Evi Papadaki
email: evi.papadaki@math.uoc.gr

Christos Kourouniotis

Tasks for enriching the understanding of the concept of linear span

Keywords: Teaching and learning of linear and abstract algebra, Teachers' and students' practices at university level, Linear span, Task-design

 along with observations about teaching and learning Linear Algebra, we present two tasks designed to enrich students' concept image regarding linear span. These tasks could be included in a problem workshop of an introductory university course on Linear Algebra. Each task is carefully created and/or selected so as to foster the ground for potential conflict factors to arise and be confronted. A preliminary evaluation shows that the tasks are well received by students and succeed in addressing certain conflicting factors.

INTRODUCTION

Linear Algebra is a subject with many applications in Mathematics and other sciences, but its teaching and learning proves to be demanding both for lecturers and students. The difficulties encountered are partly attributed to the way the subject is usually taught, as well as to students' lack of familiarity with proofs and limited knowledge of Logic and Set Theory. [START_REF] Dorier | The obstacle of formalism in linear algebra[END_REF][START_REF] Hillel | Modes of description and the problem of representation in linear algebra[END_REF]. [START_REF] Sierpinska | On some aspects of students' thinking in linear algebra[END_REF] attributes students' difficulties in Linear Algebra to their practical rather than theoretical way of thinking.

The concept of linear span seems to be quite difficult for students. Carlson (1993) states that difficulties in the notions of subspace, linear span and linear dependence / independence, if they are not addressed in time, create barriers for students. The analysis of [START_REF] Stewart | A framework for mathematical thinking: The case of linear algebra[END_REF] showed that students who were taught these concepts through formal definitions faced significant difficulties in understanding the concept of span compared to a group who were taught with emphasis on embodiment [START_REF] Tall | Building theories: The three worlds of mathematics[END_REF] and geometry. Moreover, they report that students have experienced several difficulties in linking the concept of span to the concept of a base. Finally, [START_REF] Wawro | An inquiry-oriented approach to span and linear independence: The case of the magic carpet ride sequence[END_REF] propose teaching the concept through the solution of systems of linear equations and present a teaching approach through a series of realistic mathematical activities.

The main purpose of this paper is to investigate students' understanding of the concept of linear span and to use tasks to help resolve conflict factors in the students' concept image [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF]. Based on a study of first year Mathematics undergraduates in a Greek university, we identify the misconception many students have that in a linearly dependent set each vector is in the span of the others. We use a set of design principles based on [START_REF] Sierpinska | On some aspects of students' thinking in linear algebra[END_REF] remarks about theoretical thinking and [START_REF] Harel | Three principles of learning and teaching mathematics[END_REF] principles of teaching and learning Linear Algebra, to create a set of tasks, and we present results of a preliminary evaluation of the tasks which indicate their potential to address the above misconception.

The work presented in this paper is part of the first writer's Master's thesis.

THE SETTING

The course "Geometry and Linear Algebra" is a first year mandatory course for students following the degrees in Mathematics or in Applied Mathematics at a Greek University. The course is typically taught through 4 hours of lectures and a two-hour problem workshop per week. Problem workshops are an important part in the teaching of the mandatory courses in the department. In the workshops the students are encouraged to work in groups of 5 or 6 students, on selected problems on the topics taught that week with guidance from the lecturer and a number of postgraduate or senior undergraduate students. The role of the latter is to discuss with students about the problems and the key mathematical ideas that may come up in the process. Promoting mathematical discussion among the students is a promindent element of the workshops of this course. During the semester of the study, the second writer was the lecturer of the course and the first one of the postgraduate students involved in the workshops.

During the first part of the course, students experiment with the idea of linear span in Euclidean 2-and 3-space, as an intuitive introduction to the concept. Later on, students are given a slightly modified version of the formal definition, limited to the spaces R n . The notion of linear span is usually described as the "subspace generated by the set S of vectors in R n ". In relation to the general goals of the course, students are expected to familiarize with the concept of linear span in subspaces of R n , to be able to identify its geometrical representation in the case of R 2 and R 3 and to determine if a vector is in the span of a fixed set of vectors. We note the most important aspects of the concept. Firstly, linear span is a subspace, hence it is closed under the operations of a vector space. Secondly, every element in this subspace is a linear combination of some of the vectors in S. The final aspect is also very important but sometimes overlooked. In contrast to the concept of basis, there is no limitation in the choice of the set of generators S.

A starting point for this work was a study of the written answers given by students in response to a question in the final examination for the "Geometry and Linear Algebra" course, asking them to determine whether a vector belongs to the subspace spanned by two other vectors. The findings suggested that some students may have the misconception that in a linearly dependent set of vectors, every vector can be expressed as a linear combination of the others (see [START_REF] Papadaki | Design of tasks for the understanding of the concept of linear span[END_REF]. This misconception was found to affect students' understanding of linear span and to be a potential conflict factor [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF]. We believe that examining the notion of linear span through tasks may offer the opportunity to confront such difficulties in a meaningful way. [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF] cognitive model of concept image and concept definition is used in the development of the task and to account for students' responses. According to them concept image is "the total cognitive structure that is associated with the concept" (p. 152). For each individual a concept image includes all the mental pictures (graphs, symbols, formulas etc) generated about the concept, associated properties and processes. The concept image is unique for each student and is changing over time when the student meets new stimuli. The term evoked concept image [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF] is used to describe the part of a concept image which is evoked at a specific time. Different parts of the concept image which contain conflicting aspects are called potential conflict factors [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF] and they are not evident to the individual until a stimulus causes the conflicting images to be evoked simultaneously and create confusion, in which case they are referred to as conflict factors.

THEORETICAL FRAMEWORK & DESIGN PRINCIPLES

The term concept definition is referring to "the form of words used to specify that concept" (Tall & Vinner, 1981: 152). The concept definition might be a reflection of an evoked concept image associated with the definition or a rote memorization of a given definition with little or no meaning to the student. We adopt [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF] differentiation between the personal concept definition, constructed by the individual, and the formal definition of a concept, the definition accepted by the mathematical community as a whole. The personal concept definition might contain aspects not included in the formal definition and/or ignore others. Finally, the (personal) concept definition creates its own concept image, which is part of the concept image as a whole, called concept definition image. [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF] argue that potential conflict factors can be an obstacle in understanding the formal theory, especially the ones that are in contrast with the formal concept definition. Warwo et al. (2011) investigated students' concept images of subspace and the links students create with the formal definition of a linear subspace. [START_REF] Bingolbali | Concept image revisited[END_REF] demonstrated how the construct of concept imageconcept definition can be used in socio-cultural research. They argued that although concept image is unique to the individual there are aspects that are shared among students. They link these aspects to teaching and shared experiences in the department they are studying.

In this paper we adopt the original concept imageconcept definition framework [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF] along with its more recent developments [START_REF] Bingolbali | Concept image revisited[END_REF] to design tasks that can enrich the understanding of linear span of undergraduate Mathematics students when used in situations which encourage interaction among students and tutors. We believe that this framework can be easily understood and used by mathematicians. [START_REF] Nardi | Mathematicians and conceptual frameworks in mathematics education… or: Why do mathematicians' eyes glint at the sight of concept image/concept definition[END_REF] presents evidence from discussions with mathematicians which support this idea. Therefore, we find this framework useful as a means to communicate our design and findings both to Mathematics lecturers and researchers in Mathematics Education.

In designing the tasks, we take into account [START_REF] Sierpinska | On some aspects of students' thinking in linear algebra[END_REF] remarks about theoretical thinking. To be more specific, the task should have characteristics that correspond to theoretical thinking, such as opportunities for conscious reflection, connections between related concepts or different representations and attention to contradictory thoughts. [START_REF] Harel | Three principles of learning and teaching mathematics[END_REF] emphasizes the need for curricula tailored to students' needs which aid the understanding of abstract concepts in Linear Algebra. He proposes three principles that we take into account in designing the tasks. That is, the tasks should include familiar concepts that allow connection with prior knowledge and language (concreteness principle), they should justify the need of linear span (necessity principle) and allow generalization of the key ideas (generalizability principle).

We identify the following principles based on the theoretical framework, the concept of linear span as thought in the course "Geometry and Linear Algebra" as well as the needs of our students.

1. Include key aspects of linear span: Closure under the operations of a vector space; Every vector is a linear combination of the set of generators; No limitation in the choice of the set of generators 2. Tackle potential conflict factors: The difference between linear combination and linear dependence; Modes of representation [START_REF] Hillel | Modes of description and the problem of representation in linear algebra[END_REF] 3. Promote theoretical thinking [START_REF] Sierpinska | On some aspects of students' thinking in linear algebra[END_REF]: Reflection; Connections between different representations; Attention to contradictory thoughts 4. The three principles of teaching and learning Linear Algebra [START_REF] Harel | Three principles of learning and teaching mathematics[END_REF]:

Concreteness principle; Necessity principle; Generalizability principle 5. Promote discussion: among the students; between the students and the tutor

METHODOLOGY

The aim of this work is to investigate the conflict factor identified earlier through tasks that are designed to foster the ground for this conflict to emerge and to be discussed with the students. We present data collected during a preliminary evaluation of the tasks through semi-structured interviews with seven students who had attended the course "Geometry and Linear Algebra" the previous semester. The analysis of this preliminary evaluation is expected to answer the following questions:

Can the tasks tackle this potential conflict factor? What are the roots of this conflict factor? Does the discussion around the task help students resolve their misconceptions? Do students find the tasks interesting and/or useful?

The following table summarizes the information about the seven participants.

Mathematics

Applied Mathematics 1 st Year 2 nd Year 3 rd Year 1 st Year 2 nd Year Male 0 1 0 1 0 Female 3 0 1 0 1

Prior to the interviews each student was given a folder including the task and other necessary information. The students had one week to attempt and review the tasks before the interviews. All interviews were videotaped. To ensure confidentiality each student was assigned and referred to with an alias.

ANALYSIS

The first task is based on an exercise from the book "Linear Algebra: Concepts and Methods" by Antony and Harvey (2012). Its structure was slightly altered to fit that of the course notes [START_REF] Kourouniotis | Geometry and Linear Algebra[END_REF]. It aims to create connections with prior knowledge, known processes and language under the new context and introduce to students basic ideas linked with the concept through algebraic and geometric representations of the notion. The task is divided into three interconnected sub-tasks as a scaffolding strategy to support students.

Task 1: Consider the vectors: v 1 = ( -1, 0, 1), v 2 = (1, 2, 3), w 1 = (-1, 2, 5), w 2 = (1, 2, 5) i) Show that w 1 can be expressed as a linear combination of v 1 and v 2 , but w 2 cannot be expressed as a linear combination of v 1 and v 2 .

ii) Explain what subspace of R 3 is spanned by v 1 , v 2 and w 1 . Explain what subspace of R 3 is spanned by v 1 , v 2 and w 2 . What do you observe?

iii) Show that the vectors v 1 , v 2 , w 1 and w 2 span R 3 , that is for every u = (x, y, z) there are a, b, c, d such that: u = av 1 + bv 2 + cw 1 + dw 2 Show also that every vector u ∊ ℝ 3 can be expressed as a linear combination of v 1 , v 2 , w 1 and w 2 in infinitely many ways.

The first, introductory, sub-task aims to support students' theoretical thinking in the following sub-task by limiting its focus on calculations. This task was completed by all the participants without difficulty prior to the interview. The second sub-task is expected to enrich students' image of linear span by making connections between the algebraic and geometrical representations of the concept in R 3 . It may also motivate students to seek a deeper connection between Analytic Geometry and Linear Algebra. This sub-task was completed by 5 students. Finally, the third sub-task aims to create a link between the relation of the given vectors and the number of ways arbitrary vectors can be expressed as a linear combination of the elements in the set.

Sub-task (iii) proved to be the most difficult for the participants, being completed by only 2 students before the interviews.

In more detail, the students who did not complete (ii) appeared to have trouble with methodology. The students are expected to know from the first part of the course what the geometric representation of a 1-, 2-or 3-dimensional subspace of R 3 is, therefore one has to connect this idea with the notion of linear span and check if the given vectors are linearly dependent. In both cases the students did not make this connection beforehand but the problem was quickly resolved through discussion. Apart from that, six out of the seven students found the question "what do you observe?" useful. This question was added to the task as an encouragement for reflection on the effect that different choices of vectors have on the outcome and to promote discussion. In particular, three of the students indicated that they might not have given a second thought to their result if it wasn't for this question. One of the students found the question stressful, although she had successfully answered it. Her reaction is significant to us at this point. [START_REF] Clute | Mathematics anxiety, instructional method, and achievement in a survey course in college mathematics[END_REF] found that students with higher anxiety levels can benefit more from instrumental approaches. Open questions, such as the above, are not frequent in Greek secondary education. It is therefore reasonable to assume that some students would have difficulty (and in some cases anxiety) answering this question in a problem workshop.

While discussing sub-task (ii) an unexpected observation was made by two of the students. These students interestingly replied that the span of the vectors v 1 , v 2 and w 1 is the vector space R 2 . This conflict factor is called by Wawro et al. (2011: p. 13) the "nested subspaces". Based on their evidence they hypothesized that this confusion has roots in students identifying any 2-dimensional subspace of R n with R 2 and suggested that lecturers must be aware of this as a potential conflict factor. Their hypothesis was confirmed in these cases too.

In trying to answer sub-task (iii) the biggest pitfall was following the same reasoning used in subtask (ii). This approach will not help answering the second part which requires from students to solve a system of linear equations. Despite the instructions included in the Task, four out of the five students who didn't complete (iii), tried to use the same approach as in (ii). Additionally, three of them faced a difficulty making use of the proposition "for every u = (x, y, z) there are a, b, c, d such that u = av 1 + bv 2 + cw 1 + dw 2 " and did not manage to recognize the random vector u = (x, y, z) as a parameter of the problem. Instead they identified it as another variable. In each case the task was completed with the help of the interviewer but we find that subtask (iii) required more guidance from the part of the interviewer compared to subtask (ii). The fifth student managed to solve the required linear system but she could not make a connection between the infinite number of solutions and the fact that the four vectors are more than enough to describe any vector in R 3 .

The second task was created to address potential conflict factors in relation to the notions of linear combination and linear dependence in the context of linear span.

The idea for this task was based on our goal to promote theoretical thinking and discussion. The conflict is given to the student as a statement -challenge and the goal is to find an example to support the given proposition. It is expected that students will first use a trial and error approach by reaching for appropriate vectors in their example space [START_REF] Mason | Mathematics as a constructive activity: Exploiting dimensions of possible variation[END_REF]. This approach will probably fail if students are not able to identify what are the key relations between v 1 , v 2 and w in the prοposition. If one's concept image includes conflicting ideas about the status of vectors in a set of generators, it might be difficult to find an example without careful prompting and discussion. Because of the nature of the problem, we believe that students would want to cross-examine their findings or get some guidance.

Task 2: Let v 1 , v 2 and w be linearly dependent vectors in R 3 . It is possible for w not to be in the space spanned by v 1 and v 2 although v 1 , v 2 and w are linearly dependent. Give an example. Why do you think this can happen?

Moving on to the interviews, only one student had found an example of three vectors fulfilling the requirements of the task before the interview. In four of the seven cases clear signs of conflicting images emerged. This reinforces our preliminary hypothesis that students struggle with identifying the difference between the notions of linear combination and linear dependence. Furthermore, it might be an indication that Task 2 can help potential conflict factors to emerge and be resolved in a controlled environment. The following quotations capture these observations.

Minos: So, what I thought was that I can have two vectors... which will be linearly independent that will span a plane in R 3 . I can of course... I am sure that I can find another third vector that will not belong in the plane but the relationship to be true... these three vectors to be linearly dependent.

Minos' evoked concept image of the linear span is geometric. He thinks of the span of the two vectors as a plane and he tries to find an example by checking vectors that are not on that plane. Of course, if the two vectors are linearly independent, adding a third vector that does not belong in their span will result in a linearly independent set. It seems that either this fact is not part of his concept image or his evoked concept image does not include this information because of the phrasing of the task.

In the following two quotations, the conflict can be directly connected to our preliminary findings in [START_REF] Papadaki | Design of tasks for the understanding of the concept of linear span[END_REF]. The students seem to struggle with the idea of three vectors being linearly dependent and at the same time one of them not being able to be expressed as a linear combination of the others.

Interviewer: Well, so for w not to belong in the span of the two other vectors it could not be written as a linear combination of them… Pasiphae: Yes… yes… well… But then how can they be linearly dependent? They are all together linearly dependent…

The student thinks of the two notions as equivalent. She later justifies her thinking by stating that if they are linearly dependent she can solve the algebraic equation av 1 + bv 2 + cw = 0 for any of the three vectors. Similarly, Ariadne describes her own experience with the task. It is worth mentioning that later in the interview Ariadne successfully refers to the (personal) definitions for both concepts.

Ariadne: To begin with, to me it seemed absurd at first... because… what does it tell me? It tells me that they are linearly dependent, so if I solve for w, I will find a linear combination, so based on the theory it belongs to the subspace spanned by v 1 and v 2 .

In Ariadne's case, it can be assumed that although her concept definition for linear dependence includes the information that the coefficients a, b and c are not all zero, in her evoked concept image this statement is replaced by none of them being zero.

The quotations depict two possible roots of students' difficulties with the task. That is, thinking of the linear span of two vectors as necessarily a 2-dimensional subspace or thinking of the algebraic representations of linear dependence and linear combination as equivalent.

Task 2 was thoroughly discussed with the students using different approaches based on the line of thinking of the students, but also influenced by the interviewer. The ideas portrayed in this task were discussed using an algebraic approach with four of the students and geometrically with two of them. In each interview the final example was found by the students using an informed trial and error approach. All six students reported that the discussion was very useful and Task 2 is important for understanding the concept. Three of them also said that this was the task that made them the biggest impression and four of the students suggested that it would be better if this task was presented to them in a problem workshop after a sequence of related more instrumental tasks.

Concluding, four of the students reported that they understand a notion better through examples and tasks. The way that students' concept image is formed through model examples and experience, is of course well known. What is important is the fact that the students are aware of this happening. This last observation is an indication why it is crucial to pay attention to the examples and tasks used in any course. There are students who are consciously depending on them and expect to understand the "mysterious" concepts that the lecturer is talking about through them.

RESULTS & DISCUSSION

The analysis of the interviews gave us very important information about how tasks can be improved and used in a problem workshop for an introductory course on Linear Algebra. Although all students indicated that they found the tasks useful they gave us opportunities to reflect upon their design and experiment with different tactics which can be used by tutors in an attempt to make the most out of these tasks.

Beginning with the first task, students appeared to have particular difficulty in subtask (iii). One reason might be that (iii) requires a shift in thinking and cannot be fully answered by using the same approach as in subtask (ii). In an attempt to resolve this issue we are also considering a slightly different version of this part of the task that forces students to begin with the shifted approach as follows:

Show that for every u = (x, y, z) there exist a, b, c, d such that: u = av 1 + bv 2 + cw 1 + dw 2

Conclude that v 1 , v 2 , w 1 and w 2 span R 3 . Moreover, show that every vector u∊R 3 can be expressed as a linear combination of v 1 , v 2 , w 1 and w 2 in infinitely many ways.

Another observation we made while discussing Task 1 with the students was that of "nested subspaces". This is another conflict factor we didn't take into account at first and realized it only during the interviews with the students. Our observation is in line with the hypothesis of Warwo et al. (2011).

Task 2 was fruitful both in terms of meaningful discussion and reflection. Students found Task 2 important for understanding the concept of span. We also observed manifestations of cognitive conflict which indicates that the task can be used as a means to resolve potential conflict factors. Different approaches can be used to discuss these conflicts with students (algebraically, geometrically or by trial and error). A useful tactic might be to discuss the conflicting factors using more than one representation of vectors with the same group of students.

In addition, the indications about the need of examples and tasks made by the students were of great importance. This fact depicts the necessity of well thought examples and tasks in order to help students create a coherent concept image.

This paper presents an approach on how lecturers can design tasks inspired by their observations on students' misconceptions and taking advantage of the research in Mathematics Education. The framework could be used as guidelines for tutors that are interested in developing tasks for a Linear Algebra course based on their students needs and related research. Finally, the tasks need to be tested in a problem workshop and be compared to other tasks aiming to familiarize first year Mathematics undergraduates with the concept of linear span.


Laura Broley
email: l_brole@live.concordia.ca

Nadia Hardy

A study of transitions in an undergraduate mathematics program

Keywords: Transition to and across university mathematics, teachers' and students' practices at university level, teaching and learning of analysis and calculus

In this paper, we introduce an in-progress study of the transitions students face as they advance in their mathematics courses. Previous work has discussed the changes that occur in the transition from high school to university. With regards to the knowledge students are expected to learn, however, significant similarities have been noted: to do well in introductory university courses, students can learn to solve a particular subset of tasks through routinized techniques, with limited awareness of the supporting mathematical theory. In contrast, students in advanced courses are required to work with and on that theory. The first stage of our project aims to better understand this transition by building praxeological models of the knowledge to be learned in a succession of two introductory analysis courses.

INTRODUCTION

Several studies have discussed the specific knowledge taught and learned in precalculus, calculus, and analysis courses, from different perspectives: for example, concept image and concept definition (e.g., [START_REF] O'shea | Concept images of open sets in metric spaces[END_REF], APOS theory (e.g., [START_REF] Martínez-Planell | Student understanding of the relation between tangent plane and the differential[END_REF], and the Anthropological Theory of the Didactic (ATD; e.g., [START_REF] Bergé | Le rôle de la borne supérieure (ou supremum) dans l'apprentissage du système des nombres réels[END_REF]. Our starting point is the general and relatively vague question of when in an undergraduate degree in mathematics does a student need (need in the sense of to succeed in the course) to engage in mathematical activities that may substantially, or meaningfully, lead to developing mathematical practices. We consider and frame this question within the ATD [START_REF] Chevallard | L'analyse des pratiques enseignantes en théorie anthropologique du didactique[END_REF], which provides theoretical tools for modelling any human activity or practice. The semantic distinction between these two words is essential to us. Our hypothesis is that the kinds of didactic constructs to which professors and students are exposed are decisive in fostering the emergence of practices out of collections of local, particular, and relatively short-lived activities. From the theoretical stance we take, this means the development of mathematical knowledge out of local, particular, and relatively short-lived mathematical activities.

Previous research has found that the activities proposed to students in introductory calculus courses do not necessarily encourage the development of mathematical practices. [START_REF] Lithner | Mathematical reasoning in calculus textbook exercises[END_REF] study of the exercises in undergraduate calculus textbooks used in Sweden led to the conclusion that the majority of tasks students encounter can be solved by mathematically superficial techniques such as finding and copying a similar solution outlined somewhere in the same section of the book. When working in Spanish high school calculus classes, [START_REF] Barbé | Didactic restrictions on the teacher's practice: The case of limits of functions in Spanish high schools[END_REF] observed teachers implementing mathematically incomplete practices: they solved numerous tasks in hopes of guiding students in developing solid mathematical techniques, but struggled to introduce any lasting rational discourse (i.e., theoretical block) that produced or explained the techniques. [START_REF] Hardy | Students' perceptions of institutional practices: The case of limits of functions in college level Calculus courses[END_REF], who conducted task-based interviews with students in North American college calculus courses, showed that in the absence of such a theoretical block, students construct non-mathematical reasoning to support the highly routinized practical block they develop (i.e., the techniques and corresponding types of tasks).

To describe the kinds of transitions students are expected to go through as they progress in their university mathematics coursework, [START_REF] Winsløw | Transformer la théorie en tâches : la transition du concret à l'abstrait en analyse réelle[END_REF] introduced the model depicted in Figure 1 below. The conjecture is that students encounter at least two types of transitions in the practices they are supposed to develop. The first requires them to gain some level of awareness of the theoretical block that was once absent from their exclusively practical work; the second occurs when elements of that theoretical block become part of the practical block with which they must engage autonomously. Think, for instance, of how some early university courses spend a significant amount of time in lectures elaborating previously scarce definitions, theorems, and proofs, which students may be expected to understand enough to quote in assignments or reproduce on exams. In contrast, more advanced coursework requires students to develop their own proofs, often involving the more abstract objects that were part of the theoretical block constructed in earlier courses. A recent study suggests that in the context of undergraduate multivariable calculus courses, students are not yet required to go through Transition 1: the models of the knowledge to be learned in these courses show that students are exposed to a limited practical block (Π), with no need to work with or on the corresponding theoretical block (Λ; [START_REF] Brandes | The bare necessities for doing undergraduate mutlivariable calculus[END_REF]. This said, students are indeed expected to work with and on mathematical theory when they take advanced courses later on.

A few questions arise from this:

1. What does this "work with and on a theoretical block" look like in comparison to the routinized, principally practical activity in which students seem to be engaging in introductory courses?

2. If the practices students develop in advanced mathematics courses can be modelled by the third stage shown in Figure 1, when, if ever, do students' practices reflect the second stage, and what are the mathematical activities proposed to them in such contexts?

The purpose of our study is to contribute to addressing these questions, and therefore, to the discussion of the transitions students face. To do so, we propose to model the knowledge at different stages in the didactic transposition process in two courses contained in what we will call the "analysis path" in a typical undergraduate mathematics program in North America (US and Canada). Ultimately, the goal is to reflect on the general question mentioned above: Can the activities in which students are obliged to engage lead to the development of mathematical practices (i.e., mathematical knowledge)?

THEORETICAL FRAMEWORK "Activity and practice"

As mentioned above, we have come to see the semantic difference between activity and practice as pertinent to our work. The ATD's notion of praxeology provides a fundamental model for defining mathematical practice, which, in the context of the theory, is equated to mathematical knowledge. According to the model, any practice (or piece of knowledge) can be represented by a quadruplet [T, τ, θ, Θ] involving four interconnected components: a type of tasks T, which generates the practice, the corresponding collection of techniques τ developed to accomplish T, the discourse used to describe, justify, explain, and produce the techniques (i.e., their technologies θ), and the underlying theories Θ that serve as a foundation of the technological discourse. As students progress in their studies of mathematics, they engage in numerous activities, which progressively determine the practices they develop. As a strictly hypothetical example, we could imagine students in an introductory calculus course being asked to engage in the following activities, inspired by a commonly used calculus textbook [START_REF] Stewart | Calculus: Early transcendentals[END_REF]: If these were the first activities completed by students in the corresponding sections, we could expect their actions to be localized and particular. In other words, the solutions students produce would likely be the result of their engagement in a relatively isolated act of figuring out how to solve the specific given problem. As the students participate in more activities, however, they may be exposed to tasks of the same type, and may consequently begin to develop a related practice. By the end of a calculus course, for example, students will have typically solved a large number of problems involving the calculation of definite integrals by way of various integration techniques. From this, they may have learned to recognize other activities (e.g., or

) as forming a type of task with a 2 , and therefore as requiring the same technique: making a substitution (not forgetting to change the bounds!), determining the anti-derivative of the new function, and calculating the difference of this anti-derivative evaluated at the bounds. In comparison, certain activities may be encountered by students only in insignificant (e.g., unevaluated), rare, and/or disconnected situations. The action of accomplishing those tasks may hence remain isolated and particular, never contributing to the development of practices. Activities like a 1 or a 3 , for instance, might never be encountered beyond a few recommended exercises at isolated, unique moments in the course.

Research confirms that the collection of activities given (and not given) to students play a crucial role in determining the kinds of practices they develop (and do not develop). Although students may seem to be learning mathematical practices, they may in fact be engaging in isolated activities or developing practices of a nonmathematical nature. In her research, [START_REF] Hardy | Students' perceptions of institutional practices: The case of limits of functions in college level Calculus courses[END_REF] noticed that when first-year calculus students are given activities related to slightly non-routine tasks, they often apply techniques in a mathematically unjust way. For example, when asked to compute , 20 out of 28 students factored, seven of which did direct substitution first. Her analysis of students' discourse during task-based interviews led her to conclude that the students tended to justify their techniques through perceived norms. She specifies, for example, that "it seems that students were doing substitution not to find the limit or to characterize an indetermination, but because that is 'what you do first'" (p. 351). To explain her observations, [START_REF] Hardy | Students' perceptions of institutional practices: The case of limits of functions in college level Calculus courses[END_REF] discusses how the kinds of activities to which the students were exposed led them to develop such practices, composed of a limited practical block and non-mathematical technologies. The activities in which students participated did not only relate to sets of highly routinized tasks, they also required no form of mathematical justification. Engaging in such activities, students observed patterns that led to the construction of techniques based on arbitrary lists of steps that just seemed to work; at least enough to do well on assignments and exams.

In a similar sense, we could imagine a student in our hypothetical example justifying their solution to the activity "integrate over " by saying something like: "first, find the antiderivative, then find the difference between the value at 1 and the value at -1, because that's how we always do it!" An activity such as "integrate over ", might therefore elicit the following erroneous response: Unless of course the isolated activities, a 1 and a 3 , were eventually, substantially, and meaningfully incorporated into developing the above non-mathematical practice into a practice more mathematical in nature.

"Undergraduate mathematics coursework"

As illustrated in the previous section, an anthropological perspective does not interpret students' non-mathematical practices (or knowledge) as reflecting a common misconception inspired by difficulties inherent to a given mathematical concept. Rather, it sees such practices as resulting from a concrete situation within which the student finds themselves, under the influence of institutions [START_REF] Douglas | How institutions think[END_REF]. In the ATD, the word "institution" is taken in a wide sense. For example, mathematicians work within an overarching institution that we could call Mathematical Research (MR), where their praxeologies are shaped by various shared criteria (concerning consistency, beauty, explanatory power, efficiency, etc.), but survive only if they follow the strict rules of mathematical reasoning. The students of interest to us, in contrast, are subjects of the institution Undergraduate Mathematics Coursework (UMC), which was in large part created to train potential participants of MR. This said, various conditions and constraints within UMC can require and enable a network of praxeologies that is fundamentally different from that built and recognized by MR. The non-mathematical praxeologies described in the previous section provide some examples.

To capture the transposition of knowledge as it moves from MR into UMC, Chevallard, and others (e.g., [START_REF] Bosch | Science or Magic?' The use of models and theories in didactics of mathematics[END_REF], have introduced a distinction between different types of knowledge (i.e. practice):

 Scholarly Knowledge, produced and used by mathematicians;  Knowledge to be Taught, as determined by curricula, textbooks, and professors' teaching plans;

 Knowledge Actually Taught, according to professors' actual interactions with students, e.g., in lectures;

 Knowledge to be Learned, i.e., the knowledge students are expected to develop, which is often a transposed subset of the knowledge to be taught and actually taught, with the minimal core indicated by assessment tools;

 Knowledge Actually Learned, which can only be predicted, through analyses of student work, in-class observations of students, or other specially-designed interactions with students, such as interviews or problem-solving situations.

Although a lot can happen in university lectures, the minimal knowledge students are obliged to learn to pass their courses is determined by their assignments and exams. It is not surprising that the knowledge actually learned by students is often only a transposed subset of this minimal core. Hence, if we want to know what kind of knowledge students are or could possibly be developing in UMC, then we cannot restrict our exploration to curricula, textbooks, and teachers' lecturing practices: we need to pay careful attention to the way in which students are assessed.

Students' learned knowledge in UMC may also be characterized as a progression through various sub-institutions: from secondary school to early university courses (e.g., in single and multivariable calculus), through to more advanced university courses (e.g., in real analysis, metric spaces, measure theory, and functional analysis), which may eventually lead to graduate studies and beyond. Programs can vary from school to school and from country to country. However, a common phenomenon in secondary schools seems to be that assessments focus solely on the practical block of mathematical knowledge. The teacher may be expected to know the theoretical block for explaining the material to students; but the students are typically not obliged or even invited to develop an awareness of the technology or theory, let alone how it is linked to the practical block [START_REF] Barbé | Didactic restrictions on the teacher's practice: The case of limits of functions in Spanish high schools[END_REF][START_REF] Winsløw | An institutional approach to university mathematics education: from dual vector spaces to questioning the world[END_REF]. One observed result is that many students interpret mathematical knowledge (practice) as equivalent to identifying a type of task and applying the corresponding technique [START_REF] Bergqvist | Upper secondary students' task reasoning[END_REF]. Several studies confirm that this same kind of situation can arise in early university coursework (e.g., [START_REF] Lithner | Students' mathematical reasoning in university textbook exercises[END_REF][START_REF] Hardy | Students' perceptions of institutional practices: The case of limits of functions in college level Calculus courses[END_REF][START_REF] Brandes | The bare necessities for doing undergraduate mutlivariable calculus[END_REF].

Indeed, over multiple years of coursework, students not only gain a particular view of what mathematical knowledge is, but they also develop knowledge that, when judged against the scholarly knowledge produced and used by mathematicians, is evidently non-mathematicalfrom the strategies they develop to identify tasks, to the discourses they use to justify these strategies and the techniques they choose. Nevertheless, as conjectured in the schema shown in Figure 1, a transition is expected to occur at some point: students are eventually required to develop knowledge that is completely and coherently mathematical. These circumstances lead [START_REF] Winsløw | An institutional approach to university mathematics education: from dual vector spaces to questioning the world[END_REF] to wonder about how teachers could help students accomplish such transitions. In parallel, we are inspired to validate, specify, and extend these researchers' claims by constructing praxeological models of how the different kinds of knowledge produced in the didactic transposition process progress throughout an entire undergraduate degree. In other words, we are inspired to investigate more closely the nature of the mathematical training being received by future mathematicians in the progression of their undergraduate coursework.

Of course, developing praxeological models to represent the knowledge (to be) taught and (to be) learned throughout an entire undergraduate degree is a hefty task. Within the context of our PhD project, we propose to accomplish a first stage, based on a subset of courses in one coursework path. Like in the Mathematical Research institution, Undergraduate Mathematics Coursework is divided into several subinstitutions according to domaine.g., algebra, geometry and topology, analysis, statistics, mathematical physics, or probabilityeach of which contain a grouping of courses, which can be placed in some chronological order according to their prerequisites. Having already carried out research in the early courses of an "analysis path", this is the context that seemed most appropriate for our work.

METHODOLOGY

Although our project aims at modelling different levels of knowledge that can be identified in the didactic transposition process, in this paper we discuss only the modelling of the knowledge to be learned (KTL).

Our research is conducted at a large, urban, Canadian university. The mandatory courses in the analysis path of an Honours Bachelor of Science in Mathematics include multivariable calculus (MVC) I and II, and mathematical analysis (MA) I, II, and III. Since previous work [START_REF] Brandes | The bare necessities for doing undergraduate mutlivariable calculus[END_REF] suggests that the KTL in MVC I and II is similar in nature to the KTL in calculus and pre-calculus courses, we decided to start by focussing our attention on the two courses that come next and are likely candidates for housing the transitions of interest to us: MA I and II.

As mentioned above, the KTL represents the knowledge that students are expected to develop, which can be gleaned from the various activities in which they are invited to engage (lectures, assignments, and exams), as well as the materials that frame and support the activities (course outlines and textbooks). Since the minimal core of the KTL is represented in the assessment activities students must complete on their own, we have decided to ignore what happens in lectures and focus on the activities that comprise assignments and (practice) exams.

MA I and II are institutions in themselves in that they enjoy some sort of stability.

For various reasons, course outlines and assigned textbooks tend to remain the same from year to year. The courses also maintain the same assessment structure: students complete assignments on a regular basis during the term, a midterm exam halfway through, and a final exam, with most of their mark (90% or more) concentrated in the examinations. This said, the actual activities proposed on assignments, midterms, and finals have less stability in that they can reflect personal choices of the professors assigned to teach the course in a given term. On top of this, our approach to modelling the knowledge actually learned will involve task-based interviews with students after they have passed MA I or II. Hence, we have collected the assignments and (practice) exams proposed only by the professor(s) who would be teaching those students. For instance, from the two MA I professors teaching in Fall 2017, we collected eleven weekly assignments, seven practice midterms, six practice final exams, and the actual examinations they gave to their students (these professors worked together in that they gave the same set of activities to their students).

To analyse such a collection of activities, we think about whether each activity is "isolated" or part of a "path to a practice". An "isolated" activity may occur only once in the sense that no other activities engage students in accomplishing the same type of task. Since such activities are unlikely to contribute directly to the development of a practice, we reflect on why they are proposed. In comparison, the activities that belong to a "path to a practice" typically combine with other activities to expose students to a type of task. Our goal in studying these activities is to extract a theoretical model of the praxeologies that the ideal student (i.e., the student that receives a good passing grade) is expected to develop in the course. We start by constructing punctual praxeologies related to groups of non-isolated activities.

Looking at the problem statements, we can establish the types of tasks (T) that generate the praxeologies. Determining the technologico-theoretical blocks ([τ, θ, Θ]), however, requires more data. We rely on the solutions a professor makes available to students to uncover the intended techniques, as well as portions of the expected theoretical blocks; and we complete the latter by checking the course outline and reading the relevant textbook chapters. The resulting collection of punctual praxeologies then becomes part of our data, which we use to construct more generalized praxeologies, think about how they are related to one another, and reflect on the nature of the KTL.

Eventually, we plan to put the models of the KTL for MA I and II together and compare our results with what previous researchers have found in calculus and precalculus courses. This, we hope, will allow us to discuss how the ideal student is expected to progress in the early stages of the analysis path. At the time of the INDRUM 2018 conference, we will have completed this initial theoretical stage of our project and will thus be able to share our results.

SUMMARY AND EXPECTATIONS

For a long time, mathematics students survive their courses based on developing a transformed version of a practical block, where they learn to recognize routine tasks and apply techniques to solve them in a sort of mechanical, naturalized, or normalized way, void of a mathematical theoretical block. At some point throughout an undergraduate degree in mathematics, however, the conditions for students' survivability change dramatically and possibly abruptly: they are faced with activities that require them not only to fill the void of a mathematical theoretical block, but also to develop techniques for accomplishing tasks (e.g., proofs) that involve the abstract theoretical objects that have come out of hiding. Through modelling the knowledge the ideal student is expected to develop, as well as, eventually, the knowledge students actually develop in early analysis courses, we expect our project to bring about a more detailed and concrete understanding of praxeological "transitions" that have been theorized to occur, and give us some insight into how (or if) students adapt to them.

Returning to the vague and general question that originally inspired our project, we ultimately hope to learn more about what students are actually learning throughout an undergraduate degree in mathematics. The empirical data we collect will be a contribution to largely anecdotal discussions about when (if at all) students' knowledge is invited to become, and actually becomes, coherently, completely, and complexly mathematical, just like the scholarly knowledge produced and used in the institution of Mathematical Research. The significant difference between elementary and advanced courses, as professors gain more freedom and teach topics more closely related to their field of study, leads us to predict that students are eventually required to develop mathematical practices. After all, in spite of the apparent disconnection that is often observed between university mathematics courses and mathematical research (cf. [START_REF] Broley | Levels of programming in mathematical research and university mathematics education[END_REF], the field of mathematics continues to live on, with new mathematicians emerging from the coursework that made up their mandatory professional education. In any case, through studying the principal conditions that currently shape the activities in which undergraduate mathematics students engage, we feel that we will be in a better position to discuss realistic and meaningful ways of encouraging these students to develop practices that are truly "mathematical", within the confines of educational institutions. This, we hope, will serve as complementary to the recent surge of studies (cf. [START_REF] Barquero | A bridge between inquiry and transmission: The study and research paths at university level[END_REF] aiming to explore innovative teaching approaches that question not the nature of the knowledge developed, but the dynamics of the knowledge development.

Figure 1 :

 1 Figure 1: Transitions in university mathematics coursework (from Winsløw, 2008)

a 1 :

 1 Estimate the area under the graph of from to using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate? What happens if you repeat the exercise with left endpoints? (Areas and Distances, Section 5if is convergent or divergent. If it is convergent, evaluate it. (Improper Integrals, Section 7.8)


María Trigueros
email: mtriguerosg@gmail.com

Rafael Martínez-Planell

On students' understanding of Riemann sums of integrals of functions of two variables

Keywords: Functions of two variables, APOS, integral calculus

APOS (Action-Process-Object-Schema) Theory is used to pose and test a conjecture of mental constructions that may be used to understand the relation between integrals of two variable functions over rectangles and corresponding Riemann sums. Interviews with ten students who had just finished a multivariable calculus course showed that the conjectured mental constructions are necessary.

Multivariable functions and multivariable Calculus are important in engineering and the natural sciences as a tool for modelling. Their learning has received more attention lately from the Mathematics Education community. Starting with the analysis of students' understanding about two-variable functions (for example : Trigueros and Martínez-Planell, 2010;[START_REF] Martínez-Planell | Students' understanding of the general notion of a function of two variables[END_REF] researchers have documented students' difficulties and have shown that the transition from one-variable Calculus to multivariable Calculus is far from being smooth. There are few studies in the literature that deal with students' difficulties and understanding of the integral multivariable Calculus [START_REF] Jones | Students' generalizations of single-variable conceptions of the definite integral to multivariate conceptions[END_REF][START_REF] Martínez-Planell | Students' relating integrals of functions of two variables and Riemann sums[END_REF]. In one of these few studies, [START_REF] Mcgee | A study of semiotic registers in the development of the definite integral of functions of two and three variables[END_REF] showed that a course based on lectures did not promote students' understanding, while activities introducing the use of semiotic chains and the development of synergy among representations helped students understand this concept. The research questions are: What constructions relating double integrals and Riemann sums are evidenced by students who finished a Multivariable Calculus course based on lectures? What constructions may be needed to relate double integrals and Riemann sums?

THEORETICAL FRAMEWORK

APOS theory [START_REF] Arnon | APOS Theory: A framework for research and curriculum development in mathematics education[END_REF]) is used in this study to analyse possible mental constructions by students who have already taken a course on multivariable calculus. We only summarize the main structures of this theory. An Action in APOS Theory is a transformation of a previously constructed mathematical object that the individual perceives as external in the sense that students need some guidance and are not able to justify what they do. When an Action is repeated, and the individual reflects on what he or she does, it may be interiorized into a Process. A Process is perceived as internal in the sense that it has meaningful connections to other mathematical knowledge of the individual. A Process allows the individual to imagine doing the Actions without actually doing them, to omit steps, anticipate results, and to justify the Process. Different Processes may be coordinated to form new Processes. When the individual needs to apply Actions on a Process, it can be encapsulated into an Object. When an individual shows a Process or Object conception of a mathematical notion we say that the individual "understands" the notion. A Schema is a coherent collection of Actions, Processes, Objects, and other Schemas, that the individual uses to work with problems related to some mathematical notions. Schemas are not used in this paper.

To analyse students' work using APOS Theory, a conjecture of those constructions that may be used to understand a specific mathematical notion is designed. This model, called a genetic decomposition (GD), does not pretend to be unique and needs to be tested with research data. The GD may be revised and expanded in successive cycles of research, teaching material development, and implementation. This research cycle makes it possible to use APOS theory to be better suited to future research needs to study the multivariable integral calculus.

GENETIC DECOMPOSITION

We only present a portion of the GD of integrals of functions of two variables over rectangles. Its development is based on mathematics, on the researchers' teaching experience, and data from the research literature: mainly, ideas about representation registers [START_REF] Duval | A cognitive analysis of problems of comprehension in a learning of mathematics[END_REF] described in the study by [START_REF] Mcgee | A study of semiotic registers in the development of the definite integral of functions of two and three variables[END_REF] and the ideas of "orienting pre-layer" and "product layer" described by [START_REF] Sealy | A framework for characterizing student understanding of Riemann sums and definite integrals[END_REF], which stress the need of attending to the individual meaning of the product () i f x x  and its components in the construction of integrals of one-variable functions.

The GD starts with pre-requisite constructions which include: a Process conception of two-variable functions and volume of prisms as Object.

Actions are performed on a given two-variable function in any representation with domain restricted to a rectangle, to produce the geometric representation of the restricted domain as a subset of 3D space. Actions are performed on the same function to obtain values of the function on the given domain and to represent them in the space as points and/or curves in the graph of the function. These Actions are interiorized into a treatment or conversion Process to represent the graph of the function over the given rectangle together with the rectangle so that the student can imagine the relation between function and rectangular domain as a graph in space.

Actions of evaluating the given function of two variables at a specific point of a given sub-rectangle of its domain, multiplying it by the length and width of the rectangle to form a product of the form ( , ) f a b x y  are done. These Actions are interiorized into a Process which can be coordinated with conversion Processes between different representations of function, rectangle, and given point, to imagine the product as the volume of a rectangular prism in space.

Given a continuous function in different representations defined on a rectangle, with the function simple enough so that its maximum and minimum values on the rectangle may be recognized without doing any explicit computation, the Action of obtaining an overestimate and an underestimate of the product ( , ) f a b x y  is taken. These Actions may be interiorized into a Process that enables to imagine the existence of points ( , ) ab where underestimate and/or overestimate of the product ( , ) f a b x y  are attained. This Process is coordinated with a treatment or conversion Process to draw a rectangular prism corresponding to over and/or underestimate in space. Actions are performed to change the chosen point to construct a prism that better approximates a given exact value of the integral. These Actions are interiorized into a Process that enables the recognition that for such continuous function, there is a point somewhere on the rectangle that will produce the exact value of the volume between the graph of the function and its rectangular domain.

Given two small specific positive integer numbers, n and m, the Action of subdividing given intervals [a,b] and [c,d] into subintervals of equal length both numerically and geometrically is done to obtain a subdivision of the rectangle  , and the corresponding sum, interpreting this sum geometrically, numerically, symbolically as an extended sum, symbolically using sigma notation, and verbally, may be interiorized into a Process that enables imagining forming such sums in different representations for the collection of sub-rectangles in any partition of any given rectangle.

METHOD

Ten students were chosen by their professor to be interviewed at the end of a multivariable calculus course selecting four over-average, three average, and three under-average. The course was completely based on lectures. The interviews lasted 46 minutes on average. Students answered a set of questions designed in terms of the GD and also related to what was covered during the course, and produced a written response while sharing their thoughts out-loud. The interviews were recorded, transcribed, individually analysed, and results were negotiated by the two researchers. Students' responses were analysed according to the GD, while keeping notes on unexpected responses and other difficult to classify observations. These were the questions used: 1a. The following is the complete graph of function Note that problems 1a and 1b are essentially the same in different representations. They both test the portion of the GD dealing with recognition of rectangle and function. Problem 1c gives information on the portion of the GD dealing with forming one term of a Riemann sum. Problems 1d, 1e, and 1f relate to the portions of the GD dealing with underestimate, overestimate, and exact value. Problem 1g gives information on the portion of the GD dealing with a partition and Riemann sum.

RESULTS

On function and domain of a function

Many students showed they had not constructed the concept of two-variable function. They gave evidence of considering these functions in terms of a correspondence rule, and showed difficulty interpreting functions given graphically. Moreover, these students also showed not to have constructed the concept of domain of the function. Most of them considered that the domain of a two-variable function should include information about the function, since it had to be represented in 3D space. Eight students showed difficulties similar to those of Luis, as exemplified in the following discussion with the interviewer (in Problem 1a): No, the domain is obtained from the graph. I can obtain the domain having the function but to do so I have to define the function.

After some discussion:

Interviewer: So the domain, is it only x and y or may it also include z?

Luis:

The domain may include the z.

This example shows how Luis needs a correspondence rule to determine the domain of the function. It also evidences that he considers the function itself should be part of the domain of the function. Other students showed this difficulty.

Students' responses pointed to a need to pay attention to the different representations of functions in 3D space and to have students do treatments and conversions between representations. In Problem 1a, some of them quickly represented the rectangular domain as part of the given figure in 3D space. However, when the function was given symbolically their notion of domain seemed to change. This shows that recognizing the domain of a two-variable function is a construction that needs the interiorization of Actions on functions given in different representations. These difficulties as well as the counterfactual belief of teachers that students may easily generalize concepts for one-variable functions to multivariable functions have been reported before (Martinez-Planell and Trigueros, 2012).

Area and volume

Students also showed an unexpected confusion between area and volume when they described graphs of functions in 3D-space. This difficulty surfaced in Problem 1c.

All students were able to calculate the value of that product; however, they were in trouble when explaining its geometrical meaning. Brian, for example, explained:

Brian: … this part, f(0,1) would be a point in this graph here. Change in x, change in y, I am not a hundred percent sure… that would be an area then, of the surface, or the entire function… And later: Other students, as Luis, showed confusion:

Luis:

The area of the figure, that is, the area of the function which in this case is that figure [referring to the graph in Figure 1]. Other five students showed the same confusions. The above excerpt exemplifies that a student can describe the individual components of (0,1)

f x y
 but might not be able to do the Action of putting them together to interpret it as the volume of a rectangular prism, even if they can calculate the result of the product by doing the Action of substituting the given values in the expression, as conjectured in the GD. This difficulty is possibly related to the fact that these students have not constructed space as an Object, which does not allow them to imagine what their teachers mean when they talk about a surface in space and the double integral as related to the volume under a surface (Trigueros and Martínez-Planell, 2010). As considered in the GD, these difficulties make it impossible for students to do the necessary Actions on the function restricted to a rectangle to represent the domain geometrically as a subset of space, and to interiorize the Process to imagine the relation between a restricted region on the domain and the function. The lack of all these constructions becomes an obstacle to understand other related concepts, including double integrals, as will be shown below. Students who do not show these constructions may not follow teachers' explanations; they would be confused and resort to memorization to respond to exam questions.

Only one student, Farid, gave evidence of the pre-requisite constructions described in the GD. He was successful in explaining Riemann sums and double integrals.

Riemann sums, underestimates, overestimates and double integrals

Most students had many difficulties working with problems dealing with Riemann sums and their relation to double integrals. Even after the interviewer explained to some of them that (0,1)

f x y
 was a volume and drew it, their difficulties did not enable them to make the whole construction as Brian showed in Problems 1d, 1e, and 1g:

Brian:

So the Riemann sum would be the approximation of the area [sic] under this figure [referring to the graph in Figure 1], obviously it wouldn't be as precise as the value of the integral. Let's see... so geometrically 0,1, x, y, let's draw a square here like this [he is now evaluating and drawing rectangular prisms]... 0, 1.5, maybe another square closer this way, higher... 1,1 we are still at x 1 and even higher here… like this, change in x change in y... change in y being 1/2, I don't think we get from 1 to 2 with 1/2 [He seems to believe that since 1/ 2 y  the prisms will be restricted to the region 1 1.5 y  . He might think of y  as "change in y" were the "change" is taken from the initial y value in point (0,1).], so the integral would give this area [sic] here, a figure more or less like this... Interviewer: You said area...

Brian:

[Interrupting] Volume, I mean volume, sorry... yes, volume of the integral. This would give us something more stepwise... let's see if I can draw it here like this... 1,2,3,4, like this, a series of cubes like this, stepwise, approximating, not all of this, but only this half here… [See Figure 2].

Interviewer: Do you mean the left hand part of the solid?

Brian: Yes, the left hand part of the solid would be what is approximated with this Riemann sum.

Although Brian was able to construct the meaning of volume, his construction was not right, the boxes he drew filled only the left-hand side of the rectangle. It seems that Brian could do the Actions to construct the prisms but he did not interiorize those Actions into the Process that would enable him to imagine all the constructions needed to relate Riemann sums and double integrals.

An interesting result of this experience was that even though students showed many difficulties during the interview, some of them, like Brian, showed evidence of doing some of the expected constructions during the interview. Others reflected during the interview and constructed meaning. This was the case of Victor who had considered (0,1) f x y

 as an area. When discussing Riemann sums, and after being told that this product represents a volume he explained:

Interviewer: So you drew a little box.

Victor:

Exactly a little box as we know that delta x would be 2 and delta y 1; a rectangle with width 2, eh, length 2 with 2 and height 1.

Interviewer: Then how do you compare those volumes.

Victor:

Ok, now I understand, this f(0,1) delta x delta y is only the volume up to this point, I mean up to a certain height, and then, the double integral on that same area that we put on xy is, let's say, the same box but with a height that varies with the function. Now this is it! Victor:

No, no, they are completely different, the larger is that obtained from the double integral since the height is higher.

Victor: f(2,2) delta x delta y is the box with dimensions over D…and then this is the same equation as before, but f(2,2) is higher so the volume there is larger.

Victor could do the Actions needed to compare volumes of prisms obtained from different values of the function. Another student, Farid, evidenced he could imagine forming one term of a Riemann sum, as discussed before. He also showed to have done the constructions necessary to imagine volumes of prisms and their role in Riemann sums. When comparing the volume of the prism in Problem 1d with the double integral:

Interviewer: And what does that represent? [Referring to the double integral.] Farid: That represents the volume between the surface and the plane, the domain... Interviewer: Then, how do those two numbers there compare?

After some doubts:

Farid: Represented this part [referring to the product], now, this product would be smaller, than the double integral, because this here is a, I represented it as a cube, given the value of f at that point, while this is the double integral of everything, of all the function x,y over D, so this value seems bigger Interviewer: Which one?

Farid:

The value of the double integral over D of f(x,y)dA

He was able to compare the volume of the prisms with the double integral. When he had to decide if there would be a prism with volume equal to the value of the double integral, a problem that was impossible for all the other students, he explained:

Farid:

… The thing that comes to my mind when thinking on an inequality is the sandwich theorem…that there... must exist then a value for x and y that could be named a and b that is equal to the double integral on dA.

When discussing Riemann sums with a specific partition, most students could not work with the problem, even with help from the interviewer. As was shown by Brian in the previous example, some students, including Victor, imagined drawing several prisms or boxes that shared the base, and only had different heights. Those students showed they could do the Action of changing the height of a given prism but not that of partitioning the domain into small areas of the same size. Victor could describe the sum of the prisms' volume, at first he said that the Riemann sum was always an approximation to the volume under the surface, although later he reconsidered:

Victor:

No, the Riemann sum is an approximation and if you take more subintervals, ah! If you take more subintervals, those were the squares, that one uses, the Riemann sum is a closer approximation and that approximation would be closer with more subintervals, and the double integral is the exact value.

Only Victor and Farid seemed to have interiorized the Action of forming a partition into a Process they coordinated with the Process of selecting heights for each subrectangle into the Process of calculating the volume corresponding to the prisms to approximate the volume under the surface.

DISCUSSION AND CONCLUSION

Results from this experience show that most of these students demonstrate a very limited understanding of two-variable functions and of those concepts associated to the construction of the double integral of a two-variable function and its geometrical interpretation. Only two students showed some understanding, although one of them relied mostly in memorized facts that he could use appropriately in most cases. This student seems to have constructed meaning for some of those facts during the interview. Students' responses show the importance of the predicted constructions included in the Genetic Decomposition. In this investigation we related observed difficulties with specific mental constructions in the GD that students seemed to lack. The importance of the pre-requisite constructions in learning this difficult topic was underscored. Its lack became an insurmountable obstacle to understand even the most basic ideas leading to the learning of the double integral.

A more encompassing understanding of function in different representation registers proved to be indispensable. Results indicate that students who could only perform Actions constructed a confusing network of concepts where the properties learnt about one-variable function are not well differentiated from those of two-variable functions. This inhibits their possibility to make those constructions involved in the understanding of 3D space, functions, and their domains. Fluency in operating within and across different representations plays an important role in the construction of two-variable functions as an Object, instead of considering them as simple correspondence rules containing one or more variables.

These results emphasize once more the importance of spending more time on helping students to construct the notion of two-variable function. But, even when twovariable function has been constructed as a Process, the notions of volume under the surface and the role of the Riemann sum in the construction of the double integral constitute fundamental constructions in the learning of double integrals. The genetic decomposition proved useful in determining and underscoring those mental constructions that are needed to learn double integrals with meaning. It also reveals the subtleties involved in learning the double integral. After classroom use of specially designed activities, future studies may reformulate the same interview problems and also extend them to explore other ideas of the integral calculus.

  [ , ] [ , ] a b c d  . These Actions are interiorized into the corresponding Process. Given a continuous function f defined on the rectangle, the Action of choosing a prescribed point ( , ) ij xy on each sub-rectangle of the given partition and producing the products

..

  Figure 1: (repeated in each part of problem 1 except 1b) 1d. Let

Interviewer:

  Can you tell me what does the double integral of f(x,y)dA represents? Brian: dA is the area of the function, the area of this figure, Interviewer: The area of the surface? Brian: Yes… of the surface on the given domain….

Figure 2 :

 2 Figure 2: Brian's drawing for Problem 1d, 1e, 1g (respectively)

  : I can tell you what the domain is but if I don't have a function I don't think I can tell you the exact point where each of the points in the graph is… Interviewer: So, is the graph part of the domain?

	LuisLuis: