text:
Boris Koichu
email: boris.koichu@weizmann.ac.il

Nelly Keller
email: nellyk@tx.technion.ac.il

Keywords: Problem solving, online forums in social networks, diffusion of innovations

The goal of this article is to present and theorize our more successful and less successful attempts to create and sustain problem-solving forums, in which exploratory discourse takes place. The main argument is that many implementation-related phenomena that we have encountered when working with seven high-school classes for one or two school years can be characterized and explained with the aid of conceptual tools provided by Rogers' Theory of Diffusion of Innovation. The most successful process of forming an online forum in one of the classes is presented in some detail, and the parallel processes in the rest of the classes are presented in the form of an aggregated summary. Implications for future design-based implementation research are drawn.

Implementation enterprise through the lens of a theory of diffusion of innovations: A case of online problem-solving forums Introduction

This article presents an implementation aspect of a research project entitled "Heuristic and engagement aspects of learning through long-term collaborative mathematical problem solving"1 .

The main research goal of the project (on-going until September 2017) is to produce a model of learning through mathematical problem solving, which would be attentive to cognitive, socioaffective and contextual aspects of this activity. In particular, the model is supposed to attend to interactions between variations in heuristic behaviors [START_REF] Koichu | Patterns of middle school students' heuristic behaviors in solving seemingly familiar problems[END_REF][START_REF] Koichu | On the relationships between (relatively) advanced mathematical knowledge and (relatively) advanced problem solving behaviours[END_REF] and socio-affective engagement structures [START_REF] Goldin | Beliefs and engagement structures: Behind the affective dimension of mathematical learning[END_REF] activated when high-school students collaboratively cope with challenging mathematics problems for relatively long time.

An implementation aspect of the project consists of designing and sustaining a special learning environment, in which long-term problem solving might be investigated. The intended environment comprised of a particular combination of problem-solving lessons in a classroom and out-ofclassroom work supported by online asynchronous discussion forums. Its design was strongly informed by past research on affordances of online learning environments. Past research tells us that students in online environments can actively participate in solving complex problems for 2-3 weeks almost without teacher interventions [START_REF] Moss | Knowledge building in mathematics: Supporting collaborative learning in pattern problems[END_REF] and that some of those students who tend to be silent in a classroom can actively participate in online discussions [START_REF] Schwarz | E-moderation of synchronous discussions in educational settings: A nascent practice[END_REF]. In addition, there is evidence that online discussions enable students to meaningfully use their mathematical knowledge, enhance self-regulation skills and support knowledge construction [START_REF] Nason | Fostering authentic, sustained, and progressive mathematical knowledge-building activity in Computer Supported Collaborative Learning (CSCL) Communities[END_REF][START_REF] Tarja-Ritta | Students' activity in computer-supported collaborative problem solving in mathematics[END_REF]Stahl, 2009). [START_REF] Schwarz | E-moderation of synchronous discussions in educational settings: A nascent practice[END_REF] attribute the benefits of online discussions to their unique traits, such as: fostering divergent rather than linear interactions, enabling flexible time schedules of participation in the discussions over relatively long periods of time, encouraging explicit and accurate expression of the ideas in writing. Koichu (2015ab) argues that many affordances of an online problem-solving forum stem from its fundamental characteristics of being a choice-affluent environment, that is, an environment, in which the students are empowered to make informed choices of: a challenge to be dealt with, a way of dealing with the challenge, a mode of interaction, an extent of collaboration, and an agent to learn from. In brief, past research on online collaborative problem solving presents many evidence-based cases of successfully working online forums.

However, little is known from the professional literature about how to put the forums into work and sustain them. Our experience in the aforementioned project taught us that this enterprise is truly challenging. The goal of this article is to make sense and theorize our more successful and less successful attempts to create and sustain problem-solving forums, in which exploratory rather than expository problem-solving discourse (this distinction is due to Stahl, 2009) takes place. The main argument is that many implementation-related phenomena that we have encountered when working with seven high-school classes (grades 10 and 11) can be characterized and explained with the aid of conceptual tools provided by Roger's (2003) Theory of Diffusion of Innovation.

Conceptual framework

Approach: Design-Based Implementation Research (DBIR) [START_REF] Werner | A guide to implementation research[END_REF] refers to implementation research as the systematic study of the implementation of innovations. [START_REF] Fishman | Design-based implementation research: An emerging model for transforming the relationship of research and practice[END_REF] point out that this type of research encompasses studies of fidelity, of variations in implementations as well as studies of conditions under which programs can be implemented effectively. They further refer to implementation research and to design-based research as antecedents of a new, emerging research model, which they name Design-Based Implementation Research (DBIR). The core principles of DBIR are as follows:

(1) a focus on persistent problems of practice from multiple stakeholders' perspectives;

(2) a commitment to iterative, collaborative design;

(3) a concern with developing theory and knowledge related to both classroom learning and implementation through systematic inquiry; and (4) a concern with developing capacity for sustaining change in systems. (Fishman, Penuel, Allen, Cheng and Sabelli, 2013, pp. 136-137) In addition, DBIR calls for breaking down barriers that isolate those who design and study innovations and those who study the diffusion of innovations. We find the DBIR concept and principles well-adjusted to the needs of our project.

Vocabulary: Selected elements of the theory of diffusion of innovations

The notion innovation is frequently used in the literature on implementation research as a selfexplanatory one [START_REF] Fishman | Design-based implementation research: An emerging model for transforming the relationship of research and practice[END_REF]. However, there exists a branch of the professional literature that explicitly focuses on innovations and the processes of their diffusion. In particular, Rodgers (2003) defines innovation as "an idea, practice, or object that is perceived as new by an individual or other unit of adoption. It matters little, so far as human behavior is concerned, whether or not an idea is 'objectively' new as measured by the lapse of time since its first use or discovery" (p. 11). In our case, the idea of stretching the boundaries of a classroom by means of an online problemsolving forum was an innovation because it was new to the students and the teachers.

Rodger's ( 2003) theory of diffusion of innovations meticulously characterizes the innovationdecision process, in which individuals (or other decision-making units) decide whether to accept an innovation or not. In particular, Rodgers distinguishes five stages of the process: knowledge, persuasion, decision, implementation and confirmation. The stages are briefly presented below.

At the knowledge stage, potential innovation adopters are exposed to the innovation's existence and obtain some information about how it functions. Sometimes individuals become aware of an innovation by accident, and sometimes they actively look for it in order to fulfill particular needs. It is also possible that the needs are formed as a result of one's exposure to an innovation.

At the persuasion stage, an individual forms a favorable or unfavorable attitude towards an innovation. This stage presumes affective involvement with the innovation. In particular, the individuals may mentally apply the new idea to their present or anticipated future situation. They seek to answer such questions as "what are the innovation's advantages and disadvantages in my situation?", and seek the answer mostly from their near-peers, whose opinions based on their personal engagement with an innovation, are the most convincing. There is a discrepancy between forming a favorable attitude towards an innovation and an actual decision to adopt it. Adoption of an innovation can be influenced by a cue-to-action, an event that crystallizes an attitude into overt behavioral change.

At the decision stage, an individual adopts (i.e., makes full use) or rejects an innovation. Any decision is not final however. The rejection can occur even after a prior decision to adopt; in Rodgers' terms, this phenomenon is called discontinuance. The theory distinguishes between active and passive rejection. The former type of rejection consists of considering adoption of the innovation and then deciding not to adopt it. The latter one consists of never "really" considering the use of the innovation. The decision stage frequently includes a small-scope trial. The actual sequencing of the three stages presented so far can alter. Namely, both knowledge-persuasiondecision and knowledge-decision-persuasion sequences are possible.

At the implementation stage, an individual puts an innovation into systematic use. Even though the decision has been made, the adopters may still feel a certain degree of uncertainty about the consequences of the innovation. In addition, problems of how exactly to use the innovation may emerge. Sometimes the adopters change or modify (in Rodgers' terms, re-invent) the innovation at this stage. The implementation stage can be lengthy, but it ends when the idea that has once been innovative becomes institutionalized and regularized in the adopters' normal functioning.

Finally, at the confirmation stage, an individual constantly seeks reinforcement for the decision to adopt or reject an innovation that has already been made. As a result of positive or negative messages about the innovation, the decision can be reversed. Rodgers points out that the change agents (i.e., those who influenced one's decision to adopt an innovation) have responsibility of providing supportive messages to the individuals who have previously adopted the innovation.

Methodological aspects of the project

Participants and the project's activity

Two experienced mathematics teachers and two of their 10 th grade classes took part in the first year of the project (2013-2014); five more teachers and their corresponding five classes joined the project during its second year (2014)(2015). Each participating teacher acted in the project as a member of the research group and took part in the meetings of the group. In addition, each teacher worked in contact with an additional member of the group who was responsible for the technological support and documentation of the activity. Mathematics in all participating classes was studied for five hours a week, in accordance with the Israeli high-level curriculum (see Leikin & Berman, 2016, for details). For the concerns of this article, it is enough to mention that geometry was studied two hours a week and that its study included systematic work on proving tasks.

We planned that each participating in the project class would be engaged, at least three times during a school year, in the following activity. The students cope with a series of preparatory tasks during a 90-minute lesson and are offered an especially challenging geometry problem at the end of the lesson. They then engage, for 5-10 days, in solving the problem from home in a closed (that is, available only to the students of a participating class and the members of the research group) online forum. Different technological platforms, including Google+ and WhatsApp, were tried in different classes. When the problem is solved or, alternatively, when the students give up, a 90-minute lesson is conducted in the classroom in order to get closure. The lesson consists of whole-class and smallgroup discussions, during which the students share their experiences with the problem.

Documentation of the project

Forty-two meetings of the research group were audiotaped (about 100 hours) and, in addition, documented in the protocols of the meetings (more than 100 pages). The documents produced by the group and all relevant email exchange were stored. Every member of the group was required to keep a diary. The diaries were for writing anything their authors deemed important for the project, including their thoughts and feelings in relation to the project's events. The diaries were stored in shared Google Drive of the group and were available for reading and commenting by the members.

In addition, 14 lessons were videotaped, the content of the forums was stored (more than 3000 posts), interviews with students and teachers were conducted (about 15 interviews), and the students' written feedback on different aspects of the project was collected.

The story of NK's class, which is presented below in some detail, is produced using narrative inquiry methodological tradition. As Clandinin and Caine (2008) explain, "Narrative inquiry is marked by its emphasis on relational engagement between researcher and research participants" (p. 542). This approach was chosen because we (hereafter, BK and NK) had been active members of the processes under exploration; in particular, NK was a mathematics teacher of the class. An aggregated summary of the stories in the rest of the classes is produced using a general inductive approach (Thomas, 2006), which enables researchers "to condense extensive and varied raw text data into a brief summary format" (p. 238).

Findings A (success) story of NK's class

The main events at the knowledge phase of the project in NK's class consisted of: (1) a conversation between NK and BK following BK's observation of one of NK's lessons;

(2) a conversation between NK and her students. Because of the first conversation, NK decided to take part in the project mainly because the idea to stretch the boundaries of a classroom by means of an online forum resonated well with NK's constant need to enrich her teaching repertoire in order to create valuable learning opportunities for her students. In Rodgers' terms, NK acted as a venturesome innovator who is able to cope with high degree of uncertainty about an innovation, and BK acted as a change agent. In her conversation with the students, NK acted as a change agent, and the students were potential innovation-adopters to be persuaded. NK argued that developing problem-solving skills was a strong benefit of participating in the project, and appealed to the students' curiosity to try something new and be a part of an interesting initiative. The students' reaction to the information about the project was favorable, though not exactly for the reasons that NK had presented.

The first mathematical problem of the project is presented in Figure 1. It is representative of most of the problems of the project. In particular, it looked similar to geometry problems the students were familiar with from classwork and homework. As such, the problem "invited" the students to approach it by means of mathematical ideas that worked well in the past. For instance, the students might think of including the angles, whose equality is to be proved, in a pair of triangles and attempt proving their congruence by finding some equal elements. However, such a general plan was insufficient; something else (e.g., a clever auxiliary construction) should have been invented.

Nine-Square Problem: There is a net of nine congruent squares (see the drawing). Prove that the two angles denoted in the drawing are equal. When the problem was uploaded to the Google+ forum, three students worked on it. Their threehour-long brainstorming session was unsuccessful. As a result, the forum was non-active during the next two days. The following day NK met the students at school and asked: "Why did you stop solving the problem? It is not too difficult". The students showed NK their hand-made drafts as evidence that they had tried. NK asked the students to upload their drafts to the forum and continue solving the problem together. That evening eight students entered the forum, cooperated and eventually solved the problem. Two solutions to the problem by the active participants of the forum and an additional solution by a student who was a silent observer were presented at the mathematics lesson following the forum. The students' voluntarily expressed their suggestions as to how to further run and improve the forum by the end of the lesson, In Rodger's terms, the first three students acted as venturesome innovators. Rodgers points out that this category of adopters is important for launching a new idea, but they have little influence on other individuals' decision to adopt or reject the idea. The conversation between NK and the students in school was crucially important as a cue-to-action for eight students, who acted as early adopters. Rodgers characterized this category of individuals as respectable, that is, well-integrated members of a local community whose opinion about the innovation matters for the potential adopters. The mathematics lesson, in which these eight students shared their positive experience at the forum with the rest of the class, was another crucially important cue-to-action.

The next two problems of the project were approached on the forum by about the same group of students. The students learned to share their half-baked ideas, and even developed some rules related to publishing the full solutions at the forum. In brief, they agreed that a student who obtained the full solution should not publish it early, in order to not "spoil the fun" for others. The forum was indeed exploratory rather expository in nature. Three months later, collaborative problem solving at the forum became a well-established practice for six students; most of their classmates joined the forum occasionally and constituted the early majority. It seems that each student has decided how and to which extend to use the forum. The implementation stage (three to eight month from the beginning of the project) was characterized by gradual two-directional diffusion of social and sociomathematical norms developed in the forum and in the lessons. In particular, there were several forum-like lessons initiated by the students (see Keller & Koichu, in press, for details).

The evidence of confirmation of the students' decision to adopt the innovation came from the following sequence of events. As mentioned, the Google+ forum flourished for several months, but it we have not yet mentioned that then it was deserted. NK inquired with the students about this fact and discovered that the activity moved from the Google+ to WhatsApp, a popular social network in Israel since about 2014. The students granted NK access to their WhatsApp forum, and we were happy to find there many autonomous problem-solving discussions of exploratory nature. The WhatsApp forum flourished in NK's classroom until the students' graduation in 2016.

An aggregated summary of seven stories

An aggregated summary of the conduct of the project in all participating classes, by Rodgers' phases, is presented in Table 1. The sign "+ +" in the table means that the stage is fully realized; "+" means that the stage is partially realized (e.g., the forum was active only as a trial or only few students were active); "-" means that there were no conditions for realizing the stage; "Active R" and "Passive R" stand for active and passive rejection, respectively. An empty cell means that the project did not arrive at that stage.

NK class AP class AH class ES class RN class OG class LA class

Knowledge

As Table 1 tells us, only NK's class went through all five stages, up to the point when using the online forums for problem solving stopped being an innovation and became a routine. Implementation of the project's idea at the scope comparative to NK' class occurred in one other class, and partial implementationin three classes. In two classes the project did not reach the implementation stage, despite of much effort made by the teachers and the research team.

Concluding remarks

Recalling Tolstoy's seminal assertion, happy families are all alike; every unhappy family is unhappy in its own way, we can argue that there is a unique story behind each cell of Table 1.

Unfortunately, we cannot tell these stories here due to the space constraints. In brief, sometimes school conditions or classroom norms were inappropriate for realizing the project's idea, sometimes a particular cue-to-action event did not happen at the right time or was not appropriately designed, and sometimes our decisions and actions as a research group were far from being optimal. We have also observed, more than once, the phenomena of discontinuance and of passive rejection for which we do not have convincing explanations, despite the extended data set in our possession.

We intend to continue the aforementioned project, and one of the lessons learned so far is that the DBIR concept and theories like the Rodgers' theory of diffusion of innovation should be taken seriously. Either detailed or aggregative analysis of implementation of the project idea is helpful for us as a tool for refining the roadmap of the project. In addition, we now better understand that creating conditions for implementation of an innovative pedagogical idea in a school reality should be given full attention prior to delving into a pursuit for "traditional" research questions, such as questions on cognition and affect in mathematical problem solving that have been the main research questions of the project. Based on the accumulated experience, we call for reporting and analysing not only those cases where an innovative idea is being fully implemented, but also those case where the implementation was partial or did not occur as planned. We conclude by suggesting that systematic attention to implementation issues, by means of DBIR, may have not only practical, but also fundamental theoretical significance in mathematics education.

Figure 1 :

 1 Figure 1: The first problem of the project

Table 1 : The implemented stages of the project in seven classes

 1

		+ +	+ +	+ +	+ +	+ +	+ +	+ +
	Persuasion	+ +	+ +	+ +	+ +	+ +	-	+
	Decision	Accept Passive R Accept Passive R Accept		Active R
	Implementation	+ +	+	+ +	+	+
	Confirmation	+ +

Selected findings of the project are reported in[START_REF] Lachmy | The interplay of empirical and deductive reasoning in proving "if" and "only if" statements in a Dynamic Geometry environment[END_REF],Koichu (2015ab),Keller and Koichu (in press).Thematic Working Group

Proceedings of CERME10

Proceedings of CERME10

Acknowledgment

The research project is partially supported by the Israel Science Foundation (Grant No. 1593/13; PI Koichu). We are grateful to all the participants in the project, and especially to the teachers who cooperated with us and devoted much time, thought and effort to making the project work.


domain: math shs

text:
Uffe Thomas Jankvist

Mogens Niss

The notion and role of "detection tests" in the Danish upper secondary "maths counsellor" programme

Keywords: Detection test, "maths counsellor", learning difficulties

This paper presents and discusses a specific aspect of the Danish "maths counsellor" programme for upper secondary school, namely that of detection tests. More precisely, the purpose and design of a detection test is presented, as is the prospective counsellors' use of the test. In the description, emphasis is placed on the ways in which detection tests assist in informing the maths counsellors in their work with students experiencing learning difficulties in mathematics.

Introduction

The words "test" and "testing" are omnipresent in educational research and practice in general and in mathematics education in particular. However, the actual notions covered by these terms are very diverse, as are their roles and uses. Basically, the terms mean "critical examination of a person's or a thing's qualities". There is an abundance of different purposes, goals and objects of testing, as well as a multitude of different approaches to and instruments for testing. Without attempting to outline a comprehensive theory of test and testing, one distinction is worth introducing in the context of mathematics education, a distinction between direct and indirect testing. In direct testing, the test object directly epitomises the very purpose and goal of the testing. If your purpose is to find out whether a given person can actually drive a car, a direct test consists of taking the person to trial in real car driving. In indirect testing, the test object is devised as an indirect indicator ofa probe intosomething that is not identical to the test object itself, because this "something" is either inaccessible in direct terms or too large or too complex to be fully represented by the test object. So the test object becomes a proxy fora representative ofthe underlying, but necessarily indirect, object of testing. Mathematical learning, understanding, reasoning, modelling, and problem solving are just a few examples of such underlying objects of testing, for which a wide variety of test objects are only more or less well-chosen proxies. Mathematics education research and practice make extensive use of indirect testing, alongside direct testing.

The key issue concerning indirect testing is what relationships can be established between respondents' responses to the test object and the real underlying object of testing. The generic question is: What do responses to the test object tell us about the respondent's qualities in relation to the real underlying object of testing. Since, in indirect testing, these objects are not identical it requires a non-trivial amount of clarification of concepts, of interpretation and analysis, and oftentimes of independent empirical research to account for the inferences that can be justifiedly drawn from responses to the test object onto the underlying object of testing. The "detection tests" in focus in this paper are instruments for indirect testing. So, the above-mentioned generic question has to be specified for our context. Our research question then is: In what respects and to what extent does the detection test presented below allow for the detection of students with mathematics specific learning difficulties regarding mathematical concepts and concept formation?

Due to space limitations we are not able to fully answer this question here and to fully corroborate the answer. Instead, we confine ourselves to providing some key points in an answer, that is by describing the context in which the detection tests are used; by providing an overall description of what a detection test is; and finally by means of an illustrative and authentic example. It should be noted that although the actual content of the detection tests is based on research findings and issues considered in the mathematics education literature, the notion and role of "detection test" in the sense presented here have been introduced by us and thus have not been described previously.

The "maths counsellor" programme

This section is based on [START_REF] Jankvist | Fostering an intimate interplay between research and practice: Danish "maths counsellors" for upper secondary school[END_REF]. The maths counsellor in-service teacher programme at Roskilde University [START_REF] Jankvist | A framework for designing a research-based "maths counsellor" teacher programme[END_REF] runs part time over three semesters (in total 30 ECTS -European Credit Transfer and Accumulation System), during which the upper secondary teachersideallyhave a reduced teaching load at their schools. Each semester has an overarching theme: (1) concepts and concept formation in mathematics; (2) reasoning, proofs and proving; (3) models and modelling. These themes were chosen both because they are significant to upper secondary mathematics education in Denmark, as is spelled out in the national curriculum documents, and because they epitomise key aspects of the eight mathematical competencies in the Danish KOM-project [START_REF] Niss | Competencies and mathematical learning ideas and inspiration for the development of mathematics teaching and learning in Denmark[END_REF], which constitutes the theoretical foundation of the maths counsellor programme. The teachers' work in each semester is structured in terms of three different phases: (1) to identify (i.e., detect and select) students with genuine learning difficulties in mathematics; (2) to diagnose the learning difficulties of the student(s) identified; and finally (3) undertaking intervention according to the diagnosis arrived at with respect to the individual student.

At the very beginning of each semester, the teachers are equipped with a theme-specific detection test, consisting of questions and tasks for the students in relevant classes at their schools. As will be exemplified below, these tests are developed by us and are informed by research literature regarding the specific theme. The purpose of the test is to assist the teachers in detecting students with potential learning difficulties in mathematics. Usually, each teacher detects several such students, some of whom are selected for being offered maths counselling with the aim of rectifying or reducing the observed difficulties during the semester. This typically leads to the identification of 1-4 students per class in need of, and also interested in, receiving counselling. When speaking of mathematics specific learning difficulties, we rely on our previous definition given in (Jankvist & Niss, 2015, p. 260), i.e. "those seemingly unsurmountable obstacles and impedimentsstumbling blockswhich some students encounter in their attempt to learn the subject. These stumbling blocks include, but are not limited to, a wide range of misconceptions, misinterpretations, misguided procedures, inadequate beliefs etc. with regard to established notions of mathematics. We do not include general learning disabilities, cognitive or affective disorders and the like." The purpose of the counselling is not to motivate unmotivated students, but to assist those who work hard in mathematics on a daily basis but do not succeed.

In the diagnosing phase, the participating teachersstrongly assisted by the research literature they read as part of the programme (see [START_REF] Jankvist | A framework for designing a research-based "maths counsellor" teacher programme[END_REF] employ self-constructed tasks, interviews, etc., to come to grips with the nature and origin of the students' mathematics specific learning difficulties. Taking the diagnosis as the point of departure and with support from the research literature and supervision by us, the teachers design and implement an intervention scheme for the students selected. The intervention scheme also includes steps which enable the counsellors to "measure" in what respects and to what extent the intervention has worked as anticipated for the selected students. For each semester, groups of 2-3 teachers write up a report. After the completion of the third semester, all three reports are combined into one, along with an introductory chapter. This final report forms the basis for a final oral exam at the university. The teachers who pass receive a diploma as certified maths counsellor.

What is a detection testand what is it not?

A detection test, as designed for the Danish "maths counsellor" programme, is a set of maths questions to be answered by upper secondary student classes (grades 10-12) within a time frame of 60-90 minutes without time pressure. The questions are short, both in their formulation and in the sense that they neither require lengthy procedures or computations nor longwinded explanations. Moreover, the questions do not involve conceptually complex or technically involved mathematics beyond standard upper secondary school mathematics. However, the questions are usually not routine questions either. On the contrary, many of them are deliberately posed in such a way that they break the "didactical contract" of upper secondary mathematics and require students to think and act independently. Danish upper secondary school takes three years and students usually enter at the age of 16 after having completed ten years of mandatory comprehensive primary and lower secondary schooling. Upper secondary students can choose to have mathematics for one, two or three years; three years being the advanced level. Danish upper secondary school covers three streams: general, technical, and business.

The primary purpose of a detection test is to be one among several instruments for detecting students possessing genuine learning difficulties in mathematics, within the relevant theme of the programme. So, the focus is not primarily on detecting the difficulties themselveseven though the tests do have something to offer to that end as well, because the questions in a detection test are composed such that wrong answers, individually or in combination with others, may suggest the potential presence of particular kinds of learning difficulties with a student giving these responses. As mentioned, a detection test is not meant to stand alone. When it comes to detecting students with learning difficulties, other sources of information, e.g. the teacher's prior knowledge of the students have to be taken into account as well. More precisely, a detection test may be seen as having three different roles. Firstly, in cases when the test, within a certain area or theme, points out students who by the teacher/counsellor were already suspected to have difficulties within that area, the role of the detection test is to strengthen the teacher's observations. Secondly, in cases when the test singles out students who were not already detected by the teacher, the test serves to amplify and sharpen the teacher's attention and to supplement his or her own observations of the students. Thirdly, it is also a purpose of the detection test to provide an initial support in pointing out the specific sub-domains within the test's theme, in which a detected student displays difficulties. Of course, students' test responses may not only indicate difficulties within particular mathematical topics; students' response patterns may also suggest overarching difficulties of a more principal or general nature. Thus, this third role of a detection test then typically is to provide inspiration for the following "diagnosis" (cf. later sections).

It is important to keep in mind that a detection test is not meant to be a fair test of the students' attainment levels in the subject of mathematics, neither when it comes to content knowledge, skills, and proficiency, nor when it comes to mathematical competence at large or to inventiveness or special mathematical talent. Due to the fact that detection tests are designed with a different purpose in mind, several important aspects of the usual handling of mathematicse.g. familiarity with concepts and facts, computational skills, or proficiency in solving standard routine tasksare not in focus of the tests. Similarly, the test cannot be used as a screening test in the usual sense, attempting to chart students' possession of various mathematical competencies. However, employed on a larger population of students, e.g. a year group in a given school, the test may of course be used as a screening test for the potential presence of mathematics specific learning difficulties pertaining the theme of the test, within this population, but the test is still much more focused than a general screening test for attainment level or competencies.

Even though the test contributes to singling out students with potential learning difficulties, it cannot determine, in itself, whether a given student actually possesses such difficulties. It is certainly possible to encounter poorly performing students whose erroneous answers are not due to mathematics specific difficulties, but to ill-will and shoddy job, lack of accept of the didactical contract with or in the test (e.g. because the test is not supposed to influence teachers' marks, or because the questions are of a different nature than usually encountered by the students), a bad day on the time of testing, or maybe to much more general learning difficulties (or disabilities) that manifest themselves in several subjects, not only in mathematics. To determine whether a student detected by the test actually possesses mathematics specific difficulties, supplementary means must be applied as well, not least the teacher's knowledge of the student.

Beside the fact that the test, for a student who has been "detected" by it, may provide important indications for a subsequent diagnosis of mathematics specific difficulties, the test is not a diagnostic test. It requires an independent diagnostic process to uncover the specific nature of observed learning difficulties as well as the sources actually responsible for them. Oftentimes, preliminary hypotheses concerning the nature of the difficulties, and what may have caused them, must be supplemented withor even replaced byother hypotheses as the diagnosis proceeds. This may be due to much more deeply rooted difficulties than the ones observed at first, e.g. regarding more fundamental mathematical conceptions and beliefs than those in focus of the detection test.

An illustrative example of algebraic equations and equation solving

As mentioned above, in each semester of the programme the maths counsellors are equipped with a detection test related to the theme of the semester. Hence, detection test 1 concerns mathematical concepts and concept formation (we intent to discuss detection tests 2 and 3 in subsequent publications). This test consists of some 57 questions (and sub-questions) on selected topics relevant for Danish upper secondary school. These include: concepts of number (including fractions, decimals, negative numbers, irrational numbers); percent; algebraic expressions and transformations; equations (first and second degree, with different types of numbers as coefficients and solutions, and with the unknown on both sides of the equal sign); simple functions and aspects of the coordinate system; and finally a selection of mathematical conventions such as: different symbolic notations for fractions; the equal sign; the inequality sign; minus and negative numbers. Out of the 57 questions (with sub-questions) around ten questions concern equations and equation solving. In the following we shall focus on examples of this.

As suggested by various researchers (see e.g. [START_REF] Kieran | Learning and teaching of algebra at the middle school through college levels: Building meaning for symbols and their manipulation[END_REF], students' difficulties in solving algebraic equations are of two rather distinct kinds. The first kind is related to transformation of equationsand algebraic expressionsby means of permissible operations, eventually leading to solutions. This not only involves knowing and understanding the scope and legality of the operations at issue, it is also to do with the nature and structure of the number domains implicated, the meaning of the equal sign, and the arithmetic operations involved, etc. The When a maths counsellor gives the detection test to a group of students, perhaps a larger cohort of studentssay a class or a year groupcertain patterns are likely to reveal themselves. For example questions 17 and 18 may tell us something about the students' algebraic understanding, e.g. the students' perception of how variables may and may not be denoted (anything other than x is often rejected as a variable). Questions 35a and 36a address the first kind of difficulty of solving algebraic equations, namely the operational aspect in relation to the number domains involved. Question 35a is an example of what [START_REF] Filloy | Solving equations: The transition from arithmetic to algebra[END_REF] call a "non-arithmetical equation", referring to the fact that the unknown appears on both sides of the equal sign. Question 36a may give rise to difficulties due to the appearance of the negative coefficient and division by a negative number, but also the situation of having to accept a negative number as a solution. On the one hand, questions 17, 20, and 25 may tell us about the second kind of difficulty mentioned above, i.e. knowing what a solution to an equation means, as well as about the consequences of the fact that an equation may have infinitely many solutions. On the other hand, they may also tell us something about the students' conception of equality in relation to equations and equation solving. From extensive experience, we know that Danish students have difficulties with equations that have either no solutions or any number as a solution. Question 37 addresses another aspect of the second kind of difficulty. Despite the fact that the vast majority of students are not able to correctly answer question 20, a large number of students will say that 0 is indeed a solution to the same equation in question 25. More interesting, perhaps, are those students who are able to answer that all numbers satisfy the equation 3xx = 2x, but still answer "no" to 0 being a solution. This may have to do with a belief that solutions are positive integers or be an aspect of more fundamental difficulties with 0.

To illustrate what an overview of a large student population may reveal, we provide table 1, which displays a binary ("correct-incorrect") coding of 676 Danish upper secondary students' responses from 2012 and 2013 (from all three levels and streams). For the 405 1 st year students participating in the study we may confirm that questions 20 and 37 are indeed difficult ones, since 92.8% and 85.4%, respectively, cannot answer them correctly.

As an illustration of two maths counsellors' use of the test in regard to equations and equation solving, we present the story of student Å [START_REF] Christensen | En elev med saerlige matematiklaeringsvanskeligheder[END_REF]. Student Å followed the mathematics programme at intermediate level at a general upper secondary school. The two maths counsellors spotted student Å at the beginning of Year 1, and then worked with her for three consecutive semesters, while they themselves were enrolled in the maths counsellor programme. In relation to the above questions on equations, student Å answered incorrectly on both questions 17 and 18 ("no"), she left question 20 unanswered but answered question 25 incorrectly ("no"), and left questions 35, 36, and 37 unanswered. The two maths counsellors initially interpreted this as if she had difficulties with the transformation of algebraic equations and with algebraic expressions in general, since she also gave incorrect answers to: [6] What is (a / b) • (b / a)? (Where neither a nor b is 0.) (Å: "a 2 / b 2 ".) and [50] If a = b is then b = a? (Å: "no.").

[ Based on interviews which confirmed that student Å most certainly had difficulties in solving algebraic equations, and handling algebraic expressions in general, the two maths counsellors designed a series of small interventions focusing on solving various equations, arithmetic as well as algebraic ones, etc. [START_REF] Filloy | Solving equations: The transition from arithmetic to algebra[END_REF]. Soon, however, the maths counsellors began to suspect that Å's difficulties had indeed deeper roots. Student Å found that negative numbers as well as fractions were "ugly", and on one occasion she uttered "0 that's not a number!" Sometimes student Å had difficulty at distinguishing the operations of addition and multiplication. When trying to find the difference between two numbers, she counted on her fingers. When having to find how many times 8 divides 24, she answered "four" by counting "8, 12, 16, 24", and even double checked the result by repeating the same count. Having to perform the division 24/6, she eventually gave up and replied "I don't know when the numbers are so big." It turned out that student Å had fundamental difficulties with the concept of number, including understanding of numbers, number domains and handling of numbers. She appeared only to be on safe ground when operating with small natural numbers, where calculations can be performed on her fingers. Student Å's difficulties extended to her (in)ability to correctly apply basic mathematical terms. Thus, on a later occasion she used the term "diameter" (of a pizza) as just another unit along with centimeter and millimeter.

Upon revealing the depth of student Å's difficulties and acknowledging these to be the cause of her symptomatic difficulties with equation solving, the obvious question becomes whether the detection test might have provided us with some indications of this. In hindsight, even if there were several questions that Å left unanswered, the ones she answered erroneously do seem to corroborate her subsequently revealed learning difficulties: [7] Is the numbera positive or negative or is it not possible to decide this? (Å: "negative").

[13] Which number is larger: 13/3 or 13/4? (Å: "13/4").

[15a] What is x + 0? (Å: "0x").

[26] Round off 148.72 + 51.351 to an integer. (Å: "149 + 51 = 191").

[27] Which of the following fractions are equal: 1/4, 4/16, 4/12, 2/8? (Å: "4/12 and 2/8").

[52] If a < b is b > a? (Å: "no").

[53] Is (a -1) / (b + 1) = (a / b) -1? (Å: "yes"). 55. Is (a + 3) / (a + 4) = 3/4? (Å: "yes"). 56. Is (a -1) / (b -1) = a / b? (Å: "yes"). In total, the occurrence of the above erroneous answers, together with the responses to the previous questions on equations, indicates the presence of a manifest learning difficulty syndrome with student Å.

The revelation of mathematics specific learning difficulty "syndromes"

We now return to our research question, i.e. in what respect and to what extent do detection tests allow for the detection of students with mathematics specific learning difficultieshere exemplified by concepts and concept formation regarding equations and equation solving? As we saw in the case of student Å, she most certainly was detected to possess the potential learning difficulties as suggested by the test. Clearly, the counsellors' first hypothesis concerning Å's difficulties regarding concepts and concept formation was insufficient. However, we should keep in mind that this was the first time ever that these counsellors used the instrument of an indirect detection test. Once the maths counsellors become accustomed to the instrument and skilled in using it, they tell us that they are able to make much more accurate initial hypothesesor even preliminary "diagnoses". Indeed, having experienced a case like student Å, our two maths counsellors are able to make much more qualified initial hypotheses concerning students' difficulties. Seeing the answers to the questions above on numbers, conventions, etc., these maths counsellors will no longer suspect a student "merely" to have difficulties with solving first degree equations; they will see this as a likely symptom of more deeply rooted and fundamental difficulties.

Indirect tests, such as the detection test outlined above, may mislead the interpreter of test outcomes in several ways. In the case of student Å we saw that the maths counsellors at first mistook the student's apparent difficulties for her real, more fundamental difficulties. Another example, which we have also seen time and again, is where students are perfectly able to solve algebraic equations in an instrumental manner, but do not understand the relational aspects of the operations they perform or the very meaning of the solutions they arrive at (for references, see [START_REF] Jankvist | A framework for designing a research-based "maths counsellor" teacher programme[END_REF]. This is to say that if the aim is to "train monkeys" to find solutions to equations, then this is certainly possible. Our aim with the indirect detection test is to go deeper, since "our" object of learning is more complex than to mechanically obtain a solution. Our aim is to pave the way for drawing conclusions that are much broader than what the test questions ask, taken at face value, e.g. we insert "spot probes" into aspects of students' mastery of numbers and algebraic expressions and attempt to come up with hypotheses concerning their concept of number in general: if a student comes up with this and that erroneous answer, (s)he most likely possesses such and such learning difficulties; or if, on the contrary, the student can give correct answers on this particular set of questions, then it is fair to assume that (s)he has actually grasped, in a relational manner, something significant about the entities involved.

As illustrated above, an indirect test such as a detection test may function both on an individual student level and on larger populations. For example, in the case of student Å we noticed that she answered incorrectly to question 15 and question 50 (cf. above). This we interpret as an indication of student Å not believing 0 to be a number and possibly possessing misconceptions of equality. But how special are these misconceptions for a 1 st year student like Å? The coding among the 405 1 st year students displayed in table 1 revealed an error rate of 18.3 for question 15 and 13.6 for question 50. In addition, question 14, asking what 0 • x is, which student Å answered correctly, has an error rate of 18.8 among the 405 1 st year students. This is to say that if questions 14 and 15 are taken as markers of difficulties with the number 0, and if the population of the 405 students is representative, then it might be expected that more than one sixth of the students in a class at the beginning of Year 1 will have the number 0 as a "stumbling block" in some sense. The indirect detection test may suggest the presence of syndromes, on an individual level as well as on the level of populations.

  second kind of difficulty is to do with what an equation actually is, and what it means for an object to be a solution to an equation. Detection test 1 includes the following questions, among others: [17] Are there any values of a such that a 2 = 2a? [18] Are there any values of b such that 4b = 4 + b? [20] What is the solution(s) to the equation: 3xx = 2x? [25] Is x = 0 a solution to the equation: 3xx = 2x? [35a] Solve the equation: 3x + 20 = x + 64. [36a] Solve the equation: -6x = 24. [37] For what x do we have 38x + 72 = 38x? Our purpose here is to illustrate two things: what the maths counsellor may learn from using the test on a larger population of students; and what the maths counsellor may learn about a single student from his or her answers to the test questions.

Table 1 : Binary coding of 676 student answers to selected questions from detection test 1 (coding by Morten Elkjaer Hansen as part of his master's thesis at Aarhus University, 2016)

 1

		17]	[18]	[20]	[25]	[35]	[36]	[37]
	1 st year (405)	148	285	376	193	168	149	346
	Error rate (%)	36.5	70.4	92.8	47.7	41.5	36.8	85.4
	2 nd year (196)	45	166	172	87	54	44	139
	Error rate (%)	23.0	84.7	87.8	44.3	27.6	22.5	70.9
	3 rd year (75)	12	64	49	25	25	18	43
	Error rate (%)	16.0	85.3	65.3	33.3	33.3	24.0	57.3

Proceedings of CERME10


text:
Lisser Rye Ejersbo

Morten Misfeldt
email: misfeldt@learning.aau.dk

From theory to praxis

Keywords: Design research, base-10 system, grades K-2, top-down and bottom-up project

This article is a description and discussion of a design research project in which we introduced a research idea about the influence of language on number concepts development into praxis on a school in grades K-3. Danish children have difficulties remembering the Danish number names because the Danish language resembles a primitive number concept in mathematical thinking. In the project, we renamed the numbers between 11-99 after the base-10 system. Our hypothesis was that this system would help Danish students to get a more secure concept about the base-10 system. The project lasted for three years ending in spring 2016. Our results were so convincing that the school decided to continue using the mathematical number names, and other schools that heard about the project seem to be interested in using the system as well. In other words, the project goes from being a top-down project to a bottom-up project.

Introduction

In this article, we describe a design research project, which took place in a public school in the primary grades K-3. We started with a hypothesis, and then it moved on to a design and then contact with a public school. We ran the project for three years and then worked with the school on its continuation. The main concern of our paper is that of existing and anchoring projects that have proven successful and engaged (part of) the organization with which we have collaborated.

Our research idea was provoked by the fact that the Danish number names [START_REF] Ejersbo | Danish number names and number concepts[END_REF][START_REF] Ejersbo | The relationship between number names and number concepts[END_REF] have an etymology and wording that are both peculiar and impractical. For example, in Danish, the number 73 is treoghalvfjerds (three and half-four), the number 32 is toogtredive (two and thirty), and the number 16 is seksten (sixteen). These old roots are unknown to most students. Furthermore, the number names are abbreviated. In Danish, the number 70 or halvfjerds (half-four) was once named half-four-times-twenty (in Danish halvfjerdsindstyve), but the times-twenty (in Danish sindstyve) has been lost in the counting numbers but retained in the ordinal numbers (for further explanation, see [START_REF] Ejersbo | Danish number names and number concepts[END_REF]. Another concern is the irregularity of the number names between 10 and 20, where 11 and 12 have unique names while 13 to 19 each end with a ten. Also, the two-digit numbers from 13 to 99 have an inversion property (the ones are said before the tens), and the tens have names inspired by a 20-base system.

Even though Danish number names are particularly irregular, most European languages break away from the clear regularity of the base-10 place value system. So understanding the effects of such irregularities on mathematics teaching and learning is interesting. With that motivation, we developed a hypothesis of how speaking about numbers with specific regular words that resemble the base-10 system would be beneficial for learning about the numbers in an easier way. This hypothesis has been tested with a three-year long intervention at a school in the Copenhagen area.

The project has ended, and the intervention was to a large extent successful. The data about student learning confirms that the classes where the new words for numbers system was successfully implemented have very strong performances in the areas related to number sense and arithmetic. Therefore, some of the teachers and the school organization would like to continue the work. We consequently discuss the question of how to exit intervention projects while anchoring relevant practices from the intervention into the organization.

In the paper we explore how this initial hypothesis has been activated as a way of using language as a didactical tool in a design research project. We will describe the state of the art that allowed us to develop the project idea, and how the project proceeded. Then we will briefly state our results in terms of a more well developed and tested set of hypotheses. Finally, we will describe our exit strategy for leaving a better practice at the school and reaching out to other schools and teachers.

Number concepts, base-10 and number names

We know from the research literature that there are major differences in the kind of system and regularities a language uses to describe numbers. Most European countries have an irregular naming system for numbers between 11 and 20; both the German and Dutch systems feature an inversion property of the numbers between 13 and 99, similar to the Danish one. These inversion effects were studied by [START_REF] Moeller | Early place-value understanding as a precursor for later arithmetic performancea longitudinal study on numerical development[END_REF] for two-digit numbers, showing how inversion-related difficulties predict later arithmetic performance (for an overview, see Ejersbo andMisfeldt, 2011, 2015).

Different studies (Miura & Okamoto, 1989;[START_REF] Miura | First graders' cognitive representation of number and understanding of place value: Cross-national comparison -France, Japan, Korea, Sweden, and the United States[END_REF]Miura, Okamoto, Vlahovic-Stetic, Kim, & Han, 1999) compared Japanese, Chinese, Korean, and English-speaking American first graders' (6-7 years old on average) cognitive representations and understanding of place value. The findings confirmed that the Asian-language speakers showed a preference for using base-10 representations to construct numbers, whereas English speakers favored using a collection of units. Note that a significant difference between American and Asian number names appears between the numbers 11 and 19, exactly when the base-10 system starts to use two digits. In Miura and Okamoto's (1989) study, children were asked to construct the numbers 11, 13, 28, 30, and 42 from sets of ten and unit wooden blocks. The results showed that 91% of the American first graders used unit blocks to represent the numbers on their first attempt. In contrast, about 80% of the Asian children used sets of ten blocks when representing the numbers on their initial attempt. These differences in cognitive representation were mainly ascribed to language [START_REF] Miura | First graders' cognitive representation of number and understanding of place value: Cross-national comparison -France, Japan, Korea, Sweden, and the United States[END_REF].

Learning to count and understand the base-10 system are cognitive challenges involving many small steps. We have chosen to focus on oral counting, the cardinal principle of combining a name with a cardinal value, and the combination of words for a number, its cardinal value, and the digit sign.

Children typically learn the names of numbers as a long list of words and demonstrate knowledge of the stable order principle by almost always saying number words in a constant order while emphasizing the last number [START_REF] Goswami | Cognitive development. The learning brain[END_REF]. The names are developed as sounds connected to the number of objects in the sets.

The developmental shift to understanding the number name as a cardinal value requires a qualitative shift in children's representation of numbers. The cardinal principle requires comprehension of the logic behind counting [START_REF] Goswami | Cognitive development. The learning brain[END_REF]) and the ability to judge the size of a set. It relies on a representation of quantitative information in which the coding of smaller quantities is different from that of larger quantities [START_REF] Goswami | Cognitive development. The learning brain[END_REF]). Children's conceptual understanding of numeration depends on their ability to make a connection between a number name and its cardinal value, which they learn to do by grouping and quantifying sets of objects (Thomas, Mulligan, & Goldin, 2002).

Learning how to connect the number name, its cardinal value, and the digit sign is another challenge. As discussed, two different systems must be combined with different representations.

Becoming an expert at combining these two systems means developing rapid access to an automatic use of written numbers and simultaneously being able to multitask to solve other problems in parallel. If the two systems are iconic and support each other, the child encounters less difficulty in learning this skill, as is the case for Japanese-speaking children. If the two systems are irregular and therefore conflict with each other, it is more problematic for the child to understand and remember the connection among the name, the cardinal value, and the sign. [START_REF] Duval | A cognitive analysis of problems of comprehension in a learning of mathematics[END_REF] described this situation as a conversion between registers and observed that the congruent conversions seem the easiest for students, meaning that the representation in the starting register is transparent to the target register. One obvious solution is, therefore, to use a fully regular approach to saying the names of the numbers, which means saying "one-ten and four" instead of fourteen and so on. It is possible and easy to create such a logical system for naming the numbers in Danish, and thus this became our main project idea. This reasoning helped to form the project idea of using such logical number names as a didactical tool.

The design of a research idea

Occupied by these issues of why and how different languages influence number concepts and perhaps even the ability to learn simple arithmetic, we designed a three-year project to take place at a Danish public school in the suburbs of Copenhagen.

Using design research [START_REF] Cobb | Experimenting to support and understand learning processes[END_REF], we formulated our hypotheses for empirical investigation. The hypotheses were grounded in our initial understanding of the difficulties that Danish children experience with the Danish number names. The research builds on the following two hypotheses:

1. Number names function as cognitive artifacts; hence, a concordance between spoken and written language is sensible.

2. Language constitutes concepts, which is why clear terminology seems effective in developing lucid concepts.

Project intervention

To address the question of the influence of number names on number concepts, we contacted a Danish public school that could be interested to run the project together with us. We already knew the school, which made the access easier. We were invited to a meeting with the leading team of the school, including the headmaster, together with a small group of teachers from the school. We presented our project, and the participants accepted it for one year as a start. An evaluation would decide if it should continue additionally for two more years. We decided to involve all 10 classesthree grade 2, three grade 1, four kindergarten classes-and 9 teachers in the primary section of the school. The project combined the renaming of numbers with supporting the teachers in instructing the students. In each class, 20-35% was children who had migrated from other countries, but all the children spoke Danish, and all the teaching was in Danish. The entire research project was planned to last for three years. The data consists of students' performance in classroom observations, a number understanding test, teachers' portfolios, interviews with teachers and students, and notes from collaboration with teachers. At the end of the projects, we used the national test for evaluating the students' competencies in Algebra and Numbers.

The first year

The cooperation with the teachers and the classes were only possible because of the positive attitude from the headmaster of the school. She left it to us, the researchers and the teachers, to run the project. But she and the leading team was helpful and showed interest the whole time.

An in-service course for the involved teachers was the first thing to arrange and run. At this course many questions came up and were discussed. Should the teachers always rename both the names of the numbers? How should the fractions be named? Would the student get to know the normal Danish number names? We made a lot of decisions that day and agreed that the teachers should write a log with further questions that we as researchers should answer, either by discussion or by recording answers in the log.

The participants were now ready to start the next school year with the mathematical numbers.

Kindergarten:

The Kindergarten (K) teachers were used to cooperating with each other and continued this work with the mathematical numbers. We observed the classrooms regularly and had follow-up meetings. The K teachers used both the mathematical number names and the normal number names when they named a number, or the students read a number. They also arranged joint counting for all using mathematical numbers, and they made materials for student use that helped the students to be aware of the base-10 position system, and the students became very familiar with the mathematical numbers. The parents were informed at a meeting with the kindergarten teachers only; all in all, they implemented the mathematical numbers very easily.

Grade 1: The project proceeded differently in the first-grade classrooms. All these classes had new teachers, which is normal for students in the first grade in Denmark. We were in a real-world situation with all the mess that exists there. The three first-grade classes had three different teachers who did not work together very often, and none of them continued the work done by the K class teachers. So the routines disappeared. The big difference in practice between first grade and the K class caused some chaos during the first two months. Furthermore, some of the first-grade teachers left the school or their classes during that period. But new teachers came, and during November and December the classes also worked regularly with the mathematical numbers. We were lucky that one of the newcomers believed in the project idea and was very involved with it. He became a teacher in two of the three first-grade classes, and his presence was a great benefit for the project.

Grade 2: We never observed any of these classes, but met with the teachers and discussed how they could implement the mathematical numbers in the best way. These students had already been in school for two years, and we decided that while we could not expect that they would naturally use the mathematical numbers, they should know them.

Evaluation:

In the first year we were very busy collecting data, observing the seven classes, and trying to find the best ways to implement the mathematical numbers. The teachers' log idea never caught on, so we solved any problems during our meetings with the teachers. At the end of the year, we tested all the students. The outcome of the test showed us that the student used both names for the numbers quite naturally. There was a slight tendency that the students were more secure from the spoken mathematical numbers to written numbers than from the spoken normal numbers to written numbers. In the K classes, we noticed that the students were much more secure in correctly recognizing and naming numbers between 10 and 20 than was normal for these classes.

During the year, we used the design research method as a way to evaluate the actual lesson related to how the whole project was running. We exchanged good ideas and noticed the progress and difficulties. We solved the difficulties in different ways and changed some of our means; but not our goal. We made a report of the first year with our results and data. It was positively received by the leading team, who decided to let the project continue for the next two years.

In our plan for the second year we decided not to observe in the K classes, but only meet with the K teachers. We would do a brief orientation for the new mathematics teachers in the four new first grades, and we would follow the second grades more intensively.

The second year

For each year, we expanded our research with new K classes and with that also thirds grates. The K teachers could more or less develop and repeat with their new students what they had done first year. The newcomers in the K teacher group were taught by especially one of the K teachers taking the major responsibility for informing the new teachers. As we discussed in our meetings with the teachers, the work in the K classes went very smoothly.

The four new first grades had new teachers, luckily only two teachers with two classes each. They did not know anything about the project before they chose to teach first-grade mathematics, but cooperated from the start. We met with them and introduced them to the project, and visited their classrooms several times during the year.

The three second grades were the most interesting, because the students were in their second year of the project. Their teachers were very engaged and consistent in the use of the mathematical numbers. Each time we observed the classroom we talked with the students and asked how they felt using two different names for one number. The answers were surprising:

Student 1: It is fun, and we like to use the mathematical number names, because then we always are able to remember the names.

Student 2:

We also know the cardinal at once.

Student 3:

It is a help to remember the names of the normal number as well.

We were a little overwhelmed but agreed with the students that they should explain to their parents why we used the mathematical number names. We decided that no one could do it better than them.

At the end of the year, we used the same test we had used in the first year but only with the first and second grade. We noticed again that the students' understanding of numbers was very good, both for the use of normal number names and for the mathematical number names

Due to the time we had to do the project, and how it developed in the school, we decided to minimize the observation to only the three third-grade classes, but we still met with the teachers from second grade. We did not observe the first grades or meet with their teachers. We continued with the K teachers' meetings.

The third year

We mostly concentrated our research on grade three and the kindergarten classes. The use of the mathematical number names seemed very natural for the students together with the normal number names, and they were bilingual in the numbers from 10-99. In third grade, the students were so familiar with the base-10 value system that they could transfer the knowledge to the decimal numbers, which meant that they easily answered questions correctly when asked to compare numbers like 0.4 and 0.25. In May 2016, the three classes had the national test in mathematics for third grade. Compared to the average of all the third graders in Denmark, one of the third-grade classes-the one that was observed most frequently-had an average score in Numbers and Algebra that was far above average. The other third-grade classes also showed a better result in Numbers and Algebra than the average third-graders.

The conclusion we draw was that the students using the new system showed a better understanding of the base-10 system. We saw these competencies, and met our goals, in all the classes which used the mathematical numbers. And because the national test investigates additional competencies in Numbers and Algebra, we dared to conclude that the students gained from using two names for the numbers.

During the following year, the school had a new headmaster and a new leading board, but because of the results, she decided to continue with the mathematical number names even though the research project stopped.

From Top-down to Bottom-up

With the decision that the school wanted to continue using the mathematical number names after our exit, we needed to design a plan for how it could be possible. Inspired by the research [START_REF] Jarvis | Comparisons of U.S. and Japanese first graders' cognitive representation of number and understanding of place value[END_REF][START_REF] Nielsen | Children's representation and structural development of the counting sequence 1-100[END_REF] we suggested the following plan:

1. All the teachers at the school should know that the project stopped as a research project, but that the project would continue as an intervention project with the teachers as the drivers.

2. One K teacher should be responsible for introducing the methods in the K classes for incoming teacher and for ideas to be exchanged among the K teachers.

3. A mathematics teacher should be responsible for orientation of the mathematics teachers in first grade each year and arrange a course at the beginning of the new school year, which everybody could join in.

4. There will still be access to the researchers for questions and other things; we are interested in the continuing process.

This plan was first discussed with the involved teachers who agreed to the work they should do, and then it was presented to the headmaster. She also agreed and was willing to find time for the teacher support.

We started the process with a course for mathematics teachers in the K-3 classes in August 2016. There were about 15 people at the course, which was organized and run by a K teacher, a mathematics teacher, and one of the researchers. At this course, we made a quick run through the ideas behind the project and how it had run in the previous three years. The teachers who had previously taught classes and been involved in the project exchanged ideas and views of the learning processes with the mathematical numbers. The K teacher told how she was at an in-service course for K teachers in the Copenhagen region and told about the project and how she and the math supervisor at the school videotaped how she used the mathematical number names in the K classes.

The other participants at the in-service course showed a big interest in the project. A similar course will be held again, and the math supervisor has told us that there is already a big interest in this inservice course.

Discussion

This article is less concerned with the actual results of the investigations that we conducted in the school and more concerned with the transition from an intervention driven by research curiosity to an ongoing project driven by the school itself.

The project was large with many classes involved, and we must admit that it was a little too big for only two researchers on a very low budget. We shared the work, so at times only one of us made the observations, conducted the interviews, and participated in the meetings with the teachers. Everything was documented with taping, pictures, and materials that we analyzed together; we still have data waiting for deeper analysis.

In spite of the low budget and the few researchers, or maybe because of these limitations, we saw some teachers taking over the project in an especially engaged way. One teacher, in particular, took a lot of responsibility and during a period when other teachers were out sick, he taught all the three classes in third grade. Without him, we are not sure the project would have had the success it had. It would neither be possible to run such a project without the support and interest we had from the leading team including the head master, who played an important role.

Perspectives

As it looks now, we hope that the use of mathematics number names will spread to other schools and continue to develop. Even though we officially stopped the project at a meeting with all the teachers at the school, we will continue with some kind of support if necessary. We will also stay in touch for our own sake.

Proceedings of CERME10

Acknowledgment

We would like to thank Husum Skole, Copenhagen, for financial support together with the teachers, who worked for the project.


text:
Keywords: Research findings, research results, theory to practice, implementation research

Introduction to the papers of TWG23:

Implementation of research findings in mathematics education

Uffe Thomas Jankvist 1 , Mario Sánchez Aguilar 2 , Jonas Bergman Ärlebäck 3 and Kjersti Waege 4 1 Aarhus University, Danish School of Education, Denmark; utj@edu.au.dk 2 Instituto Politécnico Nacional, CICATA Legaria, Mexico; mosanchez@ipn.mx 3 Linköping University, Department of Mathematics, Sweden; jonas.bergman.arleback@liu.se 4 Norwegian Centre for Mathematics Education, Norwegian University of Science and Technology (NTNU), Norway; kjersti.wage@matematikksenteret.no

In this introduction, we briefly present selected theoretical constructs relevant for the Thematic Working Group 23 (TWG23). We first address the topic of "implementation research" by looking into other research fields and domains where this topic is well-developed. Drawing on a taxonomy of so-called "implementation science" in health-care, we attempt to categorize the papers and posters of TWG23 according to their "implementation research aim" [START_REF] Nilsen | Making sense of implementation theories, models and frameworks[END_REF]. Using this taxonomy, we elaborate on future perspectives for the TWG by relating to ongoing discussions in mathematics education research.

A need for creating a new CERME Thematic Working Group

With almost five decades of accumulated knowledge, research findings, theoretical frameworks and experiences, the field of mathematics education research now has quite a bit to offer to the ongoing teaching and learning of mathematics in primary, secondary and tertiary education. Regardless of the well-known long and winding "journey" that research results must travel before finding an actual foothold within practice, results from mathematics education research nowadays seem to be present to a much larger extent in practice than ever beforenot to say that there is anything strange in this, but rather it is probably only natural given that the field has matured and become successively more and more established over the years.

Indeed, as researchers in mathematics education, we are frequently involved in putting previous empirical results and findings as well as theoretical constructs based on these to good use in mathematics classrooms, mathematics programs, mathematics teacher education, in-service teacher training, etc. In several countries there are ongoing developmental projects that rely heavily on previously documented research results. However, as researchers we often find ourselves in a peculiar situation when wanting to report on these activities, since such accounts do not necessarily fall under the usual paradigm of "research in mathematics education" and do not in a clear-cut manner qualify as either "empirical" or "theoretical research". To assist in closing this "gap" the purpose of creating a new thematic working group focusing on aspects and issues related to the implementation of research results and findings, is to provide a venue for discussing, collecting and advance the "implementation research" aspects of our activities.

Implementation research and its aims

In relation to research on actual implementations, mathematics education may profit from other disciplines or areas, where research on implementations is further ahead. Healthcare is one such area, where since 2006 an entire journal has been devoted to "implementation science" (the name of the journal as well). Although the journal of course publishes several empirical studies related to various aspects of implementations, it occasionally also offers theoretical studies focusing on "implementation research" itself. One such theoretical contribution is the study by [START_REF] Nilsen | Making sense of implementation theories, models and frameworks[END_REF], who proposes a taxonomy consisting of five categories of theoretical approaches in order to make "sense of implementation theories, models and frameworks" (p. 1). Nilsen describes implementation science as "the scientific study of methods to promote the systematic uptake of research findings and other EBPs [evidence-based practices] into routine practice to improve the quality and effectiveness" (p. 2). Although Nilsen focuses on the case of health care, the definition seems adaptable and applicable also to the field of teaching and learning of mathematics. As part of the background for the taxonomy, Nilsen states that "Implementation science has progressed towards increased use of theoretical approaches to provide better understanding and explanation of how and why implementation succeeds or fails" (p. 1). According to Nilsen, the theoretical approaches used in implementation science have three overarching aims:

1. describing and/or guiding the process of translating research into practice (process models);

2. understanding and/or explaining what influences implementation outcomes (determinant frameworks, classic theories, implementation theories); and 3. evaluating implementation (evaluation frameworks).

The five categories of theoretical approaches consist of those provided in parentheses following the aims above. For aim 1, process models serve the purpose of breaking down the translation process into smaller steps, stages or phases. For aim 3, evaluation frameworks serve the purpose of assisting in evaluating the successor lack thereofof a given implementation. More interestingly, perhaps, are the theoretical approaches associated with aim 2. Determinant frameworks specify types of determinants that act as barriers and enablers that influence implementation outcomes, or even specify relationships between types of determinants. Classic theories are defined as those originating from fields external to implementation science, e.g. psychology, sociology or organizational theory that can be applied to understand or explain aspects of implementation. Finally, implementation theories are defined as those which have been developed from scratch within the field of implementation science.

Implementation research aims of the papers and posters in TWG23

TWG23 received 16 papers prior to the congress. At the congress 14 papers and 1 poster were presented. In the light of Nilsen's three aims of implementation research (or science), most of the studies presented in TWG23 at CERME 10 in Dublin concerned aim 1, addressing aspects of how to adapt research results and findings to practices in schools or other learning situations. A few of the presented studies touch upon aim 3, i.e. evaluation frameworks. Although it is seldom the main focus of the studies presented, aspects related to aim 2 occasionally surface. We shall return to potential reasons for this distribution, but first we will use Nilsen's framework to categorize and briefly discuss the papers and posters in TWG23 presented at CERME 10.

In line with Nilsen's aim 1 to guide the process of translating research into practice, Ärlebäck describes and discusses the framing of, and experiences from, a project that combines research, practice, and teachers' professional development based on the tenets of the "Models and Modeling Perspective" on teaching and learning. Besides providing a general description of the methodological considerations in the project design, the paper describes how the accumulated results and experiences in the research literature on so-called model eliciting activities are used to inform the design, implementation and evaluation of activities aiming at introducing functions to grade 8 students. The focus of the paper is on the implementation, and aims to show how the teacher in question realized the offered perspective and tools in practice. The work presented by Aguilar, Castañeda and González-Polo aligns with aim 1 as it illustrates how research results generated in the field of mathematics education can be implemented in the design of mathematics textbook tasks.

In particular, it is shown how research findings related to representation registers and to the conceptualization of the concept of function as a process are used in the design of textbook tasks for upper secondary level. The poster by Chandia and Montes matches with aim 1, since they report a professional development strategy for teachers focused on improving students' and teachers' problem solving skills. The professional development strategy by Chandia and Montes incorporates research results related to the creation of professional development systems in mathematics. Bulien presents theoretical and methodological arguments for the design and implementation of a research based course for pre-service teachers aimed at clarifying and strengthen the connection between didactical and mathematical theories to in-school teaching activities. Drawing on a design experiment methodology and the theory of communities of practice, Bulien elaborates on a "Mathematics Didactics Planning Tool" for teaching in different classroom situations. Thus, this study also relates more closely to Nilsen's aim 1. Jankvist and Niss deal with the research-based design and implementation aspects of a so-called "detection test" in relation to upper secondary school students' difficulties with mathematical conventions, concepts and concept formation, in particular those related to equation solving. In a similar manner, Ahl addresses the design of a detection test related to students' difficulties with proportional reasoning. Hence, both these studies deal with Nilsen's first aim, that is, translating carefully selected research results from mathematics education into suitable test items. Based on the answers of 405 Year 1 upper secondary school students, Jankvist and Niss also address aspects of evaluation (aim 3). Another research report that corresponds to aim 1 is the one by Kjeldsen and Blomhøj. In their work, research findings on students' concept formation and the digital tutorial genre are brought to use in the teaching of a first year calculus course. They present and discuss a theory-based design and its implementation for students' productions of video tutorials aimed at supporting their understanding of the limit concept.

It could be said that this work is also related to the aim 3 delineated by Nilsen, since the study examines whether the designed learning environment supports the students' formation of key concepts in calculus or not.

The paper by Valenta and Waege touches on both aim 1 and aim 2 of Nilsen's taxonomy. The paper describes a course aimed at supporting in-service teachers' learning of ambitious mathematics teaching. The design of the course is based on a project called "Learning in, from, and for Teaching Practice Teacher Education Project" (aim 1). In addition, the particular question addressed in the paper is coupled to aim 2, since it focuses on the learning potential in the interactions between inservice teachers and course instructors during the public rehearsals that are the key innovative feature of the designed course and manifested through cycles of enactment and investigation. The theoretical paper by Nilsson, Ryve and Larsson align with Nilsen's second aim (understanding and/or explaining what influences implementation outcomes). They draw upon a systematic literature review on productive classroom practice to construct a framework for categorizing theories aiming at supporting teachers' actions in mathematical classroom practices. They do so by relating to theories and literature on educational policy research, professional development research and implementation research. Related to a larger scale early intervention program, Lindenskov and Kirsted touch upon aspects of Nilsen's aim 2. More precisely, they discuss teachers' perception of "theory" and barriers these may lead to, when implementing research results in practice. In addition, they also address aspects of the translation of theoretical constructs to the teachers as well as the suitability of these constructs provided a given context of practice. The study reported by Tamborg, Allsopp, Fougt and Misfeldt clearly falls with Nilsen's category of studies related to developing determinant frameworks (aim 2), since it investigates the of role the local supervisor (enabler) in the implementation of a mathematics teacher training program.

Amit and Portnov-Neeman's work can be related to the aims 1 and 3 proposed by Nilsen. They report on the implementation of a methodology used to teach reading and mathematics called "Explicit Teaching Method" focused on teaching students the "working backwards strategy" for solving non-routine mathematical problems; on the other hand, the effect of using the explicit teaching method as a means to learn the working backwards strategy is evaluated. Koichu and Keller position their paper as so-called design-based implementation research (DBIR) (see later).

They present an evaluation framework (aim 3) to analyze and theorize their attempts in creating and sustaining online exploratory problem-solving discussion forums using the conceptual tools provided by Rogers' "Theory of Diffusion of Innovation". Ejersbo and Misfeldt report on a design-based research (DBR) project related to developing numeracy in grades K-3. This study too focuses specifically on evaluation aspects (aim 3), not least in terms of improving the design being implemented as well as the future of the project at the local school. Kuzle's work somehow touches all three aims outlined by Nilsen. She reports on a collaborative project between educational researchers and practitioners with the goal of developing a problem-solving curriculum for grade 6 students using DBR. The curriculum was developed and implemented based on problem solving research and theory, and through the evaluation of its implementation objective and subjective factors that inhibited the full-implementation of the curriculum were identified.

Implementation of research findings in mathematics education

As seen above, a few aspects of aim 2 were touched upon in the papers and posters of TWG23, and some papers also considered aim 3. Still, aim 1 appears to be the dominant one among the reported studies. This, however, is not so strange since actual "implementation research" within the field of mathematics education must be regarded as a relatively new trend. This is of course due to the field of mathematics education itself not being much older than fifty years, but at the same time it is mature enough to have produced a sound basis of research results to actually be implemented into the practice of teaching and learning mathematics. Engaged in such implementation-oriented endeavors, researchers in mathematics education work systematically at different levels to establish evidence-based solutions to the problems and challenges faced by practitioners and learners. Whether the research carried out is empirical or theoretical in nature, implementation of research findings and results is at the core of the research activities, either in the form of evaluating and furthering actual practices or materials etc., or to deepen our theoretical understanding to facilitate, guide and support various future implementations. Hence, and as already illustrated by the papers of TWG23, implementation of research findings and results in mathematics education can take many forms and expressions. Further examples from the literature are: in the design of experiments (e.g. [START_REF] Cobb | Design experiments in educational research[END_REF] and mathematical tasks (e.g. [START_REF] Margolinas | Proceedings of the ICMI Study 22[END_REF]; as tools for professional development of in-service and pre-service mathematics teachers (e.g. [START_REF] Tsamir | Using theories as tools in mathematics teacher education[END_REF][START_REF] Sánchez | Concepts from mathematics education research as a trigger for mathematics teachers' reflections[END_REF]. The research findings and results that are implemented as part of systematic research are typically empirical results, theoretical results in terms of frameworks of different kinds, or some mixture of the two. Still, such findings and results usually fall within Nilsen's first and third aims, whereas results directly concerning aim 2 are scarcely touched upon.

As seen from the presented research studies of TWG23, implementation of research findings may have connections with research areas already existing in the field of mathematics education. One such example, although not reported in TWG23 at CERME 10, is that of lesson study. In the lesson study approach, lessons are designed and analyzed as a means to improve mathematics teaching in the classroom, but also as a means for professional development of mathematics teachers. Another existing research area, represented in TWG23, is that of task design. Hence, from the presented papers, it is clear that "implementation research" encompasses different kinds and formats (textbooks, apps, software, etc.) of didactical designs and products, stretching from task design, over teaching modules, courses, to entire programson all educational levels. Yet an example is that of design-based research (DBR), where results might take the form of a teaching module that successively and iteratively have been envisioned, designed, applied, analyzed and redesigned. The result is the final design as well as measures of how successful the design has proven to be. However, to focus on the implementation aspect of DBR means to not only focus on the end product and its success in achieving what was set out to do, but also to seriously take into account the "design phase" of the design research cycle. That is, the phase where the researcher identifies a learning problem and then uses available research results to design a (preliminary) product or tool that can help students in overcoming this learning problem. A primary concern then becomes to focus precisely on the way research knowledge is applied to generate some type of educational product. Elements of these concerns are addressed by Fishman and colleagues (2013), who forefront the implementation aspects of DBR in a research approach they call design-based implementation research (DBIR)a framework also used in a few of the papers presented in TWG23. In short, DBIR has:

"(1) a focus on persistent problems of practice from multiple stakeholders' perspectives;

(2) a commitment to iterative, collaborative design;

(3) a concern with developing theory and knowledge related to both classroom learning and implementation through systematic inquiry; and (4) a concern with developing capacity for sustaining change in systems." (Fishman and colleagues, 2013, pp. 136-137) More generally, an important aspect when implementing research findings and results into practice is to focus on what [START_REF] Burton | Methodology and methods in mathematics education research: Where is the "why[END_REF] has called the methodology of the research conducted. Burton argues that researchers in mathematics education in general pay little or no attention to explaining and motivating the rationale for the actual research design they apply to be able to draw the conclusions they report when writing up their research. This "craft knowledge" of the researcher is in a way silent. In Burton's opinion, accounts of research is full of descriptions of how results were obtained (i.e. what the explicit methods applied were), whereas elaborations on why choices were made and decisions taken in order to arrive at conclusions are rarely found. The how-question concerns the methods used by the researcher to undertake his or her research, while the whyquestion focuses on the rationale for the research design, i.e. the methodology. That more emphasis should be put explicitly on the methodology has also been put forward by for example [START_REF] Wellington | Educational research: Contemporary issues and practical approaches[END_REF], who describes methodology as "the activity or business of choosing, reflecting upon, evaluating and justifying the methods you use" (p. 22). He further argues that it is necessary to know the methodology of a piece of research to be able to impartially judge and assess it. TWG23 provides a venue and forum for researchers to discuss how to best put research results to use in practice alongside the accompanying rationale for why. In this sense, TWG23 has as one of its primary foci methodologies for initiating and institutionalizing research-based implementation designs. Over time, the activities of such a group could also make us wiser on the actual usefulness of our various research results, constructs, and frameworks.

Perspectives for the TWG at future CERMEs

Although the main focus of TWG23 seems currently to be on Nilsen's first aim, and to some extent the third aim, in time the TWG may potentially contribute much more to the second aim: in identifying determinants across various countries; in identifying relevant classical theories external to mathematics education, which may help to understand or explain implementations; and last but not least in developing homegrown implementation theories of mathematics education. This was also reflected in the evaluation of the TWG, where the question was asked: What shall be TWG23's contribution of knowledge to the field of mathematics education? The participants of the TWG collectively phrased the following "vision" for the group:

"We want to explore a wide variety of 'good examples' of implementing research findings and results (back) into practice in order to improve the teaching and learning of mathematics at all educational levels on a research-based foundation. Over time we may begin to look into the aspects of research on implementations, potential requirements for these to function, etc."

Hence, for the future of TWG23, it may be envisioned that the TWG could come to consist of a core of researchers interested in these aspects (Nilsen's second aim). But at the same time, a group like TWG23 is also a place for mathematics education researchers to go when wanting to report and discuss on any "intermediate" activities of either designing new research projects or developmental work, before the activities may result in more traditional research to be reported in other TWGs. In this sense, TWG23 also provides a forum for mathematics educators at CERME to "come and go" from one congress to another.

To put it a bit boldly, it is our hope that this TWG can assist in filling the "gap" of where to report on implementation activities in our research community, while at the same time act as a "bridge" between research and practice.

Proceedings of CERME10