Juan Pablo Lovecchio

Sébastien Rohais

Philippe Joseph

Néstor Bolatti

Pedro Kress

Ricardo Gerster

Victor Ramos

Multistage rifting evolution of the Colorado basin (offshore Argentina): Evidence for extensional settings prior to the South Atlantic opening

published or not. The documents may come

Introduction

Understanding the evolution of volcanic passive margins, especially the crustal architecture and related rifting mechanics, has been greatly enhanced thanks to high resolution and deep seismic data (Franke et al., 2006(Franke et al., , 2007;;Blaich et al., 2013;Stica et al., 2014) as well as to major advances in numerical modeling (Beniest et al., 2017;Geoffroy et al., 2015).

However, most of the proposed models were built based on 2D cross sections, without considering the basement heterogeneities on either conjugate margin in a 3D perspective. Will and Frimmel (2018) analyzed the factors conditioning the emplacement of the South Atlantic Ocean focusing on intrinsic basement heterogeneities associated with the Neoproterozoic Brasiliano-Panafrican orogeny, disregarding other heterogeneities introduced by previous rifting events.

The opening of the South Atlantic in the Early Cretaceous was only the final stage of a complex rifting process (Uliana et al., 1989) that initiated after the Mid-Permian to Early-Triassic Gondwanan orogeny (Hälbich, 1983;Hansma et al., 2016). The Central Atlantic was the locus of the Triassic rifting. Intracontinental rifts also developed in Africa and South America at this time (Zerfass et al., 2004). The Karoo rifting on Eastern Africa since the Early Jurassic (Delvaux, 2001), is associated with breakup of Africa from Antarctica and India and the opening of the Weddell Sea (Ghidella et al., 2002). In the Early Cretaceous E-W directed extension is responsible for the South Atlantic opening. Frizon De Lamotte et al. (2015) review rifting processes in the Pangea breakup and suggest that, oppositely to the active Karoo rifting, the Early Cretaceous rifting (responsible for the opening the South Atlantic) was passive rifting, related to far-field forces controlling the initiation of rifting diachronically from south to north (Nürnberg and Müller, 1991).

On the Argentinean South Atlantic margin, prerift is composed of Late Paleozoic glacial and marine successions that were involved in a Permian-Early Triassic Ventania-Cape fold belt (Pángaro and Ramos, 2012;Pángaro et al., 2016), which has been traced across the South Atlantic to the Namibian margin (Paton et al., 2016).

The Colorado and the Salado basins (Fig 4.1) are troughs oriented largely E-W, at high angle with the NNE-oriented continent-ocean boundary (COB). Magma-rich rifted margins as the southern South Atlantic are characterized by the presence of SDRs (Seaward Dipping Reflectors;Hinz, 1981;Gladczenko et al., 1997;Hinz et al., 1999;Stica et al., 2014;and Paton et al., 2017) on the continent-ocean transition (COT). We define the COT as the area with SDR development on thinned transitional crust (sensu Blaich et al., 2013). The external boundary of the COT, passing to the normal oceanic crust domain is the continent-ocean boundary (COB in Fig. 4.1).

The chronology of rifting in the Colorado and Salado basins is poorly constrained.

Hydrocarbon exploration wells were drilled on structural highs outside the main depocenters and the few wells that investigated the synrift section mostly found red beds, where datable fossils are scarce. Several hypotheses were proposed to explain the high obliquity between these basins and the NNE-oriented COB. At first, studies implied the development of both basins synchronously leading to the opening of the South Atlantic in a single and continuous rifting event. Obliquity was explained either by aulacogens (Yrigoyen, 1975, Introcaso andRamos, 1984) controlled by Paleozoic or previous sutures (Ramos, 1996), transtensional basins associated with NW-SE dextral shear systems (Keeley and Light, 1993;Tankard et al., 2009;Franke et al., 2006), or as due to structural inheritance (Urien et al., 1995;Dominguez et al., 2011;Gebhard, 2005).

More recently, two superimposed rifting stages were used to explain the obliquity: a first Jurassic event with a NE-SW oriented extension, forming the Colorado and Salado basins, and a second E-W oriented extensional event forming the South Atlantic rift in the Early Cretaceous (Macdonald et al., 2003;Gerster et al., 2011;Pángaro and Ramos, 2012;Autin et al., 2013;Franke, 2013).

In this paper, after having carried out seismic interpretation and identified cross-cutting relationships between different generations of faults, we present evidence for not only two but three independent extensional stages. An older, previously unknown event was identified in the study area. This event is evidenced by the extensional reactivation of previously compressive thrusts of the Ventania-Cape fold belt. Extensional reactivation of thrusts has been described in other locations in Gondwana (De Wit and Ransome, 1992;Aldiss and Edwards, 1999;Franzese and Spalletti, 2001;Paton, 2006).

Materials and Methods

The database made available by YPF SA comprised 25 exploration wells (Fig. 4.1) and more than 30,000 km of 2D multichannel reflection PSTM seismic lines in the Colorado and Salado basins. Petrel software was used for interpretation. Time structure maps were generated for selected horizons (Fig. 4.2) to delineate the main depocenters. Wells were tied to seismic lines using synthetic logs or VSP data when available, producing new velocity laws that were integrated into a 3D regional velocity model used to convert the time structure maps and 2D

TWT seismic sections into depth. Faults were interpreted on seismic lines and a structural model was prepared. Faults were then grouped into sets according to their orientation, dip, displaced reflectors, detachment level and cross-cutting relationships.

Results

A

4.2.

In the central area, north of the CCol depocenter, a set of NNW-striking listric faults forming tilted blocks and half-grabens, with an intra-crustal detachment level (approx. 17

Discussion

Based on the data analyzed, and by integrating the relative chronology of faulting within the regional tectonic framework, we propose a model for basin evolution with three rifting stages (Fig. 4.7).

The final stage, Early Cretaceous rifting, is the best constrained in age as it is directly associated with SDR emplacement and breakup. Neither the SDRs nor the oceanic crust have been drilled in this part of the South Atlantic. Plate kinematic models are useful to integrate data on both margins and are calibrated with oceanic crust magnetic anomalies (Moulin et al., 2010;Heine et al., 2013). SDRs have been drilled however, on the conjugate Namibian margin (Kudu well) and sedimentary packages interbedded with the volcanics were dated Barremian (McMillan, 1990). Faults grouped in FS3 are subparallel to the SDR wedges bounding reflectors and the COB and are restricted to the outer 100-200 km wide fringe of the continental crust domain (Fig. 4.7c). We interpret FS3 as genetically linked to the Early Cretaceous rifting stage, and their development in a narrow fringe as due to a concentration of strain around the incipient ridge. FS4 faults also developed as part of the Early Cretaceous rifting event, but associated with margin segmentation during SDR emplacement (see Franke, 2013).

However, the main rifting event for the area of study is older and is associated with FS2 faults forming the ECol and CCol depocenters in Colorado, and the adjacent Salado basins (Fig.

4.7b).

From the presence of a thick postrift unit preserved in ECol (between TSR2 and BU, Fig. 4.2), we infer that thermal subsidence had already started before the Early Cretaceous breakup, confirming at least two independent rifting settings. We interpret Early-to-Mid Jurassic, largely oriented N-S extension for this event and correlate it with the Karoo rifting.

Figure 4 . 1 :

 41 Figure 4.1: Structural map of the Colorado and Salado basins (offshore Argentina) and the Punta del Este basin (offshore Uruguay). A: Map showing the location of the study area. B: Total sedimentary thickness on the continental crust domain (CCD). Colorado basin depocenters: WCol (western), CCol (central), ECol (eastern). Faults have been gathered into four fault sets (FS) according to their relative chronology. SDR wedges (mapped after Franke et al., 2007; and Soto et al., 2011). COT: continent-ocean transition, COB: cont.-ocean boundary, OCD: Oceanic crust domain, FZ: fracture zone, EH: External High. The black dots show the location of the exploration wells. Yellow lines indicate the location of seismic transects presented in this contribution.

Figure 4

 4 Figure 4.2: Schematic chronostratigraphic chart depicting chronology of rifting events and the main horizons used in seismic interpretation. TSR1: Top Synrift 1, TSA1: Top Sag 1, TSR2: Top Synrift 2, TSA2: Top Sag 2, BU: Breakup Unconformity, TC: Top Cretaceous. Arrows indicate the main direction of extension for each rifting stage (respect to present day South America). Late Triassic-Early Jurassic rifting (FS1): SW-NE extension; Early-Mid Jurassic rifting (FS2): N-S extension; Early Cretaceous rifging (FS3) and SDR emplacement: WNW-ESE extension.

  structural map of the Colorado and Salado basins is presented in figure4.1. The deepest Eastern Colorado (ECol) depocenter is a half-graben controlled by a NW-striking, NEdipping master fault(Figs. 4.3, 4.5 and 4.6). In a structural section across ECol (Fig.4.3) two sets of faults were identified. Tilted blocks on the northern flank of ECol are associated with a first set of faults (FS1), dipping to the S-SW and detaching at a level within the crust (at depths of approximately 10-15 km. Paleozoic strata are preserved in the tilted blocks but the synrift deposits associated with the rotation of these blocks are mostly eroded(Fig. 4.4, Pángaro and Ramos, 2012).

Figure

  Figure 4.3: Depth-converted seismic line across the ECol depocenter (location in Fig. 4.1b): a: uninterpreted and; b: interpreted

Figure 4 . 4 :

 44 Figure 4.4: Depth converted seismic line on the northern flank of the ECol depocenter. A: uninterpreted and, B: interpreted section. Tilted blocks associated with FS1 faults are shown in detail. Note that most of the synrift associated with this rifting has not been preserved. The location of the Pejerrey (Pe) well is shown.

Figure 4 . 5 :

 45 Figure 4.5: Depth converted seismic transect across the ECol depocenter. A: uninterpreted and, B: interpreted transect. Note preserved half-grabens of rifting stage 1 on the southern flank of ECol and transection of FS1 and FS2 faults.

Figure

  Figure 4.6: Depth-converted seismic line across the Colorado basin (location in Fig. 4.1b). A: uninterpreted and, B: interpreted

  -20 km deep, Fig. 4.6) are intersected by the E-W-oriented, S-dipping, master fault forming the Central Colorado half-graben (CCol, see Fig. 4.1 for location, CCol spans south of the seismic line presented on Fig. 4.6). Similarly to the he ECol-master fault, the CCol-master fault presents a steep geometry and detachment at a deep lithospheric level (~25 km). Thick synrift seismic sequences (fault-bounded wedges with divergent reflectors towards the active fault) were interpreted for the Central and Eastern depocenters. Data quality in WCol did not allow synrift interpretation. A postrift package (reflectors passively onlapping the basement highs around the depocenters beyond the synrift-bounding faults) was also interpreted. In passive margin-related basins, the synrift-to-postrift unconformity typically marks the breakup event (Franke, 2013). The breakup unconformity (BU in Fig. 4.2) marks the onset of oceanic crust accretion and is interpreted in volcanic margins at the top of SDR wedges (Fig. 4.6). In the Colorado basin, however, a postrift package was identified between the top of the synrift (TSR2) and the breakup unconformity, tracked from the COT domain (Fig. 4.6).The NE-NNE striking faults of FS3 are restricted to a strip covering the easternmost 100-200 km of the continental crust domain. They form half-grabens on the northern flank of the Tandilia high and are responsible for structuring the external high (EH), a block formed of continental crust that separates the ECol depocenter from the COT-SDR domain(Franke et al., 2006; Gerster et al., 2011; Fig. 4.1). For depocenters formed by FS3 faults, the BU is indeed the synrift-postrift unconformity (sensu Franke, 2013). Therefore we interpret this extensive event as being part of the South Atlantic opening and largely contemporaneous with the emplacement of SDRs further east (sensuStica et al., 2014). FS3 faults may present extensional reactivation across the BU indicating extensional activity during and slightly after the breakup.

  Cross cutting of FS1 faults by FS2 faults identified in the ECol and CCol depocenters point to the presence of a previously unknown rifting stage. We associate the NW-striking, S-SW dipping listric faults with the extensional reactivation of thrusts of the Permian to Early Triassic Ventania-Cape fold belt. Seven of these former thrusts were identified in the study area(Fig. 4.7a).Figure 4.8 is a seismic transect across the former Ventania-Cape fold belt in the Colorado basin area, formed by a combination of the seismic lines presented in figures 4.6 and 4.3. The detachment for FS1 faults follows the detachment of the former thrusts and shallows towards the NE, away from the Gondwanides orogen ( Mosquera et al., 2011). Note the transection of the extensionally reactivated thrusts (FS1) with the master fault forming ECol (FS2) in the central area. A Late Triassic-Early Jurassic age for this initial rifting stage is interpreted by association with the Precuyano rifting event in the Neuquén basin (D'Elia et al., 2015) that has also been related to reactivation of Late Paleozoic structures (Mosquera and Ramos, 2006). FS1 faults do not seem to be present in the Salado basin area as this region was north of the area of influence or the Permian-Early Triassic Ventania-Cape fold belt.

Figure 4 . 7 :

 47 Figure 4.7: Tectonic evolution of the Colorado basin: a) Late Triassic-Early Jurassic rifting stage associated with FS1 faults, extensionally reactivated thrusts of the Late Permian-Early Triassic Ventania-Cape fold belt. Main reactivated thrusts are drawn in black; b) Main rifting stage (Early-Mid Jurassic) responsible for FS2 faults and the formation of the Colorado and Salado basins' main depocenters; c) Final rifting stage (Early Cretaceous) related to the South Atlantic breakup and emplacement of SDRs. FS3 faults are largely oriented NNE, parallel to the SDRs and COB; d) Location map for Fig. 4.7c; e) Location map for Fig. 4.7a and b.

Figure

  Figure 4.8: Depth-converted seismic transect across the Colorado basin. See location in Fig. 4.5b, formed by a combination of

The understanding of the complex rifting history of the southern South Atlantic segment is crucial to better model the conjugate margins. Most 2D models consider the Colorado main rifting even (Early-Mid Jurassic) as synchronous with SDR emplacement and the South Atlantic opening. We showed that it is not the case. Crustal thinning in the Colorado basin is oriented E-W, as a consequence of the Early-Mid Jurassic rifting related to N-S extension.

Models attempting to explain this 3D complexity with a single event on 2D sections oversimplify the Gondwana breakup history along the Argentinean margin.

Conclusions

The identification of three rifting stages in the Colorado basin is key to understanding the complexity of long term continental breakup processes that affected the Paleozoic


domain: math shs

text:
Andreas J Stylianides

Gabriel J Stylianides
email: gabriel.stylianides@education.ox.ac.uk

Promoting prospective elementary teachers' knowledge about the role of assumptions in mathematical activity

Keywords: Assumptions, elementary grades, prospective teachers, task design, teacher knowledge

Although the notion of assumptions is important in the discipline of mathematics and permeates (often tacitly) mathematical activity in school classrooms, instruction in the elementary school pays little attention to it. This situation is unlikely to change unless elementary teachers have a good understanding of the role of assumptions in mathematical activity and an appreciation of the pedagogical implications of that role. In this paper, we investigate how teacher education can promote this dual goal by illustrating a promising, task-based approach to supporting prospective elementary teachers to develop pedagogically functional mathematical knowledge about the role of assumptions in mathematical activity. We developed the approach in a 4-year design experiment we conducted in a mathematics course for prospective elementary teachers in the United States.

Introduction

Assumptions denote the statements doers of mathematics use or accept (often implicitly) and on which their claims are based [START_REF] Fawcett | The nature of proof[END_REF] and, thus, assumptions are fundamental to any mathematical activity at all levels. In school mathematics, however, assumptions often receive little explicit attention, especially at the elementary school level. The notion of assumptions has received also relatively little explicit attention in the mathematics education literature.

A small number of studies addressed the role of assumptions in the particular area of proving at the secondary school [START_REF] Fawcett | The nature of proof[END_REF][START_REF] Jahnke | Understanding what a proof is: a classroom-based approach[END_REF] and elementary school [START_REF] Stylianides | Proving in the elementary mathematics classroom[END_REF] levels. However, there is a scarcity of relevant research at the teacher education level. This is problematic: unless teachers are supported to develop a good understanding of the role of assumptions in mathematical activity and an appreciation of its pedagogical implications, it is unlikely that teachers will offer to their students productive learning opportunities in the area of assumptions.

In this paper, we investigate how teacher education can promote prospective elementary teachers' knowledge about the role of assumptions in mathematical activity. We focus on the design and implementation of tasks that can support Mathematical Knowledge for Teaching (MKfT).

Research background

The notion of MKfT [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF] denotes the kind of "pedagogically functional mathematical knowledge" (Ball & Bass, 2000, p. 95) that teachers need to have to be able to manage the mathematical demands of their practice. It has been noted that "there is a specificity to the mathematics that teachers need to know and know how to use" (Adler & Davis, 2006, p. 271) and that teacher education should aim to create learning opportunities for prospective teachers (PTs) that would enable them "not only to know, but to learn to use what they know in the varied contexts of Thematic Working Group 22 Proceedings of CERME10

[their] practice (Ball & Bass, 2000, p. 95). Thus mathematics courses for PTs cannot lose sight of the domain of application of the targeted knowledge (i.e., the domain of mathematics teaching).

We consider next what might be essential elements of MKfT about the role of assumptions in mathematical activity. We begin by describing two elements we identified based on consideration of the role of assumptions in the discipline of mathematics (e.g., [START_REF] Fawcett | The nature of proof[END_REF][START_REF] Kitcher | The nature of mathematical knowledge[END_REF] and mathematical analysis of classroom practice where the notion of assumptions received explicit instructional treatment [START_REF] Fawcett | The nature of proof[END_REF][START_REF] Jahnke | Understanding what a proof is: a classroom-based approach[END_REF][START_REF] Stylianides | Proving in the elementary mathematics classroom[END_REF].

 Element 1: Understanding that a conclusion is dependent on the assumptions on which the argument that led to it was based.  Element 2: Understanding that different legitimate assumptions can lead to different conclusions that, although on the surface may appear to be contradictory to each other, may nevertheless both be valid within the set of their underpinning assumptions.

These interrelated elements are central to mathematical work. The different sets of axioms in Euclidean and non-Euclidean geometries, including associated results, offer an illustration of both of these elements in the discipline of mathematics or in the upper secondary school [START_REF] Fawcett | The nature of proof[END_REF]. Similar ideas are central also to the elementary school [START_REF] Stylianides | Proving in the elementary mathematics classroom[END_REF] and are consistent with recommendations in curriculum frameworks (e.g., NGA & CCSSO, 2010, p. 6). Furthermore, the two elements are essential for mathematical knowledge that is also pedagogically functional. For example, they can allow a teacher to recognize that, when two students offer inconsistent answers to a task, this does not necessarily mean that one of them is wrong; both students may have applied sound reasoning based on different assumptions about the conditions of the task. We capture this kind of pedagogical functionality of elements 1 and 2 under a third element of MKfT:

 Element 3: Ability to recognize possible ways in which elements 1 and 2 might apply in mathematics teaching.

Regarding how to promote PTs' MKfT, in [START_REF] Stylianides | Viewing "mathematics for teaching" as a form of applied mathematics: Implications for the mathematical preparation of teachers[END_REF] we discussed and illustrated a special kind of mathematics tasks that we called pedagogy-related mathematics tasks. These tasks have two major features: (1) A mathematical focus, which relates to mathematical ideas that are important for teachers to know; and (2) A substantial pedagogical context, which is an integral part of the task and essential for its solution. Notwithstanding the importance of pedagogyrelated mathematics tasks, there is also a need for the use of other kinds of tasks in mathematics courses for teachers. In this paper, we focus on a task sequence that illustrates another approach we followed to promote MKfT. Unlike pedagogy-related mathematics tasks where mathematics and pedagogy are intertwined in the same task, this task sequence illustrates an approach to promoting MKfT in which (1) initial work on a non-contextualized mathematics task can create a productive space for pedagogical reflection and (2) follow-up instruction can foster the intertwinement between mathematics and pedagogy. We elaborate on the task sequence in the Method section.

To conclude, in this paper we address the following research question: What task sequence can offer a productive learning environment for prospective elementary teachers to develop the three elements of MKfT that we described earlier about the role of assumptions in mathematical activity?

Method

Research context

This research derived from the last cycle of a 4-year design experiment (e.g., [START_REF] Cobb | Design experiments in educational research[END_REF] in a semester-long mathematics course (3hrs per week) for prospective elementary teachers in the United States. The design experiment comprised five research cycles of implementation, analysis, and refinement of task sequences and associated implementation plans that aimed to promote PTs' MKfT. The students were undergraduates who majored in different fields and were taking the course as a prerequisite for admission to the master's level elementary teacher education program.

The task sequence we focus on was implemented toward the end of the semester and was the only one that explicitly targeted PTs' MKfT about the role of assumptions in mathematical activity. We introduced the task sequence in cycle 4 because we felt an explicit intervention was needed to adequately promote PTs' MKfT of elements 1-3. Analysis of how the task sequence played out in cycle 4, alongside our developing understanding of how things "worked," led to modifications of the task sequence, culminating in the form described below which took place in cycle 5. The task sequence

The task sequence comprised three parts. In part A we used the "Floors Problem" [START_REF] Ball | With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics[END_REF] in Figure 1. Two task features made it suitable for our goals to promote elements 1 and 2 of MKfT. First, the task conditions were ambiguous and thus subject to different legitimate assumptions; a major ambiguity concerned whether or not the person in the task had to travel directly to the second floor. Second, different assumptions about the task conditions could support different arguments for answers to the task that might appear to be contradictory but might actually be correct given their underpinning assumptions. In addition to promoting elements 1 and 2 of MKfT, we hypothesized that PTs' mathematical experience with the task in part A could offer a productive context for pedagogical reflection (cf. element 3). We capitalized on this potential of the task in parts B and C.

In part B we used two conceptual awareness pillars (or simply pillars; [START_REF] Stylianides | Facilitating the transition from empirical arguments to proof[END_REF] to which the PTs responded individually and in writing. (There was a third pillar whose discussion we omit due to space limitations and a lack of direct relevance to our focus.) The pillars are presented in the first column of Table 1 (see next section). With them we aimed to amplify PTs' mathematical learning (see pillar 1 in relation to elements 1 and 2 of MKfT) and its intertwinement with pedagogical reflection (see pillar 2 in relation to element 3). As per the definition of pillars [START_REF] Stylianides | Facilitating the transition from empirical arguments to proof[END_REF], the non-directive prompts in them could increase PTs' awareness of key realizations they might have developed (possibly in tacit form) during their work on the task.

Finally, in part C we engaged PTs in small-group and whole-class discussion around the three prompts in Figure 2 that aimed to further support PTs' reflective insights from part B and help them consider the applicability of those insights beyond the particular task. The prompts raised issues relating to elements 1-3 of MKfT. For example, in prompt 1 the first question related to elements 1 and 2, while the second question related to element 3. Each small group was asked to collectively produce a written response to each prompt prior to a whole-class discussion. Due to space constraints we report only findings from our analysis of the small groups' written responses.

1. "Conclusions are 'true' only within the limits of the assumptions on which they are based." How do you understand this statement? Do you think it is important for elementary school students to develop a sense of the role of assumptions in mathematics? Explain.

2. "Teachers should always make sure that the mathematical tasks they give to their students have unambiguous conditions." What do you think about this statement? Explain.

3. There may be situations where teachers do not realize that a mathematical task they give to their students has ambiguous conditions. What might happen in these situations and how might teachers handle the situations?

Data and analysis

The task sequence in research cycle 5 was implemented in two parallel (independent) classes of the course that were both taught by the second author. At this stage of our analysis we are using data from only one of the classes: videos and transcripts of the implementation of the task sequence in this class with 16 PTs attending on that day; these PTs' written work including their individual responses to part B and their group responses to part C; and field notes from a research assistant documenting the work of one small group. Our data analysis was guided by elements 1-3 of MKfT that we aimed to promote. Specifically, we conducted qualitative content analysis of the transcripts and field notes in part A to examine whether and how the PTs developed understandings relating to elements 1 and 2, and also of PTs' responses to parts B and C to identify themes in their responses and examine whether and how these themes corresponded to elements 1-3 of MKfT.

Implementation of the task sequence and discussion

Part A: implementation of the Floors Problem

Stylianides showed the problem statement and the building model, and explained the notation for negative numbers. He also established a common notation with the class about representing trips.

After that, he asked the PTs to work on the problem first individually and then in groups of four. He also said he would not answer any clarifying questions about the problem, an intentional aspect of our instructional design. The discussion in the small group where the research assistant was taking field notes is indicative of the way the PTs engaged with the problem (all names are pseudonyms):

Amanda At this point Stylianides passed by the small group and Monica asked him whether the trips had to start from the ground floor. Stylianides reminded her that he would rather not respond to clarifying questions and he moved on to a different group. The small group discussion continued as follows:

Victor: So… the solution of this problem depends on our assumption. As illustrated by this exchange, the decision not to respond to clarifying questions allowed for the notion of assumptions to emerge naturally in the discussion: had the instructor specified an interpretation, the PTs would not have considered alternative interpretations. The whole class discussion started with the small groups explaining their answers to the problem: 25, infinity, 15, and 51. Sherrill explained the answer of 25 by noting how each floor could constitute a separate starting point for a direct trip to the second floor. Sophie then tried to explain infinity as an answer:

Sophie Infinity as an answer was discussed again later, after the class had considered also 15 and 51 as possible answers. Overall, the whole-class discussion mirrored the previous small-group discussion. Sophie, Lindsey, and Sherrill's comments show an increased understanding of elements 1 and 2: the PTs acknowledged that the task allowed for different legitimate interpretations of its conditions and that their conclusions depended on their assumptions (element 1), while realizing that the different answers they came up with were all correct based on their underpinning assumptions (element 2).

Part B: PTs' responses to the pillars

Analysis of PTs' responses to the two pillars gave rise to the themes summarized in Table 1. Each response could receive multiple codes. The frequencies are offered to show how prominent each theme was among PTs' responses, not for generalizability. Regarding pillar 1, PTs' responses under themes 1-4 related to both elements 1 and 2 of MKfT. The responses below illustrate all themes:

Sherrill: I was surprised at the multiple assumptions made. I had not even considered the fact that answers other than 25 existed because of the way people interpreted the problem. I now realize how greatly making assumptions can alter mathematics. (themes 2, 4)

Lorri: [A]ll of our groups had different interpretations of how many ways there were […] Some of the other groups' answers […] I could understand their rationales as to why their answer was correct. Our answers were not wrong under this ambiguous problem. (themes 1, 3)

Pillars

Themes and frequencies (in parentheses) a. To encourage sensitivity to language (7) b. To increase awareness of the role of assumptions (6) c. To enhance appreciation of different possible interpretations of the same (ambiguous) text (6) d. To enhance appreciation of the interdependency between different assumptions/interpretations and different answers and solution paths (5) e. To encourage discussion, explanation, or proof (5) f. Other (3) Regarding pillar 2, PTs' responses under themes a-e showed ability to recognize how their mathematical insights from the problem could apply in teaching. Thus the responses offered evidence related to element 3 of MKfT. We offer two responses that illustrate some of the themes: Lorri: A teacher could use this ambiguous problem to get their students to understand that sometimes their answers are not wrong and that there may be other answers to a problem. Students will be able to explore their different ways of interpretation and must be open to others' interpretations […] They will understand that there will be different […] answers that will depend on the information given in the problem, which may be ambiguous. (themes a, c, d) Helen: To learn about assumptions and understand not only the importance of specifics […] but also the freedom for one to think along the terms of their own assumptions. (themes a, b, c)

Part C: responses of small groups to the discussion prompts

Overall the PTs' responses to part C were pedagogically sound and were informed by the powerful impact the Floors Problem had on their mathematical learning. Also, one can see parallels between PTs' responses in part C and the part B pillars; these were supported by our instructional design.

Prompt 1: An illustrative explanation of the first question in prompt 1 was: "You have to start with a base and then make assumptions from that. So your conclusion can only be based on those assumptions. Therefore your conclusion can be only true or false for those assumptions." Regarding the second question in prompt 1, all groups noted the importance of helping students develop a sense of the role of assumptions in mathematics. The groups justified their response with reference to the following: the importance of assumptions in the discipline; knowledge of assumptions can help students understand that different legitimate interpretations of a problem can lead to different valid conclusions; and knowledge of assumptions can highlight the need for students to explain their thinking. These justifications relate to element 3 and are underpinned by the understanding of elements 1 and 2 that was evidenced in PTs' responses to the first question in prompt 1.

Prompt 2: All groups disagreed with the statement and highlighted, in a similar manner, that the phrasing of a task should align with the teacher's goals. Here is an illustrative response: "If a teacher is giving a math test, or graded evaluation where they are looking for a specific answer then yes, the tasks should be unambiguous. However, if the teacher is trying to prove a point about assumptions or different approaches to the same problem, an ambiguous task may be beneficial." Prompt 3: Two major points emerged from the responses to prompt 3. The first point related to teachers being able to recognize that some students' approaches to a task that appear as "faulty" may be in fact mathematically sound based on unforeseen (to the teacher) legitimate assumptions. The second point related to teachers asking students to explain their thinking to see whether the students indeed operated on different assumptions rather than just focus on the final answer. Again, these justifications relate to element 3 but are also based on understanding of elements 1 and 2.

Conclusion

In this paper we investigated how teacher education can promote prospective elementary teachers' knowledge about the role of assumptions in mathematical activity, with attention to the design and implementation of a task sequence that aimed to support three elements of MKfT. We illustrated a task-based approach to promoting MKfT whereby a mathematics task, although not embedded in a pedagogical context, can nevertheless create a productive space for pedagogical reflection through the creation of a powerful mathematical experience for PTs. Analysis of data from the fifth research cycle of a design experiment in a mathematics course for PTs provided evidence for the promise of this approach and highlighted the important role of the teacher educator in implementing deliberate (pre-planned) instructional moves to help amplify PTs' mathematical learning and enhance the pedagogical functionality of their acquired knowledge. Future research can investigate whether and how the rich insights (both mathematical and pedagogical) that PTs may develop through the task sequence about the role of assumptions in mathematical activity can inform their practice.

Figure 1 :

 1 Figure 1: The "Floors Problem" (derived from Ball, 1993) in part A of the task sequence

Figure 2 :

 2 Figure 2: Discussion prompts used in part C of the task sequence

  then. I say 15… because I'm assuming that you start on the ground floor and can't pass [floor] 2 and can't change direction more than once. Victor: I say 25 because… [He was interrupted by the start of the whole-class discussion.]

  :So… can you pass the second floor and then go back to it? Or do you have to stop, because you've technically gotten there? So… you just have to look at how to get directly there? If not, it's going to be infinity!

	Beth:	That's what I did…the direct. I just counted how many up and how many down…
	Monica:	But that doesn't say that you have to come in the entrance… So you can start
		anywhere?

  : Well, we picked infinity because […] if you're saying you can go up and down and up and down as many times as you wanted before you reach the second floor, I think that the question isn't specific enough. Like they [Sherrill's group] assumed you can only go up or down one time. […] route… So really, you could do anything that you wanted to, but it just depends on what the person… what the problem is looking for, I guess. Sherrill: It's up to the interpretation of the reader... Like I read it and I just assumed that it had to be a direct route, but it clearly doesn't state that… […]

	Stylianides: So you say [referring to Sophie] if we don't assume that [what Sherrill's group
	assumed] and [we assume] you're allowed to travel up and down as many times as
	you want, then the answer would be infinity? [To the class:] What do you think
	about that? Is one of them wrong? Are both of them right? […]

Lindsey:

I think […] they just made different rules, and like, thought processes. […] You could have a direct route and that's it, no going back and forth… like, you could, when you were talking about going up and down and up and down and that's not really a direct

Table 1 : Summary of results from PTs' responses to the pillars in part B of the task sequence (n=16)

 1

	Thematic Working Group 22

Proceedings of CERME10


text:
Abdel Seidouvy
email: abdel.seidou@oru.se

Andreas Eckert
email: andreas.eckert@lnu.se

Designing for responsibility and authority in experiment based instruction in mathematics: The case of reasoning with uncertainty

Keywords: Design principles, uncertainty, responsibility, authority

come

Designing for responsibility and authority in experiment based instruction in mathematics: The case of reasoning with uncertainty

Introduction

Designing lessons and tasks has been an important part of developing theories of instruction in mathematics for over a century [START_REF] Cobb | Design experiments in education research[END_REF] and emerging from this practice is the design-research methodology. [START_REF] Plomp | Educational Design research: An Introduction[END_REF] speaks of a twofold yield in design research-projects, both producing an intervention that offers a solution to a problem in the practice and what he calls re-usable design principles. In the present study, we are aiming for the latter. Design principles are to be seen as heuristics as they do not guarantee success. The principles rather suggest design elements that could support the development of a prototype design of a task [START_REF] Plomp | Educational Design research: An Introduction[END_REF]. Design principles are guides that answer questions like "What should the lesson look like?" or "How should the lesson be developed?" [START_REF] Van Den Akker | Principles and methods of development research[END_REF]. They should connect task design with both a practical and theoretical understanding of the topic and inform the teaching practice as well as the research practice. This paper begins in the idea of developing tasks for experiment based instruction in mathematics and the special case of stochastic. [START_REF] Groth | Characterizing Key Developmental Understandings and Pedagogically Powerful Ideas within a Statistical Knowledge for Teaching Framework[END_REF], amongst others, argues that stochastic education should be researched in particular since the nature of mathematical reasoning is largely deterministic whereas that of stochastic is with uncertainty. Our task design principles are therefore not focused on the creation of new knowledge but on eliciting stochastic reasoning and developing students' understanding. In the present study, we examine design principles in relation to how meaning making is described in the semantic theory of inferentialism [START_REF] Brandom | Making it explicit[END_REF]. Our guiding research question is:

 What are the design principles that trigger students' authority and responsibility in reasoning with stochastic uncertainty? [START_REF] Plomp | Educational Design research: An Introduction[END_REF] stresses the importance of reviewing similar examples when articulating design principles for the first time. Hence, the following section will elaborate on previous examples of experimentation-oriented tasks for stochastic education in the research literature to create a deeper understanding of design principles in stochastic education. Furthermore, the theoretical background of inferentialism gives the tools to see authority and responsibility in student's reasoning connected to the design of the task.

Related works

In the Dutch tradition of RME, three principles inform task design: guided reinvention, didactical phenomenology, and emerging modelling [START_REF] Freudenthal | Mathematics as an Educational Task[END_REF][START_REF] Gravemeijer | Developing realistic mathematics education: Ontwikkelen Van Realistisch Reken/wiskundeonderwijs[END_REF]. According to the principle of guided reinvention, mathematical tasks should offer students opportunities to experience a process similar to that linked to the invention of a specific topic. On the didactical phenomenological account, task designers should consider how mathematical "thought objects" will be used by the students to structure and organize phenomena in reality. Emerging modelling emphasizes the process on progressive abstraction, from a model of a situation (experimentally real for the student) to a more general mathematical model [START_REF] Gravemeijer | Developing realistic mathematics education: Ontwikkelen Van Realistisch Reken/wiskundeonderwijs[END_REF]. Overall, in RME, mathematical tasks are often rooted in Freudenthal's vision of "mathematics as a human activity", and are designed to resemble realistic problems in context. Another school of thought close to RME is authentic teaching and learning in mathematics. The design principles of an authentic practice are used to perform certain actions and procedures, and knowledge as a tool to perform and achieve particular goals [START_REF] Dierdorp | Authentic practices as contexts for learning to draw inferences beyond correlated data[END_REF]. The underlying conjecture in authentic mathematics is that students will be motivated and engaged in rich discussions if authentic practices are used as a source of inspiration in designing mathematical tasks. [START_REF] Ainley | Connecting engagement and focus in pedagogic task design[END_REF] used purpose and utility as design principles in mathematical tasks. According to [START_REF] Ainley | Connecting engagement and focus in pedagogic task design[END_REF], purpose means that a meaningful outcome of a task is crucial for student learning. Utility refers to acknowledging the power of mathematics ideas. Chance-maker microworld [START_REF] Pratt | Making sense of the total of two dice[END_REF] is an example of a learning environment in which purpose and utility are implemented. The chance-maker is a microworld program with a series of gadgets, simulations of everyday random generators such as spinner, coins and dice. The students are challenged to find the gadgets that according to them do not work (The students were told that the gadgets were programmed) and mend them. The purposeful activity, for the students, of mending the gadgets led to the understanding of the utility of representations (connection between probability and data distribution), and the importance of the law of large number. The students are supposed to discover the relevance of mathematical ideas through realistic situations created in classrooms, real life [START_REF] Ainley | Connecting engagement and focus in pedagogic task design[END_REF] or in the microworld [START_REF] Pratt | Making sense of the total of two dice[END_REF].

Our hypothesis is that a mathematical task that can trigger students' authority and responsibility increases student awareness on the data generation process and, has the potential to develop students' reasoning with uncertainty.

Theoretical framework: The Game of Giving and Asking for Reasons -GoGAR

The framework of our research is the game of giving and asking for reasons (GoGAR); it is a metaphor used by [START_REF] Brandom | Making it explicit[END_REF][START_REF] Brandom | Articulating reasons: An introduction to inferentialism[END_REF] to describe the linguistic practices in inferentialism, which is the background theory of this study. An inferentialist view on knowledge entails giving priority to inference in reasoning in account of what it is to grasp a concept:

To grasp or understand […] a concept is to have practical mastery over the inferences it is involved into know, in the practical sense of being able to distinguish, what follows from the applicability of a concept, and what it follows from. (Brandom, 2000, p. 48) It corresponds to the practical mastery of concept and, to the increasing awareness that reasoning is central to statistics and statistics education (Bakker & Derry, 2011). "The game of giving and asking for reasons is an essentially social practice" (Brandom, 2000, p.163), and the purpose of GoGAR is to make explicit reasons that are implicit in our linguistic practice [START_REF] Brandom | Articulating reasons: An introduction to inferentialism[END_REF]Bransen, 2002).

According to [START_REF] Brandom | Making it explicit[END_REF][START_REF] Brandom | Articulating reasons: An introduction to inferentialism[END_REF], one way of understanding how reasons are made explicit in talk is in terms of interaction of inferentially articulated authority and responsibility. Authority of a claim is a process, capturing the influence of a claim in the GoGAR. However, "Authority is not found in nature" (Brandom, 1994, p.51) but is gained in taking responsibility by providing evidence for one's claims. Students are expected to make claims that are related in a certain way, justify and explain their claims. Responsibility can be defined as a quest for authority, and it also expresses the quality of requirement for performing and maintaining authority [START_REF] Hansson | Instructional responsibility in mathematics education: Modelling classroom teaching using Swedish data[END_REF]. By using GoGAR we aim to show how, independently of the quality of the reasoning, elements of task design eliciting authority and responsibility influences how students reason in the domain of uncertainty.

Method

The present paper focuses on reaching an understanding of the data generation process connected to a mathematical task (Cobb et al., 2003). The data generation process involves clarifying the significance of the phenomenon under investigation, delineating key aspects of the phenomenon that should be measured, and considering how they might be measured (Cobb and McClain, 2004, p.386).

The analytical process of this paper is that of an abductive approach [START_REF] Alvesson | Reflexive methodology: New vistas for qualitative research[END_REF]. The aim of this approach is to create an initial analytical lens to view the data through, and then allow for the emerging design principles to influence the initial theorization. The specific analytical tool is retrospective analysis by the use of constant comparisons [START_REF] Gravemeijer | Design research from the learning design perspectiv[END_REF]. Instances of data are compared to find similarities and differences related to authority and responsibility in the data.

The data set used in this analysis are transcripts of video recorded lessons in a pilot study. The purpose of the study was to document our starting point [START_REF] Cobb | Learning about statistical covariation[END_REF] prior to an initial cycle of a design experiment [START_REF] Cobb | Design experiments in education research[END_REF]. The design experiment utilizes the idea of engaging students in actual data generation and asking them to reason about the results. Our aim is that the task would trigger students to develop authority and responsibility due to their active involvement in the data generation. The empirical study was performed with a class of 20 students in grade 7 (aged 12/13) in Sweden using an experiment based task focusing on statistics. The "helicopter task" (figure 1) is a modified version of a task that originates from [START_REF] Ainley | Normalising: children's activity to construct meanings for trend[END_REF]. It involves constructing an auto-rotating helicopter out of paper and measuring five flight times per rotor length. The class dealt with the helicopter problem in two lessons. In the first lesson, the students were involved in the data generation by e.g. testing paper-helicopters of varying rotor lengths (3cm to 14cm) and measuring flight times. In the second lesson, the class interpreted and evaluated the data in groups, guided by questions provided by the teacher. This was followed by a whole class discussion. The transcripts used here are meant to illustrate the ideas developed in the paper and are chosen because they portray typical student reasoning elicited by the task while using a relativly small amount of space in the paper. The reader is then invited to evaluate the plausibility of our interpretations and thereby assess the trustworthiness of our claims as an alternative to reliability and validity more suitable for this type of research [START_REF] Lincoln | Naturalist inquiry[END_REF])

Results

The transcript elaborated on below is used to shed light on aspects of how responsibility and authority, embedded in the task, elicit reasoning with uncertainty. It is part of a whole class discussion in the second lesson where one of the groups presents their findings. The questions from the helicopter task in focus here are: Which is the best rotor length? How sure are you? (…) means that portions are inaudible and […] means that we omitted a segment because of space limitation that we feel does not add to the reader's understanding of our analysis. Table 1 contains the students' results.

[1] Teacher: Before we move on, may I ask how you noticed that it was a failure with the rotors [2] Gabriel: Because sometimes were like three, and then some were only 1 second […]

1 st 2 nd 3 rd 4 th 5 th 6 th 7 th 8 th 9 th 10 th 6 cm 2.5 2.3 2.5 2.3 2.4 1.9 2.6 2.9 2.6 1.8 7 cm 1.8 1.9 2.7 2.2 2.1 2. 4

Responsibility in justifying uncertainty

If we compare instances [2], [4], [6] and [START_REF]Because maybe you drop it from the exact same spot, maybe you drop it further down one time and another time further up[END_REF] there are similarities and differences. All four instances are parts of GoGAR about the level of uncertainty. There are differences as well, for instance how their claims are accepted in the GoGAR and how they situate the responsibility for their claims. In [2], Gabriel reasons that the failure of the rotors becomes apparent in his data, but the reasoning is underdeveloped and no one else acknowledges his authority in the following discussion. In the following lines, not included because of space limitation, instead focus is put on the definition of margin used earlier. Eric's reasoning in [4] on the other hand situates responsibility explicitly in the activity of data generation. He draws on the experience of having been a part of the data generation and claims that both the angle of the wings and the drop height could be sources of uncertainty. Jennie acknowledges Eric's authority on line [6] by referring to the action of releasing the helicopter. We call this Responsibility in justifying uncertainty. Both students and teacher recognize that there is a measure of uncertainty in the results and use various informal concepts to indicate it, for example "hard to know" on line [3] and "differences are so big" on line [START_REF]Because the differences were so big, and the times[END_REF]. Both Eric's and Jennie's giving and asking for reasons for the level of uncertainty situate responsibility in the act of data generation. Maria acknowledges that Eric's and Jennie's claims still have authority in the GoGAR on line [START_REF]Because maybe you drop it from the exact same spot, maybe you drop it further down one time and another time further up[END_REF] by relating the mathematical discussion of mean, median and uncertainty to the act of data generation.

Exercising authority by making claims

From our analysis, it appears that students in this study exercise their authority through two types of claims: terminology related claims and context related claims. While the terminology claims are related to students' previous formal knowledge, context related claims are from observations through perception. Terminology related claim: We noted that all the students in the group were given the opportunity to express themselves. As whole, the group took the chance and made a considerable number of claims in a limited time period. As mentioned earlier, making claims is one way of exercising authority. In [2], Gabriel answers to the teacher's question in [1]. A close look at Gabriel's utterance, indicates that Gabriel's authority is grounded in comparing one second to three seconds. In other words, formal logic is what lies behind his authority. In [5], Eric acknowledges Gabriel's claim [2] and undertakes it. We, therefore conclude that "Gabriel lends his authority to Eric". The same line of reasoning is observed in [START_REF]Because the differences were so big, and the times[END_REF] and confirmed in [START_REF]Because the differences were so big and the times[END_REF]. Other students in their reasoning about uncertainty in GoGAR can use these claims. In [START_REF]But we remembered that afterwards[END_REF] and [START_REF]Because the differences were so big, and the times[END_REF], concepts such as mean and median are brought to the discussion. The students' involvement in data generation creates conditions in which students can make use of their previous formal knowledge. In this case the consequence of their actions in generating data, and the uncertainty that follows activates the use of mean and median while talking about uncertainty. Context related claims: it is evident that Eric's claim in [4], "it depends on high difference", is also accepted by the class. Hence [4] is based on observations during the data generation. One argument that supports our interpretation is that (here and otherwise in the data) the whole class has reported that it was almost impossible to drop the helicopter from the same height. Eric's claim is licensed by a context. The same observation is made by Jennie in [6]. Further, Eric points out that the angle of the rotor is reason for variation and uncertainty. This observation was specific to Eric's group. However, we believe the rest of the group will use it as a premise in their reasoning with uncertainty if needed. The context created by the data generation enables students to make claims that can be used in reasoning with uncertainty.

Discussion

We organize the discussion around three questions connected to the design that seem to trigger authority and responsibility.

Which different opportunities in the task are given by the students to accept responsibility for their claims? The students reason naturally with uncertainty since they have the practical experience of manually generating data in the task to situate their responsibility. In contrast, [START_REF] Pratt | The co-ordination of meanings for randomness[END_REF] found that many students in his study based on the chance-maker microworld software had to convince themselves that the gadgets in the game were indeed random and that the mode of reasoning should be with uncertainty rather than a deterministic reasoning. As long as the students had the opportunity to influence, for example, the strength of a simulated dice throw, they were more prone to accept that it was random. One explanation could be that the students working with the chance-maker had to situate responsibility inside a black box, being the software, which led them to believe that the results were predetermined. Our analysis in [2] and [4] show how the task provides opportunity for the students to situate the responsibility of the failure in the measurement. The cause-effect opportunities created by the task initiate fruitful GoGARs in the domain of uncertainty, which also seems to fit with Pratt's (1998) findings.

Which different opportunities in the task give the students the ability to create authority for their claims? From Roth's (1996) study, we know that students show difficulties in analyzing data when they have not been actively involved in the data generation. This view is supported by [START_REF] Noss | Touching epistemologies: Meanings of average and variation in nursing practice[END_REF]. Our analysis showed in [2] [3] and [4] that the task enabled the students to act as experts. The students were able to connect data analysis and inferences to data generation while reasoning with uncertainty. The experiment based instruction, which entails manual data generation, opens up for students to take control over the process. Leaving the data generation process to students, triggers their authority and responsibility and, students are more motivated than if they would just be limited to data collection and get "the right data" [START_REF] Cobb | Principles of instructional design for supporting the development of students' statistical reasoning[END_REF]. Getting students to reflect on how their own actions influence the results is the first step in acting with authority and responsibility.

Which different opportunities in the task are given to the students to shift responsibility and authority between different domains? Designing tasks with manual data generation based on principles of authority and responsibility is not the general solution to improve statistics education. In fact, creating large data sets, suggested by [START_REF] Pratt | The co-ordination of meanings for randomness[END_REF] as being a corner stone in statistics instruction, becomes almost impossible with this focus on manual data generation during lessons. We merely suggest that there are merits to these design principles in statistics education as it naturally elicits GoGARs in the domain of uncertainty and empowers students to take responsibility and use mathematics to back up their claims. In [6] and [START_REF]But we did it with the mean which made it take longer time[END_REF] our analysis in [6] and [START_REF]But we did it with the mean which made it take longer time[END_REF] show how the task provides opportunity for the students' quest of authority as they shift responsibility between the empirical and theoretical domain.

We show that designing tasks with manual data generation elicit fruitful situations for shifting authority and responsibility between context and concept levels. Thus the students used their previous concepts to act upon the context, and also looked for evidence in the context to strengthen their authority and responsibility. In sum, creating a learning environment where students can exercise authority by using their previous formal knowledge and most importantly "our imperfect perception" can promote a fruitful GoGAR in the area of uncertainty, and in mathematics in general.

Figure 1 Instruction for the helicopter task translated from Swedish to English Table 1 Measured flight time for the helicopters in seconds

	Thematic Working Group 22
						3.1 2.3 2.3 2.8
	8 cm	2.6 1.9 2.6 2.5 2	2	2.4 2.4 3	2.9
	9 cm	3.1 3	2.9 2.9 3	1.6 1.3 1.9 1.8 2.1
	10 cm	2.2 2.6 2.4 2.8 2.4 1.6 2.5 2.3 2.2 2.2
	11 cm	2.2 2.8 2.9 2.1 2.7 1.3 1.9 1.9 1.3 2.8
	12 cm	2.8 2.2 2.6 2.4 3.3 1.3 2.3 1.8 1.7 2
	13 cm	2.6 2.2 2.5 2.3 1.7 1.7 1.6 1.8 2.4 2
	14 cm	4	3	3	3.1 2.1 2.3 1.9 2.2 1.7 2.3

Proceedings of CERME10


text:
Hulya Kilic
email: kilichul@yahoo.com

Oguzhan Dogan

Zuhal Yilmaz

Pelin Donmez
email: pelindonmez99@gmail.com

Pre-service teachers' reflections on task design and implementation

Keywords: Pre-service, middle school, mathematical tasks, reflection, teacher knowledge

In this paper, we will present a part of findings of a larger study in which we investigated both preservice teachers' pedagogical content knowledge and middle school students' mathematical skills. A total of 17 pre-service teachers worked with 7 th grade students on the mathematical tasks for a semester. The data was collected through inventories, videos of task implementations and written reflections. The analysis of inventories and written reflections revealed that the pre-service teachers had positive beliefs about using mathematical tasks in the lessons. They were able to evaluate the success or failure of a task by analyzing the implementation process. They recognized the importance of preparing appropriate tasks for students and implemented them as intended. They also noted that mathematical tasks provided them an opportunity to learn about students' mathematical understanding as well as to develop their scaffolding practices.

Introduction

The studies on mathematical tasks provide an opportunity to discuss not only students' mathematical thinking and understanding but also teachers' knowledge and skills [START_REF] Zaslavsky | Mathematics-related tasks, teacher education, and teacher educators: The dynamics associated with tasks in mathematics teacher education[END_REF]. Tasks are accepted to be one of the curricular tools that help teachers to scaffold, foster and assess students' understanding when used appropriately [START_REF] Stein | Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms[END_REF][START_REF] Watson | Taken-as-shared: A review of common assumptions about mathematical tasks in teacher education[END_REF]. Research on task design and implementation shows that "good" tasks or cognitively demanding tasks [START_REF] Stein | Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms[END_REF] have positive impacts on students' thinking and understanding [START_REF] Henningsen | Supporting students' high-level thinking, reasoning and communication in mathematics[END_REF]. Such tasks encourage students to think about what mathematical concepts are conveyed in the task, what they know about them, how they are related to other concepts and which strategies, representations or materials are helpful in terms of arriving at a meaningful solution or answer. On the other hand, designing, selecting and implementing cognitively demanding tasks are related to teachers' pedagogical content knowledge (PCK) as well as to their content knowledge [START_REF] Charalambous | Mathematical knowledge for teaching and task unfolding: An exploratory study[END_REF][START_REF] Liljedahl | Interweaving mathematics and pedagogy in task design: a tale of one task[END_REF][START_REF] Stylianides | Studying the classroom implementation of tasks: High-level mathematical tasks embedded in 'real-life' contexts[END_REF]. Teachers' decisions about whether a task is appropriate for their students in terms of its complexity or if their students possess the necessary prior knowledge or the task's goal aligns with the lesson's objectives emerge from their PCK [START_REF] Hill | Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers' topic-specific knowledge of students[END_REF]. The studies on task design and implementation showed that when pre-service teachers (PSTs) were given opportunities to develop and implement tasks, they were able to prepare "good" tasks in which they used real life context and multiple representations and they asked for explanation, interpretation or justification [START_REF] Ozgen | An investigation of pre-service mathematics teachers' skills in the development of activities[END_REF].

In this study, we aimed to investigate how PSTs' involvement in task design and implementation process influenced their views about tasks and their PCK. We used tasks as a tool to build up an environment for PST-student and student-student interactions where the PSTs observed how students worked on the tasks both individually and as a group and then joined their discussions to elicit students' thinking and support their understanding. Thus, the PSTs had opportunity to learn about students' mathematical skills and also to gain experience about how to implement a task without loss of fidelity, to scaffold students' understanding, and to manage the group work. In this paper, however, we will present some findings about PSTs' reflections on this task design and implementation process in terms of effectiveness of the tasks and their professional gains from this intervention study.

Methodology

Research setting

This study was conducted under university-school collaboration between the mathematics education department and a local middle school in Istanbul in Turkey. A total of 17 PSTs (10 of them in Fall 2015 and 7 of them in Spring 2016) and one of the 7 th grade classes (aged between 12-13) from the partner school volunteered to participate in this study1 . We arranged the students as mixed-ability groups of 4 students based on the results of an achievement test. We randomly assigned the PSTs to these groups. The PSTs worked with students in a 40-minute lesson for each week, a total of 11 weeks during the Fall and 12 weeks during the Spring.

All PSTs were senior students who volunteered to take an elective course that was carried out specifically for this study. During the first couple of weeks of the course, we talked about the design and implementation of tasks, effective ways of communicating with students and assessment of students' mathematical understanding through samples. Then, each week, prior to implementation, we discussed the tasks that they would use with the students and made revisions on the tasks when necessary. After each implementation in the school, we met the PSTs again and discussed how the implementation went. We wanted PSTs to write reflection on each implementation by analyzing the implementation videos of their own group and students' worksheets. We also asked PSTs to prepare and implement a set of tasks for their own group at the end of the semester.

The tasks

We prepared the tasks in alignment with the 7 th grade mathematics curriculum. In the Fall semester we prepared 10 tasks which were about integers, fractions, rational numbers and algebra. In the Spring term we prepared 11 sets of tasks which were about geometry, algebra and data and statistics. Each set of tasks consisted of 3 to 4 sub-tasks with some additional problems. The tasks were set up around a common theme which related to the use of mathematics in daily life. For instance, in one of the data analysis tasks, we conducted a short survey on students' preferences in the class such that we gave them a list of foods where each student would choose 5 of the foods from the list. Then we wrote down the frequencies for each food on the board and then asked them to answer questions about their preferences such as finding out the most popular food, making a bar graph of the drinks, etc. The majority of the tasks that we used were selected from the ones that we implemented in the previous years. However, we revised those tasks before the implementation based on the needs and performances of the students in our new sample.

Data collection

The data was collected through pre and post questionnaires, videos of PSTs' task implementations and group discussions, and PSTs' written reflections and assignments. One of the questionnaires [START_REF] Kayan | Preservice mathematics teachers' beliefs about the nature of teaching and learning mathematics[END_REF]) consisted of 26 5-point Likert-type items asking for PSTs' beliefs about mathematics and mathematics teaching. The other questionnaire was developed by the research team and it consisted of open-ended questions where in the pre questionnaire the PSTs were asked to write about their prior teaching experiences and their expectations from the study and in the post questionnaire they wrote about whether their expectations were met or not and what they had learned from this intervention, etc. In their written reflections, we wanted PSTs to discuss about how the implementation went. Among the other questions, we asked them to comment on whether the students were able to complete the tasks, whether the tasks achieved their goals or not and why, and what they would do as follow up. At the end of the semester, we wanted them to discuss in which tasks the students experienced the most difficulty and what their suggestions would be to revise those tasks. Furthermore, we asked them to prepare tasks for their own groups and provide the rationale behind those tasks.

Data analysis

We have not analyzed all data yet. However, because the knowledge of students' thinking and understanding, the ability of selecting or developing appropriate tasks, and using appropriate teaching strategies for particular groups of students are counted in teachers' PCK [START_REF] Hill | Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers' topic-specific knowledge of students[END_REF], we are basically looking for any instances that could be counted as an influence of task design and implementation process on the PSTs' PCK and their views about tasks. For instance, whether they paid attention to the students' earlier performances while preparing their own tasks, how accurate assumptions they were able to make about students' performances on the new tasks, etc. Yet, we did document analysis such that we analyzed the items related to task design and implementation from the questionnaires, written reflections and assignments. We attempted to figure out the frequencies of common issues that emerged from those instruments. We found out the mean scores of the items in the Likert-type questionnaire but we did not compare pre and post results because of low number of participants. We examined pre and post open-ended questionnaires, the assignments asking for making overall evaluation of the study and the reflection reports including PSTs' reflections on the implementation of their own tasks. We developed a coding scheme for the common issues that emerged from the reflection reports. For instance, when discussing the reasoning behind the success of the tasks, if the PSTs wrote that "they liked it" or "they enjoyed it" or "they had fun with it" then we coded that reasoning as "enjoyable" (see Table 1). However, out of possible 194 reflection reports, 6 of them were missing. Therefore, we coded a total of 188 reports. We achieved 90% agreement in initial coding. We discussed the discrepancies by re-reading the PSTs' reports and then we came with an agreement. Moreover, we all agreed on the common issues that emerged from the open-ended questionnaire and the assignments.

Findings PSTs' thoughts about use of tasks in mathematics

The analysis of items in the Likert-type questionnaire showed that the PSTs agreed that teachers should encourage students to be active learners (The average of 3 related items; pre ).

PSTs' reflections on the implementation of the tasks

In the reflection reports, we asked the PSTs to comment on the following questions: 1a) Were the students able to complete the task? 1b) Did the task attain its goal or not? 1c) Why did the task attain its goal or not? 2) What more would you like to do about this implementation? 3) What is your suggestion for the follow up of the task? The PSTs gave various answers to these questions. For instance, for question 1a they noted that some of the students completed the task or just one student could not finish all of them, etc. Although we coded them separately, we re-coded them in terms of whether their answers were most likely "Yes" or "No". Similarly, we defined 12 different answers for question 1c. However, we preferred to present only the most frequent ones. In Table 1, the frequencies of PSTs' answers to these questions are given. As seen in the table, the PSTs noted that in a 40-minute lesson the students were able to complete the given tasks and discuss their answers (76%). However, some of the PSTs wrote that the tasks were difficult for their students and they only answered one of the sub-tasks. For some of the tasks, the PSTs noted that although the students completed the tasks, there was no time for themselves to discuss students' answers as a group discussion.

The PSTs wrote that some of the tasks did not attain their goals (28%) mainly because of lack of students' prior knowledge (27%). In a few cases, they noted that the students did not understand the task because the text was unclear (3%). Some of the PSTs attributed the success of the tasks to use of materials (6%) and context of the tasks that attracted students' attention (8%). Furthermore, some of the PSTs wrote that the tasks attained their goal because during the group discussion the students recognized their own mistakes and learned from each other (14%). However, in 25% of the reports, the PSTs did not write anything about the reasoning behind the success or failure of the task. Furthermore, we analyzed the pattern in PSTs' perceived cause-and-effect relationship between Q1b and Q1c. Among the total of 141 responses to Q1c, the PSTs wrote that the tasks achieved their goals because students learned from each other while engaging in the task (f: 25), they used materials (f: 12) and they enjoyed it (f: 15). They noted that the tasks were not successful mostly because of students' lack of prior knowledge (f: 38).

As an answer to Q2, the PSTs noted that they were able to do whatever they wanted to do during the implementation (40%). However, some of them noted that there was not enough time to explore how students thought about the given tasks or they were unsure whether the students understood the reasoning behind the answers to the tasks or not, therefore they would like to discuss more about those issues (28%). In some of the cases, especially in the Fall term, because of students' lack of knowledge about fractions and rational numbers, the students could not finish the given tasks. In such cases, the PSTs wrote that they would like to teach about those concepts before or after the implementation if they had enough time (19%). On a total of 6% of the reports, the PSTs did not write relevant answers but summarized how the implementation went.

For the third question, 12% of the reports (not shown in the table) included irrelevant answers such as the PSTs suggested encouraging students to read more books to improve their reading skills or they criticized themselves in terms of not managing time better. In a few reports, they wrote that the tasks were difficult for the students; there should be easier tasks on the same content domain (5%). However, they mostly suggested making review exercises for students to reinforce their understanding of the tasks as well as the content domain (47%). When they realized that the students did not know much about the content, they suggested remedial lessons for them (23%).

PSTs' reflections on the tasks and the intervention study

At the end of the semester, we asked the PSTs to evaluate this intervention study including the effectiveness of the tasks and the contributions to their own professional development. We specifically asked them to determine five mathematical issues in which students had experienced the most difficulty and two tasks that they would like to revise. We also asked them to give the reasoning behind the selection of those tasks and their revisions.

According to the PSTs, among the others, the students had difficulty in the tasks which were about fractions and rational numbers (

16  f ), algebraic expressions ( 9  f ), solving equations ( 7  f ), area problems ( 7  f
), and discovering patterns ( 5  f

). They wrote that the students could not do these tasks completely because of their lack of prior knowledge ( ). Specifically, they noted that students had difficulty in fractions, rational numbers and area problems because they did not know the algorithm for four operations with rational numbers as well as the area formulas of quadrilaterals and circle. They wrote that students failed in doing operations and discovering patterns because they did not practice enough at home or did not pay attention to the operations and procedures. However, the tasks that they would like to revise were mostly about integers ( ). The PSTs wanted to make revisions to the tasks about integers not because of students' lack of knowledge but because the context the tasks was confusing for the students. Therefore, they noted that they would rephrase the text and change the order of the sub-tasks in those tasks. They decided to make the tasks about fractions and area problems easier because students did not possess the required knowledge to complete the tasks. Furthermore, in some cases the PSTs preferred to change the tasks that were done by the students but they were uncertain whether they understood them thoroughly or not.

The PSTs prepared a set of tasks for their own group of students at the end of the semester. Their rationale for their tasks was either to focus on the issues where the students experienced the most difficulty ( 13  f

) or to make an overall review of what was done during that semester ( 4  f

). In parallel with their comments about the students' mathematical difficulties, they prepared tasks about integers, fractions and rational numbers, area problems, algebraic expressions and solving equations. Some of the PSTs kept their tasks as simple as possible because of their students' poor performance on earlier tasks. Furthermore, 7 of the PSTs prepared their tasks around a common theme as we did in this study but 10 of them prepared separate problems related to the content domain that they focused on.

In the post open-ended questionnaire, the PSTs wrote that this intervention study contributed to their professional knowledge in several ways. Among the others, they noted that they practiced how to scaffold students' understanding without directly telling them the solution or answer ( ).

Discussion

The findings of this study have potential to contribute to the relevant literature that teachers learn from their own task design and implementation practices in terms of better understanding of students' mathematical thinking and how to use or tailor tasks to scaffold students' understanding [START_REF] Zaslavsky | Mathematics-related tasks, teacher education, and teacher educators: The dynamics associated with tasks in mathematics teacher education[END_REF]. Although we have not yet analyzed all data, we recognized that the PSTs were able to evaluate the task implementation process in terms of the facts related to the task itself, the organization, the students and their own professional development.

When we asked the PSTs to write their thoughts about the success or failure of the tasks, they were able to make reasonable inferences from the implementation. They recognized that the tasks were successful because the students were actively involved in the solution and discussion process (14%, see Table 1, Q1c), the materials were appropriate for the tasks (6%, see Table 1, Q1c), the tasks were aligned with the 7 th grade curriculum and they were in the role of facilitator of group discussion (28%, see Table 1, Q1c and results of post questionnaire). In only a few of the cases, they noted that the students failed to complete the task because the task was unclear for the students (3%). For the other cases, they did not blame the tasks but the students because they did not possess necessary prior knowledge that they learned in previous grade levels or in their regular mathematics lessons (27%, see Table 1,Q1c). The PSTs' interpretations revealed that they began to recognize students' mathematical understanding as well as the importance of preparing tasks both aligned with the curriculum and appropriate for a particular group of students [START_REF] Hill | Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers' topic-specific knowledge of students[END_REF]. For instance, while preparing their tasks, they preferred not to use higher cognitive demand tasks but the easier ones because some of their students did not know about the procedures required to solve given tasks, such as the rules of four operations with rational numbers. The PSTs also commented that students should revise the issues discussed in their mathematics lessons in order to understand the mathematics conveyed in the tasks (47%, see Table 1,Q3). Moreover, as we inferred from their answers in the questionnaire they appreciated the facilitator role of the teacher during the task implementation. They stated that they suppressed their feelings of telling and teaching when students could not figure out the solutions. Briefly, their reflections on the tasks and the implementations indicated that they were aware of the task implementation process which begins with selection of appropriate tasks, continues with implementation of the tasks and ends with evaluation of students' learning [START_REF] Stein | Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms[END_REF]. Because orchestration of task implementation process is involved in teachers' PCK, such awareness of the PSTs can be counted as a sign of their PCK [START_REF] Charalambous | Mathematical knowledge for teaching and task unfolding: An exploratory study[END_REF]. However, because we have not yet analyzed the implementation videos, we are not able to validate PSTs' reflections on the tasks implementation process, especially whether they were able to implement the tasks without loss of fidelity and manage the group discussions appropriately. Indeed, it is hard for teachers, even more so for PSTs, to keep the cognitive demand of the tasks such that they might have provided hints or helped students when the students did not possess the required knowledge [START_REF] Stylianides | Studying the classroom implementation of tasks: High-level mathematical tasks embedded in 'real-life' contexts[END_REF]. Therefore, we are not able to comment on their "PCK in practice" even though we could make inferences about their PCK from their written reflections.

Finally, the analysis of pre and post questionnaires revealed that the PSTs had positive beliefs about use of student-centered teaching strategies, mathematical tasks and manipulatives while teaching mathematics and such beliefs sustained and even increased throughout the study. They recognized that tasks provided an opportunity for them to elicit students' mathematical understanding and they could be used as a tool to foster students' mathematical understanding.

  they learned what kind of tasks attract students' attention more ( 3  f

  Their answers in the open-ended questionnaires were compatible with these results. The PSTs noted that doing mathematics through tasks enables students to participate in the lesson (

	x		45 . 4	), the tasks are important for students' learning and understanding (The average of 2 related
	items; pre	x		36 . 4	; post	x		61 . 4	) and manipulatives and materials facilitate students'
	understanding (The average of 3 related items; pre ( 4  f ), discover or review mathematical concepts or facts ( 09 . 4  x ; post manipulatives ( 4  f	x f	 	38 . 4 ), love mathematics  x ; post ), and use materials or 60 . 4 ). 5  f 4

Table 1 : The frequencies of pre-service teachers' answers to the questions about task implementation

 1 Were the students able to complete the task?

		No	Yes
	Fall	30 (29%)	75 (71%)
	Spring	15 (18%)	68 (82%)
	Total	45 (24%)	143 (76%)
	Q1b: Did the task attain its goal or not?
		No	Yes	No Comment
	Fall	38 (36%)	67 (64%)
	Spring	15 (18%)	63 (76%)	5 (6%)
	Total	53 (28%)	130 (69%)	5 (3%)
	Q1c: Why did the task attain its goal or not?
		No Comment	Enjoyable	Lack of Knowledge	Use of Materials	Recognize own Mistakes
	Fall	28 (27%)	10 (10%)	32 (30%)	11 (10%)	14 (13%)
	Spring	19 (23%)	5 (6%)	19 (23%)	1 (1%)	14 (17%)
	Total	47 (25%)	15 (8%)	51 (27%)	12 (6%)	27 (14%)
	Q2: What more would you like to do about this implementation?
		No	Nothing	Discuss	Teach for
		Comment	more	more	Understanding
	Fall	9 (9%)	17 (16%)	37 (35%)	29 (28%)
	Spring	1 (1%)	58 (70%)	15 (18%)	6 (7%)
	Total	10 (5%)	75 (40%)	52 (28%)	35 (19%)

Q1a:

Q3: What is your suggestion for the follow up of the task?

The school was located in a crowded neighborhood in terms of school-age children. There were 40 students in the class, average size according to national education statistics, but in the Spring semester 4 students dropped out. Since we had only 7 volunteers in the Spring semester, two researchers from the research team worked with two of the groups.Thematic Working Group

Proceedings of CERME10

Proceedings of CERME10

Acknowledgment

A part of this study was supported by The Scientific and Technological Research Council of Turkey (TUBITAK, Grant no: 215K049).