Michael Gaidoschik
email: michael.gaidoschik@unibz.it

Considerations on developmental stage models, learning trajectories and maybe better ways to guide early arithmetic instruction

Keywords: Early arithmetic instruction, counting on, computation without counting, part-whole thinking, derived facts strategies

Current learning trajectories and stage models of arithmetic development advocate the elaboration of counting on as an intermediate computation strategy. At the same time, there is empirical evidence that a considerable share of children fail to overcome this procedural approach to addition and subtraction, with damaging effects to their further mathematical thinking. This theoretical essay argues that it is fundamentally faulty to justify instructional measures by stage models that claim to capture a development without considering that its stages themselves reflect the conditions of children's learning, inter alia: the very instructional measures. As an alternative, the paper uses structure-genetic didactical analysis to delineate a teaching design that derives addition and subtraction out of part-whole thinking from the very beginning, rendering counting on superfluous.

Preliminary remarks

In this paper, "early arithmetic instruction" refers to contents and aims that are constitutive for first grade primary school in German speaking countries. Thus, I write about 6-7-year-old children who, as a rule, have attended kindergarten before; but to what extent and in which ways they have already received targeted mathematic education, will differ considerablywithin and between German speaking countriesdue to different educational plans and levels of specific competences of kindergarten staff. To enable colleagues from other nations to compare with their respective educational systems, I shortly outline just three important areas of goals for first grade arithmetic instruction that (besides others) seem to be widely accepted in the pertinent German literature (cf. Gaidoschik, 2010, pp. 207-232, for details):

First, building upon the rather heterogeneous competences they have thereto already acquired, children should (further) develop and consolidate their knowledge about natural numbers. This starts from being able to count forward and backward fluently and to establish quantities by counting, and should reach the comprehension of numbers as being composed of other numbers at least up to 10, if not 20 (part-whole concept of numbers, cf., e.g., [START_REF] Schipper | Handbuch für den Mathematikunterricht an Grundschulen. 1. Schuljahr[END_REF].

Second, and connected with the former, children should understand addition and subtraction as two arithmetic operations inverse to each other, and learn how to add and subtract fluently within the number range up to 20. More precisely: up to 20, they should learn how to compute without counting already within year one (cf., e.g., [START_REF] Schipper | Handbuch für den Mathematikunterricht an Grundschulen. 1. Schuljahr[END_REF][START_REF] Häsel-Weide | Replacing Counting Strategies: Children´s Constructs on Number Sequences[END_REF].

Third, together with all this, they should (further) develop a positive attitude towards mathematics as a worthwhile activity for which it is constitutive to deal with problems which include detecting, exploring, carrying forward, communicating and arguing about patterns and structures, not only but also in the field of natural numbers (cf., e.g., [START_REF] Krauthausen | Einführung in die Mathematikdidaktik -Grundschule[END_REF].

As far as my knowledge of international mathematics education reaches, these goals are widely accepted in other nations as well, even if institutional frameworks and time expectations may be quite different. Therefore, for some nations, the following considerations might count for children slightly younger or older than those who attend first grade in German speaking countries; but they should count for them at some time of their educational career, if they are of any relevance at all.

Some critical notes on current stage models and learning trajectories

Shortcomings of "developmental stage models" that disregard instructional influences

Within the German speaking literature, in the last decade two competing, quite similar stage models of arithmetic development have been widely noted, both developed by psychologists [START_REF] Krajewski | Early development of quantity to number-word linkage as a precursor of mathematical school achievement and mathematical difficulties: findings from a four-year longitudinal study[END_REF][START_REF] Fritz | Rechenschwäche[END_REF]. [START_REF] Fritz | Arithmetische Konzepte aus kognitiventwicklungspsychologischer Sicht[END_REF] recently claimed to have validated the latter by longitudinally testing children between 9 months before and 8 months after school-entry. Now this is not the place to discuss these models in detail (see [START_REF] Gaidoschik | Wie Kinder rechnen lernenoder auch nicht[END_REF], for a critique on a former version of [START_REF] Fritz | Rechenschwäche[END_REF]. This paper is about the question whether and in what way stage models of that kind can be helpful for mathematics educators and pedagogues in supporting children to learn arithmetic. In this vein, I outline in the following the construction of [START_REF] Fritz | Rechenschwäche[END_REF] as a paradigmatic example of a certain approach to mathematical learning.

The model in question encompasses six "levels" that its authors do not want to be interpreted as following strictly distinctly one after the other, but rather in the sense of "overlapping waves", a metaphor also used by Siegler for his models of strategy development (cf. [START_REF] Siegler | Children's thinking[END_REF]. According to [START_REF] Fritz | Arithmetische Konzepte aus kognitiventwicklungspsychologischer Sicht[END_REF], children, on Level 1, would learn how to establish quantities precisely by counting within a certain number range. On that same level, they would be able to subitize small quantities. Only on Level 2 would children relate numbers to each other by ordering them on a "mental number line". Building upon this, they now would be able to solve additions and subtractions by "counting all". "Counting on", however, would be characteristic for Level 3. Fritz et al. (2018, p. 15) state that only on that stage children would grasp cardinality and the irrelevance of order principle. Cardinal understanding would include knowing that numbers can be composed of and decomposed into other numbers. On Level 3, however, this understanding would be restricted to situations where children use manipulatives, whereas on Level 4 they would "transfer" this concept on a "symbolic-algebraic level" (Fritz et al., 2018, p. 16). However, this would not encompass the understanding of "relationality". Only on Level 5, children would use numbers to state precisely the difference between two numbers. This, together with Level 6 (bundling and unbundling), would form an indispensable precondition for understanding multiplication, division, and place value (Fritz et al., 2018, p. 17).

Fritz et al. finish their publication with "didactic implications" that remain on a rather general level, yet demand that school instruction should follow the "hierarchy of arithmetic concepts" as described by the model. As children would go through that hierarchy at different pace, this would include the need of differentiating within a class (Fritz et al., 2018, pp. 35-37).

The mathematics education community will certainly acknowledge the latter. However, before taking a closer look at how to apply such a model, let us reflect upon its epistemological status. What does it really tell us about children's thinking? [START_REF] Fritz | Arithmetische Konzepte aus kognitiventwicklungspsychologischer Sicht[END_REF] claim to capture a development within a certain time span. They describe single stages referring to children's performance in certain tasks, and interpret what children might be able or not to conceptualize on that level. From the mathematics education point of view, at least some of their interpretations lack clearness and coherence. For instance, the model is rather vague about what might distinguish the number concept of a child on Level 3 from that on Level 4, as well as on Level 4 and 5. It seems to be that the authors are more interested in differentiating levels of test performance than really of arithmetical concepts.

What is more fundamental is the model's lack of consideration of the fact that children develop their ideas while constantly dealing with inputs coming from adults as well as from other children. As for counting on, which constitutes Level 3 of the model in question: There is ample evidence that in many countries pedagogues in kindergarten and primary school, and presumably parents as well, encourage children and in fact show them how to solve addition problems by counting on instead of counting all (cf. [START_REF] Gaidoschik | Didactics as source and remedy of mathematics learning difficulties[END_REF]. They certainly do so because they think this to be conducive. I will gather some evidence that casts doubt on this assumption in the following paragraph. What has to be stated at this point, is a fundamental theoretical shortcoming of any model of learning development that disregards the possibility that what the model claims to be "normal" or even "necessary" within the presumed "nature of children" might rather be the consequence of circumstances that are man-made.

Counting on: A "natural" stage that still needs to be pushed by learning trajectories?

There is a logical inconsistency in this shortcoming that becomes even more apparent in the case of "The Learning Trajectories Approach" [START_REF] Clements | Learning and teaching early math. The learning trajectories approach[END_REF] or stage models that are directly formulated to guide teacher's actions, such as the "Mathematics programme of study" (UK Department of Education, 2013) or the "Number framework" (New Zealand Ministry of Education, NZME, 2008NZME, , 2012)). All three guidelines stress that eventually children should learn to add and subtract fluently without counting. The "Number framework" is especially clear in that point:

It is important for you to recognise that part-whole thinking is seen as fundamentally more complex and useful than counting strategies. One reason is that counting methods are strictly limited, whereas part-whole methods are more powerful. Counting strategies are an inadequate foundation for these ideas, and this means that for counters, many advanced number ideas are inaccessible. Therefore, your major objective is to assist students to understand and use partwhole thinking as soon as possible. (NZME, 2008, p. 7) Nevertheless, all three guidelines recommend that teachers actively work out counting on as a shortcut strategy for addition, and analogue counting strategies for subtraction with their classes. Thus, if a teacher who follows the "Number framework" notices a child to add by counting all using materials (Stage 2), s/he should work with that child so that it first reaches Stage 3 ("Counting from One by Imaging") and then Stage 4 ("Advanced Counting"). For that purpose, the teacher should orchestrate activities so that children learn to put "the bigger number first" and count on when confronted with an addition problem (NZME, 2012, p. 36). Only after having reached that stage, the teacher should guide children further to Stage 5 where they should "develop the idea" that "addition and subtraction problems can be solved by part-whole strategies instead of counting" (NZME, 2012, p. 37).

I did not find an explicit justification for this recommendation in the pertinent publications I reviewed, but it seems to be that the authors deem it indispensable to intentionally guide children from one stage to the next and not skip a single of these stages as defined by the underlying stage model.

Thus, these models extract stages of "development" from the evidence that many children progress from counting all over counting on to finally computing without counting. Nota bene: Many children progress in that way; yet, a considerable number stick to counting strategies in higher grades [START_REF] Cowan | Basic calculation proficiency and mathematics achievement in elementary school children[END_REF][START_REF] Hopkins | The prevalence and disadvantage of min-counting in seventh grade: problems with confidence as well as accuracy?[END_REF], and doing so forms an essential part of learning difficulties [START_REF] Gaidoschik | Didactics as source and remedy of mathematics learning difficulties[END_REF]. Then, there is evidence that the share of children who finally quit counting on strongly relates to instruction. In nations with a tradition of not encouraging children to count on, but rather use derived facts strategies already at an early stage, this share seems to be much higher [START_REF] Geary | Development of arithmetical competences in chinese and american children: influence of age, language, and schooling[END_REF]. Intervention and field studies indicate that putting the focus of early instruction on derived facts strategies instead of counting on, will reduce the number of children who cling to counting for computing (e.g., [START_REF] Steinberg | Instruction on derived facts strategies in addition and subtraction[END_REF][START_REF] Gaidoschik | Empirische Befunde zum Lehren und Lernen auf Basis einer Fortbildungsmaßnahme zur Förderung nicht-zählenden Rechnens[END_REF]. Notwithstanding all this, the cited models postulate a quasi-natural progression:

"Children follow natural development progressions in learning and development. As a simple example, they learn to crawl, then walk, then run […] They follow natural developmental progressions in learning math, too […]." (Clements & Sarama, 2009, p. 2) Thus, on the one hand, the authors regard counting on as part of a "natural development". On the other hand, they take their model as a justification to instruct children how to count on. However, doing so is a didactic decision, and by no way "natural" nor necessary, as I will argue in the following. Presumably, the authors would deny intervening in this way in a supposedly inevitable process to be logically inconsistent and put forward their endeavor to "promote children's growth" (Clements & Sarama, 2009, p. 5), so that what is "natural" may move forward without avoidable impediments. Yet, as stated above, research indicates that in fact it may impede children when teachers quite artificially guide them to count on for addition and subtraction problems [START_REF] Gaidoschik | Didactics as source and remedy of mathematics learning difficulties[END_REF].

An alternative approach: Structure-genetic didactical analysis

Whereas learning trajectories and stage models suggest teachers stick to what is supposedly inherent in the "nature" of children, [START_REF] Wittmann | Structure-genetic didactical analyses -empirical research of the first kind[END_REF] "structure-genetic didactical analysis" advises instruction to follow the "nature of mathematics". Of course, Wittmann not at all advocates disregarding what empirical research indicates about the ways children learn mathematics, the difficulties they meet, and the misunderstandings that may arise based on their "prerequisite knowledge" as well as on what happens in the classroom. He explicitly rejects the '"broadcast' method of transmitting knowledge from the teacher to the student" that he ascribes to "the traditional 'subject matter mathematics'" dominant in German mathematics education for a long time (Wittmann, 2015, p. 17). Therefore, his idea is not that we would only need to unfold the logic of any subject matter to children in order to secure they grasp it step by step. Still he insists that what a subject matter "is about and how it should be introduced in the classroom, cannot be decided by means of empirical methods imported from psychology, but should be based on a sound mathematical and epistemological analysis" (Wittmann, 2015, p. 7).

In the following, I will try to delineate in brief what, in my understanding, such an analysis would imply for the subject matter in focus of this paper, i.e. the introduction of addition and subtraction. In doing so, I take into account the pertinent empirical research, but without considering immutable what might rather be consequence of a certain type of instruction that we could at least try to change.

Deriving addition and subtraction out of part-whole thinking: A sketch

"Emerging from the [mathematical] subject" (Wittmann, 2015, p. 6), addition and subtraction are inverse operations, with the laws of commutativity and associativity being central for addition. Applying these laws allows for powerful computation strategies. Subject matter analysis as well as empirical evidence indicate that in order to grasp these concepts, it is crucial to think numbers as composed of numbers. Young children encounter numbers in many ways, but counting is dominant in the first years. For a child who uses numbers primarily to count quantities, numerical part-whole relations are not at all obvious; rather, the child might take any number as a single entity, so that e.g. six is conceptualized as being before seven, rather than being a part of it (cf. [START_REF] Gaidoschik | Wie Kinder rechnen lernenoder auch nicht[END_REF].

Therefore, if early arithmetic instruction strives to help children acquire a solid fundament of addition and subtraction, it seems sensible first to focus on activities that foster part-whole understanding of numbers. Only on that basis, we can and should convey addition and subtraction as two ways of operating with parts and wholes that supplement each other. Following that order, which is of course not dictated by stage models, but advocated by subject matter analysis, we have a good chance to render computation by counting superfluous from the very beginning [START_REF] Gaidoschik | Didactics as source and remedy of mathematics learning difficulties[END_REF].

In more detail: In order to be able to explore and internalize part-whole relations, when quantities exceed subitizing (see below), a child needs to know how to determine the number of the quantity by counting. Therefore, one main-topic (A) of early arithmetic activities certainly should be counting as a means to establish precisely how many of whatever you have. This includes the targeted working out of the "counting principles" described by [START_REF] Gelman | The child's understanding of number[END_REF]. Any first-year classroom or group of kindergarteners will be quite heterogeneous in that respect. As a basis for the steps to follow, all children should at least consolidate counting skills and knowledge up to 10, the basis of our number system; most children will exceed that number by far, so there is a need for differentiation.

At the same time (B), it is important to build upon what children, partly independently from their counting competences (cf. [START_REF] Fritz | Arithmetische Konzepte aus kognitiventwicklungspsychologischer Sicht[END_REF], have already acquired in the field of subitizing of quantities up to four and perceptual subitizing of bigger quantities structured in a way that allows for establishing their number without counting (cf. [START_REF] Clements | Learning and teaching early math. The learning trajectories approach[END_REF]. This could start with numbers up to five (number of fingers on a hand) or six (maximum of dots on one side of a die). Yet, we would gain little for the following if a child identifies these quantities with a glance but without noticing them as composed (four as two and two or three and one, five as four and one or three and two, etc.). Thus, activities should focus on the structured perception of these quantities. [START_REF] Schultz | Teile-Ganzes-Denken über Zahlen und Operationen: Herausforderung und Leitidee des Anfangsunterrichts[END_REF] delineate how this could form the starting point of targeted elaboration of part-whole thinking: We may request children to construe operations of joining, separating, missing addend etc. as just different interpretations of dot structures, for example, "I see five dots altogether. If I take away the outer four, the central one will remain." We may then encourage the children to transfer these nuclei of number knowledge to real-world situations: "I've got one Euro. If mummy gives me four more, I will have five." Note that the intention of such early activities is not that children solve these problems by counting, but by referring to the part-whole interpretation of structured dots, first supported by seeing the dots, then by knowingly imagining their structures [START_REF] Schultz | Teile-Ganzes-Denken über Zahlen und Operationen: Herausforderung und Leitidee des Anfangsunterrichts[END_REF].

A third "construction site" (C) of early activities, once again building upon the rather diverse competences that children will already have acquired in that field, is comparing quantities and numbers. As a precondition for further arithmetic, children should learn and/or consolidate that two quantities are the same number if it is possible to establish a continuous one-to-one correspondence between their items; no counting is needed for that. On that basis, they should learn that "one more" means that a mapping of the items results in one item left for the bigger quantity [START_REF] Gaidoschik | Rechenschwächen vorbeugen -Erstes Schuljahr: Vom Zählen zum Rechnen[END_REF].

I use the metaphor "construction sites" to indicate that we should work with children in all three Fields A to C more or less concurrently, which is, bearing on German speaking countries, within the first weeks of grade one. Certainly, we may concentrate activities on only one site for some days, but switch to another the next day or even combine working on all sites on the same day. This is sensible, as many children in single areas will need quite a lot of practice over a longer period to reach the level required as a basis for next steps. Yet, to achieve desirable automatisms, e.g. in counting forward and backward, it would not help to stick to one single theme for too long in a row, but rather provoke tedium. More importantly, the sites relate to each other. We should link counting to comparing, to help children realize that each number in the sequence is "one more" than its predecessor (and not only "the next number", see above). Working on perceptual subitizing should include comparing, etc. Such as the three sites merge into one another, the next important step is only an extension of the above-mentioned exercises: advanced activities that fit for exploring and internalizing part-whole relations, but now for all numbers at least up to 10. [START_REF] Gaidoschik | Rechenschwächen vorbeugen -Erstes Schuljahr: Vom Zählen zum Rechnen[END_REF] describes in detail how the use of fingers which are not being counted nor unfolded one by one, but moved and perceived in groups simultaneously, could form a starting point for first acquiring mental pictures and finally automatized knowledge of at least some important compositions of all numbers up to 10. Thus, children could first learn how to "stretch out eight fingers without counting", then describe how they would do so without really doing it ("I need one full hand and three more fingers"). Then, if already available to the child (see below), s/he could use symbols to record the action (5+3=8), think about what could be done subsequently ("So this is eight fingers. What if you take away the one full hand?"), and record that action as well (8-5=3). Already at that stage, children can and should try to apply this knowledge to real-world situations, for instance by inventing word problems. In that way, the acquisition of part-whole knowledge is (almost) from the beginning closely linked to addition and subtraction.

Note that I am not talking about writing equations from the very start, but operating with real, then imagined material. Fingers are very useful "material" at the beginning, as they allow to experience clearly and convincingly the difference between using structure-knowledge (take away five means take away one full hand at once) and counting (take away quite tediously one single finger after the other). In the following, for some purposes, ten-frames are useful, then twenty-frames and 100 dot arrays. Also with these materials, yet, it is paramount to work out how to see and use structures, how to relate other numbers to 5 and 10; only this enables to operate without having to count one by one.

Fingers might be most convincing to work out the five-plus-x structures of the numbers 6 to 10. Another indispensable structure that should be in the focus of early activities are doubles and halves (6 as 3+3, 8 as 4+4 etc.). Therefore, a next step could be to concentrate on these very structures, and again connect them with tasks that elicit their application in solving addition and subtraction tasks. Once more, it will turn out that counting is superfluous to solve these tasks. Presumably, some children might still apply counting strategies; yet, in this context there is high probability that the teacher or other children will convince these children that there is a better, much easier way, by thinking about and applying knowledge of the part-whole structure of the numbers involved.

Closing remarks

This is not the place to unfold a detailed guideline for teaching early arithmetic. The aim of the above sketch is to illustrate the paper's main message: Structure-genetic didactical analysis allows developing a teaching design that avoids engaging children in using counting on for computation from the very beginning. It does so in full consideration of pertinent research on how children develop their thinking under certain conditions. However, it strives to change these conditions where analysis shows that they may lead to unnecessary delay, if not continuing problems. Research indicates that the widespread tradition to foster counting on is such an unfavorable condition. Structure-genetic didactical analysis opens an alternative in line with the needs of the children as well as of the subject matter. Of course, an indispensable next step would be to implement these ideas in classrooms, and evaluate and further develop them based on reflected experiencea clear case for educational design research. I have started such research on a small scale, yet this paper is deliberately not about the ongoing pilot study. I had rather put the basic concept up for international discussion. After all, it would not make sense to trial on larger scale a design that does not withstand careful scrutiny.


Sofia Graça
email: soffiagraca@hotmail.com

João Pedro Da Ponte

António Guerreiro
email: aguerrei@ualg.pt

Rational number operations: What understandings do children demonstrate?

Keywords: Rational numbers, fractions, fractions operations

This qualitative and interpretative study aims to identify grade 5 students' knowledge of rational number operations in order to ascertain their rational number sense and operation sense before and after a teaching experiment. Data was gathered using two tests and four individual semistructured interviews. The results show that, in the pre-test, children performed rational number operations based on their whole number knowledge. In the post-test, after the teaching experiment, they showed conceptual understanding of the operations, demonstrating rational number sense and operation sense.

Introduction

Rational numbers are important mathematical entities, with an important place in the mathematics curriculum, particularly in early years. They involve complex ideas, which represent conceptual obstacles for children [START_REF] Behr | Teaching rational number and decimal concepts[END_REF]. Mathematics education researchers have identified difficulties in children's work with these numbers, for example, regarding operations [START_REF] Hansen | Children's errors in mathematics[END_REF]. Teaching methods used in schools, typically oriented towards the application of rules, lead students to perform sequences of procedures without understanding their meaning [START_REF] Braithwaite | A computational model of fraction arithmetic[END_REF], for example, obtain equal denominators and then add or subtract numerators; multiply numerators and denominators on fraction multiplication; and "invert and multiply" on fraction division. Consequently, children do not develop rational number sense [START_REF] Lewis | Lesson study with mathematical resources: A sustainable model for locally-led teacher professional learning[END_REF] nor operation sense [START_REF] Huinker | Examining dimensions of fractions operations sense[END_REF]. Thus, this study aims to identify grade 5 students' knowledge of rational number operations in a pre-test, before a teaching experiment, and in a post-test, after that teaching experiment, in order to ascertain their rational number sense and operation sense.

Rational number operations

General aspects

Students often demonstrate not understanding what happens when they add, subtract, multiply or divide fractions because the teaching of this topic is typically based on the presentation of rules that they do not understand. Usually, they apply their knowledge of integers on rational number operation tasks, and they operate with rational numbers as if they were integers. Thus, children make mistakes that demonstrate misunderstandings in their conceptual understanding: "strategy errors", for example, applying the operation to numerators and denominators, independently; and "execution errors", for example, incorrect performing a calculation procedure or failing to change numerators or denominators when attempting to obtain equivalent fractions [START_REF] Braithwaite | A computational model of fraction arithmetic[END_REF].

Rational number understanding involves rational number sense, that is, understanding: (i) that a fraction is a number; (ii) partitioning fractions; (iii) the meaning of denominators; (iv) knowing what is the whole; (v) fraction size; and (vi) fractions can represent quantities greater than one [START_REF] Lewis | Lesson study with mathematical resources: A sustainable model for locally-led teacher professional learning[END_REF]. Operation sense is another component of rational number sense. According to [START_REF] Huinker | Examining dimensions of fractions operations sense[END_REF], it includes the following understandings: (i) meanings and models for operations, e. g., rectangular, circular or area models, used to represent the operations; (ii) ability to recognize and describe real-world situations for specific operations; (iii) meaning of symbols and formal mathematical language; and (iv) knowledge the effects of operation on numbers. In the perspective of [START_REF] Brown | Algebra students' difficulty with fractions: An error analysis[END_REF], gaps in the conceptual understanding of rational number operations are the cause of poor student performance in these operations. Both number and operation sense represent a way of thinking about numbers and their operations. Problem solving allows children to explore and understand rich mathematical situations, in which operations arise. Thus, problems can more easily assign operations meaning and promote meaningful learning. The use of models can help children to understand the conceptual basis of each operation. As they establish a relationship between models and algorithms, children can understand what each step of an algorithm represent in practice [START_REF] Behr | Teaching rational number and decimal concepts[END_REF].

Addition and subtraction

Fraction addition and subtraction are often taught based on the use of procedures, such as, "denominators have to be equal, and then you can add or subtract the numerators". However, students often do not understand the need of common denominators or the role of equivalent fractions, which constitute fundamental understandings in these operations [START_REF] Behr | Teaching rational number and decimal concepts[END_REF]. This procedure-based approach does not promote the conceptual understanding of fraction addition and subtraction. Together with the prior knowledge of whole numbers, it leads children to make common mistakes, because they confuse the procedures. In addition, children have more difficulty understanding questions involving subtraction than involving addition. Sometimes, they do not identify the correct operation [START_REF] Behr | Teaching rational number and decimal concepts[END_REF]. [START_REF] Braithwaite | A computational model of fraction arithmetic[END_REF] refer that most children's difficulties are associated with the nature of denominators. When they are equal, children usually just add or subtract numerators; when they are different, they usually make independent whole number errors, that is, apply the operations to numerators and denominators (strategy errors). [START_REF] Behr | Teaching rational number and decimal concepts[END_REF] point out that the use of models can help children to understand the involved concepts when they add or subtract fractions.

Multiplication

Multiplication is usually viewed as successive addition of "equal groups". However, in the rational number field, this is an inappropriate idea because the product can be smaller than both factors and this is not compatible with an additive procedure [START_REF] Greer | Multiplication and division as models of situations[END_REF]. Tasks with a "multiplicative relation" meaning can be useful on learning rational number multiplication so that children overcome this misunderstanding. The use of models allows children to understand how the unit change during the problem and that the product is not always greater than each of the factors [START_REF] Behr | Teaching rational number and decimal concepts[END_REF]. Children multiply both fractions but they do not understand how and why the algorithm works.

Division

The division operation is often linked to sharing situations. In these situations, the goal is to get the unit value. However, learning division only based on this meaning leads students to the incorrect idea that division always decreases or that the dividend should be greater than de divisor [START_REF] Greer | Multiplication and division as models of situations[END_REF]. The meaning "measure" of division is another way to think about this operation, requesting the number of groups instead the unit value. This meaning allows a more robust understanding of division. The use of models makes it easier for children to understand this meaning of division and allows them to verify how many times the divisor go into the dividend. The common denominator algorithm is an interesting strategy to help children understand this situation and how the algorithm works because it represents the way children reason about this meaning of division.

Research methodology

This is a pre-test-intervention-post-test study (carried out in 2017/18). Participants are four grade 5 students from a public school in Portugal, from a disadvantaged socioeconomic environment. They were selected because of their ease in oral communication and the fact that their performances on tasks were representative of the remaining class, as they had different performance levels. A pretest, applied before a teaching experiment, involved different rational number concepts. Data analysis in this paper focuses on fraction addition and subtraction because these are contents already learns in previous years, so, children already had this knowledge. A post-test was applied after that teaching experiment (in the following lesson) and also involved different rational number concepts. However, the focus were all four operations because multiplication and division operations were addressed during the teaching experiment since multiplication and division were new contents in this grade. The children solved the tests, individually, for 50 minutes, and then they were individually interviewed in order to explain their reasoning in each question. The interviews took place in the same day they solved tests and the following day.

The teaching experiment lasted six weeks, consisting of eighteen tasks (45 questions in total). The tasks involving all four operations were the last eleven ones. Addition, subtraction, multiplication and division, in problem solving situations, were addressed separately, in this specific order. These tasks involved 27 questions (13 questions involved the use of models). The teaching experiment followed an exploratory approach [START_REF] Ponte | Teachers' professional practice conducting mathematical discussions[END_REF], that is, tasks were presented to children and they worked individually, followed by a group discussion and synthesis. To solve the tasks, in an initial phase, the children were guided to use (circular or rectangular) models, provided in the task statements. The actions performed in the models were accompanied by the corresponding numerical sentence (numerical sentences changed as the models changed), so that the children understood the relationship between them. At a later stage, they could choose the solution strategy they preferred. The goal was to progress from the use of models to symbolic manipulation with understanding. On fraction addition, the children had two circular models to represent the quantities to be added. The denominators of fractions and the number of divisions in the models were equal, so, they just had to add the number of pieces. Later, the denominators of fractions and the number of divisions were different, so, they had to make divisions in the models because they could not add pieces with different sizes. Lastly, models had no division and the denominators of fractions were different so, the children had to make their own (equal) divisions to get the result. Subtraction of fractions followed the same procedure. Circular and rectangular models were used to promote understanding about these operations, allowing to understand the need of equal denominators and the role of equivalent fractions.

On fraction multiplication (meaning "multiplicative relation"), a rectangular model was used. This model allowed the children to understand the role of different fractions when they are multiplied, for example, the role of operator and the change of the reference unit throughout the problem. When children are asked to solve × , they find a half because is the initial quantity and then they find of that part. Faction division focused on the "measure" meaning and the common denominator algorithm was addressed. The rectangular model was effective in understanding this meaning of division and allowed to conceptually understand the algorithm. For example, "Luísa had 2 meters of tape to put in hats and each hat takes meters. In how many hats can she put tape?" The children represented the dividend in models by shading and then they could verify how many are in 2 .

For data analysis, the interviews were all audiotaped, transcribed and analysed in depth, as well as the children solution strategies in both tests. These two methods of data collection allowed us to analyse students' rational number sense and operation sense according to the [START_REF] Lewis | Lesson study with mathematical resources: A sustainable model for locally-led teacher professional learning[END_REF] and [START_REF] Huinker | Examining dimensions of fractions operations sense[END_REF] categorizations, respectively, before and after the teaching experiment. These methods also allowed to ascertain how they evolved from pre-to post-test, and the role of the models in the development of children conceptual understanding.

Results

Pre-test

A fraction addition operation was involved in the question "Pedro, João and Gonçalo started running a path with 6 km in a park near their school but none of them got to the end. Pedro stopped after , João run only and Gonçalo . What fraction represents the travelled distance by the three friends together?" Although most children had identified the implicit operation in the story problem, their performance was quite weak. They often added numerators and denominators independently, to get the result (strategy error), and presented the result . When they were asked if it would be possible to add fractions with different denominators, they demonstrated not understand the question. Just one child, Ana, tried to use a rectangular model to support her reasoning, but it was not correctly built and she left the question unanswered. In a fraction subtraction question, "What is the difference between the boy who ran the longest distance and the boy who ran the shortest distance?", again, the children subtracted numerators and denominators independently, to get the result (strategy error). Ana (Figure 1), again, tried to use rectangular models to reason about the question, however, these were not correctly built. She could not identify which operation should use.

Post-test

In the post-test, an addition question was posed to children "Clara, Maria and Gabriela started a path. Clara stopped to rest after of the path, Maria stopped after and Gabriela after . What fraction represents the distance travelled by the three friends together until the moment they stopped to rest?" Most of children were successful both in identifying the operation and applied the correct algorithm. They demonstrated to understand the situation through the following statements "Here below not everything was the same size. I made it to be and I started here, with 10. […] Then, I added all parts and left the bottom number which represents the size of the parts that are here!" (Ana), or "The whole course had five parts and all together made seven of these parts, seven bits of !" (David). Just one child, Pedro, did not correctly add the numerators after obtaining equivalent fractions. He obtained the fraction and said it was equal to one unit (Figure 2), making an execution error. During the interview, Pedro identified and corrected his error and, after the new result , he wrote that "together they did more than a whole path!" or "I think this is so because this boy made six. These only made four!" Fraction subtraction question was posed to children "In the end of the course, they decided to eat sweets from a bag with kg. Together they ate kg. How much did they leave in the bag?" Most children identified the correct operation and used it as solution strategy.

They demonstrated understanding its meaning saying that "It's a subtraction operation! If they eat

[Kg] from the whole bag, we have to get that part […] but we can't do three minus one because the parts are not equal and they have to be! We need equal fractions!" (David). Just one child, Pedro, tried to solve this question using a model (Figure 3). He justified that "each friend ate , so, since it is one of three, I counted one because it represents what they ate, and it is one of three because they were all the sweets they had!" Thus, he referred that six sweets were left, because it is the number of unshaded circles. Pedro considered the initial quantity as a continuous entity, as shown by the model that he tried to use, but considered the withdrawn part as a discrete entity. We do not know how much candy is in the bag but its weight.

The child considered continuous and discrete quantities as measured with the same unit.

The division as measure was underlying in the following question: "Maria is making chocolate cakes for her sister's birthday. She has 2 kg of sugar and each recipe of chocolate cake takes kg of this ingredient. How many chocolate cake recipes can she make?" Most of children had a good performance in this question. They identified the correct operation to be used in the problem, using the common denominator algorithm as solution strategy and demonstrated to understand its meaning "Here, I divided to see how many fit in 2 […] she can make eighteen cakes!" (David).

Only Ana did not identify the correct operation in this situation. She added both amounts and justified her choice saying that, "she wanted to make a cake with the whole amount, so, I made an addition!" This child also demonstrated difficulties regarding fraction simplification.

The following fraction multiplication question (multiplicative relation) was posed to children: "To decorate a cake, Maria put chocolate in of the whole cake and coconut in of the part that had chocolate. What fraction of the whole cake has both decorations?" Only David identified and used multiplication operation as solution strategy. He justified that "Maria put coconut over the chocolate part, so, I multiplied because it's one part of another! is the shaded parts twice!" When asked about the relationship between the operation and the model, he said "2 is the number of pieces and 6 is the size of the pieces. It's just a little bit.

[…] So, we have to multiply numerators and denominators!" The remaining children had a poor performance in this question. Nara subtracted both quantities, however, during the interview, she said that " means there was half of the cake, but … well… it was in that part [ ]! I think it was not to subtract, it was to multiply!" So, she demonstrated some understanding about this meaning of multiplication. Pedro and Ana attempted to solve the problem using circular models but they demonstrated some difficulties in understanding this question. They also demonstrated misunderstandings related to divisions in the models. Horizontal and vertical divisions were made in the circular model. Pedro, after representing on his model, said: Although Pedro was unable to solve the problem correctly, during the interview he showed better understanding about the question as well as its relation with the fraction multiplication.

Discussion and conclusion

The children who participated in this study had already learned fraction addition and subtraction in previous years. However, in the pre-test, their performance in questions involving these operations were very weak. They added and subtracted numerators and denominators to get the result, which constitutes a common mistake especially when denominators are different [START_REF] Braithwaite | A computational model of fraction arithmetic[END_REF]. This solution strategy demonstrates lack of understanding that a fraction represents a single number.

It also shows misunderstanding related to meaning of the denominator, since they added and subtracted fractions with different denominators, for example, on fraction addition ( + + = ).

These mistakes show that these children were unaware of the relationship between numerator and denominator, as in the study of [START_REF] Hansen | Children's errors in mathematics[END_REF], reflecting weaknesses in their rational number sense. Children's solutions also show that they did not conceptually understand these operations. Similar results were found in the study of [START_REF] Brown | Algebra students' difficulty with fractions: An error analysis[END_REF], which suggests that the poor performance is a consequence of gaps in the conceptual understanding of operations. These children did not understand the meaning of the operations and they have very limited knowledge of models to represent them, as shown in Figure 1. They also did not reason about the results obtained, which they did not understand to be inadequate in relation to the situation. Thus, their operation sense also had serious shortcomings. Their responses showed that they learned to solve operations applying rules (that they generalized in a wrong way from whole number operations) without understanding their meaning.

In the post-test, the children showed a better understanding of rational number addition and subtraction, compared to the pre-test. On these operations, they demonstrated to understand the use of equal denominators and what they represent, which was possible due to the use of models during the teaching experiment. The children know that it is not possible to add or subtract fractions with different denominators, as reported by Ana. That is, they understood that a fraction represents a number and not two independent numbers. This is an important understanding of student's rational number sense. Knowledge concerning to the meaning of the denominator, another component of rational number sense, was also observed, for example, "I kept the value down, which tells the size of the parts that are here!" (Ana). Justifications such as "The whole course had five parts and together they travelled seven of these parts. Seven bits of !" (David), showed that these children know the fraction size and that a fraction can represent quantities larger than one. Regarding operation sense, they began to understand the different meanings of operations and to assign meaning to the symbols. The reasoning "Here I divided to see how many times this [ ] fits in 2 " (David), was possible due to the use of models. He showed that he conceptually understood this meaning of division. He reflected in the quantities and how they are related and also identified the appropriate operation. The children also demonstrated understanding the effect of operations on numbers when they analysed the reasonableness of results: "I think it is this way because this boy made six. These only made four!" (Pedro) or " are the pieces shaded twice.

[…] It's just a little bit!" In general, these children demonstrated conceptual understanding of formal mathematical language. The division operation as measure and the multiplication operation as multiplicative relation, used in this teaching experiment, allowed a deep understanding of these operations. The children overcome the misconception that multiplication always increases and division always decreases. This was possible due to the use of models during the lessons, especially in an initial phase, as recommended by [START_REF] Behr | Teaching rational number and decimal concepts[END_REF]. Therefore, we conclude that this teaching experiment based on the use of models, allowed the students to construct the algorithm from the models, and led them to develop some components of their rational number sense and operation sense. This study represents an alternative way of approaching rational number operations, promoting students' conceptual understanding, which often does not occur.

Figure 1 :

 1 Figure 1: Ana's solution

Figure 2 :

 2 Figure 2: Pedro's solution (i)

  Figure 3: Pedro's solution (ii)

"

  't forget the coconut was in the chocolate part! Pedro: Ah! I think I had to split this part [ ]… To put the coconut on top because it is "of"! Its of !

Acknowledgment

This study is supported by FCT -Fundação para a Ciência e a Tecnologia, through a doctoral grant to Sofia Graça (SFRH/BD/130343/2017).


Robert Gunnarsson
email: robert.gunnarsson@ju.se

Ioannis Papadopoulos

Pairing numbers: An unconventional way of evaluating arithmetic expressions

Keywords: Arithmetic, Order of operations, Precedence rules

We have analyzed the written solutions to four arithmetic expressions comprising addition/subtraction and multiplication of 4 to 7 numbers from 235 students (112 in Greece and 123 in Sweden,[11][12]. Beside the order-of-operations and sequential (left-to-right) calculations, several students seem to calculate in an unconventional way by pairing numbers. That is, there are students who combine the numbers two-and-two, seemingly regardless of the operations, and then operate on these pairs when evaluating an expression. In our data, pairing seems to be as frequent as sequential calculations. We identify the characteristics of three qualitatively different ways of pairing. The different ways seem to be applied independent of the length of the numerical expression.

Introduction and theoretical background

In mathematics, we use conventions for arithmetic expressions to be unambiguously perceived. However, to design successful teaching on how to perceive and calculate arithmetic expressions, we need to first learn what ideas are present among the students. In this report, we present a study of those ideas and can show that evaluating numerical expressions by number pairing seems to be more significant in students' ideas than previously reported, and that it can emerge in different types.

Without the mathematical conventions like, e.g., the rules for the order of operations, there would be different ways in which numerical expressions can be evaluated. Accordingly, it has been shown that students who have not yet internalized these conventions use different ways in their calculations [START_REF] Blando | Analyzing and modeling arithmetic errors[END_REF][START_REF] Glidden | Prospective elementary teachers' understanding of order of operations[END_REF][START_REF] Gunnarsson | Arithmetic expressions with multiple operations -How to solve it?[END_REF][START_REF] Headlam | An investigation into children's understanding of the order of operations[END_REF]Libenberg, Linchevski, Sasman, & Olivier, 1999). This might result in errors such as exchanging a number with another number, exchanging an operation with another operation, or, simply, making computational errors. [START_REF] Blando | Analyzing and modeling arithmetic errors[END_REF] call these errors "careless" and "substitution errors". But, there are also errors that are more related to the structure of the expression. [START_REF] Linchevski | Structure sense: The relationship between algebraic and numerical contexts[END_REF] showed that students, while being inconsistent in the computational aspects, can be very consistent in these structural errors.

There has been substantial research devoted to students' handling of arithmetic expressions from perception and cognitive points of view. It has been shown in several studies that the mere proximity of two numbers can result in that these are calculated first, regardless of the operator in between [START_REF] Gómez | Symbol spacing and its influence in school children's calculation of mixed-operation arithmetic expressions[END_REF][START_REF] Jiang | Spatial factors influence arithmetic performance: The case of the minus sign[END_REF][START_REF] Landy | Proximity and precedence in arithmetic[END_REF][START_REF] Rivera | Persistent perceptual grouping effects in the evaluation of simple arithmetic expressions[END_REF]. More generally, it has been shown that the visual appearance of the mathematical notation can cause students to disregard mathematical rules [START_REF] Kirshner | Visual salience of algebraic transformation[END_REF]. A particular effect of the visual appearance was pointed out by [START_REF] Linchevski | Structure sense: The relationship between algebraic and numerical contexts[END_REF]. They showed that some students can dissociate two parts of an expression, often induced by an operation sign (typically minus), like, e.g., when is evaluated as .

In addition, there are several educational research studies showing that students can perceive the structure of numerical expressions in different ways. [START_REF] Kieran | Children's operational thinking within the context of bracketing and the order of operations[END_REF] has shown that the rule 'to be calculated first' can be perceived as 'to be put to the left'. Conversely, it appears as if students perceive what is read first (i.e., to the left, if that is the reading order) should be calculated first. At least, several studies describe that students calculate expressions sequentially from left to right [START_REF] Blando | Analyzing and modeling arithmetic errors[END_REF][START_REF] Headlam | An investigation into children's understanding of the order of operations[END_REF][START_REF] Liebenberg | Focusing on structural aspects of numerical expressions[END_REF].

There have also been brief reports that some students pair up numbers when processing arithmetic expressions [START_REF] Gunnarsson | Arithmetic expressions with multiple operations -How to solve it?[END_REF][START_REF] Liebenberg | Focusing on structural aspects of numerical expressions[END_REF][START_REF] De Villiers | Order of operations: The math and the myth[END_REF]. While de Villiers ( 2015) mentions it as one type of erroneous calculation (without further discussion) where order of operations should have been used, [START_REF] Liebenberg | Focusing on structural aspects of numerical expressions[END_REF] describe students who break up expressions in parts. As example, they find students calculating by grouping numbers in pairs and then processing these pairs as . However, we have noted that, apart from the example above, the expressions used in most previous studies are quite short and contain two (or maximum three) different operators. Typically, the numerical expressions are of the form , occasionally . The exception found in the study of [START_REF] Liebenberg | Focusing on structural aspects of numerical expressions[END_REF] with longer expressions (up to seven numbers, thus six operations), may have enabled them to observe a breaking-up-into-parts effect. Hence, in this study we would like to explore this effect further by using long expressions.

Therefore, with the idea that teaching arithmetic conventions must take its point of departure in students' actual ideas on how calculations should be completed, the aim of this study is to further explore students' idea of pairing when calculating arithmetic expressions. In short, we would like to find what forms of pairing mechanism occur when students evaluate arithmetic expression. This study is part of a larger project to design teaching to improve students' arithmetic computational competence.

Method

This study engaged 112 Greek students and 123 Swedish students, 11-12 years old (grade 5 in Sweden, grade 6 in Greece). In total students from nine different schools (five Greek schools and four Swedish schools) were included. The schools were chosen from mixed background to represent all achievement levels. According to the curriculum and the teaching traditions in the two countries, the students should have been taught the rules for the order of operations prior to the study. The students were given a written test and sufficient time (no time limit) to answer it. The test comprised several tasks, all typeset with the equation editor of a conventional word-processor, to avoid unintended influence from the visual appearances of the expressions. The test was aimed to test students' understanding of arithmetic conventions. Four of the tasks were specifically designed to explore students understanding of the order of operations by expressions of different lengths. These tasks are shown in Figure 1. The data, comprising the students' written answers to the testtheir written evaluations of the expressions in Figure 1 was subject to a qualitative content analysis as described, e.g., by [START_REF] Leedy | Practical Research -Planning and designing[END_REF]. In the analysis, the correctness of the answers was not considered, instead the underlying mathematical idea were in focus for each solution of a numerical expression. In the first step we were looking for well-known ideas, like following the order of operations, sequential calculations (left-to-right), detachment (from [START_REF] Linchevski | Structure sense: The relationship between algebraic and numerical contexts[END_REF], and signs of pairing. Hence, data, for each task separately, was first divided into different groups depending on what ideas seemed to have governed the students while solving the task. In some cases, the students used different ways throughout the process of evaluating a single expression. Then, the different groups were combined based on the main underlying mathematical idea the solutions seemed to be based on. This resulted in a few different categories of foundational mathematical ideas.

In the second round of analysis, we focused only on the solutions that have a character of number pairing. Here, no fixed categories were defined prior to the categorization. The categories of ways of pairing numbers emerged from the data in the analysis process. Data were examined separately by the two researchers and discussed to resolve discrepancies. We did not see any significant difference in the data of Swedish and Greek students. Hence, in this study we have not separated the data of the groups. The numbers below are the combined total numbers for the entire data set.

Results and discussion

First, we note that only a minor part of the errors in the students' solutions is due to arithmetic errors. We do find the same type of errors as reported in previous studies (like, e.g., by [START_REF] Headlam | An investigation into children's understanding of the order of operations[END_REF], such as changing operators (e.g., multiplication is replaced by addition), changing numbers and simple arithmetic errors. But, the major part of the operations explicitly described in the data is correctly calculated. Even in the cases of changing numbers or simple arithmetic errors we find that it is easy to trace back the intention of the operationthe actual underlying mathematical idea. This is in agreement with the findings of [START_REF] Linchevski | Structure sense: The relationship between algebraic and numerical contexts[END_REF]. Hence, we can conclude that to a large extent the data (the result of the evaluated numerical expressions) originate from students' ideas on how numerical expressions should be evaluated.

Second, we find three main 'ideas' that seem to influence students' solutions: order of operations, sequential calculation and pairing. In our data, we find students who follow the convention and use the (correct) order of operations, and we find students who use a sequential calculation (left-toright), see Figure 2. As an example, a student that would have calculated the second expression (b in Figure 1) according to is considered to have followed a left-toright, sequential, strategy. This does not mean that the specific student is always following that strategy, but at least did so in this specific expression. Some students indeed are consistent in their calculations, in the sense that they calculate all four expressions in the same way, or with the same apparent idea behind the calculation. Most of the students who are consistent have used the order of operations. But there are also students who consistently have used a sequential calculation on all four expressions. For instance, two students consistently used one specific type of pairing (which we denote 'entangled pairing' further down in the report). In total, we find that the order of operations was used in 47% of all expressions, sequential calculations in 12% of the expressions, and that pairing seemed to be the main idea in 16% of the expressions in the students' solutions.

Typically, students' difficulties with numerical expressions from other aspects of precedence, like e.g. detachment (as from [START_REF] Linchevski | Structure sense: The relationship between algebraic and numerical contexts[END_REF] or visual salience [START_REF] Kirshner | Visual salience of algebraic transformation[END_REF], are included in the category named "other". In task c, , we note that many students evaluate the expression as . We believe this is directly related to the detachment-effect reported by [START_REF] Linchevski | Structure sense: The relationship between algebraic and numerical contexts[END_REF]. However, the number of solutions that seem to be based on detachment is also the reason why the "other" category is relatively much more frequent for the (c) expression, as shown in Figure 2.

Additionally, as will be shown further down (Fig 4 c), there are examples in the data where students have used a pairing strategy in the first term , but then, seem influenced by detachment and used that in the continuation of the calculation. Hence, it could be a combined pairing and detachment effect. In Figure 2, these examples have been counted as "pairing".

Particularly, in the data from the first task we found several students' responses that could be described as due to pairing. Although, we cannot exclude the possibility that there are other ideas that have influenced the students' behavior and made them evaluate the expression as .

When we explored the mathematical ideas, we found that the pairing mechanism can come in qualitatively different forms.

Pairing overall

The first type, and perhaps the easiest to separate from the others, is a mechanism of pairing throughout the entire expression. It is a pairing that goes overall. This occurs when the pairs span across different operations. It seems as when one pair of numbers has been "used", the attention is directed to the next pair. No consideration is taken to the precedence of operations. Instead, it seems that numbers are the focus of the attention. Figure 3 shows three different examples where students have used pairing overall to evaluate the expressions. The brackets were made spontaneously by the students. This type was particularly evident when there was an even number of numbers in the expression, as in the expressions ac (cf. Figure 1). We also find examples where students seem to have started with the pairing approach in the expression d with odd number of numbers. One such example is shown in Figure 3(c). Here, the student has paired all the numbers, except the last, which could not form a pair in the same way. However, we also find examples in the data of students' solutions that indicate that single numbers are explicitly included in the later step of the calculation, as, e.g., the student who explicitly writes , with brackets inserted by the student. Hence, the student has indicated pairing as a mechanism that is applied overall the entire solution.

Pairing within terms

Another type of pairing-mechanism observed in the data is pairs formed within the terms, that is, not across different operations. To some extent, this preserves the order of operations, not breaking the rule that multiplication should precede addition/subtraction. In that respect, this category appears to be some intertwining of the order of operations and the pairing mechanism. Figure 4 shows three examples of pairing within terms that we attribute to this category. Pairing within terms is particularly evident when three factors are to be multiplied. Then many students seem to invent some pair where one number is disregarded, as in Figure 4(a), or added/multiplied in the end, similar to the pairing mechanism reported by [START_REF] Liebenberg | Focusing on structural aspects of numerical expressions[END_REF]. However, some students used the numbers twice, as in Figure 4(b,c). Hence, in the latter two examples, there is a double counting of one of the numbers (the centre factor of the term). In Figure 4(b), the number "4" is used both in and in , separately. In Figure 4(c), the number "3" is double counted. To us, it appears as if this type of pairing can be such a strong idea in the students' minds that they form pairs regardless if the numbers are used more than once, as in Figure 4(b,c).

Unfortunately, our test did not include any term with four factors. We did not foresee this alternative. If such example would be used, we hypothesize that one could find students who paired the factors within that term without the invention of double-counting.

Entangled pairing

In our data, we find students who are operating with each adjacent pairs of numbers. So far, we have only seen two students do like this, but, on the other hand, they are very consistent in this strategy. In this type of pairing mechanism, the operations are subordinate. No consideration is taken to what operations there are in the expression. In the end, the pairs are added. Examples of this type of idea of how numbers should be paired is shown in Figure 5. Hence, compared to the "pairing overall"-strategy, this type of pairing disregards the operations even more, as pairing is conducted across all operations. In addition, similarly to the "pairing within terms"-type of strategy, double counting of numbers does not seem to be a problem for these students. In this "entangled"-strategy all numbers (except the first and last) are operated with twice. In the end, the results of each operation are added

Concluding remarks

Apart from [START_REF] Liebenberg | Focusing on structural aspects of numerical expressions[END_REF] and a short example by de Villiers ( 2015), previous research on students' solutions to numerical expressions have focused on arithmetic errors and on sequential (left-to-right) strategies. However, our data suggest that pairing is a much more abundant idea in students' evaluation of numerical expressions than previously shown, at least for longer expressions. We find that 16%, as an average over all students' solutions, are solved with pairing being the underlying mathematical idea. Hence, it seems that the idea of pairing could be as frequent as the idea of sequential calculations.

Moreover, we find three qualitatively different types of pairing ideas. We find ways of pairing bound within terms, and ways of pairing applied in the expressions overall. In addition, we have shown that expressions with an odd number of numbers in the terms can trigger an apparent adaption of the pairing to the order of operationsthe double counting of numbers within terms. The origin of pairing seems a deeply rooted idea. We find that despite being taught the order of operations, a considerable amount of the students apply the idea of pairing nevertheless. In addition, as previous research has shown, the mere proximity of numbers can trigger these to be calculated first [START_REF] Gómez | Symbol spacing and its influence in school children's calculation of mixed-operation arithmetic expressions[END_REF][START_REF] Jiang | Spatial factors influence arithmetic performance: The case of the minus sign[END_REF][START_REF] Landy | Proximity and precedence in arithmetic[END_REF]. Hence, pairing could be closely related to deep cognitive processes. Nonetheless, pairing is a persistent and strong idea and needs to be considered and handled when designing teaching on how to perceive and calculate arithmetic expressions.

Figure 1 :

 1 Figure 1: The four different numerical expressions used in this study (in the Greek version of the test, the expressions were numbered , , , , respectively)

Figure 2 :

 2 Figure 2: Strategies used to evaluate the different numerical expressions

Figure 3 :

 3 Figure 3: Three examples of students' answers where pairs have been formed across the entire expression without consideration of what operation should precede the other ("pairing overall")

Figure 4 :

 4 Figure 4: Examples of student answers indicating pairing considered within terms

Figure 5 :

 5 Figure 5: Three examples of where each adjacent pair of numbers are evaluated in an entangled way.In the end, the results of each operation are added