text:
Paul Andrews
email: paul.andrews@mnd.su.se

Constantinos Xenofontos
email: xenofontos.c@unic.ac.cy

Beginning teachers' perspectives on linear equations: A pilot quantitative comparison of Greek and Cypriot students

Keywords: Linear equations, comparative research, Cyprus, Greece, teacher education

In this paper we examine aspects of beginning primary teachers' understanding of linear equations. First-year teacher education students on a programme in Cyprus (12 Greek and 21 Cypriot) were shown a solution to the equation x + 5 = 4x -1 comprising four rows of mathematically correct algebra but no commentary. They were asked to explain, in writing, the solution to α friend who had missed the lesson in which such equation solving processes had been taught. Analyses found almost all students, irrespective of nationality, writing about knowns and unknowns before offering a 'change the side and change the sign' rule. However, a major difference was that Cypriot students' accounts typically included an objective for the equation solving process, which the Greek students' did not.

Introduction

The topic of linear equations occupies an important position in students' learning. It "stands on the border between mathematics as concrete and inductive and mathematics as abstract and deductive", offering "one of the first authentic opportunities for them to connect their understanding of arithmetic to the symbolism of mathematics" (Andrews & Sayers, 2012, p.476). Yet, it is a difficult topic to teach well, because when learning arithmetic, learners typically come to see the equals sign as an instruction to operate [START_REF] Kaput | Algebra in the early grades[END_REF]. This operational perspective (McNeil & Alibali, 2005) creates few problems with respect to arithmetical equations, with the unknown in one expression, because it supports a process of operation reversal [START_REF] Herscovics | A cognitive gap between arithmetic and algebra[END_REF][START_REF] Kieran | The learning and teaching of school algebra[END_REF]. However, non-arithmetical equations -the unknown in both expressions -requires a relational [START_REF] Kieran | The learning and teaching of school algebra[END_REF] understanding of the equals sign as an assertion of equality between two expressions [START_REF] Alibali | A longitudinal examination of middle school students' understanding of the equal sign and equivalent equations[END_REF][START_REF] Filloy | Solving equations: The transition from arithmetic to algebra[END_REF] in order that they can operate on the unknown as an entity. In short, students find equation solving problematic because operational perspectives on expressions like 3x+1 prevent their being construed as objects subject to, in relational terms, operations themselves [START_REF] Kieran | Algebraic thinking in the early grades: What is it?[END_REF]. Furthermore, to compound students' difficulties, teachers' practices frequently collude in the maintenance of an operational perspective on the equals sign [START_REF] Haimes | The implementation of a 'function' approach to introductory algebra: A case study of teacher cognitions, teacher actions, and the intended curriculum[END_REF][START_REF] Harel | Attention to meaning by algebra teachers[END_REF][START_REF] Stephens | What "counts" as algebra in the eyes of preservice elementary teachers[END_REF], highlighting a need to evaluate the equations-related understandings that beginning teachers bring to their courses. This reflects the aim of this pilot study and an atypical approach, which is described below.

Perspectives on the teaching and learning of linear equations

Typically, research on equation solving has focused on approaches to the solution of nonarithmetical equations, not least because their solution poses few conceptual difficulties. In this respect, the most widely criticised is redistribution, a rote-learned, change the side, change the sign procedure (Nogueira de [START_REF] Nogueira De Lima | Procedural embodiment and magic in linear equations[END_REF], focused on transposing the equation so that the unknown finishes on the left-hand side and a value on the right [START_REF] Filloy | Solving equations: The transition from arithmetic to algebra[END_REF]. The unknown's arbitrary leftwards movement perpetuates operational conceptions of the equals sign and fails to support students' understanding that such movement does not change the equation's equality [START_REF] Capraro | Algebraic equations: Can middle-school students meaningfully translate from words to mathematical symbols[END_REF]. Other approaches, like trial and improvement, support an understanding of the relational nature of the equals sign and the role of the unknown in context [START_REF] Knuth | Middle school students' understanding of core algebraic concepts: Equivalence & variable[END_REF]. However, while it may be an appropriate initial strategy in a teaching sequence, it is inefficient and does not support the learning of general equation solving strategies [START_REF] Filloy | Solving equations: The transition from arithmetic to algebra[END_REF]. Other approaches(see Dickinson and Eade, 2004;[START_REF] Fong | Solving algebraic word problems[END_REF], present the equation as two rows of mathematical objects, one laid on top of the other, as in the representation of 2x + 10 = 4x + 2 shown in Figure 1. Here, the authors claim, students can see easily how it reduces to 2x + 2 = 10; an equation is amenable to an operation reversal procedure. However, while such approaches be procedurally helpful and support students' understanding of equations as manipulable objects, they may hinder students' understanding of the invariance of the solution. Moreover, despite Dickinson and Eade's (2004) optimistic arguments otherwise, they remain problematic with respect to negative coefficients [START_REF] Marschall | Polish teachers' conceptions of and approaches to the teaching of linear equations to grade six students: An exploratory case study[END_REF].

Finally, in studies of teachers' unprompted approaches to equation solving, the balance scale has been the most widely reported, being the approach of choice in case studies from, for example, Canada [START_REF] Haimes | The implementation of a 'function' approach to introductory algebra: A case study of teacher cognitions, teacher actions, and the intended curriculum[END_REF], Finland, Flanders and Hungary [START_REF] Andrews | Teaching linear equations: Case studies from Finland, Flanders and Hungary[END_REF], New Zealand [START_REF] Anthony | Solving linear equations[END_REF] and Poland [START_REF] Marschall | Polish teachers' conceptions of and approaches to the teaching of linear equations to grade six students: An exploratory case study[END_REF]. Here the solver manipulates, through addition or subtraction, weights on scale pans, while keeping the scales in balance. Its advocates argue that it helps students see the equation as a whole and not an instruction to operate [START_REF] Warren | Young children's ability to use the balance strategy to solve for unknowns[END_REF]. Moreover, it supports an understanding of the need to do the same to both sides [START_REF] Anthony | Solving linear equations[END_REF] and underpins the symbolic foundations of later algebraic formalisms [START_REF] Andrews | Opportunities to learn in the Budapest mathematics classroom[END_REF]. Systematic attempts to evaluate the balance's efficacy have shown that it helps students to understand the principles of equations, solve non-arithmetical equations with understanding, particularly from the perspective of doing the same thing to both sides [START_REF] Araya | The effect of analogies on learning to solve algebraic equations[END_REF][START_REF] Warren | Young children's ability to use the balance strategy to solve for unknowns[END_REF] and facilitates students' acquisition of an appropriate vocabulary [START_REF] Vlassis | The balance model: Hindrance or support for the solving of linear equations with one unknown[END_REF]. Its critics argue that it cannot represent negatives in anything but a contrived way [START_REF] Pirie | The equation, the whole equation and nothing but the equation! One approach to the teaching of linear equations[END_REF], a criticism supported studies showing teachers simulating the tying of helium filled balloons to scales to counter the weight of objects in the scale pans [START_REF] Anthony | Solving linear equations[END_REF].

In this paper we explore how students following an initial primary teacher education programme in Cyprus construe non-arithmetical linear equations. Due to Greek being the language of instruction, the programme includes both Cypriot and a high proportion of Greek students.

Methods

Shortly after the start of their course and before they had been exposed to university mathematics teaching, students were shown a solution to the equation x + 5 = 4x -1 and asked to write a short account of how they would explain it to someone who had missed the lesson in which it was introduced. The solution, with no additional narrative, was presented as follows

x + 5 = 4x -1 5 = 3x -1 6 = 3x 2 = x A non-arithmetical equation was used for several reasons. Firstly, it could not be solved by means of a reversal of operations. Secondly, it should expose, in ways that an arithmetical equation could not, the underlying equations-related conceptions students bring to their courses. Thirdly, it would uncover the extent and depth of their equations-related procedural knowledge as, at each step, they would need not only to interpret and explain the solver's hidden thinking but decide what would need to be made explicit to the unknown learner. It was believed that this would be a more effective means of uncovering students' familiarity and understanding of the topic than a conventional test item and expose any pedagogical predisposition they bring to their course, as a result of their previous schooling.

Analysis

Students' transcripts were subjected to a constant comparison analysis [START_REF] Fram | The constant comparative analysis method outside of grounded theory[END_REF], whereby a transcript was read and re-read to identify different equations-related understandings. This was followed by the next transcript being read and re-read in order to find further evidence of the original codes and any new ones not seen in the first. If new categories were found then the earlier transcript was re-read in case they had been previously missed. This process continued for all 33 transcripts, a number typically thought sufficient to achieve categorical saturation, and yielded seven categories of understanding, which can be seen in Table 1.

With respect to demonstrating the emergence of these categories, we turn to Ekaterini, a female Cypriot student, who wrote that To solve this exercise we have to first set apart the known from the unknown numbers. The known numbers are the ones that don't include a letter, as for example, 5 and -1. Unknown numbers are the letters or the numbers that are accompanied by a letter, for example, 4x and x.

Ekaterini's comment about separating the known from the unknown implies an implicit objective for equation solving; in essence, solving an equation entails precisely that. In the same sentence can also be seen evidence of her awareness of the unknown and its role in equation solving. In the second and third sentences she goes further and defines an unknown. She then wrote (her parentheses):

To separate the two, the known numbers should be on one side of the equation, for example, on the left side, as we solved it in the class, while the unknown numbers should be on the other side. Later, I added 5 + 1 (whose sign has changed because it moved to the other side of the equation) and subtracted 4xx (again the sign changed because x has moved to the other side of the equation and when there is x or y alone this means 1x or 1y). Finally, I reached 6=3x and so I divided 6 by 3 so that x is equal to 2.

Throughout this paragraph runs a rote procedure for equation solving invoking two simple instructions. The first is that knowns must be moved to one side of the equation and unknowns to the other. The second is that when an object moves from one side of an equation to the other its sign changes. Finally, having achieved this objective, an understanding of the role of inverse operations is invoked to divide the total of the knowns by the coefficient of the unknowns. Within this procedure, as with many of the students' suggestions, is evidence of flexibility in that it does not matter to which side of the equation which type of object travels, implying that it may be a matter of convenience.

Articulating an awareness of the unknown 31

Defining what is meant by an unknown 15

Offering an implicit objective for equation solving 18

Offering an explicit objective for equation solving 1

Offering a rote procedure for equation solving 29

Articulating an understanding of the role of inverse operations 26

Offering an unspecified process for equation solving 4

Table 1: The seven categories of understanding elicited from the data

With respect to the remaining two categories, only one student offered an explicit objective for equation solving. In this case, Irene, a Cypriot female, wrote that the "question in this equation is to find 'x' and what value it has". Finally, four students wrote of an unspecified procedure. For example, Moira, a Greek female wrote, "I would tell the student that we separate the known from the unknown numbers and then make the calculations". From her comment we inferred three perspectives on equation solving; an implicit objective tied to separating the known from the unknown, an awareness of the unknown and an unspecified procedure. The seven categories of response, along with their respective frequencies, can be seen in Table 1.

Results

The figures of Tables 2 and3 show which of the seven categories identified by the constant comparison analysis were found in the accounts of the Greek and Cypriot students respectively. From these can be inferred both similarities and differences. With respect to the former, several similarities were identified. Firstly, with a single exception in each country, students' explanations showed an explicit awareness of the unknown. Secondly, a very high proportion of students from both countries -only four students did not -offered a rote procedure for solving the equation and of these, all focused on the mantra, change the side change the sign. Thirdly, only five students did not write in ways indicative of their understanding inverse operations, although this was typically seen with respect to explaining how the solution is reached from the point where 6=3x, as in Chloe's comment that "after reaching 6=3x, we divided 6 by 3 so that x will be by itself". Fourthly, around half of all students in each group defined what they meant by an unknown The figures of Tables 2 and3 show also that Greek students' data yielded fewer codes per student, 3.17, than their Cypriot counterparts mean of 4.10. Indeed, at the upper end of the spectrum the accounts of seven Cypriot students, A, D, H, O, Q, S and U, yielded five categories of response, compared with that of just one, F, Greek student. At the lower end, the accounts of four Greek students, A, D, H and J, yielded only two categories each, compared with zero Cypriot students. These differences were statistically significant in two ways. Firstly, a t-test showed differences in the mean number of codes were unlikely to be due to chance (t = 2.95, p = 0.006). Secondly, a chisquare test performed on the data in Table 4 confirmed that variation in the number of codes yielded by each of the Greek students and Cypriot students respectively were unlikely to be due to chance (χ2 = 9.15, df = 3, p = 0.027). This difference in the codes, we argue is likely to be a consequence of differences in how the two systems introduced their students to linear equations; a possibility warranted by, for example, evidence that Cypriot students tended to specify objectives in their accounts in ways that their Greek colleagues typically did not. Finally, of the eight students whose accounts yielded five codes, seven yielded the same five. That is, they indicated an awareness of and defined the unknown; they offered implicit objectives and a rote procedure alongside an awareness of inverse operations related to division. In short, the most complete responses were typically the same.

Number

Discussion

The aim of this pilot study, the quantitative analyses for which are presented in this paper, was to explore the equations-related understanding primary teacher education students bring to their courses and, in so doing, evaluate the effectiveness of a simple to implement tool for later comparative use. In this instance comparison was made possible by the fact that courses in Cyprus are taught in Greek, making them accessible to Greek students. The results are methodologically encouraging but, acknowledging the fact that all respondents were prospective teachers, mathematically worrying, albeit with some qualifying strengths.

Methodological encouragement stems from the evidence that students responded positively to the invitation and produced written accounts sufficient to expose their perspectives on or conceptions of linear equations. It was also encouraging that the tool was able to discriminate between the two cultural groups in its highlighting similarities and differences in students' accounts that, we infer, reflect systemic differences in the ways in which linear equations had been experienced by these two sets of students as learners of school mathematics. In short, the tool proved fit-for-purpose.

The mathematical disappointment derives in part from the lack of any evidence of students holding a relational [START_REF] Kieran | The learning and teaching of school algebra[END_REF] conception of the equals sign, in that nothing said by any student indicated an understanding of the equals sign as an assertion of equality between two expressions [START_REF] Alibali | A longitudinal examination of middle school students' understanding of the equal sign and equivalent equations[END_REF][START_REF] Filloy | Solving equations: The transition from arithmetic to algebra[END_REF]. Mathematical disappointment also derives from the very high proportion of students in both countries who seemed to construe equation solving as a rote process of 'change the side and change the sign'. That is, the majority of students appeared to have a procedural rather than a conceptual perspective on equations in which symbols are moved around "with a kind of additional 'magic' to get the correct solution" (Nogueira de Lima & Tall, 2008, p.4). Indeed, even those students whose accounts yielded the most categories of response presented procedural perspectives with implicit objectives and rote procedures. However, in contrast with international research showing the balance as teachers' preferred representation [START_REF] Andrews | Teaching linear equations: Case studies from Finland, Flanders and Hungary[END_REF][START_REF] Anthony | Solving linear equations[END_REF][START_REF] Haimes | The implementation of a 'function' approach to introductory algebra: A case study of teacher cognitions, teacher actions, and the intended curriculum[END_REF][START_REF] Marschall | Polish teachers' conceptions of and approaches to the teaching of linear equations to grade six students: An exploratory case study[END_REF], no reference to the balance was made by any student. Indeed, with the exception of their implicit awareness of inverse operations, which we discuss below, nothing written by any student indicated a narrative based on performing the same action to both sides of the equation.

Despite the negatives, there were some interesting positives. The majority of students, particularly the Cypriot, indicated an awareness of the role of inverse operations. In every case this occurred at the point in the solution where the step connecting 6=3x to 2 = x was discussed. Here, students indicated, albeit implicitly, an awareness that division was the inverse operation to invoke, insights that seem to confound the mechanical procedure of 'change the side, change the sign'. Also, despite the highly procedural nature of their accounts, only two students, one from each country, did not demonstrate an awareness of the unknown. That is, students not only used an appropriate vocabulary but were generally aware of the function of the unknown in the equation solving process. Indeed, around half of all students from both groups offered a definition, typically implicitly, of the unknown, as in Carissa's account in which she wrote, "I separated the known from the unknown numbers. Unknown x + 5 (known) = unknown 4x -1 (known)".

Finally, Cypriot students' accounts yielded more response categories than their Greek colleagues, typically due to their tendency to offer objectives, again implicit, which their Greek colleagues did not. We speculate that such a difference may reflect cultural teaching norms; while the evidence of these students' accounts indicates the outcome of procedural teaching this particular finding suggests that Cypriot teachers may warrant their procedural approaches to a topic in ways that their Greek colleagues may not. In sum, the responses from both sets of students indicated deep-set procedural perspectives on linear equations. It was clear that they had understood the task in that all their explanations were valid but, with both the Cypriot and the Greek curricula advocating that students learn a relational mathematics, the implications of this study for teacher education are profound.

Figure 1 :

 1 Figure 1: A representation of 2x+ 10 = 4x + 2

Table 3 : Distribution of Cypriot students' equations-related insights

 3

Table 5 : Distribution of total number of codes per student per country

 5

			of codes per student
		2	3	4	5
	Greek	4	3	4	1	12
	Cypriot	0	5	9	7	21
		4	8	13	8	33

Proceedings of CERME10

. 2 4 3 2 4 5 3 2 3 2 4 4 3.17 Table 2

: Distribution of Greek students' equations-related insights

With respect to differences, only two Greek students offered any sense of objective, albeit implicit, for the equation solving process compared with 17 of the Cypriot, of which only one, student F, offered an explicit goal.

Explicit objectives 1 1 5 4 4 5 3 4 3 5 3 3 4 3 4 4 5 4 5 4 5 4 5 4.1


domain: math shs

text:
Introduction to the papers of TWG11:

Comparative studies in mathematics education

Paul Andrews, Eva Jablonka, Miroslawa Sajka and Constantinos Xenofontos As with earlier CERMEs, TWG 11 adopted an eclectic perspective in its interpretation of comparison as referring to any study that documents, analyses, contrasts or juxtaposes cross-cultural or crosscontextual similarities and differences across all aspects and levels of mathematics education. In this way the TWG aimed to encourage critical but supportive discussions around a number of predetermined and emergent themes.

A recurrent but very productive aspect of this working group has been the relatively small number of paper presentations. This year ten papers and three posters created space not only for colleagues to share their research in detail but allowed participants to engage in lengthy and inclusive discussions on the nature of comparative mathematics education research and the means by which it can be meaningfully and rigorously undertaken. As is shown below, the different paper contributions fell naturally into four themes, each reflecting significant substantive and methodological variation.

Firstly, two papers, with very different foci and methodological conceptualisations, framed a discussion on the importance of identifying an appropriate unit of analysis. This issue, while of importance in all research, is particularly foregrounded in comparative research. One of the two papers, Sajka's eye-tracking examination of different response groups' visual attention to the statement of a mathematical problem showed that such groups respond in culturally conditioned ways. The second of these two papers, Clarke, Mesiti, Cao and Novotna's study of the culturallylocated variation in the vocabulary of the typical school mathematics classroom, foregrounded the importance of understanding that much mathematic didactics vocabulary may be culturally unique. Importantly, both studies exposed hitherto unconsidered demands with respect to what is actually being analysed.

A second theme could be found in the three papers construed as having a tacit focus on beliefs. Firstly, Andrews and Xenofontos presented a quantitative analysis of Cypriot and Greek initial teacher education students' understanding of a hypothetical solution to a non-arithmetical linear equation. They found considerable emphases on rote solution methods although Cypriot students were more articulate on the matter than their Greek colleague. Secondly, Koljonen's case study of one Swedish teacher's deployment of the teacher guide associated with a translated Finnish textbook showed how teachers' responses to such materials, irrespective of any instructional material they contain, are interpreted by culturally-determined expectations of what is appropriate. Thirdly, Nosrati and Andrews interviewed Norwegian and Swedish upper secondary students about their experiences of school mathematics and found a dominant perception that the purpose of school mathematics is to prepare students for a successful economic engagement with the world. All three studies highlighted well the extent to which mathematics classroom participants' perspectives are informed by culturallylocated beliefs.

A third theme, also reflected in three papers, concerned mathematics teacher knowledge. In the first of these, Kingji-Kastrati, Sajka and Vula used an extant test to examine Kosovar and Polish teacher education students' knowledge of fractions. In the second, Tchoshanov, Quinones, Shakirova, Ibragimova and Shakirova used a battery of TIMSS-derived test items to examine differences in US and Russian lower secondary teachers' content knowledge. Both studies highlighted substantial differences in the content knowledge of the groups under scrutiny. Finally, Xeonfontos and Andrews examined Cypriot and Greek students' didactical explanations of the same hypothetical solution to a non-arithmetical linear equation as discussed above. In this case, Cypriot students' explanations were more didactically robust than their Greek colleagues. All three papers confirmed the extent to which mathematics content knowledge is a not the culturally independent body of knowledge assumed by international studies of student knowledge.

The fourth theme drew on two papers framed by the anthropological theory of didactics (ATD). Firstly, Modeste and Rafalska's drew on ATD's of didactical transposition, or the transformation of academic knowledge to that knowledge taught, to highlight differences and similarities in the presentation of algorithmics in the curriculum materials of Ukraine and France. Secondly, Asami-Johansson, Attorps and Laine exploited ATD's concept of praxeology, which provides the methods for solving a domain of problems (praxis) and a structure (logos) on the discourse those methods, to compare the practices of case study teacher educators in Japan, Finland and Sweden. The two studies, in addition to highlighting substantial differences between the cultural groups under scrutiny, showed how different elements of ATD can be productively employed in cross-cultural studies.

Finally, three posters, each with different foci and methodologies, were presented. Haara, Bolstad and Jenssen, as preparation for a later Norwegian study, presented a research review on mathematical literacy in school. Istúriz, González-Ruiz, Diego-Mantecón, Recio, Búa, Blanco, González and Polo reported on an Erasmus project in which students in different countries develop activities to integrate art into STEM activities. Finally, Tesfamicael, Botten and Lundeby presented a comparative analysis of Norwegian and Ethiopian textbooks presentation of relations and functions.

Overall, the papers and posters presented to the group reflected not only cultural diversity but also methodological pluralism. For example, studies included those that were informed by a priori theoretical assertions and those that were not. There were equal numbers of qualitative and quantitative studies focused on a range of aspects of children's and teacher education students' learning of mathematics. All studies confirmed the extent to which mathematics and its teaching and learning are culturally normative.

Proceedings of CERME10


E Bauer

R Belkhou

S Cherifi

A Locatelli

A Pavlovska

N Rougemaille

Magnetostructure of MnAs on GaAs revisited

The ferromagnetic to nonferromagnetic (α-β) phase transition in epitaxial MnAs layers on GaAs(100) is studied by x-ray magnetic circular dichroism and x-ray magnetic linear dichroism photoemission electron microscopy in order to elucidate the nature of the controversial nonferromagnetic state of β-MnAs. In the coexistence region of the two phases the β phase shows a clear XMLD signal characteristic of antiferromagnetism. The nature and the possible causes of the elusiveness of this magnetic state are discussed.

I. INTRODUCTION

While the ferromagnetism (FM) of the hexagonal α phase of MnAs [Fig. 1(a)] has been firmly established for nearly 100 years, 1 the magnetic state of the orthorhombic β phase [Fig. 1(b)], usually described as paramagnetic (PM), has been the subject of considerable controversy.

The controversy started with the suggestion of Guillard 2 that the β phase is antiferromagnetic (AFM) because of the increase of the susceptibility with temperature, dramatically violating the Curie-Weiss law characteristic for paramagnetism. Subsequent neutron diffraction studies, however, could not detect AFM long range order, 3 thus apparently eliminating Guillard's proposal. In the search for an explanation of the anomalous magnetic behavior a number of proposals have been made. Bean and Rodbell 4 developed a thermodynamic theory within the molecular field approximation involving a strain dependence of the exchange energy with which they could explain the first-order FM-PM α-β transition. Goodenough et al. [START_REF] Goodenough | Proceedings of the International Conference of Magnetism[END_REF] introduced a model in which the crystal changed from a high spin state in the α phase to a low spin state in the β phase which allowed them to give a qualitative explanation of the temperature dependence of the susceptibility. Earlier Kittel [START_REF] Kittel | [END_REF] had already suggested an exchange inversion model in which the abrupt volume change during the phase transition is connected with a change of the sign of the exchange constant via exchange magnetostriction. None of these models could explain all aspects of the two phases and of the transition between them. For example, neutron diffraction studies showed that the β phase is not a low spin phase. 7 Other neutron diffraction data indicated that there are spin correlations within distances of less than 20 Å (Ref. 8) but not with AF coupling between nearest neighbors as suggested earlier. 9 Recent polarized neutron scattering experiments 10 found magnetic correlations which indicate ferromagnetic short range order in the β phase and also in the high temperature paramagnetic hexagonal γ phase. Nevertheless, it was widely accepted that the β phase is paramagnetic. All these studies were made on bulk material. A new phase in the study of MnAs started when Tanaka et al. 11 succeeded to grow epitaxial MnAs layers on GaAs(100). Soon other groups also grew and studied these layers and one of them 12 made an important observation: when the films were cooled in a magnetic field ("field cooled") from high temperatures below the α-β transition temperature their hysteresis curve was asymmetric, i.e., it showed an exchange bias typical for FM/AFM interfaces. This was attributed to strain-stabilized AFM β-phase inclusions in the α phase. The exchange bias field H E which is the measure of the asymmetry of the hysteresis curve approached a constant value at low temperature but went rapidly to zero in the α-β transition region. The exchange bias in fieldcooled layers was recently confirmed again at room temperature. 13 Most of the work on the epitaxial MnAs film, however, was concentrated on the ferromagnetic α phase, which is now well understood, mainly due to the many multimethod studies of the Däweritz/Ploog group as documented in an extensive recent review by Däweritz. 14 In nearly all of this work including a very recent study 15 the β phase was considered to be paramagnetic. Only recently, stimulated by the theoretical work to be discussed next, experiments have been interpreted as supporting an AFM β phase. 16,17 of the magnetic state of the β phase, only the most recent and relevant ones will be considered here. They use different methods and models and come to different conclusions, thus keeping the discussion of the β phase alive. The first one 18 compares three AFM phases with different spin distributions with the FM and PM phases and comes to the conclusion that a random arrangement of AFM planes has the lowest energy for the lattice parameters of the β phase. The second 19 one considers the same AFM phases and the FM and PM phases and finds that the PM phase has the lowest energy. The PM phase is also favored by the third one, 20 which uses a quite different approach. Thus, at present it appears that from theory the β phase is most likely paramagnetic, at least in the bulk. This does not exclude an AFM state in epitaxial layers that are highly strained so that exchange magnetostriction could favor the AFM coupling, as suggested by Kittel 6 long ago.

II. EXPERIMENT

The most direct way to search for AFM is x-ray magnetic linear dichroism (XMLD) spectroscopy. In the case where FM and AFM regions coexist a microscopic technique is useful in order to separate the signals from the two regions. This is achieved by XMLD photoelectron emission microscopy (XMLDPEEM). Linear dichroism (LD) is proportional to the quadrupole moment of the charge distribution which in AFM materials depends on the direction of the magnetic axis via the spin-orbit coupling. In this case the magnetic contribution to the LD signal is proportional to 3(Ā•P)-1, where Ā and P are unit vectors in the directions of the magnetic axis and of the electric field, respectively. In XMLDPEEM 2p core electrons are excited by the incident linear polarized light from a synchrotron radiation source into empty d states. The secondary electrons produced in the filling process of the core hole are used then for imaging. By rotating P the direction of Ā can be determined. The 2p (L 3 ,L 2 ) absorption edges have a structure due to final state multiplet coupling between the 2p core hole and the 3d valence band hole, as illustrated in Fig. 2 for the L 3 edge of Mn in MnAs. The XMLDPEEM images shown below (Figs. 5 and6) are difference images between images taken from the two energies indicated in Fig. 2, normalized by the sum of these images. In the present study both magnetic imaging methods were combined with low energy electron microscopy (LEEM) and low energy electron diffraction for structural characterization.

More about these imaging techniques can be found in the corresponding chapters in Ref. 21. The experiments were performed in the two spectroscopic photoemission LEEM (SPELEEM) systems at the nanospectroscopy beamline of the synchrotron radiation facility ELETTRA in Trieste, Italy. One of the instruments, which has a 60° beam separator, is described in Ref. 22; the other one has a 90° beam separator but is otherwise similar. The properties of the light source can be found in Ref. 23.

The samples were prepared in the laboratory of Däweritz by solid-source molecular beam epitaxy at 200-250 °C in the so-called A orientation in which the MnAs (1-100) plane is parallel to the substrate surface and the MnAs [11-20] direction parallel to the GaAs [110] direction. The films were annealed at 400 °C and capped with As before removal from the vacuum system for transfer to the SPELEEM in which they were decapped. Numerous films with different thicknesses and orientations have been studied with XMCDPEEM after the first experiments. 24 Some of the results, which have contributed considerably to the understanding of the ferromagnetic α phase, are described in Ref. 14 and references therein. The XMCDPEEM studies reported here were done only at two thicknesses, 300 and 120 nm in the 90° beam separator SPELEEM instrument.

III. RESULTS

We begin with a brief overview of some of the results from our previous XMCDPEEM studies of the α phase that are relevant for the present study. In the coexistence range of the two phases the strain relaxation causes the formation of stripes whose periodicity depends linearly on thickness in the thickness range studied (20-500 nm) and whose relative width depends on temperature. Figure 3 shows an example from the intermediate thickness range. The black and white regions are FM with opposite magnetization perpendicular to the stripe direction; the gray regions give no FM signal and are supposed to be "PM." In the thinnest films, which have a simple FM domain structure, the β fraction changes approximately linearly with temperature as, shown in Fig. 4 for a 40 nm thick film. In thicker films, in which the domain structure is more complicated, the change is also more complicated, as shown elsewhere. In the present study the samples could not be cooled. Thus, only experiments from room temperature upwards could be made. Figure 5 compares the XMCDPEEM and XMLDPEEM images of 300 and 120 nm thick films. The comparison of the right and left images clearly shows the 1:1 correlation between the bright AFM stripes (right) and the zero intensity stripes between the FM stripes (left). The XMLDPEEM images were taken with vertical polarization, i.e., with the E vector of the light, which has a glancing angle of incidence on the sample, nearly perpendicular to the sample surface. The signal seen in the bright "PM" stripes, therefore, is a measure for the out-of-plane component of the AFM magnetic axis Ā. A similar but weaker signal is obtained for horizontal polarization, which is a measure for the in-plane component of Ā. This means that Ā is neither in-plane nor perpendicular to the surface but tilted and closer to the film normal.

FIG. 4. Dependence of the "paramagnetic" fraction of a 40 nm thick MnAs layer on GaAs upon temperature. The data were taken from LEEM images in which the stripe structure produces a phase contrast due to the height differences between ferromagnetic and paramagnetic stripes.

In the images shown in Fig. 5 the plane of incidence of the photon beam was normal to the stripe direction, which gives only partial information on the direction of Ā. Its full characterization can be obtained by rotating the sample 90°. This was done with the 120 nm thick film. Analysis of all four component images gave tilt angles of Ā of about 40° to the film normal and to the magnetic easy axis of the FM stripes ( [11-20]). The small XMLD signal and signal-to-noise level in the difference images of the 120 nm thick film cause a large uncertainty in these values, so that they should be considered as preliminary. The temperature dependence of the AFM signal was studied in both films and is illustrated in the XMLDPEEM images of Fig. 6 for the 300 nm thick film, together with the corresponding XMCDPEEM images. Again the correlation between the images on top and on the bottom is clearly seen. In the XMLDPEEM images the intensity in the AFM stripes decreases with temperature simultaneously with the decrease of the width and contrast of the FM stripes, i.e., with the magnetic field acting on them. Again, the low signal-to-noise ratio causes large error bars, but to a first approximation the AFM signal decreases linearly and disappears together with the FM regions at the α-β transition temperature. The temperatures shown in Fig. 6 have a small systematic error and should be considered only as relative values. It is interesting to note that some very weak stripe contrast can be seen at much higher temperatures (about 100 °C) which is not understood at present and needs further study. It should be noted that the films were not field of the FM stripes which is predominantly in-plane perpendicular to the stripes.

IV. DISCUSSION

The XMCDPEEM results reported in the previous section clearly show that the "PM" stripes in the coexistence region of the β and α phases are AFM with a tilted magnetic axis, even in the absence of an applied magnetic field during cooling. However, the observation that a clear AFM signal is seen only in the presence of the α phase suggests that the PM phase becomes AFM only in the presence of a magnetic field. The magnetization studies in which an exchange bias was observed only upon field cooling but not in the absence of an applied field 12,13 point in the same direction. The apparent discrepancy between the two types of studies can be explained as follows. Without an applied field during cooling the two sides of the FM stripes have with equal probability the magnetization pointing to the right and to the left, as illustrated in the 10 XMCDPEEM images shown in Fig. 7. Accordingly the alignment of the AFM regions at the FM/AFM boundary induced by the magnetic field of the FM stripes will also be equally probable in both directions. As a result the asymmetry of the interface pinning, which is necessary for exchange bias to occur, will be lost. During cooling in an applied field all FM stripes are magnetized in the same direction and as a consequence also all AFM regions at the FM border, producing the necessary asymmetry. In XMLDPEEM no interface pinning is needed, only a preferred direction of the AFM domains. The fact that these domains must be small is evident from the absence of long range order as determined by neutron diffraction. The work in Ref. 9 suggests domain sizes of the order of 10 Å. The spin structure within these regions and their boundaries can be postulated at present only on the basis of the recent density functional calculations. 18,19 Both calculations consider the same three spin configurations for the AFM phase. Counting the two Mn atoms in the basal plane of the β-phase unit cell as 1,2 and the corresponding two Mn atoms in its midplane as 3,4 (see Fig. 1), the spin configurations are (A) ↑↑↓↓, (B) ↑↓↑↓, and (C) ↑↓↓↑. All three AFM configurations have a somewhat higher energy E (lower binding energy) than the FM configuration (↑↑↑↑) in their respective optimized structure. Both calculations clearly favor in-plane AFM coupling over in-plane FM coupling and differ only in the energies of the two in-plane coupling configurations: 19, slightly favoring configuration B with AFM coupling within the basal plane and FM coupling perpendicular to it. However, the thermal energy at 350 K is 30 meV, so that configuration C with AFM coupling in both directions has also a finite probability.

E C > E B > E FM + 45 meV in Ref. 18 while E C > 2E B = E FM + 35 meV in Ref.
From these data the following picture emerges of the AFM structure if it exists: it consists of domains with spin configuration B separated from each other by domain walls with spin configuration C which may be considered as stacking faults of configuration B. This is similar to the disorder proposed in Ref. 18. As far as the "if" is concerned it must be pointed out that Ref.

18 concludes that β-MnAs is AFM, based on the considerably higher energy of the PM phase, while Ref. 19 concludes that β-MnAs is PM, based on the result that at the experimental lattice dimensions E AFM is considerably larger than E FM and comparable to E PM which has a minimum at these dimensions. The AFM configuration is favored only at lattice distortions that are considerably larger than the experimentally observed one. The discrepancy between the two conclusions has been attributed to the assumption of zero magnetic moment of the Mn atoms in the PM state in Ref. 18. 25 Thus from the point of the theory, β-MnAs must be considered to be PM at present, at least in the bulk. It could be argued that the large strain in epitaxial layers on GaAs could be the cause of AFM. However, precise synchrotron x-ray diffraction experiments 26 have shown that the misfit strain is relaxed by misfit dislocations and that it decays within a very thin interface layer. It certainly cannot cause AFM in a 300 nm thin film. Misfit strain as the cause of AFM is also excluded by the measurements of the exchange bias 12 which was found to be negligible in the thinnest films. One possible explanation that could reconcile present theory and experiment is that the positions of the Mn atoms in the basal plane in the epitaxial films are different from those in the bulk. The calculations of Ref. 19 show that the main driving force towards AFM is the exchange coupling J between inequivalent Mn atoms within the Mn zigzag chains with distance d [see Fig. 1(c)]. While the exchange coupling between neighboring basal plane layers and that between equivalent in-plane Mn atoms J 1 changes little with distortion and remains FM with increasing distortion (because the atomic distances change only little), J becomes AFM already at 75% distortion, where 100% distortion corresponds to equilibrium distortion of the β phase. If the distortion in epitaxial layers should be larger than in the bulk, AFM may be stabilized in them even if the bulk is PM. A larger distortion could possibly be induced by thermal strain. Film and substrate have different expansion coefficients, in particular, along the b axis of β-MnAs. Upon cooling from the growth temperature (250 °C) to room temperature thermal stress-induced shifts of the Mn positions along the b axis [Fig. 1(c)] could occur. However, they would have to be rather large in order to produce the lattice distortion needed for AFM to be favored according to Ref. 19. In thick films the thermal stress is partially relaxed by cracks perpendicular to the b axis.

Another possibility for reconciliation of theory and experiment which does not involve different Mn basal plane positions in the bulk and in epitaxial layers is the fact that theory does not take into account lattice vibrations. Their importance is illustrated by the observation that anomalous changes in the lattice dynamics, which show strongly anisotropic vibration amplitudes with maximum amplitude approximately perpendicular to the zigzag chains, are more important than the static changes considered in the calculations. 27 Because the coupling constant J between the nearest neighbors in the chains is important for AFM coupling, these vibrations should have a significant influence on the coupling. Kittel 6 has already pointed out that the indirect spin-spin coupling via phonons plays an important role when the magnetocrystalline anisotropy is high, which is the case in MnAs.

The disappearance of AFM upon the transition to the high temperature γ phase may be caused by the breakup of the magnetic coupling by thermal fluctuations because the thermal energy at the transition temperature (110 meV) is significantly larger than the coupling constants. This may be aided by the proliferation of magnetic domain walls, causing the disintegration of the domains. The breakup of the coupling is accompanied by the transition from anisotropic to isotropic vibrations, 27 which is an indication of the intimate relationship between spin-spin and spin-phonon couplings.

V. SUMMARY

We have shown unambiguously with XMLDPEEM that the β phase of epitaxial MnAs layers on GaAs(100) is antiferromagnetic at least in the α-β coexistence region, even in the absence of an external applied field during cooling. A comparison with the literature suggests that an applied field during cooling leads to a uniaxial alignment of nanoscopic antiferromagnetic domains in which the Mn atoms are coupled antiferromagnetically within the basal plane and ferromagnetically perpendicular to it. Antiferromagnetism is present only on the nanoscale so that the β phase may also be considered to be superparamagnetic similar to the state of nanoscale ferromagnets. There are still many open questions, in particular, concerning the magnetic state between the α-β and the β-γ transitions in the absence of long range ferromagnetic order, concerning the size of the magnetically ordered regions and concerning the details of the influence of an applied field during cooling. Future x-ray magnetic dichroism and polarized x-ray scattering studies without and with an applied field and an improved polarized neutron scattering work over a wider temperature range will hopefully answer these questions.

FIG. 1 .

 1 FIG. 1. Structure of MnAs. (a) Hexagonal α phase (NiAs type). (b) Orthorhombic β phase (MnP type); the corresponding unit cell dimensions of the α phase are indicated by dashed lines. (c) Mn atomic positions in the β phase and interatomic distances in the basal plane. The zigzag chains discussed in the text are indicated by thicker lines. Open circles, Mn; black circles, As in the unit cell; and gray circles, As atoms in neighboring unit cells.

FIG. 2 .

 2 FIG. 2. Mn L 3 (2p 3/2 ) absorption spectrum. The secondary electron yield is plotted as a function of the exciting photon energy. The lines indicate the photon energy used for the two images whose difference is displayed in the XMLDPEEM images (Figs. 5 and 6).

24

 24

FIG. 3 .

 3 FIG. 3. XMCDPEEM images from MnAs on GaAs(100) illustrating the dependence of the stripe period and of the magnetic domain structure at room temperature upon film thickness; (a) 120 nm, (b) 180 nm, and (c) 300 nm. Field of view is 5 µm.

FIG. 5 .

 5 FIG. 5. XMCDPEEM (left) and XMLDPEEM (right) images of 300 nm (top) and 120 nm (bottom) thick MnAs layers on GaAs(100) at room temperature. Field of view is 5 µm.

FIG. 6 .

 6 FIG. 6. Temperature dependence of the AFM (top) and of the FM (bottom) contrasts and fractions in a 300 nm thick film. The temperatures are from left to right 33, 42, and 48 °C. Field of view is 5 µm.

FIG. 7 .

 7 FIG. 7. Temperature dependence of the domain structure of a 180 nm thick MnAs film on GaAs(311) during cooling from left to right. The same irregular domain pattern is observed on GaAs(100). Field of view is 5 µm.

ACKNOWLEDGMENTS

One of the authors (E.B.) wants to thank L. Däweritz for providing the samples and S.

Sanvito for clarifying correspondence. He also acknowledges support by Office of Naval

Research under Grant No. N-000140210922 and by NATO.


M Ochoa

Q Pian

R Yao

N Ducros

And X Intes
email: intesx@rpi.edu

Assessing patterns for compressive fluorescence lifetime imaging

Keywords:

A novel hyperspectral single pixel system was used to compare different compressive basis patterns for intensity imaging, lifetime imaging, and FRET quantification. Six popular basis patterns were compared experimentally in a phantom containing two fluorescent dyes. The basis patterns that performed best for lifetime quantification were used to measure FRET occurrence in well-plate samples with varying acceptor-donor ratios. The ABS-WP approach using Haar patterns and the compressive sensing approach with Hadamard Ranked patterns displayed the best overall performances at a 50% compression ratio.

Fluorescence lifetime imaging has been widely used to investigate biosamples at the macroscopic and microscopic levels [1][2][3][4][5]. Fluorescence lifetime is an intrinsic property of fluorophores that depends on its molecular environment and not on its concentration [6]. Its imaging is particularly useful to quantify Förster Resonance Energy Transfer (FRET), which is widely employed as a nanoproximity assay for biomedical applications [7][8][9]. Recently, macroscopic hyperspectral singlepixel imaging has been proposed as one avenue to spatially image FRET occurrence in vitro as well as in vivo in preclinical models [10]. Single-pixel imaging exploits spatially compressed optical signals to recover 2D images from a few 1D measurements [11]. Mathematically, it consists of inverting the linear equation to solve for the sample plane s m ΔtPs,

Intensity imaging using Hadamard and Fourier patterns have been compared in Ref. [14]. In this Letter, our contribution is:

(1) to compare patterns (e.g., wavelets) that have not been considered so far, and (2) to assess fluorescence lifetime and FRET quantification. Herein, we experimentally investigate six different approaches and assess their ability for 2D imaging as well as to quantify monolifetime and NIR FRET interactions, which represents a challenging biexponential application. The hyperspectral single-pixel system employed in this study is described in Ref. [10] and depicted in Fig. 1(a). In brief, the system operates in transmission with two digital micromirror devices (DMD, D4110, Digital Light Innovations, Texas). The illumination DMD, where compressive basis patterns are implemented, is coupled to a Ti:Sapphire pulsed laser (Mai Tai HP, Spectra Physics, California) that can be tuned from 690 to 1020 nm. Light emitted by the sample is collected through the wide-field detection DMD, which is coupled to a 16-channel detection PMT spectrophotometer (MW-FLIM, Becker and Hickl GmbH, Germany) via a fiber bundle. Note that this double-DMD architecture is not required for 2D single-pixel imaging but implemented to also be able to perform optical tomography [17]. A time-correlated singlephoton counting unit (TCSPC) records the fluorescence decays over the 16 spectral channels in parallel. The system is controlled through LabView. All patterns are generated using MatLab, considering 32 × 32 images that are converted to eight-bit integers and resized to 672 × 672 to match the DMD physical resolution. The patterns are divided into positive and negative parts for further processing after data collection. Herein, we used only 512 patterns out of the 32 × 32 1024 patterns, resulting in a 50% compression ratio. The list of basis patterns to be compared is: (a) Hadamard Normal, (b) Hadamard Sign, (c) Hadamard Ranked, (d) Speckle, (e) Fourier, and (f) Haar. The Hadamard patterns are generated using a natural sequence through the Hadamard Matlab function [START_REF] Ryser | Combinatorial Mathematics[END_REF][START_REF] Sylvester | [END_REF][START_REF] Pian | Time-Resolved Hyperspectral Compressive Single-Pixel Wide-Field Optical Imaging[END_REF]. We consider three different Hadamard pattern orders: Normal uses the default sequence provided by the function (see Visualization 1) and Sign are the normal patterns arranged by positive and negative parts (see Visualization 2). The patterns are further arranged from low to high spatial frequency to yield Ranked Hadamard patterns (see Visualization 3). Speckle patterns are obtained from uniformly distributed random multiplicative noise using the imnoise Matlab function [15,[START_REF]Mathworks, image toolbox: user's guide (r2012a)[END_REF] where m represents the measured data, P the illumination patterns, and Δt the integration time for each pattern. The imaging performance depends on the choice of compression patterns as well as the strategy for selecting a subset for measurements [12]. Multiple basis patterns such as Hadamard [13], Fourier [14], speckle [15], and wavelet [16] have been proposed. with a mean of 0 and variance of 10 (see Visualization 4). Fourier patterns are accomplished by cosine function generation. The x∕y frequencies change from 1 to 8 with eight phase shifts (see Visualization 5). Haar patterns are taken from a Haar wavelet basis that contains 16 scaling functions [16] (see Visualization 6). To investigate the performances of these various basis patterns in the case of monolifetime, we prepared fluorescence phantoms with RPI letters: R and I contained AF750 (ThermoFisher Scientific, A33085) fluorescent dye at 4.16 μM concentration prepared with PBS 1X; letter P contained HITCI (Sigma Aldrich, 252034) dye at 40 μM concentration prepared with 100% ethanol. The volumes of the RPI letters are 149, 124, and 81 μL, respectively. Using a piecewise homogeneous phantom allows estimation of the noise of the reconstructed image.

Seven identical phantoms were prepared: one per basis patterns and one for measuring lifetime ground-truth values with a gated-ICCD system [START_REF] Abe | [END_REF]23]. To minimize evaporation effects, we filled each phantom with the same stock solutions of the dyes before measuring each basis patterns. Moreover, because lifetime is considered independent of concentration, minimal evaporation changes should not compromise its quantification. The system illumination and detection schemes were calibrated with the six different pattern sets. The calibration process involved matching the field-of-view of the illumination and detection DMDs. In addition, the patterns were projected on the image plane and recorded using an external NIR CCD to be later used during the image reconstruction step [Fig. 1(a)]. Each pattern accounted for 0.5 s exposure time, and the excitation wavelength was 740 nm. A 780 nm (Semrock FF01-780/12-25) filter with spectral range 774-786 nm was added to the detection optics, and the PMT detection range was defined from 715 to 778 nm with central wavelength at 742 nm. The field of view (FOV) was 35 mm × 35 mm, and the optical power density was ∼5.49 mW∕cm2 . Using constant parameters for each basis patterns, we acquired a data cube composed of one time point spread function (TPSF) per pattern and per detection channel. The TCSPC unit produced each TPSF over 256 time channels. We first evaluated the 32 × 32 intensity images recovered by the different approaches for different compression ratios. For all basis patterns, we considered increasing compression ratios (∼50%, 60%, 70%, 80%, and 90%) by decreasing the number of patterns. While most approaches select patterns prior to acquisition [14], adaptive strategies select patterns during acquisition [16,24]. For Hadamard, Fourier, and Speckle basis, patterns were chosen a priori, and a linear solver reconstructed the images from the raw data. We used the so-called TVAL3 algorithm [START_REF] Li | Compressive Sensing for 3d Data Processing Tasks: Applications, Models And Algorithms[END_REF], which promotes piecewise constant solutions. For Haar wavelet patterns, we implemented an adaptive basis scan by wavelet prediction (ABS-WP) using the SPIRIT toolbox [START_REF] Rousset | Spirit_toolbox[END_REF]. The Haar patterns were selected among the Haar basis during acquisition, and the image was simply recovered through fast inverse wavelet transforms of the raw data. The images reconstructed for a 50% compression ratio are shown in Fig. 2(a), and the image reconstruction quality for increasing compression ratios is plotted in Fig. 2(b).

Intensity images are computed by integrating the hyperspectral time-resolved data cube over the detection channel located between ∼760 and 765 nm and over the 256 time channels. Before every experiment, a ground-truth image was measured using the NIR CCD camera described in Fig. 1. The 616 × 619 CCD image is resized to 32 × 32 to match the size of the singlepixel reconstructed intensity images. Both the ground-truth and the recovered intensity image are normalized to their maximum value. We used the structural similarity index (SSIM) metric to measure the similarity between the ground-truth and the recovered images

SSIM 2μ x μ y c 1 2σ xy c 2 μ 2 x μ 2 y c 1 σ 2 x σ 2 y c 2 , (2)
where μ x and μ y are the local means, σ x and σ y are the standard deviations, σ xy is the cross-covariance, and the c 1 and c 2 terms are regularization constants for luminance and contrast [START_REF] Wang | [END_REF]. Variable x constitutes the output 32 × 32 reconstructed image, and y is the ground-truth 32 × 32 CCD image. In this case SSIM was calculated globally. Overall, the best intensity images are recovered by the ABS-WP approach, no matter the compression ratio. Ranked Hadamard patterns provided similar results to the ABS-WP approach, up to a compression rate of 70%. Normal Hadamard reconstruction shows large artifacts, which are related to the measurement of high-frequency patterns leading to low signals that are highly affected by noise.

The SSIM values for Sign Hadamard and Fourier patterns are similar for compression ratios below 70%. Finally, speckle patterns yielded lower SSIM values than the other basis patterns, except for the normal order Hadamard that led to the worst image quality. Second, we exploited the time domain data to quantify lifetimes. Each TPSF from the 12th spectral channel of the recovered data cube was fitted to a monoexponential decay convolved with the instrument response function (IRF), i.e., mx, y, t IRFt exp -t τx, y ,

where mx, y, t is the TPSF recovered at pixel x, y, and τ is the fluorescence lifetime. The IRF was measured after each basis measurement with a wide-field illumination pattern. Each pixel of the TPSF is denoised with a one-dimensional low-pass filter and normalized. The fluorescence decay was then tail-fitted using a Matlab built-in optimization interior-point algorithm. Initial values of 1 (ns) with 0.05 (ns) boundaries were provided for letter P and 0.5 (ns) with 0.05 (ns) for letters R and I, where the letters are segmented from the ground-truth images shown in Fig. 2(a). To deliver accurate fits with low residuals, initialization values were provided to the optimization algorithm per each letter. The obtained lifetime maps are displayed in Fig. 3(a). To quantitatively compare the performance of each compressive basis patterns, we also computed, in each letter, the mean lifetime and standard deviation values as summarized in Fig. 3

(b).

To compute the descriptive statistics, the lifetime values on each letter were ordered and 60% of them used, that is 60 pixels for R, 52 pixels for P and 25 pixels for I. By imaging the additional RPI phantom on a gated ICCD system, mean values of 0.50 0.017 (ns) were obtained for letters R and I containing AF750 dye and 0.92 0.018 (ns) for letter P containing HITCI dye. The lifetime range for each letter is evaluated, and a closer relation of the mean value and standard deviation to ground-truth are therefore desired. In terms of mean values for letter P containing HITCI dye, the closest mean relation is accomplished by ABS-Haar, which also displays the smallest standard deviation range after the Hadamard Sign. The standard deviations of the other basis patterns partially contain the ground-truth range for this letter. On the other hand, ABS-Haar and Ranked Hadamard better approximate the ground-truth mean for letters R and I, which contain the same dye AF750 and therefore should ideally yield the same lifetime mean. For these letters ABS-Haar, Fourier, and Hadamard Ranked display the smallest standard deviation ranges. Even though normal Hadamard encloses the ground-truth values, its range for letter P also encloses values present in the lifetime range of letter R and vice-versa. Overall, the ABS-Haar and Hadamard Ranked basis patterns showed the most accurate lifetime estimations for all letters (i.e., closer means to ground truth and smaller standard deviation for both fluorophores). To further compare these basis patterns, we conducted an experimental study focusing on one of the most challenging lifetimebased applications, NIR FRET. As the ranked and sign Hadamard, ABS Haar, and Fourier were the best at quantifying lifetime on the RPI phantom, the quantification of FRET was restricted to these basis patterns. The FRET sample contained AF700 Mouse IgG1 (ThermoFisher Scientific, MG129) donor dye and AF750 goat antimouse IgG (ThermoFisher Scientific, A-21037) acceptor dye with respective concentrations of 50 and 100 μg/mL. A PBS 1X solution was used to prepare the desired concentrations. Six wells were filled with 250 μl of different acceptor-donor combinations. The wells described in Fig. 1(b) for well 1 contain AF700 only, well 2 and well 3 PBS only, and wells 4 to 6 contain ratios of AF750 to AF700 of 1∶1, 2∶1, and 3∶1, respectively. The PBS and AF700 only wells were used as experimental control; therefore, FRET interaction is only expected in wells 4 to 6. Four well-plate samples were prepared from stock solutions. Samples were excited at 695 nm and exposed for 1.5 s per pattern. A 715 nm long pass filter (Semrock, FF01-715/LP-25) was added to the detection optics. The third channel, which was used for reconstructions (∼720 to 730 nm), would detect minor acceptor bleedthrough. All the experimental parameters were kept constant for all patterns. Hadamard Ranked, Hadamard Sign, and Fourier were reconstructed through TVAL3 and Haar patterns through the ABS-WP approach. A biexponential fit yielded the amplitudes and lifetimes of the FRETing and non-FRETing populations. At each pixel, we assume:

mt IRFt a F exp -t τ F a N F exp -t τ N F , (4)
where a F , a N F are the amplitudes of the FRETing and non-FRETing donor and τ F , τ N F their respective lifetimes. Amplitudes are normalized such that a F a N F 1. Initial values of 1 0.05 (ns) and 0.35 0.05 (ns) for long and short lifetime were provided. Figure 4(a) shows the FRET donor fraction (a F ) percentage maps for wells 1-4 for the different basis patterns. For increasing acceptor to donor ratios, the FRET interaction should increase linearly [START_REF] Abe | [END_REF]. Although well 1 contains a 0∶1 ratio and no FRET interaction is expected, relative amplitudes of ∼15% and ∼85% are estimated. The correction of this inherent type of bias, which has already been reported in Refs. [10,[START_REF] Abe | [END_REF]28], falls outside the scope of this study. recovering the lower FRET donor fraction spatial distribution in its entirety). It is important to mention that FRET quantification experiments are limited by the variability of FRET interactions from sample to sample; however, similar groundtruth linear trends have been previously reported [10,[START_REF] Abe | [END_REF]28]. In summary, all basis patterns investigated herein were successfully implemented in a single pixel paradigm for lifetime imaging. In all cases, spatially resolved TPSFs could be reconstructed. Though, when spatial accuracy and lifetime quantification were assessed, the ABS-WP approach and ranked Hadamard strategy outperformed the other compressive basis patterns. The ABS-WP approach and ranked Hadamard strategy provided very similar performances for a compression rate at 50%, both for intensity and lifetime-based quantification. Ranked Hadamard performed better for FRET quantification. Considering that the excitation wavelength, exposure time, power density, detection range, and FOV were kept constant, comparing raw TPSFs for the highest photon counts between the ABS-WP approach and ranked Hadamard reveals that ABS-WP based on Haar patterns lead to more noisy TPSFs with a lower number of photon counts, as displayed in Fig. 4(b). This is most likely the reason why ranked Hadamard performs better in the case of biexponential fitting with short components (NIR FRET). However, it is important to note that the ABS-WP approach performed better at higher compression ratios for intensity reconstructions. Because employing higher compression ratios is critical to enable the implementation in single-pixel methodologies for bed side applications, this finding elicits the need for further refinements and investigation on the performances of the ABS-WP approach at very high compression ratios. The ABS-WP approach implemented herein used the Haar wavelets, but further evaluation of ABS-WP wavelets like biorthogonal CDF [29] should be carried out and compared to the ranked Hadamard methodology. Additionally, in this study, the ABS-WP approach was implemented a posteriori (the whole Haar basis patterns were acquired and then pattern prediction was performed postacquisition). Further study should focus on seamless experimental implementation of on-the-fly prediction and compare its computational burden to the fixed pattern basis as obtained with ranked Hadamard. It is expected that the implementation of the ABS-WP approach will then lead to robust and very fast imaging protocols for lifetime imaging, especially in the case of monolifetime applications. Overall, after selection of the appropriate illumination basis patterns and the reconstruction approach, compression ratios higher than 50% should be explored to improve both resolution and acquisition times.

Fig. 1 .

 1 Fig. 1. (a) Hyperspectral single pixel imaging platform [10] and (b) RPI phantom and FRET wellplate.

  Figure 4(c) displays the mean values of each well and their standard deviation range. ICCD ground-truth values for the FRET samples are also included. A linear trend within ground-truth values is observed for the four sets of patterns, with ABS-Haar displaying the least accurate 0∶1 ratio estimation and linearity. Fourier patterns on the other hand result in a smoother FRET Donor Fraction (FD%) map (though without

Fig. 4 .

 4 Fig. 4. (a) FD% maps per basis patterns. (b) Highest raw data TPSFs. (c) FD% mean and standard deviation to ground-truth.

(a) 32 × 32 pixels ground-truth (GT) and reconstructed intensity images at 512 patterns and (b) SSIM in relation to ground-truth.

(a) Lifetime maps. (b) Mean lifetime and standard deviation within letters R (red), P (blue), and I (black); the horizontal lines indicate ICCD ground-truth values.

Acknowledgment. We thank Sez-Jade Chen for the advice on experimental sample design and ground-truth validation protocols.

Funding. National Institutes of Health (NIH) (R01 CA207725, R01 EB19443); Agence Nationale de la Recherche (ANR) (ANR-17-CE19-0003).


text:
Tanja Hamann
email: hamann@imai.uni-hildesheim.de

"Sickened by set theory?" -About New Math in German primary schools

Keywords: New Math, primary, Germany, set theory, reform

Purpose and question of the study

In the Federal Republic of Germany, public remembers so-called New Math mostly as a curious episode during which set theory has preceded numbers in primary mathematics education, a reform that has been controversially discussed and finally taken back, having been branded a failure. Thus, a closer look at the history of this reform is required.

As in any educational reform, changes concerned all different components of the educational system. Keitel (1980, p. 449) summarizes three levels that are crucial for the course of any reform, namely a subject's educational scientific community, educational administration and curricula alongside their implementation in textbooks. Neglecting practical classroom experience, this list stucks to a more theoretical view, whereas [START_REF] Fend | Schule gestalten: Systemsteuerung, Schulentwicklung und Unterrichtsqualität [Shaping school: System control, school development and educational quality[END_REF] constitutes a multi-layer model of educational system upon the levels administration (politics and curriculum), school (teachers) and lesson (pupils and parents), putting more emphasis on practice and social reception. In both concepts textbook production is seen as being closely linked to curricular decisions, nevertheless it is also stated thatparticularly within the German New Math movement -schoolbooks served as "instruments of innovation" inside the classroom (Keitel, Otte, & Seeger, 1980, p. 73). For these reasons, the project is based on a multilayer model of schooling comprising the following levels: scientific theories of education, curricula, textbooks and implementation in classroom. Former analysis of the reform in Germany has mostly emphasized on curricular aspects [START_REF] Damerow | Die Reform des Mathematikunterrichts in der Sekundarstufe I: Eine Fallstudie zum Einfluß gesellschaftlicher Rahmenbedingungen auf den Prozeß der Curriculum-Reform[END_REF][START_REF] Keitel | Entwicklungen im Mathematikunterricht [Developments in mathematical education[END_REF][START_REF] Zumpe | Der neue Mathematikunterricht in der Grundschule: Eine kritische Analyse auf dem Hintergrund bildungspolitischer Zielsetzungen Anfang der 70er Jahre[END_REF], has mainly been focused on secondary education and dates back from when New Math curricula were still mandatory, causing those accounts to criticize the concepts and implementations rather than bringing them in line with long-term historical development.

From this derives the main purpose of the project, which is to describe ideas and concepts (with regard to aims, contents, principles, methodical suggestions) leading to the reform of mathematics education in West German primary schools and compare them to how they were implemented into methodical concepts. Three different 1st grade textbooks and associated teacher handbooks from the 1960s and 1970s were chosen as exemplary key sources: alef by Bauersfeld et. al., which developed from the country's only long-term classroom project (Frankfurter Projekt), Wir lernen Mathematik by Neunzig & Sorger, which was the first New Math textbook for primary level being published and which is explicitly based on the concept of Z. P. Dienes, and Mathematik für die Grundschule by Fricke & Besuden, which originated from a former numeracy textbook based on the operative principle and therefore on the results by J. Piaget.

Results

It shows that the courses differ when it comes to basic curricular decisions as well as to the relation between mathematics, sets and arithmetic. Bauersfeld et. al. have created a non-linear course that is based on fundamental logical (relations, sets, transformations) and geometrical concepts. Here, arithmetic is founded on mathematics; sets serve as one example of a basic mathematical notion.

Fricke & Besuden subordinate all their subject matter decisions to the learning of operative thinking. Thus, mathematics becomes a means for this, and arithmetic as well as set theory, logic and geometry serve as examples for mathematical thinking. Neunzig & Sorger's course, however, is aiming at a mathematical foundation of arithmetic, which is solely based on sets.

Especially the latter, which largely narrowed the original ideanamely to replace pure arithmetic by propaedeutic mathematics from the startto teaching set theory in advance of numbers and numeracy, was widely disseminated and thus influential for the course of reform. The question occurs what might have caused this development. One result of further investigation of concepts of numeracy education up to the 1960s at the German Volksschule is that such an approach could be brought in line with long-term tradition of German primary arithmetic education.

Thematic Working Group 12Proceedings of CERME10

Proceedings of CERME10


text:
Maurice Oreilly
email: maurice.oreilly@dcu.ie

Multiple perspectives on working with original mathematical sources from the Edward Worth Library, Dublin

Keywords: History of mathematics, original sources, cooperative learning Thematic Working Group 12

In the spring semester of 2016, the author worked with twenty undergraduate (BA) students on original sources from the Edward Worth Library (1733), in Dublin. The end goal was to produce an online exhibition of the mathematical works from that library. The approach to this collaborative work is given, set in the context of a more general framework of the pedagogical value of working with original sources for teaching the history of mathematics. Examples of feedback from the students are given, as is an outline of how the exhibition itself was eventually shaped. The conclusion reflects on the learning gained by all involved in the collaboration.

Introduction

This paper outlines the process and product of working, with twenty students and two other collaborators, on the mathematical works in the Edward Worth Library (EWL). This collection opens a window on two centuries of the development of mathematics in England from the 1530s to the 1730s, a period of momentous cultural change in Irelandwith the expansion and consolidation of English political power and the emergence of an Anglo-Irish Protestant eliteand scientific blossoming, culminating in the work of Isaac Newton and his circle, with very strong influences from continental Europe, especially from France and the Netherlands. The work, on original sources from EWL, of the author with his students and the librarian is outlined, as is the subsequent work in preparing an online mathematical exhibition, launched in November 2016.

The Edward Worth Library and my encounter with it

The Edward Worth Library was founded in 1733 by its benefactor, Edward Worth (1676-1733), a medic who had a passion for fine books and who left some 4500 volumes to Dr Steevens' Hospital, in Dublin. Unusual in Ireland for this period, about one third of his collection comprise texts of medical and scientific interest. Of these, in turn, about 9% are mathematical. The collection is housed in a single room with glazed cabinets designed specifically for that purpose. The books themselves are in impeccable condition due, in no small way, to the fact that they were used very little during the 283-year history of the library [START_REF] Mc Cormack | The Enigma of Worth[END_REF].

I have taught a one-semester module on the history of mathematics every two years since 2008. A few weeks after completing teaching such a module in 2012, I visited EWL on 15 th June, for the first time, in the company of Professor William (Bill) F. McComas, visiting Fulbright Professor (to Dublin City University, DCU) from the University of Arkansas. Combining his interest in the history of science and mine in the history of mathematics, EWL gave the impression of an Aladdin's cave of early modern European learning. We were enthusiastically welcomed by the librarian, Dr Elizabethanne Boran. With a permanent staff of one (the librarian herself) and some visiting scholars and interns from time to time, it is challenging to open the doors to the public, although small specialist conferences are accommodated as much as possible. The librarian has overseen the publication of online exhibitions that enabled the treasures of the library to be appreciated by a wider readership (Edward Worth Library, n.d.).

On 6 th May 2014, the librarian welcomed the first group of my students (undergraduates in their second of third year of a three-year BA) to visit EWL. She had put on display works I had chosen by Viète, Harriot, Descartes, L'Hospital, Wallis, Ward, Huygens and Newton, dating from 1615 to 1732. The reaction of the students to viewing the original sources was very positive, however it was too late in the academic year for this cohort of students to engage with these original sources in any meaningful way. Their reaction, together with my own growing familiarity with scholarship in the HPM community on the importance of working with original sources, prompted me to seek an opportunity to work with EWL in a more significant way in 2016. The librarian was keen to extend the scope of the online exhibitions at EWL, and it seemed realistic to build a mathematical exhibition around the work of the next cohort of students (twenty, including two visiting from the USA). Necessary preparatory work on my part was to become familiar with the online EWL catalogue and derive from it a (provisional) list of the mathematical works in the collection. From this list (of 109 works), I identified for which ones the author had a biographical entry in MacTutor [START_REF] O'connor | MacTutor history of mathematics[END_REF], and found online versions of about a third of the texts, using EROMM, for example. Searching for so many original sources was time-consuming, yet very rewarding; it was an exercise I had previously no reason to undertake on such a scale. At this stage, I considered myself ready to lead the students in the EWL Project. This paper gives an overview of how the project itself was conceived and implemented.

Original sources and teaching History of Mathematics

Much is said in the literature about the use of original sources in teaching the history of mathematics, and this is not the place to provide a comprehensive overview. Kiernan identifies the challenge facing both teacher/lecturer and students:

The task of presenting original works of mathematics to a class of undergraduates may seem like a daunting task. It requires much preparation. How does one become confident that one can do this? Well, you must remember that you are not alone in this task. Attending any conference with themes on the history of mathematics will remind you that there are many others who have the same interest as you. (Kiernan, 2010, p. 412) Because I, the author, am also the lecturer and so a crucial player in the work described in this paper, it is appropriate to describe, in the first person, how I engaged with the tasks involved. Glaubitz is quite clear that working with original sources is quite a different kind of work from that which students usually think of when they 'do' mathematics:

Of course, on the part of the teacher it involves quite some preparation, but this is always the case when you are going to try something new. What is more, the study of original sources requires teachers and students to be prepared to dive into some strange and unknown realm of thinking, to appreciate cultural and historical contexts andlast but not leastto deal competently with written text that is more extensive than the word problems they are used to in mathematics. (Glaubitz, 2010, p. 351) If working with original sources is, indeed, 'something new', it "[provides] context, motivation and direction for students' mathematical endeavours" (Barnett, 2012, p. 336). Thus, such work has a strong potential to open up new perspectives for students:

By reading historical sources students can be acquainted with episodes of past mathematics where other meta-discursive rules governed the discourse. (Kjeldsen, 2010, p. 52) Even when students do not make explicit reference to unfamiliar 'meta-discursive rules', appreciation of such rules can often be noticed indirectly in the 'surprise' they express in their reflections on working with original sources.

Designing the EWL project

The EWL Project was designed as an integral part of a 5-credit (ECTS) module on the History of Mathematics (having a strong, but not exclusive, emphasis on the development of algebra). Credit for this module comprised two main components: 70% for a terminal examination and 30% for continuous assessment (CA). Roughly one third of the examination and 17 marks of the 30 allocated to CA were associated with the EWL Project. Thus, in effect, a substantial 40% of the total credit for the module was assigned to this project. What were the salient aspects of the project and how was it presented to the students?

The semester began on 1 st February (2016) and it was important to arrange a visit to EWL early in the semester to allow the students to become familiar with material that would be very foreign to their experience. In advance of the visit, I prepared a one-page overview (with web links, indicated in bold, below):

It is a great privilege to be invited to view a selection of the mathematical works at EWL of which there are about 109 in total. These span a period of almost two centuries, from Cuthbert Tunstall's De arte supputandi libri quatuor (1538) to Isaac Newton's posthumous Arithmetica universalis (1732), published the year before the establishment of EWL and the death of its founder.

According to the entry in MacTutor, Tunstall (1474-1559) had a prominent career as a diplomat (at the court of Charles V in Aachen, for example) and bishop (eventually of Durham). His De arte supputandi (1522) was the first printed work published in England devoted entirely to mathematics. This was not an original work, but was based on Luca Pacioli's Suma (1494).

Newton (1643-1727), arguably the greatest English mathematician, worked in a broad range of disciplines including theology, optics, mechanics, algebra and (especially) the calculus. EWL has no fewer than 16 works attributed to Newton. There are several other works in EWL by others associated with Newton (such as Barrow, 'sGravesande and van Musschenbroek), all of whom (along with Newton himself, of course) are presented in the EWL online exhibition on Newton.

The 109 books are written in four languages: English (21), French [START_REF] Barrow | 1677) of the remaining 37: 16 were born before 1640 (and were living during Worth's lifetime, but were much his senior) 16[END_REF], Latin (64) and a combination of Greek and Latin (9). The authors of 74 of these (and some of the authors, such as Newton, have more than one book in EWL) have biographical entries in MacTutor. EWL expects to prepare an online exhibition on Mathematics at EWL in summer 2016. You are invited to contribute to this by writing a page on one of the mathematical works in EWL.

In advance of the visit to EWL on 25 th February, take a look at one of the seven 'big' exhibitions already online at EWL (Newton, Botany, Alchemy & Chemistry, Infectious Diseases, Astronomy, Dr Steevens' Hospital and Aldines) or one of the eight smaller exhibitions (e.g. Looking at the Moon or Surgery). Think about the following:

1. How do you find navigating these webpages? 2. What features of the exhibition (you chose) do you find attractive? 3. What do you find exciting/daunting about the prospect of writing a page for the Mathematics at EWL exhibition?

This page set the scene for the project. Its subsequent development required much attention to detail, taking on board, to a greater or lesser extent, the reactions of the students as observed in their submitted work, contributions to a forum (in a virtual learning environment), and queries by email or verbally.

On the day of the visit, 16 of the twenty students made it, along with a colleague (Fionnán Howard) who played a crucial role later in editing work for the online exhibition. The students were given hard copy of the following:

 A list of the twelve works of which nine were put on display by the librarian  A page of text or an illustration from ten of these  Chapter 3 ("How are mathematical ideas disseminated?") from Stedall's The History of Mathematics -A very short introduction [START_REF] Stedall | The history of mathematics: A very short introduction[END_REF] Students were asked for their reflections on visit, and responded enthusiastically and on a wide range of aspects. Different perspectives on the library itself included (with student identifiers given in [square brackets]):

On arrival to the Edward Worth Library (EWL) it was just how I would have pictured it. Not only had the books all been perfectly preserved but so had the library itself. [8] I expected the library to be quite big and look similar to any other library. My expectations were quite incorrect. [START_REF] Barrow | 1677) of the remaining 37: 16 were born before 1640 (and were living during Worth's lifetime, but were much his senior) 16[END_REF] I felt as if I went back in time [12] In each case, the student's surprise, even sometimes delight, is evident. On the nine books put display, comments included:

I was surprised that we were allowed to view some texts up close given their iconic meaning.

[16]

The books themselves were written in a number of different languages, mainly Latin and French, with only a small amount in English, which again highlights how things have changed.

[10]

The most astonishing thing I saw was the quotient rule in a book from 1696. It was bizarre seeing it in such and old context. [2] There is a real sense here of situating a rule, familiar from school, now in the 'exotic' period of the calculus textbook (L'Hospital, 1696). In her introduction the librarian had made a very reasonable conjecture on the relative plainness of the binding of almost all of Worth's mathematical books, and one student picked up on this:

Worth didn't seem to pick these books for their binding or appearance alone, but their content [7] Some students were thoughtful, as if for the first time, about how mathematics emerges and is communicated:

It quickly became apparent to me that to study any subject during this time period you would need a huge amount of devotion and intellect. You would need to be fluent in various languages to even access three quarters of the books, you would need to travel to find the books and you would need an in-depth knowledge of the subject to comprehend them.

[2]

I was curious about the various languages used in the books and wondered whether every mathematician needed to be fluent in order to understand and broaden their mathematical knowledge.

[6]

The visit gave me a much better and clearer insight into the HoM module, it put into context how the maths we do and formulas we take for granted did not just fall from the sky, but were sought after and achieved by great minds. [12] Other students articulated hermeneutical sensibilities about the foundations of mathematics:

Seeing some of the diagrams and equations that were printed in the math books we viewed (though in a different language and confusing at times) made me very aware of how long math has been around. That may sound childish at first, but it was extremely eye-opening to realize that though we may now solve problems using different methods, the foundational concepts and ideas are similar. [9] I suppose because these books were collected such a long time ago, we assume that they had a complete different idea of maths, but in fact this is where our understandings stemmed from. The visit to the Edward Worth library made me completely believe and understand that. [START_REF] Barrow | 1677) of the remaining 37: 16 were born before 1640 (and were living during Worth's lifetime, but were much his senior) 16[END_REF] Others again drew attention to the language in which the works were written:

I was also reminded how much I take it for granted that English is the principal language of science today.

[7]

I was taken aback by how easily I could comprehend what was in the books even though they were written in different languages. [18] Overall the response to viewing original sources close up and in the 'intense' atmosphere of EWL was strong and clearly expressed. After giving these initial informal impressions of their visit, students were asked to work on the detail of the EWL Project in two phases. The first phase was introduced as follows:

The overall aim of this project is to produce material that will contribute to an online exhibition of the Mathematical works at EWL. Specifically, you are asked to choose one mathematical book from the collection and explore it in whatever way you can.

Several approaches that students might have adopted were suggested, without being prescriptive. These included (along with many others): finding out about the author (from the web), about the historical context in which they lived and about who influenced their mathematical development.

Students were encouraged to make plenty of rough notes, to be explicit about why they had chosen a particular work, and to relate it to their prior knowledge and readings. Moreover, the librarian sent a message of encouragement with some notes on the availability of English translations of books in other languages using the British Library's English Short Title Catalogue (ESTC). She also recommended paying attention to the preface of their chosen book (if available in English) so as to get an idea of the context of the book and what the author considered important, and to identifying the most striking illustrations in the book (if any).

The four students who could not make the visit had the opportunity to read the feedback of those who did, and compose questions for them. In this way, they were integrated into the work of the EWL Project alongside their peers.

Not surprisingly, some books were chosen by more than one student: seven chose Wilkins (1691), three chose L'Hospital (1696) and 2 chose Tunstall (1538). Having reviewed the work of phase one and given feedback, I nudged students to choose unique books, so that, in the end, 19 distinct books were 'chosen' (with two opting for Wilkins).

In phase two, students were required to review and polish there work, taking my feedback into account, reviewing features of the existing EWL online exhibitions and distilling their work to produce an engaging, informative and reliable piece of between 800 and 1200 words.

Later, in a short 'capstone' exercise, students were asked to identify the two readings they found most helpful for insights into the history of algebra, and give reasons why? Many of them chose readings related to their work on the EWL Project, indicating its strong significance in their view.

Preparing the online mathematics exhibition

Once the students' work was complete, the next challenge was to devise the structure for the online exhibition. The librarian, Elizabethanne Boran, had much experience in this matter and, together, we decided to use the headings (Arithmetic, Algebra, Geometry, Conic Sections and Infinities) in Ward's The Young Mathematicians Guide (1719) as categories for five sections, and to augment these by four more, namely Probability, Applications, Notation and Communities. These nine sections were then to be introduced by an opening section entitled 'What is Mathematics?' A decision was made to incorporate the students' project work to the greatest extent possible (within an appropriate editorial rubric). Each of these thematic sections included links to the works of featured mathematicians. A full list of these mathematicians is given in Table 1 (presented chronologically, with reference to the life of Edward Worth himself).

Number represented in EWL who died (or were born) in a particular period List of those chosen for the EWL Exhibition In this table, three (indicated by *) are included for their editions of Euclid. The exhibition itself can be seen online [START_REF] Oreilly | Mathematics at the Edward Worth Library[END_REF].

Conclusion

This paper has outlined the design and implementation of collaborative work within a module dedicated to the history of mathematics, drawing significantly on original sources. The result of this work has led to the production of an artefact external to the module, the online mathematical Thematic Working Group 12 exhibition at EWL. It is evident that much was learned by all those involved, students, librarian and lecturers, giving clear testimony to the value of engaging deeply with original sources, in this case the mathematical works collected by the medic, Edward Worth, in the late 17 th and early 18 th centuries. It is hoped that many will view the exhibition and thereby enjoy the work of those who created it.

Table 1 : List of those chosen for the EWL Exhibition

 1

Proceedings of CERME10

Acknowledgment

I acknowledge the work of collaborators, Dr Elizabethanne Boran, Fionnán Howard and the twenty BA students (named on the online exhibition). I would like to dedicate this paper to Professor Jan van Maanen who was the first to alert me to the wealth of insight to be enjoyed through working with original sources.