Eric Albin

Ronnie Knikker

Shihe Xin

INFLUENCE OF WALL RADIATION IN 2D CAVITIES HEATED FROM BELOW

Keywords: Rayleigh-Bénard, wall radiation, convection, bifurcation

This study investigates numerically the effects of wall radiation on Bénard cells in cavities heated from below using Chebyshev spectral methods. Bifurcation theory is also used to describe and analyse the cellular flow solutions obtained in a square and a rectangle cavities filled with air.

As wall radiation modifies the flow via the adiabatic vertical wall condition, the motionless and thermally stratified solution no longer exists at low Rayleigh numbers and is replaced by a very weak 2 × 2 cellular solution. This 4-cell weak flow undergoes pitchfork bifurcations at higher Rayleigh numbers: both perfect and imperfect pitchfork bifurcations are observed in the presence of wall radiation. Bifurcations to cellular flows with a reflection symmetry become imperfect because the cellular flows with the opposite rotating directions are no more equivalent. Therefore the resulting convective and radiative Nusselt numbers of the two bifurcated branches are no longer the same on the top or bottom wall. Note also that Nusselt numbers (either convective or radiative) of any bifurcated branch at one Rayleigh number are different on the bottom and top walls with wall radiation. Results obtained are analysed and some recommendations are given to describe Bénard flows with wall radiation.

Introduction

Heat transfer in a cavity heated from below has been widely investigated [START_REF] Gad | Effect of surface radiation on RBC in cavities heated from below[END_REF]. Nevertheless, few studies with wall radiation have been conducted and the effect of wall radiation is less understood in the litterature. Mizushima et al. [START_REF] Mizushima | Sequential transitions of the thermal convection in a square cavity[END_REF] used bifurcation diagrams to describe various thermal and flow regimes in such a cavity without radiation. Such bifurcation analyses are not used in the past studies with radiative walls.

In [START_REF] Ridouane | Effects of surface radiation on natural convection in a rayleigh-benard square enclosure: steady and unsteady conditions[END_REF], Ridouane et al. describe various steady-state patterns and unsteady solutions in a square cavity with radiative walls. They describe different flow solutions and heat regimes in various nested tables. For instance, they analyse bi-cellular flow regimes ascending at the middle of the cavity, but do not study the reverse solution descending at the middle. Also, the given convective and radiative Nusselt numbers are not associated to the top or bottom walls. The effect of vertical wall emissivity (0 V 0.85) and of cavity aspect-ratio (1 AR 10) on Rayleigh-Bénard convection is investigated in Gad and Balaji [START_REF] Gad | Effect of surface radiation on RBC in cavities heated from below[END_REF]. The numerical solutions are computed using FLUENT to solve steady incompressible Navier-Stokes equations with radiative walls and a transparent medium. They observe jumps from unicellular to bi-cellular flows at some critical Rayleigh numbers, but they do not address the problem of multiple solutions nor the issue of flow bifurcations. Some preliminary studies are presented by Xin et al. [START_REF] Xin | Méthode de collocation chebyshev pour le couplage entre la convection naturelle en cavité remplie d'air et le rayonnement pariétal[END_REF][START_REF] Xin | Méthodes d'ordré elevé pour le couplage de la convection naturelle avec le rayonnement de surface en cavité remplie d'air[END_REF][START_REF] Langenais | Etude numérique de l'influence du rayonnement pariétal sur la convection de rayleigh-bénard en cavité 2d remplie d'air[END_REF]: the coupling of natural convection with surface radiation in a cavity using a Chebyshev spectral method is presented and validated and the first bifurcation at low Rayleigh number is studied for a square cavity and a twice longer cavity in [START_REF] Langenais | Etude numérique de l'influence du rayonnement pariétal sur la convection de rayleigh-bénard en cavité 2d remplie d'air[END_REF].

The present study continues the work performed in [START_REF] Langenais | Etude numérique de l'influence du rayonnement pariétal sur la convection de rayleigh-bénard en cavité 2d remplie d'air[END_REF], adopts bifurcation analyses used in [START_REF] Mizushima | Sequential transitions of the thermal convection in a square cavity[END_REF] and suggests, for wall radiation cases, that results of Nusselt numbers should be presented for both the top and bottom walls or at least for the specified wall (either top or bottom wall).

The paper is organized as follows. After recalling the governing equations of the cavity problem with wall radiation, the bifurcation theory is first used to describe the different flow regimes. Heat transfers at walls are therefore analyzed using Nusselt numbers. Some concluding remarks and recommendations end the paper.

Governing equations

A cavity of length L and height H is heated from below with a constant high temperature T h at the bottom wall and cooled from top with a constant low temperature T c at the top wall (see Fig. 1). Two aspect ratios are considered in this paper: a square cavity (L = H) and a rectangular cavity (L = 2H). The cavity is filled with air considered as a Newtonian transparent medium of density ρ, kinematic viscosity ν, thermal conductivity k, specific heat c p , thermal diffusivity α = k/(ρc p ) and thermal dilatation coefficient β. In order to make comparisons with the results of Gad and Balaji [START_REF] Gad | Effect of surface radiation on RBC in cavities heated from below[END_REF], the considered fluid properties are T 0 = Tc+T h 2 = 309 K, g•β ν•α = 8.0736 • 10 7 and Pr = ν α = 0.71 with g = 9.81m • s -2 . Natural convection flow in such a cavity is governed by the Navier-Stokes equations and depends strongly on the Rayleigh number Ra = gβ∆T H 3 να where ∆T = T h -T c is the differential temperature in the cavity. In presence of wall radiation, it also depends on a radiation to conduction parameter Nr = σ(T 4 h -T 4 c ) • H/(k∆T ) with σ = 5.67 • 10 -8 W • m -2 • K -4 being the Stefan-Boltzmann constant. The Rayleigh number is changed by varying ∆T with Nr 4σT 3 0 H/k = 7.765 (i.e. H = 3.14cm to match [START_REF] Gad | Effect of surface radiation on RBC in cavities heated from below[END_REF] with T c = T 0 -∆T /2 and T h = T 0 + ∆T /2). In this study, Navier-Stokes equations of the incompressible flow under the Boussinesq approximation are solved :

∂u ∂x + ∂v ∂y = 0 (1a) ∂u ∂t + u ∂u ∂x + v ∂u ∂y = - 1 ρ 0 ∂P ∂x + ν ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 (1b) ∂v ∂t + u ∂v ∂x + v ∂v ∂y = gβ(T -T 0 ) - 1 ρ 0 ∂P ∂y + ν ∂ 2 v ∂x 2 + ∂ 2 v ∂y 2 (1c) ρ 0 c p ∂T ∂t + u ∂T ∂x + v ∂T ∂y = k ∂ 2 T ∂x 2 + ∂ 2 T ∂y 2 (1d)
A no-slip wall condition is forced at the walls (u = v = 0). The bottom and top walls of the cavity are isothermal walls (T (y = -H/2) = T h and T (y = H/2) = T c ). The two vertical walls are adiabatic in the sense that radiative transfer balances local heat conduction:

-k ∂T ∂x qc +q r = 0 on left wall and -k ∂T ∂x qc -q r = 0 on right wall

The radiative heat flux q r is positive when wall radiates more towards other walls than it is irradiated. It is computed from radiosity J using a high-order method with integration by parts proposed in [START_REF] Knikker | High-order numerical implementation of surface radiation for the coupling with natural convection in an air-filled square cavity[END_REF]:

q r = 1 - (σT 4 -J) (3)
The wall radiation coupled to natural convection is implemented using the Chebyshev collocation method and validated by Xin et al. [START_REF] Xin | Méthodes d'ordré elevé pour le couplage de la convection naturelle avec le rayonnement de surface en cavité remplie d'air[END_REF][START_REF] Knikker | High-order numerical implementation of surface radiation for the coupling with natural convection in an air-filled square cavity[END_REF]. For the present Rayleigh-Bénard configurations, the governing N-S equations are discretised on a 61 × 61 grid for the square cavity and a 121 × 61 grid for the rectangle cavity. This high-order coupling method allows to use in terms of wall radiation a resolution of 20 points on the vertical wall without affecting the accuracy of results. The use of spectral methods allowed us to check the amplitude of spectral coefficients and ensure that the spatial resolution used is sufficient. In order to calculate unstable solutions and establish bifurcation diagrams, a steady-state solver is also used [START_REF] Xin | Méthodes d'ordré elevé pour le couplage de la convection naturelle avec le rayonnement de surface en cavité remplie d'air[END_REF].

results

To evidence the influence of wall radiation, simulations are carried out with and without wall radiation in a cavity of aspect ratio AR=1 or 2. Without radiation, the presented results are valid for any cavity height. In presence of wall radiation, the cavity has a height of 3.14cm, a vertical wall emissivity V = 0.10 and a horizontal wall emissivity H = 0.85 and corresponds to the configuration simulated by Gad & Balaji [START_REF] Gad | Effect of surface radiation on RBC in cavities heated from below[END_REF]. In the following, the results are presented in a dimensionless way using the cavity height H as a reference length, ∆T as a reference temperature and U ref = α • Ra 1/2 /H as a reference speed.

Flow regimes

Bifurcation diagrams established using the x-velocity near the top left cavity corner are presented in Fig. 2 for both cavities. At small Rayleigh numbers without radiation, the well-known stable case is obtained: the fluid is at rest and a stratification of temperature is observed. Wall radiation induces a weak flow near the vertical walls with a 2 × 2 cell pattern. These solutions at low Rayleigh are denoted S 0 . For larger Rayleigh number (up to Ra = 16000 for the square cavity and 8000 for the rectangle cavity), many flow patterns are found like mono-cellular up to 4-cell flows. To make easier the following discussion, flow solutions are named S ± X where X is the number of flow-cells and ± indicates the flow direction at point D (i.e. close to the top left corner).

When Rayleigh number increases gradually in a square cavity, the flow solution S 0 undergoes a perfect supercritical pitchfork bifurcation towards stable mono-cellular flows denoted S ± 1 . Solution S + 1 is ascending along the left wall whereas S - 1 is descending. Wall radiation delays the onset of monocellular flows: a critical Rayleigh number of 2585 is observed without radiation while a value of 2908 is obtained with wall radiation. In the rectangle cavity without radiation, S 0 bifurcates with a perfect pitchfork towards bi-cellular flows at Ra c = 2013. With wall radiation, the pitchfork bifurcation becomes imperfect and occurs at a slightly higher value of Ra c = 2104. This means that wall radiation keeps the flow symmetry of S 0 (the 2×2 cell pattern observed at lower Rayleigh number) and makes the solution S 0 to bifurcate preferentially towards the ascending solution near the vertical walls (S + 2 ) rather than the solution branch S - 2 . Figure 2 also shows unstable flows obtained by the steady-state solver. In the square cavity without radiation, the unstable solution S 0 bifurcates around Ra = 6743 towards unstable bicellular solution S ± 2 with a perfect pitchfork bifurcation. Wall radiation makes again imperfect this pitchfork bifurcation at Ra = 7142. Solutions S ± 2 then become stable after a perfect subcritical pitchfork bifurcation at Ra = 11290 without radiation. In the case with wall radiation S 2 solutions bifurcate respectively at Ra = 11817 and 11844 for the S + 2 and S - 2 branches. This difference of the lower and upper branches comes from the fact that S + 2 and S - 2 are no longer equivalent after the imperfect bifurcation.

The aspect ratio modifies flow solutions. A greater number of flow patterns is found in the rectangle cavity. Without wall radiation the first bifurcation of S 0 branch is a perfect supercritical pitchfork type and the bifurcated solutions S ± 1 possess two cells. While with wall radiation the first bifurcation of S 0 branch is an imperfect pitchfork type. In both cases, unstable S 0 solutions undergo other bifurcations at higher Rayleigh numbers. At Ra = 16000, flows with one, two or three vertical cells are found in the cavity (S ± 1 , S ± 2 , S ++ 3 and S -- 3 ). It is seen that wall radiation keeps almost the same bifurcation diagrams as those observed in the case without radiation and that in the case of wall radiation the bifurcations take place at slightly higher critical Rayleigh numbers. Nevertheless, it should be noted that all the pitchfork bifurcations are perfect and supercritical without radiation and that the pitchfork bifurcations linking S 0 -S ± 2 , S ± 0 -S ± 4 , etc. are made imperfect by wall radiation.

Heat transfer

The heat flux crossing the cavity walls is characterised by radiative and convective Nusselt numbers averaged over the bottom and top walls:

Nu C = q c k ∆T H with q c = 1 L L 0 q c dx (4a) Nu R = q r k ∆T H with q r = 1 L L 0 q r dx at bottom wall 1 L L 0 (-q r ) dx at top wall (4b)
The reference heat transfer used for both convection and radiation is pure a conductive flux. Fig. 3 plots the bifurcation diagrams using the convective Nusselt numbers for both cavities without radiation. Convective Nusselt numbers are the same on the top and bottom walls. The heat transfer rate of one bifurcated branch is the same as the other one. Fig. 4 shows averaged convective Nusselt numbers in the presence of wall radiation on both the top and bottom walls. It depicts that the bifurcated branches related to any imperfect pitchfork bifurcation (branches S ± 2 and S ± 4 for example) are no longer equivalent and that the averaged convective Nusselt numbers are not the same on the same wall. On the branches bifurcated from a perfect pitchfork bifurcation (branches S ± 1 and S ± 3 for example) convective Nusselt numbers are the same on the same wall. The convective Nusselt numbers on the bottom wall of a square cavity presented in Fig. 4a are in good agreement with the results of [START_REF] Gad | Effect of surface radiation on RBC in cavities heated from below[END_REF]. To reach this agreement, the cavity height was set to 3.14cm (i.e. N r 7.765). The radiation to conduction parameter was not explicitly provided by [START_REF] Gad | Effect of surface radiation on RBC in cavities heated from below[END_REF], but it is was deduced from other parameters in the article. Also, it is guessed that their Nusselts are given at the bottom wall because the agreement would be less satisfactory at the top wall.

Radiative Nusselt numbers are represented at the bottom wall in Fig. 5. As stated before, radiative Nusselt numbers on the bifurcated branches related to an imperfect pitchfork birfurcation are different on the same wall. This is the case for S ± 2 and S ± 4 , while on those related to a perfect pitchfork birfurcation (S ± 1 and S ±+ 3 ) the same radiative Nusselt is observed. Values of convective and radiative Nusselt numbers of stable flows at Ra = 2000 and 16000 for the square cavity with wall radiation are given in Table 1. Stable solutions are given at Ra = 2000 or 8000 for the rectangle cavity. The table highlights that convective or radiative Nusselt numbers

Nu bottom C + Nu bottom R = Nu top C + Nu top R (5)
The sum of convective and radiative Nusselt numbers are null on the left and right walls since they are imposed by the adiabatic condition. Note also that radiative Nusselt numbers are not equivalent for S + 2 and S - 2 whereas those of S + 1 and S ++ 3 are equivalent to those of S - 1 and S -- 3 .

Nu left R Nu right C Nu right R Nu top C Nu top R Nu bottom C Nu bottom R AR = 1
S 0 -3.9•10 -4 3.9•10 -4 -3.9•10 -4 3.9•10 -4

Conclusions

Simulations have been conducted in a square and a rectangle cavity to investigate the effect of wall radiation on Rayleigh-Bénard convection and heat transfer. Both the cases with wall radiation and without radiation have been investigated and the bifurcation diagrams were established. In both cases many cellular flow branches are encountered at moderate Rayleigh numbers (Ra 16000 for the square cavity and Ra 8000 for the rectangle cavity). Weak cellular flows are observed even at very low Rayleigh numbers with wall radiation.

It is observed that the bifurcation diagrams are only slightly modified by wall radiation: the corresponding critical Rayleigh numbers increase slightly. Wall radiation makes that convective (and radiative) Nusselt numbers are different on the top and bottom walls. In terms of bifurcation nature, wall radiation introduces a new phenomenon as it makes certain pitchfork bifurcations imperfect. In these cases, the bifurcated branches are no longer equivalent in terms of flow and heat transfer: convective (and radiative) Nusselt numbers are no longer the same on the same cavity wall.

The present work leads to the following recommendations to describe Bénard convection with wall radiation and more generally natural convection with wall radiation:

-a radiation to conduction parameter (or the cavity height) should always be given since wall radiation depends on the cavity length; -convective and radiative Nusselt numbers should always be associated to a wall because they are different on the top and bottom walls; -they should also be associated to a solution branch as the bifurcated branches can be not equivalent.

Figure 1 :

 1 Figure 1: Geometry of the simulated cavity. Horizontal walls are isothermal and vertical walls are adiabatic. Points A, B, C and D are located at ±0.1H from the corners.

FigFigure 2 :

 2 Fig. 2a Square cavity, no radiation

Fig

  Fig. 3a Square cavity Fig. 3b Rectangle cavity

Fig

  Fig. 4a Square cavity: bottom wall

Fig

  Fig. 5a Square cavity

  not the same on the bottom wall nor on the top wall in presence of wall radiation. For comparison, convective Nusselt numbers are equal to 1, 2.46 and 2.0346 for respectively S 0 , S ± solutions in the rectangle cavity without radiation. These values indicate that wall radiation decreases convective Nusselt numbers except for S 0 and S - 2 solution. The total heat transfer is of course enhanced by wall radiation. It can be checked that the total heat transfer is the same at the top and bottom walls:

	2.2 2.4		stable unstable Gad & Balaji 2010						2.4 2.2		stable unstable
	2										2
	1.8										1.8
	1.6										1.6
	1.4										1.4
											1.2
		0	2000	4000	6000	8000	10000	12000	14000	16000	1	0	1000	2000	3000	4000	5000	6000	7000	8000
			Fig. 4c Square cavity: top wall					Fig. 4d Rectangle cavity: top wall
	2.2 2.4		stable unstable								2.2 2.4		stable unstable
	2										2
	1.8										1.8
	1.6										1.6
	1.4										1.4
	1.2										1.2
	1	0	2000	4000	6000	8000	10000	12000	14000	16000	1	0	1000	2000	3000	4000	5000	6000	7000	8000
			Figure 4: Convective Nusselt numbers on the cavity walls with wall radiation.
	1 2 in the square cavity without radiation. They are equal to 1, 1.8699, 2.2467 and 2.0803 for and S ± S 0 , S ± 1 , S ± 2 and S ±± 3

are

Table 1 :

 1 Convective and radiative Nusselt numbers of stable solutions with wall radiation (S 0 solutions are given for Ra = 2000 and other solutions S ± i are given at Ra = 16000 for the square cavity and 8000 for the rectangle cavity).

	4.36
	4.34		stable unstable							4.94		stable unstable
											4.92
	4.32
	4.3										4.9
	4.28										4.88
	4.26										4.86
	4.24
											4.84
	4.22
	4.2										4.82
	4.18	0	2000	4000	6000	8000	10000	12000	14000	16000	4.8	0	1000	2000	3000	4000	5000	6000	7000	8000
				Figure 5: Radiative Nusselt number on the cavity bottom wall.
	case		Nu left C

shihe.xin@insa-lyon.fr


Sarah Mathieu-Soucy

Claudia Corriveau²

Nadia Hardy

Exploration of new post-secondary mathematics teachers' experiences: preliminary results of a narrative inquiry

Keywords: Teachers' and students' practices at university level, Preparation and training of university mathematics teachers, Narrative Inquiry, Becoming a teacher

This paper reports on a pilot study that has taken place during the winter semester of 2017, in the context of a larger project whose goal is to contribute to studying the transitions from being a university mathematics student to becoming a postsecondary mathematics teacher. With very scarce literature on these transitions at this specific level, this pilot study acts as an exploration into new teachers' significant experiences that may be involved in shaping their relationships with mathematics, and its teaching and learning. We conducted a narrative inquiry with three new post-secondary mathematics teachers who were interviewed on a regular basis during a semester. Those interviews provided an insight into the new teachers' experiences by pointing out themes that are relevant to them. We conclude with a discussion on what remains to be achieved to conduct this research.

INTRODUCTION

This paper presents a pilot study that took place during the winter semester of 2017. It addresses the transition from being a university mathematics student to becoming/being a post-secondary mathematics teacher. More precisely, the context is set in cegep institutions (general and vocational colleges), the first step in postsecondary education in the province of Quebec, Canada. The focus of this study is on new cegep mathematics teachers who, with an education mainlysometimes exclusivelyin mathematics, negotiate the transition from being a mathematics student (undergraduate or graduate) to teaching at post-secondary level. We start with a brief description of cegep institutions, followed by a discussion on the literature on becoming a teacher at post-secondary level. The theoretical framework and the object of study are then described followed by the methodology and the main results. We conclude with the next steps in the study and our expectations for the results.

A Few Words on the Context of Study

Cegeps are general and vocational colleges. Two-year pre-university programs are offered in sciences, arts or social sciences; three-year technical programs, such as nursing, computer science, building engineering technology, etc., are also offered. All programs include compulsory general courses, such as philosophy, French, English and physical education and multiple programs offer mathematics courses. Some technical programs include mathematics courses that are specific to the field of study. Those courses are offered mostly by mathematics instructors, even if they are not very familiar with the profession their students are aspiring to do. Pre-university programs in sciences and social sciences offer two calculus courses (differential and integral calculus) and one linear algebra course, slightly different from one program to the other, depending on the institution. Social science students are required to take a quantitative methods course. Some institutions also offer optional mathematics courses for science students, such as multivariable calculus, probability and statistics or discrete mathematics. Therefore, from advanced calculus for science students to quantitative methods for social science students, as well as mathematics applied to computer science, cegep mathematics teachers are required to teach a wide array of courses, to a wide range of programs and students.

To teach in a cegep institution, individuals are (officially) required a 3-year undergraduate degree in mathematics or connected field (Conseil Supérieur de l'Éducation, CSÉ, 2000). However, each institution can add other requirements, such as a master or a doctorate in mathematics. A handful of universities in the province offer a one-year graduate certificate in cegep teaching, unique for all disciplines taught in cegep. Institutions will sometimes see it as an asset when hiring although cegep instructors who have completed one of these certificates qualify them as far from the reality of cegep teaching (CSÉ, 2000).

This context is quite different than that of elementary and secondary institutions. To teach at these institutions individuals are officially required a 4-year degree in education, which sometimes include minimal formal mathematics training. Also, cegep institutions are often a place of transition from secondary to university education; students are introduced to more formal mathematics and some level of autonomy is expected from them (schedule less structured, no mandatory attendance to classes).

Teaching at Post-Secondary Level

Little is known about becoming a mathematics teacher at post-secondary level (e.g. [START_REF] Speer | How do mathematicians learn to teach? Implications from research on teachers and teaching for graduate student professional development[END_REF]. More and more research about post-secondary mathematics education is being conducted, arguing this level is key to the training not only of future teachers but also scientists, engineers and mathematicians [START_REF] Hodgson | The mathematical education of schoolteachers: Role and responsibilities of university mathematicians[END_REF]. However, many recognize the need for change in the way the mathematics are taught at this level. Most research emphasizes the mistakes and negative traits of university teaching and claim the inadequacy of the existing training programs on creating change and improving post-secondary teaching, while only few seem to offer concrete solutions (e.g. [START_REF] Beisiegel | Being (Almost) a Mathematician: Teacher Identity Formation in Post-Secondary Mathematics[END_REF][START_REF] Belnap | Putting TAs into context: Understanding the graduate mathematics teaching assistant[END_REF][START_REF] Speer | Connecting teaching beliefs and teaching practices: A study of teaching assistants in reform-oriented calculus courses[END_REF][START_REF] Speer | Mathematics teaching assistant preparation and development[END_REF]. About this, Beisiegel's work (2009) focused on how graduate students "developed a sense of themselves […] as post-secondary teachers of mathematics" (p. 2) during a teaching assistantship. The author looks closely into the graduate students' lives and their journey into becoming teachers. She concludes that existing training is inadequate and puts forward the need to investigate closely the experiences of these students becoming new teachers in order to effectively study post-secondary teaching and training (2009, p. 25). This recommendation is the starting point for our work with new cegep teachers who were not so long-ago university mathematics students. Our goal is to better understand their experiences in their life as new mathematics teachers and in the transition from being a mathematics student to becoming a post-secondary teacher.

The challenge -and partially the trigger for our pilot study -is that the literature on post-secondary teaching in general, and in cegep in particular, is very scarce. Looking at research on elementary and high school teachers, we see that it is heavily based on the education they received, namely a degree that aims at training them as teachers (e.g. [START_REF] Ambrose | Integrating change in prospective elementary school teachers' orientations to mathematics teaching by building on beliefs[END_REF][START_REF] Ensor | From preservice mathematics teacher education to beginning teaching: A study in recontextualizing[END_REF][START_REF] Franke | Understanding teachers' self-sustaining, change in the context of professional development[END_REF]. However, post-secondary mathematics teachers are mainly, if not solely, trained to become mathematicians. Therefore, we cannot transpose or extend the results encountered in the literature about teachers' training for elementary and secondary levels to the group we are interested in. With this lack of background on the transition under study, we chose to conduct a pilot study to inform our main research project.

THEORETICAL FRAMEWORK AND OBJECT OF STUDY

The focus on experience of this study led us to consider Dewey's philosophy (1938) to frame it. For Dewey, one learns through experience and experiences shape how one goes about the world and about new experiences (1938, p. 35). Dewey also sees being faced with challenging experiences as a key aspect of life, growth and change: "growth depends upon the presence of difficulty to be overcome by the exercise of intelligence" (1938, p. 79). Indeed, faced with a familiar context, one can have an idea of how to act, and the consequences of those actions, based on knowledge acquired through past experiences. However, more reflection is needed in an unfamiliar context, where one could have to connect many different, apparently unrelated, experiences in order to know how to act and the related consequences (1938, p. 68). This results in new knowledge that could be applied to one's action per respect to future experiences.

Because of the importance of new and challenging experiences in life, growth and change, we chose them to be the focus of our study. In the context of our research, becoming a mathematics teacher at post-secondary level is seen as a trigger for new and possibly challenging experiences. And because a situation may be new and challenging for some but not for others depending on one's past experiences, it is important to put the focus on the experiences lived by the new teachers that are challenging and educational for them, and not all experiences that are lived or that might be assumed challenging and educational by others. We call those significant experiences.

However, this approach opens the door to a broad and varied spectrum of experiences that may be significant in the transition under study. In our project, we focus on significant experiences that may shape new teachers' relationships with mathematics and its teaching and learning. We will use the abbreviation RWMTL for the remaining of this paper to refer to the relationships under study. These RWMTL entail the visions, opinions, beliefs, attitudes and feelings about mathematics, and its teaching and learning.

In this sense, we hypothesize that new teachers possess a RWMTL, developed through years of doing, learning, being taught, and sometimes teaching mathematics, and that these RWMTL play an important role in their becoming post-secondary mathematics teachers. This hypothesis is supported by the work of [START_REF] Speer | Connecting teaching beliefs and teaching practices: A study of teaching assistants in reform-oriented calculus courses[END_REF] and [START_REF] Beisiegel | Being (Almost) a Mathematician: Teacher Identity Formation in Post-Secondary Mathematics[END_REF]. Speer's work emphasizes that graduate mathematics students have beliefs about mathematics and mathematics education, and that their RWMTL play a role on how new teachers will act in their new profession, how they will deal with, interpret and react to what happens to them. In particular, she claims that teaching assistants' beliefs, especially about mathematics and undergraduate students, have a significant impact on how they teach and interact with students. These RWMTL are influenced by the education they received in university and by the implicit teacher training they received during their time as students (Beisiegel, 2009, p. 42). It creates a relationship, conscious or not, with education through their experiences as students. Indeed, [START_REF] Beisiegel | Being (Almost) a Mathematician: Teacher Identity Formation in Post-Secondary Mathematics[END_REF] claims that graduate mathematics students' life experiences in a mathematics department, shape their "views of the discipline and teaching" (p. 43), as well as how they view what and how they should become. She argues that the experiences those students have can play an important role into how they define their role as mathematics teachers: "it appears that in the lives of mathematics graduate students there exists a complex and intricate interplay between the structures that they encounter, their feelings about mathematics and themselves and their ideas of their future role as mathematics instructors or professors" (p. 43). This last quote reinforces our hypothesis that the RWMTL play an important role in individuals becoming postsecondary mathematics teachers. In general terms, the goal of our main study is to investigate these RWMTL; in particular, our goal is to identify the nature of the significant experiences that shaped new teachers' RWMTL. Incidentally, the goals of the pilot study this paper is reporting on were to come up with a list of themes relevant to, or in relation with, the significant experiences of the new mathematics cegep teachers, in order to support and guide the main study later on.

METHODOLOGY

Our methodology is based on ideas brought forward by narrative inquiry ("NI", [START_REF] Clandinin | Narrative Inquiry: Experience and story in qualitative research[END_REF]. NI argues that the essence of human experience happens narratively and choosing this method means acknowledging that people make sense and give meaning to their lives narrativelythey lead storied lives (Clandinin, 2013, p. 13). Furthermore, NI takes into account the wholeness of someone's life while allowing the researcher and the subject to collaboratively investigate and distinguish what makes it unique and specific.

NI was developed specifically to understand and inquire into experiences, in relation to the people who have them and the physical and social context where they are [START_REF] Clandinin | Engaging in Narrative Inquiry[END_REF]. This is to emphasize how past experiences and social and individual matters influence how someone lives an experience. In our case and for the research project, we use NI to determine what significant experiences shape new teachers' RWMTL. In the context of the pilot study we are reporting on, we use NI to shed a first light into these significant experiences; collaboratively working with the subjects in thematising them, their nature and their shaping role.

With this in mind, we built open and semi-structured interviews with broad questions, so teachers would account for what is actually relevant or important in their journey of becoming teachers. This gives a chance for the unexpected to arise.

The Process of Investigation

Weekly meetings were planned with three cegep teachers during a whole semester (January to May 2017), but we ended up meeting with each of them 15, 10 and 4 times respectively over the whole semester. The meetings, which lasted between 30 minutes and 90 minutes, were audio-recorded. The first teacher was starting his third year as a teacher, had completed a master in mathematics and a one-year certificate in pedagogy. The second participant was starting his second year of teaching, had completed a master in mathematics and had also completed a one-year certificate in pedagogy. The third participant was teaching his second course while pursuing their second year of Ph.D. in mathematics. They were asked to share stories of events they lived in the week prior to the meeting, which they identified as significant for their RWMTL, and of reflections they made about these events. After each meeting, an account was written in the form of a story of the meeting, using as much as possible the words of the teacher, with only slight changes for clarity purposes. Those accounts constitute the data that was analyzed.

RESULTS

Emerging themes

The pilot study served as a way for us to learn about significant experiences of new teachers in relation with mathematics, teaching and learning. Our goal was to circumscribe areas, contexts or topics, we call them themes to be concise, which seemed to play an important part into their lives, thoughts and reflections. We wanted to find themes that seemed to foster experiences that were significant for them. This list is not meant as "a list of understandings" but rather as a list of words to help us think to understand the stories (Clandinin, 2013, p. 39). Of course, those ideas will not be final as we do want to stay open to what will come up as we meet new people next year, for the main study, and learn about their experiences.

First, as we considered the teachers' relationship with teaching, two categories arose: "being a teacher" (1) and "teaching to students" (2). The former (1) addresses the very personal aspects of the journey into becoming a teacher. It includes themes such as the new teachers' expectations of the profession of teaching mathematics in a cegep institution. This unfolds in reflections in relation to whether or not they attended cegep as students, the lifestyle they expect to have while holding such a position and the expectations arising from their memories of their own teachers. Another theme that came up is about being part of an institution and being part of a community of teaching, who respectively involves reflection on the role of cegeps in the society, and integrating in a team of established teachers. Finally, the ability to teach, which includes their reflections in relation to their own and others' abilities, and the initial training they should have or want to have, were topics that played a role in their everyday life and reflections.

The latter subdivision (2) of the themes in relation with teaching that we found addresses themselves as teachers in relation with their students. This includes the role they (should, can) have in their classroom, whether it is to pass on knowledge, make it interesting or getting them prepared for a job or university. They also reflected on the assessment of the level of difficulty of concepts or problems as very central in their daily life, whether it comes from their own judgment or from a formal source such as ministerial specifications. In the same line of thought, reflections on the assessments were playing a huge role in their experiences with questions such as how many, when, weight of each, and also how to prepare the students for them and how to assess exactly what needs to be assessed. Finally, their teaching method, using technology for example, and the ability to adjust to the students, to their ways of thinking and being, were present in the reflections expressed.

As we considered the teachers' relationship with learning, the following themes emerged. First, the expectations of the level of the students, as far as what they should know from high school or from previous cegep courses, and the expectations in relation to the students fulfilling their role in the classroom, such as participating in class and doing their homework, were central topics for the teachers. On another note, the teachers found challenging to manage the students' expectations in regard of their teacher, as far as the act of teaching and the level of the material they were expecting. Finally, the teachers mentioned choosing the attitude towards the students as being a challenge, especially when it comes to differences between countries and provinces, since many teachers are not from Quebec, where the relationship between a teacher and its students might be different according to different cultural norms.

As far as the teacher's relationship with mathematics, we did not come up with a list of themes as we found that our data did not allow us to do so. Indeed, as we met the participants over and over, we realized that it was not spontaneous for the new teachers to talk about mathematics when asked about their daily lives. They favoured talking about teaching and their students. We conjecture that they have a number of things to think about other than the mathematics, such as the themes mentioned above, which seem to take much more work to be mastered or managed than to master the mathematics they are teaching. In other words, it made us realize that for (these) new teachers, it is difficult to talk about mathematics spontaneously when questioned about their experiences as new teachers; considerations in relation to teaching and learning seem to take all their time. Among the three participants, only one would address mathematics when discussing their everyday activities; the individual still in the midst of completing their Ph.D. in mathematics. This made us think that the new teachers that were full-time, not actively conducting research in mathematics nor taking mathematics courses, seemed to have in the front of their mind matters that were farther from mathematics and closer to teaching and (their students') learning. The two other participants, when asked directly, struggled to talk about mathematics for more than a few moments.

Finally, we found multiple themes what were present across mathematics, teaching and learning. Indeed, it seemed that some institutional aspects specific to cegep were having an impact on the teachers experience at multiple levels. First, as mentioned in the introduction of this paper, cegep institutions offer science, social science and technical programs. The teachers expressed multiple times that there were challenges on multiple levels to teaching to different audiences. For example, in a differential calculus class for science, a teacher has to teach to future engineers and future doctors, two groups who do not have the same interests and goals. Also, this same teacher may teach in the same semester differential calculus for social science students (the course covers almost the same material), who have again very different interests, goals and aspiration, to study psychology or economics for example, in relation with mathematics and, for some of our participants, very different ways of thinking. Finally, that same teacher may have to teach mathematics to technical programs, where the goal is for the students to learn the mathematics they would need to apply in their field. Therefore, the teachers we met expressed how difficult it was to navigate teaching to those different audiences: how to teach them according to their expectations and aspirations, how they learn best, and to determine what kind of mathematics, and what in the mathematics, they really needed.

Another key aspect across mathematics, teaching and learning is the fact that cegeps offer courses during the day and courses at night as part of continuing education. Indeed, teaching in continuous education meant that the teacher was not assigned an office space, and was not active in the department (they were not necessarily invited to department meetings). This also meant that they were often left on their own to discover the ways of the institution, would rarely come across other instructors and seemed to lack opportunities for asking advice. It is also important to mention that new cegep teachers often have to teach at multiple institutions during the first few years, sometimes even during a semester, before they can be guaranteed work at one place. This isolation and constant change seemed to be heavy for teachers who enjoy working with others, reflect through discussions on teaching, get advice from more experienced colleagues or just be active in their workplace. The students are also different, as most day-students are young adults who go to school full-time and, night-students are older and work full-time during the day. And again, to try to adapt to all those aspects turned out to foster a number of significant experiences for our participants.

CONCLUSION: WHAT'S NEXT

This pilot study helped us understand and circumscribe themes that were at the heart of becoming a teacher in cegep institutions.

However, through the 29 interviews we conducted, very few experiences were complete enough to be talked about in terms of how they played a role in the new teachers' lives, mainly because the reflection in relation to an experience was often missing or incomplete. Indeed, from our data, we found that the reflective experience can take a long time, longer than the academic term in which the experience happened. This made us realize that our goal of understanding changes in the RWMTL could not be tracked over this short period of time. Indeed, our pilot left us believing that reflection on recent experiences is often not mature enough for teachers to be comfortable verbalizing it and sharing it with us. Also, we were unable to know if the recent experiences would play a significant role in the transition under study on a longer term. Therefore, to be able to hear about significant experiences, we need to go another route. Our conjecture is that by introducing a theme first and asking participants to share past experiences in relation to this theme, we could be able to grasp some of these significant experiences and the role they play in shaping new teachers' RWMTL. Indeed, by asking the participants to talk about experiences that have shaped their RWMTL in relation to a specific area, context or topic, we hope to hear about the experience and the reflection associated to it. As [START_REF] Dewey | Experience and Education[END_REF] would say, the quality of an experience comes mostly from what it opens to, i.e. the resulting tools and how they help understanding and acting towards new experiences (p. 27). This takes time, since we think about our past experiences differently with time, reflection and probably other experiences. We therefore want to understand how the new teachers evolved in this transition, how their way of living and acting towards new experiences evolved with time and new experiences. As Clandinin (2013) said, we, the participants and the researchers, "are always interpreting [our] pasts from [our] present vantage points" (p. 46).

The pilot interviews also revealed that the participants had a hard time focusing on experiences specifically related to mathematics and often tended to discuss experiences related to class management, institutional constraints, etc. This lead us to design new protocols with the hope of instilling the norm that the goal is to talk about experiences with mathematics, and its teaching and learning.

To conclude, the pilot study helped us in creating protocols for the main study that would guide our endeavour to understanding some of the experiences that shape new cegep teachers' RWMTL, and more precisely, what kinds of experiences are significant as they transition from being mathematics students to cegep teachers, all while staying open to the new lives we will meet as the main data collection unfolds.


Megan Wawro
email: mwawro@vt.edu

Michelle Zandieh²

Kevin Watson

Delineating Aspects of Understanding Eigentheory through Assessment Development

Keywords:

In this report, we share insights we have gained from developing an assessment for documenting students' understanding of eigentheory. We explain the literature and theory that influenced the assessment's development and share question examples. We frame our results in terms of three eigentheory settings ( , and eigenspaces) and four interpretations (numeric, algebraic, geometric, and verbal). Results from our analysis include students' reasoning being influenced by setting, insights into students' struggle with understanding eigenspaces, and the importance of making connections between and across various interpretations.

INTRODUCTION

Linear algebra is particularly useful to science, technology, engineering and mathematics (STEM) fields and has received increased attention by undergraduate mathematics education researchers in the past few decades [START_REF] Dorier | On the teaching of linear algebra[END_REF][START_REF] Artigue | Mathematics thinking and learning at post-secondary level[END_REF]Rasmussen & Wawro, 2017). A specifically useful group of concepts in linear algebra is eigentheory, or the study of eigenvectors, eigenvalues, eigenspaces, and other related concepts. Eigentheory is important for many applications in STEM, such as studying Markov chains and modeling quantum mechanical systems; however, research specifically focused on the teaching and learning of eigentheory is a fairly recent endeavor and is far from exhausted.

As part of our ongoing research program analyzing students' understanding of eigentheory (e.g., [START_REF] Watson | Knowledge about student understanding of eigentheory: Information gained from multiple choice extended assessment[END_REF][START_REF] Wawro | Meta-representational competence with linear algebra in quantum mechanics[END_REF], we created an assessment instrument focused on the multifaceted and interconnected nature of eigentheory. The purpose of this paper is to describe insights have we gained about students' conceptual understanding of eigentheory as a result of developing, using, and refining this assessment instrument.

THEORY AND LITERATURE REVIEW

We ground our work in the Emergent Perspective [START_REF] Cobb | Constructivist, emergent, and sociocultural perspectives in the context of developmental research[END_REF], which assumes that mathematical development is a process of active individual construction and mathematical enculturation. In this paper we focus on the former by analyzing mathematical conceptions that individual students bring to bear in their mathematical work (Rasmussen, Wawro, & Zandieh, 2015). The literature on the teaching and learning of eigentheory points to several aspects important to students' conceptual understanding. Here we summarize that literature by highlighting what we found to be important aspects for building a working model for understanding eigentheory. [START_REF] Thomas | Eigenvalues and eigenvectors: Embodied, symbolic and formal thinking[END_REF] found that students struggle to coordinate the two different mathematical processes (matrix multiplication versus scalar multiplication) captured in the equation to make sense of equality as "yielding the same result," an interpretation that is nontrivial or even novel to students [START_REF] Henderson | Symbol sense in linear algebra[END_REF]. Furthermore, students have to keep track of multiple mathematical entities (matrices, vectors, and scalars) when working on eigentheory problems, all of which can be symbolized similarly. For instance, the zero in refers to the zero vector, whereas the zero in is the number zero. This complexity of coordinating mathematical entities and their symbolization is something students have to grapple with when studying eigentheory. [START_REF] Thomas | Eigenvalues and eigenvectors: Embodied, symbolic and formal thinking[END_REF] also posit that this struggle to coordinate may prevent them from making the needed symbolic progression from to , which is central to determining the eigenvalues and eigenvectors of . In their genetic decomposition of eigentheory concepts, [START_REF] Salgado | Teaching eigenvalues and eigenvectors using models and APOS Theory[END_REF] posit that students need to interpret the procedure of finding eigenvectors and eigenvalues of as determining the solution set of the homogeneous system of equations created by the matrix equation . [START_REF] Harel | Three principles of learning and teaching mathematics[END_REF] posits that the interpretation of "solution" in this setting, the set of all vectors that make the equation true, entails a new level of complexity beyond solving equations such as

Literature on student understanding of eigentheory

, where , and are real numbers. When considering the notion of eigenspace in particular, [START_REF] Salgado | Teaching eigenvalues and eigenvectors using models and APOS Theory[END_REF] found that students struggle to coordinate the number of eigenvectors corresponding to a given eigenvalue with the dimension of the space spanned by the eigenvectors. Thus, understanding eigentheory not only involves coordinating mathematical processes and entities but also equations and solution sets.

In addition, students have to make sense of instructors' frequent movement between geometric, algebraic, and abstract modes of description, but this may be challenging [START_REF] Hillel | Modes of description and the problem of representation in linear algebra[END_REF]. In fact, [START_REF] Thomas | Eigenvalues and eigenvectors: Embodied, symbolic and formal thinking[END_REF] found that students in their study primarily thought of eigenvectors and eigenvalues symbolically and were confident in matrix-oriented algebraic procedures, but the majority had no geometric or embodied views. In contrast, other researchers have shown how exploration through dynamic geometry software [START_REF] Çağlayan | Making sense of eigenvalue-eigenvector relationships: Math majors' linear algebra-geometry connections in a dynamic environment[END_REF][START_REF] Gol Tabaghi | Using dynamic geometry software to explore eigenvectors: The emergence of dynamic-synthetic-geometric thinking[END_REF][START_REF] Nyman | Those do what? Connecting eigenvectors and eigenvalues to the rest of linear algebra: Using visual enhancements to help students connect eigenvectors to the rest of linear algebra[END_REF], geometric interpretations of a linear transformation [START_REF] Zandieh | An example of inquiry in linear algebra: The roles of symbolizing and brokering[END_REF], or real-world contexts [START_REF] Salgado | Teaching eigenvalues and eigenvectors using models and APOS Theory[END_REF] can help students develop conceptual understanding of eigentheory. We similarly agree on the importance of understanding eigentheory concepts in multiple ways and navigating between various interpretations, and we incorporate this complexity in our model of student understanding of eigentheory.

Working Model of Understanding Eigentheory

Regarding what it may mean to have a conceptual understanding of eigentheory, our current working model is a network of connections within and across three main settings of how eigentheory is framed. The three sets of relationships that are pertinent are: (1) relationships indicated by the eigen-equation ;

(2) relationships indicated by the homogeneous form of the eigen-equation ; and

(3) relationships indicated by a linear combination of eigenvectors. Within the first two settings, what is most frequently the focus of inquiry is one particular eigenvector for either form of the eigen-equation. However, when considering the collection of all that satisfy either eigen-equation, one arrives at the eigenspace of associated with . The relationships between vectors in the same eigenspace are the focus of the third setting. For instance, if and are eigenvectors of with eigenvalue , then all vectors that are a linear combination of and (i.e., for scalars and ) are also eigenvectors of associated with . Furthermore, reasoning about relationships in this third setting almost certainly involves reasoning about either the first or second setting as well. Each of these settings includes entities and relationships between those entities that may be realized in various ways. We organize this variability in our model according to four main interpretations: graphical, numeric, symbolic, and verbal.

THE EIGENTHEORY MCE ASSESSMENT

The development of the Multiple Choice Extended (MCE) assessment instrument for eigentheory grows from our prior work in student understanding of span and linear independence in which we developed the MCE-style question format [START_REF] Zandieh | Extending multiple choice format to document student thinking[END_REF]; questions begin with a multiple-choice element and then prompt students to justify their answer by selecting all statements that could support their choice (see Figure 1). This format was inspired by existing conceptually-oriented assessment instruments in undergraduate mathematics and physics (e.g., [START_REF] Carlson | The precalculus concept assessment: A tool for assessing students' reasoning abilities and understandings[END_REF]Hestenes, Wells, & Swackhamer, 1992;[START_REF] Wilcox | Multiple choice assessment for upper-division electricity and magnetism[END_REF].

Development of the Eigentheory MCE involved multiple steps. First, we compiled a database of questions about eigenvectors, eigenvalues, and related concepts from research on student understanding of eigenvectors and eigenvalues (e.g., Gol Tabaghi & Sinclair, 2013; [START_REF] Salgado | Teaching eigenvalues and eigenvectors using models and APOS Theory[END_REF][START_REF] Thomas | Eigenvalues and eigenvectors: Embodied, symbolic and formal thinking[END_REF], online resources for clicker and classroom voting on linear algebra [START_REF] Cline | MathQUEST/MathVote[END_REF], and previous linear algebra homework assignments, exams, and interview protocols used by research team members (e.g., [START_REF] Henderson | Symbol sense in linear algebra[END_REF]. Second, we used research results regarding students' understanding of eigentheory from the literature, as well as our own teaching experience and theoretical thinking, to determine which questions seemed to address important aspects of what it may mean to have a conceptual understanding of eigentheory. Third, the most promising questions were edited into the MCE format, which involved moulding the problem into a multiplechoice question and developing six corresponding justification choices that required students to reason within and between various eigentheory settings and interpretations. Fourth, through multiple rounds of administering the assessment to students, analysing the data, and subsequent refinement, we arrived at the current Eigentheory MCE. It contains six questions, each with six justification choices; five questions are in Figure 1 (the sixth is omitted because of space limitations). The MCE questions were created to elicit student thinking about eigentheory within and across the settings and interpretations within our working model of understanding eigentheory. For example, the stem of Question 1 is a numeric interpretation; its given justifications for students to choose as true and relevant, we see that justification (i) is a symbolic interpretation in the setting, (iii) is a symbolic interpretation in the setting, and (v) is a geometric interpretation in the setting (Figure 1). As students choose justifications that support their answer to the main question, they are prompted to reason about eigenvectors and eigenvalues within and across multiple settings and interpretations.

METHODS

The Eigentheory MCE was given to two introductory linear algebra classes taught by the same instructor at a large, research-intensive public university in the United States at the end of Spring Semester 2016. The course utilized the Inquiry-Oriented Linear Algebra (http://iola.math.vt.edu) curricular materials and Lay (2012) as its textbook. One class (of 29 students) received the MCE with given closed-ended justifications (see Figure 1), and the other class (of 28 students) received an open-ended version where students had to write their own justifications for their multiple-choice answer; we refer to these as Class C and Class O, respectively. Students had 20-25 minutes to work on the assessment. All student work referred to in this paper is labelled "B#."

Analysis of the closed-ended MCE consisted of entering the data into spreadsheets and looking for trends such as: (a) common sets of justifications that students selected or did not select; (b) how selecting certain justifications may have influenced students' multiple choice selection; and (c) instances in which we would have expected students to select what we viewed as related justifications, but they did not. We used Grounded Theory [START_REF] Glaser | The discovery of grounded theory: Strategies for qualitative research[END_REF] to characterize the concepts students brought to bear in their justifications in the open-ended MCE, coding independently and discussing our results as a team to find emerging themes. Finally, we compared students' responses across questions and across classes to discover further insight into student understanding of eigentheory.

RESULTS

We include four insights into student understanding of eigentheory discovered from our MCE data analysis. These selected results are organized by settings (sections 1-2) and interpretations (section 3) from our working model of understanding eigentheory.

Reasoning about relationships within or

We found that as students respond to an MCE question, they seem to situate it within a particular setting, perhaps the setting they are most familiar or comfortable with, regardless of the setting in which the question was initially written. Furthermore, a student's chosen setting can lead to different ways of reasoning about a problem. We present two illustrations of this from Class O: B65's justifications for Q1 and Q2, and B66's justifications for Q1 and Q3. First, in Figure 2, B65 seemed to situate both problems within the setting. On Q1, B65 explained that multiplying the vector by the matrix resulted in six times that vector. On Q2, B65 explained that needed to be a scalar multiple, and thus the only possible vector would be one along the same line as the vector , namely the vector . In both cases B65 emphasized that for an eigenvector, multiplying by the matrix yields a multiple of the original vector, thus working within the setting.

Second, in Figure 3, B66 seemed to situate both problems in the setting. On Q1, we infer this student first found the matrix , multiplied each vector from the MCE question by it, and chose the vector that was mapped to the zero vector. Then, on Q3, although what s/he actually writes is idiosyncratic, we can infer s/he was still reasoning with the homogeneous equation, imagining the vectors and being mapped to zero by the matrix , and thus the vector would also map to zero. In both Q1 and Q3, B66 emphasized an action on the eigenvectors to produce the zero vector, seemingly invoking the setting. We note that the stems of Q1, Q2, and Q3 are not written so as to elicit student reasoning within a particular setting. This allows for use of the open-ended assessment to measure a student's preferred setting or for the closed-ended assessment to measure whether students can interpret the problem in either setting

Reasoning about Eigenspaces

The previous section provides examples of the relationships involved in the first two settings: to . In this section we attend to the eigenspace setting, which focuses on the relationships involved with scalar multiples or linear combinations of eigenvectors. An eigenspace, like any vector space, is closed under addition and scalar multiplication; thus, a linear combination of vectors in an eigenspace is also an eigenvector with the same eigenvalue as the other vectors in that eigenspace. When asked about eigenspaces, students may draw on these facts and/or may work within one of the previous two settings to derive these principles.

For Q3, only six (of 28) students in Class O circled the correct answer (a) that is an eigenvector with eigenvalue 2, five chose (b) an eigenvalue of 5, and 16 chose (c) that was not an eigenvector. Approximately half the students in each group used the phrase " is a linear combination of" as part of their justification (3 for (a), 3 for (b), 7 for (c)). Sample justifications using "is a linear combination of" are given in Figure 4. Very few students in Class O gave justifications that brought in the relationship between eigenvectors and eigenvalues described in the first two settings. One such student was B66, described in the above section. In Class C, however, 13 (of 29) students chose justification (iv) (symbolic ), 11 of which correctly selected answer (a). Because a much higher percentage of Class C circled (a) than in Class O, it is possible that this justification served as a hint that helped some students choose the correct answer. On the other hand, this MCE option allowed us to test whether students recognized the relevance of this set of relationships for the given question.

Although some students who answered (c) used the phrasing "linear combination," their arguments focused more on the linear independence of the vectors. The answers students gave for (c) include: "Eigenvectors must be linearly independent from each other so if is a linear combination of and then it cannot be an eigenvector," [B58], and "Because they all correspond to the same eigenvalue they all must have unique eigenvectors and is a linear combination of and and therefore not unique and not an eigenvector of " [B79]. These answers focus on eigenvectors as necessarily being linearly independent or unique. This focus may come from students remembering that eigenvectors of distinct eigenvalues are linearly independent or that textbook solutions often give an eigenspace basis as the final answer, which may explain students thinking there are only finitely many eigenvectors for an eigenspace.

In Q5, eigenspaces were represented geometrically, and students who completed it were rather successful in selecting the correct multiple-choice answer (14/21 in Class C and 21/26 in Class O). However, many students still focused on finite numbers of vectors. On Q5, reasons given by some students to support the correct choice (b) similarly focused on finite numbers of eigenvectors: "Matrix already has 3 eigenvectors so there's no room for a 4 th " [B59], and " is a linear combination of and , and there are already 3 eigenvectors for 3 dimensions, so cannot be an eigenvector of " [B66]. We conjecture these students may have been conflating the total number of possible eigenvectors (infinite) for a 3x3 matrix with the number of linearly independent vectors needed to create the bases for the 1-and 2-dimensional eigenspaces. Alternatively, B58's justification for Q5 focuses on dimension: "In a 3x3 matrix there can only be 3 dimensions to the eigenspace. and together span the entire space of so there cannot be another eigenvector of besides and ." We conjecture grasping the difference between finiteness of dimensions and infiniteness of eigenvectors may be particularly vital for understanding eigenspaces. No, because is a linear combination of the two vectors. is a linear combination of and Both and are scalar multiples of their previous form so the resultant vector will be an eigenvector as well Since it is a linear combination of the other eigenvectors, it would also be an eigenvector. NOTE: Typed versions are used here to improve readability of students' handwritten justifications

Reasoning Across Interpretations

We conclude our results by discussing various aspects of students reasoning across interpretations and the ways in which the MCE afforded that. In particular, we focus on symbolic versus geometric interpretations of eigentheory. On Q1, as noted in the previous section, a majority of students in Class O wrote at least one equation (symbolic interpretation), but none wrote anything geometric in their justifications. This could be an indication that students might favour algebraic reasoning over geometric reasoning when justifying their answers to eigentheory questions, even though the classes used the IOLA curriculum which specifically introduces eigenvectors and eigenvalues geometrically. On the other hand, it could be that the numeric interpretation that Q1 was written in did not elicit geometric interpretations from students in their open-ended justifications, or that students see symbolic justifications as more acceptable to the teacher or the broader math community than geometric ones. In a more direct way of assessing students' ability to see connections to the geometric interpretation, the closed-ended MCE gives students the geometric justification choice (v) on Q1, and roughly half (14/29) of the students in Class C selected it. Furthermore, over 80% of the total students from both classes answered the multiple choice stem of Q1 and Q2 correctly (48/57 and 51/57 respectively), demonstrating some ability to reason both numerically and geometrically about eigenvectors and eigenvalues. Because the wording of the MCE questions and justifications makes use of the four different interpretations from our working model, we are better able to assess students' understanding of the symbolic, numeric, geometric, and verbal interpretations in eigentheory, both within and cross settings.

DISCUSSION

Research on student thinking often relies on students' written work on mathematics problems as evidence of how students make sense of or reason about particular content. Our research here is no exception, with student work on the MCE revealing a variety of ways that students understand aspects of eigentheory. However, the MCE's closed-ended justifications extend a written question's ability to examine connections between settings and interpretations that students might not have initially considered or felt the need to include in their justifications. For instance on Q1 in Class O, four students wrote some form of as part of their justification, ten wrote some form of , four wrote some form of both equations, and ten students did not write either equation. In contrast, on Q1 in Class C, 23 students selected both justifications (i) ( symbolic) and (iii) (symbolic ), and only one student selected neither. Hence, when students were forced to consider the two eigentheory settings (i) and (iii), the large majority was able to see both as true and relevant. As other researchers have pointed out the importance of understanding both equations in eigentheory, it is significant that the MCE may give new insight in students' understanding of connections between these two settings.

We do see some potential limitations of the MCE. First, it is more time consuming to take than a simple multiple-choice test, and this affects the number of questions that can be asked. The MCE can also be cognitively taxing because students must consider each justification to determine its truth and relevance. Third, scoring MCE results can be complicated. We hope that further refinement and use of the MCE, as well as developing possible scoring systems, will continue to broaden and deepen the mathematical community's understanding of how students reason about eigentheory.

Figure 1 :

 1 Figure 1: Questions 1-5 from the Eigentheory MCE

Figure

  Figure 2: B65's reasoning within the setting.

Figure 3 :

 3 Figure 3: B66's reasoning with the setting.

  v is a linear combination of y & z Since the value 2 already causes y & z to equal 0, adding a multiple to it will not change that

Figure 4 .

 4 Figure 4. Example of four students' open-ended justifications for Question 3


Reinhard Hochmuth
email: hochmuth@idmp.uni-hannover.de

Discussing Mathematical Learning and Mathematical Praxeologies from a Subject Scientific Perspective

Keywords: Curricular and institutional issues concerning the teaching of mathematics at university level, transition to and across university mathematics, subject scientific approach, mathematical praxeologies, real numbers

. The main common concern of ATD and the subject-scientific approach is to conceptualize and analyse "objects" like "institutionalized mathematical knowledge" and "university" not as conditions that cause reactions but essentially as meanings in the sense of generalized societal reified action possibilities. The link of both approaches is illustrated by the issue of "real numbers" in the transition from school to university: Hypotheses are derived for further actual-empirical research, which intrinsically incorporate content-and subject related perspectives as well as societal and school-related findings.

INTRODUCTION

This paper contributes to an ongoing major research project that describes and analyses form and content of advanced mathematics and its teaching and learning from a subject scientific point of view. This approach is grounded in "Critical Psychology", framed by [START_REF] Holzkamp | Grundlegung der Psychologie[END_REF] (see [START_REF] Tolman | Critical Psychology: An Overview[END_REF] and Schraube & Højholt (2015) for English written introductions). Recently this theory becomes internationally more known in the mathematics education community due to [START_REF] Roth | A cultural-historical perspective on mathematics teaching and learning[END_REF], who assessed "German Critical Psychology" as a further development of the culturehistorical activity approaches by [START_REF] Leontjev | Activitiy, consciousness and personality[END_REF] and [START_REF] Vygotsky | Mind in Society: The development of Higher Psychological Processes[END_REF]. It's beyond the scope of this paper to describe and analyse in which respects "German Critical Psychology" differs and goes beyond culture-historical activity theory. Instead the paper intends to point out its compatibility and (partial) complementarity with ATD as well as to illustrate its potential relevance for further research concerning university mathematics education.

Main features of "Critical Psychology" and its subject-scientific point of view are well elaborated psychological categories (roughly: basic notions, see for details (Holzkamp, 1985, pp. 28)) for describing and analysing cognitive and emotionalmotivational dimensions of subject [individual] related experiences, in particular thoughts, actions and learning, in a way that major societal aspects are inherently be incorporated. It aims (besides others) to provide individuals with analytic tools for their self-reflection of problematic experiences and situations to reveal their inherent dependencies and circumstances, thus allowing individuals to achieve a more reflective learning. Within this framework, there is so far a lack of research that relates to mathematical learning in general and in university in particular. "Critical Psychology" provides points of contact for incorporating research results concerning the societal and historical genesis of knowledge and reference structures as well as institutionally framed (e.g. school, university, study courses) external and internal transposition processes [START_REF] Chevallard | La transposition didactique. Du savoir savant au savoir enseigné[END_REF]. ATD's praxeological analyses could principally inform any psychological or sociological theory considering teaching and learning. Already in [START_REF] Castela | When Praxeologies Move from an Institution to Another One: The Transpositive Effects[END_REF] a link between ATD and cultural historical activity theory is discussed, in particular between Roth's concept of crossing boundaries between different socio-cultural contexts and the issue of inter-institutional transitions. This basic idea is in the following taken up in a broader sense.

In view of the ongoing major research project, this contribution has the status of an intermediate step presenting programmatic ideas about linking ATD with the subject scientific approach. This combination might in particular be fruitful for research connecting detailed analyses of mathematical practices with a complex vision of learners and teachers in a way that (with respect to both sides) their intrinsic societal mediatedness is systematically incorporated. Though this rather pretentious goal can easily be formulated (at least by using abstract notions, which would require a lot of pages to be embedded in a coherent theory and to be explained in detail [START_REF] Holzkamp | Grundlegung der Psychologie[END_REF][START_REF] Holzkamp | Lernen: Subjektwissenschaftliche Grundlegung[END_REF]), the following lines also indicate that there is still a way to go combining both approaches in actual-empirical research (Holzkamp, 1985, pp. 509).

The structure of the paper is organized as follows: In the first two sections we introduce some notions from ATD and the subject scientific approach. Then we discuss the link of both approaches. Finally we illustrate the link and some of its aspects and opportunities considering the issue of "real numbers" in the transition from school to university: After an ATD-orientated overview about the nowadays typical treatment of real numbers in German secondary schools and considering various options of extending this discourse in the transition from school to university , we discuss subject scientific related aspects taking into account societal and school-related findings and how they might contribute to validate mathematical and didactical practices. This theoretical analysis exemplarily demonstrates how the link intrinsically integrates content-and subject-related perspectives and leads to hypotheses for actual-empirical research projects.

SOME NOTIONS FROM ATD

ATD [START_REF] Chevallard | Fundamental Concepts in Didactics: Perspectives Provided by an Anthropological Approach[END_REF]Winslow, Barquero, Vleeschouwer & Hardy 2014) aims at a precise description of knowledge and its epistemic constitution. Its concepts make possible to explicate institutional specificities of knowledge and related practices. An underlying conviction of this approach is that cognitive-oriented accesses tend to misinterpret contextual or institutional aspects of practices as personal dispositions. A basic concept of ATD are praxeologies represented in so called "4T-models (T,τ,θ,Θ)" consisting of a practical and a theoretical or logos block. The practical block (know-how, "doing math") includes the type of task (T) and the relevant solving techniques (τ). The logos block (knowledge block, discourse necessary for interpreting and justifying the practical block, "spoken surround") covers the technology (θ) explaining and justifying the used technique and the theory (Θ) justifying the underlying technology. Praxeologies give descriptions of mathematics by reference models that are activity oriented (techniques, technologies). The interconnectedness of knowledge is modelled in ATD by means of local and regional mathematical organizations that allow contrasting and integrating practical and epistemological aspects in relation to different institutional contexts. Therefore ATD is in particular helpful for analysing institutional realizations of mathematical knowledge within different learning contexts, e.g. the use of mathematics in signal theory [START_REF] Hochmuth | Überlegungen zur Konzeptualisierung mathematischer Kompetenzen im fortgeschrittenen Ingenieurwissenschaftsstudium am Beispiel der Signaltheorie[END_REF].

More than 15 years ago Chevallard has introduced the additional notion of "scale of levels of codeterminations" that in the meanwhile has become rather important in ATD analyses [START_REF] Bosch | Twenty-Five Years of the Didactic Transposition[END_REF]. The hierarchical sequence of levels covers civilisation, society, school, pedagogy, discipline, domain, sector, theme and subject (in the sense of topic). Each level provides some kind of framework, within among others actions on lower levels are possible, supported or hindered and praxeologies are in a certain sense embedded. In Barbé et al. ( 2005) is shown, for example, how general didactic restrictions for teaching mathematical topics in school can affect teachers' practices and their established praxeologies, in particular the shaping of the practical and the logos block and the relations between them. To mention one further example, [START_REF] Job | Empirical positivism, an epistemological obstacle in the learning of calculus[END_REF] argue that smoothing the transition-gap from calculus to analysis shows at least the tendency to blur the distinction between the different discourses in school and university, which tends to reinforce an empirical positivist attitude by students as an epistemological obstacle to learning (ditto; p. 641). Generalizing their arguments, one might say that there are issues relating to general world views (society) that affect institutionally settled praxeologies. Moreover, [START_REF] Chevallard | La transposition didactique. Du savoir savant au savoir enseigné[END_REF] introduced the notion indicating the relation of a position x (a typical position of an individual) within an institution I to a praxeology o. The "scale of levels of codeterminations" underlines that the institution and, with the institution, the position x and the praxeology o have to be considered as being societal situated, i.e., that in their analyses specific emphasize also has to be put to societal assignments that are related to societal mediation processes. The subject scientific perspective, which is introduced and discussed in the next both sections, allows to further specify positions x keeping the significance of societal and institutional mediatedness (in a materialistic sense, see for example [START_REF] Arndt | Unmittelbarkeit[END_REF]).

SOME NOTIONS CONCERNING THE SUBJECT SCIENTIFIC APPROACH ("CRITICAL PSYCHOLOGY")

"Critical Psychology" claims to present a scientific discussable and criticisable elaboration of basic psychological concepts (categories). The starting point is a historical-empirical investigation of general historical-specific characteristics of relations between societal and individual reproduction as well as its dialectic mediatedness. One of the central subject related categories is "action potence", which is the potence to ensure the disposal about the subject's individual living conditions together with others (Holzkamp, 1985, pp. 239).

Within the context of this paper there are three important points to notice: First, the actual historical-specific form of subjectivity is characterized by the "possibility relation" regarding the societal reality, which includes in particular the basic experience of intentionality. Second and connected to the first, the specific modality of subjective experiences comprises the discourse form "reasoning discourse": "I" speak about my "own" actions in terms of subjective reasonable (not necessarily "rational") activities and of premises in the light of "my" life interests. A third crucial point is that the "human's relationship to the environment is almost always mediated. […] Categories of psychology like learning, emotion, motivation and cognition cannot fail to be significantly altered by the fact of our existence's social mediatedness. The most important mediation category is meaning." (Tolman, 1991, pp. 14-15) These three interrelated issues are combined in the assertion that conditions are given to "me" in terms of meanings in the sense of generalized societal action possibilities and that reality aspects, which are relevant for "me", denoting again the generalized subject standpoint, become premises for "me" in the light of "my" life interests. Therefore, subject scientific considerations are essentially given by meaningpremises-reasons-relations, which a priori situate experience and activities of the (individual) subject "within the world" Accordingly, Holzkamp (1985, pp. 342) figured out the level of subjective action reasoning as the main subject specific level: It represents the level with respect to which individual experiences and activities (e.g. learning) has to be reconstructed and analysed.

Via the specific notion of meaning, human activities, like teaching and learning, are intrinsically thought as societal mediated. This implies that any analysis of subjective actions requires the reconstruction of subjectively relevant conditions in the sense of generalized action possibilities and the consideration of their societal mediatedness. Since meanings appear (via objective-subjective premises) as the medium within which subjects' reasoning discourses are grounded, their study is a prerequisite for describing and analysing related cognitive, motivational and emotional processes as aspects of subjects' activities. But, although meanings in the indicated sense are rather relevant for acting and thinking, they do not determine them. Instead, they represent action possibilities that might become premises in the light of subjectively perceived "life interests".

THE LINK BETWEEN ATD AND THE SUBJECT SCIENTIFIC APPROACH

The position x within an institution can be (re-)interpreted as the "position" and/or "situation of life" from the subject point of view, which includes intentionality, the modus of reasoning discourse and societal mediatedness. As an "element" (a "position x") of an institution a subject is typically not confronted with the whole world but only with a local "situated" section represented by those meanings typically produced and reproduced within the institutional context. Hence, institutional contexts provide specific frameworks for premise-reasoning-patterns. In this sense ATD's praxeological analyses contribute to concretisations of meaningpremise-reasoning-patterns that are typical within an institution at the position x. In particular the concept of praxeologies allows capturing substantial aspects of mathematical practices in such a way, that they can be injected as facets of action related meanings, i.e., they can be (re-)interpreted as generalized societal action possibilities, which were potentially reflected in subject related reasoning schemes as premises and/or reasons. In this sense praxeological analyses can be seen as one nontrivial first step within subject scientific research projects: They might inform microanalyses of task solution processes by exploring institutional established practices. They are relevant for describing and analysing related activities, since they appear as institutionalized medium, within subjective action reasoning grounds. With respect to premises-reasoning-patterns the technological dimension of praxeologies, i.e. the justification and validation of techniques, is of specific importance. But, see above, praxeologies do not determine subjects' activities, since there is an unconscious-conscious step by subjects of selecting, neglecting or highlighting facets of praxeologies in view of their evaluation of "life interests" and how they are perceived by them at "position x" in the "institution I" in view of all prospects addressed by "the scale of level of codeterminations". Thus, the latter is rather relevant for both, the analysis of meanings (essentially by ATD) and the analyses of premises-reasons-relations (essentially by the subject scientific approach). In fact, both strands can't be seen as totally separated but as dialectically interrelated, since institutionalized practices live through subjects' [individual] activities.

THE ISSUE OF "REAL NUMBERS"

In this section we give first a short overview of the nowadays typical treatment of real numbers in German secondary schools in grade eight or nine. Because of the space limitation a detailed praxeological analysis can't be presented.

The Treatment of Real Numbers in German Schools

The treatment of real numbers in German schools presumes that rational numbers are known and can be represented by ratios, decimal fractions and points on the number line. Moreover it is presumed that students are able to switch between those representations. In particular basic calculations should be understood and can be executed with respect to the different representations. The typical starting point for the introduction of real numbers is the observation or proof (sometimes!) that there are quadratic equations like without rational solutions. Next it is observed (but typically not proven) that one can find approximations by proper decimal fractions that fulfil those equations up to an arbitrary chosen error. On the other hand it is (geometrically) argued that there is a magnitude x, the length of the square diagonal, satisfying that corresponds to a certain point on the number line. The intuitive conviction about the existence of those points on the number line supports the idea, that (somehow converging) infinite sequences of approximating proper decimal fractions give a (unique!) final finite result, a number, that can be represented by a non-terminating decimal fraction. This type of discourse justifying the existence of infinite-finite objects (i.e., infinite processes giving in a certain sense a finite result) has in particular been considered in Lakoff & Nūñes (2002) as basic for the whole analysis and denoted as "basic metaphor of infinity". The new objects of non-terminating and non-periodic decimal fractions are called irrational numbers and build, together with the already known rational numbers, the set of real numbers. Moreover, the calculation rules that are known for rational numbers are assumed to be also true for all real numbers. Whereas in former years one can at least find Descartes' geometrical arguments for explaining multiplication and division for general real numbers such arguments are nowadays missing. Corresponding to the sketched treatment of real numbers there are nearly no tasks that are related to structural aspects of real numbers or that enforce to reflect arguments of the discourse concerning limits or the existence of points or numbers respectively. Instead the tasks focus on various isolated techniques that are locally established, for example approximation techniques like interval bisections and the Heron algorithm or the use of calculation rules.

In terms of the 4T-model the established mathematical praxeologies can be characterized as essentially punctual (or at most local) with isolated types of tasks and corresponding isolated techniques, where the tasks can be solved without referring to superordinated technological aspects, i.e., there are praxeologies with technologies having in particular weak connections to for .The technological and theoretical discourse remains (so far it is represented at all) mostly implicit and essentially incomplete. These observations blend with those presented by González's et al. ( 2013) institutional analysis and with results from a qualitative study by [START_REF] Bauer | Mentale Repräsentationen von Irrationalzahleneine Analyse von Schülerinnenaufsätzen[END_REF]. For corresponding results considering prospective secondary mathematics teachers we refer to [START_REF] Sirotic | Irrational numbers: The gap between formal and intuitive knowledge[END_REF] and for an actual rather detailed study, which problematizes in particular the use of the number line and investigates the knowledge of fresh French university students see [START_REF] Durand-Guerrier | Conceptualization of the Continuum, an Educational Challenge for Undergraduate Students[END_REF].

Real Numbers in University Education and Potential Foci for Transitions

First of all it is interesting to notice that mostly the treatment of real numbers in university is either done axiomatically (courses for math majors) or (more or less) skipped (courses for engineers and natural scientists), but does generally not intend to connect or complete school praxeologies, i.e. for example: showing the one-to-one correspondence of number line and decimal fraction views [START_REF] Kirsch | Zur Behandlung der reellen Zahlen im Oberstufenunterricht[END_REF]; discussing geometrically the completeness of the number line [START_REF] Artmann | Der Zahlbegriff[END_REF]; showing how to add and multiply non-terminating and non-periodic decimal fractions.

For improving this situation one might think of transition measures that are adjusted to study courses for, e.g., math majors, prospective grammar school teachers or engineering students. A general scheme suitable for describing and analysing desirable transitions is given as follows [START_REF] Hochmuth | A General Scheme for a Heterogeneous Manifold of Transitions[END_REF]:

where o represents a praxeology and s a student within the institution S (school), ω a praxeology and  a student within the institution U (university) in relation to one or several blocks of the praxeology and is the power set symbol. The scheme works as a heuristic tool and allows to express that techniques, technologies or theories of o might be differently relevant for the relation of S within U to a (perhaps new) praxeology (see [START_REF] Biehler | Relating different mathematical praxeologies as a challenge for designing mathematical content for bridging courses[END_REF] for a slightly more restricted scheme).

Applying the scheme we illustrate next various transition foci: At first the focus might be on techniques and technologies of o such that related skills are improved, but tasks, techniques and technologies are only slightly extended, for example: ordering of square roots and decimal fractions; applying calculation rules for simplifying terms. This might be important for all above mentioned study courses. Secondly the focus might be on technologies and their further development (possibly also their theoretical embedding), for example: knowing, that square roots like or can only be approximated by finite decimal fractions; justifying real exponentials and powers [START_REF] Winsløw | Klein's double discontinuity revisited: contemporary challenges for universities preparing teachers to teach calculus[END_REF]. This might in particular be important for prospective grammar school teachers. Thirdly the focus might be on techniques and the replacement of technologies by technologies that are strongly and systematically embedded in real analysis, for example: constructing the set of real numbers by Dedekind cuts or Cauchy sequences; starting with axioms for ℝ and identifying natural and rational numbers within this new set of objects. This might in particular be important for math majors. Within the ATD-framework each case could be analysed, explored and specified in greater details and in view of the scale of level of codeterminations. We skip further details in this paper.

Links to the Subject Scientific Approach

From the subject scientific point of view the analysed praxeologies represent meanings in the sense of institutionalized action possibilities. Students' reasoning and activities ground in those praxeologies but select, neglect or highlight them in view of an evaluation of their "life interests" and how they are perceived in view of the institutions S and U, corresponding to positions s and , and in particular aspects related to levels of codetermination. It is well-known that in the transition a lot of issues play a role, see e.g. [START_REF] Gueudet | Investigating the Secondary-Tertiary Transition[END_REF]. In the following we will discuss two different but complementing issues that are specifically linked to "Critical Psychology".

The first issue refers to the level "society" and in particular to Dowling's sociological analysis of myths, exemplarily the myth of "reference" [START_REF] Dowling | The sociology of mathematics education: Mathematical myth/pedagogical texts[END_REF]. Within our frameworks Dowling's myths can be (re-)interpreted as technological aspects of mathematical praxeologies in school that are related to the societal significance of educational processes. With respect to the latter Dowling differentiates between assignments concerning general societal aspects and those concerning their historic specificity. Considering the historic specificity the exchange-value aspect comes into play, which somehow undermines the use-value aspect and establishes a problematic mixture of both. This fits to the observation that actual school introductions of real numbers typically refer to "real world" problems like doubling the area of a quadratic piece of chocolate. Such references dominate the justification of introducing real numbers although treating the "real world" problems does not require "exact" solutions and, moreover, algebraic extensions would be sufficient to resolve this issue. Following Dowling, the myth of "reference" is not only a didactical issue that relates to illustrative introductions, but possibly leads to problematic technological and theoretical ideas, which do not disappear by establishing new, for example, axiomatic praxeologies, instead they possibly survive and constitute a strong epistemological (and motivational) obstacle (similar to the epistemological obstacle considered in [START_REF] Job | Empirical positivism, an epistemological obstacle in the learning of calculus[END_REF]) for students' learning of university mathematics, in particular for future grammar school teachers. This could in particular happen, since Dowling's myths might dominate students' general view of their "situation of life" and therefore their accentuating of meanings.

The second issue relates to the organization of learning in school and to the "schooland-exam system" mentioned for example in [START_REF] Chevallard | Teaching mathematics in tomorrow's society: A case for an oncoming counter paradigm[END_REF]. Partly because of this issue [START_REF] Holzkamp | Lernen: Subjektwissenschaftliche Grundlegung[END_REF] introduced the notion of defensive learning, a learning which primarily intends to prevent negative consequences. An important aspect of this notion is the opposition between ostensive and conceptual thinking that represents, according to [START_REF] Holzkamp | Grundlegung der Psychologie[END_REF], the historic-specific societal concretization of the cognitive aspect of human activities: Ostensive thinking is essentially characterized by taking things as they appear to be and, in terms of the 4T-model, by strongly focusing on locally situated technical and technological issues, which blend with the above described praxeological organisation of "real numbers" in school. Again, corresponding "ostensive" students' views on their "situation of life" and related meanings-premises-reasoning-patterns might let transition measures' intentions fail and in particular the incorporation of technological school-blocks within technological university-blocks, which results in new isolated praxelogies with new but still weak technologies. It is an empirical open question how this tendency is amplified by actual initiatives aiming to reduce transition problems by establishing a "university-and-exam system".

FINAL REMARK

The hypotheses derived in the last section illustrate the necessity that an analysis of measures supporting students in the transition from school to university have systematically to take into account both approaches, the praxeological and the subject scientific as well as, with respect to both, the scale of level of codeterminations. The established link between ATD and the subject scientific approach facilitates theoretical and actual-empirical studies factoring in systematically aspects, which are intrinsically connected to the institutional and societal level and have impact both on institutionalized praxelogies and subjects' meaning-premises-reasoning-patterns.