Cecilia Kilhamn
email: cecilia.kilhamn@edu.uu.se

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

Algebraic thinking in the shadow of programming

Keywords: Algebra, algebraic thinking, programming, computational thinking

This paper calls attention to how the recent introduction of programming in schools interacts with the teaching and learning of algebra. The intersection between definitions of computational thinking and algebraic thinking is examined, and an example of a program activity suggested for school mathematics is discussed in detail. We argue that students who are taught computer programming with the aim of developing computational thinking will approach algebra with preconceptions about algebraic concepts and symbols that could both afford and constrain the learning of algebra.

Introduction and background

In the wake of introducing programming into school mathematics curricula [START_REF] Mannila | Computational thinking in K-9 education[END_REF] we have identified the intersection of algebraic thinking and computational thinking as an important area to explore (Figure 1). Specifically, this intersection has come to the fore in Sweden, where programming has been inserted into the national mathematics curriculum as part of the core content of algebra. Computational thinking (CT) is a fairly new concept in educational research, first introduced by Papert in 1996. The term involves the kind of thinking skills needed to understand and capitalize on computers. Since Wing launched CT as a didactical term in 2006, researchers in computational science as well as mathematics education have attempted to define it. For example, [START_REF] Hoyles | Revisiting programming to enhance mathematics learning[END_REF] describe CT as incorporating four central thinking skills: decomposition, pattern recognition, abstraction and algorithmic thinking. Programming, on the other hand, can be seen as a problem-solving activity that can be used to address the different aspects of CT (Manilla et al, 2014). Although CT is generally considered to encompass more than programming, teaching programming requires the use of CT (Hickmott, 2017). Moreover, programming is a feature of CT that does not necessarily involve writing code in any particular computer language (Bocconi, Chioccarello & Earp, 2018). Programming is thus a more inclusive term than coding, and seen as an activity in which students develop computational thinking. In this paper, we limit the discussion to aspects of CT that can be developed through programming. The aim of this paper is to raise the question of how programming in schools, with the goal of developing CT skills, may potentially interact with, afford or constrain students' development of algebraic thinking. A contemporary international movement in school development is to implement programming in the curriculum. This has been done in various ways [START_REF] Mannila | Computational thinking in K-9 education[END_REF]. For instance, in England, "Computing" was introduced as a new subject in the curriculum, while Finland and Sweden adopted a blend of cross-curriculum and single subject integration with the strongest link to mathematics [START_REF] Bocconi | The Nordic approach to introducing computational thinking and programming in compulsory education[END_REF]. Unlike other countries, Sweden included programming in the mathematics curriculum in close connection to algebra through all grade levels. The curriculum change was released in August 2017 and was taken in effect throughout the country by August 2018. There is extensive literature from the last three decades tackling issues of teaching and learning programming at university level [START_REF] Grover | Computational thinking in K-12: A review of the state of the field[END_REF], but except for the early work on LOGO programming there are few studies of teaching programming in the early school years [START_REF] Blikstein | Pre-college computer science education: A survey of the field[END_REF][START_REF] Hickmott | A scoping review of studies on computational thinking in K-12 mathematics classrooms[END_REF]. Furthermore, most educational research about programming concerns the learning of programming per se, rather than learning mathematical ideas through programming. One exception is the ScratchMaths research program, which in light of the recent curriculum changes in England, set out to explore the relationship between computational thinking and mathematical thinking in general, identifying mathematical ideas such as place value, proportional relationships, coordinate systems, symmetry and negative numbers in ScratchMath activities [START_REF] Benton | Bridging primary programming and mathematics: Some findings of design research in England[END_REF]. We wish to broaden the discussion about "learning to program" or "programming to learn" by raising questions about how learning (and knowing) programming influences the learning of algebraic thinking. Until now, research on computational thinking and algebraic thinking has run on separate tracks, but the Swedish case offers a great opportunity to investigate the intersection of these two research domains in the wake of programming activities.

Programming activities in Swedish schools

The description of programming in the Swedish revised curricula and on-line material provided by the government,1 focuses in grades 1-3 on the use of symbols to construct and follow stepwise instructions either without computers (unplugged) or with simple robots such as Bee Bots or screen equivalents such as Lightbot. 2 In grades 4-6 algorithms are created in visual programing environments such as Scratch. 3 and in grades 7-9 other programming languages are introduced. In the visual environments, an algorithm is often described as a "recipe" or a "function", and if variables are introduced, they are mostly non-numerical. While a function is described as a correspondence between two sets in abstract algebra, a function in school algebra is commonly interpreted as a co-variation of numerical values possible to represent in a graph [START_REF] Usiskin | Conceptions of school algebra and uses of variables[END_REF]. If students in the early grades meet functions as a correspondence between non-numerical input and output variables in programming activities, they will have experiences that may challenge the school algebra conception of function where one quantity is said to vary depending on another quantity, represented in a cartesian graph. Due to the limited space of this paper, we leave to the side further investigations of early programming activities in relation to algebra and focus our discussion on programming in text-based environments commonly included in later grades.

Algebraic thinking and programming

As algebraic thinking (AT) slowly made its way to the early school years during the last decades, the definition of what algebra is and what comprises algebraic teaching has become wider and more inclusive [START_REF] Kieran | Teaching and learning algebraic thinking with 5-to 12-year-olds[END_REF]. A broad definition of AT based on work with young children is suggested by [START_REF] Radford | The emergence of symbolic algebraic thinking in primary school[END_REF] who describes AT as analytically dealing with indeterminate quantities using culturally and historically evolved modes of symbolizing. Although this definition excludes operating on specific numbers, it includes generalized arithmetic such as analyzing properties of addition and multiplication or analyzing the structure of an algorithm (for example using the distributive property to work out the partial products in a multiplication algorithm). We set out to investigate if this definition of algebraic thinking excludes or includes aspects of computational thinking that are at the heart of programming. We note that in a contemporary state-of-the-art publication written by members of an ICME topic study group on the teaching and learning of early algebra at ICME 13 [START_REF] Kieran | Teaching and learning algebraic thinking with 5-to 12-year-olds[END_REF] there is no discussion at all about the connections between the learning of algebra and computer programming. The word "programming" is never mentioned and "computer" appears only in one paragraph (out of 17 chapters), where Mason claims that computer languages have emerged from algebra, and that these "expressive and manipulable languages […] invoke algebraic thinking and algebraic awareness" (Mason, 2018, p. 335). [START_REF] Kaput | What is algebra? What is algebraic reasoning[END_REF] describes early algebra in terms of three content strands including: i) the study of structures; ii) the study of functions; and iii) the application of a cluster of modelling languages both inside and outside of mathematics. In addition, symbolizing is described as core aspects of algebra (ibid). We find that all three content strands could also be aspects of computer programming. As an example, constructing code to generate all odd numbers presupposes reflection on what an odd number is and how it could be described and represented. Such an exercise is a study of structure, modelled in a specific programming language. Although the study of structure in algebra (e.g. generalized arithmetic) is emphasized in more recent work (e.g. [START_REF] Kieran | Teaching and learning algebraic thinking with 5-to 12-year-olds[END_REF], such structures have not been related to structures that becomes visible when breaking down complex calculations into small steps in a computer program. Although structure seems to be fundamental in both domains, little has been said in previous research literature about the study of structures embedded in computer algorithms in relation to the study of structure described as an aspect of algebraic thinking.

Investigating the intersection between computational and algebraic thinking.

For the purpose of comparison, we have tried to illustrate similarities between two definitions of CT and two definitions of AT by aligning them in Table 1. The definitions are chosen as examples of contemporary definitions in the respective areas, proposed by researchers often cited in the literature. We see that both CT and AT deal with structure, generalization and symbolization in various ways, describing culturally developed organized symbol system where structures can be studied and generalizations made. Questions that arise are if structure, symbolization and generalization represent the same thing in the two domains, and if the syntax and semantics of programming languages are aligned with or divert from corresponding algebraic symbolism. Surprisingly, the definitions of CT do not explicitly mention functions or variables, which are fundamental concepts for AT (see Table 1). In programming activities, these concepts appear up front, where an algorithm in a program is often described as a function, and variables are omnipresent. Perhaps they are simply taken for granted in CT. Algorithmic thinking is not mentioned as part of AT, although it involves features that are fundamental in algebra, such as the ability to analyze a problem, specify it precisely, and find adequate basic actions [START_REF] Futschek | Algorithmic thinking: The key for understanding computer science[END_REF].

Further, Futschek mentions the ability to construct a correct algorithm using the basic actions, to think about possible special and normal cases, and to improve the efficiency of an algorithm. We see such abilities as prominent when algebra is used as a problem-solving tool where the basic actions are for example rules of equation solving or symbolic manipulation. Finally, according to Futschek, "algorithmic thinking has a strong creative aspect", which undoubtedly should be true for algebraic thinking as well. Can we assume that students who develop their algorithmic thinking skills through programming also develop their algebraic thinking skills? Could algebra teaching explicitly relate to and build on algorithmic thinking? Note that algorithmic thinking in CT concerns the how and why of algorithms, as opposed to what [START_REF] Hiebert | What research says about the NCTM Standards[END_REF] calls algorithmic reasoning, which is to recall a solution algorithm based on "rote thinking". Next, we give an example of a programming activity in order to highlight some points of intersection with AT, in particular the issue of symbolization, taken from government-provided teaching materials available on line in connection to the revised curriculum in Sweden [START_REF] Rolandsson | Aktiviteter, del 4. I Matematikundervisning med digitala verktyg II, åk 7-9[END_REF].

An example of differences in syntax and semantics in algebra and programming

In this activity, the students are going to write a short computer program inspired by the algorithm of "Sieve of Erastothenes", i.e., an algorithm for finding prime numbers. Examples are given in two programming languages: Python and Javascript. The latter is given in Figure 2 below. By using this algorithm, one can find out if a given integer is a prime number or not. We have highlighted six places in the code that illustrate some crucial passages regarding syntax and semantics. These places are circled and marked from 1-6 in Figure 2 and will be discussed next.

Let us begin to consider the equal sign. At the places 1, 2, 3 and 5 in Figure 2 the equal signs all represent assignments of different values to variables. For instance, at place 2 the program declares the variable a and assigns it the value 2, i.e., gives the variable the value 2. This usage of the equal sign differs from how it is normally used in algebra, where the equal sign is a symbol for an equivalence relation, and different from arithmetic, where pupils tend to interpret the equal sign as an operator symbol ( equals ) rather than a relation [START_REF] Kieran | Concepts associated with the equality symbol[END_REF].

In algebra, it would be meaningless to write , since it is not true for any value of . However, in programming the expression makes sense since the equal sign is understood as the assignment "add 1 to the value " (see place 3 in Figure 2). This is often used when a program needs to loop through a range of consecutive integers, as in Figure 2 when the program loops from 2 to the input value stored in the variable "number".

Let us now consider place 4 in Figure 2, where we have two consecutive equal signs (==). This is the only place in this code where the equal sign is not used as an assignment. Instead, == is used as a relational operator that tests if two entities are equal. In our example the program tests if the result of the modulo operation "number % a" is equal to 0, i.e. if the remainder is equal to 0 after dividing the input value "number" with the loop variable . Here, the double equal sign is used in a similar way as the relational equal sign in algebra.

Many instructions in programming, such as for and if, have built-in checks for truthfulness that closely corresponds to the algebraic equality without including an equal sign in the syntax. An example of this can be seen at place 6 in Figure 2 where the instruction if(check) should be understood as 'if the variable check is equal to "true", then…'. Perhaps one might argue that in programming the relational equality sometimes is applied although it is invisible in the code, similar to the convention in algebra where we write as [START_REF] Hewitt | Young students learning formal algebraic notation and solving linear equations: Are commonly experienced difficulties avoidable?[END_REF]. Let us now demonstrate some differences regarding the meaning of the variable concept in programming and algebra. First, in programming variables can be used to hold non-numbers such as in places 1 and 5 where the variable check holds the Boolean value "true" and then "false". Second, in programming a variable can change value during the execution of the program. This is illustrated at the places 1, 3 and 5 in Figure 2. As already mentioned, at place 3 the variable increases with 1 for every execution ( ). This means that if, for instance, is equal to 3, the computer calculates the value 3+1 and then is assigned the new value 4, i.e. changes value from 3 to 4. As we are about to see this differs from the algebraic context. In algebra, the concept of a variable is broad and has been further categorized as unknown, variable (in a more nuanced meaning) and placeholder [START_REF] Ely | Unknown, placeholder, or variable: What is x?[END_REF]. The term unknown corresponds to a determinate quantity in an equation that remains to be solved. Clearly, an unknown cannot change its value as it is predetermined from the equation. The term variable is meant to correspond to a varying quantity, typically and in the equation . It can be perceived that the variables in this equation can change value, but the change is related to different cases of the problem. E.g. "Case 1: we assume is 1, then will be 2", and then "Case 2: we assume is 2, then will be 5". Within each of these cases we allow the variable to have a determinate value which will make into an unknown (that can be easily resolved). Observe that we cannot assign the value 1 and later derive another value for within the same case, that would constitute a contradiction ( and imply

) and force us to eliminate this case. The same argument can be applied for the term placeholder. This is different from programming where instructions are executed in order, one after the other. When an instruction assigns a new value to a variable it does not constitute a new case, instead we are just changing the value recorded in a specific place in the memory of the machine (referred to by the variable).

Discussion

We have shown above how programming activities can interact with students' development of algebraic thinking in different ways. One of these concerns symbolization. In algebraic thinking, the use of culturally and historically evolved modes of symbolizing [START_REF] Radford | The emergence of symbolic algebraic thinking in primary school[END_REF] is described as a prominent feature. Notice the plural, indicating that algebraic thinking is not exclusively bound to alphanumerical symbols and conventional algebraic syntax, and could thus hypothetically incorporate also programming languages. While many programming languages use similar, but not equivalent, symbol systems, there is no common syntax. For example, in some languages the short hand notation ++ (the increment operator) is used instead of . Nevertheless, the differences between the established algebraic symbol system and new programming languages could potentially entail difficulties for students learning algebra.

In order to clarify some of these differences we have, from the example in Figure 2, compiled three categories regarding the meaning of some operations (the semantics) and the symbolic representations of these operations (the syntax) that appear in algebra and programming: 1) Different symbols represent the same meaning: e.g., the modulus operation, represented by in algebra and in programming; 2) The same symbol represents different meanings: e.g., the equal sign (=) representing relational equality in algebra and assignment in programming; 3) Symbols with no corresponding meaning in the two domains: e.g., approximately equals ( ) in algebra and increment with one (++) in programming. These differences could afford the development of algebraic thinking through contrasting examples and awareness of accuracy, or constrain it if the teacher is unaware of the different experiences students have. In particular, the equal sign in algebra is already a cause for much didactical effort in helping students switch from an operational to a relational meaning [START_REF] Kieran | Concepts associated with the equality symbol[END_REF][START_REF] Kieran | Teaching and learning algebraic thinking with 5-to 12-year-olds[END_REF]. When programming introduces yet another meaning, this could cause more confusion. In contrast, the use of different symbols to represent the same meaning could help students understand that the meaning of a mathematical concept does not depend on the symbol used to represent it.

The meaning of variable is another issue found in the intersection of CT and AT. The use of nonnumerical variables (such as true/false) in programming affords a wider understanding of the concept of variable than the traditional use of variables as quantities, and does not correspond to the school algebra idea that letters always represent numbers.

The focus on algorithms and operational execution of instructions in programming may interact with the focus on structure in algebra. As shown in Table 1, both CT and AT value the study of structure, decomposition and pattern recognition. Approaching questions about structure from two perspectives could serve as an affordance for developing AT if differences in structure are noticed, and discussed. Algorithms are important in programming but absent in descriptions of AT. In school mathematics, the idea and use of algorithms have changed greatly since the introduction of digital tools in the 1980's when traditional algorithms were replaced by an increased emphasis on number sense and conceptual understanding. The fact that the term algorithm was removed from the description of arithmetic in the Swedish national curriculum in 2011 and reinserted in 2017 as an aspect of the core content of algebra in connection to programming, implies a shift of emphasis from a procedural use of algorithms, to a conceptual understanding in terms of "algorithmic thinking" [START_REF] Futschek | Algorithmic thinking: The key for understanding computer science[END_REF]). An open question for further empirical research is, if this newly awoken interest in algorithms will induce a study of the structures embedded in algorithms, and as such afford the development of algebraic thinking in terms of increased focus on structure and generalizations, or if it will tip over towards a more procedural approach to mathematics as a whole, thus working as a constraint to AT.

We argue that future algebra students who are taught programming in school, will encounter working with variables, structure and patterns, and engage in simplifying, generalizing and symbolizing in computer related contexts. We therefore urge for an awareness of what these experiences might do to how algebra is conceptualized and talked about in school, and what preconceptions teachers need to be aware of. We trust that empirical research into the intersection of AT and CT, as programming evolves in school practices, will highlight possible pitfalls and supply algebra teaching with new input.

Figure 1 :

 1 Figure 1: The intersection of CT and AT

Figure 2 :

 2 Figure 2: An algorithm for deciding if a given integer is a prime number

Table 1 : Some features of contemporary definitions of CT and AT that are found in the intersection Features in the intersection Computational thinking (CT) Algebraic thinking (AT) Grover & Pea 2013 Hoyles & Noss 2015 Radford 2018 Kaput 2008

 1

	Structure	Structured problem	Decomposition.	Deals with indeterminate	The study of structures
		decomposition	Pattern	quantities in an
		(modularizing)	recognition.	analytical manner
	Symbolization	Symbol systems		Using idiosyncratic or	Acting on symbols within an
		and representations		specific culturally and	organized symbol system
				historically evolved	through an established
				modes of symbolizing	syntax
	Generalization and	Abstractions and	Entailing	Deals with indeterminate	Systematically symbolizing
	abstraction	pattern	abstraction	quantities in an	generalizations
		generalizations.		analytical manner
		Conditional logic.
	Functions and				The study of functions,
	variables	*	*		relations and joint variation
	Algorithms	Algorithmic	Algorithmic
		notions of flow of	thinking	**	**
		control

* there is no mention of functions or variables in these definitions of CT ** there is no mention of algorithms in these definitions of AT

https://larportalen.skolverket.se/#/moduler/1-matematik/Grundskola/Lärare%20i%20matematik

https://www.bee-bot.us; http://lightbot.com

http://scratched.gse.harvard.edu


N Verdin

S Charton

H Roussel

D Maurel

Taylor-Couette contactor as miniaturized column for solvent extraction pilot tests

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

Introduction & motivation

[1] Lanoë, J.-Y., (2002), Lanoë, J. (2002). Proceedings ATALANTE 2012 [2] Nemri, M., Climent, E., Charton, S., Lanoë, J.-Y., Ode, D. (2012), Chem. Eng. Res. Des. 91, pp.2346Des. 91, pp. -2354 [3] [3] Nakase M., Takeshita K. (2012).. Procedia Chemistry 7, pp. 288-294 [4] Amokrane, A., Charton, S., Sheibat_Othman, N., Becker, J., Klein, J.-P., Puel, F., (2014) Can. Jal. Chem. Eng., 92(2), pp. 220-233 eferences Scale-reduction is a strong issue for R&D studies, especially in the nuclear industry. In this aim, a miniaturized liquid-liquid extraction column based on Taylor-Couette flow has been designed and tested [1]. The device consists in two concentric cylinders. The inner cylinder (OD 17 mm) is rotating while the outer one (ID 20 mm) is fixed. The column is operated at counter-current and the emulsion of the two immiscible liquid phases is provided by the shear-stress prevailing within the counter-rotating vortices. It is able to operate with the same separation performances as a 10 times higher industrial pulsed column (operated at some m 3 /h) but with a flow-rate as low as some 100mL/h, thus involving minimal amounts of solvent and of radioactive materials.

The main objective of this study is to investigate the ability of numerical simulation to predict the wall effects on the emulsion achieved in

  Taylor-Couette flows. Many possible configurations in Liquid-Liquid Extraction R&D studies are tested: direct (O/W) or inverse (W/O) emulsion, high viscosity. W) Low viscosity oil (O1) High viscosity oil (O2) Kinematic viscosity ν (m 2 .s -1 Regardless of the oil phase viscosity  Droplets of non-wetting oil achieved (optimal)  Rings of wetting oil at the walls (unadapted) When wall properties are optimal  Emulsification hindered by viscosity (larger droplets) -Good agreement between numerical results and experimental observations: a wetting dispersed phase forms annular rings (local flooding) while a non wetting one allows efficient emulsification -VOF simulations are time consuming (approx. 150h / case on 8 processors) and Drop Size Distribution (DSD) cannot be predicted -Coupled CFD-PBE simulation (Euler-Euler + DQMOM), based on previous work on pulsed column [4] are currently implemented to predict the DSD h = 3    Parametric Study The study focuses on contact angle θ and the viscosity of the organic phase ν Taylor-Couette contactor as miniaturized column for solvent extraction: numerical study of liquid-liquid emulsions Nicolas VERDIN, Sophie CHARTON, Hervé ROUSSEL, Didier MAUREL CEA, DEN, DTEC, SGCS, F-30207 Bagnols-sur-Cèze, France However, in this small gap configuration, surface effects are increased and might be detrimental for the process efficiency. Preliminary tests, based on 2 columns with different wall properties (hydrophilic / hydrophobic) were made. Bad emulsification and local flooding were highlighted in some cases. Effect of viscosity (continuous phase)  Requires (very) high rotation speed (same Re)  Overcomes wall effect (no effect of the contact angle) When wall properties are optimal  Emulsification more difficult with viscous oil • Wall interactions Wall Adhesion option Contact angle of the dispersed phase θ = 20° (non wetting drop)  = 110° (wetting drop)




Per Nilsson
email: per.nilsson@oru.se

Andreas Eckert
email: andreas.eckert@oru.se

Color-coding as a means to support flexibility in pattern generalization tasks

Keywords: Pattern generalization, visualization, color-coding, time-limitation, flexibility

This study investigates how color-coding can support processes of flexibility in figural pattern generalization tasks. A lesson from a Grade 8 class serves the case for our investigation. The lesson is part of a larger research project, which is based on the iterative research methodology of design experiments and involves a total of six lessons, distributed over two classes (three lessons in each class). The study shows how coloring can encourage students to move from recursive strategies, like successive addition, and support processes of flexibility in linking algebraic expressions and the meaning of n to visual structures of an expanding figural pattern.

Introduction

In the development of algebraic thinking, students need to learn to see connections between the structure of a pattern and how the pattern can be described by a general algebraic expression [START_REF] Rivera | Visual templates in pattern generalization activity[END_REF]. It is crucial to discern a pattern unit when making such connections, around which a pattern structure can be modeled in relation to the place order of the elements in a pattern series [START_REF] Nilsson | Flexibility and coordination among acts of visualization and analysis in a pattern generalization activity[END_REF].

Based on a constructivist methodology, educational research in mathematics has significantly contributed to our developing understanding of how students may obtain algebraic generalizations involving figural and numerical patterns. However, our knowledge is less extensive on how teaching can support the learning of pattern generalization [START_REF] Rivera | Visual templates in pattern generalization activity[END_REF]. In other words, if students need to learn to connect visual and symbolic elements effectively in relation to a structural unit, there is a need for further investigations on learning activities that can support that coordination to take place [START_REF] Rivera | Visual templates in pattern generalization activity[END_REF]. In this study we aim at making a contribution to such investigations by investigating how color-coding in task design (Watson & Ohtani, 2015) can support students' pattern flexibility in the visualization of structures in figural pattern generalizations (Figure 1). We address the research question: How can color-coding be implemented in task design to support flexibility in students' visual structure reasoning, in order to reach a general expression of a figural pattern?

Theoretical background

Individuals see patterns differently depending on how they perceive the structure of the pattern and conceptualize pattern-units [START_REF] Rivera | Visual templates in pattern generalization activity[END_REF]. Investigating 8-13 year old students' generalization of linear pattern [START_REF] Stacey | Finding and using patterns in linear generalising problems[END_REF] found that the constant difference property was largely recognized and used as a unit to find the n th element by a recursive process of successive addition of this property. Stacey also found that some students were using proportional reasoning, according to the constant difference property. However, when adopting proportional reasoning, Stacey saw that students may tend to ignore additive elements of a pattern; a significant number of students used a direct proportional method, determining the n th element as the n th multiple of the difference. For instance, the constant difference is three in Task 5 (Figure 1). Applying a direct proportional method of the n th multiple of the difference results in the generalization 3n, which excludes the extra dot that must be added.

Successive addition of the constant difference property is limited to linear patterns. To develop general expressions of patterns where the difference between subsequent elements in the pattern is not constant (e.g. quadratic patterns) one needs to adopt visual structure reasoning (cf. [START_REF] Rivera | Visual templates in pattern generalization activity[END_REF][START_REF] Stacey | Finding and using patterns in linear generalising problems[END_REF]. In Task 5, the constant difference is three and, applying the strategy of successive addition, the number of dots for the fourth figure is found from 4+3+3+3. If we generalize this strategy we reach the expression 3n+1. However, in visual structure reasoning we define the unit by partitioning the figures in appropriate parts, in relation to the place order of the figures [START_REF] El Mouhayar | Variation in strategy use across grade level by pattern generalization types[END_REF]. In Task 5 we can visually structure the third figure in threes. In other words, the third figure of the pattern is composed by three threes plus one extra dot. Similarly we find out that the fourth figure of the pattern consists of three fours, plus one extra dot. Obviously we come to the same general expression, 3n+1, as in the additive situation. However, important to note is that n has a different meaning in the two situations. In successive addition n gains meaning as the number of constant differences, related to the place value of the n th element in the series, whilst in visual structure reasoning n is a structural feature of a figure.

Generalization tasks are often grouped in near generalization tasks and far generalization tasks [START_REF] El Mouhayar | Variation in strategy use across grade level by pattern generalization types[END_REF]. Near generalization tasks involve finding the value of a step that is close to previous steps and far generalization tasks consist of determining the value of a step that is relatively far from given steps. Previous research reports that while near generalization tasks are accessible to a majority of the students, solving far generalization tasks is not. In the present study we will use both near and far generalization tasks, to challenge pattern flexibility, towards the formulation of a general expression, which is based on visual structure reasoning.

It has been observed that, once students have fixed on a pattern in a certain way, it can be hard for them to give up their perception [START_REF] Lee | An initiation into algebraic culture through generalization activities[END_REF]. The key to success of seeing an algebraically recordable pattern seems to be at the first stage of pattern perception, where a certain flexibility is necessary to hit on a mathematically recordable pattern [START_REF] Zazkis | Generalization of patterns: The tension between algebraic thinking and algebraic notation[END_REF]. Visualizations Miriam was thinking according to the picture. Pick the expression you believe best matches her picture.

of a pattern can provide strong support for the symbolic representation of a pattern [START_REF] Healy | Visual and symbolic reasoning in mathematics: Making connections with computers? Mathematical Thinking and Learning[END_REF]. Perception, or visualization, in pattern generalization is however not a passive process [START_REF] Nilsson | Flexibility and coordination among acts of visualization and analysis in a pattern generalization activity[END_REF]. Getting insight to a problem situation includes an operative, analytical activity [START_REF] Deliyianni | A theoretical model of students' geometrical figure understanding[END_REF][START_REF] Duval | Geometry from a cognitive point of view[END_REF], like structuring a figure into appropriate parts and/or transforming a figure into another figure [START_REF] Rivera | Visual templates in pattern generalization activity[END_REF]. In the present study we will investigate how a teacher can use color-coding as a visual means for supporting flexible, visual structure reasoning, to reach a general expression for a figural pattern.

Method Design research methodology

This study is part of a larger research project that investigates teaching and learning of pattern generalization. The project follows the methodology of design experiments [START_REF] Cobb | Design experiments in educational research[END_REF], involving a lesson series of three consecutive lessons. The project team consists of two researchers, the authors of the paper, and two teachers (Teacher A and Teacher B). Teacher A taught a class of 13-year old students and Teacher B taught a class of 14-year old students. All lessons were developed and analyzed according to prospective, reflective and retrospective analyzes [START_REF] Cobb | Design experiments in educational research[END_REF]. Each lesson was conducted twice: First in Teacher A's class and then in Teacher B's class. The changes made to Teacher B's teaching were based on reflective analyses of what happened in the corresponding lesson of Teacher A's teaching. In the present study we focus exclusively on Lesson 1.

Lesson 1 involved seven tasks, distributed through Socrative1 , and aimed to support flexibility in visual structure reasoning in pattern generalization. All tasks in Lesson 1 were situated in the task context of the growing chair-pattern (Figure 1) [START_REF] Rivera | Visual templates in pattern generalization activity[END_REF]. In Lesson 1, a time-limited far generalization task and color-coding were at the heart of the lesson. We focus, particularly, on Lesson 1B (Teacher B's class). However, the results of the study are based on a retrospective analysis of both Lesson 1A and Lesson 1B, and the relationship between them. All lessons were video-recorded from three different positions; one placed at back of the classroom, one at front of the classroom and one zooming in on a specific group.

The answers of each task were made available to both the teacher and the students, which were intended to reveal students' reasoning and create an opportunity for whole class discussion where students could react on each other's reasoning. No color-coding was added to the first four tasks.

The first task was a near generalization task. Here the students were asked to figure out the number of dots in the 4 th figure of the chair-pattern. The second task was a semi-far generalization task, where students were asked to figure out the number of dots in the 9 th figure. The third task and follow up question were far generalization tasks about the number of dots in figure 1000. In the third task the students had a limited amount of time. They were asked to reflect on whether it was possible for them to complete the task within a minute with the method they had been using in the second task. Until now the students were expected to reason according to the constant difference property, since that was how they had solved similar tasks in the past. Task four asked students about the meaning of n in pattern tasks. The task was supposed to challenge the students to start to reflect on how n can be connected to place order in a pattern. Figure 1 shows the fifth task, with a color-coded visual structure, designed to stimulate students' visual structure reasoning. The sixth and seventh task flipped task 5 around and asked the students to color-code different versions of algebraic expressions. First the students were asked to apply n+(n+1)+n on the seventh figure; then, in task seven, they color-coded for example (n+1)+(n-1)+(n+1). The tasks, and their relationship, aimed at furthering the students' flexible reasoning; to be flexible in ways of discerning visual structure units by reversing and expanding their reasoning in task 5. To open up for whole class discussions, the students were asked to share their color-coding by publishing them in Padlet2 .

Method of analysis

The analysis aimed to investigate how color-coding can support flexibility in students' visual structure reasoning, in order to reach a general expression of a figural pattern.

The analysis followed in chronological order, according to the sequence of the seven tasks. In the first step, we coded and grouped the data according to the ways in which students distinguished and built their reasoning on a pattern unit and connected a pattern unit to n, to the place order of the figure in question. Next, we investigated how they changed their way of reasoning about the chairpattern during the lesson. Particularly, we looked in detail at how color-coding supported students to move from the strategy of successive addition to discern a unit in the visual structure of a pattern, connected to n, i.e. to the place order of the figure in question.

Result and analysis

Successive additive reasoning and n as a symbol for place order of any figure in the pattern

In the first task, 75% of the students claimed that the fourth figure contained 13 dots. Many students discerned three as the constant difference property of the pattern and used this property in successive addition. In order to challenge successive addition and to prompt proportional reasoning the teacher introduced Task 2 and 3. In Task 2 the students were asked to determine the number of dots in the ninth figure. While the students were working on the task, the teacher walked around in the classroom and listened to the groups. He noted that most students conclude that the ninth figure contains 28 dots and that they reached this by successively adding up by three. In order to challenge successive additive reasoning, the students were asked figure out the number of dots in the 1000 th figure within one minute, based on the strategy they used in Task 2. This task stimulated monitoring processes [START_REF] Stylianou | On the interaction of visualization and analysis: the negotiation of a visual representation in expert problem solving[END_REF]; it challenged the students to reflect on their strategies and how efficient they were. The students submitted their individual answers in Socrative and about 40% of the students believed that they could not determine the number of dots in figure number 1000 within one minute. Strategies that the student found too time-consuming were, for instance, to draw all figures and count the dots or to successively add threes. The formula 3n+1 had not yet been expressed by the teacher or any student. But from students' group-work the teacher observed that some students expressed aspects of such reasoning. The teacher asked Charlie to explain:

Charlie: Because, as it [the pattern] constantly increases by three, you need to multiply thousand by three and then plus one.

Charlie gives further reasons for his strategy by showing how it works in an example were the number of dots of a figure is known to the class or can be visually confirmed:

Charlie:

For instance, if we look at figure 3 [within Figure 1 above], it [the pattern] constantly increases by three, and we take three times three it will be nine and then, as it is ten [dots] there [pointing at the board], you need to add one more. And, then it is the same with the other; thousand multiplied by three plus one.

After the three first tasks we have reason to believe most students in the class came to understand that they can multiply the number of a given figure with three and then add one to reach the number of dots of any figure. However, a general algebraic expression involving n had not yet been formulated. In Task 4 the students were asked to post the meaning of n in Socrative. The most common posts were expressions of 3n+1 and that n stands for any figure. The teacher asked, "Do we agree, everyone who agrees raise their hands, that n stands for any figure?" Almost all students raised their hands. That the students had a sense of n, as a symbol that stands for any figure, was assumed to be necessary for moving on to Task 5. Task 5 introduced color-coding as a means for supporting students to discern a unit in the visual structure of a figure, in relation to the place order of the figure in question.

Supporting structural flexibility by color-coding

The symbol n was now understood by most students in the class, as a symbol for place order of any figure in the pattern. However, up until now, n was seen as a variable, determining how many constant differences one should multiply to reach the number of dots in any figure. Tasks 5, 6 and 7 were designed to change perspective on n. The tasks were designed to stimulate students' pattern flexibility by pushing them to make sense of n in the visual structure of the expanding chair-pattern.

The students posted individual answers on Task 5 in Socrative. The result were rather even between alternative b) and c) (Figure 1). The students were then asked to discuss in their groups and to explain to each other how they were reasoning. After the group-discussion the students were asked to post an individual answer again and now the majority of students chose alterative b). George was asked to present his reasoning at the whiteboard: According to flexible processes, George then said that alternative c), displaying n+1+n+n, is the same as alternative b). The only difference is the order of the signs. He implicitly manipulated the expression by visualizing a bracket in the expression (n+1)+n+n.

A number of students answered a), displaying 2•n+n+n. The teacher asked why alternative a) was chosen. Charlie responded:

Charlie: It works for the first figure but not for the other.

We believe this is a relevant interpretation made by Charlie and, supported by his observation, we also suggest that deciding on alternative a) is based on calculations rather than on visual structural reasoning like the one George showed an example of. The color-coding implemented in the design of Task 5 helped George to externalize and show in public his way of reasoning. The design supported George to make sense of n in the visual structure of the pattern and to communicate his meaning making to the class by visually linking parts of the expression to the corresponding parts of the figure.

The class turned to Task 6, to color the seventh figure of the pattern according to n+(n+1)+n. In Task 5 the class observed and followed how George linked an algebraic expression to the colorstructure of the pattern figures. Task 6 aimed at letting students make sense of and generalize this idea according to a new figure. The students worked in pairs. As they finished, they posted and made their solutions public at the whiteboard. The group-solutions witnessed the fact that several students had changed perspective of the meaning of n, to see n in the structure of the figures. The teacher picked out Rikki's solution to Task 6, where the link between the algebraic expression and the coloring was evident and proper according to the task (Figure 2). For some reason the teacher did not succeed in presenting Rikki's own solution on the board. Instead, he showed an uncolored picture of the three first figures of the pattern and asked Rikki to guide him through the coloring of (n+1)+(n-1)+(n+1) in the second figure of the pattern (Figure 3). That the teacher failed in up-loading Rikki's solution became something positive from a teaching perspective. Now Rikki was forced to give details of her reasoning so the teacher was able to understand how he should color the figure. Moreover, in the dialogue between the teacher and Rikki, the rest of the class was provided an opportunity to follow, in detail, how the solution was constructed.

Concluding discussion

In this paper we addressed the research question: How can color-coding be implemented in task design to support flexibility in students' visual structure reasoning, in order to reach a general expression of a figural pattern? In answering the research question, we have shown how coloring in pattern generalization can stimulate processes of flexibility [START_REF] Stacey | Finding and using patterns in linear generalising problems[END_REF] in linking algebraic expressions and, particularly, the meaning of n to visual structures of expanding figural patterns.

The present study sheds new light on task-design to support reflection on visual structures [START_REF] Rivera | Visual templates in pattern generalization activity[END_REF] and pattern flexibility [START_REF] Nilsson | Flexibility and coordination among acts of visualization and analysis in a pattern generalization activity[END_REF]. In pattern generalization, the first stage of pattern perception is crucial, where a certain flexibility is necessary to hit on a mathematically recordable pattern [START_REF] Zazkis | Generalization of patterns: The tension between algebraic thinking and algebraic notation[END_REF]. To move beyond linear pattern generalizations one needs to discern a unit from the visual structure of a pattern, connected to the place order of the pattern-figures [START_REF] El Mouhayar | Variation in strategy use across grade level by pattern generalization types[END_REF]. The present study shows how color-coding can support such processes of flexibility by turning students' attention to the relationship between an algebraic expression and visual structures of the pattern. The color-coding prompted the students to change perspective of n, from a number variable (how many threes to multiply) to a visual unit in the structure of a figure. In particular, the analysis shows how implementing coloring in pattern generalization task, supported students in discerning and making sense of n in the relationship between n in the structure of a general expression and n in the structure of a figure.

The role of far generalization tasks, and the added condition of time-limitation, has not been the main focus in the present study because of limited empirical evidence. However, the analysis indicates that it might be worth re-considering the role of far generalization tasks in the learning of pattern generalization. Previously, research has focused on students' difficulties to account for far generalization tasks and how students find them more difficult than near generalization tasks [START_REF] El Mouhayar | Variation in strategy use across grade level by pattern generalization types[END_REF]. Near generalization tasks can be handled by recursive reasoning, which is not as applicable to far generalization tasks. Recursive reasoning necessitates that the preceding value of the pattern is known. So, to determine the value of the 1000 th figure, one needs to know the value of figure number 999. The present study also shows that students find a far generalizing task more difficult than near generalization task due to recursive reasoning, coming in the form of successive addition. Challenging students to a far-generalization task did not support multiplicative, proportional, reasoning in a direct sense. However, the analysis shows how it made students reflect on and examine limitations of a recursive, additive approach and come to be responsive to proportional reasoning. Hence, instead of just accounted for students' difficulties, the far generalization task provided the teacher a context for exposing the limitation of successive additive reasoning and for highlighting and emphasizing multiplicative structures. In the present study the teacher created such a context by making Charlie's reasoning in Task 2 public.

The outcome of the present study calls for further research in the realm of task and lesson design (Watson & Ohtani, 2015). Still many questions are open on a patterning approach to algebra. We need to know more about tasks that challenge students' flexible reasoning and engage students in visual structure reasoning. We encourage research to further explore coloring as principles for designing tasks in authentic practices of algebra teaching. The current study also indicates the need to carefully sequence tasks in pattern activities, in order to encourage processes of reflection and flexibility that move students' understanding towards multiplicative, structural, reasoning, which can be used to handle patterns that go beyond linear patterns.

  figure 1 figure 2 figure 3 1 + 2 + 1 2 + 3 + 2 3 + 4 + 3

  would figure this out first [pointing to (n+1) in the expression]. First I was thinking n plus one. It is this [pointing to the two blue dots in the first figure of the pattern]. This is the blue, one could say. n plus one. And, both of them [pointing to the two red dots in the first figure] are n.

  taking n+1 [saying something audible] and, then I draw that in the middle [The teacher points to the vertical line of eight dots that Rikki has colored in blue]. n is the number of the figure, thus seven [the teacher is nodding in confirmation]. But then it was plus one so, then I did eight. And then it was n on the other sides. Seven down and seven up [the teacher moves his hand over the picture to illustrate down and up of the seven dots Rikki is referring to]. In Task 7 the students were working in pairs and used Paint to color other expressions in the pattern. The teacher noted that Rikki solved the first subtask of Task 7, coloring (n+1)+(n-1)+(n+1).

Figure 3 :

 3 Figure 3: Rikki's color-solution to Task 7 Figure 2: Rikki's color-solution to Task 6

Socrative is an online student response system where teachers post questions that show up on students' devices and allows them to post answers. Answers become instantly available to the teacher, which s/he can choice to make available to the students.

Padlet is an online virtual pinboard that allows students to publish contributions that can be instantly available for the whole class on the white board or on students' own devices.


Hervé Roussel

Sophie Charton

Design, development and testing of miniature Liquid-Liquid Extraction contactors for R&D studies in nuclear environment

An innovative miniature stage-wise contactor:

the "Next Generation" mixer-settler

Individual stages arranged on CEA-patented rails [START_REF]Patent FR 2 831 075 A1, Extracteurs liquide-liquide montés en batterie[END_REF] constituting modular batteries, all settlers facing the same direction in order to facilitate interphase observation Mixer volume : 6 mL Settler volume: 20 mL Flowrate (A + O): up to 500 mL/h Four-stages battery Individual stage Proper choice of mixing chamber materials for emulsion type control (stainless steel for oil in water or Teflon for water in oil)

Optimized W-shaped aqueous weir (right) to achieve a smooth continuous flow over a wide range of flowrates (20 mL -500 mL)

Micropump for flowratecontrolled internal recycling at each stage (light or heavy phase)

1 cm 1 cm
Miniature Rushton-type turbine for combined pumping and emulsification in the mixing chamber This turbine provides high pumping effect

The miniature mixer-settler must achieve the same functions as larger ones, but the design must be adapted to the tiny size (dominant role of wall effects)

1 cm
The stage efficiency is high (≈90% compared to a theoretical stage) and, as expected, growths with the residence time and the agitation speed This mixer-settler meets the major expectations and is under optimisation to improve its useability in high activity environment This particular flow occurs in the gap between two concentric cylinders (the inner one rotating, the outer one being fixed). Above a critical speed, liquids move in contra-rotating vortices. It creates high agitation while axial mixing remains very low [START_REF] Nemri | Experimental and numerical investigation on mixing and axial dispersion in Taylor-Couette flow patterns[END_REF] .

When applied to a biphasic system, the Taylor-Couette flows creates the emulsification needed for mass transfer [START_REF] Davis | Liquid-liquid extraction between rotating cylinders[END_REF] . The Taylor-Couette column is suited for testings with radioactive solutions:

• the gap between the cylinders is narrow (≈ 2 mm)  small flowrates ≈ 100 mL/h • HETS (height equivalent to a theoretical stage) is much lower as in pulsed column  Taylor-Couette column is smaller as the equivalent pulsed column. A Taylor-Couette column can achieve the same separation as a 10 times higher pulsed column, while working with 10 times lower flowrates.

 Conclusion



  Mass transfer efficiencyTest system: extraction of nitric acid by a monoamide-based solvent

  study for nuclearized miniature mixer-settlers (F. Dutilh, CEA/MAR/DTEC/SDTC/LTAP) column (75 cm) Pulsed are widely implemented for nuclear fuel treatment, but their reduction to laboratory scale is extremely challenging: • diameters below 15 mm are not achievable  lower limit for flowrates ≈ 1 L/h • the pulsation generates high axial dispersion  pilot columns must be nearly as high as industrial ones (≈ 8 m)  Taylor-Couette flow  Taylor-Couette extraction column



  Mass transfer efficiencyTest system: extraction of nitric acid by a TBP-based solvent)


Norbert Oleksik
email: oleksik@dmuw.de

Transforming equations equivalently? -theoretical considerations of equivalent transformations of equations

Keywords: Grundvorstellungen, aspects, mathematical concepts, mathematics education

Although algebraic transformations of equations are commonly used to identify solutions of equations, it remains unclear, which transformations are explicitly considered to be equivalent transformations, and which are not. Therefore, equivalent transformations of equations are analyzed, by using the notions of concept image, concept definition, aspect, and Grundvorstellungen, as well as their relations to each other. Different formal concept definitions of equivalent transformations of equations are examined to get a first impression of possible characteristics of concept images. As a further approach, Grundvorstellungen are derived from aspects of the equivalence of equations and put in relation to the theory of concept images. Different contradictions can be identified using these notions, which are possible sources of misconceptions.

Equivalent transformations of equations

Learners of mathematics are being introduced to equations in primary school. Simple arithmetic equations are to be solved by adding, subtracting, multiplying, and dividing numbers. In secondary schools, with the shift from arithmetic to algebra, variables are expressed by the use of letters. This is usually when pupils are introduced to more formal techniques of solving equations. They are transformed to other equivalent equations, so that a formula can be applied, or the solution is apparent (as in the case of the equation . Since equivalent transformations of equations are a central tool for solving equations, the concept of equivalent transformations of equations should be well defined and known to every mathematician and teacher of mathematics. When considering the following two equations, the question whether the initial equation was transformed equivalently provides room for discussion:

Squaring the expressions of an equation is usually not considered to be an equivalent transformation of an equation (e.g. Vollrath & Weigand, 2007, p. 214), although it may lead to an equivalent equation, as in the example above. In order to analyze why some transformations are considered to be equivalent transformations, while others are not, even though they share the feature of leading to an equivalent situation, the theory of concept image and concept definition by [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF] is used. This theory will then be related to Grundvorstellungen, which has a long tradition in Germany, due to the works of vom Hofe (2016), using a model provided by [START_REF] Greefrath | Aspects and "Grundvorstellungen" of the concepts of derivative and integral[END_REF], which offers a further approach to the concept image. Subsequently the geometric model for teaching transformations of equations as described by [START_REF] Filloy | Educational algebra[END_REF] is being discussed in regard to the following theoretical considerations.

Concept image and concept definition

An individual's view on a mathematical idea is not only shaped by the subject matter itself but also by further experiences. Therefore, [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF] differentiate between mathematical concepts and cognitive processes which resemble an individual's understanding of the concept. For the latter the term concept image is used "to describe the total cognitive structure that is associated with the concept" (Tall & Vinner, 1981, p. 152). When looking at the two equations above one may associate functions, variables, or something else with them. Given that within literature squaring is not considered to be an equivalent transformation of equations, it seems rather unlikely that the concept of equivalent transformation of equations is associated with the two equations above. It may even be declined to consider this as an equivalent transformation. Which leads to the assumption that the individuals concept image of equivalent transformations does not contain the operation of squaring. The question that naturally raises, from this little theoretical discourse, is how equivalent transformations of equations are defined. For this matter the term concept definition is used, which is "a form of words used to specify that concept" (Tall & Vinner, 1981, p. 152). A concept definition of an individual is referred to as personal concept definition and is not necessarily correct or existent (Tall & Vinner, 1981, p. 152). In contrast to the possibly erroneous personal concept definition, the formal concept definition is "accepted by the mathematical community at large" (Tall & Vinner, 1981, p. 152). It should be mentioned, that one can be able to state a "correct formal concept definition whilst having an inappropriate concept image" (Tall & Vinner, 1981, p. 153). It is the concept image that gives meaning to the concept definition (Greefrath et al., 2016, p. 103), while the definition may be just a memorized sequence of words. Prior to planning an empirical investigation of individual concept images and personal concept definitions of the concept of equivalent transformations of equations, it seems wise to examine the concept of these transformations theoretically by comparing different formal concept definitions.

Formal concept definitions of equivalent transformations of equations

Equivalent transformations of equations can be defined as transformations which do not alter the solution set (Vollrath & Weigand, 2007, p. 242). In this definition no further information is given about the characteristics of a transformation. The key is the property of not losing or adding any solutions of (or to) the equation, which is crucial when it comes to solving equations. This would lead to the conclusion that the example above is an equivalent transformation of equations. Walz (2017, p. 361) defines equivalent transformations of equations in contrast to transformations which possibly alter the solution set, which justifies the claim of squaring not being an equivalent transformation. When taking this condition into account, which is rarely made explicit, there are only few types of transformations left to be considered as equivalent transformations of equations. Adding (and subtracting) numbers, as well as multiplying (and dividing by) numbers other than zero always, lead to equivalent equations (Langemann & Sommer, 2018, p. 172). This demand for generality of not altering the solution set can only partially be granted when it comes to adding or multiplying expressions containing variables. One might accidently divide by zero and lose one or more solutions:

Another definition highlights the reversibility of the transformation and states that transformations, which can be reversed, without changing the meaning are equivalent transformations (Langemann & Sommer, 2018, p. 172). Since it is not explicitly stated what meaning refers to in this context, this definition appears rather doubtful. Considering that an equation containing variables determines a set of elements for which it becomes a true statement (Kirsch, 1987, p. 106), it can be assumed, that meaning refers to the solution set as well. [START_REF] Drouhard | Symbols and language[END_REF] translate the German term "Bedeutung" (with respect to Frege), as denotation (instead of meaning as done in the translation above) as a subdomain of meaning and refers also to the truthfulness of equations (pp. 232-234). Therefore, it can be assumed that this is the case in this definition. None the less it is interesting in the manner that it highlights reversibility as a property of equivalent transformations.

When comparing these different definitions, it appears crucial, that the solution set does not change when an equivalent transformation is applied. This is the defining property of an equivalent transformation, in the first definition discussed. The second one extents it by demanding a certain generality and claiming it never alters the solution set. Although the third definition is not quite clear, it highlights another characteristic: equivalent transformations have to be reversible. It is noteworthy, that all of these definitions relate, more or less explicitly, to the solution set.

Aspects and Grundvorstellungen an additional approach to the concept image

Given that (formal) concept definitions are the basis of concept definitions (Greefrath et al., 2016, p. 103) predictions about possible individual concept images can be derived from them. In order to draw further conclusions of possible characteristics of concept images the construct is approached by using the idea of Grundvorstellungen (GV) as an addition to the concept definition, which may also be part of a concept image (Greefrath et al., 2016, p. 103). Greefrath et al. (2016, p. 101) defined the term in the following manner: "A Grundvorstellung of a mathematical concept is a conceptual interpretation that gives it meaning" (for a more detailed discussion, see vom [START_REF] Vom Hofe | Grundvorstellungen" as a category of subject-matter didactics[END_REF]. Unlike the concept image, a GV relates to aspects of mathematical concepts (Greefrath et al., 2016, p. 101), while the concept image describes the whole cognitive structure that is associated with a mathematical context. The term aspect of a mathematical concept is defined as "a subdomain of the concept that can be used to characterize it on the basis of mathematical content" (Greefrath et al., 2016, p. 101). Because of these relations, identifying and examining aspects of equivalent transformations of equations offers new insights into possible characteristics of a concept image. Since the equivalence of equations appears to be the defining property of the equivalent transformations of equations and therefore characterizes it, it will be examined in greater detail in the following section.

Aspects of equivalence of equations

Equivalence of two equations means that they share the same solution set (Kirsch, 1987, p. 108). Another possibility to determine the equivalence of two equations is by transforming one equation into the other using valid transformations (Oldenburg, 2016, p. 11). Since this takes into account transformational activities, the focus will be on this aspect of equivalence, which from here on will be referred to as transformational equivalence. This aspect of equivalence requires defining valid transformations. For this matter, two similar but different types of rule sets are described in the following.

Balance rules

One type of rulesets for transforming equations are the balancing rules, which are expressed in regard to expressions (Malle, 1993, p. 219):

The main feature of this ruleset is, that the two expressions on each side of the equal sign is being modified in the same way. The procedure is often visualized, by using scales, which stay in balance when the same amount of weight is added to or withdrawn from both sides. This is the reason why Malle (1993, pp. 220-221) refers to them as balancing rules. It should be noted though, that these rules can be visualized in various other ways as well.

Elementary transformation rules

Another set of rules are the elementary transformation rules, which are again expressed in regard to expressions (Malle, 1993, p. 219):

( ( These rules may appear very similar to the balancing rules but require three expressions to perform a transformation of an equation. The given expressions of an equation therefore determine what kind of transformations can be performed. This may be seen as an advantage for choosing this ruleset over the balancing rules when teaching [START_REF] Malle | Didaktische Probleme der elementaren Algebra [Didactical problems of elementary algebra[END_REF].

In contrast to the elementary transformation rules, the balancing rules allow working with additional expressions, while the elementary transformation rules only allow using the given expressions. The latter may be considered as special kinds of balancing rules, which require fewer calculations [START_REF] Malle | Didaktische Probleme der elementaren Algebra [Didactical problems of elementary algebra[END_REF] as can be seen in the following example:

Grundvorstellungen and aspects of equivalent transformations of equations

An aspect is the basis of an individual GV, while at the same time the GV gives the subject meaning to the aspect (Figure 1). When applying this idea to equivalent transformations of equations, two different GVs can be derived, from the aspects described previously (Figure 1). It should be noted, that any GV that is derived from the balance rules can also be derived from the elementary transformation rules, given that they can be considered a shorter version of the balancing rules, which cover only certain cases. Using elementary transformation rules, the transformation may be perceived as a transposition of the expression according to the "change side -change sign rule" (Kieran, 1992, p. 400). This means that you exchange the operator in front of an expression with the inverse operation, while moving the expression to the other side of the equation. This idea of transposition appears to be coherent with the findings of [START_REF] Henz | Does bodily movement enhance mathematical problem solving? Behavioral and neurophysiological evidence[END_REF], in which their hypotheses, that algebraic manipulations take place in an algebraic symbol space (and is linked to movement) is empirically supported. The scheme of this rule can be represented in the following manner (Malle, 1993, p. 218):

Balancing GV

The balance rules may be interpreted as "doing the same thing on both sides". This can be represented visually by a scale that keeps the balance or as a more abstract scheme: . Overly generalizing a scheme, may lead to errors in performance (Malle, 1993, pp. 172-175). In this case it may lead to the misconception, that squaring two expressions of an equation always leads to an equivalent equation. This demonstrates how problematic generalizations of a scheme may be, if no further knowledge is being acquired about when it is appropriate to apply a certain rule.

Example: geometric model and GVs

So far models for teaching equivalent transformations of equations have only been mentioned, without discussing them in greater detail. One well known model is the balance model using scales which highlights that operations should be carried out on both sides of an equation and therefore may foster the balancing GV. In addition to this model, [START_REF] Filloy | Educational algebra[END_REF] propose a geometric model for visualizing linear equations and algebraic transformations of these equations (p. 100). Within this model "algebraic expressions are represented as lengths and areas of rectangles (x and its coefficients are lengths, and their products and the independent terms are areas)" (Filloy et al., 2008, pp. 100-101) (see Figure 2). The purpose of this model within the study is to teach early algebra learners how to transform an equation, with the variable occurring on both sides of the equation, such as (with being positive integers and ) into the equation [START_REF] Filloy | Educational algebra[END_REF]. This can be done by coloring and comparing areas (see Figure 2). Ideally followed by the conclusion, that the two left over areas must be of the same size [START_REF] Filloy | Educational algebra[END_REF]. Considering that the colored areas can be faded out, while focusing on the white areas, it seems like this approach embodies a subtraction of from both sides of the equation and therefore leads to the assumption that this model fosters the balancing GV ("doing the same thing on both sides"). At the same time the transformation appears to rather fit the elementary transformation rule than the balancing rules. Taking a closer look at the examples provided by [START_REF] Filloy | Educational algebra[END_REF] from their interviews, it is noticeable that the interviewee appears to move the expression from the left side of the equation, to the right side, while changing the sign (e.g. Figure 3). Therefore the, geometric model seems to rather foster the transposing GV than the balancing GV according to the data samples provided (Filloy et al., 2008, pp. 108-110;Filloy & Rojano, 1989, pp. 22-23).

Summary and discussion

Squaring is usually not considered an equivalent transformation of equations, although it may lead to an equivalent equation. Since this seems to be contradictory it raises the question, what does one consider to be an equivalent transformation of equations? In order to approach this question, the cognitive structure, that leads to certain assumptions, is described as concept image. Different formal definitions of equivalent transformations of equations were examined, all of which contained the idea of not altering the solution sets. Since this demand is fulfilled by squaring in the first example, this might not be what individuals consider as defining property. What seems more suitable in this case is the demand of generality, that an equivalent transformation does always maintain the solution set.

The second approach to identify factors that influence the concept image is, by using the notions of aspects and GVs. Considering that the equivalent transformations of equations should lead to equivalent equations, the transformational equivalence is examined in greater detail. This aspect requires defining valid operations, in this case the elementary transformation and the balance rules. These two rule sets are the basis of the transposing GV and the balancing GV. Using these rules, it can be argued that squaring is not covered by them and therefore might not be captured within the aspect of transformational equivalence. Then again it can be considered using the balancing GV, that it is an equivalent transformation of equations, even then when it is not, because "you do the same thing on both sides". Judging in which cases a transformation leads to an equivalent equation, requires the ability to recognize the equivalence of equations. Discussing the geometric model of equations and transformations of equations, it was pointed out that it is not necessarily the model that indicates a certain GV, but the individual's interpretation and use of the model. Rather than formulating a GV for every model, the approach used within this paper suggests deriving them from aspects of a given concept. This seems useful, given that different models may affect our transformation processes in the same manner. However, models cannot be abandoned, if we want to avoid that algebra learners simply perform meaningless transformations. Therefore, the GVs which are described in a rather schematic manner need to be linked to a model. It is rather the reasoning accompanying the model, than the model itself which is important.

Considering that algebraic transformations of equations are located at the transition from arithmetic to algebraic thinking (Filloy et al., 2008, p. 96), it seems like very little attention has been paid to this topic in recent research. Simply addressing it with reference to a certain model or a certain set of rules does not seem to be sufficient. More importantly is the link between models, the underlying concept and the individual, which needs to be addressed with closer regard in the future. It appears useful to examine the concept of equivalent transformations of equations in greater detail. As the discussion above shows, definitions of the concept do not necessarily grasp what is meant by equivalent transformations. But if we do not make explicit what we mean, when we talk about equivalent transformations of equations, it carries the risk that this implicit information is not acquired by the learner. Does it not make learning and teaching more difficult? The considerations presented within this article can help identifying sources of misconceptions and give an insight to educators about limitations of definitions, rules and models describing equivalent transformations of equations.

Figure 1 :

 1 Figure 1: Relations between aspects and GVs (extracted from Greefrath et al., 2016, p. 103) applied to equivalent transformations of equationsTransposing GV

  Figure 2: in the geometric model, with being colored (Filloy et al., 2008, p. 102)

Figure 3 :

 3 Figure 3: Example of a learner operating on the unknown, prior to being introduced to formal techniques of transforming equations (Filloy & Rojano, 1989, p. 110)All in all, it is not possible to draw a general conclusion on what GV is fostered by the use of the geometric model. Especially when considering that the data examined here was not intended to answer which GV it fosters. None the less it should be noted that GVs are not necessarily a question of model, but of what is made of it by the individual. It is possible that another person understands this model as subtracting the same expression from both sides of an equation.

Figure 4 :

 4 Figure 4: Relations between aspect, Grundvorstellung, concept definition, concept, and concept image (by Greefrath et al., 2016, p. 103) applied to equivalent transformations of equations


Eder Pinto
email: pinto.eder@gmail.com

Bárbara M Brizuela
email: barbara.brizuela@tufts.edu

María C Cañadas

Eder Pinto

Representational variation among elementary school students. A study within a functional approach to early algebra

Keywords: Representation, generalization, linear functions, functional thinking

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

Introduction

Functional thinking-which has been found to facilitate the introduction of algebra in the early grades-focuses on the relationships between two or more covarying quantities [START_REF] Blanton | A learning trajectory in 6-year-olds' thinking about generalizing functional relationships[END_REF]; these relationships can be expressed through different representations. Representations, which form an integral part of how students think about functions, "can denote and describe material objects, physical properties, actions, and relations, or things that are far more abstract" (Goldin & Shteingold, 2001, p. 4). Therefore, representations help to structure and expand students' thinking [START_REF] Brizuela | Multiple notational systems and algebraic understanding: The case of the "best deal" problem[END_REF]. Broadly, our interest is focused on types of representations used by elementary students when working with covarying quantities. Some researchers have described how elementary school students working with problems that involve single linear functions use, represent, and understand the relationships involved in a given problem (e.g., [START_REF] Brizuela | Multiple notational systems and algebraic understanding: The case of the "best deal" problem[END_REF]. The originality of this study lies in the exploration of the types of representations used by elementary school students when working with problems involving different types of linear functions. More specifically, this paper analyzes the answers given by eight third-and eight fifth-grade students participating in a CTE, and their answers in semi-structured interviews, after the CTE. The research question is: how do students' representations vary when working with different types of linear functions? Based on this research question, we define two specific aims to describe the type of representations used by these students when: (a) solving problems which involved three types of functions: y=x+a; y=ax; and y=ax+b; and (b) answering questions regarding specific values and when asked to generalize the relationship between variables.

Background

The relationship between generalization and representation is central: both are intrinsic to algebraic thinking and consequently to functional thinking. According to [START_REF] Kaput | Algebra from a symbolization point of view[END_REF], representations-a socio-cultural vehicle used to generalize-enable students to build and complete the ideas that help them reason about general statements and compress multiple instances into the unitary form of a single statement that symbolizes the multiplicity. Thus, generalization is the "act of creating that symbolic object" (p. 20).

Several studies highlighted the role of representations in generalizing the relationship between two variables. According to some findings, 8 to 10-years-old students, instructed to use algebraic notation, represented the linear relationship y=x+a by mainly using algebraic notation rather than natural language [START_REF] Carraher | Early algebra is not the same as algebra early[END_REF], while other studies reported that fifth graders (10 to 11-years-old) spontaneously used algebraic notation to generalize a problem that involve y=mx+b [START_REF] Pinto | Generalization in fifth graders within a functional approach[END_REF]. Another study distinguished between the representations used by students when working with specific values and when generalizing, concluding that students who do not use algebraic notation when they represent a generalization do "not yet have a representational means to compress multiple instances into a unitary form that could symbolize these instances" (Blanton et al., 2015, p. 542). Further research is therefore needed on how the representations used by students vary when working with different linear functions, and how they differ depending on whether they are working with specific values or generalizations.

The types of representations that can be used by elementary school students to solve problems involving linear functions include: (a) natural languageoral; (b) natural languagewritten; (c) pictorial; (d) numerical; (e) algebraic notation, (f) tabular; and (g) graphic [START_REF] Carraher | Early algebra is not the same as algebra early[END_REF]. Considering the suggestions of early algebra literature, we stress the role of natural language because it is considered as a useful scaffold to understand symbolic representations [START_REF] Kaput | Towards a theory of symbol use in mathematics[END_REF][START_REF] Radford | Gestures, speech, and the sprouting of signs: A semiotic-cultural approach to students' types of generalization[END_REF], and helps to broaden students' understanding about functions, improving their abilities to solve problems [START_REF] Macgregor | The effect of different approaches to algebra on students' perceptions of functional relationships[END_REF].

Method

This study forms part of a broader project that explores functional thinking among elementary school students in Spain.

Students

Two groups were intentionally selected for the first year of the study: 24 third-grade (8 to 9-yearsold) and 24 fifth-grade (10 to 11-years-old) students. The students had not worked previously with such problems. Then, we interviewed eight students in each group to obtain a deeper understanding of how they responded to problems involving relationships between two variables.

Data collection: CTE and interviews

The data were collected during the CTE and individual interviews. The CTE and interviews objectives were to: (a) explore how students relate the variables involved in a problem involving a linear function; (b) introduce different types of representations to express functional relationships; and (c) explore students' generalization when working with functional thinking tasks.

A four-session CTE was designed for each grade during the last term of 2014/2015 period, with each session lasting approximately 60 minutes. Each CTE session was divided into three parts. First, we introduced the context of the problem, highlighted the representations given and asked the students questions about specific values to ascertain whether they had understood it. Second, the students were given individual worksheets with questions about specific values and generalization related to the problem. Third, the researchers led a classroom discussion around the responses to some of the questions on the worksheets. During the 2015/2016 period, eight students from each grade were interviewed in two 30-minute interviews. The interviewees were deliberately selected to include children who had performed differently during the CTE and it provided a way to more closely detail individual students' representations from the CTEs. Table 1 shows the general context of problems posed, types of functions, and types of representations introduced during the CTE and interviews.

Timing

Problem posed CTE Third Session 1. María and Raúl are brother and sister. María is the elder. We know that María is 5 years older than Raúl (y=x+5) (natural languagewritten and table ). Sessions 2 and 3. Carlos earns 3 euros for each T-shirt he sells (y=3x) (natural languagewritten, table, and graphic). Session 4. Different corridors are composed of white and grey tiles. All the tiles are square and of the same size and are to be laid in the following pattern: (y=2x+6) (natural languagewritten, and pictorial) [START_REF] Küchemann | Algebra[END_REF].

Fifth Session 1. Carlos earns 3 euros for each T-shirt he sells (y=3x) (natural languagewritten and table ). Session 2. Carla earns 3 euros for each T-shirt she sells, while Daniel earns double that amount for each T-shirt and has saved 15 euros (y=3x; y=2x+15) (natural languagewritten and table ). Session 3. A grandmother tells her grandson that she has some money to give him and proposes two deals (y=2x; y=3x-7) (natural language -written) (Adapted from [START_REF] Brizuela | Multiple notational systems and algebraic understanding: The case of the "best deal" problem[END_REF]. Session 4. Different corridors are composed of white and grey tiles. All the tiles are square and of the same size and are to be laid in the following pattern: (y=2x+6) (natural language written and pictorial) (Adapted from [START_REF] Küchemann | Algebra[END_REF]. Interviews Fourth Interview 1. It costs 2 euros to enter a car park and 1 euro per hour to park there (y=x+2) (natural language-oral, algebraic notation). Interview 2. Elsa conducts a train. Three passengers get on at each stop (y=3x+1) (natural languageoral, algebraic notation).

Sixth

Interview 1. A geometric pattern with a different number and arrangement of points (y=4x+1) (natural languageoral, algebraic notation, and pictorial). Interview 2. Two telephone rates have different costs (y=10x; y=5x+60) (natural languageoral and algebraic notation).

Table 1: Problems posed in CTE and interviews

The problems proposed involved different linear functions (y=x+a, y=ax, and y=ax+b); some of them were selected from previous studies and others were designed by the research team, considering different types of linear functions. The questions concerning each problem were sequenced according to the inductive reasoning model of [START_REF] Cañadas | A proposal of categorisation for analysing inductive reasoning[END_REF], which structures questions from specific values to generalization. For instance, the fourth task for third and fifth grade included questions regarding:  Specific values. For example, "How many grey tiles do they need for a corridor with 10 white tiles?", and  Generalization. For example, "The workers always lay the white tiles first and then the grey tiles. How can they calculate how many grey tiles they need in a corridor where they've already laid the white ones?"

The data analyzed in this paper were the students' CTE and interview worksheets and the transcriptions of the video-recorded interviews.

Data and analysis categories.

The first step was identifying the types of representations used by the students. Figure 1 gives an example of each type of representations1 used by students when they answered in different tasks. The representations used by the students to solve problems were first identified and then analyzed considering: the type of linear function in each problem (y=a+x, y=ax, and y=ax+b), and the students' answers when working with specific values and when generalizing.

Results and Discussion

We present the findings for the types of representations used by each group of students, considering our research questions. As the data in Table 2 shows, the students tended to use the representation introduced in the problem itself, regardless of the type of linear function involved. Specifically, oral and written natural language prevailed in this group of eight students. Mario's response during the first interview to a problem with a function of the type y=x+b is an example.

Third and fourth grades

Interviewer: Let's suppose I don't know how many hours I'm going to park, but I tell you that I'm going to be there for x hours (…). How can I know how much it's going to cost me?

Mario: (…) Well, if you're there for x hours, you add 2. For all the hours [the car is parked] you add 2.

As exemplified in the above extract, the prevalence of oral language is an essential element in learning to recognize and understand a function [START_REF] Macgregor | The effect of different approaches to algebra on students' perceptions of functional relationships[END_REF]. The use of a numerical representation was also observed at least once in all the problems involving specific values. This makes sense, since students were being asked about specific, and not general values. Numerical representation was as frequent as natural language in students' answers to problems with functions of the form y=ax+b.

During the interviews one year later, some students spontaneously arranged the values of the variables in tabular form. Figure 2 shows how Susana arranged the data to explore the specific values in a problem of the type y=mx+b (interview 2). As Figure 2 shows, Susana arranged the specific values in two columns separated by a dash (-). She listed the number of stops on the left and the number of people on the train on the right. This table, according to [START_REF] Martí | Tables as cognitive tools in primary education[END_REF], shows that "data be organized (categorization, establishing correspondences) in a certain spatial layout" (p. 134). Susana's table is a novel way to represent, relate and understand the values involved in the problem. Spontaneous tabular representations were observed during the interviews in the problems involving functions such as y=x+a and y=ax+b.

The eight students used different representations depending on whether they were working with specific values or generalizing. During the CTE, the variety of representations was wider when they were exploring the relationship with specific values (written natural language, numerical, algebraic notation) than when generalizing, when they used written natural language only. One possible explanation for this difference may lie in the complexity inherent in generalizing the relationship between two variables, as suggested by other authors (e.g., [START_REF] Radford | Gestures, speech, and the sprouting of signs: A semiotic-cultural approach to students' types of generalization[END_REF]. Nonetheless, natural language would appear to be the way these students explain general rules. During the interviews, the students used the representation present in the problem to generalize the relationship, whereas for specific values they used numerical and tabular representations as well. We can see in Table 3 that these students tended to use the same type of representation as introduced in the problem. Significantly, in the tiles problem (CTE, y=mx+b type function), two of the students used algebraic notation spontaneously. Camila's answer to the first question associated with that problem is reproduced in Figure 3.

Fifth and Sixth Grades

1. How many grey tiles do they need for a floor with five white tiles? They need 16 grey tiles. Formula: (X x 2)+6=16. X = number of grey tiles Camila used a representation not introduced in the problem. Although some of the questions posed in the earlier sessions involved the use of algebraic notation, here the student used it to express the general rule, although she was answering a question about a specific value. This type of representation was also observed in the interviews, specifically in four students' answers to the problem in the second interview, in which the functions were of the type y=ax and y=ax+b. That spontaneous use of algebraic notation differed from the findings for the third-and fourth-grade students and suggests that some of the sixth graders were quicker to adopt the use of algebraic notation to express general relationships between two variables.

Most of the students in fifth and sixth grades generalized using (oral or written) natural language or algebraic notation, whereas for specific values they used a wider variety of representations (oral and written natural language, pictorial, numerical, algebraic notation and tabular). Numerical representation prevailed in these students' solutions to the specific value questions. Once again, this make sense given that were being asked about specific values.

Conclusions

This paper seeks to shed light on how the representations used by intermediate and upper elementary school students (8 to 12-years-old) vary when solving problems that involve different types of linear functions. Specifically, in both groups, when the students generalized the functional relationship, they used the same type of representation as introduced in the problem (natural language or algebraic notation). The variety of representations was broader when they worked with specific values. These findings reveal that while they were aware of different types of mathematical representations (used when working with specific values), when generalizing they only used two. As noted by other authors (e.g., [START_REF] Brizuela | Multiple notational systems and algebraic understanding: The case of the "best deal" problem[END_REF], this highlights the importance of teaching representations in elementary school to enable students to gradually assimilate them as they are constructing meaning for different types of representations when generalizing and grasping the meaning of functional relationships.

Concerning this study' specific objectives, no major differences were observed in the representations used in one type of function or another by either third-and fourth-or fifth-and sixth-graders. That may be because both groups had participated in a CTE in which they solved problems involving different linear functions. Tabular representations appeared spontaneously in some of the third/fourth-grade students' answers, whereas algebraic notation appeared among the fifth/sixth-graders. Significant use of numerical representation was found in both groups and it was consistent with what the problem asked them. The major difference between the two groups was in the types of representations used when working with specific values and when generalizing. When generalizing, the third-and fourth-graders used a narrower variety of representations than the older group of students. This seems to suggest that in this sample, the students in the higher grades of elementary school had more resources from which to draw when expressing general rules.

Figure 1 :

 1 Figure 1: Examples of types of representation in students' answers

Figure 2 :

 2 Figure 2: Susana's answer when working with specific values

Figure 3 :

 3 Figure 3: Camila's answer to the tile problem (CTE, session 4)

Table 2 : Representations used by third-and fourth-graders

 2 Table 2 lists the frequency for each type of representations used by students in CTE and interviews, considering different linear functions and when working with specific values and when generalizing.

				CTE		Interview
		Representation	y=x+a	y=ax	y=ax+b	y=x+a	y=ax+b
	Specific values N-oral				8*	7*
		N-written	2*	4*	5*
		Pictorial
		Numerical	4	7	6	1	7
		Algebraic	4*	5*
		Tabular				1	2
		Graphic
	Generalization	N-oral				7*	6*
		N-written	5*	2*	2*	1
		Pictorial
		Numerical
		Algebraic				6*	5*
		Tabular
		Graphic
	* = representation introduced in the problem itself

Table 3 : Representations used by fifth and sixth graders

 3 Table3lists the types of representations used by the eight students in this group, when solving the problems posed.

				CTE			Interview
		Representation	y=x+a	y=ax	y=ax+b	y=x+a	y=ax	y=ax+b
	Specific values N-oral				1	1*	1*
		N-written	1*		5*
		Pictorial			3*	5*
		Numerical	5	2		2	4	4
		Algebraic			1
		Tabular				6*	3*	4*
		Graphic
	Generalization	N-oral				4*	3*	3*
		N-written	3*	4*	7*
		Pictorial
		Numerical		1	1	1	1	1
		Algebraic	2*	3*	2	5*	3	4*
		Tabular

Which were not mutually exclusive.

Acknowledgment

This study was funded by the Spanish Ministry of Economy and Competitiveness under National R&D Project EDU2016-75771-P; the first author benefited from a PhD. grant awarded by the Chilean Government through the CONICYT, folio 72160307-2015.