terms of mathematical content and level of difficulty. These four forms (F1, F2, F3 and F4) were administered to a probabilistic sample of 1647 students attending grade 8.

Analytic strategy.

Our four mathematics achievement tests were administered by means of a spiraling process (according to which different forms are administered to different students within each classroom) in order to randomly assign forms, in this way each form was administered to approximately 400 students. When using this design, differences between group-level performance on administered forms are taken as direct indications of the differences in difficulty between forms [START_REF] Kolen | Test equating, scaling, and linking[END_REF] and it is particularly adequate to make comparable answers given by different subgroups of students. In addition, all achievement tests were equated to make estimations directly comparable. Both item and person parameters estimated via the Rasch model [START_REF] Rasch | Probabilistic models for some intelligence and attainment tests[END_REF] were scaled in an empirical range equal to [-4; +4] logits, where 0.00 logit does not mean absence of ability but it is the ability-difficulty level in relation to which students have a probability equal to 0.5 of encountering an item successfully.

In the result section, we provided a summary of psychometric characteristics of the item D9 in F1, F2, F3, and F4. In order to compare, male and female performances, we provided a visual display of the set of observed means for each person factors level (i.e., for boys and girls) across each of the class interval present in the item-trait test-of-fit specifications. Each level is plotted in relation to the Item Characteristic Curve, i.e. the theoretical curve estimated by the Rasch model according to which no factor other than students' intrinsic ability can explain the probability of a correct answer. At the bottom of the table, we reported the distractor plots drawn for males and female, separately, in order to explore their answer behavior in relation to each answer option.

Results.

The table below (Table 1) reports on differences in item functionality in F1, F2, F3, and F4. For each of them, in addition to information about fit (i.e., Weighted MNSQ) always close to 1 and thus good, the distribution of correct and wrong answers provided by students is available in order to provide a first glance on the differences between booklets.

Booklet F1

Booklet F2 Item formulation shows little variation comparing booklets F1 and F2 and it's the same also comparing booklets F3 and F4. As already explained, in both versions the question intent is the same but the inversion of factors into the multiplication might change the influence of the misconception on decimal numbers and the answering strategy activated by students to solve the item might change. Indeed, we observe a lower percentage of correct answers (Table 1) in the first version, included in booklet F1 and F2, than in the second one, included in F3 and F4.

To analyze gender differences, we provided a visual display of the set of observed means for each person factors level (i.e., for boys and girls) across each of the class interval present in the item-trait test-of-fit specifications. Each level is plotted in relation to the Item Characteristic Curve, i.e. the theoretical curve estimated by the Rasch model according to which no factor other than students' intrinsic ability can explain the probability of a correct answer. Distractor plots drawn for males and female, separately, showed differences in answer behavior between boys and girls in relation to each answer option (Table 2). In particular, option B is more attractive for low-ability male students relative to females with the same ability level. Instead, it is evident that females are more attracted by option D, which is strictly related to the misconception, since they give the result of 4×5 instead of 4×0.5. Little differences can be disclosed also in relation to option A, more attractive for low-ability girls and all these evidences are confirmed also by percentage reported in Table 2. Most of these differences are more remarkable in F2 (Table 3), with a clear advantage of males relative to females. Nevertheless, by exploring Distractor Response Curves drawn by gender, response patterns for the set of distractors associated with D9 presented in F1 and D9 presented in F2 are similar. The main difference between them can be disclosed into the distractor plot drawn for males in Table 3: high-ability boys are not attracted by any distractors, while distractor D is attractive also for high-ability females. Both in booklet F3 and F4 (Table 4 and5), the item D9 proposes the inversion of factors, i.e. 0.5×4 in place of 4×0.5. The analysis of answers to the item D9 in booklet F3 confirms an overall advantage of males. Differently from previous cases, higher differences are observable especially at medium-ability level along the latent trait (around 0.0 logit). Nevertheless, differences observed for low ability students are the most result of this analysis: from -1.5 to -0.5 logit all differences are in favor of females. This evidence is strongly confirmed by analyzing F4 data: the probability of a correct answer is 20% higher in female than in male group, from -1.50 to -0.5 along the latent trait. In both cases, distractor analysis shows interesting dissimilarities. The most interesting differences between boys and girls can be observed in F4. In particular, option B is attractive for males and not attractive for females, while option D is more attractive for girls especially at the lower level of the latent trait, confirming the evidences emerged in F1 and F2. Conclusions.

Males Females

In this paper, we compared two versions of the same item with the purpose of analyzing a specific misconception concerning multiplication with decimal numbers. The results showed interesting evidence that will be investigated in further studies from a qualitative point of view. The analysis showed that our new formulation (0.5×4) is simpler (in terms of item difficulty) than the first one (4×0.5) and this is particularly due to the improvement of the performances of girls. This result is coherent to the model of multiplication as a repeated sum proposed by [START_REF] Fischbein | The role of implicit models in solving verbal problems in multiplication and division[END_REF] or by [START_REF] Sbaragli | Il ruolo delle misconcezioni nella didattica della matematica[END_REF]. In particular, our new formulation seemed to affect low-ability girls positively suggesting that they are more prone than boys to the effect of misconception. Moreover, empirical findings showed a sharper difference between boys and girls at the higher ability level, unless boys confirmed their superiority in encountering this task independently on its formulation at the top level of the ability distribution. Furthermore, in both versions, distractor D (i.e., product equal to 20) which is strictly related to the misconception, is always preferred by females. In particular, we observed that the inversion of the two terms of a multiplication has a huge impact on students' answering behavior especially on females: strong gender gap emerges in favor of males in the first version (4×0.5) for students scaling at the upper-tail of the latent trait and in favor of females in the second version (0.5×4), especially at the lower tail of the ability distribution. In the second form the influence of the misconception appears lower and (probably for this reason) the item is easier since students can calculate the product by using the implicit model of multiplication as repeated addition and this particularly helps low-ability girls.

  by gender and Analysis of Variance by Gender for item D9 administered in booklet F1 (top-left) -DIF-plot (top-right) -Distractor Plot of males (bottom-left) and females (bottom-right) Males Females Table 3: Responses by gender and Analysis of Variance by Gender for item D9 administered in booklet F2 (top-left) -DIF-plot (top-right) -Distractor Plot of males (bottom-left) and females (bottom-right)

Table 4 :

 4 Responses by gender and Analysis of Variance by Gender for item D9 administered in booklet F3 (top-left) -DIF-plot (top-right) -Distractor Plot of males (bottom-left) and females (bottom-right) Males Females Table 5: Responses by gender and Analysis of Variance by Gender for item D9 administered in booklet F4 (top-left) -DIF-plot (top-right) -Distractor Plot of males (bottom-left) and females (bottom-right)

Table 1 : Different formulations of the item D9

 1

Table 1 . Item functionality in F1, F2, F3 and F4

 1

	item:19(D9)


Núria Gorgorió
email: nuria.gorgorio@uab.cat

Lluís Albarracín
email: lluis.albarracin@uab.cat

Anu Laine
email: anu.laine@helsinki.fi

Impact of access requirements on the mathematical knowledge of students admitted to Primary Teaching programs: a microcomparative study

Keywords: University access requirements, primary teaching degree, background mathematical knowledge

We present a comparative micro-study analyzing whether the changes made to the admission requirements to the primary teacher training programs of the Universitat Autònoma de Barcelona and the University of Helsinki in the academic year 2017-18 have had an impact on the background mathematical knowledge of those admitted to the programs. We are interested in finding elements that provide information for university admission policies. Our results show that significant changes only took place in Barcelona and these were a consequence of the variations in the academic profile of the students admitted. We conclude that working with a context-bound instrument, while making comparison difficult, directs the researchers' attention to the need for an in-depth theoretical discussion about the mathematical knowledge of candidates to initial teacher education programs and a methodological discussion on how to evaluate this knowledge.

Introduction

Our study is among those that attempt to determine what factors influence the development of mathematical knowledge for teaching during teacher education. According to [START_REF] Blömeke | Assessment of teacher knowledge across countries: a review of the state of research[END_REF], some of these variables are the institutional characteristics of each program, the idiosyncrasies of the national and social context, and the individual characteristics of the students attending the programs. The survey we present here is a small part of an emerging comparative study that aims to contribute to the discussion of university policies regarding access to primary teacher education programs while promoting a theoretical debate about what the desirable mathematical knowledge of students admitted to these programs would be (for more details about the project, see e.g. [START_REF] Albarracín | The mathematical knowledge of students entering primary teaching degrees and its relationship with university admission criteria and requirements[END_REF][START_REF] Gorgorió | Fundamental Mathematical Knowledge: progressing on its definition and concretization[END_REF]. We understand that the admissibility criteria and the requirements for access to educational programs, as institutional particularities, contribute to defining the individual characteristics of their students. We assume that the mathematical knowledge with which students access primary teaching programshenceforth, background mathematical knowledgeinfluences the development of the knowledge needed to teach mathematics during their education.

As educators of primary school teachers, we are interested in finding elements that provide information for university admission policies so that requirements can be established to ensure a minimum of mathematical knowledge among admitted students. Comparative studies between different national realities that evaluate the mathematical knowledge of the candidates in relation to the admission criteria can help to explain the success differences between the different training programs, thus identifying variables that may otherwise remain hidden in national studies and hence remain unquestioned.

Since 2014, at the Universitat Autònoma de Barcelona, Cataloniahenceforth BCNwe have been evaluating the background mathematical knowledge of students admitted to the primary teaching program through a specific test. In the 2016-17 academic year, we decided to also administer the test to the students admitted to the Master of Education degree at the University of Helsinki, Finlandhenceforth HELto study the results comparatively, taking the entrance requirements into account. We then observed that students at HEL performed significantly better than students at BCN.

In this paper, we present a comparative micro-study analyzing whether the changes introduced in the 2017-18 academic year regarding requirements for admission to the primary teacher training programs in Catalonia and at the University of Helsinki have had an impact on the background mathematical knowledge of those admitted to the programs. The changes are of a very different nature and have to do with the idiosyncrasies of the respective university and school systems. For the students admitted to a teaching degree in Catalonia, and in particular to BCN, September 2017 was the first time they had to pass a specific mathematics test. At HEL the change consisted in increasing the proportion of candidates for whom not only the VAKAVA-test and suitability test but also the mathematics and Finnish grades of the matriculation examination were considered for admission.

We exploited quantitative data collected in the academic years of 2016-17 and 2017-2018, i.e. before and after changes took place in the admission criteria both at BCN and HEL. Here we discuss how the changes introduced have impacted students' performance on the test, comparing the performance across years, both within and between universities. We are especially interested in seeing whether the differences occur as result of the changes in the academic profile of the admitted students prior to their University studies.

Admission to a primary teaching degree at BCN and HEL

The recent introduction of the requirement to pass a specific mathematics test for admission to the primary teaching degrees in Catalonia justifies the presence of BCN in the research study. Since the results of the first PISA Study were published in 2000, there has been widespread international recognition of the success of Finland's educational model. Even though this success is due to many factors, the most important aspect is believed to be the quality and competence of its teachers (Salhberg, 2011). For this reason, it seemed logical to include a Finnish program in this study. In addition, since the policies and admission requirements to the primary teaching program in Helsinki are very different from the Spanish ones, we consider it interesting to compare HEL and BCN.

Since 1979, primary teachers in Finland need to have completed a Master of Education degree (300 ECTS). After compulsory education, youngsters can enter the job market directly or instead attend either university-preparatory Upper Secondary School or Vocational School. At Upper secondary school, students can choose to do either intermediate mathematics or advanced mathematics. At the end of Upper Secondary School, students take the Finnish Matriculation Examination, Ylioppilastutkinto in Finnish (henceforth YLIO), which is the only national-level test in Finland. Each examinee is required to complete at least four exam papers, only one of them (mother tongue) being mandatory. The remaining three are chosen by the student and may include one in mathematics, either intermediate or advanced. Graduates of Vocational School may also gain admission to university. In Finland, students interested in becoming primary teachers, after passing the YLIO, must go through a special two-stage application process. The first stage involves taking an exam (the so-called VAKAVA testi) based on a set of education-related readings. Those who perform well on this exam go on to the second stage, a suitability test that assesses their motivation to study and suitability to work as teachers. Universities in Finland agree on the number of new admissions in cooperation with the Ministry of Education. At HEL, fewer than 10% of applicants are accepted annually for the primary teacher education program. In 2016-17, 48 students (40%) were selected based on the suitability test alone and 72 students (60%) were selected based on the suitability test and their grades in the YLIO in 4 subjects -Finnish and 3 optional subjects. In 2017-18, only 24 students (20%) were selected based on the suitability test alone, and 96 students (80%) based on the suitability test and the grades in the YLIO in Finnish and mathematics. Thus, the number of students at HEL who took a mathematics test in the YLIO increased by at least 20 percentage points.

In Spain, primary teachers are required to have completed a Degree in Primary Teaching (240 ECTS). Access to primary teaching degrees can be gained via the Baccalaureate (2 years of noncompulsory education) or through a vocationally-oriented Professional Cycle. Those who study Baccalaureate may choose a track containing mathematicseither mathematics or mathematics for social sciencesbut some tracks do not contain any mathematics subjects. The professional cycles from which students gain access to the primary teaching programs do not have mathematics courses. Those that have completed the baccalaureate have to pass the Pruebas de Acceso a la Universidad (henceforth PAU) to be admitted to any university program. Four of the exam papers of the PAU are universal for all students across Spain, but this is not the case for mathematics, which is an optional subject. Students who have completed a Professional Cycle may also gain access to university. Therefore, applicants to teacher education programs in Spain will not have taken any mathematics courses in post-compulsory education unless they have taken one of the two tracks of the Baccalaureate that include mathematics. It is important to point out that the number of student places in primary teaching degrees in Spain bears no relationship to the actual number of teaching jobs in the market. At BCN, around two thirds of the applicants were admitted for the academic years 2016-17 and 2017-18 1 . In the academic year 2016-17, there was no specific requirement for admission to Primary Teaching programs in Spain outside the universal procedure described above. However, in Catalonia, starting in the 2017-2018 academic year, a specific complementary entrance examinationwhich includes a mathematics testwas implemented.

1 http://universitats.gencat.cat/ca/altres_pagines/informe_i_estadistiques/informes_i_estad_pre/

Data collection

Instrument

The instrument we used to collect our data was the mathematics test that had been designed and used to pilot the specific complementary entrance examination in Catalonia mentioned above that had the administrative purpose of regulating access to primary teaching degrees in Catalan universities. The mathematics test was intended to ensure a minimum mathematical content knowledge of admitted candidates (for a detailed description of how the test was created and validated see [START_REF] Albarracín | The mathematical knowledge of students entering primary teaching degrees and its relationship with university admission criteria and requirements[END_REF]. The test evaluated disciplinary knowledge of a conceptual and procedural nature.

The test had 25 questions, with an open-answer format, related to the prescribed curriculum of Spanish compulsory education -Numbers and Arithmetic, Space and Shape, Relations and Change, Magnitude and Measure, and Statistics and Randomness. The weight given to each content area was related to the actual presence of said contents in the school practice. Thus, there were 7 items on numbers and arithmetic (procedural and conceptual exercises with natural, integer, fraction and decimal numbers), 5 items on magnitude and measure (concepts of length and area, calculation of perimeter and area, and units change, in both decimal and sexagesimal systems), 5 items on relations and change (3 items on proportionality, 1 patterns, and 1 graph interpretation), 4 items on space and shape (2 conceptual on plane geometry, 1 visualization, 1 representation) and 4 items on statistics (1 reading and 1 construction of bar charts, 1 interpretation of a pie chart, all of them procedural, and 1 on conceptual measures of central tendency). Students had 90 minutes to answer the test and no calculators were allowed.

The following are examples of the items in the test:  4-A product is on sale. According to the label, the normal Price is £125. The sale price is £100. What percentage of discount has been applied?  7-Which prime number can be made by subtracting two multiples of 7?  14-What is the surface area of a square with a permiter of 32 cm?  16-Which of the following figures must have equal diagonals? a) square; b) rhombus; c) rectangle; d) trapezium  18-How many cm are in 7.8 km?  24-What number would we add to the list in order for the mean to equal 7? {2, 6, 6, 6, 8, 8, 9} From the beginning we were aware that the test, as it had been defined, was going to be culturally context-bound to the idiosyncrasies of the educational system in which it had arisen. Although the curricula in Finland and Spain are not so different, the emphasis placed on each of the subjects and the way they are taught in class is. We have purposely included some of the items of the questionnaire as examples that will serve to illustrate how the questionnaire was bound to the context. For example, it was already expected that item 7 shown above would harm students' results in HEL, since, although they work on divisibility, the term "prime number" is not as familiar to them. Another example would be item 16 since it includes the idea of a rhombus, which is rarely used in Finlandit is not a category in itself in the classification of quadrilaterals that schoolchildren learn, contrary to what happens in Spanish school mathematics lessons, where the rhombus gives rise to a category in itself. A full discussion of how the test was context-bound can be found in [START_REF] Albarracín | The mathematical knowledge of students entering primary teaching degrees and its relationship with university admission criteria and requirements[END_REF].

Participants and data collection

In HEL, the test was taken by the 116 students starting their primary teaching education program in 2016-17 and 2017-18, at the very beginning of their courses in August. In BCN, the 254 students that initiated their teaching education in 2016-17, and the 276 that did so in 2017-18, took their test in February, at the beginning of the second semester. Neither in HEL nor in BCN had students taken any university class on mathematics or mathematics education at the time of the test.

Students had to answer the test on a worksheet, without using a calculator and in a maximum time of 90 minutes. The answers were marked, assigning 1 point when the answer was completely correct (meanings, calculations, units, etc.) and 0 points if it was not. The students' grades in the exam are part of our data, the other group being the information they gave us about their previous academic trajectory, allowing us to gain information on which (if any) mathematics courses they had taken as part of their pre-university education.

Results

Reliability of the instrument

First, to test the reliability of the data obtained in the test, we calculated the Cronbach's alpha, both overall and for each campus, as an internal consistency estimate of the reliability of the test scores.

The results that we obtained (α BCN•16•17 = 0.8157; α HEL•16•17 = 0.8106; α BCN•17•18 = 0.7963; α HEL•17•18 = 0.7579; α Overall = 0.7985), in all cases higher than the reference value of 0.7, were a suitable indicator of the internal consistency of the instrument, meaning that the different questions of the test evaluate the mathematical knowledge with which students arrive at the teacher education programs, in a coherent way and from a disciplinary perspective. In this way, we consider that the instrument sufficiently and consistently approaches our idea of background mathematical knowledge. As can be seen in Table 2 for 2016-17, students at HEL obtained significantly better results than students at BCN. It also shows that at HEL there was no significant change in performance between 2016-17 and 2017-18. However, results at BCN were significantly better in 2017-18, to the extent that such results cannot be statistically distinguished from the results in HEL of the same year.

Differences in performance

Impact on the academic profile of students admitted to BCN

We note that there are only significant differences between the results before and after the implementation of a new requirement at BCN, this being the only change affecting entry conditions at BCN. In [START_REF] Albarracín | The mathematical knowledge of students entering primary teaching degrees and its relationship with university admission criteria and requirements[END_REF] we had found that the academic profile of students prior to entering university generated different results. Therefore, it is interesting to explore what aspects have changed in the profile of admitted studentswe have no information about those who applied for admission but were not admitted.

We labelled BCN students who had only taken the compulsory mathematics BCN•COM and those who had followed the Baccalaureate track containing mathematics BCN•BAC. Table 3 shows the proportion of students and the mean of the scores obtained in the test by the students in each subgroup, as well as the standard deviations. In both the 2016-17 and 2017-18 academic years, students who had studied mathematics during Baccalaureate obtained better test results than those who had not. In addition, the 2017-18 academic year increased the number of students in the first group by 10 percentage points. In the 2017-18 academic year, students who had studied mathematics during Baccalaureate obtained better results than the same group in the previous academic year, but by a few thousandths we cannot say that this difference is significant. However, the difference is significant between 2016-17 and 2017-18 for those who had not studied mathematics during their post-compulsory education.

Discussion

In our contribution we analyzed the impact of changing the criteria and requirements for admission to the teacher education programs of the Universitat Autònoma de Barcelona and the University of Helsinki. In both cases, the change implies a greater demand in relation to mathematics. However, we see that while in Barcelona the impact is significant, in Helsinki it is not so.

The results from BCN show that the introduction of the mandatory requirement to pass a mathematics test in order to gain access to the teachers' education program has generated a change in the academic profile of admitted students and their mathematical background knowledge has improved significantly compared to the previous academic year. We found that the number of students who continued to study mathematics after completing their compulsory schooling has increased by 10 percentage points. Our knowledge of the Catalan university system allows us to state that the only variable that could have generated this change in the academic profile of the students is the introduction of this new requisite. In addition, those who dropped out of mathematics at the end of compulsory schooling have also proven to have a better background mathematical knowledge than those in the same group the previous year. From informal conversations with the students, we dare to say that they have prepared for the new specific mathematics test that was part of the new entrance requirements.

However, we did not observe any significant changes in Helsinki. We believe that this is essentially due to the fact that the instrument used to collect our data was bound to the Catalan context. Although access to the primary teaching program in Helsinki had increased its mathematical demands, the terminology and formalism of the test that we used are still elements that have little weight on the education of Finnish students. In Hel, the new entrance requirement implies that a greater number of admitted students have studied secondary mathematics. However, the mathematics of intermediate or advanced courses of upper-secondary in Finland are of little help to answer test questions linked to an approach that is characteristic of another national context.

We find it interesting to compare data from two cultural realities as different as the university systems of Finland and Catalonia. These results provide us with ideas that a study restricted to only one country would not generate, and these ideas can contribute to decision making. In particular, the changes introduced in Helsinki in the 2017-18 academic year are a consequence of the analysis of the results obtained by its students in the 2016-17 academic year. In Catalonia, our study can help to support the changes introduced that are beginning to be questioned by the media. We plan to incorporate data from other European universities into our study in the near future.

We are aware that working with a context-bound data collection instrument generates dilemmas, such as those suggested by [START_REF] Clarke | The validity-comparability compromise in cross-cultural studies in mathematics education[END_REF], that appear when an international comparative study is carried out. In spite of this, researchers working together who are outsiders with respect to each other's data, cultural contexts and national realities leads to richer and more nuanced interpretations. In addition, working with a context-bound instrument, while it makes comparison difficult, directs the researchers' attention to the need for an in-depth theoretical discussion about what the desirable mathematical knowledge would be for candidates to initial teacher education programs. An agreement on the theoretical level on what basic knowledge an entrance examination should evaluate would help to agree on the characteristics of the examination on the methodological level, to avoid it being strictly tied to a context.

Table 1 : Centrality and distribution of marks A

 1 Table1presents the basic statistics that describe the centrality and distribution of the scores in the test by the students in HEL and BCN in 2016-17 and 2017-18 academic years. series of mean comparison t-tests allows us to establish the significance of differences (Table2)

	Variable	HEL•16•17	HEL•17•18	BCN•16•17 BCN•17•18
	N	116	116	254	276
	Mean	17.233	17.595	15.996	17.750
	SE Mean	0.405	0.368	0.302	0.264
	StDev	4.363	3.963	4.812	4.389
	Minimum	7.000	6.000	3.000	4.000
	Q1	14.000	15.000	12.000	15.000

Table 2 : T-tests comparison of score means across universities

 2

Table 3 : Proportion of students that had taken mathematics after compulsory education, and the mean scores of each subgroup, with standard deviations

 3 Table 4 shows the results of the T-tests comparing the mean scores of the subgroups.

		Percentage of students	Mean score (max.	Standard
		in each subgroup	25)	deviation
	BCN•COM•16•17	42.52%	13.611	4.488
	BCN•BAC•16•17	57.48%	17.760	4.263

Table 4 : Results of the T-tests comparing the mean scores of the subgroups depending on whether they had taken mathematics after compulsory school

 4

Acknowledgment

The research presented has been developed under the auspices of the research grants EDU2013-4683-R and EDU2017-84427-R -Spanish Ministry of Science, Innovation and Universities.


Francis Mairet

Olivier Bernard
email: olivier.bernard@inria.fr

Twelve quick tips for designing sound dynamical models for bioprocesses

Twelve quick tips for designing sound dynamical models

for bioprocesses

Since a model should be tailored to the objectives which will depend on applications and environment, a universal model representing any possible situation is probably not the best option.

Here are twelve tips to develop your own bioprocess model. For more details on bioprocess modelling, the readers could refer to [START_REF] Dochain | Automatic control of bioprocesses[END_REF]. More tips concerning computational aspects can be found in [START_REF] Sandve | Ten simple rules for reproducible computational research[END_REF][START_REF] Osborne | Ten simple rules for effective computational research[END_REF].

1/15

Tip 1: Define your objective and the application context Years of high school learning about how to set-up mechanistic models based on the fundamental F = m.a relationship of mechanics, or on the Ohm law have corrupted our minds.

It took centuries to identify the corpus of laws supporting today's physical models. Fig 1 recalls that, previously, there used to be some "less accurate" predictive models that have been forgotten. At present, models in these fields, even if empirical, are excellent approximations and -at least for those we studied at school-always ended-up in rather simple, often linear and mathematically tractable models. The complexity of biological systems requires a more open viewpoint, where different models of the same process can be useful and complementary. Therefore, before writing equations, one must first clearly define the model objective. The model can be designed for numerous reasons, among which prediction of future evolution, understanding of the process behaviour, estimation of unmeasured variables or fluxes, operator training, detection and diagnosis of failures, optimization and control. Tip 2: Adapt your modelling framework with your objective, your knowledge and your data set When developing a model, it is crucial to keep in mind the objectives of the model and the framework for its application. A model targeting the understanding of some metabolic processes inherently requires the user to embark on the details of the cell metabolism [START_REF] Baroukh | Mathematical modeling of unicellular microalgae and cyanobacteria metabolism for biofuel production[END_REF][START_REF] Baroukh | Dynamic metabolic modeling of heterotrophic and mixotrophic microalgal growth on fermentative wastes[END_REF].

Predicting the impact of meteorology on outdoor microalgal processes means that light and temperature must be included somewhere in the model. A model for on-line control can be more straightforward (often because it will benefit from on-line information on process state). So, keeping in mind the model objective, one has to choose which variables to include, but also the type of model: deterministic versus stochastic, homogeneous versus heterogeneous (in terms of space or phenotype). The available data set or data that can be provided by the experimental set-up will also constrain the choice of model complexity. Parameters should be calibrated at some point, or at least reasonably determined from the experimental information. Model complexity can first be measured by the number of state variables (variables with dynamics) together with the number of parameters and stay compatible with the objectives and data. Tip 3: Take care with dimensions, intensive and extensive properties This tip seems very basic, but, in our opinion, it is worth emphasising. The dimension of the model equation should be checked. Particular care should be taken between intensive and extensive variables [START_REF] Fredrickson | Formulation of structured growth models[END_REF]. This is particularly true when dealing with a metabolic model.

A metabolite concentration could be expressed per unit of culture volume or intracellular volume. The concentration dynamics should then include the dilution by the reactor feeding 3/15 or by cellular growth, respectively. Moreover, the kinetics of intracellular reactions should depend on intracellular concentrations, not culture concentrations. In several studies, it remains unclear. Tip 4: Do not assume gas concentrations equilibrate with atmosphere Assuming gas concentrations equilibrate with the atmosphere is a common mistake. If we measure the dissolved CO 2 concentration in a glass of water in equilibrium with the atmosphere, it will be proportional to P CO 2 , the CO 2 partial pressure at the interface (i.e. in the air): [CO 2 ] = K h P CO 2 where K h is Henry's constant at the considered temperature and salinity. At steady state, there is no more gas exchange between the atmosphere and liquid phase.

If algae are developing in the glass, the CO 2 concentration will be lower, because the algae permanently consume it. As a consequence, there is a permanent flux of CO 2 from air to water, with a flow rate

Q CO 2 = K L a(CO 2 -K h P CO 2 )
which will balance the consumption of CO 2 by the algae. Now the concentration of CO 2 is lower than K h P CO 2 , its natural equilibrium value without algae.

Tip 5: Check the mathematical soundness of your model

A mathematical analysis of your model may help to detect potential errors, limitations and drawbacks in model design, and to better apprehend the process. Whenever possible, one should check mass conservations, check the boundedness of the variables (in particular their positivity), and study the asymptotic behaviour of the model. This last point could be, for some models, particularly challenging. It is essential to keep in mind that nonlinear dynamical models are complex mathematical objects with potentially weird behaviours, 4/15 including limit cycles, chaos or abrupt change in behaviours after bifurcation when one of the model parameters has been slightly modified [START_REF] Strogatz | Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering[END_REF]. Mathematicians spend months trying to understand and prove the behaviour of systems of low dimension, e.g. with "only" three state variables. The mathematical complexity is breath-taking when considering standard bioprocess models. Often, the properties of these models are hardly suspected, and Pandora's box stays closed. Even the number of equilibria that can be produced is rarely discussed. Adding new features or including more realism into a model extends the risk of unexpected model behaviours.

The objective is to determine whether the trajectories of your system converge towards an equilibrium (a global equilibrium, or different equilibria depending on the initial conditions), if they present sustained oscillations (limit cycle) or even show a chaotic behaviour. These properties should be in line with the behaviour of your bioprocess, otherwise the model should be revised.

Tip 6: Be aware of structural identifiability

Most of the parameters in physical modelling have a clear meaning and can be directly measured on the process. Also, physical models are often linear. The theory of linear systems and their identification has received much attention, indirect identification of a tenth of parameters can be accurately carried out by modern algorithms [START_REF] Walter | Identification of parametric models from experimental data[END_REF][START_REF] Ljung | System identification[END_REF]. For the biological systems, which are in turn nonlinear and described by rough approximations, more modesty is required.

Theoretical identifiability of the parameters is a complex mathematical property [START_REF] Walter | Identifiability of parametric models[END_REF], which is often characterized by cryptic (but accurate) mathematical formulations. In a nutshell, this theoretical mathematical property states that a parameter value can be uniquely determined by (nonlinear) combinations of measurements and their derivatives (with respect to time) at any order. More simply, a unique set of parameters can produce a given model output. With non-linear models, it is possible that two sets of parameters can produce exactly the same results. To illustrate the non-identifiability pathology, we present 5/15 in Table 1 two illustrative astonishing examples for trivial models.

The first example is unfortunately not so rare. It consists in representing an inhibition kinetics (from substrate S) with a product of Monod and a hyperbolic inhibition term. A numerical example is given in Table 1 (Example 1), where two parameter sets produce exactly the same values. Parameters here are only locally structurally identifiable.

The second example in Table 1 uses a trivial logistic equation (x is the biomass) modified to deal with mortality rate (which is obviously a very bad idea). Here, an infinity of parameters provide the same biomass dynamics, they are structurally not identifiable.

These two examples also demonstrate that it is useless to attribute a biological meaning to a non-identifiable parameter. In the first case, what was, in turn, the inhibition constant: K i or K s ? In the second example, is K the carrying capacity of the medium? Perhaps more problematic when using an automatic algorithm for parameter identification, non-identifiable parameters will kill any approach. Especially if it is a global approach, any optimisation algorithm will oscillate between several of the possible solutions, or average them, and often will never converge.

In general, assessing identifiability for complex dynamical models is very challenging. This is a reason why modellers must refrain from embedding too many processes into a model, and privilege lower complexity models when only a limited set of measurements is available for validation.

Tip 7: Double check numerical implementation

If your model has been implemented only once, then it probably contains at least three mistakes. We know this is not true for you, but it is for most of the people. So if the model was right, after a rapid change in one of the equations for testing the effect of one factor, it would become wrong because eventually the test is not removed. There are strict coding rules and use of validation tests [START_REF] Duvall | Continuous integration: improving software quality and reducing risk[END_REF], but they are rarely respected for model development because the model implementation is generally not carried out by computer scientists. Also, the way models are implemented can highly differ, and some computer languages may be 6/15

K i = 1, K s = 2. μ = 1, K i = 2, K s = 1.
max. growth rate inhibition constant affinity constant

φ(S) = 2 S (S+1)(S+2) General case μ, K i , K s . μ K i Ks , K s , K i .
max. growth rate inhibition constant affinity constant

φ(S) = μ S S+Ks K i S+K i Example 2: logistic growth with mortality ẋ = μ(1 -x K )x -Rx Numerical values μ = 2, K = 1, R = 1. μ = 3, K = 1.5, R = 2.
max. growth rate carrying capacity mortality rate

ẋ = (1 -2x)x General case μ, K, R. μ + θ, K μ+θ μ , R + θ.
max. growth rate carrying capacity mortality rate

ẋ = μ(1 -x K )x -Rx
In Example 1, two different parameter sets produce the same value of the function φ(S). In Example 2, an infinite number of parameter sets can produce the same dynamics ẋ for an arbitrary value of θ. The parameters meaning (as often claimed) does then not make any sense.

7/15 more difficult to cross check. Excel ® is an excellent tool for displaying data and for simple computations, but it is not an appropriate tool for simulating complex models since it is almost impossible to cross-check implementation. Some graphical languages also have these drawbacks when a connection to a wrong node can corrupt the result while being almost impossible to detect.

One way of reducing the risk of error is a double implementation, with two different computer programmers and two different languages. This has been the case for the models used in wastewater treatment, ADM1 for anaerobic digestion [START_REF] Batstone | The iwa anaerobic digestion model no.1 (adm1)[END_REF] and ASM1 for activated sludge [START_REF] Henze | Activated sludge models ASM1[END_REF]. The first comparison between different implementations revealed to be quite quaint. Also, simple case studies must help to check simple theoretical properties (positivity of variables, mass conservation, etc...) that must be respected.

Tip 8: Pay attention to practical identifiability

The cost criterion to be optimised (typically the sum of squared errors) is generally nonconvex, and many local minima perturb parameter identification. In practice, it is often not possible to get an accurate estimate of parameters from the data sets. The most efficient algorithms are generally limited to three parameters to be determined per measured quantity (assuming a reasonable sampling over time). The weird consequence is that fitting a model to a set of data is generally possible, but that does not mean that the estimated parameters are reasonable. Whenever a parameter has a clear meaning, the validity of the identified value must always be checked, and bounds can be added during the identification process. Multiple algorithm initialisations are also strongly recommended. Collecting informative data is also key for practical identifiability, which means data corresponding to high sensitivities of the model outputs with respect to parameter variations (cf. Fisher information matrix [START_REF] Walter | Identification of parametric models from experimental data[END_REF]). As a matter of illustration, it is not possible to estimate a parameter related to growth inhibition if substrate concentration is always too low to trigger inhibition.

Finally, a literature review is an essential resource for parameter values, in particular for algorithm initialisation. Nonetheless, exotic chimaera can appear when picking up 8/15 parameters from different papers! Tip 9: apply the "divide and conquer" strategy to identify your parameters

Do not try to get all your parameters at once, through a never converging optimization algorithm and rather identify subsets of parameters. In many cases, after simple algebraic manipulations some parts of the model can lead to relationships between some measured quantities and eventually provide some combinations of the parameters. For example, the pseudo-stoichiometry can often be identified independently of the reaction rates after some straightforward transformations [START_REF] Bernard | On the estimation of the pseudo-stoichiometric matrix for macroscopic mass balance modelling of biotechnological processes[END_REF]. Some working modes do considerably simplify the model, and are often an opportunity to extract such relationships. For example, during a phase when nutrients are nonlimiting, the Michaelis-Menten kinetics can be replaced by constants. Similarly, if different equilibria can be observed for various inputs, they would probably lead to very interesting relationships between some of the model parameters [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF].

Tip 10: determine parameter and model uncertainties

Assessing measurement uncertainty propagation is of utmost importance to assess model accuracy. This first means that the experimental data must be associated to the variance of their measurement error. There are different strategies to compute not only the parameter values but also their confidence intervals. This is straightforward when parameters are deduced from linear relationship, but is can also be estimated in a more complex case thanks to the covariance matrix of parametric errors [START_REF] Walter | Identification of parametric models from experimental data[END_REF]. The strong scientific added value is that the simulation scheme will predict not only outputs but also the confidence intervals derived from the covariance matrix of prediction errors, or from Monte Carlo simulations.

9/15

Tip 11: Validate the model with data not used for identification When observing the vast diversity in bioprocess models, only a few of them have been appropriately validated. First, because it is not possible to validate a model, a model can only be discarded when it is not compliant with experimental records [START_REF] Vatcheva | Experiment selection for the discrimination of semi-quantitative models of dynamical systems[END_REF]. However, assuming a relaxed use of the "validation" term, it would mean that the model has been proven accurate for a large variety of cases. In particular for cases significantly different from the learning data set (data that has been used for the calibration). This ideal situation is very difficult to meet in practice, and most of the time the validation datasets only differ by some initial conditions, or by a single different forcing variable. If the model has enough parameters, it can probably fit a calibration dataset nicely with only a few points. However, it will exhibit abysmal performances for cross-validation. For larger calibration data sets, the fit will probably less successfully highlight the quality of the model, but prediction capacity might be highly enhanced. The plot will not look that nice, but the model will definitely be more powerful and relevant.

Claiming that the model is valid is, therefore, an act of faith, and a very weak scientific assertion. As running experiments takes time and is money consuming, the number of experiments is by essence limited. As consequence, it becomes clear that the conditions for which the model has been validated must be clearly stated. Knowing the "model validation domain" will in itself be precious for future model use. Also, providing data sets for which the model did not do its job is intrinsically useful, although rarely done.

Often, the question is instead to choose the best model among a few candidates. A more complex model, with more parameters, will mechanically better fit the data. However, that does not mean it is more correct, it just means it is more flexible. The Akaike criterion [START_REF] Burnham | Multimodel inference: understanding AIC and BIC in model selection[END_REF] is a good option to compare the performance of two models of different levels of complexity.

However, the only real criterion to assess the predictive power of a model, and therefore to compare model performances is cross-validation, assessing the model with data which were not used for calibration (and data whose dynamics are significantly different from 10/15 the calibration data set). Additionally, the candidate models can even be used to find the experimental conditions that will allow to differentiate them better [START_REF] Vatcheva | Experiment selection for the discrimination of semi-quantitative models of dynamical systems[END_REF].

Finally, models can include the effects of different factors which often have been studied

separately. The models then gather these effects classically by multiplying the different terms or using Liebig's law of minimum. Validation experiments could be the last chance to test possible interactions between these factors and find the best way to combine their effects in the model. More generally, what makes the success and the efficiency of a model, is not limited to the biology it embeds and to the realism of its predictions. A model is inexorably associated with a set of tools to calibrate it, estimate which are the most sensitive parameters, optimise a criterion, determine the input which maximizes productivity etc... The associated toolbox to make the model applicable and efficient is probably at least as necessary as the model itself. Great models can have complex structures or behaviours, which eventually make their use more tricky. For example, the outstanding Geider model [START_REF] Geider | A dynamic regulatory model of phytoplanktonic acclimation to light, nutrients, and temperature[END_REF] is in turn rather challenging to calibrate, and specific methods dedicated to its calibration are needed [START_REF] Smith | Quantitative comparison of photoacclimation models for marine phytoplankton[END_REF].

Even simpler models, such as the Hinshelwood model [START_REF] Hinshelwood | Chemical kinetics of the bacterial cell[END_REF] for temperature, advantageously predicts a mortality rate [START_REF] Serra-Maia | Influence of temperature on Chlorella vulgaris growth and mortality rates in a photobioreactor[END_REF], but calibrating this model often turns into a nightmare [START_REF] Grimaud | Modeling the temperature effect on the specific growth rate of phytoplankton: a review[END_REF].

Keeping two different modelling approaches can significantly help in this case, by using the toolbox of one of the models to manage the other one. Typically, using a temperature response model from [START_REF] Bernard | Validation of a simple model accounting for light and temperature effect on microalgae growth[END_REF] as a gauging device makes the calibration of Hinshelwood's model 11/15 much less painful. Providing all these kinds of information on your model should promote its adoption by the community.

Conclusion

Modelling in biology is a question of choices and trade-offs. The striking difference between two different modellers is often the choice in model complexity. Extensive tests, using cross-validation datasets or based on Akaike criteria may reveal that one model has a better prediction capability than the other, but in other circumstances, it might be the opposite. Our culture has contributed to hatch the illusion of a unique and universal model behind nature. However, even if this idea were right, we are far from having discovered it.

Also, always trying to run after such universal representation of nature, inexorably leads to models whose complexities do not match the available measurements and our capability to validate the model. So, why should we keep a unique model? Why not use a series of models of increasing complexity? Surrogate models consist of a simplified version of a simulator, which is easier to handle mathematically, resulting in more straightforward use for optimisation or control. The surrogate model can be derived and calibrated from the most complex model, but the opposite is also true. A simplified model, with limited accuracy, can provide bounds for a more detailed model. Also, a complicated model can be simplified into different sub-models depending on the environment and the limiting factor (nutrients, light or temperature). Working with a set of coherent models should not necessarily increase difficulty, it creates a consistent framework that can prove to be very useful for different purposes, from model calibration and process optimisation, up to advanced control.

Fig. 1 .

 1 Fig.1. Medieval theory of the canon ball trajectory, from Walther Hermann Ryff (1547)[4]. The canon ball trajectory was an assemblage of circular arcs and segments. Models in physics are now excellent approximations, but they have sometimes been improved during century-long periods. In biology, we are still at the dawn of model development.

Table 1 .

 1 Analysis of two simple examples with identifiability issues.

		Parameter	Parameter	False claim	Function
		set #1	set #2	parameter meaning
	Example 1: substrate uptake with inhibition		φ(S) = μ S S+Ks	K i S+K i
	Numerical	μ = 2,
	values

  Tip 12: Share codes, tips, tools, and model limitations More and more journals require this, and it is to be welcomed. Providing your model -with all the files necessary to reproduce your simulations (including parameter values, initial conditions etc...) -will favour its dissemination within the scientific community. Your model would thus be further validated with new data sets. Additionally, it promotes error checking, helps the reader if some model details in the manuscript are unclear, and removes any suspicion of fraud.

Acknowledgements

The authors are grateful to Jacob J. Lamb and J. Ras for their help with improving the English. O. Bernard is thankful to ENERSENSE for welcoming him during his sabbatical.

12/15


Kirsti Hemmi
email: kirsti.hemmi@edu.uu.se

Madis Lepik
email: mlepik@tlu.ee

Lars Madej
email: lars.madej@edu.uu.se

Kajsa Bråting
email: kajsa.brating@edu.uu.se

Joakim Smedlund
email: joakim.smedlund@abo.fi

la diffusion de documents scientifiques

Keywords: early algebra, comparative analysis, mathematics curriculum, primary education

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction to early algebra in Estonia, Finland and

Sweden -some distinctive features identified in textbooks for Grades 1-3

Introduction to early algebra in Estonia, Finland and Swedensome distinctive features identified in textbooks for Grades 1-3

Background

Early introduction to algebra has been in focus in recent research, and several researchers point out that it is beneficial to start working with important ideas within algebra already in primary school (e.g. [START_REF] Blanton | The development of children's algebraic thinking: The impact of a comprehensive early algebra intervention in third grade[END_REF]. For example, an intervention study by [START_REF] Blanton | The development of children's algebraic thinking: The impact of a comprehensive early algebra intervention in third grade[END_REF] shows that students can improve their understanding of so-called big ideas of algebra with an appropriate teaching approach and relevant material. International comparisons show that there are great differences between approaches to the teaching and learning of algebra in different countries (e.g. [START_REF] Cai | The development of students' algebraic thinking in earlier grades[END_REF]Leung, Park, Holton & Clarke, 2014). These comparisons have mainly concerned countries with significantly different cultures and school systems. In contrast, the present study investigates the early introduction to algebra in the three neighbouring countries of Estonia, Finland and Sweden, where the school systems are quite similar. For example, all three have nine years of compulsory school with no tracking. Moreover, national steering in the three countries is relatively weak, with no approval process for curriculum materials that are commercially produced. We deem that the similarity of the school systems facilitates the identification of specific features concerning the views on early progression in algebra in these countries. In international evaluations, there are differences between the countries in students' learning outcomes, making it interesting to discuss different approaches to early algebra in relation to this variation. While Estonian and Finnish students' results in international comparisons have been relatively good, algebra has not been the strongest area in Finland (Yang Hansen et al., 2011). In Sweden, students' results in algebra have been low since the 1960s regardless of the variation in other mathematical topics (Hemmi et al., 2018).

Curriculum materials are produced within certain educational traditions, and may therefore be shaped by national perspectives on the specific school subjects (cf. [START_REF] Andrews | The curricular importance of mathematics: A comparison of English and Hungarian teachers' espoused beliefs[END_REF]. In all three countries, mathematics textbooks are an important part of classroom work. Therefore, in this paper, we exemplify and discuss typical characteristics of the approaches to early algebra in Estonia, Finland and Sweden as identified in our textbook analysis concerning Grades 1-3 (ages 7-9 years).

We have also included textbooks produced by Finnish Swedish1 authors in the analysis, as there are differences between the two language groups in Finland when it comes to students' learning outcomes, as well as teachers' relation to mathematics textbooks (e.g. Pehkonen, Hemmi, Krzywacki & Laine, 2018). We acknowledge that the textbooks can be used by teachers in various ways when designing and enacting mathematics lessons (e.g. [START_REF] Lepik | Using textbooks in the mathematics classroomthe teachers's view[END_REF]Röj-Lindberg, Partanen & Hemmi, 2017;[START_REF] Kilhamn | When does a variable vary? Identifying mathematical content knowledge for teaching variables[END_REF]. However, we deem these kinds of studies important as we learn how textbook authors interpret the introduction of early algebra in different culturaleducational contexts. The following research question guides the study: What are the specific characteristics of early algebra in textbooks series produced in four different contexts?

Relevant research and theories

Few comparative studies address the progression of algebra in different countries' curricula. [START_REF] Cai | The development of students' algebraic thinking in earlier grades[END_REF] study how algebraic concepts are developed and represented in the curricula of the US (Investigations curriculum), China, Singapore, South Korea and Russia (Davydov curriculum). The study by [START_REF] Cai | The development of students' algebraic thinking in earlier grades[END_REF] shows that the three Asian countries build informal equation solving on the use of inverse operations and the related doingundoing. In the Russian Davydov curriculum for Grades 1-3, children develop algebraic thinking by exploring and comparing quantities before the study of arithmetic and, in contrast to the US Investigations curriculum, letters are used from the very beginning [START_REF] Cai | The development of students' algebraic thinking in earlier grades[END_REF]. Both the Russian and the US curricula address real-life problems, but in fundamentally different manners. In the former, they are carefully planned and sequenced to help students develop a theoretical understanding of mathematical concepts and an ability to analyse problem situations [START_REF] Cai | The development of students' algebraic thinking in earlier grades[END_REF]. Rather than proceeding inductively from the concrete to the abstract as Investigations does, the Davydov curriculum develops students' ability to see the same abstract relationships across different concrete contexts. Investigations engages students in mathematical problems embedded in authentic contexts and these applied problem-solving activities require US students to explore problems in depth, construct their own strategies and approaches utilizing a variety of tools (e.g., manipulatives, computers, calculators), and communicate their mathematical reasoning through drawing, writing, and talking [START_REF] Cai | The development of students' algebraic thinking in earlier grades[END_REF].

Based on recent research on early algebra and the language of learning progression, [START_REF] Blanton | The development of children's algebraic thinking: The impact of a comprehensive early algebra intervention in third grade[END_REF] have identified five so-called big ideas connected to algebraic thinking. These big ideas consist of: 1) equivalence, expressions, equations and inequalities; 2) generalized arithmetic; 3) functional thinking; 4) proportional reasoning; and 5) variables. In a recent study, we used the big ideas as the basis for an analytical tool in order to analyse the algebraic content in the current Swedish curriculum for mathematics as well as two Swedish textbook series for Grades 1-6 [START_REF] Bråting | Development of algebraic thinking: opportunities offered by the Swedish curriculum and elementary mathematics textbooks[END_REF]. In the study reported in this paper, we look more in-depth at Grades 1-3 and the first two big ideas. The firstequivalence, expressions, equations and inequalitiesincludes relational understanding of the equal sign; representing and reasoning with expressions, equations, and inequalities; as well as relationships between and among generalized quantities. The second big ideageneralized arithmeticinvolves reasoning about structures of arithmetic expressions (rather than their computational value) as well as generalizations of arithmetical relationships, which include fundamental properties of numbers and operations (e.g., the commutative property of addition) and inverse properties of operations [START_REF] Blanton | The development of children's algebraic thinking: The impact of a comprehensive early algebra intervention in third grade[END_REF].

Method

We chose the textbooks to be analysed based on our knowledge of their popularity in each region.

In order to examine the variation within different contexts, we chose two textbook series each in Estonia (Matemaatika by Koolibri and Matemaatika by Avita), Finnish Finland (hereafter FinFin) (Kymppi and Tuhattaituri), and Sweden (Eldorado and Matte Safari). Only one series has been produced in the Finnish Swedish context (hereafter FinSwe), Lyckotal. In the data analysis, we identified the sections connected to the big ideas of equivalence, expressions, equations, and inequalities as well as generalized arithmetic. The insider(s) in each region conducted the analysis according to our collective interpretation of the categories based on our thorough discussions, and the analyses were then discussed by the whole research team. The analysis was qualitative, identifying the character of the texts and their order/place of appearance throughout the three grades in order to understand the progression of this big idea within various topics. The only quantification that took place in the analysis was connected to the identification of possible dominating topics. For example, if a certain kind of task occupied a great number of pages in a textbook or occurred regularly throughout the grades, it was marked in the analysis protocol. While it is not possible to report the results of the entire study within the scope of this paper, in the following we will describe, exemplify, and discuss the most striking differences and similarities displayed in the textbooks from the four regions.

The results with illustrative examples Estonia

Addition and subtraction are introduced simultaneously in Grade 1, with subtraction introduced as the inverse of addition. In Grade 2, multiplication and division using whole numbers are introduced in a similar manner. Students are to find missing values using the inverse property of the operations (Figure 1). They should learn the names of the components and results of the four arithmetic operations in order to be able to reason about informal equation solving using the inverse property of the operation at a more general level. In addition, the trial and error strategy is introduced for searching the value of an unknown. At this stage, various symbols (box, blank, letter) are widely used in the textbooks to represent a variable or an unknown. Children are expected to learn to use letters to express the relationships between quantities and determine the correct order of operations in expressions (parentheses, multiplication/division and addition/subtraction) from Grade 2. The notions of equivalence and inequality of two expressions (incl. letter expressions) are introduced and practiced using the formal signs (=, <, >) from the beginning. Rather than seeing equations as simply objects to manipulate, in Estonian primary textbooks from Grade 2 the equations describe relationships between varying quantities that arise from contextualized situations. Children are required to schematically represent the internal quantitative relationships in word problems and to write expressions of these relationships (Figure 2). This process is sequenced by the authors so that the same abstract relationship appears in different concrete contexts. Textbooks also systematically guide students in composing their own word problems for given expressions.

Finland

In the FinFin textbooks, equalities and inequalities are presented with formal signs (>, <, =) in parallel from the very beginning of Grade 1 and are used consistently throughout the three grades. First, only numbers are compared, and later during Grade 1, expressions are compared. This approach is different from the FinSwe style, in which the equal sign is introduced in Grade 1 while the signs for inequalities are only briefly introduced and practiced. Similar to the Estonian textbooks, one of the FinFin textbooks (Tuhattaituri) introduces the inverse property of addition and subtraction and guides students in using it in informal equation solving during Grade 1. The same is done with multiplication and division in Grade 2. This is not the case in the FinSwe textbooks, however, where equations start to appear at the beginning of Grade 1 in connection with the partition of specific numbers, for example with the number 5 in Figure 3.

The inverse property of addition and subtraction is introduced very briefly in the FinSwe textbooks during the first school year using so-called number families, whereby the student is tasked with writing out different expressions with the numbers 10, 3 and 7 in the shape of a triangle with + and signs included. Hence, the students are expected to see how the members of this "family" are connected to each other via the operations. In the FinFin context, the priority rules are presented in Grade 2 and students learn to systematically calculate the value of expressions with two operations, stepwise marking every new phase below the previous expression. In Grade 3, students are to handle expressions with three operations in a similar matter, as in Figure 4a, where the first line includes two divisions and one addition. For Grade 3 the FinSwe textbooks state that the multiplicative operation is done before the additive, but all calculations are done in a straight line, i.e., not each below the previous as in the Finnish textbook Kymppi, shown in Figure 4a. The commutative properties of operations are presented during Grade 1 (addition) and Grade 2 (multiplication) in the FinFin textbooks. The formal names of the operations are used from the beginning, and in Grade 3 students are to choose the correct expressions for sentences, for instance "Subtract from the quotient of 18 and 2 the product of 2 and 3." A special feature of the Finnish textbook is its repeated exercises from the very beginning in solving systems of equations in an informal manner (see Figure 4b).

Sweden

What characterizes the Swedish textbook series, especially Matte Safari, is the emphasis on open number sentences (see Figure 5). Inverse expressions occur in the context of number families, but the connection between addition and subtraction is barely visible in practice. As in the FinSwe textbooks, no explicit connections are used to guide students in applying the inverse properties in equation solving. We found only one example in a single textbook (Eldorado) in which students were to check their solutions using the inverse property. Both textbook series present the commutative property of addition and state that subtraction is not commutative. Multiplication is introduced in Grade 2, and the commutative property of multiplication is shown in both textbook series. The inverse relation between multiplication and division is only presented in Eldorado. Expressions are addressed in both textbook series, but more often in Eldorado. In both series, the expressions go "both directions", i.e. students are expected to write an expression from a real-world problem or a picture, but also to create their own text problem for a given expression. This is similar to the Estonian and FinSwe textbook series. An example of this is the task "Draw and write text problems" in Grade 1, in which students are to create text problems for the expressions 5+3+2 and 12-5. Priority rules are not considered in either of the two Swedish textbook series; and neither are inequalities, even though one of the textbooks guides students in using the symbols for "greater than" and "less than" to express which of two numbers is larger or smaller. However, this appears only one single time and in the other textbook these symbols are not used at all. The relational meaning of the equal sign is emphasized in both textbook series, especially Matte Safari. For example, students are asked to write = or ≠ between expressions like 9 -2 and 4 + 4 or 7 + 3 and 2 + 8. There are also fully open number sentences, such as _ + _ = _ + _, in line with the FinSwe textbooks. Neither of the textbook series introduces formal equation solving. Yet, one of the textbooks introduces "the symbol x" as a placeholder in Grade 1, using the sentence "The symbol x has a value which makes the equality true", and lets students find the value of x through inspection (Figure 6). However, this is not followed up in Grade 2 or 3.

Discussion and conclusion

Our analysis reveals both differences and similarities between the four regions. Next, this will be discussed in light of relevant research.

The Estonian textbooks and one of the Finnish textbook series frequently use inverse properties of operations in connection with informal equation solving, which is similar to how informal equation solving is prescribed in the Singaporean and Chinese curriculum documents [START_REF] Cai | The development of students' algebraic thinking in earlier grades[END_REF]. Meanwhile, the Finnish Swedish and the Swedish textbook series use inverse properties of operations in the context of "number families" (Figure 3), but not explicitly in connection with informal equation solving. Instead, they both frequently use open number sentences to introduce students to equation solving. In Sweden, this approach might be connected to the emphasis on the meaning of the equal sign, which is explicitly pointed out in the current Swedish mathematics curriculum for Grades 1-3 [START_REF] Bråting | Development of algebraic thinking: opportunities offered by the Swedish curriculum and elementary mathematics textbooks[END_REF]. In all textbook series from the four regions, the handling of expressions appears to varying degrees and students are expected to write expressions based on real-world problems. However, it is only in the Estonian textbooks that students are expected to create letter expressions and equations, which is in conformity with the Russian Davydov curriculum in which letters are used from the very beginning [START_REF] Cai | The development of students' algebraic thinking in earlier grades[END_REF]. The Swedish and the FinSwe series guide the students in finding real-world problems for expressions by means of writing and drawing. This approach is analogous to the US Investigations curriculum, in which students are required to explore problems in depth and construct their own strategies using a variety of tools [START_REF] Cai | The development of students' algebraic thinking in earlier grades[END_REF].

We know that both Estonian and Finnish students have had outstanding results on PISA (Programme for International Student Assessment), in which concrete real-life problems are solved. Both Estonian and FinFin textbooks focus on the creation of quite complicated expressions in Grades 1-3, which might enhance students' structural understanding of problems. This is something to delve into more deeply in further studies. Perhaps the most striking result is the difference we found between the Finnish language contexts. Although they follow the same national curriculum, the FinFin textbooks contain elements similar to the Estonian textbooks, while the FinSwe textbooks seem to have been influenced by Swedish traditions or vice versa. In fact, there seems to be a greater difference between the two Swedish textbook series than between one of the Swedish series and the Finnish Swedish textbook series. There is no government auditing of the textbooks in the three countries, and we are seeing an increased exchange of curriculum material between countries (e.g. [START_REF] Hemmi | Challenging traditional classroom practices: Swedish teachers' interplay with Finnish curriculum materials[END_REF]. In addition, some publishing companies have emerged across the countries. This might lead to greater variation between textbook series within single contexts, but also similarities across various contexts, making our understanding of the connections between curriculum materials and student outcome within a certain context more difficult.

Figure 1 .

 1 Figure 1. Division as the reverse of multiplication in Estonian Grade 2 textbook, (Matemaatika by Koolibri)

Figure 2 .

 2 Figure 2. Modelling word problems in Estonian Grade 2 textbook (Matemaatika by Koolibri)

  No explicit connections are made to guide students in using the inverse properties in equation solving. In the FinFin textbooks, the informal equation solving proceeds step-by-step by first focusing only on equations in which the same term of the operations is missing, while in the FinSwe books, diverse and sometimes open equations start appearing quite early in Grade 1. For example, students are asked to create two different expressions that are equal to each other during the first school term by filling out empty spaces (e.g. _ + _ = _ + _), or with expressions in which the operations are missing (e.g. 4 _ 3 = 5 _ 2).

Figure 3 .Figure 4 .

 34 Figure 3. Partition of the number 5 in the FinSwe textbook (Lyckotal)Contrary to the FinFin textbooks, at no point during Grade 1 are students asked to check their solutions by using the inverse operation. Neither the FinFin nor the FinSwe textbooks use letters for variables or unknown quantities during Grades 1-3; instead, they use gaps or pictures such as flowers and hearts. Thus, in the textbooks from the two language contexts students are expected to write the expressions of operations based on pictures or real-world problems from the beginning. The FinSwe textbooks also guide students in finding real-world problems for expressions, which we do not find in the FinFin textbooks.

Figure 5 .

 5 Figure 5. Practicing of open number sequences (Matte Safari)

Figure 6 .

 6 Figure 6. Examples of equations in the Swedish Grade 1 textbook (Eldorado)

About 5% of the Finnish population has Swedish as their mother tongue, and have the right to receive education in Swedish.

Acknowledgment

This work was supported by the Swedish Research Council [Grant number 2015-02043].