text:
Nick Pratt
email: nick.pratt@plymouth.ac.uk

Julie Alderton

It's different, it's difficult, it's unknown': Letting go of levels

Keywords: Elementary school mathematics, assessment, Foucault, governmentality. Thematic Working Group 21

This paper explores primary teachers' accounts of their responses to major changes in the curriculum and assessment system in England, which has recently re-designated expected standards of achievement and progress. Analysis is informed by Foucauldian poststructural understandings of power/knowledge and truth to examine how they reorganise their practices as mathematics teachers within a policy context which continues to compel schools to focus on performance. By means of a small-scale empirical study, we identify the tensions created when the 'rules of the game' change and how technological assessment tools require and enable teachers to reproduce levels and labels to categorise pupils. Our aim in undertaking this analysis is not to compare teachers' assessment practices to an ideal, beyond policy, but to illustrate how government-driven changes to assessment are insufficient to change underlying discourses of performativity which ultimately shape practice.

Introduction

The examination [assessment] combines the techniques of an observing hierarchy and those of a normalizing judgement. It is a normalizing gaze, a surveillance that makes it possible to qualify, to classify and to punish. It establishes over individuals a visibility through which one differentiates them and judges them. (Foucault, 1977, p. 184) The quote from Foucault begins to set out both our substantive interest and the theoretical stance, namely, an interest in mathematics assessment from a sociological perspective. Our focus is on assessment not simply as a technical activity to improve pupil outcomes, but as a mechanism through which teachers manage their professional selves; the way in which mathematics assessment is used as part of their ongoing professional identification and as the basis, and evidence, of their success. Our starting point is the claim that in English primary (5-11) schools, assessment, and the curriculum alongside which it takes place, plays a majorperhaps the majorrole in influencing teachers' actions. There are many reasons why this is the case but, as Pratt (2016a) argues, in essence they revolve around the marketized and high-stakes, accountable nature of the English system and the 'performativity' [START_REF] Ball | The teacher's soul and the terrors of performativity[END_REF] this manifests in teachers' work.

The changing context of English mathematics education

It is difficult in a short paper to describe fully the complex landscape of an education system and how it is changing and we refer the reader to Pratt (2016a) and [START_REF] Keddie | Children of the market: performativity, neoliberal responsibilisation and the construction of student identities[END_REF] for more detailed discussions of English primary schools. However, in summary the system is based in a neo-liberal, neo-conservative framework which affords an increasingly marketized, competitive and accountable approach to school improvement. This has led to a strong discourse of 'progress', since it is the change in pupils' levels of attainment across each year which has been the key measure against which schools, and individual teachers, have been judged. In turn this leads to a strong discourse of control, a belief that pupils' progress is predictable and controllable across time; and therefore of teachers' responsibility for learning outcomes obtained through their teaching [START_REF] Pratt | Playing the levelling field: teachers' management of assessment in English primary schools[END_REF].

However, over the last 18 months, both the curriculum for mathematics and the assessment system have been reformed. The new primary national curriculum (NC) (DfE, 2013) stipulated increased expectations in mathematics with more challenging national tests. Perhaps most importantly for teachers, previous NC 'attainment levels' have been superseded by an 'expected standard' set for 2016, at a higher level than in 2015 (DfE, 2016). The rationale for this change is described in the final report of the government's Commission on Assessment Without Levels (McIntosh, 2015, p. 5), as follows:

Despite being intended only for use in statutory national assessments, too frequently levels also came to be used for in-school assessment between key stages in order to monitor whether pupils were on track to achieve expected levels at the end of key stages. This distorted the purpose of in-school assessment, particularly day-to-day formative assessment. The Commission believes that this has had a profoundly negative impact on teaching.

Too often levels became viewed as thresholds and teaching became focused on getting pupils across the next threshold … Depth and breadth of understanding were sometimes sacrificed in favour of pace.

Guidance specifies that the majority of pupils should move through the programmes of study of the NC at broadly the same pace (DfE, 2013), crucially replacing previous advice to accelerate high attaining children through new content. At the classroom level, 'progress' through the curriculum has been replaced by 'progress' within it; and a new language of 'mastery' has sprung up to describe this, which "denotes a focus on achieving a deeper understanding of fewer topics, through problemsolving, questioning and encouraging deep mathematical thinking" (McIntosh, 2015, p. 17). Progress measures of pupils and schools across key stages are also calculated differently. Monitoring progress by levels and sub-levels has been replaced by a value-added measure. Pupils' results at the end of key stage 1 and key stage 2 (at ages 7 and 11) are compared to the achievements of other pupils with similar attainment nationally, and a new 'floor' standard requires that at least 65% of pupils meet the expected level in mathematics (and English), or that a school achieves sufficient progress scores (DfE, 2016). Schools not achieving the floor standard will be scrutinised through additional inspection and may have their freedom curtailed. Indeed, the Commission notes that "with freedom, however, comes responsibility" (McIntosh, 2015, p. 10) and "recognises that the transition to assessment without Attainment Targets and levels will be challenging, and that schools will have to develop and manage their assessment systems during a period of change" (p.16). However, it justifies this on the basis of "a much greater focus on high quality formative assessment as an integral part of teaching and learning"; the raising of "standards" in line with neoliberal policy. As we have previously pointed out, in a performativity culture such as this one, assessment has become a means by which teachers gain and maintain professional capital (after Bourdieu, see Pratt, 2016a).

Theoretical Framework

To understand the effect of changes to the 'rules of the game' of assessment, we draw on Foucault, particularly his notion of governmentality [START_REF] Foucault | Discipline and Punish: The Birth of the Prison[END_REF] that surrounds English education [START_REF] Ball | Foucault, Power, and Education[END_REF][START_REF] Llewellyn | Problematising the pursuit of progress in mathematics education[END_REF]; the notion that dominant discourses become normalized to such an extent that (teacher) subjects consent to particular action and hence come to govern themselves. (Note, discourse here refers to "a group of rules proper to discursive practices … [which] define the ordering of objects" (Foucault, 1972, p. 49) and is more than just language.) Our aim is to make visible the ways in which assessment discourses normalize certain practices and relations between teachers, school systems and pupils, rendering them common-sense, irrevocable and changeresistantbut not to judge these against some ideal version of practice. In theorising these forms of governmentality in and through assessment, two related ideas are in play: power/knowledge and truth. Power, according to Foucault, is enacted, not held by individuals, and is not exercised simply as an obligation or a prohibition on those who 'do not have it'; it invests them, is transmitted by them and through them; it exerts pressure upon them, just as they themselves, in their struggle against it, resist the grip it has on them. (Foucault, 1977, p. 27) We emphasise that this can be a good or bad thing; power can liberate and is not oppressive per se, but either way, 'power produces knowledge' (ibid). 'Experts' in a field (teachers in their classroom settings, but also senior managers in the school as a whole, policy makers and children as 'expert pupils') produce knowledge through their language and activity which positions and exerts pressure in terms of the way it influences what can and cannot be said and done. In this sense, it forms a 'game of truth'. For Foucault, truth is not something to be found outside of relations. Rather it is something produced through such relations so that "each society has its regime of truth, its 'general politics' of truth: that is, the types of discourse which it accepts and makes function as true" (Foucault, 1980, p. 131). Thus, the question is not what the truth 'is', but how things come to be taken as true; how this is used in order to make manifest and exert power relations. This is the truth which does not belong to the order of what is, but to the order of what happens … a truth which is not found but aroused and hunted down: production rather than apophantic. This kind of truth does not call for method, but for strategy. (Foucault et al, 2008, p. 237) It is through this theoretical lens that we return to mathematics assessment, and the following questions: how do teachers respond to the changes that a new curriculum and assessment system impose; and in doing so, how do they re-organise the economy, and politics, of truth in assessment practices in order to (re)empower themselves as experts?

Methodology

The project involved extended semi-structured interviews with primary teachers in 9 different schools (12 teachers in total) in the first year after the removal of levels. Teachers and schools were chosen purposively to reflect a range of ages, experience, school types and locations, but in this paper we draw on just three of the participating teachers -Ann, Jill and Mike, all working in state schoolsin order to keep the analysis manageable. Mike and Jill are in their late 20s and both are coordinators of mathematics in their schools and are both on a programme of training to develop leadership in 'mastery' of mathematics. They work in a village and an urban school respectively; Jill has been teaching for 8 years and Mike for 7. Ann is in her late 30s, has been teaching for 19 years and in her current, town, school for 5 of these. She is a class teacher, but not a specialist in mathematics. Data from all the interviews were analysed thematically in relation to the substantive and theoretical framework -teachers' assessment practices, as we understood them in relation to power/knowledge and truths. Whilst we can only present a small set of data we have selected this carefully, ensuring that teachers' views, though sometimes individual, are never contradictory of the data set as a whole. Our aim is not to claim that the specifics are generalizable to every teacher beyond, or even within, the data set. Rather, the analysis is of the system of governmentality and the dominant discourses that constitute it. We think it offers a trustworthy and useful analysis in this sense, meaning that it is likely to be generalizable to other teachers in terms of the way in which their work becomes problematized, even if not in terms of how individuals are able to respond. All our work conformed to the ethical procedures of the British Educational Research Association and were approved by our employing institutions.

Analysis -Reproducing the truth

The DfE's Commission on Assessment without Levels is very clear over the point of their removal.

Removing the 'label' of levels can help to improve pupils' mind-sets about their own ability. Differentiating teaching according to pupils' levels meant some pupils did not have access to more challenging aspects of the curriculum. (McIntosh, 2015, p. 15) Interestingly, this critique itself illustrates Foucault's central point about governmentality, namely that it is through labelling that subjects are categorised, normalized and objectified. They 'become' their labeland act accordingly in the common-sense, normal(ized), way that this affords. Whilst removing the language of levels is well-intentioned in order to remove such labels, we noted above that teachers' work takes place in a culture of performativity with dominant discourses of control and responsibility. Central to governmentality, they require teachers to 'know' what their pupils can and cannot do so that they can take responsibility for 'filling the gaps' in their knowledge by "identifying specific 'corrective' activities to help them do this" (ibid. p.17). These, then, become questions of truth, of what pupils 'actually' and 'really' know. However, as Foucault notes, a truth statement is "contingent on the instruments required to discover it, the categories necessary to think it, and an adequate language for formulating it in proposition" (Foucault et al., 2008, p. 236). The language of levels may have gone, but the imperatives for control remain and so a new language is needed for teachers with which to think and speak it. Our interviews suggest that the language of 'mastery', codified through other national continuing professional development programmes, has offered teachers such an alternative, so that:

For every child you can click on an objective and say whether you are working towards it, achieved, secure, or greater depth. (Ann)

Basically we have developed a system throughout the year. So, we haven't bought a system in. We've simply developed our own system as a school where we've given the children a grade of either 1, 2, 3 or 4. (Mike) When we were talking, as a school, what we were going to put for our levels, we said "what shall we call them?" We've got to have things and labelling them "emerging, developing, secure, exceeding". (Jill) Ironically then, the notion of mastery which was meant to take teachers away from codifying and levelling has provided alternative "types of discourse which it [the system] accepts and makes function as true" (Foucault, 1980, p. 131). Classification continues, but with new levels. What is significant in terms of governmentality is that, despite the best intentions, this replacement is inevitable since it is founded in the performative discourse which underpins pedagogic activity. In English primary schools this performance is measured by 'progress'; in the past meaning the movement up levels and sub-levels of attainment. Although the removal of levels has meant that there might be a new official understanding of it -that "progress can involve developing deeper or wider understanding, not just moving on to work of greater difficulty" (McIntosh, 2015, p. 12)it has not removed the imperative of being able to make it demonstrable as the way in which schools are judged. In other words, knowing 'where pupils are' is still central to "the status of those who are charged with saying what counts as true" (Foucault, 1980, p. 131) and is not therefore optional. Foucault (1980, p. 131) has pointed out that the political economy of truth is characterised by, amongst other things, the form of scientific discourse, economic and political demands and the ways in which it is diffused and consumed amongst different organisations. Each school in our study has made use of some form of tracking system, either commercial software or a spreadsheet of some sort, as a technology for capturing data and in different ways teachers are looking for these technologies to help them seek the truth about the progress of their pupils. In each case, there are two technologies at work. Firstly, a tracking system recreates labels: you've got all the statements and you can say whether the children are working towards it, expected for it, or exceeding for it, or something. Then it breaks it down into them being, for each year group, they are beginning to access or beginning plus, working towards or working at plus, secure, secure plus. There are six basic, what would have been sub-levels. (Mike) But, he notes, "it can't generate something that tells you your child probably is secure or probably is working at" and "it's not comparing your children to anyone else. It's not saying anything." Whilst teacher judgement is "fine and good" it does not seem to represent a sufficient truth for the accountability purposes to which it is to be put. Mike's school has therefore turned to commercially produced online tests. These give him "beautiful data" and whilst it also serves a formative purpose in identifying "gaps" it "provides a comfort blanket" because "it gives you a standardised score and it's based against however many thousand children from around the country".

Hunting for truth with technology

Whilst Mike has turned to comparative statistics to produce knowledge of progress, Jill agrees that numbers and labels mean that "it somehow feels like it's clearer, but if it's not well-defined that's quite dangerous, really". Rather than seeking a truth in statistics though, Jill is committed to the idea of illuminating pupils' mathematical understanding and somehow mapping this onto the new labels so that they can say, "these children are where they should be and these children aren't … so that the gaps that they have got [can be] filled". Rather than comparisons to other pupils nationally, Jill's plan is to exemplify for colleagues a truth about what each label (developing, secure etc.) looks like in terms of the objectives from the curriculum that pupils can achieve. In this way she hopes that "it would be very clear where the children were and where their next steps were more clearly" and that "within the following year's teaching you can see that clear progression, and then that becomes a way for teachers to show progress". Jill's belief seems to be that professional judgement, evaluating pupils' understanding against exemplar materials, will, in time, allow teachers to learn what the new levels "feel like".

The rationale for the removal of levels and a focus on mastery was, in part, based on the assertion that "too often … teaching became focused on getting pupils across the next threshold instead of ensuring they were secure in the knowledge and understanding defined in the programmes of study" (McIntosh, 2015, p. 5). We have illustrated how levels have been recreated by teachers to serve the function of performativity, yet this is not to say that the idea of refocusing on pupils' understanding of the curriculum was not welcomed and encouraged by this move. Mike notes that alongside the security of knowing how their pupils rank against others "we are thinking about 'OK they are working at expected levels or just below but what are their gaps and how am I going to fill their gaps?" Jill claims that "I think the move away from levels has been absolutely fantastic" because it allowed them to "take the time to sit back and actually think about the underlying maths". Ann also welcomes the focus on ensuring that "gaps are filled" and considers this as central to pupils' success. However, in her experience it was just a lot that had to be covered and part of it was because there were gaps that I needed to go [over]. So for example my class didn't have a very good understanding of decimals, so rather than teaching thousandths and all of what was in the year 5 curriculum, I've had to go right back to the start and doing tenths. And that is your year 3 and year 4 objectives. (Ann) This has led to her being reluctant to say that any child is secure and to her "feeling that almost, as a teacher, you've failed", with her confidence being affected as a result. The school uses a system called School Pupil Tracker Online (SPTO) which, unlike the other systems, is meant to calculate whether pupils are emerging, developing or secure, but Ann does not trust its output.

I just experimented with 'what if I made that [objective] mostly achieved?'. And by doing that I could see that it was literally one little click turns that level up. … I didn't like the fact that just one click sent that judgement over, particularly when it didn't look like it was right. She notes that even if the company that runs the software alters this in the coming year "it sounds like the standard is going to slightly change every single year, which just makes it completely confusing. How can you work towards something that you don't know what it is?" This lack of clarity over the truth of her pupils' learning is leading to some tension for Ann.

So within what I do with the children I see progress but I don't always see it in what I've got on paper, on SPTO. The progress isn't always reflected there … I thought I was a good maths teacher, maybe I'm not, because of what's coming out … In some ways I'm almost fighting against it and saying 'you will not do this to my confidence' [laughs], yeah.

A new normalizing gaze

We noted above that one intention of removing levels was to avoid labelling pupils in ways that prevented access to the curriculum. As the quote that begins this paper makes clear, however, from Foucault's position any examination "combines the techniques of an observing hierarchy and those of a normalizing judgement" which "establishes over individuals a visibility through which one differentiates them and judges them" (Foucault, 1977, p. 184). Foucault's use of normalizing here is two-fold. On the one hand it points to the standardisation and categorisation of pupils; their allocation into categories, in this case 'emerging', 'secure' etc. which are then used to define normal, and hence abnormal, and to take remedial action. Mike refers to "the ones who haven't quite got there" and Jill to those who are "where they should be" and those that are not. On the other hand, it refers to the notion of making this categorisation 'normal' practice; common-sense, inarguable, defining what can and cannot be thought and said. Thus, although removing levels is meant to avoid differentiating pupils and restricting their access to the curriculum, the need to track progress makes such differentiation necessary. To speak of progress is to speak of changes in category as the only "type of discourse which [society] accepts and makes function as true" (Foucault, 1980, p. 131); "a truth provoked by rituals, captured by ruses, seized according to occasions" (Foucault et al., 2008, p. 237). Such rituals create a practical tension in the idea of normalisation. As Mike notes, It's that challenge we're set of trying to keep together and moving forward together but having children still working at a greater depth but closing the gap for the ones that are lower.

For those already 'succeeding' as secure, mathematics involves a range of activities. Mike describes "10 children who we saw as working at greater depth and they worked in groups with teaching assistants and had some really different kind of problem solving". However, Jill points out that the governmentality around floor targets means that for "the children who are almost secure but not quite, there is a real push to get them [over the threshold]".

Yes, but I think the secure one [is key] at the moment. I think at the moment with the new system it's different, it's difficult, it's unknown. I think it's that 'where are for secure?' (Jill).

Hence, whilst the change in the curriculum structure is meant to ensure that children move together through the content, the manner in which assessment inevitably "establishes over individuals a visibility through which one differentiates them and judges them" (Foucault, 1977, p. 184) means that the way in which they experience the subject is far from equal.

Discussion

Our analysis suggests that although superficially things might look different and teachers may feel that their practice has changed, this appears to be largely a reconstruction of the same dominant discourses in new language. Whilst the specific practices of governing might have been altered, the fundamental forms of governmentality have not and teachers are in the process of reconstituting much of what they had before. We recognise that the recent changes have opened up opportunities for discussion, collaboration and reflection within and between schools and made teachers pause and take stock of assessment in ways that feel positive to them. However, they have also reproduced pressures and tensions which can work to deflect attention away from questioning the responsibilities of policy makers and the implications for the teaching and learning of mathematics in the new system. Whilst there is a significant impact on teachers' day-to-day teaching and assessment practices, and how these are evaluated, the performative role of the teacher remains largely the same. There does seem to be more consideration of pupils' development in mathematics; though this is produced in particular ways: an atomised curriculum and filling in gaps. There are signs too that far from alleviating the problem of access to the curriculum for all children, there is a new normalizing gaze; one that focuses teachers' efforts on an even slimmer tranche of pupils who might just be normalizedliterally, to the middle of the normal distribution. Similarly, only those who are 'secure' in their 'knowledge' of the subject get access to a rich version of mathematical problem solving. These points raise questions about the way in which such tightly managed forms of assessment affect pupils' relationships with the subject and about the equity of pupils' access to the curriculum. The nature of these authoritative discourses of progress, control and responsibility that make up performativity, and the version of mathematics and assessment produced within them, appear difficult for the teachers in our study to identify. All schooling operates within policy and its incumbent discourses and can never be free of it, however the value of a Foucauldian analysis is in making such discourses visible to those responsible for making changes to the assessment system.

Proceedings of CERME10


domain: math shs

text:
Olande Oduor
email: olande@lnu.se

Sweden; oduor. Linnéuniversitetet

Analysing students' graphicacy from a national test

Keywords: Assessment, examinations, graphical artefacts, misleading diagram, competencies

The present study explores a task containing graphical artefacts from the Swedish national test (Nationella provet -NP) for a sub-sample of grade 9 students' solutions. The sub-sample comprising of 115 students' solutions to the task is closely analysed, using an analytical construct founded on "identification" as well as the "critical-analytical" approach to problem solving. Based on this construct it is observed that a sizable number of students' solutions follow a visual strategy with strong reliance on everyday forms of expression. Given the nature and purpose of NP we posit that students use the methods and tools that reflect general school practice. The analysis used in this study is perceived as problematising the assessment of competency from high stake tests, and the educational setting in general.

Introduction

National tests generally play a vital role in the evaluation of educational systems, informing educational reforms and policy implementations as well as serving as a local comparative instrument [START_REF] Eurydice | National Testing of Pupils in Europe: objectives, organisation and use of results[END_REF]cf Skolverket, 2016). This implies that results from national tests can be used to explore the effects of curriculum documents on teaching and learning. Some researchers, (e.g. [START_REF] Boesen | Assessing mathematical creativity:comparing national and teacher-made tests, explaining differences and examining impact[END_REF]) have suggested that centrally administered tests can influence instructional practices, for instance by guiding the time set aside to teaching specific topic strands in the mathematics classroom. Given the orientation towards competencies in the mathematics classroom, it is also envisaged that the Swedish national test (Nationella provet -NP) might be a suitable indicator of teaching and learning of mathematics competencies. Thus, national tests can be perceived as having the potential to provide insight on students' fluency with mathematical concepts, as well as mathematics competencies (NCTM 2000;Niss & Højgaard, 2011;[START_REF] Skolverket | Curriculum for the compulsory school, preschool class and the recreation centre[END_REF]; see also [START_REF] Sáenz | The role of contextual, conceptual and procedural knowledge in activating mathematical competencies (PISA)[END_REF].

The goals of the study

The aim of the present study is to gain insight into the strategies and approaches that some students at the end of the compulsory school in Sweden employ, as they interact with a mathematics task containing graphical artefacts. This study is based on written solutions from NP. The purpose is to explore the strategies and tools used to solve this task, as well as to determine the potential provided by a model focusing on mathematical tools and forms of expression as a means of exploring students' mathematics competency.

Background and theoretical construct

Several methods have been used to analyse students' responses to test items (e.g. [START_REF] Åberg-Bengtsson | Separating the quantitative and analytic dimensions of the Swedish Scholastic Aptitude Test[END_REF][START_REF] Goodchild | An exploratory study of mathematics test results: what is the gender effect[END_REF]. In some of these studies the focus has been on general performance with microanalysis done at topics strand levels. While there are models proposed for examining the response to tasks (e.g. [START_REF] Gal | Adults' Statistical Literacy: Meanings, Components, Responsibilities[END_REF][START_REF] Gal | Assessing statistical knowledge as it relates to students' interpretation of data[END_REF][START_REF] Ben-Zvi | Junior high school students' construction of global views of data and data representations[END_REF][START_REF] Watson | Statistical literacy: A complex hierarchical construct[END_REF], these seem to suggest general skills needed to solve the task with a bias towards statistical literacy. [START_REF] Friel | Making sense of graphs: critical factors influencing comprehension and instructional implications[END_REF] developed a three-tier model entirely devoted to graphical artefacts: without loss of generality this can be collapsed into two tiers (cf. [START_REF] Gal | Assessing statistical knowledge as it relates to students' interpretation of data[END_REF][START_REF] Bertin | Semiology of graphics: diagrams networks maps[END_REF][START_REF] Olande | Students' narratives from graphical artefacts: Exploring the use of mathematics tools and forms of expression in students, graphicacy[END_REF]. While there are models for analysing students' test results, there is a dearth of studies exploring the use of subject specific tools and forms of expression in students' solutions.

In the present study an analytical construct (see [START_REF] Olande | Students' narratives from graphical artefacts: Exploring the use of mathematics tools and forms of expression in students, graphicacy[END_REF], focusing on the application of mathematics tools and forms of expression while solving test items, is employed. In this construct response to test items is perceived as being oriented towards identification and critical-analytical approaches. Items eliciting the identification approach are largely "self-evident" and as such, problem solvers might not need to unpack their mathematical skills entirely in order to solve the problem. On the other hand, items seen as demanding a critical-analytical approach require focused engagement from the problem solver. For example, this could be in the form of a critical analysis of underlying factors in the task, evaluation and selection of appropriate tools for interacting with the task, as well as reporting the solution with relevant subject specific forms of expression.

The construct guiding the present study borrows from a socio-semiotic paradigm where the emphasis is on artefacts as a means of coming to know. [START_REF] Radford | The ethics of being and knowing[END_REF] posits that the investigation of students' interaction and use of semiotic means of objectification is a methodological way of accounting for learning (see also Radford, 2003;[START_REF] Vygotsky | Mind in society: The development of higher psychological processes[END_REF]. It is recognised that a sign or symbol does not exist in isolation but is always bound with intentions, motives and the objects of action [START_REF] Roth | The dawning of signs in graph interpretation[END_REF][START_REF] Olande | Graphical artefacts: Taxonomy of students' response to test items2014[END_REF]. Thus, in a critical-analytical approach, "being critical" encompasses more than visually interrogating a graphical artefact, but also includes the means and the tools used in the sense making process. The assumption about what can be perceived as a non-hierarchical path to cognition is significant for the analytical framework: i.e. it is access to tools and forms of expression that is perceived as largely determining a problem-solving trajectory.

The research questions

Thus, the concern of the study is to outline and analyse tools and forms of expressions used by students as they interact with a task from NP. This task was picked from section C of the national test, a section that requires students to justify in one way or another how they arrive at their solutions. The task (figure 1) was selected for further analysis given that it provided a combination of visualidentification as well as critical-analytical components. This way of assessing the task is different from the marking process employed by graders while awarding credit to students' solutions. The marking scheme used by graders did not indicate assessment of diverse solutions provided by students but largely gave written statements as guidelines. Thus item a) scored full credit when the solution contained the expression TRUE with corresponding justification such as making a comparison based on the sizes of the shaded areas. For item b) partial credit was awarded where the solution contained the expression FALSE with corresponding justification indicating an understanding that the pie charts express different quantities e.g. "Australia has more medals than Spain". Full credit was awarded where the solution, in addition to the general statement FALSE, explicitly made comparisons based on mathematical forms of expression e.g. computation with fractions. A reliability check conducted by an independent entity [START_REF] Skolinspektionen | Kontrollrättning av nationella prov I grundskolan och gymnasieskolanredovisninng av regeringensuudrag Dnr[END_REF] from a representative sample of Swedish students for section C of the test, indicated that for items scoring grade G -(for the purpose of this study partial credit) in 51% of the cases the graders gave higher grades than the assessors. In 27% of the cases there was correspondence in the credit award. For items scoring VG -"full credit" in 37% of the cases the graders' credit award was higher than that of the assessors, with 41% indicating correspondence in the credit award.

From a competency perspective [START_REF] Skolverket | Curriculum for the compulsory school, preschool class and the recreation centre[END_REF], this task might be considered to focus on developing competencies in the following: using and analysing mathematical concepts, use of appropriate mathematical methods to solve problems, and the use the of mathematical forms of expression to discuss, reason and give account of questions, calculations and conclusions. However, given that the task explicitly called for a justification of the student's given solution, it was generally perceived as eliciting a critical-analytical approach. The success rate for item 9a was 78% for both girls and boys, while the success rate for item 9b was 47% and 55% for girls and boys respectively. The research questions are outlined as:

1. What range of tools and forms of expression are made manifest as students interact with the task containing graphical artefacts?

2. How does tool selection and use impact on the characteristics of the task solution provided?

Task analysis

Students' solutions to task No. 9 were closely analysed with respect to: a) forms of expression (see Table 1 for typical examples of student responses) mathematical, everyday, graphical or one-word and b) tool use/sign of tool accessibilitymathematical operators and symbols. With regard to tool use seven approaches to task solving were identified, namely: i) visual comparison of graphical artefact, ii) critical -questioning the production of the graph iii) fraction iv) proportion v) percent vi) division vii) multiplication and vii) other solutions. Based on aspects of identification and critical-analytical approaches [START_REF] Olande | Students' narratives from graphical artefacts: Exploring the use of mathematics tools and forms of expression in students, graphicacy[END_REF] and aspects of [START_REF] Sfard | Thinking as Communicating: Human development, the growth of discourses, and mathematizing[END_REF]cf. Schleppegrell, 2007) categories of mathematical discourse, the author developed a coding scheme and analysed the students' solutions. The focus was on the forms of expression and mathematical tools -concepts used in working out a solution, rather than the correctness of students' solutions. The correctness or otherwise of the item solution was pegged on the test score awarded by test graders within the framework of the test situation. Significantly, within this coding scheme the perception of the forms of expression was as follows: 1) everydaythe use of causal expressions wherein aspects of the obviousness of solution are embedded. In Table 1, while the type of language used in the solution is everyday, it is apparent that there are quantities being compared: "Britain obtained most gold" and "Australia more silver" 2) mathematicalthe use of mathematical concepts and methods in the solution 3) graphicalthe use of illustrations in an attempt to amplify the visual aspects of the task. In the illustration given in Table 1 above while the solution for item b) did not receive credit award from the graders, the solution appears to be appealing to the visual faculties in a comparison exercise. Zooming in on gold or silver, the student seem to be posing the question can't you see they are different?

Results and analysis

The success rate for task No. 9 seems to indicate that students did not have as much difficulty with aspects of item 9a as compared to item 9b. This can be explained in part by the nature and the array of tools needed for effective interaction with the different items. Thus, the different forms of expression of the items were analysed. For this item, a majority (72%) of the students' solutions used everyday forms of expression, of these 89% provided successful solutions. While only 15% of the students used mathematical forms of expression, the success rate was relatively high at 95%. Students providing a one-word response to the task gave the majority of unsuccessful solutions. For this item it was observed that more than 50% of the students used visual comparison and/or the comparison of totals.

The general pattern for forms of expression for item 9b was no different for item 9a (see Table 2). It is noteworthy that students providing one-word answers and those using graphical strategies recorded rather higher rates of incorrect responses. Evidently, students' results indicate that the use of forms of expressions reflects one way or another on the success rate of their solutions. Consequently, it is of particular interest to explore the subject specific tools and forms of expression used in solving the task, and how these might have impacted on the quality of the response provided. This analysis is conducted for item 9b (see figure 2) From figure 2 it is shown that most of the solutions awarded full credit included the application of mathematical symbols and calculation. For these solutions the predominant tools of manipulation/calculation were fraction, division and percent. The solutions providing incorrect responses seem to be largely based on visual strategies.

From the results it is apparent that students using mathematical tools and forms of expression had a relatively higher success rate. The predominant tools used in solving the task were fraction, division and percentage. The frequency with which fraction and division are used is not entirely unexpected since these tools are closely related; the same applies to some extent to percent. It could be suggested that the task in some way elicits the use of fraction: It is not uncommon for mathematics teachers to use pie charts in the teaching of, or in the introduction of, the topic strand of fraction. From this approach there is a natural connection to percentage; a circle divided into two gives two halvesthis is often perceived as 50-50 (%). The results also indicate that some of the students using these tools (fraction, percentage) experienced some difficulty in application. It is worth noting that the students using multiplication obtained full credit. These students seem to employ a tool that might not be considered "self-evident" for the task. This necessitated using the tool in a creative manner, thus indicating a higher level of confidence and "procifiency" in using the tool.

For the students scoring full credit for the items, it is observed that most of them are proficient in the use of item specific tools namely, fraction, division and multiplication. From the students' solutions it seems that only a small number indicate some aspect of interrogating the purpose of the graphics. Indeed there seems to be a general scarcity of solutions communicated using subject specific forms of expression, that is, mathematical language.

Discussion

The present study sought to apply a construct focusing tools and forms expression in solving mathematics tasks containing graphical artefacts. The purpose was to outline the usage of tools and forms of expression, and the quality of the solution thereof. A deeper analysis of task No. 9 provided more insight into strategies and tools used in interacting with graphical artefacts. It is observed that it is in part the grasp of the tool in use that determines the quality of the solution given. Based on the level of confidence in tool use, it is possible for the test taker to interrogate the task from different perspectives. Based on what can be considered as overlapping tool use (see figure 2), the results are perceived as suggesting that "reading" a graphical artefact can be a complex undertaking that might involve reading the graphreading within the graphreading beyond the graph (cf [START_REF] Friel | Making sense of graphs: critical factors influencing comprehension and instructional implications[END_REF]. In the case of this task, there is indeed a different array of tools available to the students as they solve the task. However, it is communication using subject specific forms of expression that appears to be wantingthis might be an indicator that as much as the tools are available and "visible" to some of the students, the competency to apply and organize the same to produce a sound solution is a major challenge. This was observed in the case of students indicating knowledge of appropriate tools needed to solve the task, but apparently lacking the necessary skills to effectively apply the same in a problem-solving situation. Thus the observation made in the present study underscores the importance of having a solid foundation in the use of mathematical tools and forms of expression (concepts) in different settings. The importance of the use of subject specific forms of expression is also observed. For item b) there was higher correspondence in credit award between the graders and assessors as compared to item a) which did not elicit the use of the subject specific forms of expression as such. The analytical framework employed in the present study also helped to identify the strengths and weakness in students' written solutions, thus providing valuable indicators for developing classroom practice.

Given the interest in, and the focus on mathematics competencies in the Swedish mathematics classroom, the present study can also be perceived as drawing attention to the practice of assessment: if the concept of mathematical competencies implies imparting aspects of such skills as mathematicians use in the processes of mathematisation (cf Niss & Højgaard, 2011;[START_REF] Sfard | Thinking as Communicating: Human development, the growth of discourses, and mathematizing[END_REF], then assessment practice might need to be refocused to examine such use of tools and forms of expression that enable the learner to understand, and to participate in, activities within the mathematics community.

Figure 1 :

 1 Figure 1: Task No. 9 from NP Sweden

Figure 2 :

 2 Figure 2: Frequency of tool use and credit award

Table 1 : Forms of expression identified from test task

 1

	Mathematical	Everyday	Graphical	One-Word
	Britain ≈ 5/12 ≈ 0,40 40%	True: Britain obtained		False
	gold (18,8 medals)	most gold in its
	Australia ≈ 35/120 ≈ 0,30 30% gold (13,8 medals)	diagram and Australia more silver.	Yes! Britain's bit is larger
	Answer: true		Yes Spain's bit is larger

Table 2 : Forms of expression identified from item 9a

 2

		Everyday Mathematics	One-word Graphical
	Form of expression	0.72	0.15	0.11	0.02
	Correct responses	0.89	0.95	0.32	1.00
	Incorrect responses	0.11	0.00	0.60	0.00
	No response	0.00	0.05	0.08	0.00

Table 3 : Forms of expression identified from item 9b

 3

		Everyday Mathematics	One-word Graphical
	Form of expression	0.50	0.32	0.13	0.04
	Full credit	0.29	0.91	0.00	0.25
	Partial credit	0.43	0.04	0.13	0.00
	Incorrect responses	0.27	0.02	0.79	0.75
	No response	0.00	0.05	0.08	0.00

Proceedings of CERME10


Mac Schwager
email: schwager@mit.edu

Anderson Dean

Daniela Rus
email: rus@csail.mit.edu

Data-Driven Identification of Group Dynamics for Motion Prediction and Control

A decentralized model structure for representing groups of coupled dynamic agents is proposed, and the Least Squares method is used for fitting model parameters based on observed position data. The physically motivated, difference equation model combines effects from agent dynamics, interactions between agents, and interactions between each agent and its environment. The technique is implemented to identify a model for a group of three cows using GPS tracking data. The model is shown to capture overall characteristics of the group as well as attributes of individual group members. Applications to surveillance, prediction, and control of various kinds of groups of dynamical agents are suggested.

Introduction

We wish to model groups of coupled dynamic agents, such as flocks, swarms, and herds, using measured data from those agents. For example, we would like to be able to use the trajectories of people in a crowd to develop dynamical models that capture the behaviors of the crowd as a whole. This is a prohibitively complicated problem in general, however, we provide a practical solution by restricting our attention to a special model structure. We propose a difference equation model that is decentralized and nonlinear, though it is designed to be linear-in-parameters. We use the Least Squares method to fit model parameters to position data from a group of agents. Such a model may then be used, for example, to predict future states of the group, to determine individual roles of agents within the group (e.g. leaders vs. followers), or, ultimately, to control the group. This could help ranchers to manage overgrazing by livestock, or allow animal behavioral scientists to determine social roles within animal groups. In the case of people, the ability to model group behavior has numerous applications in surveillance, urban planning, and crowd control. Also, this technique can be used to design controllers for groups of engineered agents to mimic the behaviors of natural agents.

The large body of work on designing and analyzing models of flocks, swarms, and similar decentralized dynamical systems ( [START_REF] Jadbabaie | Coordination of groups of mobile autonomous agents using nearest neighbor rules[END_REF][START_REF] Reynolds | Flocks, herds, and schools: A distributed behavioral model[END_REF] and references therein) does not incorporate learning from data to tune such models. However, other related learning problems have been investigated. For example, in [START_REF] Correll | System identification of self-organizing robotic swarms[END_REF] a system identification technique is used to model global properties of a swarm of robots over time using observed data from the robots. There also has been considerable activity in learning behavioral models of individual natural agents. In [START_REF] Oh | Data-driven MCMC for learning and inference in switching linear dynamic systems[END_REF] and [START_REF] Pavlovic | NIPS*2000), chapter Learning Switching Linear Models of Human Motion[END_REF], system identification is carried out on switching linear systems to learn models of the honey bee waggle dance and human hand motion, respectively, and in [START_REF] Delmotte | Data-driven generation of low-complexity control programs[END_REF] a technique is used to find Motion Description Language (MDL) codes from observed ants.

Our work introduces two specific innovations. The first is in applying system identification to learn the parameters of a decentralized dynamical system from data. The second is in the model structure itself, which uses a novel, non-gradient field to express the interaction between each agent and its environment. We demonstrate the technique by fitting a model to GPS data from a group of three free-ranging cows. The model is validated by testing the whiteness of the residual error, and by comparing global statistics of simulations verse the actual data. We also show how the resulting model can be used to control simple robots. Recently, the technique has been validated with data from ten cows, though we do not present those results here.

Model Description

We consider a linear-in-parameters model structure with three naturally distinct parts to describe the motion of coupled physical agents moving over a plane surface. Firstly, each agent is given internal dynamics to enforce the constrains of Newtons laws. Secondly, a force3 is applied to each agent from its interaction with each of the other agents in the group. Thirdly, a force is applied to each agent as a function of its position in the environment. All remaining effects are modeled as a white noise process.

Individual Agent Dynamics

Supposing a group of m agents, the proposed model structure for an individual agent i ∈ {1, . . . , m} can be written in state-space, difference equation form as

x τ+1 i =     1 0 ∆t 0 0 1 0 ∆t 0 0 a i 0 0 0 0 a i     x τ i +     0 0 0 0 1 0 0 1     m ∑ j=1, j =i f i j (p τ i , p τ j ) + g i (p τ i ) + w τ i . (1)
Agent i's state

x τ i = [e τ i n τ i u τ i v τ i ]
T consists of its East position, North position, Eastern component of velocity, and Northern component of velocity after the τth iteration, and its position is given by, p τ i = [e τ i n τ i ] T . The time step ∆t is given by t τ+1 -t τ , and we assume it is constant for all τ. The term a i represents damping, a i = 1 for zero damping, and |a i | < 1 for stable systems. The function f i j (p τ i , p τ j ) determines the coupling force applied by agent j to agent i. The function g i (p τ i ) represents the force applied by the environment to the agent at point p τ i . Finally, w τ i is a zero-mean, stationary, Gaussian white noise process uncorrelated with p j ∀ j used to model the unpredictable decision-motive processes of agent i. Nonholonomic constraints which are often present in mobile agents, such as people, cattle, or automobiles, are neglected in this treatment, though they could be incorporated with an increase in the complexity of the model structure. Note that the force terms are only applied to affect changes in velocity in accordance with Newton's second law.

Agent-to-Agent Interaction Force

Dropping the τ superscripts for clarity, the form of the agent coupling force f i j (p i , p j ) is given by

f i j (p i , p j ) = θ 1 i j - θ 2i j p j -p i n i j , (2)
where n i j = (p jp i )/ p jp i is the unit vector along the line from p i to p j (henceforth, • will denote the 2 norm). This is the simplest of a family of force laws commonly used in computational models of physical, multi-body systems (e.g. kinetic gas models). The important feature of this family is that an agent is repulsed from its neighbor at close distances and attracted to its neighbor at far distances. To see this property clearly, examine the magnitude of (2) given by f i j = θ 1 i j -θ 2 i j / p j -p i , and shown in the left of Figure 1. After some manipulation, the sum of f i j over all neighbors j can be expressed as

∑ j =i f i j = φ f u i φ f v i θ f i , (3)
where

φ f u i = (e 1 -e i ) p 1 -p i • • • (e m -e i ) p m -p i -(e 1 -e i ) p 1 -p i 2 • • • -(e m -e i ) p m -p i 2 φ f v i = (n 1 -n i ) p 1 -p i • • • (n m -n i ) p m -p i -(n 1 -n i ) p 1 -p i 2 • • • -(n m -n i ) p m -p i 2 , and θ f i = θ 1 i1 • • • θ 1 im θ 2 i1 • • • θ 2 im T ,
permitting a slight abuse of notation since the indices j = i are excluded from the above vectors. This notation will be useful in what follows.

Environment-to-Agent Interaction Force

The agent's preference for certain paths in the environment is modeled as a nonlinear mapping from each point on the plane to a force vector felt by the agent. To this end, two networks of Gaussian basis functions are used, one for each of two perpendicular force components.

In particular, the function g i (p i ) can be written

g i (p i ) = θ u i1 • • • θ u in θ v i1 • • • θ v in    φ g 1 (p i ) . . . φ g n (p i )    , (4)
where

φ g ik (p i ) = 1 σ ik √ 2π exp(-p i -γ ik 2 2σ 2 ik
), and k ∈ {1, . . . , n}. Each Gaussian is centered at γ ik , with standard deviation σ ik , and its strength is represented by the un- known parameters θ u ik for the Eastern component, and θ v ik for the Northern compo- nent. Gaussian basis functions were chosen for their familiarity; the objective being to demonstrate the modeling approach with a minimum of complications. A number of other basis functions types could be used, including wavelets, sigmoidal functions, or splines.

It is important to note that a vector-field parameterized in this way is not a potential gradient. A potential gradient field cannot admit circulation around closed paths. 4 We introduce a non-gradient parameterization to enable circulation, as one can imagine agents intending to traverse closed orbits on the plane. For example, a cow may have a routine of passing between a water source, a shaded tree, and a grassy patch in a periodic fashion.

Figure 1, on the right, shows a plot of an example force-field parameterized in the above way. The arrows show the forces induced by the field, the black dots show the centers of the Gaussian functions, γ ik , and the red curve shows the path of an agent over the vector field. The swirling patterns evident in the vector field would be impossible if it were a gradient field.

The expression in (4) can be put into a different form to match that of (3). In particular

g i (p i ) = φ gu i φ gv i θ g i , (5)
where

φ gu i = [ φ g 1 • • • φ g n • • • 0 • • • ], φ gv i = [ • • • 0 • • • φ g 1 • • • φ g n ],
and

θ g i = [ θ u i1 • • • θ u in θ v i1 • • • θ v in ] T .
Although somewhat awkward, this form will become useful in what follows.

System Identification with Least Squares Fitting

The model that was described in Section 2 can be manipulated into a form so that its parameters can be fitted using the Least Squares method for system identification [START_REF] Ljung | System Identification: Theory for the User[END_REF]. We assume that only position measurements, p τ i , τ = 1, ..., N, are available to perform the fitting. We can eliminate u i and v i from the dynamics in (1) to provide a second order equation in the position only. Notice that from (1) we can write

p τ+1 i = p τ i + ∆t u τ i v τ i , (6)
and

u τ+1 i v τ+1 i = a i u τ i v τ i + m ∑ j=1, j =i f τ i j + g τ i + w τ i . (7)
We can solve [START_REF] Pavlovic | NIPS*2000), chapter Learning Switching Linear Models of Human Motion[END_REF] for [u τ i v τ i ] T and substitute into the right hand side of [START_REF] Reynolds | Flocks, herds, and schools: A distributed behavioral model[END_REF]. We then substitute the result back into the right hand side of (6), shifting time indices appropriately, to obtain the desired expression

p τ+2 i = p τ+1 i + (p τ+1 i -p τ i )a i + ∆t m ∑ j=1, j =i f τ i j + g τ i + w τ i .
We can use the above expression to formulate a one-step-ahead predictor. First, define the combined regressor vectors φ τ

u i = [ (e τ+1 i -e τ i )/∆t φ τ f u i φ τ gu i ], and φ τ v i = [ (n τ+1 i -n τ i )/∆t φ τ f v i φ τ gv i ]
, and a combined parameter vector

θ i = [ a i θ T f i θ T g i ] T
. By taking the expectation conditioned on the positions, substituting (3) and ( 5) for ∑ j =i f i j and g i , respectively, then making use of the combined regressor and parameter vectors we get

pτ+2 i = p τ+1 i + ∆t φ τ u i φ τ v i θ i , (8)
where pτ+2 i is the expected value of p i after τ + 2 time steps, given positions up to τ + 1, and w τ i drops out in the conditional expectation. The Least Squares method is now implemented to find the optimal model parameters. Specifically, we wish to find the parameters, θ i , to minimize the mean squared prediction error over all available time steps. The mean squared prediction error can be written

J i = 1/(N -2) ∑ N-2 τ=1 (p τ+2 i -pτ+2 i ) T (p τ+2 i -pτ+2 i ).
Substituting into J i with (8) and ( 6) yields

J i = ∆t 2 N -2 (Y i -Φ i θ i ) T (Y i -Φ i θ i ), (9)
where

Y i = [ u 2 i . . . u N-1 i v 2 i . . . v N-1 i ] T , and Φ i = [ φ 1 u i T . . . φ N-2 u i T φ 1 v i T . . . φ N-2 v i T ] T ,
and u τ i and v τ i are obtain from [START_REF] Pavlovic | NIPS*2000), chapter Learning Switching Linear Models of Human Motion[END_REF]. The least squares problem is then formulated as θ * i = argmin θ i J i (θ i ). Following the typical procedure for solving the least squares problem we find that

θ * i = [Φ T i Φ i ] -1 Φ T i Y i . ( 10
)
The right hand side of (10) consists entirely of measured data while the left hand side is the vector which represents the optimal parameters of the model. We assume that the data are rich enough that the matrix inversion in (10) is possible. The deep implications of this invertibility are discussed in [START_REF] Ljung | System Identification: Theory for the User[END_REF]. The myriad merits and deficiencies of Least Squares fitting compared with other learning methods will not be discussed in this work.

The resulting residual error in the fitting process is now used to define ŵi , so that

w τ i = u τ+1 i v τ+1 i - φ τ u i θ * i φ τ v i θ * i .
If the "true" system dynamics are represented by the fitted model, we expect to find that w i is zero-mean, stationary, Gaussian white noise. Specifically, for perfect fitting, E[w i (t)w T i (t + τ)] = δ (τ)Q i , where δ (τ) is the Kronecker delta function. Therefore, the "whiteness" of w i can be used as an indicator of the goodness of fit that has been achieved. For simulation purposes, we would assume w i is white noise and design Q i to be equal to the empirical covariance of w i .

Modeling a Group of Cows

The method presented above was used to model the dynamics of a herd of cows. Data collected from actual cows were used for fitting the model parameters and for evaluating the resulting model.

Data were collected in a single trial from three cows wearing GPS devices. For each cow, data consisted of approximately 3100 GPS position entries collected at 1Hz. The data for all animals were artificially synchronized to a common clock using a standard linear interpolation. The characteristic time scale of cow dynamics is considerably longer than 1 second, thus such an interpolation is expected to have a negligible effect on modeling results.

Least Square Results

The data were used to find model parameters as described in section 3. The panels in Figure 2 show the force coefficients F i j (p i , p j ) for the three cows. For each cow the two plots show the interaction force with each of the two other cows in the herd. Note that the forces are not necessarily pair-wise symmetric, that is, F i j = F ji in general.

The environment-to-agent vector fields are shown in the panels of Figure 3 for the three cows. The black dots show the centers of the Gaussian basis functions, γ ki , the blue arrows show the direction and magnitude of the force felt by a cow at each point, and the red dots show the position data used for regression. The Gaussian centers were spaced over an even grid containing the trajectory of the cow. If the trajectory did not enter the grid region of a Gaussian function, it was dropped from the network. This primitive pruning algorithm was used for simplicity; more complex algorithms could be employed. The Gaussian widths were chosen to be 2/3 the length of the grid space occupied by the Gaussian.

The ability of the model to capture the relevant qualities of the data was investigated by examining the covariance function of the residual error, w i , in comparison with the covariance function of the velocity as calculated by [START_REF] Pavlovic | NIPS*2000), chapter Learning Switching Linear Models of Human Motion[END_REF]. The top of Figure 4 shows the empirically calculated auto-covariance of the Eastern component of the velocity for Cow 1, and the bottom figure shows the empirically calculated auto-covariance of the corresponding residual error. The dotted lines indicate a 90% whiteness confidence interval. This means that, for each point on the auto-covariance function, a stationary Gaussian white noise process would have produced an empirical auto-covariance within the dotted lines with probability .9. Notice that the strong temporal correlation in the velocity has been greatly reduced in the residual, indicating that the model has captured a good deal of the predictable qualities of the data. Furthermore, a large majority of the residual auto-covariance points lie inside the confidence interval, giving good reason to believe that it is a "nearly white" signal. The plots for the other cows are excluded in the interests of space, though the results are similar. whiteness confidence interval, meaning that a Gaussian white noise process would have generated an empirical auto-covariance inside the interval with probability .9 at each point. Clearly the residual error is "more white" than the original velocity.

Herd Simulation

Simulation experiments were carried out with the models fitted in Section 4.1. We simulated a group of three simple mobile robots controlled to have the dynamics in (1) with the parameters found in section 4.1. The difference equations were numerically solved in a Matlab environment.

The trajectories of the robots from a typical simulation are shown in the left side of Figure 5 laid over a schematic showing the fences of the paddock where the actual cow data were recorded. The trajectories of the simulation are similar to those of the real cows. What is perhaps more surprising is that the simulated robots track the fence lines, as did the real cows. This tendency is captured solely through the agent-to-environment force field (described in section 2.3), as the model has no direct knowledge of where fence lines may lie. Furthermore, statistics were gathered for the simulated robots and compared with those from the cow data. Figure 6 shows

Conclusions

In this paper we presented a method to generate models of groups of dynamical agents (e.g. flocks, herds, and swarms) using observations of the agents' positions over time. We formulated a physically motivated difference equation model, and used the Least Squares system identification method to fit the model to data from a herd of three cows. Many avenues remain open for the extension of this work. Most importantly, the method should be tested on data from much larger groups, though such data is difficult to collect. Recently, we have validated the technique with data from ten cows with results very similar to those presented here, however data from an order of magnitude more animals would likely reveal collective behaviors not seen in smaller herds. It would also be interesting to compare this model structure with others, to use different learning algorithms, to explore recursive learning algorithms for on-line learning, or to examine analytical properties of the learned models.

Fig. 1 .

 1 Fig.1. The magnitude of the agent-to-agent interaction force is shown on the left for θ 1 = θ 2 = 1. On the right, the vector field representing the force felt by an agent at each point on the plane is shown for an example agent trajectory. The swirling patterns evident in the field are made possible by a novel parameterization.

Fig. 2 .

 2 Fig. 2. The agent-to-agent interaction forces are shown for the three cows. Each curve represents the size of the force imposed by one cow on another as a function of the distance between the cows. A positive value is attractive while a negative value is repulsive.

Fig. 3 .

 3 Fig. 3. The environment-to-agent force field is shown for the three cows. The black dots are the centers of the Gaussian functions and the red curve shows the data used for regression, marking the cow's path over the region.

Fig. 4 .

 4 Fig. 4. The empirical covariance function for the Eastern component of the velocity is shown in the top plot, and for the error residual in the bottom plot. The dotted lines indicate a 90%whiteness confidence interval, meaning that a Gaussian white noise process would have generated an empirical auto-covariance inside the interval with probability .9 at each point. Clearly the residual error is "more white" than the original velocity.

Fig. 5 .

 5 Fig. 5. The left picture shows trajectories of a team of simulated robots controlled to behave like a herd of cows. The robots use dynamical laws generated from the procedure described in this work. Their positions are laid over the fence lines of the paddock where the original data were collected, though they have no direct knowledge of fence positions. The right picture shows the actual cow data over the same time window.

Fig. 6 .

 6 Fig. 6. The bar charts compare various statistics for the actual cow data and the simulated robots. The top charts show the mean and standard deviation of the distance from one cow to the other two cows in the group. The bottom charts show the mean and standard deviation of the speed of each cow.

In this work the term "force" is used in a metaphoric sense. When we talk of a "force" we are referring to the intention of the agent to accelerate in a particular way using it's own motive mechanisms.

proof: Let Ψ (p) be a potential function and V (p) = -grad(Ψ ) its gradient field. Then curl(V ) = curl(-grad(Ψ )) = 0, thus by Green's Theorem, s V ds = A s curl(V )dA = 0, where s is any closed curve on the plane, and A s is the area enclosed by s.

ACKNOWLEDGMENTS

This work was done in the Distributed Robotics Laboratory (DRL) at CSAIL-MIT and was supported in part by NSF grant IIS-0513628 and the MURI SWARMS project. We are grateful for this support. We thank the DRL lab members for stimulating feedback and discussions. We thank Iuliu Vasilescu and to Carrick Detweiler for designing and building the hardware used to collect the animal data. The cattle data was collected in collaboration with Dr. Dean Anderson on the Jornada Experimental Range under the New Mexico State University Institutional Animal Care and Use Committee (IACUC), Number 2007-001. We thank the staff at the US Department of Agriculture -Agricultural Research Service, Jornada Experimental Range (USDA-ARS-JER) for their support in using their facilities and animals.


text:
Ann Marie Gurhy
email: amgurhy@eircom.net

Using assessment for learning to enhance mathematics education in the primary school: Irish students' perspectives

Keywords: Assessment for learning, mathematics, student voice, student engagement

Previous research suggests that assessment for learning (AfL), when used effectively, can greatly enhance student learning and achievement. However, students' views regarding AfL are often overlooked. This paper reports on one part of a broader study that investigated the interplay between lesson study, continuing professional development in AfL and mathematics teaching and learning. The paper explores the views of students in one Irish Primary School regarding their use of AfL practices during mathematics lessons over the course of one academic year. Evidence suggests the use of AfL strategies and techniques enhanced children's mathematical confidence, and improved their engagement with, and attitudes to, mathematics. By the end of the intervention, children readily used the language of AfL, engaged in AfL practices and played a more active role in their own learning and assessment, identifying the use of self-and peer-assessment as a highlight.

Background and focus of the project

Assessment, as argued by [START_REF] Gardner | Quality assessment practice[END_REF], is a "hot topic" across the entire education spectrum and rarely out of the limelight (p.103). In particular, assessment for learning, with its emphasis on learning as opposed to measurement has in recent years, according to [START_REF] Chappuis | Clear purpose: Assessment for and of learning[END_REF], "garnered the lion's share of assessment attention and established a pretty good name for itself" (p.21). This paper is about assessment for learning (AfL) in mathematics which is conceptualised using the following second generation definition generated by the Third International Conference on Assessment for Learning in New Zealand in 2009 which states: Assessment for Learning is part of everyday practice by students, teachers and peers that seeks, reflects upon and responds to information from dialogue, demonstration and observation in ways that enhance ongoing learning. (Klenowski,p.264) This definition clearly captures the key tenets of AfL, foregrounds classroom practices, highlights the notion of AfL as a bridge between teaching and learning [START_REF] Wiliam | Embedded formative assessment[END_REF], and views teachers and students as the primary agents of educational change [START_REF] Lysaght | An instrument to audit teachers' use of assessment for learning[END_REF]. Research evidence suggests that focusing on the use of day-to-day AfL is one of the most powerful ways of improving learning in mathematics classrooms and can result in significant learning gains [START_REF] Wiliam | Keeping learning on track: Classroom assessment and the regulation of learning[END_REF]. Additionally, various studies have also linked AfL to increased student motivation and self-esteem (e.g., [START_REF] Clarke | Active learning through formative assessment[END_REF], enhanced self-regulated learning and metacognitive abilities (e.g., [START_REF] Andrade | Classroom assessment in the context of learning theory and research[END_REF], and better student-teacher relationships (e.g., [START_REF] Clarke | Outstanding formative assessment: Culture and practice[END_REF]. In the Irish context, government policy emphasises the centrality of AfL in teaching and learning although few teachers have received assessment-related continuing professional development (CPD). The Department of Education and Skills (DES, 2011a) has highlighted that AfL is not used sufficiently widely in Irish schools and concerns have also been raised about teacher assessment literacy. Regarding mathematics, data from the 2009 National Assessments of Mathematics and English Reading (DES, 2010b), school inspections (DES, 2010a) and international reports (PISA, 2009) have suggested Irish students are underperforming. Indeed, [START_REF] Hislop | Applying an evaluation and assessment framework: An Irish perspective[END_REF], chief inspector with the DES, argues that it was Ireland's poor performance in PISA 2009 that precipitated publication by the Irish government of a strategy aimed at improving standards of literacy and numeracy in Ireland: Literacy and Numeracy for Learning and Life; The National Strategy to Improve Literacy and Numeracy for Children and Young People 2011-2020(DES, 2011a). This strategy is one of the most significant documents pertaining to education in the Irish context in recent years, and it is especially pertinent to this research since it has particular implications for numeracy, assessment and CPD. The Literacy and Numeracy Strategy increased the amount of time allocated to literacy and numeracy at all class levels, and set out ambitious improvement targets in English and mathematics as measured on standardised tests to be achieved by 2020. Compulsory standardised testing in English and mathematics changed from two to three points in the primary cycle (second, fourth and sixth classes), with mandatory annual reporting of aggregated results to the DES to facilitate collation of a national picture of achievement. Additionally, schools must use these results as part of "robust self-evaluation" (p.40) and to prepare three-year improvement plans for the promotion and improvement of numeracy and literacy. Results also have to be given to Boards of Management and parents. Nevertheless, while various scholars (e.g., [START_REF] Leahy | From Teachers to Schools: Scaling up Professional Development for Formative Assessment[END_REF] agree that AfL, when used effectively, is a warranted strategy that can improve student learning and achievement, and Irish government policy emphasises the importance of using AfL in teaching and learning, apart from some small-scale studies (e.g., [START_REF] Lysaght | From Balkanisation to Boundary Crossing: Using a Teacher Learning Community to Explore the Impact of Assessment on Teaching and Learning in a disadvantaged School[END_REF], research into AfL in the Irish context remains sparse, particularly in the area of mathematics. Furthermore, there has been little or no research investigating what is going on in the hearts and minds of learners while engaging in AfL practices. This study seeks to address this gap in the field by specifically investigating the following research question presented as a hypothesis:

The use of AfL strategies and techniques, and the adoption of AfL principles, would enhance children's mathematical confidence, and improve their engagement with, and attitudes to, mathematics.

Methodology

This is a practitioner action research mixed methods explorative case study, which operated within the pragmatic paradigm. Over the course of one academic year, it investigated the impact of AfL practices on the teaching and learning of mathematics at fourth-class level in one primary school in the Republic of Ireland. Specifically, it explored how the use of AfL principles, strategies and techniques affected students' attainment on standardised mathematics tests and their dispositions towards mathematics. Additionally, the research investigated the potential of lesson study (LS) as a vehicle of collaborative professional learning in AfL and considered the impact engaging in LS had on teachers' skills, knowledge, and use of AfL, and their beliefs towards AfL as a form of assessment.

A key part of this study was the use of peer-to-peer learning as a vehicle of continuing professional development (CPD). Meeting after school on twenty-three occasions over the course of the intervention the teachers learned about AfL strategies and techniques on a phased basis before implementing them in their mathematics lessons prior to the next meeting. The AfL strategies used were as follows: learning intentions and success criteria; questioning and classroom discussion; feedback; self-and peer-assessment. Additionally the teachers learned about and implemented more than twenty AfL techniques, for example, rubrics, think-pair-share, two stars and a wish, fist-to-five, learning logs, ABCD cards and comment-only marking (For further details see [START_REF] Wiliam | Embedded formative assessment[END_REF].

Site selection and research participants

This research project took place exclusively in the school where I teach, Scoil na nAingeal (pseudonym), from September 2012 to June 2013. It is a vertical, urban, girls-only Primary School in the Republic of Ireland with an enrolment of 438 students and an all-female staff. The profile participants included all students enrolled in fourth class for the academic year 2012-2013, 51 girls, the average age of whom was ten years in September 2012. Classes were pre-formed, intact groups and so, according to [START_REF] Creswell | Research Design: Qualitative, quantitative, and mixed methods approaches[END_REF], random sampling was not considered appropriate. Three teachers participated in the project, the two fourth-class teachers and one member of the Special Educational Needs (SEN) team.

Data collection and analysis

Both quantitative and qualitative data were collected to decide whether to accept or reject the above research hypothesis. This aided triangulation, enhanced the study's findings and enabled better understanding of the research problem. The quantitative data were collected using one scale of an instrument developed by the researcher called the Attitude to Mathematics Questionnaire (ATMQ). This scale was labeled the ATMQ-TIMSS since it used statements 8a-8h of the Trends in International Mathematics and Science Study (TIMSS) 2007 Grade Four Student questionnaire verbatim. These statements examined "students' general attitudes towards mathematics" and "their self-confidence in learning mathematics" (Mullis, Martin & Foy, 2008, p.173). Leaving the TIMSS questionnaire unchanged facilitated comparative analysis with national and international data, and ensured the reliability of this scale (median reliability coefficients across all TIMSS countries at fourth grade was 0.83, Mullis et al., 2008, p.401). Fifty children completed the ATMQ-TIMSS pre-and postintervention. Scoring was done using a four-point Likert scale, with response options ranging from 'Agree a lot' to 'Disagree a lot'. Raw data from the ATMQ-TIMSS were coded or categorised, recorded and prepared in Microsoft Excel and then imported into the software package, SPSS 21, where dependent (paired) samples t-tests were conducted in order to compare the scores of the same participants at Time 1 and Time 2, to ascertain whether or not the intervention had an impact [START_REF] Pallant | SPSS Survival manual: A step by step guide to data analysis using IBM SPSS[END_REF].

The qualitative data comprised transcripts and video from focus groups (FG) interviews, teachers' learning logs (TLL), students' learning logs (LL), and the researcher's journal. In addition to aiding triangulation, these data supplemented the quantitative data by facilitating a more in-depth analysis of students' views about using AfL strategies and techniques in their learning of mathematics. [START_REF] Braun | Using thematic analysis in psychology[END_REF] six-step approach to thematic analysis was used to guide analysis of each data item individually and subsequently the complete qualitative data set.

Results and discussion

Quantitative data

Results from analysis of the ATMQ-TIMSS indicated that there was a statistically significant difference between pre-test (M = 2.03, SD = 0.73) and post-test scores (M = 1.56, SD = 0.42; t (49)= 5.09, p < .001), i.e. post-test scores indicated more positive attitudes towards mathematics (mean values in post-test scores were closer to 1 = Agree a lot). The magnitude of the difference between the pre-and post-test means can be interpreted as being large (eta squared = .35). Table 1 presents combined 'agree' percentages for pre-and post-results for each statement in the ATMQ-TIMSS scale.

Table 1: ATMQ-TIMSS Scale

A more detailed exploration of the eight statements from the scale revealed that following the intervention the combined 'agree' percentages scores for the five positively-worded statements (a, b, d, f, h) had increased by between 12 and 20 percentage points while the percentage of students agreeing with the three negatively-worded statements (c, e, g) had decreased by between 14 and 20 percentage points. This suggests that children believed the intervention had positively impacted their general attitudes towards mathematics and their self-confidence in learning mathematics. Specifically, regarding the three statements which measured students' general affect towards mathematics (d, g, h), results indicated that following the intervention almost 100% of the participants agreed a little or a lot with these three statements (Table 1).

Note. % Agreeing = Agree a lot +Agree a little

Italicised text highlights statements that were recoded.

For discussion purposes, percentage scores for the three PATM (Positive attitude towards mathematics) statements and the four SCM (Self-confidence in learning mathematics) statements were amalgamated to provide a composite percentage score for these scales which could then be compared with TIMSS data.

*Regarding statement b, Clerkin (personal communciation, April 15, 2015) suggests it was probably originally intended to be part of the PATM scale but following factor analysis was found not to represent positive affect in the same way as the other scale items and so was excluded.

Qualitative data

Five main themes were identified following the qualitative analysis but due to space limitation these are discussed only briefly here.

Enjoying the AfL journey

When describing their experience of using AfL practices in mathematics, the children regularly used words such as 'fun' and 'enjoyment'. Their enthusiasm regarding using AfL can be succinctly summarised by Maria's comment: "I love doing AfL and I would like to continue doing it" (LL, 13/02/2013).

Growing positivity and self-confidence in mathematics

It is clear from the data that at the end of the study students reported a growing positivity towards and increased self-confidence in mathematics that they attributed to their use of AfL. Some believed that using AfL practices made mathematics easier and increased their liking of mathematics: "It makes maths so much more fun, 'cause like in third class I used to hate maths and then like now that there's all these different strategies, it just makes maths so much easier" (Sophie, LL, n.d.).

A changed classroom dynamic

Scholars suggest (e.g., [START_REF] Hayward | Assessment and learning: The learner's perspective[END_REF], engaging in AfL practices in the spirit in which it is meant impacts learning and teaching and acts as a leverage for change in classroom practices, and roles and relations. Children in this study, while recognising the teacher as overall guide and arbiter in the classroom, also identified teachers as learners. Additionally, they were beginning to monitor their own learning and to evaluate their progress. For example, Chloe commented, "I think it's helped me, that it's not letting the teacher correct all your work, that you, you kind of have to check it, and you have to, because there's some silly mistakes that you could make" (FG1). Others displayed an increased confidence in their ability to assess, both themselves and others: "I think that when you correct it yourself, or for your friend, you know what you're correcting, you know why you're correcting, what you did wrong, so you know what you're doing" (Emma, FG1). This suggests that through the process of engaging in AfL strategies and techniques over the course of one academic year, these students were becoming more independent learners, accepting the responsibility this brought, and were moving towards self-regulated learning.

Peer-and self-assessment: -a highlight for children

Using peer-and self-assessment was undoubtedly a highlight for children participating in this study as this comment illustrates: "I love self-assessment and peer-assessment" (Hollie, LL, n.d.). The following comment from Ruby demonstrates that by the end of the intervention the children had developed a good understanding and appreciation of peer-and self-assessment:

I thought the self-assessment was excellent because we were judging ourselves and could learn from our mistakes. Peer-assessment was brilliant for your partner or pair could judge your work and spot mistakes that you might not have spotted yourself. (Ruby, LL, 12/06/2014)

The children believed that self-and peer-assessment not only enhanced their learning but that it was also fun. Similar to research by [START_REF] Topping | Peers as a source of Formative Assessment[END_REF], trusting your peers was mentioned by a number of participants as integral to good peer-assessment practice. For example, Hollie remarked: "I think peer-assessment is the best because you get to like trust your friends more, so they'll be more honest with you in the future" (FG1).

Unexpected insights

The qualitative data also revealed some unexpected insights about students' perspectives regarding AfL. The children revealed they used the learning intentions and success criteria from their mathematics lessons to help them when doing their mathematics homework. Additionally, they described how they appropriated the AfL strategies and techniques to help them learn in other subject areas. Many children reported they liked using rubrics since they scaffolded the assessment process. Maria wrote: "My favourite thing about the AfL was using the rubric" (LL, 12/06/2013) and later explained why:

I like using the rubric because when we were first going to do peer-assessment I was like 'oh God, what will I say was wrong?' and 'I don't know what to do here', but then you showed us the rubric, and I was like 'oh, it's ok', ... with the rubric, it tells you what you're supposed to assess, (FG2)

Conclusions

While acknowledging the limitations of this study, such as the fact that it takes place in a single school and employs a case study strategy with convenience rather than non-probability sampling, it nevertheless contributes new insights regarding AfL and mathematics, especially from students' perspectives. It demonstrates that children in primary school are capable of engaging in AfL practices, including peer-and self-assessment, and want to do so. Findings suggest that using AfL practices enhanced children's mathematical confidence, and improved their engagement with, and attitudes to, mathematics.

This study is important since, to date, little empirical research has been done into the effects of AfL practices on students' mathematics learning in the Irish context at primary level. Additionally, this study can supplement research done by academics regarding AfL since it provides a practitioner researcher's perspective of the field, thus "inside-outside" [START_REF] Cochran-Smith | Inside/Outside: Teacher research and knowledge[END_REF]. Notwithstanding, perhaps the most significant contribution made by this research is that it provides a unique opportunity to listen to, and contemplate, the voice of young learners as they discuss their experiences of using AfL practices in their mathematics learning.

Proceedings of CERME10