Ysette Weiss
email: yweiss@uni-mainz.de

Rainer Kaenders
email: r.kaenders@uni-bonn.de

History of mathematics and current developments in education

Keywords: Democracy, history of mathematics, educational reforms

With the introduction of so-called educational standards, the German government has mandated by law that German schools, universities and teacher training colleges use the language of competences as a universal language to describe, plan, test and develop teaching and learning processes. This shift to output orientation resulted in a break with the formerly internationally recognized educational tradition of the Enlightenment. In order

Introduction

The present paper responds to a desire that several of our colleagues had expressed at the International Conference on the History of Mathematics Education (ICHME) in Utrecht in the Netherlands in 2017: To place current developments in German mathematics education in a historical context. The reasons for this request were that these colleagues realised similar developments sometimes, possibly with a time shift, in other European countries, such as output and competence orientation, the introduction of educational standards, central tests and global assessments, or the economization, centralization and digitalization of the education system. The study of the history of mathematics and mathematics education seems to support a critical view of these developments. It was pleasant to find congeniality. However, it immediately rose questions which of these issues constitute a common concern. What precisely is going on with these developments? Another aspect of this debate inspired by our lively discussions was whether and how the study of history can support the teaching of critical judgement and the safeguarding of democracy and human values in pedagogy.

In the meantime, some new perspectives such as a better understanding of the origins of competence orientation in psychology (cf. [START_REF] Ryan | Intrinsic and Extrinsic Motivations: Classic Definitions and New Directions[END_REF][START_REF] Gelhard | Kritik der Kompetenz[END_REF] emerged from the study of the developments in mathematics education at the end of the last millennium, and from the authors' attempts to incorporate them into seminars on mathematics education. The reforms and changes we are going to discuss are, however, not specific to mathematics education. Therefore, this discourse should be relevant to the history of education, pedagogy, sociology and psychology.

However, the changes in German education policy at the beginning of our millennium were accompanied by a shift in the subjects and theoretical foundations of educational sciences towards applied psychology, empirical research and the notions of evidence and measurability. The latter had an impact on school development and beliefs and convictions about the superior importance of factors that contribute to good teaching and "effective" learning. Still, if there were to be a lecture about the history of current reforms in education, it would probably be attended by only a few people. Most students put educational sciences on a par with a general methodology (cf. [START_REF] Jahnke | Rede an die Hörerinnen und Hörer der Vorlesung, Einführung in die Mathematikdidaktik, Universität Potsdam im SS[END_REF], and therefore, above all, want to receive the latest and empirically proven approaches in these subjects in order to be able to adapt themselves to the requirements of their future working situation. Here the economization of the university system and the new role of students as customers and future employees plays its part.

Why should the history of mathematics education be taught to future teachers and in which form can it be incorporated in mathematics education? Learning from history does not automatically mean that history prevents us from repeating mistakes. Politicians are not supposed to be historians: Historical situations never completely recur, and therefore the future cannot be predicted from even the most profound knowledge of the past. However, on a small scale with limited demands, it is quite possible to learn from history [START_REF] Geiss | War da was? -Historische Bildung im Output-Zeitalter[END_REF]. The recognition of constellations and gradient patterns occurring over time plays an essential role in this. Even though it is not possible to transfer causal connections, the study of structural components, which recur and make up these constellations and patterns, can certainly contribute to sharpening political judgement. However, the tightrope between showing such patterns and indoctrination through the political or even ideologically influenced production of time references is extremely narrow [START_REF] Bergmann | Geschichte als Steinbruch? Anmerkungen zum Gegenwartsbezug im Geschichtsunterricht[END_REF].

Because of this, and the highly political significance of the reforms, we are going to work backwards: We start with a description of the current situation and ask the students to find differences with the practices, school subjects and events of the past, which look at first sight very similar or carry similar names to those of today. Thus, the formation of analogies does not arise through our study of historical sources, but is rather questioned by these sources.

In the implementation of educational policy requirements, teachers in Germany have a great deal of freedom in the design and application of these requirements through the legally guaranteed freedom of methods [START_REF] Gasser | Methodenfreiheit -Theoretische Grundlagen zu einem Aspekt des Professionalisierungsproblems der Volksschullehre[END_REF]. Dealing with the history of mathematical teaching can help to appreciate existing structures, to include experiences from the history in change processes and to relativize socalled "new approaches".

Through taking a historical perspective on the development of mathematics education and related educational policy, our goal is to support our students' need to question the reasonableness and necessity of political reforms and to shape them as responsible future teachers.

Some features of educational reforms

The study of the history of educational reforms and their theoretical foundations is particularly relevant today. German students have experienced several reforms during their school time. The theoretical foundation of these reforms has not yet taken place. Keywords for these reforms are output and competence orientation, the introduction of educational standards and central tests, the abolition of the orientation classes and pre-school education, the reduction of upper secondary classes by one year, the digitalization of learning environments, the restructuring of secondary schools, and the overall present inclusion1 .

In particular, these political reforms have been pushed through with tremendous pace. Therefore, it is certainly worthwhile to engage in reforms, which had been prepared and installed during half a century; e.g. the Meraner Reform was discussed widely and implemented in small steps [START_REF] Schubring | Der Aufbruch zum "funktionalen Denken": Geschichte des Mathematikunterrichts im Kaiserreich[END_REF]. However, the current reforms have a different character. The abolition of the orientation level 2 and pre-school education, the reduction of upper secondary classes by one year, the restructuring of secondary schools, the shift from special schools for specific disabilities to integrated/ inclusive forms of schooling, the shift from the three-tier school system towards a comprehensive school: All these can be seen as structural reforms of the school system. Looking back at the history of German schools, it strikes us that every of the former changes in the education system was related to just one type of school and perhaps their related types, and was prepared for and carried out over a period of 200 years. The current reforms, however, took place almost simultaneously in a period of just 20 years and involved all school forms at once.

It is worth studying the history of different school types and of preschool education separately and to investigate their links to teacher training and assessment development for teachers and students [START_REF] Leschinsky | Schule im historischen Prozess: zum Wechselverhältnis von institutioneller Erziehung und gesellschaftlicher Entwicklung[END_REF]. There is a large body of literature now starting from original sources like school archives, commission reports and resolutions, as well as secondary literature, which studies these reforms from the point of view of institutional history [START_REF] Müller | Sozialgeschichte und Statistik des Schulsystems in den Staaten des deutschen Reiches[END_REF]. Of course, the study of these sources relates institutional aspects to the biography of its main actors and of the study of political and economic contexts. It is noteworthy that it is often difficult to find historical sources with plans, programs and resolutions, but if one does find them, one knows the names of the main institutional actors. This is not the case with the present documentation of reforms. Here, the authors hide behind huge organizations and their programs. It is hard to find out who is responsible.

Another approach when looking at the history of reforms is to see the teaching community as a community of practice. When doing so, the importance of associations, societies, clubs, unions etc. should also be taken into account. One can also study reforms as the history of concepts, ideas and value systems as they develop in communities.

In order to describe the current changes in mathematics education as a change of value systems from input 3 to output orientation in terms of economization with a measurable and an only functional notion of education, we here give a short historical overview of where the notion of competences comes from.

What is the dispute about competence orientation all about?

The concept of education systems, which is based entirely on economic aspects, has a long tradition in the OECD. The conference documents and the results of the discussion at the "OECD 2 Grade 5 and 6 of secondary school were supposed to be an orientation period in order to decide about the type of school.

3 Input orientation is often misleadingly referred to as the lack of qualification goals and results as well as interpreted evaluations see (Ladenthin, 2011, p.1).

Conference in Washington", which was also decisive for the entire public discussion of educational and educational issues, showed unequivocally in 1961 the limited economic and technical view of human development and progress: "'It goes without saying that the educational system must be an aggregate of the economy, it is just as necessary to prepare people for the economy as real assets and machines. The educational system is now equal to highways, steel works and chemical fertilizers'. Thus the claim can be made 'without blushing and with good economic conscience that the accumulation of intellectual capital is comparable to the accumulation of real capitaland in the long range may outmatch it.'" (Graupe & Krautz, 2014, p. 3) Earlier, Graupe and Krautz explicated: "The same conference volume states that, with regard to developing countries, it would be 'nothing short of cutting a million people loose from a way of life that has constituted their living environment for hundreds or thousands of years. Everything achieved by these countries' schools and education until now has served social and religious aims which have primarily allowed for resignation and spiritual comfort; things that completely go against any economic sense of progress. Changing these century-old approaches may perhaps be the most difficult yet also most important task for education to accomplish in developing countries.'" (loc. cit., p. 2) Here, human development is reduced to economic growth and technical progress. Socio-historical, cultural and educational aspects are not only ignored, but also presented as disturbing and negative. In essence, the views expressed by the OECD have not fundamentally changed since then, but the causes of the need for unifying reforms are now general concerns, such as the growing globalization of the economy and the new requirements of a technological society. 50 years on, the OECD has in its own words, "become central, providing indicators of educational performance that not only evaluate but also help shape public policy." (Gurría, 2011, p.318). The introduction of competence orientation is not a particularly German phenomenon. However, it seems that in Germany it leads to more radical changes in the educational system than elsewhere. The pretext for a radical change in education policy was the "moderate performance" of German students in the PISA test 2003, which was not in line with social expectations. In the media and by politicians it is referred to as the "PISA shock".

Although the OECD had already tested in mathematics decades before, and the TIMS study had already taken place (since 1995), the second PISA test in 2003 was suddenly taken as "the truth" by German politicians and media about the German educational system and as an indicator of what students "really" know. The political goal became to improve the result of tests such as PISA or the TIMS study. Competence orientation promises, among other things, to be able to do this and turn education into a manageable system: "Educational standards with their reference to student competences are explicitly formulated in a way that allows them to be checked with the help of corresponding questions or tests. This measurability characterizes them nationally and internationally, and with all due modesty, it is this characteristic that makes it possible to determine at certain points in time whether and to what extent students are adequately prepared for life or whether there is a need for optimization." [Blum et al. 2006, p. 9

, translation by the authors]

As a measure against the PISA shock, in 2004 the German Conference of Ministers of Education adopted the so-called "educational standards" (Bildungsstandards) and reorganized the curricula on the basis of competence orientation.

However, is 'competences' really just a new word for something akin to learning goals? Has that not always been around? And, who can object to schools and universities at least formulating the goal of ensuring that graduates are fully competent when leaving? Would anyone not want them to apply this knowledge meaningfully and use it to solve "inner-mathematical problems" as well as ones? Do the practitioners and theoreticians of mathematical doctrine focus on an enemy who is not actually a threat? Is it not good that requirements are standardized so that they can be taught systematically? This critical discourse may appear to large parts of a general mathematical audience as splitting hairs. The introduction of competence orientation as a universal and legally prescribed paradigm for the description and design of learning processes, however, has a very specific impact on the mathematical culture in teaching and research. The definition, which is based on the German competency orientation, goes back to Weinert:

Competences in this context are the cognitive abilities and skills available to or learnable by individuals in order to solve specific problems and the associated motivational, volitional and social readiness and ability to successfully and responsibly use the solutions in variable situations. (Weinert, 2002, pp. 27-28, translation

by the authors)

The introduction to this definition in the above-cited text is rarely added:

In this context, the OECD has repeatedly suggested that the ambiguous concept of performance should generally be replaced with the concept of competence.

It is vital for mathematics teachers in schools and universities to understand these developments more thoroughly and to engage in a humanities discourse with pragmatic consequences that cannot be fundamentally clarified by empiricism and that only partially takes place within the mathematical culture. In competence orientation, we are dealing with a fundamental change in our understanding of learning. Is it about learning to understand something, or is it about convincing others on the basis of measurable output that I have understood something?

The conceptual system of competence orientation derives from applied psychology [START_REF] Gelhard | Kritik der Kompetenz[END_REF]. For a long time it was used for the selection and adaptation of workers who were meant to meet specially defined psychological requirements in the workplace, such as patience, accuracy, speed, etc. Although competence orientation with regard to teaching was promoted on the initiative of the OECD (cf. Weinert, 2002, p. 27) and by pedagogical psychologists and educationalists working predominantly on a quantitative empirical basis, there is still no unequivocal empirical evidence to date that suggests that the competence orientation currently implemented by the state has a positive effect on the knowledge and skills of high school graduates or new students.

As of now the concept of competence, the credo of the testing industry, based on a definition by Franz Weinert, has become the central concept of the transformation of our entire education system.

It has evolved from a psychological selection tool into the guiding principle for quality control of industrial production of human capital, as the OECD has promoted for decades. After all -and this cannot be emphasized enough -competences are a psychological instrument. Modeling, collaborating, arguing and even moral competences (Weinert 2002, p. 28) etc. are elevated to context-free problem-solving activities. As they do not have to do justice to any context, they become observable and measurable psychological categories.

Are there developments in the past akin to competence orientation and educational standards?

If we look at competence orientation as a promise of salvation, as a concept of being able to acquire skills applicable to any subject without learning to be an expert in that subject, then we cannot find similar developments during the last 200 years: Skills and knowledge are traditionally closely related to content, either from the perspective of vocational training or from the perspective of humanistic education.

The existence of very general concepts guiding educational reforms leads us to the reform of school geometry, driven by the slogans such as "Neue Geometrie" (new geometry) or "Los von Euklid" (away from Euclid), both in the 19 th century. We may also think of the Meraner Reform and its motto of "Erziehung des funktionalen Denkens" (Education of functional thinking) or the "Neue Mathematik" (New Maths movement) (Schubring, 2014, pp. 241-257), In all these cases, the principles for the reforms were inspired by developments in mathematics such as projective and other non-euclidian geometries, descriptive and analytic geometry in the 19th century and functional analysis, algebra, logic and set theory, and probability theory at the turn of the last century.

If we think about the pretext of the reform the so-called "PISA shock", it sounds quite similar to the "Sputnik shock" on the eve of the New Maths reforms. The Sputnik shock however, was part of world affairs whereas PISA was enacted especially to get an influence on national educational systems.

We traced the development of the notion of competences back to its original use as a psychological selection tool and its modern use to limit the notion of "Bildung" to practical usefulness and functionality. The discourses about role, extent and place of applications and modeling in todays mathematical education have also a history and can be retraced to discussions during the twenties, fifties and eighties. The current so-called modeling problems in A level tests became part of German school mathematics in the context of the PISA shock and its subsequent reforms to improve the poor performance of students in the field of modeling competences.

As we can see, there are various ways to explore the history of mathematics education: On the one hand, we went back to the roots of current educational reforms; on the other, we looked for similar patterns in the past.

Conclusion

We did not answer the question of how to continue the discourse, which started at ICHME, but it inspired us to look for ways to include the history of mathematics education in teacher education. The current gradual shift in the fundamental principles of the German education system make reflecting on these foundations even more important.

Notably the von Humboldt Bildungsideal is built on two notions: the autonomous individual and the cosmopolitan or Universalist (Weltbürger)that is, a universally interested person who cares about the important questions of humankind. The university should beboth for students and professors a place for autonomous individuals to become such a Weltbürger. Student teachers, who are about to become responsible experts for Bildung at school, not only need to come in contact with these ideas, but should also be given opportunities to work on their own Bildung and personal development.

Political activities of the Government to implement the UN Convention on the Rights of Persons with Disabilities.

Acknowledgment

Type any acknowledgment here.


Elena Gonzalez-Blanco

Salvador Ros

María Pablo Ruiz

Luisa Díez

Helena Bermúdez Sabel

Agustín Caminero
email: accaminero@scc.uned.esml.diezplatas

Martínez Cantón

Luciana Ayciriex
email: luciana@linhd.uned.es

Pablo Ruiz Favo

Maria Luisa Díez Platas

Helena Bermudez

Clara I Martínez Cantó

Poetry and Digital Humanities making interoperability possible in a divided world of digital poetry: POSTDATA project

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

ABSTRACT

POSTDATA (Poetry Standardization and Linked Open Data) aims at bridging the digital gap among traditional cultural assets and the growing world of data. It is focused on poetry analysis, classification and publication, applying Digital Humanities methods of academic analysis -such as XML-TEI encoding- [START_REF] Dombrowski | TEI and Project Bamboo[END_REF], (Flanders &Hmali, 2013) in order to look for standardization, as well as innovation by using semantic web technologies [START_REF] Cigarrán-Recuero | Assessing semantic annotation activities with formal concept analysis[END_REF] to link and publish literary datasets in a structured way in the linked data cloud. The advantages of making poetry available online as machine-readable linked data are threefold: first, the academic community will have an accessible digital platform to work with poetic corpora and to contribute to its enrichment with their own texts; second, this way of encoding and standardizing poetic information will be a guarantee of preservation for poems published only in old books or even transmitted orally, as texts will be digitized and stored; third: datasets and corpora will be available and open access to be used by the community for other purposes, such as education, cultural diffusion or entertainment.

POSTDATA will be materialized in the creation of a digital semantic web-based platform for poetry analysis and edition, to study, publish and share digital collections in a virtual research environment using digital humanities open standards combined with traditional philological academic analysis. The environment will be open to any language and type of poetry and accessible for multiple users with different profiles, and it will provide access to digital resources on poetry linked together through data repositories. These data will be subsequently indexed by a search engine and finally provided simultaneously on a Website interface, an API and a single point of access to enriched and open information in accordance with RDF formalism.

The proposal is based in three pillars: Semantic modeling, poetry lab and infrastructure deployment.

a) Semantic Modeling: Linked Open Data (LOD)

The effort of gathering data with an encyclopedic spirit was the origin of poetical repertoires (Horváth 1991). Since then a huge number of them has been developed (see repertories and digital database references). Interoperability among poetic repertoires is not simple, as there are not only technical issues involved, but also conceptual and terminological problems: each repertoire belongs to its own poetical tradition and each tradition has developed an idiosyncratic analytical terminology in a different and independent way for years. There are only a couple of studies which deal with some of the above mentioned aspects (Bootz & Szoniecky 2008 and Zöllner-Weber 2009), but there is not a conceptual model of ontology referred to metrics and poetry yet. The result of this uncoordinated evolution is a bunch of varied terminologies to explain analogous metrical phenomena through the different poetic systems, whose correspondences have been hardly studied (González-Blanco & Seláf 2014, González-Blanco et al. 2014a and2014b). These technologies have shown to be flexible enough to reflect poetic needs, but it is necessary to establish a common conceptual frame to order and classify the philological information in two levels: first an abstract level with general metadata for describing all the classes (such as "poem", "stanza", "line" or "accent") and their properties (such as "has number of syllables" or "has rhyme"), and second an individual "thesaurus" or set of terms to build controlled vocabularies to name each particular phenomenon in any different literary corpora.. As no previous model of such a poetic conceptualization existed before, this model will be one of the main and most innovative contributions of the project.

To achieve POSTDATA goals it has been necessary to face some challenging situations. On the one hand, in order to propose a standard of poetry data model it has been necessary to face the interoperability problems between existent poetic repertoires. This has been solved by creating a common data model for the analysis of European poetry. For this purpose, a metadata application profile (MAP), a semantic model in the Linked Open Data (LOD) framework, has been built and at the time of writing, the validation process is finishing. This MAP will allow the communication of existing data that couldn't be shared before expanding the frontiers of knowledge and research. Once a European poetry Map is developed, the data will be available in machine-readable format that allows us to apply the linked-data paradigm,Figure 1 Figure 1. Excerpt of Postdata Model.

b) Poetry Lab: Natural Language Processing (NLP) and computational linguistics

There have been several interesting approaches to poetry analysis and generation from a computational linguistic perspective using automated linguistic analysis or text mining (Gervas, 2015). Nevertheless, there are two issues that are not easy to overcome: first, most teams that use these technologies are mainly formed by computer engineers with a limited knowledge of linguistic and literary issues, and second, the linguistic rules applied to poetry analysis are often imperfect, as poets usually break the "rules". This is especially frequent when dealing with Medieval or ancient texts, as the exception to the rule are quotidian.

Poetry Lab it is a set of tools ranging all the different levels of poetry scholarship, from the most formal processes related with scansion to the most cognitive ones like metaphor understanding or other related with knowledge and subjective perception involving IA techniques.

On the other hand, the automatization of poetic analysis is , as well, a really challenging situation. Since the poetic analysis is diverse ( different languages, different poetry tradicitons). Poetry Lab is a space where researchers would be able to implement the most up-to-date language technologies and computational methods to process poetry data. Since no set of tools to address basic poetry issues existed before, the Poetry Lab will contribute to the researcher community and users by democratizing technology and user experience.

At this moment tree tools are available in Postadata Poetry Lab ,Figure 2: 1. ANJA devoted to automatic enjambment analysis in Spanish 2. SKAS a first step in metrical Spanish scansion. (lexical syllabification recognizing stressed and unstressed syllables) 3. HISMETAG a Entity recognition framework for Medieval Spanish [START_REF] Cai | Cost Action Interedition www.interedition.eu[END_REF]. Nevertheless, there is not a single platform devoted to poetry analysis, edition, visualization and publication, user-friendly and based on a linked open data system, such as the one proposed in this project. The third pillar of this proposal is focused on the creation of a digital platform for poetry edition oriented to different kind of users: scholars with academic purposes who want to work on critical digital editions, nonexperienced uses that want to read, share and learn more about poetic traditions and also companies who will use this resource for different application in fields like education, psychology, tourism or cultural purposes. It will have the interoperable capacity that allows us "recycling" and integrating previously existing tools that have been developed by other research teams at previous projects. Innovation lies in the application context for this combination of tools, which specifically oriented to poetry analysis.

CONCLUSIONS

As it has been described, the state of digital poetry world is still very idiosyncratic, unconnected and uneven from the technological readiness levels point of view. POSTDATA project aims at becoming a reference in terms of philological digital humanities standardization, interoperability by using linked open data and a digital platform user-centric which will go far beyond research to enhance the digital poetic user experience. This project is based on a crowdsourcing philosophy, as it will create a virtual environment in which scholars and users may add, analyze, publish and reuse poems and data.

Figure 2 .

 2 Figure 2. Poetry tools: http://prototipo-postdata.linhd.uned.es

Acknowledgment

Authors would like to acknowledge the support of European research project ERC-2015-STG-679528 POSTDATA, research project (2014I/PPRO/ 031) from UNED. Furthermore, authors thank the staff of LINHD ─ the Digital Humanities Innovation Lab at UNED


Maria Afrooz
email: mafrooz@mathematik.uni-kassel.de

Rita Borromeo Ferri

Desirable difficulties while learning mathematics: Interleaved practice based on e-learning

Keywords: Desirable difficulties, interleaved learning, blocked learning, e-learning, geometry

Focus and rationale of the poster

The focus of the poster is the presentation of the project about interleaved practice based on elearning with the focus on geometry. The poster is intended to provide an overview of the theoretical constructs, the indication of the content, the study design and possible implications.

Indication of the theoretical framework of the study reported

The theoretical framework of the study refers to the theory of desirable difficulties, which is a new approach from empirical research of learning according to the idea: "To make things hard, but in a good way" (Bjork & Bjork, 2011, p. 55). The learning is made difficult in the short term, but in the long term the learning development improves. Research shows [START_REF] Rohrer | Interleaved practice improves mathematics learning[END_REF][START_REF] Bjork | Memory and metamemory considerations in the training of human beings[END_REF] that desirable difficulties lead to a more sustainable learning success.

The poster focusses on the "interleaved learning", which represents a desirable difficulty. Interleaved practice is defined as an alternating and arrangement of several different learning contents, e.g., A, B, C [START_REF] Lipowsky | Wünschenswerte Erschwernisse beim Lernen[END_REF], see Figure 1). The positive effect of interleaved practice is explained by the cognitive theories of learning psychology, because the sustainable learning results from a higher cognitive demand and a better memory performance [START_REF] Bjork | Making things hard on yourself, but in a good way: Creating desirable difficulties to enhance learning[END_REF][START_REF] Sweller | Cognitive load theory[END_REF].

In contrast to the interleaving is the "blocked learning", which represents one kind of the traditional learning in mathematics lessons. Blocked learning, in turn, facilitates learning by teaching one learning content at a time before moving on to another, e.g., A, B, C (see Figure 2). The possible consequence of blocked practice is that the learners can not remember the first topics after the learning the last learning contents [START_REF] Bjork | Making things hard on yourself, but in a good way: Creating desirable difficulties to enhance learning[END_REF].

Indication of and justification for the content

The described project aims at investigating, whether the learning achievements of interleaved learning students differ from blocked learning students by the use of e-learning. Digital media are an important part of today's society and also used in school education for information, communication and networking processes. In the change of society, traditionally structured teaching must also be adapted [START_REF] Bönsch | Die neuen Sekundarschulen und ihre Pädagogik. Grundstrukturen und Gestaltungsideen[END_REF]. Using e-learning enables an innovative and current method for school education and promotes self-regulated learning (Blaschitz, Brandhofer, Nosko & Schwed, 2012).

Implications for existing research in the area

Possible practical implications for the existing research in e-learning seems to be the development of an e-tutorial for educational settings at school. One goal of the empirical research in didactics of mathematics should be the promotion of sustainable learning in mathematics school lessons. If interleaved learning proves to be a successful new teaching concept in geometry, the few studies on interleaved practice based on e-learning (e.g., [START_REF] Ziegler | Delayed benefits of learning elementary algebraic transformations through contrasted comparisons[END_REF] have to be expanded in various learning contents of mathematics to explore the effects of interleaved practice further.

Figure 1 :

 1 Figure 1: Interleaved Practice with the learning contents A, B, C

Figure 2 :

 2 Figure 2: Blocked Practice with the learning contents A, B, C


Eitan Ben-Haim

Anat Cohen
email: anatco@post.tau.ac.il

Michal Tabach
email: tabachm@post.tau.ac.il

Types of graphic interface design and their role in learning via mathematical applets at the elementary school

Keywords: Educational design, graphic design, schematic graphics, cognitive load, mathematical applets

The research goal was to test the suitability of the character of the graphic design on the student's achievements while engaging with mathematical applets for elementary grades. This quantitative research compares two pairs of mathematical applets, which differ only by the type of graphic design; in the extent of detail and the amount of distraction, while the mathematical problems are identical. The first applet has animated graphics based on designs by Matific, a repository of mathematical applets. The second applet was designed specifically for this research in a schematic and visually simple version of the same activities. Students in the schematic graphics group made fewer mistakes and needed less time to complete the activity than students in the animated graphics group. In addition, students with lower mathematic ability succeeded better in the schematic group. No differences between the two groups were observed regarding the level of students' enjoyment.

Introduction

Gamification is an effective learning strategy in mathematics education at the elementary level. Moreover, a clever integration of educational computer games during the learning process contributes to the student's learning, and provides positive experiences (Wouters & Van Oostendorp, 2013). During integration of mathematical computer games both cognitive processes and motivational implications on the learning are taken into account.

The cognitive load theory and the dual-coding theory [START_REF] Kalyuga | Incorporating learner experience into the design of multimedia instruction[END_REF] depicts the learning process as two independent learning channels, a visual and a textual channel. The decoding of each channel is processed separately in our minds, and only then can be processed together to create a unified meaning alongside prior knowledge. The decoding and processing procedures strain the working memory, which has a limited load capability [START_REF] Cook | Visual representations in science education: The influence of prior knowledge and cognitive load theory on instructional design principles[END_REF]. The split between the two channels can ease the learning process if the two channels are supplementary. However, an ineffective split may cause additional load on the working memory and therefore decrease learning efficiency. Learners who face difficulty with the learning material, and those who have less prior knowledge, are more likely to be "sensitive" to the dual channel split. Prior studies have shown that such learners are less capable of utilizing visual content that accompanies textual content [START_REF] Tsai | Visual behavior, flow and achievement in game-based learning[END_REF][START_REF] Mason | Do fourth graders integrate text and picture in processing and learning from an illustrated science text? Evidence from eye-movement patterns[END_REF].

Aside from the cognitive aspect, another perspective on the effectiveness of learning by computer games, is through the motivational influence it may have on the learner. Enjoying the learning activity may lead the learner to identify with the learning goals and values [START_REF] De Lope | Design methodology for educational games based on graphical notations: Designing Urano[END_REF][START_REF] Chang | Differential effects of learning games on mathematics proficiency[END_REF][START_REF] Sedighian | Can educational computer games help educators learn about the psychology of learning mathematics in children[END_REF]. Graphic design which is consistent with the learner's interests can help the learner relate to the game. Relatedness, as used by [START_REF] Ryan | Intrinsic and extrinsic motivations: Classic definitions and new directions[END_REF], is crucial for achieving intrinsic motivation.

Many studies compare computer games to traditional teaching and identify three advantages of digital technologies: increase entertainment, enhancing motivation of the students, and augmenting their ability to feel involved and active in a learning environment that is individually fitted to their needs (Wouters & Van Oostendorp, 2013;[START_REF] Giannakos | Enjoy and learn with educational games: Examining factors affecting learning performance[END_REF][START_REF] Chang | Differential effects of learning games on mathematics proficiency[END_REF][START_REF] De Lope | Design methodology for educational games based on graphical notations: Designing Urano[END_REF]. However, there is a lack of research comparing learning from different computer games. A gap exists between developers, who are not familiar with the design domain, and designers who are not proficient in the educational and pedagogic area of expertise. This gap causes the challenge of creating digital educational materials consistent with theoretical knowledge in the field [START_REF] De Lope | Design methodology for educational games based on graphical notations: Designing Urano[END_REF]. Learning applets can be compared with many different parameters. This research focuses on one issue: how to design the interface. Specifically, should one use schematic or animated graphics. This research provides a partial and specific response to the lack of empirical work in this area.

The study

The present research stems from a broader study which examines possibilities of better utilizing web-based games in the mathematics classroom, given the practical need for a framework for teachers' choice of mathematical applets, and their integration within the sequence of lessons. Informed by the cognitive load theory and the dual-coding theory, the goal of the presented research is to test the suitability of the character of the graphic design on the student's achievements, in mathematical applets for elementary grades. Three research questions have been studied: (a) How does the learning process differ in learning from animated graphics versus schematic graphics? (b) In what ways are these differences affected by students' proficiency in mathematics? (c) What are the differences in students' enjoyment between graphic designs? Due to space limitation, in this paper we present only the findings relating to the working time and the level of accuracy of the answers (for more details See [START_REF] Ben-Haim | Types of graphic design and their role in learning-games interface in mathematics education at the elementary age: schematic graphics vs. animated graphics[END_REF].

Methodology

This research defined two types of graphics: "animated graphics" and "schematic graphics", which differ in the extent of detail and the amount of distraction (Figure 1). This quantitative research compares two mathematical applets. The two applets differ only by the type of graphic design, while the mathematical problems are identical. The first applet has animated graphics based on designs by Matific, a collection of educational math applets which is in use by many schools in Israel. The second applet was designed specifically for this research in a schematic and visually simple version of the same activities. 287 fifth and sixth grade students (50.9% males, 49.1% females) from three elementary state schools in the center of Israel participated in the activity to collect data for this research. All the children were randomly divided into two equal groups: animated graphics group vs. schematic graphics group. The schematic group employed simple and abstract graphics, based on geometric shapes and solid colors (with no shading or hues). In contrast, the animated group employed more complex and colorful graphics, including simple animation. In the animated graphics there is the appearance of depth and space. The elements are figurative and include expressive characters creating an accompanying narrative.

This research monitored students' activity during their play, which enabled the identification of different levels of success between the two groups, differences in working times, and comparison of the students' reports of their degree of enjoyment from the activity. We examined the differences in the learning processes in the two groups. In addition, the students from both groups were divided into two levels based on their proficiency in mathematics. The differences between the groups in performance throughout the activity were compared using statistical tests (t-tests, ANOVA and chisquare) to identify statistically significant differences. Privacy of the children was protected in this research. The schools' names were replaced by numbers, and students' names were not mentioned.

Findings Differences in the learning process between the two groups

The research results show evidence that students in the schematic graphics group made fewer mistakes than students in the animated graphics group. Figure 2 shows the percentage of students who answered correctly on the first, second or third attempt (light blue and light orange refer to the percentage of students that answered correctly on the first attempt). As can be seen, for all three questions included in the applet, the schematic group (blue) answered with fewer attempts (meaning fewer mistakes) than the animated group The differences between the groups are greater in the first and third questions, which were more difficult than the second. Further results show that the schematic group needed less time to complete the activity. Notably, these differences between the groups were observed on the first game of the applet, but not the second.

Differences between the two groups and the students' proficiency in mathematics

The difference between the two graphic groups is even greater among the students with weaker math achievements (Figure 3). Students with lower mathematic ability succeeded better in the schematic than in the animated graphic group. Further results arise from two measures by which the students are classified as having low-or highscores in mathematics: according to the student's school's performance, or according to the student's answer to a final question which was identical in the two groups. Some differences were found, though not always consistently, in distinguishing between low-and high-performing students. In some cases, the animated group was more challenging to low-performing students than to highperforming students. Also, low-performing students in the animated group, required significantly more time than other students.

Differences in students' enjoyment between graphic designs

Contrary to expectations, no differences were observed regarding the level of enjoyment that students reported at the end of the activity with the applet. Students from both graphic groups, and both performance levels reported similarly on how much they liked the applets. These findings show no advantage for the animated graphics design with regards to satisfaction.

Discussion

The findings of this research show differences between the two graphic groups. Students in the schematic graphics group made fewer mistakes and needed less time to complete the activity than students in the animated graphic group. In addition, students with lower mathematic ability succeeded better in the schematic group. These findings consistent with the literature on cognitive load theory and on the dual-coding theory [START_REF] Kalyuga | Incorporating learner experience into the design of multimedia instruction[END_REF]. In some cases, the animated group was more challenging to low-performing students than to high-performing students. Also, low-performing students in the animated group, required significantly more time than other students. These results are consistent with professional literature claiming that low-performing students and students with lower prior knowledge, encounter difficulties integrating visual and verbal inputs [START_REF] Cook | Visual representations in science education: The influence of prior knowledge and cognitive load theory on instructional design principles[END_REF][START_REF] Mason | Do fourth graders integrate text and picture in processing and learning from an illustrated science text? Evidence from eye-movement patterns[END_REF]. This means that graphic design of mathematical applets must be adapted to the mathematical knowledge of the learners. Students may find the activity harder when the graphic design is complex, if the activity generates significant cognitive load on the working memory. These findings lead to the conclusion that complex graphics may cause inhibitive cognitive load depending on the learning situation and on the student's performance. Notably, no differences between the two groups were observed regarding the level of students' enjoyment.

The advantage of gamification in the learning process is in its ability to hide the real necessity to learn mathematics (that is remote to the child), and present a goal that the learner can understand and relate to, which also leads the learner to achieve the real goal of learning mathematics [START_REF] Sedighian | Can educational computer games help educators learn about the psychology of learning mathematics in children[END_REF]. Gamification may lead to learning via inner motivation, out of interest and curiosity with a clear goal in mind. The design of learning materials should speak in a familiar tone and be relevant to the learner's world [START_REF] Hui | Learning mathematics through computer games[END_REF].

The common misconception regarding digital learning is to think of technology as if it were "chocolate covered broccoli" (Bruckman, 1999, p.75). The advantage of digital gamification is not simply as a means of up grading conventional learning through external motivation [START_REF] Habgood | Motivating children to learn effectively: Exploring the value of intrinsic integration in educational games[END_REF]. In order to truly leverage digital integration, the applet must be embedded in the learning process, and must stimulate an active intellectual activity [START_REF] Sedig | From play to thoughtful learning: A design strategy to engage children with mathematical representations[END_REF]. The visual graphics of mathematical applets must act as the starting point for understanding the problem in a given exercise, leading the learner to insight and solutions [START_REF] Van Garderen | Visual-spatial representation, mathematical problem solving, and students of varying abilities[END_REF]. To do this, the graphics must integrate between the game-play of the applet, its plot and the learning content (Wouters & Van Oostendorp, 2013).

The philosopher Marshall McLuhan saw the medium as the source of change in society and coined the phrase "the medium is the message" (McLuhan & Fiore, 1967). In his view, it is not the content transmitted through the medium, rather than the medium itself that comprises the meaning, it is the medium itself that significantly influences the society. In our opinion, this insight echoes in the context of integrating applet-games into the classroom. Applets excite the learner simply by the use of the screen, rather than by the displayed content. It is our belief that as educators, we must not "take a step back" when the enters the classroom. Rather, we must insist on taking an active role directing and navigating the digitally enhanced learning process.

This research was performed with the attempt to maintain objectivity as much as possible, in the collection and analysis of the data. However, there are some limitations in this research. From practical considerations, the studied population was limited, and comparison between the groups was based on only two applets among a large pool of applets. Hence the studied population is not representative of Israeli students, and the activities are not representative of general mathematical learning games, nor of Matific applets in particular. It should be noted that, from among the three schools, only students from school 1 were not familiar with Matific before data was collected. While this research advances the understanding of development and design of digital learning interfaces in math education, further research is needed to extend these preliminary insights, to enhance the benefits of digital learning, and to sharpen teachers' guidelines for implementation.

Conclusion

The results of this research assist us in formulating indicators for teachers, that will enable them to address both personal/experiential aspects of the learner, as well as cognitive aspects. In light of this research, three guidelines have been deduced which could assist developers, designers and teachers to reach a more effective integration of mathematical applets in the elementary school environment:

(1) Different graphic designs are suitable for different audiences and for different learning situations. Complex and animated design can cause difficulties if the learning situation is challenging; (2) A schematic graphic design, with few unnecessary details, is more appropriate for applets that tend to have a high cognitive load on the learner; (3) The design must consist of a narrative that connects between the interests of the learner and the content of the learning material. Figure 4 presents the conceptual chart flow that enables practical application of the research findings. As shown in figure 4, the teacher must first identify the characteristics of the existing graphics in the applet in question. Two parameters need to be taken into account: (1) complexity of the graphics, and ( 2) to what extent the graphics serve a pedagogical role. Graphic complexity is rich in contains animations, describes facial expressions and may contain a third dimension, similar to the animated group in this study. In contrast, schematic graphics portray minimal detail often represented abstractly. The role of graphics covers a spectrum from pedagogical to motivational. Pedagogical graphics are integrated with the learning content and complete that content by visualizing it. Motivational graphics merely motivate and do not clarify the learning material.

In the second step, the teacher may examine the applets suitability to the target audience. Figure 4 depicts the questions that can be asked in the process and portrays two possible scenarios, each leading to a different outcome with respect to a different type of graphic design. Applets identified as having "overwhelming" graphics should lead to closer attention to the following indicator in overall decision making. The three scroll bars at the bottom of figure 4 express different aspects for the given audience. The right-hand side (blue) indicates difficulties that the learners encounter. The left-hand side (orange) indicates less challenging learning situations. The two scenarios depict clear-cut cases, where the three markers clearly lead to one side or the other. To conclude, schematic graphics is preferable when encountering challenging learning situations.

The research conclusions can assist developers and designers of instructional materials in decision making regarding graphics in relation to learning content. Applets built on a linear continuous sequence of activity can design a gradual increase of the graphic load, so that the learner in the early stages of the applet will gradually adapt to the new interface. As the learner progresses throughout the activities, the graphics may unfold into a more complex environment, without having a negative effect on the learner due to the familiarity with the interface. Another rule of thumb regards applets addressing the learning stages of instruction in which the graphics may maintain simplicity and a schematic abstract approach, as opposed to the stages of practice, assessment, evaluation, or recall in which the graphics may consist of a more complex design. This implies that the design team and the pedagogic team must work together in planning educational applets.

While the animated group presents a more attractive design than the schematic group, the former does not incorporate the learning content within the visual design. This is a common major disadvantage among applets. Developers and designers must manage the split between the applet plot and the learning material in order to truly utilize the advantages of digital gamification. This study provides rules of thumb on considerations regarding future development and design of mathematical applets.

Figure 1 :

 1 Figure 1: Screenshots of two applets from the two types of graphics (translated into English -originally in Hebrew)

Figure 2 :

 2 Figure 2: Percentage of students according to success in answering during the applet

Figure 3 :

 3 Figure 3: Percentage of attempts to answer during the first applet, comparison of groups and of students' proficiency

Figure 4 :

 4 Figure 4: A conceptual chart for selecting graphics according to the characteristics of the students

Acknowledgment

This study was funded by the chief scientist of the Israel Education Ministry.


Chantal Buteau
email: cbuteau@brocku.ca

Eric Muller

Kirstin Dreise

Joyce Mgombelo

Ana Isabel Sacristán
email: asacrist@cinvestav.mx

Students' Process and Strategies as They Program for Mathematical Investigations and Applications

Keywords: Programming, Instrumental Genesis, Strategies, University Mathematics Education

This paper focuses on the process of university mathematics students engaging in a sequence of programming-based mathematical project tasks as part of a course. Data of this naturalistic research was collected mainly through four student projects and semi-structured individual interviews. The analysis led to narratives of students' development process (instrumental genesis) in which enacted strategies (instrumented actions) are highlighted. In this paper we discuss a participant's development process. Results suggest that the student, after 1 course, has appropriated programming as instrument for creating a tool for pragmatic purposes; however, not yet as instrument for mathematics investigations and applications, i.e., as an object-to-think-with (Papert, 1980).

Introduction

There is a resurgence of interest in integrating computer programming-more broadly, computational thinking (CT)-in education (e.g. in UK; [START_REF] Benton | Bridging primary programming and mathematics: Some findings of design research in England[END_REF]. This interest reflects how scientific fields, including mathematics, have developed computational counterparts [START_REF] Weintrop | Defining computational thinking for mathematics and science classrooms[END_REF], as well as the rise and need for proficiency in computational practices as 21st century skills. We see a crucial need to understand how students can be empowered to participate in such computational thinking, now an integral part of the mathematics and broader community.

In our work, we want to better understand how students come to appropriate programming as an instrument for mathematics investigations and applications "as mathematicians do". This appropriation involves students developing fluency (e.g. enacting strategies to progress forward) as they are (in the process of) engaging in this kind of mathematical work. This includes, not only becoming skilled at programming, but also designing and interpreting the mathematical work facilitated, or made possible, by programming. In this proposal, we discuss the case of a student's fluency development, analysed from the first-year data that was collected as part of a five-year study that examines how postsecondary mathematics students learn to use programming as a CT instrument for mathematics. It is a naturalistic (i.e., not design-based) research that takes place in a sequence of three project-based mathematics courses (called 'MICA') implemented in the mathematics department at Brock University (Canada) since 2001, where undergraduate mathematics majors and future mathematics teachers learn to design, program (e.g. in VB.net), and use interactive computer environments to investigate mathematics conjectures, concepts, theorems, or real-world applications [START_REF] Buteau | Integration of programming in the undergraduate math program at Brock University[END_REF][START_REF] Muller | Learning mathematics through the design and implementation of exploratory and learning objects[END_REF].

Conceptual Framework

Our work is framed by various interrelated concepts reflected in the literature on CT, CT in mathematics, and in mathematics education [START_REF] Buteau | Computational thinking in university mathematics education: A theoretical framework[END_REF]. [START_REF] Wing | Computational thinking benefits Social Issues in Computing[END_REF] defines CT as "the thought processes involved in formulating a problem and expressing its solution(s) in such a way that a computer-human or machine-can effectively carry out" (p. 5), while [START_REF] Hoyles | Revisiting programming to enhance mathematics learning[END_REF] consider CT as abstraction, algorithmic thinking, decomposition, and pattern recognition. CT is an underlying process to computer programming described e.g. by [START_REF] Weintrop | Defining computational thinking for mathematics and science classrooms[END_REF] as understanding, modifying, and writing codes (e.g., in Python or C++). [START_REF] Brennan | New frameworks for studying and assessing the development of computational thinking[END_REF] identify key dimensions for developing CT, namely computational concepts (e.g. iteration), computational practices (e.g. debugging projects or remixing others' work), and computational perspectives (e.g. a designer's evolving views about what could be programed).

CT has changed the nature of contemporary research in mathematics; for example, we now see computer-based proofs and new domains of research related to mathematics and computation such as bioinformatics. The European Mathematical Society (2011) recognizes this emerging way of engaging in mathematical research: "Together with theory and experimentation, a third pillar of scientific inquiry of complex systems has emerged in the form of a combination of modelling, simulation, optimization and visualization" (p. 2).

Based on their literature review and interviews with experts who use CT, [START_REF] Weintrop | Defining computational thinking for mathematics and science classrooms[END_REF] outline what they believe to be the integral CT practices for mathematics and science, in particular encompassing this third pillar [START_REF] Broley | Legitimate peripheral) computational thinking in mathematics[END_REF], across four main categories throughout which programming is an underlying practice: data practices, modelling and simulation practices, systems thinking practices, and computational problem-solving practices. E.g., the latter practice involves interpreting and preparing problems for mathematical modeling, assessing different approaches, developing modular solutions, and creating computational abstractions.

In the field of mathematics education, CT is not new; indeed, it has a 45-year legacy that started with the LOGO programming language [START_REF] Papert | Mindstorms: Children, computers, and powerful ideas[END_REF] and extended into the theory of constructionism. Studies of constructionism in undergraduate-level mathematics education show how programming supports students' understanding of mathematical concepts and contributes to the development of critical thinking skills (e.g., [START_REF] Wilensky | Paradox, programming and learning probability[END_REF]. As for students' processes when engaging in using programming for mathematical investigations and applications, we proposed a model based on insightful reflections on MICA student experiences [START_REF] Buteau | Student Development Process of Designing and Implementing Exploratory and Learning Objects[END_REF] which was later refined through a literature review [START_REF] Marshall | Learning by designing and experimenting with interactive, dynamic mathematics exploratory objects[END_REF]; see Figure 1. We use [START_REF] Lave | Situated learning: Legitimate peripheral participation[END_REF] concept of "legitimate peripheral participation," which describes how learners enter into a community of practice and gradually take up its practices, to understand how undergraduate students learn mathematics through CT activities. Based on this idea, mathematics is not knowledge to be acquired but rather is a process of participation through which the student gradually gains membership to a community (of mathematicians). We thus focus on how students create and use computer tools to engage in opportunities to participate peripherally in practices considered to be integral to the mathematical community as outlined by [START_REF] Weintrop | Defining computational thinking for mathematics and science classrooms[END_REF], i.e., in the four CT practice categories. Thus, our work is focussed on how students (newcomers) engage in computational thinking for mathematics as mathematicians (elders) do. In fact, the proposed development process of a student engaging in programming for mathematical work as summarized in Figure 1 (removed of its numbering) seems to align with the process by which pure mathematicians use programming in their research work (Broley, Caron & St-Aubin, 2018).

In this proposal, we focus on this process and the strategies that students enact when engaging in CT-based mathematics work. In the instrumental approach framework (Rabardel, 1995(Rabardel, /2002)), the process is related to a student's instrumental genesis [START_REF] Artigue | Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work[END_REF]: the process by which the student transforms an artifact into an instrument through schemes of usage and action. The instrumental genesis is a twofold process: instrumentalization -directed towards the artefact-, and instrumentation -directed towards the user-concerning "the development and appropriation of schemes of instrumented action which progressively take shape as techniques that permit an effective response to given tasks" (Artigue, 2002, p. 250). These schemes of instrumented action consist of technical and conceptual components. According to [START_REF] Drijvers | One episode, two lenses[END_REF], techniques may be considered "as the observable part of the students' work on solving a given type of tasks (i.e., a set of organized gestures)" (p.27), e.g., strategies used by a student at a certain step/cycle (see Figure 1) when engaging in a given programming-based mathematical task; and the schemes "as the cognitive foundations of these techniques that are not directly observable, but can be inferred from the regularities and patterns in students' activities"(p. 27). As [START_REF] Bozkurt | Instrumental genesis of a preservice mathematics teacher: instrumented actions for perpendicular line construction in a dynamic geometry environment[END_REF] summarize:

An artefact is initially not meaningful to the user until he or she develops associated schemes of instrumented action to use the artefact for achieving a task, and effectively turning the artefact into a useful mathematical instrument. (p.44) E.g., turning (VB.net) programming into an instrument for mathematics investigation or application as mathematicians do, which we call 'a CT-instrument for mathematics'. In this proposal we discuss first results commencing to address our research question: How do post-secondary students come to appropriate programming as a CT-instrument for mathematics?

Methodology

Our research uses iterative design methods whereby some parts (participant recruitment and data collection) were designed in a way that would be least intrusive to (or constrained by) the natural learning environment. The study follows students' development over the course of (and beyond) their MICA I-II-III courses as they engage in 14 exploratory object (EO) mathematics project tasks, each resulting in an interactive environment and a report of mathematical findings [START_REF] Muller | Learning mathematics through the design and implementation of exploratory and learning objects[END_REF]. This paper focusses on a first analysis of data collected from the first year of our research. It thus draws from MICA I students where six (among 46) participants were recruited (voluntarily). In the MICA I course, there are 4 EO project tasks (which count for 71 % of students' final grades): 3 assigned individual ones, and a fourth one where students select the topic and can work in pairs or individually. Two-hour weekly labs progressively integrate through guided exercises the learning of computational concepts (variables, loops, etc.) in a mathematical context, whereas two-weekly hour lectures introduce the math background needed for engaging in the EO tasks [START_REF] Buteau | Integration of programming in the undergraduate math program at Brock University[END_REF].

Data from the participants included each participant's 4 project assignments (EO and report) and individual semi-structured interviews after completion of each of these EO tasks. The design of the interview guiding questions was informed largely by the students' development process model (Figure 1). In addition, data collected included weekly post-laboratory session online reflections (answering guiding questions) and an initial baseline online questionnaire before the beginning of MICA I course. Lab session and EO assignment guidelines were also collected and were complemented by an informal understanding of the MICA I course from the research team members' experiences in different capacity (researcher, instructor, course approver, former student).

All of the collected student data were analysed through thematic analysis techniques. Codes were developed based on categories informed by the conceptual framework (and associated literature). Each participant's interview and lab reflection data was coded individually by two coders, followed by a thematic analysis done jointly by the two coders. Themes were consolidated among the six participants' analyses, and led to sixteen themes regrouped in five main meta-themes, one of which concerned strategies. Narratives were composed, following the model steps from Figure 1, of each participant's process of engaging in their four EO projects, highlighting their enacted strategies, and overall summary. We now present preliminary findings by illustrating one of the six participants' development process with a focus on her use and development of instrumented techniques, i.e., enacted strategies (in italic, below), as a way to gain insights into her instrumental genesis towards appropriating VB.net programming into a CT instrument for mathematics.

Findings: the case of Hannah

Hannah (pseudonym) is a mathematics and computer science co-major, and thus had programming experience (in Java) prior to her MICA I course. In her first assigned EO project -exploring a conjecture of their choice about prime numbers or hailstone sequences-, Hannah struggled with the initial step of conjecturing (steps 1-2 in Figure 1); once she was passed this, she incrementally designed and programmed (steps 2-3) with ease her interactive environment (EO). When the time came to interpret the mathematics output of her program (steps 5-7), Hannah struggled and provided only data examples without mathematical explanation. In the EO 2 project on a RSA encryption application, Hannah reported feeling more satisfied and confident, finding it overall easier, in particular since she could start right away with designing and programming (i.e., step 1 was provided by the assignment guidelines). However again, her focus on the mathematics (step 5-7) seems to have been limited. We elaborate next on Hannah's engagement in her third EO task.

Hannah's development process and enacted strategies in her third assigned EO project task

Students are then asked to design, program, and use an interactive environment to explore, graphically and numerically, the behavior of a dynamical system based on a two-parameter cubic [START_REF] Buteau | Integration of programming in the undergraduate math program at Brock University[END_REF]. It follows an introduction of discrete dynamical systems in lectures, guided computer lab activities to program an EO to explore the logistic function system (quadratic, 1 parameter), including an instructor guided exploration of the system as the parameter is changed. At this time, students have been introduced to all basic computational concepts for mathematical work: variables, interface design, loops, conditionals, events, sequencing, and graphing. This assignment has a significant component on mathematical investigation. For Hannah, it is a critical learning opportunity since she struggled with the mathematical components of the previous two EO projects.

Similarly to EO 2, the assignment guidelines provided the focus of the EO project (step 1 in Figure 1). Hannah voices that she did not research the related mathematics (step 2) when beginning the assignment because the lectures and labs were sufficient; however, in reflection, she believes she would have needed more time spent on the math concept to grasp it. In fact, Hannah missed some lectures due to illness and felt unable to catch up with the mathematical concepts. As in the past assignments, Hannah understands the programming concepts and is able to start designing and programming her project (step 3) in lab time by remixing from the two previous labs on the logistic function dynamical system. She used the computational practice of incremental steps with testing to complete the programming (programming cycle)see Figure 2 for a screenshot of her EO:

Hannah: the first step was setting up everything. … we have to setup the graph... regardless of … 'a' and 'b'… we're converting it to pixels instead of x and y's… So, we had to make sure that was working first before you could plug in anything. but struggled with understanding the cobweb:

Hannah: one of the challenges was, um, the, um, trying to figure out if it actually does converge. Because, um, the way we visualize it is, um, for each of the points. So, if you have 10 points, uh, 10 sequence numbers that you have, you draw lines between them. But, because we don't have like, sort of like, the starting point of the line and the ending point, you don't know if it is going that direction, which direction are going.

In the final part of the assignment, the report, Hannah's lack of understanding becomes clearer. Although her program works, her report does not reflect understanding; in particular, she uses the program to generate an example, rather than elaborating calculations and drawing it by hand as asked (similar to how she completed her EO 2). Her report is lacking in explanations or conclusions to communicate her mathematical knowledge (step 7) and the discussion portion of the report is missing. This EO project could have been a great positive learning experience for Hannah to use programming for mathematics (learning) as this assignment was heavier on the mathematics investigation than previous ones. However, Hannah missed the opportunity partly due to external factors (illness).

Hannah's overall trend over the 4 EOs

Throughout the course, Hannah's focus seems to remain on programming rather than moving on to programming for mathematics, as revealed through her general approach to the 4 EO assignments.

Hannah finds difficulty in identifying on her own, a mathematics topic relevant to a programmingbased approach (step 1). She prefers to be given a conjecture (as in EO 2) and avoids creating her own when possible (as in EO 4). Once the mathematics problem or conjecture is sorted out, Hannah becomes much more comfortable. Most of the time, she researches thoroughly and uses resources to be well prepared for a solution before starting to program. She wants to understand the mathematics a priori and then program it, rather than using programming to understand (her general approach to 'interpreting and preparing problems for mathematical modeling' practice). This method causes difficulty in EO3 as programming for investigation is an intended way for students to understand. Hannah uses her previous programming experience to help her complete the design and programming (step 3) with ease and little help from others: she works incrementally and uses functions and modules to structure her program (her approach to 'developing modular solutions' practice). Overall, Hannah seems to treat the course as a programming class for designing mathematics calculators rather than a mathematics class using programming as a tool for understanding and investigation, i.e. as an "object-to-think-with" [START_REF] Papert | Mindstorms: Children, computers, and powerful ideas[END_REF]. This perspective seems to limit her ability to dive in and explore new mathematical concepts deeply. In the final interview, Hannah said she intends and is looking forward to take the MICA II course. It will be interesting to see how (and if) she adapts her approach, enacting new strategies, in this more mathematically demanding course.

Conclusion

We have described Hannah's development process (including her strategies) as she engages with designing, programming, and using mathematics EOs. We observed that Hannah seems to appropriate programming as instrument for creating a tool for pragmatic purposes -i.e. as a calculator-however, not yet as instrument for mathematics investigations and applications -i.e., as an object-to-think-with [START_REF] Papert | Mindstorms: Children, computers, and powerful ideas[END_REF]. The findings provide insights on Hannah's development of fluency and instrumental genesis -i.e. how far she has come, through MICA I course, to appropriate (VB.net) programming as a CT instrument for mathematics investigations and applications. The transition from an instructionist to a constructionist approach to learning mathematics seems to have remained a challenge for Hannah, even with her previous programming background. In the near future and with the current trend, it is expected that students will arrive in mathematics department with programming skills. The case of Hannah can thus be considered as of a typical 'student' of the near future, and as such, provides insights of how these typical students' instrumental genesis may develop.

The first year student data analysis led to refine further the development process model (see step 2 and step 3* in Figure 3) to better capture the different strategies enacted by students. By highlighting how Hannah's strategies are enacted in the development process model, our study provides a beginning of how we can explore the role of strategies, as techniques, in students' instrumental genesis, leading next to study the related instrumented action schemes. The next steps, in our 5-year research, are also to bring together all of the thematic analysis findings and the development narratives for a deeper insight into students', including Hannah's, instrumental genesis, and to continue follow our participants into their upper-year MICA courses. It will also include the examination of instructors' adoption of learning environment (i.e., instrumental orchestration).

Figure 1 .

 1 Figure 1. Development process model of a student engaging in programming for a mathematical investigation or application (Marshall & Buteau, 2014)

Figure 2 .

 2 Figure 2. Screenshot of Hannah's third EO project about the 2-parameter cubic dynamical system Hannah tested the mathematics output of her program against other students' work and known examples from class (step 4); e.g., Hannah says: Hannah: I used the, um, same values that, um, [the instructor] put in the labs. And, um, I checked, I compared the sequence and saw the math was working and then the graph and how it looks. However, she had difficulty in investigating the given math topic or understand its meaning (steps 5 & 6), although she did investigate the cubic function by systematically varying its parameters: Hannah: I left 'b' fixed at zero and I increased 'a' a little bit and little bit and I see how it affected the curve….So, yeah, I played around with [a and b] until, um, until I got the fixed points.

Figure 3 .

 3 Figure 3. Hannah's summarized enacted strategies during the development of her 4 EO projects

  Hannah developed multiple methods of checking her mathematical work including using online resources, calculating by hand, and comparing known examples. Because of her heavy preparation (in step 2) and strong programming skills (in step 3) there are not often many bugs to solve (step 4). When attempting to use the program to investigate the math problem (step 5) beyond the creation of the project, Hannah seems to lack confidence. She often isn't sure how or what to conclude from the program output. She doesn't seem to know how to use the program for investigation and wants to understand first, then "explore" later to confirm what she knows rather than to discover new ideas. Her limited investigations make it hard to integrate the results with her math knowledge (step 6). She finds some results noteworthy, but not necessarily what is intended as part of the investigation and her insights, rather than mathematical, are more often related to programming. Lastly, when communicating her results in her written report (step 7), her descriptions are often short and not thorough in explanations. She provides examples that are either too simple to demonstrate knowledge or uses her program to generate an answer without further explanation of the process. Figure3summarizes Hannah's enacted strategies over the course of her 4 EO tasks.

Acknowledgement

This work is funded by SSHRC (#435-2017-0367) and received ethics clearance (REB #17-088).