text:
Anna-Katharina Roos
email: anna-katharina.roos@mathematik.uni-wuerzburg.de

'Grundvorstellungen' and their application to the concept of extreme points

Keywords: Grundvorstellung, calculus, extreme point, misconception

In this article, we discuss the German construct of Grundvorstellungen and highlight the connection with mathematical aspects. After recalling the definition of these terms, we compare it with similar notions used in the literature such as "concept image/concept definition", and "metaphors", pointing out some common features and differences. Moreover, we use Grundvorstellungen and aspects for the analysis of the concept of extreme points. Finally, we briefly sketch how it may be used for the discovery and interpretation of students' misconceptions.

Motivation

There exists a vast literature on problems, mistakes, and misconceptions of mathematical topics in school. In order to discuss these misconceptions in connection with "correct" ideas, several approaches have been proposed, such as the construct of so-called "Grundvorstellung" (or GV, for short), which is mainly used in German literature (e.g. vom Hofe, 1995;Greefrath et al., 2016b). In this article, we briefly recall this construct and compare it with similar constructs appearing in the literature. In particular, we show how the GV construct can be applied to extreme points, with an emphasis on the transition from school to university. To this end, we give a short overview over GVs for extreme points, which helps us to classify students' problems concerning extreme points.

Theoretical background

Let us first explain the notion of Grundvorstellungen. Grundvorstellungen originate from the traditional German didactical approach of subject-matter didactics and have a long tradition in German mathematics education (see vom [START_REF] Vom Hofe | Grundvorstellungen mathematischer Inhalte[END_REF]. Vom Hofe extended the subject matter didactics approach by his understanding of Grundvorstellungen: he suggested to additionally take into account the learners' perspectives (see Straesser, 2014, p. 569). As a consequence, he distinguishes between GVs from a normative viewpoint, on the one hand, and GVs from a descriptive viewpoint, on the other hand. Vom Hofe (1992, p. 347) recalls the main characteristics of this construct: It gives meaning to a concept by connecting it with already known facts or experiences. It brings along the development of corresponding representations and the ability to use the concept. Scrutiny reveals that it is useful to introduce different types of GVs, which read as follows: A primary GV is a GV which gives meaning to a concept by connecting it with concrete experiences and objects from everyday life. A secondary GV is a GV which gives meaning to a concept by connecting it with ideas and representations of other mathematical concepts (see e.g. vom Hofe & Blum, 2016, p. 234).

To clarify the connection between GVs and the mathematical content of a concept, Greefrath et al. introduce a distinction between aspects and GVs of a mathematical concept: "A Grundvorstellung of a mathematical concept is a conceptual interpretation that gives meaning to it." (Greefrath et al., 2016a, p. 101). However, an aspect of a mathematical concept is a part of this concept, which refers to its mathematical content, like its definition, theorems related to the concept, its properties, or connections to other concepts. It is a part of this concept, which can be used to characterize it (see Greefrath et al., 2016a, p. 101).

Another distinction is given by Greefrath et al. (2016b, pp. 18-19) who introduce the notions universal and individual GVs: A universal GV works as a normative guidance for teachers and answers the question what students should think of a concept, i.e. a universal GV refers to vom Hofe's normative viewpoint. Universal GVs refer to mathematical aspects. However, there is no one-to-one correspondence between GVs and aspects: One GV can refer to several aspects, and one aspect can be the basis for several GVs (see ibid., p. 17-18). On the other hand, an individual GV answers the question what a special student actually thinks of a concept, i.e. an individual GV is related to vom Hofe's descriptive viewpoint. Individual GVs can be detected by watching students solving problems or by analyzing their explanations with regard to certain questions. Therefore it is a GV that an individual person possesses regarding a certain concept.

For our purposes-to find reasons for students' misconceptions-it is necessary to introduce yet another kind of GV which we call partial GV. A partial GV is an idea that gives meaning to a concept in a limited context. It also goes back to mathematical aspects of a concept, and thus to a normative viewpoint. However, it loses its generality through restrictive premises. Depending on the goal, it could also serve as a (preliminary) educational objective. An example is the idea "multiplication makes bigger" which is correct only in certain cases, e.g. the natural numbers (larger than 1). To highlight the differences between these types of GVs, we want to add the following point to the description of a universal GV: "Universal" means that it works in every given context. Consequently, we categorize an individual GV either as a universal GV, as a partial GV, or as a part of them. The following graphic puts the characterization of the GVs in a nutshell: The notion Grundvorstellung could be translated, loosely speaking, as "basic idea", "basic notion" or "basic mental model" (see vom Hofe & Blum, 2016, p. 226;Greefrath et al. 2016a, p. 101). However, we decided not to translate the notion to not mix it up with other (mainly in Germany used) constructs like fundamental ideas, universal ideas, central ideas, big ideas, leading ideas (for further details see [START_REF] Vohns | Fundamental Ideas as a Guiding Category in Mathematics Education -Early Understanding, Developments in German-Speaking Countries and Relations to Subject Matter Didactics[END_REF].

GVs are especially interesting regarding misconceptions. These "erroneous conceptions" have different names (preconceptions, alternative conceptions, misconceptions) and are defined in various ways in the literature (see e.g. [START_REF] Leinhardt | Functions, Graphs, and Graphing: Tasks, Learning, and Teaching[END_REF][START_REF] Hammer | Misconceptions or P-Prims: How May Alternative Perspectives of Cognitive Structure Influence Instructional Perceptions and Intentions[END_REF]. We just recall the main facts: They are a repeatable, robust, "well-formulated system of ideas" (Leinhardt et al., 1990, p. 5) and they are cause for errors, although they do not need to constitute a complete theory (ibid., p. 5-6). If one takes a closer look at these, one can categorize some as so called epistemological obstacles (see e.g. [START_REF] Brousseau | Les obstacles épistemologiques et la didactique des mathématiques[END_REF], or overgeneralizations, like in our background as partial GVs used in the wrong context. Vom Hofe (1996, p. 259) describes the importance of GVs regarding misconceptions as follows: Firstly, GVs as a normative guidance should help during the learning process so that individual ideas become individual GVs and not misconceptions (see vom Hofe, 1996, p. 259). In this respect, GVs work as a prevention of misconceptions. Secondly, GVs should work as a plausible alternative to already existing misconceptions. If learners already have misconceptions they need not just be dissatisfied with the former idea, but also need to be given an intelligible, plausible and fruitful alternative by the teacher (see Posner et al., 1982, p. 214). In this respect, GVs work as a remedy for misconceptions.

The purpose of this article is to give a survey on universal and partial GVs at university level that refer to aspects of the concept of extreme points. To begin with, let us briefly recall the role of GVs in the existing literature.

GVs and concept image/concept definition

One of the most important and popular works on the properties of concept image, concept definition, and their distinction is due to [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF]. A concept definition consists of the words used to specify a concept. The definition of a concept given by the mathematical community is called formal concept definition, whereas the definition given by an individual is called a personal concept definition. A concept image consists of all non-verbal connections to and connotations inferred by an individual regarding a certain concept. These connections and connotations may include all kinds of representations. There exists some literature describing similarities and differences between concept image/concept definition and GVs: vom Hofe and Blum (2016, p. 237) state that GVs refer to the concept image from both a normative and descriptive level. Greefrath et al. write: "A concept image may contain several individual Grundvorstellungen that conceptualize different perspectives of that concept. Individual Grundvorstellungen are central components of a valid concept image […]." (Greefrath et al., 2016a, p. 103) However, for some learners, also misconceptions may be part of their concept image. The individual GVs can either be partial GVs or universal GVs. As Greefrath et al. write: "These Grundvorstellungen give meaning to mathematical concepts that may be studied with respect to various aspects. Each of these aspects may be expressed with one of the various [formal] concept definitions that one reads in textbooks. Thus, a [formal] concept definition is a specific realization of an aspect." (Greefrath et al., 2016a, p. 103) The personal concept definition could either be similar to the formal concept definition or differ from it. It also has a relationship to the concept image.

GVs and metaphors

Apart from concept image/concept definition tools, metaphors have also been used to describe mathematical cognitive processes (see [START_REF] Lakoff | Metaphorical Structure of Mathematics[END_REF][START_REF] Lakoff | Where Mathematics Comes from[END_REF]. First, we recall some facts about the concept of metaphors used by Lakoff and Núñez. They analyze the (conceptual) structure of mathematics and therefore use results of metaphor theory of cognitive linguistics. This theory is about how mathematics is constructed. To describe mathematics, they distinguish several kinds of metaphors, the two crucial ones being grounding metaphors and linking metaphors, which are defined as follows:

"Grounding metaphors ground mathematical ideas in everyday experience." In addition, "[...] linking metaphors allow us to link one branch of mathematics to another. For example, when we metaphorically understand numbers as points on a line, we are linking arithmetic and geometry." (Lakoff & Núñez, 1997, p. 34) Let us now point out some similarities and differences between the concepts of GV and metaphor. Both GVs and metaphors reveal several perspectives of one concept through a detailed analysis of it. GVs and metaphors suggest using this analysis for educational purposes. Grounding metaphors resemble primary GVs, since they connect everyday life experiences with mathematics. Linking metaphors resemble secondary GVs, because they connect different domains within mathematics. However, a fundamental difference is the aim of these theories: finding of GVs through a subject matter analysis aims at making the most important aspects of a concept comprehensible and giving them meaning, respectively. Referring to metaphors in the sense of Lakoff and Núñez (1997, p. 31), this educational aim is named "peripheral" for the theory of metaphors. The discussion about metaphors tends more toward revealing the structure of mathematics and unconscious connections to non-mathematical concepts as mentioned above. Metaphors are used to explain our linguistic vocabulary when talking about mathematical contents.

There are also connections with other theories like the theory of conceptual fields of [START_REF] Vergnaud | The theory of conceptual fields[END_REF], which also works as a framework for organizing didactic situations. For lack of space we do not go into detail, but just remark that an exact analogue to the GV construct does not exist in the literature, but constructs which exhibit some similarities with GVs.

Aspects and GVs of extreme points

After defining the concept of GVs and comparing it with other concepts from the literature, we are now going to use it for the concept of extreme point, which is the major topic of this article. The following GVs were identified by both discussions with lecturers of analysis courses and a subject matter analysis of the concept of extreme point.

Aspects of extreme points

We will discuss three aspects of the concept of a (local) extreme point:

The aspect of "largest/smallest value"

Extreme points are the points (x,y) with the largest/smallest y-value with respect to a neighbourhood of x.

This aspect refers directly to the definition of an extreme point. It illustrates the connection of an extreme point (of a graph of a function) with the maximum/minimum of a (totally ordered) set: extreme points are the points, whose y-values coincide with the maximum/minimum of the range (on a given domain). Already children have some experience with this aspect.

Non-mathematical examples:

 Searching for the highest point (of a mountain)  Searching for the best object of category (fastest car, most expensive house, highest building)

Mathematical examples:

 Interpretation and discussion of graphs: where is the highest/lowest point?

The aspect of "largest/smallest value" is the most fundamental one for the concept of extreme point.

The aspect of "derivative zero"

Extreme points are located at points where the derivative becomes zero.

In this connection, however, one has to be careful and take into account two important points. First, there are points, which are not extreme points, although the derivative becomes zero at these points. Second, since this aspect refers to the necessary condition for the existence of (local) extrema, one has to impose two additional premises: the point must be an interior point of the domain, and differentiability of the function is required. By dropping one of these premises, one can easily find examples of extreme points where the derivative does not become zero (e.g. boundary points of an interval).

Non-mathematical examples:

 From a physical viewpoint: searching for the points where the velocity becomes zero.

Mathematical examples:

 The algorithm for the identification of extreme points most frequently taught at school: find the zeros of the first derivative. Afterwards, check the sign of the second derivative at these zeros to detect maxima and minima.

The aspect of "change of monotonicity"

Extreme points are located at points where the sense of monotonicity of a function changes, i.e., from increasing to decreasing or vice versa.

It is, in fact, easy to see that a change of monotonicity always implies the existence of an extreme point. The converse, however, is false: not every extreme point induces a change of monotonicity (consider again extreme points at the boundary of an interval).

Non-mathematical examples:

Change of the direction of movement:

 Activities: mountain climbing (change of uphill to downhill), bike riding (change from pedaling to idling).

Mathematical examples:

 At school (premise: differentiable functions): a table of sign changes for the first derivative (distinguishing extreme points from saddle points).

GVs of extreme points

Let us now study three GVs of the concept of (local) extreme points of a real function (defined on a nondegenerate interval). The first GV is universal, the second and third are partial.

Universal GV

The GV of "largest/smallest value"

This GV refers directly to the aspect of largest/smallest value and may be interpreted in a concrete geometrical way: extreme points are the points with the largest/smallest y-value with respect to a certain neighbourhood. The students should:

 Identify extreme points through analyzing y-values of points of a function graph algebraically or graphically.

 Demonstrate the connection between a maximum/minimum of a (totally ordered) set and an extreme point of a function by projecting (pieces of) the graph onto the y-axis. Once the students understand this GV correctly, they can specify different subcategories of extreme points, such as those sketched in the synoptic Table 1.

This GV is universal and should thus be emphasized when introducing the concept of extreme point. For applications, however, it is not that useful: it would take an infinite long time to analyze every point with regard to the value of its y-component, because the domain of definition is a continuum. However, there exist another two GVs, which are more helpful in applications.

Partial GVs

The GV of "slope zero"

This GV reflects the fact that the graph of a function has slope zero at an extreme point. Since this idea is intimately related to the aspect of "derivative zero", it should be assumed that the function is differentiable at the (interior) point in question. As this is usually the case for functions dealt with at school, it is mainly consolidated as a consequence of math classes. The GV of "slope zero" calls upon the GVs of the derivative of a function like the GVs "Tangent slope" or "Amplification factor" (see Greefrath et al., 2016a, pp. 106-113):

 Tangent slope: slope zero entails the existence of a horizontal tangent at the extreme point in question that approximates the graph locally. This means that the graph is also almost horizontal.

 Amplification factor: slope zero implies here that a change of the independent variable x leads to almost no change in the dependent variable y.

The reason, why this is just a partial GV is the fact, that neither the argument "extreme point implies slope zero" nor the converse argument "slope zero implies extreme point" is true. The GV "change of monotonicity"

If you look at the graph of a function and see that it is strictly increasing up to a certain point x0, and then strictly decreasing, you could intuitively argue that x0 is a (local) maximal point. This idea exhibits an analogy to the curvature of a graph at inflection points; if a function changes at some point from left curvature to right curvature, or vice versa, this is an inflection point. As far as differentiable functions are concerned, passing from an increasing piece to a decreasing piece can be interpreted as the transition of the first derivative from the upper half-plane to the lower halfplane. The GV of "change of monotonicity" goes back to a specific "dynamical" experience of the students: if you regard the graph as trajectory of a moving particle, the movement changes in the ydirection precisely at minimal or maximal points. The students should:

 Partition the graph of a function into monotonically increasing and decreasing pieces and identify extreme points as boundary points of the partition intervals.

The reason, why this is just a partial GV is the fact that it works only "in one direction", because the logical converse is again false: not every extreme point leads to a change of monotonicity.

Discussion

These GVs play an important role when analyzing mistakes of students in their first semester at university concerning the concept of extreme point. The partial GVs especially can lead to mistakes when their restrictive premises are not considered. Our study aims, on the one hand, at finding mistakes and misconceptions of mathematics students after their first analysis course and, on the other hand, at giving insight into possible reasons for these mistakes. It turns out that the concept of GVs helps to find normative education objectives of extreme points, and it supplies-used in a descriptive way-an orientation of what ideas students have of extreme points.

Figure 1 :

 1 Figure 1: Connection of universal, partial and individual GVs

Table 1 : Subcategories of extreme points

 1

Proceedings of CERME10


domain: math shs

text:
Anatoli Kouropatov
email: anatoliko@gmail.com

Tommy Dreyfus
email: tommyd@post.tau.ac.il

Combining two theories in the design for learning about integrals

Keywords: Design of instruction, Abstraction in Context, proceptual thinking, integration

We show how a combination of two theories, Abstraction in Context and Proceptual Thinking, served as basis for design decisions in the framework of a research study about learning the integral concept in high school via constructing knowledge about accumulation.

Introduction

Designing learning units involves decisions about the transition from syllabus to curriculum [START_REF] Dreyfus | Processes in the Transition from Syllabus to Curriculum[END_REF]; Mathematics Education offers home-grown theories supporting this transition; curriculum designers use these theories when design is a main goal.

Here, we present a case study of using theories for the design of a didactical toola learning unit intended for students' construction of a specific mathematical notion. This study is part of a larger research project [START_REF] Kouropatov | The Integral Concept in High School: Constructing Knowledge about Accumulation[END_REF] offering a didactic approach supporting high school students in acquiring a conceptual understanding of the integral. For this purpose, we asked ourselves: a) What does "a conceptual understanding of the integral" mean? and b) How can we support students in acquiring such understanding? Answering question (b) required design. The design was preceded by a thorough didactical-mathematical analysis of approaching integration via the idea of accumulation [START_REF] Thompson | The concept of accumulation in calculus[END_REF]. A tight interrelationship developed between the design process and the relevant theories, Abstraction in Context and Proceptual Thinking. In this paper, we exhibit this relationship via the need for and effect of these theories in the design process and hence the contribution of these theories to a basis for the transition from syllabus to curriculum.

Theoretical frameworks and their influence on the design

We follow [START_REF] Tabach | Computerized environments in mathematics classrooms: a research -design view[END_REF] in distinguishing  a pre-design stage involving considerations before starting the actual development and research work with students;  an initial design-research-redesign stage of sporadic and isolated activities, and the observation, data collection, and analysis of their implementation with a few students;  a final stage that comprises further redesign, for the creation of a coherent, complete taskbased curriculum, and its implementation )limited, in our case, to four pairs of students(.

These stages indicate what to do but not how to do it. Design decisions require theories. For our study, we have adopted Abstraction in Context (AiC) because its roots in constructivism and in activity theory make it suitable to design for and analyse the construction of abstract knowledge during the learning process [START_REF] Hershkowitz | Abstraction in context: epistemic actions[END_REF]. We have also adopted the theory of Proceptual Thinking (PT) because of the proceptual nature of the mathematical notions in focus, in particular accumulation [START_REF] Gray | Duality, ambiguity & flexibility: A proceptual view of simple arithmetic[END_REF]. The theories are compatible but different: PT deals with how students see mathematics; AiC deals with how students acquire knowledge.

Theoretical considerations at the pre-design stage

AiC as basis for design

AiC takes abstraction to be a learner's activity of vertical (in the sense of Freudenthal) reorganization of previous mathematical constructs in order to arrive at a new (to the learner) construct. Abstraction leads from an initial, vague first form, which may lack consistency, to an elaborated form [START_REF] Davydov | Types of generalization in instruction: Logical and psychological problems in the structuring of school curricula[END_REF]. The activity is interpreted in terms of epistemic actions performed by a learner, or a group of learners, for a specific purpose, in a particular context. The context includes the social setting as well as the learners' personal background, in particular the previous mathematical constructs resulting from previous processes of abstraction. 'Reorganization' includes establishing new connections between previous constructs, making mathematical generalizations, and discovering new strategies for solving problems. 'Vertical' implies building a new level of abstraction on top of a previous level. For the researcher, the question arises how to support and unveil the processes by which the students' new constructs may emerge as a vertical reorganization of previous constructs in the current context. AiC argues that for this purpose, it is crucial to carry out an a priori analysis. This theoretical content analysis aims at identifying the elements of knowledge (mathematical facts, notions, claims, strategies, representations, etc.) that together constitute 'learning a concept' and have a didactical perspective, namely can be constructed by learners in a suitable context using appropriate didactical tools. The results of the a priori analysis are descriptions of the elements of knowledge that belong to the world of mathematics but may be linked to the current context, and operational definitions that constitute descriptions of observable student behaviour: utterances or actions that provide criteria for assessing whether a student's constructing action corresponding to the said knowledge element has occurred.

AiC has originally been proposed in the framework of a curriculum development project, in which abstraction was a central concern, and [START_REF] Hershkowitz | Abstraction in context: epistemic actions[END_REF] have already then expressed the hope that it will be useful not only for analysing students' processes of abstraction but also for designing sequences of activities supporting students in such processes. At the pre-design stage, AiC requires a sequence of activities, each intended toward the construction of an appropriate element of knowledge, while the sequence is hierarchically structured as imposed by verticality. In other words, AiC helps us find a structure of the subject knowledge that is appropriate for implementation in the design. For example, based on the a priori analysis of the mathematical content, we decided that the unit should be designed as a four level vertical structure of the following conceptual components: Approximation in the context of given geometrical objects; Accumulation Value (the definite integral) in the context of given analytical objects or situations; Accumulation Function (the definite integral with varying upper bound); and Integration-Differentiation interplay, mainly the Fundamental Theorem of Calculus (FTC). Further analysis of each level provided the vertical structure of the elements of knowledge intended to be constructed by the students. We present two of the four levels in some detail.

For didactical reasons and based on verticality, Approximation was interpreted as Geometrical Shapes Approximation (GSA) with the following three knowledge elements:

APG "General approximation": The size of a given object can be approximated by replacing the given object with known objects;

APR "Refined approximation": The approximation can be made more precise by decreasing the size of the replacing objects and increasing their number;

APL "Approximation limit": The size of a given object can be determined as precisely as one wants by continued refinement.

The corresponding operational definitions are that we will say students have constructed APG if they explicitly (verbally and/or graphically) replace a given object with known objects;

APR if they explicitly (verbally and/or graphically) refine the approximation by decreasing the size of the replacing objects and increasing their number;

APL if they explicitly (verbally and/or graphically) identify a value as the exact size of a given object by continued refinement.

Knowledge elements for the other three conceptual components were similarly described and defined [START_REF] Kouropatov | The Integral Concept in High School: Constructing Knowledge about Accumulation[END_REF][START_REF] Kouropatov | Learning the integral concept by constructing knowledge about accumulation[END_REF]. These descriptions and definitions constitute the framework for the design of activities in the learning unit. In other words, by means of the description of the elements of knowledge, AiC informs the decisions of what should be designed and in what hierarchy it should be organized. AiC does not inform how to design for each notion by means of micro-tasks. For that purpose we used PT.

PT as basis for content design [START_REF] Gray | Duality, ambiguity & flexibility: A proceptual view of simple arithmetic[END_REF] defined the notion of procept as an aggregate of three things: process, concept (or object), and symbol. For example, the symbol 0 ()

x f t dt
 is meant to evoke both the process of accumulation (integration) and the concept of accumulation function (integral), with the cognitive combination of all three, process, concept, and symbol, being called a procept. This stance has crucial didactical implications: students might first meet a process; later, a symbol is introduced for that process and/or its product, and this symbol takes on the dual meaning of the process and the object created by the process. Proceptual Thinking is then defined as the ability to switch one's focus between these dual roles of the symbols as is useful and efficient in the current context, for example solving a problem. Someone who has the ability to think in this way may be described as versatile [START_REF] Tall | Encouraging versatile thinking in algebra using the computer[END_REF]. Versatility includes a global picture of a concept as well as the ability to break it down into a process, seeing each stage as part of the whole concept. According to [START_REF] Hong | Versatile understanding in integration[END_REF] versatility is critical for comprehension of the integral concept.

We see the integral as a multilevel, hierarchic procept, which is composed of (in the sense of AiC, and hence intended to be constructed by students from) other procepts including function, graph, approximation, sum, and accumulation; hence, we continue the pre-design stage by  using the result of the above a priori analysis in order to identify and describe the main subprocepts of the integral procept;  identifying the hierarchic structure of the integral as an aggregate of procepts.

The main didactical flow of ideas was derived directly from the procept hierarchy of the mathematical notion of the concept of the integral. In particular, the didactical goals are: to create an opportunity for the learners to carry out a process that is meaningful for them (e.g., to

approximate an unknown area of some shape by accumulating the known areas of small parts of this shape); to give the learner the possibility to internalize this process as a concept (e.g., by quantifying the process, by discussing the characteristics of this process); to introduce the learner to the common mathematical symbol as encapsulation of the completed process and the internalized object; these considerations became the leading considerations of the initial stage of design. In other words, PT allows us to answer the question of how the learning activities should be designed.

We present two examples from the learning unit that show how we took into account the proceptual nature of the intended elements. The first one, is the initial activity for introducing Approximation via GSA. And the second one is from the middle of the unit, and is intended to lead students to constructing the concept of Accumulation Function. As mentioned above, these two concepts, together with the Accumulation Value and the FTC, are the four components of the vertical structure of the suggested design of the unit.

Regarding GSA, students carried out the process of approximating the length of a given (sketched) curve (interval, semicircle, non-standard curves) using a ruler, compass, protractor, square paper (with two different mesh sizes) and calculator. Then, students discussed the "quality" of the resulting approximation and were asked to refine it (for example by using more sophisticated measurements) with the intention to lead to internalization of this process as a concept. Finally, students were asked to find the length as precisely as possible (the existence of such a value was taken as intuitively obvious). This "process, concept, existence" triad constitutes the GSA procept according to the above analysis.

Regarding the notion of accumulation function, the activities offered students opportunities to carry out the process of co-variational change of the accumulation value according to the value of the right end-point of a certain sub-interval (using approximation or algebraic considerations); students dealt with a table and/or graph and/or verbal representation of this change with the intention to lead to internalization of this process as a concept; finally, the symbol ( ) ( )

x a A x f t dt  
was introduced.

This "process, concept, symbol" triad constitutes the accumulation function procept.

Theoretical considerations at the initial stage of design

The influence of AiC and PT on the design of the unit could, in principle, best be demonstrated by the design of activities about the procept of accumulation, the central notion of the learning unit.

Because of space limitations, we concentrate instead on a small part of this: When describing a process of accumulation, one should know "how to start accumulating" -in other words, how to calculate an initial quantity. Then, one should know how to calculate further pieces of the accumulating quantity. The general answer to this problem is approximation. Here, the proceptual nature of approximation is particularly important: We see approximation as a process, and the result of this process, of calculating (as accurately as required) some unknown value (length, area or volume) by using known values.

While approximation refers to many kinds of quantity, verticality suggests a sequence of activities that starts with concrete geometrical shapes (lines, 2-dimensional and 3-dimensional shapes) followed by geometrical shapes that are given analytically (using elementary functions) in a coordinate plane (space); only then, more general quantities, given analytically, are considered.

Such a sequence allows students to construct their knowledge, starting in a concrete context of geometrical drawings and bodies that is intuitively clear to them, and where all quantities (i.e. length, area, volume) have positive values. This context requires relatively little previous knowledge and allows for rather linear and smooth vertical reorganization. Next, follows a more formal context of analytically given objects or situations (all quantities still having positive values). And finally, students are asked to deal with general quantities. Practically, approximation may be made by measurement, by geometrical consideration (with known formulas), or by algebraic considerations (analysing some algebraic term). In light of these considerations we have designed a sequence of activities, which we now, following an a priori analysis, interpret as focusing on (i) GSA with its three elements of knowledge 'General Approximation' (replacing the given object by known objects), 'Refined Approximation', and 'Approximation Limit'; and (ii) parallel elements of knowledge for 'Analytical Shape Approximation'.

Similar considerations apply to the concept of accumulation function. We see the accumulation function as a process of change (e.g. the change of accumulating area beneath the graph of the function while "the upper bound is moving") and the product corresponding to this process (e.g., a graph of this process demonstrating the ability to characterize it). This approach to the accumulation function immediately leads to the following conclusions: for constructing the accumulation function element of knowledge, students should know (even if only intuitively) that if we change the upper end-point of some sub-interval of the function domain, the appropriate value of the given function and the accumulation value of the given function will also change; and they should know how to characterize the process of the changing of the accumulation value. In light of these considerations, we have designed a sequence of activities, focusing on the Accumulation Function element of knowledge via its component elements (not specified in this paper).

On the basis of the above we claim that the combination of AiC and PT allows us to make decisions regarding the design of activities for the learning unit.

Theoretical considerations at the final stage of design

As a result of the two previous stages (pre-design and initial design) we developed a sequence of activities that were organized according to the above four component vertical structure. Each of its four components constitutes a hierarchical procept that is vertically composed of sub-procepts. We interpret this whole structure as the procept of Integral.

At the final stage of the design we analysed the developed activities with the purpose of avoiding inconsistent usage of terms, symbols, and visual representations. Another important issue we took into account at this stage was adaptation of the unit to students' previous knowledge. Thus, for example, at the previous stages we had used the number e for some of tasks. We recognized that this notion is not familiar to the students, so at the final stage of the design certain activities have been changed (e.g., by using π instead of e).

The implementation of the unit was organized in the form of learning sessions of pairs of students with in the presence of a researcher. The time interval between the sessions was typically between one and two weeks. We considered that for every part of the unit, students need some introductory and some summarizing activities. These activities aimed to provide a smooth flow of the learning process and were developed (in the form of a short discussion that was led by researcher) at the final stage of design.

As a result of the final stage, we created a task-based curriculum unit introducing the concept of integral via the idea of accumulation with a fair measure of internal coherence. This unit has been implemented with four pairs of students.

AiC as a tool for design evaluation

An essential component of AiC is the nested epistemic actions model for describing and analysing, at the micro-level, processes of abstraction by which learners construct new knowledge. The model uses the three epistemic actions of Recognizing (previous constructs as relevant in the present situation, R), Building-with (the recognized constructs to achieve a local goal, B), and Constructing (assemble and integrate previous constructs so that a new construct emerges by vertical mathematization, C). In processes of abstraction, R-actions are nested in B-actions, and R and Bactions are nested in C-actions.

Following [START_REF] Dreyfus | The nested epistemic actions model for abstraction in context -Theory as methodological tool and methodological tool as theory[END_REF] the core of the method is the analysis, utterance by utterance, of transcripts to identify R, B and C epistemic actions as building blocks of abstraction. The RBC methodology helps making processes of knowledge constructing observable. This claim is based on empirical results regarding many content areas including integration [START_REF] Kouropatov | Learning the integral concept by constructing knowledge about accumulation[END_REF].

RBC analysis of the learning sessions has been successfully used for evaluating the design of the activities by identifying problems with the implementation; this evaluation has uncovered instances where the design (or micro-design) of activities or their sequencing needed to be improved. We present two examples.

The first example concerns the concept of approximation limit (APL) referred to above. The RBCanalysis of the performance of one pair of students (A and B) supplied empirical evidence about students' constructing processes of APG and of APR but not of APL, which is a crucial component of approximation. Therefore, the design of the activity for the following groups of students has been refined in a way that supports the constructing process of APL. The elaboration consisted of adding questions leading the students to intuitively distinguish between overestimates (decreasing to the exact value) and underestimates (increasing to the exact value) of the approximated value. The revision was successful in the sense that all following student pairs succeeded in constructing APL.

The second example relates to the issue of lacking previous constructs assumed by the design. For example, when constructing the procept of approximation via GSA, students M and N demonstrated a lack of previous constructs such as identifying coordinates of points on a graph, or calculating lengths of segments. For example, in the activity of finding the length of a quarter-circle, the students quickly recognized the relevance of approximation. They replaced the curve with a set of chords but then got stuck because they didn't know how to calculate the chord-lengths. The idea of choosing the segment endpoints according to some division of the given interval was new them and outside their current grasp. The teacher's intervention was needed and was locally helpful. So, we can argue that there was a need for an additional element of knowledge that our design did not take into account: division of the given interval creating an appropriate division of the graph.

Conclusions and further questions

We presented a case study of using theories for design decisions; this case dealt with learning the integral concept in high school via constructing knowledge about accumulation. The theories were most significant but not the only resource for decision making. The decisions were inspired by theory (e.g., in the case of verticality of the structure of the elements of the intended knowledge), by practical experience (e.g., in the case of assumption regarding intuitive accessibility of some elements of knowledge for students), or by both (e.g., in the case of building the system of subprocepts of the procept of the integral or in the case of designing the sequence of learning activities). However, we argue that in the process of designing the learning unit on integrals for high school students, the theories have been interwoven and have played crucial roles in the process of development and implementation of the unit at all stages: the pre-design, the initial, and the final stages, as well as for fine-tuning the design after its evaluation.

The theories that we adopted for the purpose of the design are Abstraction in Context (AiC) and Proceptual Thinking (PT). These theories were adopted on two levels: AiC on a cognitiveepistemological level with the purpose of coming to design decisions regarding the nature and the structure of knowledge be learned (at a macro-level, which seems to be efficient in a more general context); PT on a didactic-implementation level with the purpose of coming to design decisions regarding how to help learners to achieve this knowledge (at a micro-level, which seems to be efficient in the context of mathematics procepts). The role of the theories differed from stage to stage: AiC was more essential at the pre-design stage while PT was more fruitful at the initial stage. However, the synergy of the theories was more influential than their diversity: Our design aims at supporting students in constructing proceptual knowledge of the Integral that we interpret as a hierarchical procept that is vertically composed of sub-procepts. Our research allows us to follow how students acquire a proceptual view while they construct their knowledge. We speculate that students' behaviour that is coherent with the suggested operational definitions (in terms of AiC) can be interpreted as evidence for the acquisition of a proceptual view (in terms of PT).

Following the research, we find ourselves in a better position to pose two relevant yet unsolved problems, a practical and a theoretical one. The practical problem is how to optimally profit from theory when designing instruction. The findings of the research show that the AiC and PT frameworks can be used for development and evaluation of an important instructional instrumenta learning unit. What about other instructional instruments, such as homework assignments, tests, and so on? Could we also use these theories for the design of such instruments? Additional research and experiments are needed in order to suggest the adaptation of the discussed theories for these types of instrument. The theoretical problem concerns the consistency of theories: Could we have used other theoretical frameworks instead or in addition to AiC and PT, what consistency issues would have arisen, and how different a design would have resulted?

Proceedings of CERME10


text:
Heather Lynn Johnson
email: heather.johnson@ucdenver.edu

Evan Mcclintock
email: evan.mcclintock@ucdenver.edu

Peter Hornbein
email: peter.hornbein@ucdenver.edu

Amber Gardner
email: amber.gardner@ucdenver.edu

Daniel Grieser
email: daniel.grieser@ucdenver.edu

When a critical aspect is a conception: Using multiple theories to design dynamic computer environments and tasks to promote secondary students' discernment of covariation

Keywords: Secondary school mathematics, instructional design, computer simulation, learning theories

We explicate how we used different theories of learning to design dynamic computer environments and tasks to promote secondary students' discernment of covariation-central to students' study of fundamental mathematical ideas such as rate and function. Using Marton's variation theory, we designed task sequences to foster students' discernment of the critical aspect of covariation. Using Piaget's constructivist theory, we defined the critical aspect, covariation, in terms of students' conceptions of a relationship between attributes whose measures vary. Using Thompson's theory of quantitative reasoning, by quantities we mean attributes of objects that students can conceive of as being possible to measure. We provide data to demonstrate how a student's discernment of covariation advanced during her work on a task sequence. We discuss implications for the design of dynamic computer environments and tasks focused on the mathematics of change and variation.

By drawing on more than one theory of learning, researchers can combine tools and lenses to investigate complex phenomena (e.g., [START_REF] Cobb | Putting philosophy to work: Coping with multiple theoretical perspectives[END_REF][START_REF] Sfard | On two metaphors for learning and the dangers of choosing just one[END_REF][START_REF] Simon | Amidst multiple theories of learning in mathematics education[END_REF]. [START_REF] Cobb | Putting philosophy to work: Coping with multiple theoretical perspectives[END_REF] recommended that researchers "act as bricoleurs by adapting ideas from a range of theoretical sources" (p. 29). [START_REF] Sfard | On two metaphors for learning and the dangers of choosing just one[END_REF] argued that researchers should not assume that theoretical "patches of coherence" somehow would combine to form a single, unifying theory of learning. Yet, using multiple theories can pose challenges, particularly if researchers view theories as competing, rather than complementary [START_REF] Simon | Amidst multiple theories of learning in mathematics education[END_REF]. To address challenges, it is useful for researchers to take into account the grain sizes of different theories [START_REF] Kieran | Frameworks and principles for task design[END_REF][START_REF] Watson | Parameters for practice and research in task design in mathematics education[END_REF].

By distinguishing between the grain sizes of theories, researchers can more effectively interpret and use theory for task design purposes [START_REF] Kieran | Frameworks and principles for task design[END_REF][START_REF] Watson | Parameters for practice and research in task design in mathematics education[END_REF]. Broadly, grain sizes include grand theories (e.g., Piaget's constructivist theory), intermediate theories (e.g., Marton's variation theory), and domain specific/local theories (e.g., Thompson's theory of quantitative reasoning). Furthermore, it is useful for researchers to acknowledge interrelationships between theories of different grain sizes. Theories of smaller grain size depend upon or address particular aspects of theories of larger grain size [START_REF] Watson | Parameters for practice and research in task design in mathematics education[END_REF]. For example, Thompson's theory of quantitative reasoning depends upon Piaget's constructivist theory to define quantities in terms of students' conceptions. By drawing on theories of different grain sizes, researchers can adapt and interpret grand theories for task design and implementation (e.g., [START_REF] Cobb | Putting philosophy to work: Coping with multiple theoretical perspectives[END_REF][START_REF] Kieran | Frameworks and principles for task design[END_REF][START_REF] Thompson | Didactic objects and didactic models in radical constructivism[END_REF][START_REF] Watson | Parameters for practice and research in task design in mathematics education[END_REF].

When researchers engage in task design, they should make explicit how their theory choice informs their task design [START_REF] Watson | Parameters for practice and research in task design in mathematics education[END_REF]. Using Marton's variation theory (2015), we designed task sequences to engineer opportunities for students to discern critical aspects central to fundamental mathematical ideas, such as rate and function. We posit that one such critical aspect is covariation. Using Piaget's constructivist theory (1985), we define covariation in terms of individuals' conceptions. Using Thompson's theory of quantitative reasoning (1994Thompson's theory of quantitative reasoning ( , 2002)), we articulate the conception; by covariation we mean a conception of a relationship between attributes whose measures vary. Students' conceptions of covariation impact their understanding and use of function [START_REF] Thompson | Variation, covariation, and functions: Foundational ways of thinking mathematically[END_REF].

We build on the work of researchers who have designed dynamic computer environments and tasks to foster students' study of the mathematics of change and variation (e.g., [START_REF] Kaput | The mathematics of change and variation from a millenial perspective: New content, new context[END_REF][START_REF] Saldanha | Re-thinking covariation from a quantitative perspective: Simultaneous continuous variation[END_REF][START_REF] Thompson | Didactic objects and didactic models in radical constructivism[END_REF]. In this paper, we explicate how we used theories of different grain sizes to design dynamic computer environments and tasks to promote secondary students' discernment of covariation. To avoid remaining only in the abstract, we provide data to demonstrate the utility of our approach in fostering students' discernment of covariation.

Theoretical and conceptual framework

We use Piaget's constructivist theory to orient our research. We foreground students' conceptions to explain how researchers might design dynamic computer environments and tasks to foster students' development of difficult to learn mathematical ideas such as function and rate. We focus on students' mental operations, which refer to actions that individuals can enact in thought or in the physical world [START_REF] Piaget | The equilibration of cognitive structures: The central problem of intellectual development[END_REF].

We use Marton's variation theory to guide the design of our dynamic computer environments and related task sequences. Broadly, [START_REF] Marton | Necessary conditions of learning[END_REF] argued that instructional designers should develop task sequences that provide students opportunities to discern critical aspects. The task sequence should involve patterns of variation, then invariance in the critical aspects [START_REF] Marton | Necessary conditions of learning[END_REF]. We draw on Piaget's constructivist theory to orient our interpretation of the critical aspect that we intend for students to discern. By critical aspect, we mean a conception. The critical aspect-covariationrefers to a conception of a relationship between attributes whose measures vary.

When using variation theory it is important for instructional designers to determine if the critical aspect is comprised of a single aspect or of interrelated aspects [START_REF] Marton | Necessary conditions of learning[END_REF]. For example, suppose a designer intends for students to discern the color blue, which students could conceive of as being comprised of a single aspect. A task sequence should begin with variation in color and invariance in some unrelated feature (e.g., blue ball, green ball, red ball), then invariance in color and variation in the unrelated feature (e.g., blue ball, blue block, blue cone), and then variation in both. In contrast, if a designer intends for students to discern the depth of blue color, students would need to conceive of interrelated aspects (depth, color) that comprise the critical aspect. In this case, the task sequence should include variation and invariance in each interrelated aspect (e.g., different colors of the same depth, then different depths of the color blue), then move to variation in both aspects. A conception of covariation necessitates a conception of a relationship between interrelated aspects (attributes whose measures vary). For example, in a situation involving the varying height and distance of a car in a turning Ferris wheel, the height and distance are the interrelated aspects, and students' conceptions of a relationship between measures of height and distance (covariation) is the critical aspect.

We found Marton's variation theory and Piaget's constructivist theory to complement each other for the purposes of our task design. From a constructivist perspective, we do not assume that a relationship between attributes whose measures vary (covariation) is something that is "out there" for students to perceive. [START_REF] Marton | Necessary conditions of learning[END_REF] argued that researchers should not assume that students already attend to the critical aspect prior to encountering a task sequence; therefore, task sequences should include variation in critical aspects (contrast) prior to variation in noncritical aspects. To foster students' discernment of covariation, we incorporated variation in the types of interrelated aspects (height, width, distance) prior to variation in the representation of those aspects.

We use Thompson's theory of quantitative reasoning to explain what we mean by the attributes whose measures vary-the interrelated aspects comprising the critical aspect of covariation. Drawing on Piaget's constructivist theory, [START_REF] Thompson | The development of the concept of speed and its relationship to concepts of rate[END_REF] defined quantities in terms of individuals' conceptions of attributes of objects. Therefore, quantities are not "things" that exist in the physical world. Following [START_REF] Thompson | The development of the concept of speed and its relationship to concepts of rate[END_REF], we claim that an individual conceives of some attribute as a quantity, if the individual can conceive of the possibility of measuring that attribute. For example, we would claim that a student conceived of "height" as a quantity if the student provided evidence of envisioning the possibility of measuring the height of some object.

We selected Thompson's theory of quantitative reasoning because we found it to be useful for interpreting Piaget's constructivist theory. Accordingly, the mental operations on which we focus are quantitative operations, which involve actions on attributes that students can conceive of as measurable, or in other words, actions on quantities [START_REF] Thompson | The development of the concept of speed and its relationship to concepts of rate[END_REF]. Specifically, we focus on students' conceptions of covariation, which entail the quantitative operations involved in forming and interpreting relationships between attributes whose measures vary. For example, a student conceiving of covariation could form and interpret relationships between the varying measures of height and distance for a Ferris wheel car traveling around one revolution of a Ferris wheel.

The diagram in Figure 1 illustrates how we used different theories to inform our task design. We placed Marton's variation theory at the top to foreground our intention to design a task sequence to foster students' discernment of a critical aspect comprised of interrelated aspects. We placed Piaget's constructivist theory in the center to communicate how we used this grand theory to define the critical aspect, covariation, in terms of individuals' conceptions. We placed Thompson's theory of quantitative reasoning at the base to show how we used this local theory to explain what we mean by covariation-a conception of a relationship between attributes whose measures vary.

Ferris wheel dynamic computer environments

Using Geometer's Sketchpad software, Johnson developed two dynamic computer environments for use with the task sequence. The environments consisted of a Ferris wheel animation and linked graph, each of which students could control separately or in conjunction. The environments related either the height of a Ferris wheel car from the ground or the width of the car from the center to the distance traveled around one revolution of the wheel (Figure 2 shows height and distance). See [START_REF] Johnson | Task design: Fostering secondary students' shifts from variational to covariational reasoning[END_REF] for more details about the environments. The Ferris wheel environments contained three affordances particularly relevant to our use of variation theory. First, students could vary each of the interrelated aspects (e.g., height and distance) individually by dragging or animating the dynamic segments on the vertical and horizontal axes. Second, the environments included different interrelated aspects-height and distance (Figure 2), and width and distance (not shown). Third, the environments included variation in the axes used to represent the interrelated aspects on the Cartesian plane (e.g., height and distance represented on the vertical and horizontal axes [Figure 2], then horizontal and vertical axes, respectively [not shown]).

Our use of Piaget's constructivist learning theory and Thompson's theory of quantitative reasoning informed our choices about the types of quantities to include on each of the axes. Specifically, we included quantities measurable with linear units, because it is less difficult for students to conceive of using linear units to measure quantities (see also [START_REF] Piaget | The child's conception of movement and speed[END_REF]. Furthermore, [START_REF] Thompson | Didactic objects and didactic models in radical constructivism[END_REF] recommended students use their fingers as tools to represent change in individual quantities. In the Ferris wheel environments, students could use either their fingers or the dynamic segments on the vertical and horizontal axes to represent change in individual quantities.

The Ferris wheel task sequence

Purpose and setting

We view tasks as problems designed for particular audiences and settings (see [START_REF] Sierpinska | Research in mathematics education through a keyhole: Task problematization[END_REF] Johnson designed the Ferris wheel task sequence to provide students opportunities to discern covariation. In a small neighborhood school in an industrial region of a large U.S. city, Johnson conducted a series of small group interviews with five ninth grade students (~15 years old), enrolled in an introductory algebra course. Interviews occurred approximately once per week. During the interviews, students completed the Ferris wheel task sequence (see Table 1). Johnson designed the task sequence for a small group interview setting; however, teacher/researchers could adapt the tasks for use in different settings (see [START_REF] Johnson | Investigating functions with a Ferris wheel[END_REF].

Variation and invariance in the Ferris wheel task sequence

To foster students' discernment of critical aspects comprised of interrelated aspects, [START_REF] Marton | Necessary conditions of learning[END_REF] recommended that instructional designers begin with task sequences containing variation in individual interrelated aspects, then variation in the both interrelated aspects, against a background of invariance. In the Ferris wheel task sequence, we intended for the situation of a turning Ferris wheel to provide a background of invariance. Furthermore, [START_REF] Marton | Necessary conditions of learning[END_REF] recommended variation in features (or dimensions) of those interrelated aspects. We provided two types of variation in features: the type of interrelated aspects (width, height, or distance), and the representation of each aspect on the Cartesian plane (horizontal or vertical axis). Table 1 shows the Ferris wheel task sequence, including variation in interrelated aspects and representations on the Cartesian plane.

Task Interrelated aspects Representation on axes on

Covariation and quantity in the Ferris wheel task sequence

When the critical aspect is a mental operation, instructional designers should provide students opportunities to engage in activities in thought as well as action. Each task in the Ferris wheel task sequence contained five parts: (1) Explain what the interrelated aspects measure in the Ferris wheel situation;

(2) Sketch a graph relating both aspects; (3) Use dynamic segments to represent change in individual aspects (e.g., height or distance); (4) Predict a car's location on the Ferris wheel given only dynamic segments representing the changing individual aspects; and (5) Compare the computer generated graph to the sketch in (2). Through each of the five part tasks, Johnson provided students multiple opportunities to discuss and represent their thinking about how the interrelated aspects (height and distance or width and distance) were changing individually and together. For example, students sketched a graph relating the interrelated aspects prior to viewing any facets of the dynamic graph. Furthermore, Johnson provided students opportunities to discuss and show the possibility of measuring interrelated aspects of the Ferris wheel situation (e.g., "height" represents the vertical distance from the car to the base of the Ferris wheel, see Figure 2).

A case of a student's discernment of covariation

We use the work of one student-Ana-to demonstrate the promise of this design approach for fostering students' discernment of covariation. Ana's work demonstrates the range of reasoning of all five students who completed the Ferris wheel task sequence. Building from Ana's work, we present a case of a student's discernment of covariation.

We share data from Ana's work for Part 2 of Tasks 1 and 3: Prior to viewing the dynamic graph, sketch a graph relating both aspects. We selected data from Part 2 to illustrate how Ana's sketches changed prior to viewing aspects of the dynamic graph. We attribute the changes in her sketches to changes in her conceptions of a relationship between varying measures of height and distance.

Figure 3 shows Ana's written work for Part 2 of Tasks 1 and 3. For Part 2 of Task 1 (left), Ana drew the curved graph, labeling it "height," and then drew the line graph, labeling it "distance" (Figure 3, left). When asked what her labels meant, Ana stated: "This (points to the curved graph) would be the graph shape if we were dealing with the height, and this (points to the line graph) would be the shape if we were dealing with the distance." For Part 2 of Task 3, using one continual motion, Ana sketched a single graph (Figure 3,right). When asked to explain her thinking, she stated that the "distance keeps on going," but the height will reach "a certain amount," and then "it goes back down." To illustrate, she drew an arrow along the left of the distance axis. Next, she drew a small, darkened segment on the graph, and two arrows extending along the graph. (conceived of a relationship between attributes whose measures vary). Furthermore, Ana's discernment of covariation was not limited to the aspects of height and distance. She also demonstrated discernment of covariation when working with width and distance in Tasks 2 and 4.

Discussion/Implications

When critical aspects involve interrelated aspects, [START_REF] Marton | Necessary conditions of learning[END_REF] recommended that instructional designers develop task sequences that include different backgrounds. In our task sequence, we used only a Ferris wheel situation, and we recommend that researchers designing task sequences to foster students' discernment of covariation also include different situations. However, we provide our recommendation with a caveat-the different situations should include interrelated aspects measurable with linear units. For example, if we were to design a task sequence for a filling bottle situation, we might ask students to relate the height of the liquid in the bottle to the diameter of the liquid in the bottle. Our caveat stems from our use of Piaget's constructivist theoretical perspective. It is less difficult for students to conceive of the possibility of using linear units to measure attributes (e.g., [START_REF] Piaget | The child's conception of movement and speed[END_REF]. Therefore, we recommend that task sequences designed for students to discern covariation (a critical aspect involving interrelated aspects) should include interrelated aspects measurable with linear units. Researchers have shown that even successful university students have difficulty using graphs to represent relationships between height and volume in a filling bottle situation (e.g., [START_REF] Carlson | Applying covariational reasoning while modeling dynamic events: A framework and a study[END_REF]. If students have difficulty conceiving of the possibility of using a three-dimensional unit to measure volume, it may impact their discernment of covariation for situations involving such attributes.

By using theories of different grain sizes, we were able to employ multiple, compatible lenses to engage in task design that looked both across and within the sequence of tasks. By guiding our variance and invariance of interrelated aspects, Marton's variation theory informed design across the task sequence. By fostering our choices about the kinds of aspects to vary, Thompson's theory of quantitative reasoning informed our design within tasks in the sequence. The ability to view a task sequence from different perspectives-in our research, looking both across and within-is a productive result emerging from the use of multiple theories to do compatible explanatory work to augment the design of task sequences intended to foster students' discernment of critical aspects.

Figure 1 :

 1 Figure 1: Relationships between the different theories we used to inform our task design

Figure 2 :

 2 Figure 2: Ferris wheel dynamic computer environment, distance and height

Figure 3 :

 3 Figure 3: Ana's graphs in Part 2 of Task 1 (left) and Part 2 of Task 3 (right) For Part 2 of Task 1, we interpret that Ana represented individual variation occurring in the measures of height and distance. Not only did she sketch two graphs, she labeled the actual sketches, rather than the axes. We use Ana's work for Part 2 of Task 1 to demonstrate that Ana did not enter the Ferris wheel task sequence already conceiving of a relationship between the varying measures of height and distance, or in other words, conceiving of covariation. Moving forward to Part 2 of Task 3, Ana used a single graph to represent a relationship between the varying measures of height and distance. Not only did she sketch a single graph, she annotated the graph to show how the single graph represented variation in the measures of both height and distance. Therefore, we claim that Ana demonstrated discernment of covariation during her work for Part 2 of Task 3

Table 1 : Ferris wheel task sequence: Variation in interrelated aspects and representation

 1

Thematic Working Group 17Proceedings of CERME10

Proceedings of CERME10


text:
Daniela Behrens
email: daniela.behrens@uni-bremen.de

Angelika Bikner-Ahsbahs

The perspective of indexicality: How tool-based actions and gestures contribute to concept-building

Keywords: Digital tool, indexicality of gesture, gesture-of and gesture-for, modes of dragging

In this paper we apply the theoretical perspective of indexicality to gesture use on a digital place value chart on the iPad and show that this perspective allows for explaining how mathematical meaning is accumulated linking specific gestures to the actions performed on the digital devise. Thus, practical dragging leads to structural dragging via operational dragging, resulting in a shift of the representational function of gesture (gesture-of) to the epistemic function of gesture (gesturefor).

Introduction

De Freitas and [START_REF] De Freitas | New materialistic ontologies in mathematics education: The body in/of mathematics[END_REF] have proposed to adopt a new materialistic view in research on learning mathematics, specifically for technology use. In this respect, the indexicality of signs is relevant [START_REF] Sinclair | The haptic nature of gesture -Rethinking gesture with new multitouch digital technologies[END_REF]. This paper investigates the added value of the theoretical perspective of indexicality while adopting it to the analysis of technology-based epistemic processes in the design project DeciPlace, to understand more deeply how acting based on a multi-touch surface contributes to building mathematical knowledge.

DeciPlace is a design-based research project. Its main goal is to develop a task sequence for conceptualizing decimal fractions as structures in small groups of students by the use of a digital place value chart (DPC) on the iPad [START_REF] Behrens | How a digital place value chart could foster substantial understanding of the decimal place value system[END_REF][START_REF] Behrens | The dragging gesturefrom acting to conceptualizing[END_REF]. The core approach is to act tool-based with the DPC. Thereby, conceptualizing is not considered as a pure individual cognitive process but as a collective communicative process of constructing mathematical structures in an embodied and multimodal way (see [START_REF] Krause | The mathematics in our hands: How gestures contribute to constructing mathematical knowledge[END_REF]. For this kind of learning, the instrumental approach (see [START_REF] Drijvers | Integrating technology into mathematics education: Theoretical perspectives[END_REF], which is often used for research on technology learning settings, fails to attain insight into the epistemic process in all its aspects. Our focus is on tool-based acting and interacting. However, the way this contributes to knowledge construction is not yet understood in-depth. In this paper, we will first present the tool, then outline what we mean by indexicality and finally apply this perspective in the analysis of some episodes from the DeciPlace data corpus to show how the adoption of this perspective deepens insight into tool-based collective learning.

The digital place value chart (DPC)

The digital place value chart on the iPad (designed by [START_REF] Ladel | Designing a technology based learning environment for place value using artefact-centric activity theory[END_REF] itunes-App: "Place Value Chart") can represent a number on three different levels in parallel: In the bottom row of the chart tokens can be displayed by tapping on the screen. In the upper row the name of the place value as well as the number of tokens in each column of the place value chart is indicated. Additionally the standard notation of the represented number can be displayed above the chart. When the user drags a token to the next (after next) column to the right, the token is de-bundled automatically into ten (hundred) tokens and so on (see Figure 1). The other way around, by dragging a token to the next (after next) column to the left, either nine (99) tokens move along with the dragged token bundling together to one token in the new column orif there are not enough tokens to come along withthe dragged token slides back to its origin. Hence, in contrast to traditional place value charts (paper and pencil or material tokens) this digital version de-bundles and bundles (if possible) automatically, while the represented value is kept constant, and it gives feedback if the intended manipulation is impossible.

Recent results: Three modes of dragging

In the analysis of activities on the DPC by a pair of grade 6 students who were introduced to the extension of the place value system from natural to decimal numbers, we noticed a shift in the students' activities from dragging-actions on the iPad to dragging-gestures (incorporated by the characteristic movement to the right or left), becoming more and more independent from the representation on the screen during the course of interaction with the DPC (see [START_REF] Behrens | The dragging gesturefrom acting to conceptualizing[END_REF].

Using an epistemic analysis based on ideas developed by [START_REF] Krause | The mathematics in our hands: How gestures contribute to constructing mathematical knowledge[END_REF] we were able to distinguish three modes of dragging:

 Practical dragging comprises actions of dragging tokens performed directly on the digital place value chart, when students use the function of bundling or de-bundling by dragging with a practical aim without scrutinizing the underlying principle.  Operational dragging can be observed when students are able to foresee the result of bundling or de-bundling or when they want to test something by dragging a token intentionally in the chart, so that they use both transformations to fulfil a particular goal. This mode of dragging can be manifested either as a direct act of dragging in the digital place value chart or as a dragginggesture referring directly to the chart.  Structural dragging becomes apparent, when a particular mathematical structure is being described generally and the movement of dragging is represented in a gesture from left to right or vice versa performed independently from any concrete representation.

Aim of this paper

Using these three modes of dragging, we were able to describe the epistemic role of gestures in processes of building the decimal fractions' concept, which is mainly based on the principle of bun-dling and debundling (see [START_REF] Behrens | The dragging gesturefrom acting to conceptualizing[END_REF]. However, we were not able to understand in detail how these three modes of dragging contributed to processes of learning based on the digital artifact. In this paper, we will address this topic by answering the following research question:

How do actions and gestures "regarded as indices" contribute to conceptualizing the decimal fractions' concept based on the digital place value chart on the iPad?

Describing the theoretical approach: Actions and gestures regarded as indices

In this paper, we focus on the connection between the digital place value chart and a pair of individuals interacting with each other with regard to the device. To examine this interaction we focus on signs which are produced in the setting, such as gestures, inscriptions, tokens on the display, the artifact itself and so on.

The students' collective epistemic process is manifested in their actions (based on the tool), their verbal utterances and other semiotic resources. These actions can be analyzed in a multimodal way based on the concept of the semiotic bundle [START_REF] Arzarello | Semiosis as a multimodal process[END_REF], which consists particularly of gestures, speech, inscriptions and relations among each other. To emphasize the influence of tool-based actions on speech, gesture and inscriptions, we adapt the perspective of the indexicality of actions on multi-touch devices described by [START_REF] Sinclair | The haptic nature of gesture -Rethinking gesture with new multitouch digital technologies[END_REF]. This perspective draws on Peirce's notion of semiotics, "in which signs (icons, indices and symbols) differ in terms of the nature of the relationships between the signifying sign and the signified" (Sinclair & de Freitas, 2014, p. 355). According to Peirce, a sign is defined by a triadic relation between sign, object and interpretant:

A sign, or representamen, is something which stands to somebody for something in some respect or capacity. It addresses somebody, that is, creates in the mind of that person an equivalent sign, or perhaps a more developed sign. That sign which it creates I call the interpretant of the first sign. The sign stands for something, its object. It stands for that object, not in all respects, but in reference to a sort of idea, which I have sometimes called the ground of the representamen. (Peirce, 1932, 2.228, emphasis in the original)1 .

As a consequence, a sign comes into being when there is an individual who produces an interpretant according to the relation between the sign and the object. This relation distinguishes a sign to be an icon, an index or a symbol (Peirce, 1994, p. 239).

While an icon is characterized by producing the idea of resemblance of sign and object in individuals and symbols are defined to be conventionalized signs, an index refers to its object not so much because of any similarity or analogy with it, […] as because it is in dynamical (including spatial) connection both with the individual object, on the one hand, and with the senses or memory of the person for whom it serves as a sign, on the other [START_REF] Peirce | Elements of logic[END_REF], 2.305, cited in Sinclair & de Freitas, 2014, p. 355f.).

By this, indexical signs "show something about things, on account of their being physically connected with them" [START_REF] Peirce | What is a sign?[END_REF](Peirce, [ ] 1998, p.5;, p.5;cited in Sinclair & de Freitas, 2014, p. 355).

Sinclair and de Freitas (2014) emphasize that also the action that resulted in the emergence of another sign may be included in the concept of indexical signs:

For instance, the chalk drawing of a parallelogram on a blackboard is often considered to be an iconic reference to a Platonic conception of parallelogram, but it is also an indexical sign that refers to the prior movement of the chalk. This latter indexical dimension is usually not emphasized in the semiotic study of mathematical meaning making, since we tend to focus on the completed trace and dislocate it from the labour that produced it. (Sinclair & de Freitas, 2014, p. 356) Taking this assumption into account, we can further assume that every process of producing a sign is an indexical sign referring to the sign and the sign itself refers indexically back to the effort which produced it.

They conclude that "indexation becomes part of an experience that exceeds itself, and is thus selfgenerative" (p. 359). Thus an action on multi-touch screens leaves traces, hence, these traces as well as a hand gesture may refer to the original action when this gesture is produced in a similar way to the action on the device. Taking up this theoretical perspective we want to examine the above described modes of dragging with respect to "how they function as indexical, material actions" (p. 360) trying to explain how dragging movements contribute to build the concept of decimal fractions.

Applying the theoretical approach: Indexicality in dragging modes

We are re-analyzing episodes of our design-study DeciPlace [START_REF] Behrens | The dragging gesturefrom acting to conceptualizing[END_REF] in order to answer the above posed research question. This way we will investigate the added value of this indexicality perspective.

As described above, practical dragging takes place when tokens are dragged directly in the digital place value chart from one column to the other either to the right or to the left without observable intention. The digital place value chart reacts to this practical dragging of one token to the next column on the right by de-bundling this token into ten tokens. Likewise, the DPC can bundle ten tokens to one in the next column, when one token is dragged into the next column on the left. Bundling and de-bundling by dragging keeps the value of the decimal number the same. This way, dragging can be linked to bundling or de-bundling in a material way being performed as an action of dragging from right to left or vice versa on the DPC. According to Sinclair and de Freitas (2014) an action of dragging leaves tracese.g. the new arrangement of tokens within the chartwhich refer back to the original action of dragging and vice versa.

In the following scene, two students are asked to find different representations for the number 101 in the place value chart:

1 Bella: I'll just try (drags a token from hundreds to tens within the digital place value chart on the iPad (in the tens' column ten tokens emerge), see Figure 2) Woah 2 Hanna: Ten and One. In this scene of practical dragging the newly emerged bunch of ten tokens in the tens' column can be seen as an iconic sign representing ten tens and therefore the number 100. Additionally the bunch of ten tokens refers back to the action of dragging a token from the hundreds' column to the tens' column on the DPC and at the same time to the DPC's reaction by letting the token explode into ten tokens without changing the represented number (concept of de-bundling). Thus, a new sign emerged by indexation linking the action of dragging on the DPC with a visual representation of de-bundling. This in turn produced a reaction of astonishment expressed by Bella ("Woah"), again referring back to the DPC's reaction on her dragging.

When the students get more and more familiar with the DPC in phases of practical dragging, they may apply movements of dragging intentionally for a specific purpose, for example while making a conjecture or predicting what will happen in the case of dragging directly in the DPC. Operational dragging can take place as an action of dragging on the surface of the device and also as a gesture directly above the surface referring to the DPC but without touching it (the so-called potential level, see Krause, 2016, p. 134-139). Because of the material link between dragging actions and dragging gestures both being performed by a similar movement from left to right or vice versa, dragging gestures may refer to previous actions of dragging and what is already linked with them. In this respect, operational dragging being conducted as a gesture on the potential level of reference is materially linked to the performance of dragging in the DPC and at the same time linked to the traces which are potentially and materially produced by that.

In the following situation the place value "tenth" is just introduced by the interviewer adding the new column named "tenth". The students are asked to find further representations for the number 4. At first, the students tap in four tokens into the ones' column of the DPC.

3 Bella:

Can I drag over one (moves her right hand at the bottom of the iPad from the ones' column to the tenths' column; see Figure 3) and see what gets out? 4 Interviewer: Try. 5

Bella:

(drags a token from the ones' to the tenths' column, where ten tokens emerge; see Figure 4) 6 Hanna: Ohh. 7

Bella: So ten are (1 sec.) one one (pointing to the ones' column) are ten tenths (moves her hand to the right flexing and extending her index finger pointing to the tenths' column). Although both dragging-gestures are executed more or less equally, in relation to speech they function differently. The first dragging-gesture seems to focus on the movement of dragging to the right from ones to tenths specifying "what" and "where" (Krause, 2016, p. 125) to "drag over" (a wording frequently used by the students). In contrast, the second dragging-gesture adds the way by which the insight that "one one are ten tenths" (line 7) was gained, referring back again to the experiences and assumptions made by the action of dragging rightwards on the DPC just before (line 5). Similar to the notion of model-of and model-for (van den Heuvel-Panhuizen, 2003, p. 14) we have identified the development from a gesture-of (representing the action of dragging) to a gesture-for (representing the procedure of arriving at this particular conclusion) mediated by operational dragging.

Structural dragging is done when a dragging-gesture is conducted in the gesture space without visible references to any concrete representation of a place value chart. This was used particularly when describing the concept of bundling within the digital place value chart, which is a main step of conceptualizing the decimals' structure. Assuming that the characteristic movement from left to right or vice versa indicates the material link between dragging-actions and dragging-gestures, we can con-sider structural dragging to be an indexical sign on the traces left by practical and operational dragging including all experiences, intentions, and conjectures made before.

During the whole design experiment, the two students established a shared context where they observed the other student dragging and negotiated shared answers to the tasks. In the first situation here, Bella performs the dragging (line 1), while Hanna sums up the emerged result (line 2). In the second scene Hanna reacts to Bella's dragging on the chart (line 5) and the chart's reaction by astonishment (line 6) and is therefore likewise involved in conceptualizing de-bundling. Thus, a dragging-gesture of one student can also be taken as an index to previous dragging-gestures by the other student. This way, both students and the device constitute an ecology of tool-based interaction to build the concept of decimal fractions.

Discussion: Reflections and consequences

Applying the theoretical perspective of indexicality we have reconstructed how the action and the gesture of dragging can accumulate more and more aspects about bundling and de-bundling. This process of mutual reference between indexical actions and gestures brought forward the conceptualization of bundling and de-bundling as the basis for the concept of decimal fractions. At the same time the dragging-gestures detached more and more from the concrete dragging-actions on the DPC and became shared signs by enriching indexical references.

Similar to the shift from model-of to model-for (van den Heuvel-Panhuizen, 2003, p. 14), a shift from gesture-of to gesture-for was observed, that is: A dragging gesture first represents the action of dragging (gesture-of), later the dragging-gesture is used as an epistemic means to structure the description of the base-ten structure (gesture-for). Operational dragging can be considered as an intermediate state. It produces a change of view from dragging tokens to bundling and de-bundling as the underlying concept. Hence, adding indexicality to gesture analysis may improve our understanding in how epistemic processes progress.

Whether or not this theoretical perspective keeps being fruitful for tool-based learning in general can only be answered by further empirical research. The main issue will be how this perspective can be fruitfully linked with local theories and models for learning specific contents, such as expanding natural numbers to decimals fractions. For that, we will apply the indexicality perspective to additional data from design experiments with another 15 student pairs in the DeciPlace project, attempting to prove our results and gain further insight into the role of tool-based dragging-actions andgestures for contributing to conceptualize the decimal fraction's structure.

Figure 1 :

 1 Figure 1: De-bundling a token from ones to tenths in the digital place value chart

Figure 2 :

 2 Figure 2: De-bundling as dragging from hundreds to tens

Figure 3 :

 3 Figure 3: Dragging-gesture to the right from ones to tenths above the chart

Proceedings of CERME10

This refers to Peirce in terms of '(section/page)', where

2.228 stands for 'volume 2, paragraph 228'. Thematic Working Group 17 Proceedings of CERME10