text:
Dié Gijsbers
email: d.gijsbers@gymnasiumbeekvliet.nl

Birgit Pepin

Context based tasks on differential equations to improve students' beliefs about the relevance of mathematics

Keywords: Mathematics education, task design, relevance of mathematics, differential equations, differentiation by interest

Motivated and talented mathematics students are not always convinced about the relevance of mathematics. More insight into applications of mathematics can be beneficial for students in terms of preparing them for their future study and career. Using design research a particular intervention had been developed which differentiated by student interest, in order to improve students' beliefs about the relevance of mathematics. The students selected were those studying advanced mathematics at upper secondary school in the Netherlands. The intervention had been designed to teach differential equations through tasks with science-, medicine-, or economics-related contexts. The results show that the students appreciated the context-rich tasks, which provided them with insights into how mathematics can be applied in other sciences and contributed to the improvement of their beliefs about the relevance of mathematics.

Introduction

Students' beliefs about the relevance of mathematics has been a topic of much research over the last decades. However, in most studies the main reason for studying students' views on the relevance of mathematics has been the assumption that a positive notion of the importance of mathematics contributes to a positive attitude towards learning mathematics [START_REF] Schoenfeld | Explorations of students' mathematical beliefs and behaviour[END_REF]. Hence, most research studies conducted in this area have aimed at understanding and improving students' attitudes towards mathematics, and ultimately improving their performance in mathematics at school [START_REF] Farooq | Students' attitude towards mathematics[END_REF][START_REF] Mohamed | Secondary Students' Attitude towards Mathematics in a Selected School of Maldives[END_REF].

However, we start from the premise that already motivated and talented students can also benefit from a clear view on the usefulness of mathematics for their future education and career. Dutch secondary school students in an advanced mathematics course (aimed at improving students' algebraic skills and elaborating the connection between mathematics and other sciences) praised the course as being challenging and fun, but they also mentioned that it was not clear to them how the mathematics would be useful for their future study ( [START_REF] Elst | De motivatie voor de keuze voor en de beleving van Wiskunde D[END_REF].

In our research project a design research approach has been used to develop and intervene with tasks aimed at simultaneously teaching a new (for students) mathematical concept (differential equations) and improving students' beliefs about the relevance of mathematics in general, and in particular with respect to their future study and career ambitions. The students sampled were those taking the advanced mathematics course (in their final year before university entry) in Dutch upper secondary schooling.

The mathematical topic of the designed intervention was the theory of analyzing, solving and interpreting first order differential equations. Differential equations are an important subject in any undergraduate university curriculum in a broad range of domains, such as engineering, physics, biology and economics, which makes it a suitable topic for an intervention with a focus on the relevance of mathematics.

To improve the odds that the real-life problems posed in the designed tasks would appeal to the students, the designed intervention differentiated by student interest. Students were offered the opportunity to choose between different real-life problems, whilst ensuring that regardless of their choice they learned the same mathematical concepts of differential equations.

The research question of the study was the following: How does a learning strategy based on differentiation by interest for teaching ordinary differential equations (to upper secondary students in a "strong mathematics" course) improve the view of students on the relevance of mathematics for their future study and career?

In the subsequent section, we provide an overview of the relevant literature. Next, the research design including the context of the study and the data collection strategies used are described. Finally, we provide a discussion of the results and our conclusions.

Literature

Students' beliefs about the relevance of mathematics is considered as one of the factors that can play an important role in their attitude and motivation. Several questionnaires measuring the attitude of students towards (learning) mathematics use a scale for measuring students' beliefs about the usefulness of mathematics: for example, the Attitude scale towards Math [START_REF] Martinot | Measurement of affective goals: validity and national norms of the Attitude Scales towards Mathematics (ASM)[END_REF] contains a scale named Relevance of Mathematics.

Most studies using these surveys do not emphasize the improvement of students' beliefs about the relevance of mathematics, as they are aimed at students' attitudes towards mathematics. However, a recent study of first year university students in engineering focused on improving perceptions of the relevance of mathematics in engineering. In this study [START_REF] Flegg | Students' perceptions of the relevance of mathematics in engineering[END_REF] has described the use of context-based learning by applying mathematics to real-life problems as a promising approach.

The teaching of differential equations has undergone some major changes over the past decades in favor of more contextualized, problem-based education, and a less traditional, analytical approach [START_REF] Boyce | New Directions in Elementary Differential Equations[END_REF]. This was in line with our planned design to incorporate real-life problems in the assignments. At university level several initiatives have reported good results using this new teaching approach (e.g. [START_REF] Huber | Teaching Differential Equations with Modeling and Visualization[END_REF] and the development of new course material using Realistic Mathematics Education (e.g. [START_REF] Rasmussen | Locating starting points in differential equations: a realistic mathematics approach[END_REF]. In a comparison of a traditional textbook with a textbook that incorporated discipline-specific perspectives to teach the mathematical knowledge to engineering students, [START_REF] Czocher | Contextual Learning in Math Education for Engineers[END_REF] conclude that a contextual approach is also more in line with recommendations from the research literature.

Task design is widely recognized as an important, albeit complex activity that is at the core of mathematics education. The term 'task' is used to describe a wide variety of student activities aimed at learning mathematics (Watson & Ohtani, 2012). In our research tasks are guided group assignments about a real-life problem.

The study

The context

The Dutch education system consists of eight years of primary education, and 4-6 years of secondary education (depending on the level of education). The highest level of secondary education is the pre-university education called VWO (voorbereidend wetenschappelijk onderwijs) with a duration of six years, which provides students access to university. For every student at VWO level mathematics is a mandatory course. However, in the last three years students can choose between two different mathematics courses: "wiskunde A" (mathematics A) and "wiskunde B" (mathematics B), the latter being the more mathematically demanding course, which is obligatory for technical and engineering studies at university.

In 2007 an advanced mathematics course called "wiskunde D" (mathematics D) was introduced to offer challenging and engaging mathematics, and where the relevance of mathematics and its connection to other sciences should become clearly visible (cTWO 2007). This course is aimed at students with an interest in sciences and engineering, and it includes mathematical topics, which are part of every first year university curriculum: e.g. complex numbers; analytic geometry; and differential equations. Students with 'wiskunde B' are offered the opportunity to take this advanced course in mathematics in addition to their regular course.

It might be expected that students who take this advanced course in mathematics are convinced about the relevance of mathematics, which is likely to fuel their obvious motivation to learn mathematics by taking this advanced course. However, studies on the implementation of 'wiskunde D' tell a different story. In a study by [START_REF] Elst | De motivatie voor de keuze voor en de beleving van Wiskunde D[END_REF] students praised the course as being challenging and fun, but they also mentioned that it was not clear to them why the mathematics in the course was useful for their future study and career. According to a study by [START_REF] Cheung | De implementatie van wiskunde D[END_REF] teachers of 'wiskunde D' stated that the course was well suited as a preparation for a future study in a technical or engineering environment, but they also stated that the curriculum did not emphasize enough the applications of mathematics and the connections to other sciences [START_REF] Cheung | De implementatie van wiskunde D[END_REF].

To be able to understand the theory of ordinary differential equations, secondary school students require almost all basic mathematical knowledge taught in secondary education as pre-knowledge. Hence, this topic is scheduled near the end of the final (6th) year, to make sure all major mathematics/'wiskunde B' and mathematics/'wiskunde D' topics have been covered. It can also be expected that at that time the students have a good idea of their intended future study. This might motivate the students to choose tasks with real-life problems associated with their future study.

The real-life problems in the study by [START_REF] Flegg | Students' perceptions of the relevance of mathematics in engineering[END_REF] all had an engineering context, which was in line with students' study (of engineering at university). However, from our experience secondary school students taking the 'wiskunde D' course are not all interested in engineering. Students typically also enroll for medical or economics studies after graduating with 'wiskunde D'. Hence, only focusing on real-life problems in engineering would have been a too narrow approach. To give students a good view of applications of mathematics to solve real-life problems, the learning strategy should offer differentiated instruction, based on students' interest.

Beside this "practical" reason to give students a choice of which problems they wanted to investigate, research shows that differentiated instruction based on students' interest supports their autonomy and improves students' motivation for the task at hand [START_REF] Katz | When Choice Motivates and When It Does Not[END_REF]. Differentiation is commonly used to accommodate the different learning styles of the students; however, it can also serve to accommodate other differences between the students, such as their interests and plans for their future study and career [START_REF] Tomlinson | Differentiating Instruction in Response to Student Readiness, Interest, and Learning Profile in Academically Diverse Classrooms: A Review of Literature[END_REF].

Research design and data collection strategies

For the development of the intervention, a module consisting of tasks on real-life problems, a design research approach was chosen. The design process comprised of three phases: a preliminary phase; a iterative development phase; and a final evaluation phase. The preliminary phase included a context analysis and a literature review. The learning goals for teaching differential equations were defined, based on the five strands of mathematical proficiency [START_REF] Kilpatrick | Adding it up[END_REF].

In the second phase a partial prototype of the module was developed. In March 2015, a small pilot test was conducted, with a prototype consisting of only three tasks. In a second design cycle a module was developed, which comprised of 14 tasks covering five different types of differential equations. Validation of separate tasks was done (1) by expert appraisals from university experts in mathematics education, focusing mainly on relevance and consistency of the design; and (2) by secondary school teachers focusing on the consistency and expected practicality of the design.

The designed intervention was carried out from January to March 2016. Three classes of 'wiskunde D' students of two different schools in the Netherlands, in total 49 students, participated. The three classes all had a different teacher, one being a researcher in this study.

The data collection strategies for the intervention included -two student surveys (before and after the intervention) asking the students about their future study plans and their views on the relevance of mathematics, using the 8 question scale Relevance of Mathematics (Relevance scale) from The Attitude scale towards Math [START_REF] Martinot | Measurement of affective goals: validity and national norms of the Attitude Scales towards Mathematics (ASM)[END_REF]; -a student survey after the intervention to evaluate the module and the tasks;

-student interviews after the intervention about the module and the relevance of mathematics in general, and in particular for their own future study and career;

-the video recording of a teacher meeting about their experiences during the intervention;

-the collection of exam results after the intervention at one school. These results were compared to the results of the 'wiskunde D' students of the previous exam years, who were taught the same theory of differential equations but in the traditional classroom setting.

The intervention

The intervention module comprised of five tasks, each covering a different type of differential equation. Of each task up to three different versions were developed, which applied the same mathematical concept to entirely different contexts. Prior to the first task students were given time to read short descriptions of each problem and were given the opportunity to make, for each of the five subsequent tasks, a choice which context (real-life problem) appealed to them most.

The different contexts for each of the tasks consisted of: a science/engineering related problem; a biological/medical problem; and an economical/social problem. Regardless of students' choices of contexts for the five tasks, the students worked with and learned the same mathematical concepts during their work on these parallel tasks. The students were guided through the process of modelling the problem and exploring the mathematical model analytically, graphically and numerically. After solving the mathematical problem the students were asked to interpret the results within the context of the chosen assignment. Table 1 gives an overview of the 14 tasks.

Results

40 out of the 49 participating students filled in the survey about the module. Responses to most questions were measured on a five-point scale, ranging from "I strongly disagree" to "I strongly agree". The students were generally quite positive about the whole module. Asked to grade the whole intervention on a scale of 1 to 10, they rated the module 6.5 on average. Interestingly, this mean score differed greatly between the three groups, with one group scoring surprisingly lower than the other two (see Table 2). Additional comments in the survey from this group suggest that they lacked proper guidance of their teacher. Hence, the practicality and effectivity of the intervention can substantially benefit from a good description of the role of the teacher as coach, helping the students to advance in their assignments. In the teacher meeting the same recommendation was made.

Overall, 28 out of 40 students agreed (of which 6 strongly) with the statement "The tasks gave me a good view on how mathematics is applied in other sciences", and only 3 students disagreed with this statement. The negative statement "I think that these tasks were not very realistic" was disagreed by 29 students (of which 12 strongly) and only two students agreed with this statement. The survey also contained open questions asking students what they liked about the module and what in their opinion should be improved. Positive points were working in groups (11 times) and the application of mathematics in other sciences (15 times) with statements like: "it was fun to experience that math is useful", "clear applications of math" and "you get a better view on how mathematics can be applied".

Some negative comments were about ICT problems encountered during the intervention (e.g. "slow laptops"). However, the majority of the feedback was about the way the mathematical concepts of differential equations were introduced to them. For example, they complained that having to read the theory by themselves made it harder to grasp the concepts. Some also missed the "traditional" teacher-led instruction, where the "basic" theory is explained by the teacher before the students start working on the assignments.

The student interviews were conducted by the teacher/researcher (one of the authors). In the interviews the students voiced the same concerns about the lack of teacher instruction and the ICT problems. However, they were positive about the tasks and the application of mathematics to reallife problems, as the extract shows.

Interviewer: Did the tasks have some added value? To give a lot of people the idea what can be done with mathematics. That there is mathematics behind everything. Not a lot of students would have realized that before, I think.

Comparison of the 'wiskunde D' exam results of the 36 students from school A with the results on a similar exam by the 27 'wiskunde D' students of the previous year did not show a significant change in the grades. As can be seen in Table 3, the students of 2016 scored slightly better than their peers in 2015 but that was to be expected as their average grade before the exam was also better.

The intervention was not intended to improve the exam results, and the findings showed that the dual focus on both the mathematical concepts of differential equations and the relevance of mathematics to other sciences did not affect the grades of the students in a negative way. From the Relevance of Mathematics scale, conducted before and after the intervention, we obtained some promising results. Only 33 out of the 49 students completed both the pre and post survey, as shown in the last column of table 2.

A one-sided paired-samples t-test was conducted to compare the answers of the 33 students on 8 items of the Relevance of Mathematics scale before and after the intervention. The result of this test for the whole group of 33 respondents did not show a significant improvement with a p-value of 0.19. But the same test on the 18 results from the high response group gave a p-value of less than 0.02 indicating a significant positive change in their response to the questions about the relevance of mathematics. Due to the low response rate of the two other groups we were not able to get any significant results from these separate groups.

We also conducted the paired t-test on the answers of the 17 students who had the lowest scores on the pretest. They had scored relatively low on their beliefs about the relevance of mathematics, all with an average score between 2.38 and 3.63 for the 8 items on the five point Likert scale. On the same test after the intervention their average scores ranged between 2.63 and 4.63. 12 of these 17 students scored higher on the test after the intervention resulting in a p-value of 0.002, which indicates a strong significant positive change in these students' beliefs about the relevance of mathematics. This result indicates that students who do not already have strong beliefs about the relevance of mathematics benefit most from the context-based tasks.

Conclusion

The intervention had positive effects on selected student views about the relevance of mathematics without affecting the examination results. Providing students with purposefully designed tasks where mathematics is applied to real-life problems cannot only challenge their assumptions about the relevance of mathematics, but also improve their awareness of its usefulness.

Teacher professional development is a topic for further study that will be the focus of a new design cycle of the module. Applications like a teacher meeting, where the goals of the module are explained and an extension of the teacher manual with guidelines how to introduce the module and how to coach the students during the group assignments, are expected to contribute to a better understanding of the role of the teacher during future interventions.

  think so. I think it adds quite a lot. It helps… Interviewer: It helps? In what way? Student 1:

Table 1 : Overview of the real-life problems used in the module

 1

	Task Differential	Scientific/engineering	Biological/medical	Economic/social
	equation	problems	problems	problems
	1	Nuclear disaster	Bacterial food poisoning Forged paintings
	2	Mixing water problem Intravenous infusion	Advertising effect
	3	CO poisoning	Estimating time of death Price indexing
	4	Oil production	The Ebola epidemic	Population growth
	5	Skydive	Blood alcohol content

Table 2 : Respondents and rating of the module by group

 2

	Schools	Group	Group size #Respondents	Rating	#Respondents
				survey	(mean)	Relevance scale
	School A	1	19	18	7.1	18
		2	17	14	5.5	7
	School B	3	13	8	6.9	8
	Total		49	40	6.5	33

Table 3 : Exam grades of the students in 2015 and 2016

 3

Proceedings of CERME10


domain: math shs

text:
Andreas Eckert
email: andreas.eckert@lnu.se

Agency as a tool in design research collaborations

Keywords: Collaboration, agency, interaction, design research

The aim in this paper is to shed light on interactional aspects of researcher and practitioner collaboration in design research in mathematics education. Symbolic interactionism is used to gain understanding of interactional aspects as it has potential to take both individual and social aspects of the interaction into account. Aims and agencies are in focus of the retrospective analysis of the collaboration between two researchers and two practitioners as they collaborate to develop instructional design. The analysis show how referring to authoritative disciplines as the mathematics community influence agency and therefore has great potential to influence how the negotiation of meaning progress and participants acts. I argue that agency could be viewed as an indirect tool that has the potential to direct the collaboration when designing tasks based on what aim different actors put in the foreground.

Introduction

Collaboration is at the core of design research in mathematics education. A key characteristic is that it is research conducted with researchers and practitioners in real-world settings [START_REF] Plomp | Educational Design research: An Introduction[END_REF]. This collaboration is often between one or more researchers and practicing teachers. It is essential that this team collectively has the competence to develop the design, conduct the lessons, and perform the retrospective analysis [START_REF] Cobb | Design experiments in education research[END_REF]. This means that we have different actors in the activity, each with its interpretation of the aim and purpose, as well as of the actual activity and the mathematics involved. Since each actor has a specific set of competences and a vital role to play in the collaboration to develop the design, the question is how their differences influence the discussions during meetings and by extension the design? [START_REF] Mcclain | Working Both Sides[END_REF] explores this interplay of differences in the classroom interaction between students who performed the resulting tasks and the teacher who orchestrated the discussions. She views it as interplay between the students' contribution, the task and what she calls the proactive role of the teacher. One key aspect of the interplay is where the authority over the topic, the agency, lies. Agency is thought of as a capacity to act in social interaction. [START_REF] Mcclain | Working Both Sides[END_REF] emphasizes that it is important for students' learning that agency shifts between different actors in the activity.

The discussion in this paper focuses on the developing phases of design research. How does this shifting of agency that [START_REF] Mcclain | Working Both Sides[END_REF] identifies in the design research classroom appear in the development of tasks in between lessons? The aim of this paper is to further understand how the aim and agency fluctuate between the participants and its influence on how the negotiation develops as researchers and teachers collaborate to develop instructional design. Results could be viewed as a contribution to the mathematics design research methodology discourse of how collaboration between researchers and practitioners can support the design process.

Previous research

Design research in mathematics education is described as a research design that is interventionist; iterative; process, utility and theory oriented; and has involvement of practitioners [START_REF] Plomp | Educational Design research: An Introduction[END_REF]. The idea is to develop tasks and activities, test them in real classrooms, evaluate the outcomes and then revise the design in an iterative process involving practitioners from the field. All in all, the research design enables the team to pursue multiple goals in the same project. The aim is to understand the processes involved in utilizing the developed tasks from the point of view of a chosen theory, design usable material for users in real life context and lastly to contribute to further development of theory. [START_REF] Plomp | Educational Design research: An Introduction[END_REF] calls it the twofold yield of design research, producing both research-based intervention and knowledge about interventions in the form of theory.

Beside the global aims of design research, an intention to develop theories and instructional designs, there are local aims as well. The instructional design has an aim, an intention to stimulate learning, in the form of a hypothesised learning process and goal for the subjects [START_REF] Cobb | Design experiments in education research[END_REF]. The socalled Hypothetical learning trajectory (HLT) is defined by [START_REF] Simon | Reconstructing mathematics pedagogy from a constructivist perspective[END_REF] as follows:

The hypothetical learning trajectory is made up of three components: the learning goal that defines the direction, the learning activities, and the hypothetical learning processa prediction of how the students' thinking and understanding will evolve in the context of the learning activities. (p. 136) [START_REF] Cobb | Individual and collective mathematical development: The case of statistical data analysis[END_REF] argues that the learning goal of a HLT should be from a group perspective rather than an individual one. [START_REF] Simon | Reconstructing mathematics pedagogy from a constructivist perspective[END_REF] amongst others talk about a prediction of individual learning processes and thinking whereas [START_REF] Cobb | Individual and collective mathematical development: The case of statistical data analysis[END_REF] call this line of thinking highly idealized at best. Instead, he proposes a focus on collective mathematical development in the classroom community. A HLT then consists of an "envisioned sequence of classroom mathematical practices together with conjectures about the means of supporting their evolution from prior practices" (p. 9).

Global and local aims are both individual interpretations as well as objects for the research team to negotiate. In this paper, it is assumed that this process is a social interaction within the team. Interaction is described with the theoretical background of symbolic interactionism [START_REF] Blumer | Symbolic interactionism: perspective and method[END_REF]. It proposes that humans act according to the meaning that objects have to them and that the meaning of objects arise out of social interaction. This is an interpretative process where humans constantly interpret others' actions and the meanings they indicate before acting themselves [START_REF] Blumer | Symbolic interactionism: perspective and method[END_REF]. It means that participants of the research team interact according to how they interpret the local aim of the design as well as their own global aims and interpretations of the mathematics in question. [START_REF] Voigt | Negotiation of Mathematical Meaning and Learning Mathematics[END_REF] calls this the negotiation of meaning in the mathematics education context. It is a negotiation because the actors contribute to a discussion based on their interpretation of what is being discussed and at the same time re-evaluate their own understanding, thus creating a negotiation of what is viewed as the community's meaning of the objects.

As global and local aims and the design itself are negotiated, participants position themselves through their contribution to the negotiation. [START_REF] Burr | Social constructionism[END_REF] talks of the capacity to take up positions for one's own purposes and that agency lies in responsive actions in interaction. In the mathematics classroom context, [START_REF] Mcclain | Understanding the Role of the Institutional Context in the Relationship Between Teachers and Texts[END_REF] defines agency as "authority over both the mathematics being taught and the sequencing and presentation of that content" (p. 63).

Combined, they frame the agency concept for this paper. For the purpose of this paper agency is viewed as involving one's intentions, sense of responsibility, as well as one's expectations of recognition and reward in taking a particular action. As the participants of a research team interact in the development of the design, they act according to their interpretations and their positioning in a community, for example as a representative for the mathematical community, and act with the authority of that discipline. Participant act according to different agencies as fits their purposes and evaluate its impact on the negotiation, also known as the dance of agency [START_REF] Pickering | The Mangle of Practice[END_REF]. As the negotiation progress, agency shift between the participants and within them. Shifting agency enables the participants to contribute in different ways and from multiple perspectives, for example as mathematicians, practitioners or researchers. The result is an effect on the design of the HLT in line with different actors' fluctuating aims and their agency to contribute according to those aims.

Method

The data used here is generated from video recordings of a small-scale teaching experiment involving probability with students from year 5 and 6 in a Swedish elementary school. The aim of the task was for the students to become able to discuss matters of relative frequency data and the law of large numbers in a probability context. Relative frequency is a way of analysing data from repeated random events, such coin flips, where the number of observations of each outcome is divided by the total number of events. The law of large numbers then states that as the sample size increases, the likelihood of a difference between the relative frequency and the actual probability of the event decrease. Thus, can the relative frequency be used as a measure of probability. The research team consisted of the author of this paper, a senior researcher (here called Paul) and two teachers (here called Karen and Tilly). The whole process was initiated by the two teachers who felt that they needed inspiration and experience in teaching probability, which they had never done before. The balance of numbers provided a sense of balance between researchers and teachers, which later has been recognized by [START_REF] Stephan | Conducting classroom design research with teachers[END_REF] to be an important factor to highlight teachers' unique knowledge in design research collaborations. The work was organized as such that after an introductory meeting, the two researchers drafted a design proposal in line with the requested topic. That design was further developed by a discussion within the team, which was video recorded, and then initiated by one of the teachers in the classroom. Minor adjustments were carried out between the two teachers' lessons and major changes of the lesson sequence were carried out after both teachers had used each lesson plan. A total of 5 lessons were designed, although one of the teachers divided the last lesson into two because of time management issues.

The activity and hypothetical learning trajectory

The task design originated from a teaching experiment by [START_REF] Brousseau | An experiment on the teaching of statistics and probability[END_REF], where the students were asked to investigate a chance event with an unknown sample space. The aim was to introduce basic principles of the Law of large numbers from probability as well as a frequency perspective on probability theory. We used an opaque soda bottle containing an unknown amount of small coloured balls (neither the students nor the teachers knew the content of the bottles) during the first lesson. When the bottle was turned over, the colour of one ball was revealed while remaining inside the bottle. Thus, creating a constant but unknown sample space. The activity was presented as a race in the first lesson with three contestants on a six-step track. As one of the three colours was observed on a bottle turn, that colour advanced one step down the track. The students were asked to guess which colour would first get six observations during each race. Based on the topics discussed by the students in the first lessons, the following three lessons made use of a transparent bottle with a visible sample space. Here the students were asked to discuss chance, random variation, sample space, sampling and the law of large numbers. The importance of the sample space was highlighted in the second lesson because of ideas discussed in the first lesson. The students got to return to the opaque bottle in the last lesson(s) and again, in an organized manner, investigate the unknown sample space from the first lesson with the use of the law of large numbers. By producing a large enough sample, they could reason about the sample space in the opaque bottle by translating the relative frequency of each outcome into the probability of that outcome. Overall, one class needed a total of five lessons and the other class six lessons, to reach an agreement about the unknown sample space in the opaque bottle.

Method of analysis

The analysis of the transcript in the forthcoming section is inspired by retrospective analysis from the design research methodology. It is based on open inquiry and constant comparison [START_REF] Glaser | The discovery of grounded theory: strategies for qualitative research[END_REF] where you retrospectively analyse and compare small instances of data from the whole set with one another to gain insights into the processes [START_REF] Gravemeijer | Design research from the learning design perspective[END_REF]. Trustworthy accounts of possible meanings can be developed by immersing oneself in the social setting, using participant observation, alongside systematic coding of data in retrospective analysis [START_REF] Cobb | Participating in Classroom Mathematical Practices[END_REF]. Instances of active contribution to the negotiation, utterances by the teachers or researchers, are coded with open codes. These instances are then compared to each other to find differences and similarities in their actions as they indicate the participant's interpretations, aims and agency. When looking at longer sequences, patterns are sought after, especially how participants' aims and agency influence the development of the negotiation. Short excerpts used for constant comparisons are presented in the text and expanded upon. Tied to those excerpts are expansions on the continuing interactions not shown in the transcripts due to space limitations.

Results

The excerpts from the transcripts presented here are all from a meeting with the two researchers and the two teachers between Karen's first lesson and Tilly's first. The purpose was to engage in a minicycle to evaluate the initial design and revise it before Tilly used it in her classroom. One of the researchers, the author of this paper, was present during Karen's lesson and the discussion utilized their experiences as a main source of data to analyse. In the first episode, the two teachers discuss Karen's experience during the first lesson. She had asked the students to reflect on the notions statistics, chance and probability at the beginning of the lesson and then proceeded to carry out the design outlined here in an earlier section. Notice how the focus shifts from being about implicit aspects of the activity to being about students connecting knowledge.

Karen:

What fascinated me was that their engagement induced the use of the concepts that we highlighted and reconnected to what we did at the beginning. Karen acts with an interest in the activity as an eliciting factor for a student's development of concepts. Tilly indicates that she is interested in how Karen carried out the activity in the classroom and both act as if their main aim is to further develop the task. Tilly then subtly indicates that she has shifted her focus towards the students' learning process in the second utterance. She acts as if her interpretation of the aim has shifted towards the global aim of contributing to theory on students learning processes. The negotiation takes off in another direction, initially being about gaining understanding of the design aspect towards being about understanding aspects of learning. The global aim of understanding the students' learning in respect of making connections as well as developing language is pursued long after this extract ends. The following episode picks up this chain of events further into the meeting. Karen admits that the development of the lesson had made her unsure of how she interpreted the three concepts statistics, chance and probability. We start off with her reading her own notes from what she found out from a dictionary after the lesson.

Karen:

"Not be able to calculate in advance. Statistics, summarization of information, nah, Probability, the chances of getting" for example blue.

Paul The initial statement was copied from a dictionary and Karen acts as if she trusts and places the authority within that community. At least in the case of chance and probability, she relies on the dictionary and achieves agency with the use of it. The group's prior negotiation of meaning of statistics makes her less confident in the case of statistics; she indicates that she gives primacy to the group's interpretation. Paul questions this authority altogether. Tilly seems to pick up on Paul's questioning and probes the nature of Paul's agency; is he acting from a mathematics education researcher perspective, or a mathematical community perspective or something else? Paul continues by acting as if there are better interpretations of these concepts by questioning the wordings. He later on continues to negotiate the meaning of these concepts by means of examples and more mathematically precise definitions and thereby achieves agency by referring to the mathematical community. This exchange impacts the negotiation towards being even more focused on language.

It also becomes apparent that Paul's agency influences how Karen and Tilly use technical terms in the remainder of the session. Paul's aims and agencies remain in focus as they are given more space and remain unquestioned as the work progresses. The following episode is from the later parts of the meeting. It shows the impact of Karen's and Tilly's interaction in the first episode regarding the topic but also how Paul is left as an authority. Notice how Paul remains unopposed even though his claims and sentences are incomplete just as Karen's were in the previous episode.

Paul: My question about statistics springs from that fact that statistics is a rather large subject… A large topic so to speak

Karen: Mm

Paul: And I think that the curriculum sort of… Even in our … our curriculum contains what this student is saying the ratings of a TV-show. That you compare… often just think about observations, or we usually say frequencies, frequency tables and so on, so you limit the whole field of statistics to what one might call data collection, frequencies and such.

Tilly: Mm

Paul: What do you think, we could think about beginning to establish this type of concepts like… What is the frequency of blue? What is the frequency… How many observations of blue? How many observations of red? So, you insert this type of technical terms to become more precise. Specify a little bit more. That is, I imagine, a part of learning, that you learn to… You use a language and start to become a little bit more precise.

Paul acts as if he wants to shift focus to the local aim of the activity. He offers an alternative, or additional, learning goal in students developing their language through the activity. He pushes his agenda by referring to the authoritative mathematical community and therefore achieving agency. Both Karen and Tilly accept Paul's agency and leave his claims unopposed and instead adjust their use of language after the episode to fit Paul's. Paul's agency also results in a shift of focus in the following interaction. The continuing negotiation still involves interpretations of language but also aspects of the design and how language development can be anticipated and sequenced in a HLT.

To sum up the analysis, I exemplify how rearranging the aims of the activity, placing the aim of understanding students' learning in the foreground instead of the aim to develop the design, can have huge impacts on the course of the developmental process. Karen's and Tilly's interaction in the first extract refocused much of the remaining discussion towards negotiating the meaning of language use and development. An example of the dance of agency in designing educational activities has also been offered. Especially how some agency has higher authority and thus also more impact on the negotiation process.

Discussion

Shifts of agency emerged in the presented episodes. Karen and Tilly both seemed to mostly rely on personal agencies in their actions, trusting in their professional experience in teaching for learning and language development. Paul on the other hand was perceived to rely on agency achieved from highly regarded disciplines as the mathematics and mathematics education research communities when he contributes with examples and language. It corresponds with results from when [START_REF] Pickering | The Mangle of Practice[END_REF] studied mathematicians in their work. He saw that mathematicians often relied on their personal agency as they created initial ideas but "surrendered" to that of the discipline, as they needed to resort to following standard procedures of (for example) proofs. A similar phenomenon was observed in the episodes. Karen and Tilly relied on their personal agency until Paul referred to the mathematics community; Karen and Tilly then surrendered to the formalized language of the discipline to a greater extent. Further on it resulted in worksheets handed to the students using this formalized language and thus re-evaluating the HLT in light of formal use of concepts. I suggest that it emphasizes how the dance of agency [START_REF] Pickering | The Mangle of Practice[END_REF] is a principle to regard in collaboration between researchers and practitioners as it has the potential to be an indirect tool to guide the negotiation process in different directions.

Shifts in aims emerged in the data. Karen initially acts in line with the global aim of developing aspects of the design but soon shifts to align with Tilly to negotiate the meaning of a student's language development. Paul later uses the mathematics community to achieve agency to shift focus towards developing the design once again and then sits back to evaluate its impact. [START_REF] Blumer | Symbolic interactionism: perspective and method[END_REF] argues that not only individual aims should be regarded but also that of the group. When multiple actors interact over a period of time a joint action is formed. It is a social construct that extends from being merely the sum of all actions, for example it also has its own aim that is negotiated by its participants. One might look at Karen's, Tilly's and Paul's shifts in aims as attempts to negotiate what aim should be in focus or in the foreground while the others remain in the background. It becomes apparent in the topics discussed that the negotiation of aims for the joint action has impact.

In the extended data, patterns emerge of how shifts in the joint aim redirect the following interaction until it was renegotiated. The result was an added activity in the HLT that was meant to challenge students' ideas of chance. Reasons for why the participants chose to contribute in this way did not emerge from the existing data but one could speculate whether it has something to do with the respective role of the actors, being from a tradition of research or education. It presents a way forward to further advance our insight in the collaboration between researchers and practitioners in design research in mathematics education.

Implications

As [START_REF] Stephan | Conducting classroom design research with teachers[END_REF] highlights the importance of working with small groups of teachers instead of just one in design research collaborations, the dance of agency is yet another tool to create purposeful design research collaborations. In the case of Paul, agency is used to steer the negotiation of meaning towards more mathematically aligned use of vocabulary. In another setting, it is possible to rely on shifting agencies to empower teachers in the collaboration by placing it in the domain of mathematics teachers. There is also the possibility to put focus on the research agenda when discussions tend to steer away. One could also consider the interpretation that it has the potential to shift the power relations, creating an (even bigger?) imbalance between participants, making participants less likely to make substantial contributions to the development of the task. The conclusion is that conscious achievement of agency can be used as a tool for researchers in design research collaborations to manoeuvre the discussion and shape the HLT to fulfil different aims.

Proceedings of CERME10

Thematic Working Group 22Proceedings of CERME10


text:
Zenon Borys
email: zenon.borys@gmail.com

Jeffrey Choppin
email: jchoppin@warner.rochester.edu

Tensions between resource perspectives and trends in the design and dissemination of digital resources

Keywords: Digital curriculum, teacher practices, curricular resources

Teachers are increasingly using digital resources to design lessons. We describe three perspectives for describing teachers' interactions with digital resources, perspectives that denote different assumptions with respect to teacher agency and the connections between capacity development and resource use. This paper examines the tensions between these conceptions of teachers' interactions with digital resources and the ways other actorsincluding policy makers, curriculum developers, and purveyors of online contentframe the purpose for and development of digital resources. Our analysis suggests that the assumed role of the teacher differs across different sets of actors and different visions related to the design and development of digital curriculum resources. The implications relate to the opportunities for teachers to transform digital resources to suit their purposes and to develop and grow professionally as a result.

Teaching is design work: Teachers actively interpret and mobilize resources to attain pedagogic and curricular goals. Moreover, teachers increasingly use, and are expected to use, digital resources to design lessons. However, there are tensions between education technology discourses, curriculum design trends, education policies, and the work teachers do with digital resources. As a case in point, in Sweden a public/private partnership endeavor created a repository of resources and lessons for teachers that was little used because the design of the repository did not take into account how teachers actually take up digital resources, illustrating the lack of alignment between teachers' professional practices, education policy, and the design of instructional resources (CERI, 2009). As Remillard (2012) notes, teachers do not simply interpret authors' intentions as they engage with curriculum resources, they engage with the artifact itself (p. 114). Given that teachers' work with resources can promote teachers' professional development and meaningful experiences for students, it is important to understand how tensions between teachers' practices and the assumptions about teaching and teachers embedded in digital resources potentially constrain teachers' opportunities to develop design capacity.

To illustrate how digital resource design and dissemination limit teachers' agencyand thus their roles as designerswe examined tensions between conceptions of teachers' interactions with digital resources and the ways other actorsincluding policy makers, curriculum developers, and purveyors of online content (commercial publishers, large philanthropies)frame the purpose for and development of digital resources. These actors constitute the largest source of resources and information regarding digital resources and thus their influence merits attention from the research community. The ways digital resources are framed and promoted by policy makers and other actors may be at odds with the actual work teachers do and the needs of their students.

Conceptualizations of teachers' work with digital curriculum resources

The meaning of the term resources depends on the perspective of the researcher and what is being researched [START_REF] Ruthven | From design-based research to re-sourcing 'in the wild': reflections on studies of the co-evolution of mathematics teaching resources and practices[END_REF]. Researchers frame the relationship between teachers and their use of resources in three ways: resources function as tools that mediate the act of teaching; resources function as artifacts (instruments) that signify outcomes of processes; and resources function as objects whose creation is a key component of teachers' professional work [START_REF] Remillard | Examining resources and re-sourcing as insights into teaching[END_REF]. These are elaborated in more detail below.

The most common framing in teachers' design work is that resources are tools that mediate teachers' work, in line with sociocultural notions of tool mediation (e.g., [START_REF] Wertsch | Mind as Action[END_REF]. The resource's function is primarily to help/aid/assist teachers in the work of teaching. Teachers' use of resources allows them to improve practices they already engaged in or, sometimes, to engage in new practices altogether. Often, the affordance or benefit of the resource is located within the resource and teachers perceive the affordance or benefit and apply it to their local context. [START_REF] Kasten | Using dynamic geometry software in the mathematics classroom: A study of teachers' choices and rationales[END_REF], for example, showed that teachers selected digital applets to help present topics in new ways. Also, [START_REF] Duncan | Teachers' views on dynamically linked multiple representations, pedagogical practices and students' understanding of mathematics using TI-Nspire in Scottish secondary schools[END_REF] demonstrated that teachers made their classroom practices more student-centered because the resources they were using automatically provided students feedback and gave students more agency over their interactions with content. These teachers crafted lessons that capitalized on this shift in the student-content relationship. In these two examples, the benefits or affordances were thought of as residing in the resource.

Considering resources as an artifactrather than as a tool -connotes place and time: teachers' use of resources is inseparable from when they are used and teachers' purposes for using resources. The theory of instrumental genesis [START_REF] Rabardel | People and technology[END_REF] considers three factors: the impact of the design of tools on how they are used (instrumentation); the users and their previous experiences with using the tool (instrumentalization); and the purposes and goals users assign to tool use (schemes of utilization). Acknowledging the different factors that impact a how teachers take up tools makes the processes associated with using the tool rather than the tool itself the object of study. Rabardel defined an instrument as an artifact coupled with the schemes of utilization users assign to it. It is important to note that the theory of instrumental genesis is focused on the resource in use, not the resource itself. The use of the term instrument emphasizes a process of transformation where the focus is on how teachers use resources to attain goals, not some innate quality of the resources. This is especially important when considering teachers' work with resources because teachers increasingly draw from a variety of sources when crafting instruction and use resources differently over time according to their context (e.g. [START_REF] Gueudet | Teaching Mathematics with Technology at the Kindergarten Level: Resources and Orchestrations[END_REF].

The third conceptualization is framing resources as objects, where a key component of teaching is to create resources. Document genesis [START_REF] Gueudet | Towards new documentation systems for mathematics teachers?[END_REF]) is a theory focused on teachers' creation of resources for teaching. Stemming from the instrumental approach, document genesis explains how, in the creation of resources, teachers develop schemes of utilization that are comprised of teachers' knowledge and processes for using resources. Document genesis offers a window into teachers' evolution of practices with their resources. Since teachers often reuse resources from previous years, the documents from one year become resources for the next year and are embedded with teachers' experiences and modifications [START_REF] Gueudet | Towards new documentation systems for mathematics teachers?[END_REF]. Often, when resources are considered tools, teachers are not producers: their work is of application and applies to one instance of enactment without connection to future use. However, when treated as objects of design, the focus is on how teachers create resources to meet their local contextual challenges and on how the teachers' understanding and use of resources evolves over time.

In a complex landscape where teachers weave together various resources in the design of lessons, treating resources as an object of teachers' design work provides researchers with a theoretical grounding to make sense of and explain interactions in online spaces, such as resource repositories. The resources being downloaded and shared by teachers are not solely resources (tools) to mediate teaching: their creation and transmission constitute important facets of teaching. For example, Trglaova and Jahn (2013) examined how teachers improved resources posted to a repository based on the feedback they received from other teachers. Also, this framing allows researchers to situate teachers in broader collectives and to determine how the impact of belonging to and participating in collectives has on teachers' practices (Guedet & Trouche, 2012).

These perspectives on resources suggest different roles for the teacher with respect to how digital resources get taken up. In the first, resource as tool, the resource is considered to have innate qualities that teachers can employ to varying degrees in their lessons but not substantially revise. The latter two perspectives outline a more active role for the teacher, suggesting not simply a mediating role between resource and student, but a transformative role for the teacher with respect to the design and use of the resource. More generally, thinking about teachers' design processes positions teacher capacity as an important goal in the design and use of digital resources; consequently, perspectives on digital resources that minimize teachers' role have potential impact in terms of agency and ultimately the development of teacher capacity. Below, we explore how trends in the design and dissemination of digital resources may be in tension with conceptualizations of active roles for teachers with respect to the design and use of digital curriculum resources. We consider how different actors influence the design and dissemination of digital curriculum resources, and that these efforts may neglect the role of the teacher. An aim is to show that there are potentially conflicting values between perspectives focused on teaching as design and perspectives that promote characteristics of digital resources that constrain teachers' roles and their ability to be responsive to their local context.

Trends in the design and dissemination of digital resources

In this section, we articulate broad trends in the design and dissemination of digital curriculum materials and then connect those trends to the perspectives that emphasize the design role of teachers. We characterize these trends by focusing on the actors who emphasize particular perspectivesand exercise considerable influencerelated to the design and dissemination of digital resources. We focus on the following three broad sets of actors because of the considerable influence they exercise over the design and dissemination of digital curriculum materials: designers, policy makers, and purveyors of online content (e.g., commercial publishers, for-profit educational websites, large philanthropic or corporate organizations). Included in the group of corporate entities are philanthropies and corporations not previously engaged in educational publishing (e.g., the Gates Foundation, Amazon, Mark Zuckerberg's funding efforts), who strive to influence both content as well as delivery mechanisms for that content, especially in the U.S.

These groups make a number of claims about the potential transformative features of digital curriculum materials. We focus on three features to highlight the roles of the sets of actors identified above:

 Content can be more relevant and interactive;  High quality content can be inexpensive and widely accessible; and  Content can be customized to meet the needs of individual students.

We selected these features because they are the primary focus of the design, dissemination, and publicity efforts of the actors described above. Below, we describe how these features are emphasized by the various actors and how they are in tension with the perspectives on resource use, with implications for how teachers get positioned as active designers who can develop increasing capacity to design and enact curriculum materials.

Content can be more relevant and interactive

Advocates claim that content in digital materials has the potential to be more interactive and relevant, as it can be updated and revised to fit the local context. In terms of interactivity, digital texts can be flexible with respect to navigation (e.g., hyperlinks) and with respect to creating documents with resources and materials from a range of sources, including the web [START_REF] Zhao | Curriculum, digital resources and delivery[END_REF]. Other kinds of interactivity include the use of sliders or buttons to manipulate parameters in a model to investigate problems or phenomena (Dede, 2000). More powerful forms of interactivity involve the use of tools with flexible purposes in open working environments, such as curriculum programs developed in Israel and Korea (cf. [START_REF] Lew | Developing and implementing "smart" mathematics textbooks in Korea: Issues and challenges[END_REF][START_REF] Yerushalmy | Inquiry curriculum and e-textbooks: Technological changes that challenge the representation of mathematics pedagogy[END_REF]. In general, interactivity can be conceived in terms of the choices users can make to influence their engagement with the content. We focus on the set of actors we call designers to highlight how interactive features are incorporated into digital resources. We refer to designers as those who conceptualize features of digital materials based on research on learning and learning systems. Of the sets of actors described above, the curriculum resources designed and disseminated by designers are the most aligned with teaching as design perspective. These resources offer the greatest flexibility in terms of adaptation by teachers and in terms of generating interactions with students that provide opportunities to understand how student thinking develops. Designers emphasize learning experiences that augment or enhance what is possible in paper curricula. Moreover, designers emphasize the development of tool-rich workspaces that enhance interaction, communication, and exploration. Designers incorporate ubiquitous access to tools that allow for dynamically linked representations and the ability to record and curate work (Confrey, 2016;[START_REF] Lew | Developing and implementing "smart" mathematics textbooks in Korea: Issues and challenges[END_REF][START_REF] Yerushalmy | Inquiry curriculum and e-textbooks: Technological changes that challenge the representation of mathematics pedagogy[END_REF]. Confrey, Lew, and Yerushalmy emphasize that workspaces should provide access to a suite of tools that learners strategically select as they engage in complex problems. These workspaces facilitate the use and manipulation of multiple representations, including symbols, in ways that are intuitive and that communicate increasingly formal inscriptions of the mathematics. Furthermore, these workspaces should allow students to store and curate their work, for future reference for themselves and external audiences.

In terms of developing teachers' design capacity, and thus align with the artifact and documentation perspectives described above, designers emphasize more complicated and idiosyncratic learning paths for students in terms of deviating from a rigid demarcation and flow of mathematical activity (Confrey, 2016). Curriculum materials differ from open tools, such as Sketchpad, Cabri, or Mathematica, in that they are intended to provide structure by bounding and sequencing mathematical activity. Integrating rich problems and work spaces provides opportunities for the kind of complex activity that involves non-linear processes (unproductive approaches may precede more productive approaches), complex interactions of tools and representations, and the collective negotiation of the viability and validity of solution paths. Such complex activity can disrupt well-defined lesson structures and allocations of time (both duration and synchronicity)(c.f. Ritella & Hakkarainen's [2012] discussion of chronotype), interrupting the potential flow of a lesson, with implications for following a prescribed scope and sequence of mathematics. Managing these non-linear activity flows calls for more prominent roles for teachers and entails developing new forms of capacity in terms of understanding curriculum progressions and coordinating (orchestrating) multiple tools and artifacts in the workspace.

The work of designers focuses on the interactive and flexible features of digital resources, while other actorspolicy makers and purveyors of online content (mostly commercial interests and philanthropies operating from a neoliberal perspective)emphasize different features, explored below. The different features emphasized by these other actors have implications in terms of the roles and capacities envisioned for teachers.

Free and open digital content

Policy makers have pushed digital content that is freely available and open source. They argue that this would make high-quality content accessible to low-resource high needs districts. Internationally, there has been a push for Open Education Resources (OER) for nearly a decade now: "The open educational resource (OER) movement aims to break down such barriers [from proprietary systems] and to encourage and enable freely sharing content" (OECD, 2007). Recently, the US Department of Education launched an initiative designed to encourage districts to adopt open resources and to share their efforts and experiences with others, in part to make access to high-quality instructional resources more equitable (USDoE, 2016). The use of open resource content, however, can be timeconsuming and the resources themselves are of uneven quality. There is little quality control with respect to content, and much of it requires little interactivity or minimal educative features for explaining the design rationale to teachers. Furthermore, these efforts assume that teachers have considerable capacity to evaluate, select, and sequence content chosen from a variety of sources. Given that much of these efforts are aimed at low-resourced districts, especially in the U.S., there is an assumption that teachers can use the materials without modifications, which aligns with the resource as tool perspective. Recently, there have been efforts to curate content in ways that provide quality control and articulate curriculum progressions (Confrey, 2016); however, these efforts have yet to be coordinated with the larger policies for open resources and their impact on teacher design capacity is not yet determined.

Customizing content for individual learners

Policy makers, commercial publishers, and large philanthropies have emphasized the promise of digital content to be customized to meet the needs of individual learners. Customization has been discussed in a variety of ways. This can be achieved through systems that emphasize mastery learning, in which software dictates the sequencing of content for a learner based on the learner's performance on skills-based assessments (e.g., [START_REF] Means | Lessons from five years of funding digital courseware: Postsecondary success portfolio review[END_REF]. Or, it could involve personalizing the software settings so that the user has control over video and audio as well as the presentation of the text. A third way is for the teacher to make content selections within a program so that different students would see different content. Mostly, however, the personalized systems entail diagnostic assessments administered through online programs that dictate the pacing and sequencing of content [START_REF] Choppin | A typology for analyzing digital curricula in mathematics education[END_REF]. These efforts, largely funded and publicized by large philanthropic or corporate entities, push to embed digital content in comprehensive learning management systems that include data reporting and classroom management systems. They also emphasize adaptive programs based largely from the mastery learning perspective (e.g., courseware funded by the Gates Foundation). Some educational websites, either for-profit or philanthropyfunded, create collections of lessons, sometimes developed by a small group or by larger author groups (e.g., Khan Academy, sofatutor.com). While these programs may eventually involve sophisticated adaptive systems and customized learning tools that allow learners to explore content in complex workspaces, the initial versions of these programs typically entail low-level content and reporting of percentage of correctly answered multiple-choice questions [START_REF] Choppin | A typology for analyzing digital curricula in mathematics education[END_REF], characteristics that constrain opportunities for learning. Furthermore, the programs minimize the role of the teacher, impacting development of teacher capacity.

Implications of tensions between perspectives

Teachers' interactions with digital resources involves agency: in the act of designing, teachers take up digital curriculum resources, interpret their purposes, and transform them to respond to their pedagogical purposes and their evolving understanding of how those resources can be used to engage students. This assumes that teachers are designers, actively interacting with and transforming curriculum resources to engage students. The three perspectives describing teachers' interactions with digital resources denote different assumptions with respect to teacher agency and the connections between capacity development and resource use. The assumed role of the teacher differs across the perspectives of the different sets of actors that relate to the design and development of digital curriculum resources.

The analysis illustrates that different perspectives on the design and dissemination of digital resources have implications for the opportunities teachers have to understand, adapt, and transform digital resources and thus develop capacity as a result. The designer perspective offers the greatest possibility of alignment between views of teachers as designers and the affordances of digital curriculum materials, though much of their focus and expectations is on creating a medium for rich student interaction rather than informing teacher use and capacity development. Other actorssuch as policy makers, commercial publishers, large philanthropieshave different perspectives, and motives, for designing and disseminating digital curriculum. The programs associated with these perspectives show little sensitivity to the complexities involved with how teachers take up digital curriculum resources, and minimize the role of teachers to transform digital resources to suit their purposes. Many of these programs emphasize personalized learning, which entails little interaction with teachers, and little flexibility on the teacher's part to construct learning experiences for students. These entities exercise great influence on which programs are available, and their impact on teacher autonomy and teacher capacity needs to occupy the attention of researchers.

Proceedings of CERME10

Acknowledgment

This research was funded by U.S. National Science Foundation, DRL #1222359.


text:
Birgit Pepin

Sean Delaney
email: sean.delaney@mie.ie

Sebastian Rezat
email: srezat@math.upb.de

Andreas J Stylianides

Introduction to the papers of TWG22: Curricular resources and task design in mathematics education

Keywords: Curriculum resources, task/curriculum design, design principles, teacher collective work, design capacity

In this report the themes and points for discussion of the Thematic Working Group 22 on Curricular Resources and Task design are briefly explained and summarized. These related to design principles and task characteristics, also of digital curriculum resources, and perspectives on the development of digital resources were identified. At the same time the role of teachers in task design, whether as "designers" or "partners in task design" or as mediators of tasks designed by others, and of course the role of students working with the tasks/resources, were acknowledged as crucial issues. Concerning teachers, pre-service or in-service teacher education, and generally teachers' work in collectives, were perceived as stimulating contexts for the design of and work with curriculum resources: how can teachers develop design capacity and knowledge when working with curriculum resources, and which affordances (and constraints) are provided by digital resources? Concerning students, discussions evolved around issues related to how digital resources can provide feedback for enhanced student learning.

Background

The TWG 22 received 25 paper and poster submissions: 17 submissions were accepted as papers, 6 as posters, and 2 were withdrawn (for various reasons: lack of funding; lack of time). The contributing authors came from 17 different countries.

The sessions were organised into five paper sessions organised under four themes, one poster session, and one report & discussion session. For each session we now briefly present the authors and titles of the individual presentations, followed by a summary of themes discussed in the sessions.

Session 1 & 2

Theme: Issues linked to the design of tasks and resources The issues identified in these two sessions related to four main themes. First, participants discussed the relationships between design principles and task characteristics. This is particularly important, if the design is expected, for example, to afford conceptual change (see Cohen-Eliyahu), or to support student reasoning competency involving aspects of authority and responsibility (see Seidouvy & Eckert) or to enhance students' beliefs about the relevance of mathematics (see Gijsberg & Pepin), or to facilitate noticing and visualization when developing flexible multiplicative reasoning skills (see Brocardo, et al.). It is also pertinent for tasks that were re-designed from paper and pencil to etasks (see Nagari Haddif). Second, perspectives on the development of digital resources were identified, and tensions between conceptions of teachers' interactions with digital resources and the ways other actors (e.g. policy makers, curriculum developers) frame the purpose for and development of digital curriculum resources (see Borys & Choppin). A third point of discussion was the role of teachers in task design, whether as "designers" or "partners in task design" or as mediators of tasks designed by others. In fact, it was concluded that often teachers were left out, or insufficiently considered, in relation to their role of mediating tasks. A fourth point of discussion was related to the role of students working with the tasks/resources, which was addressed in the majority of the papers presented in these two sessions.

Session 3 (poster session)

In this session all posters were presented in order to provide additional opportunities for contributors to participate in this TWG.

-

Session 6

Theme: Issues linked to teacher and student use of resources/curriculum materials at primary level

• Daina: From textbook to classroom: a research on teachers' use of pedagogical resources in the context of primary school in the French

• Delaney: Children's performance on a mathematics task they were not taught to solve: A case study

• Gaio: Programming for 3rd graders, Scratch-based or Unplugged?

• Rezat: Students' utilization of feedback by an interactive mathematics e-textbook for primary level

The issues identified in this session related to four main themes. First, although not directly addressed in all papers of this session, the choice, design and use of digital as compared to nondigital curriculum resources were discussed. This was seen as particularly relevant, as often digital curriculum resources are combined with traditional materials (see Gaio), such as textbooks. Second, participants discussed the use of mathematical tasks/curriculum resources to promote particular learning goals (see Diana), in particular as textbooks often do not provide information on how to organize the teaching of particular activities. This links to the third point, the enactment/ implementation of tasks depending on teacher goals (see Delaney). A fourth point related to task design and feedback, that is ways in which teachers or curriculum developers can provide effective feedback, i.e. feedback that actually influences the development of mathematical concepts (see Rezat).

Session 7

In session 7 the following themes were identified as overarching issues:

• Task design: what does task mean; what does design mean?

• Teacher design capacity & the role of the teacher in "design" (incl. implementation)

• Digital resources/tasks/curriculum materials

• Plurality of theoretical frameworks & clarity

• Operationalization of theoretical frames

• High inference claims & evidence

Whilst the first three have been addressed under previous sections, the last three deserve a special mention. It was noted that a plurality of theoretical frames was used. This is not an issue in itself, but it becomes a problem, when the diversity of theoretical frameworks diverts from, and sometimes covers, the problem addressed, that is obscures the clarity of the research. Linked to that selected participants would have wanted a better, or clearer, description of how the theoretical frames were actually operationalized. Moreover, it was mentioned that too often the researchers made high inference claims based on insufficient evidence. This was seen as a shortcoming of such research. Overall, participants emphasized the positive and constructive atmosphere in the group, where criticism was welcomed as a vehicle for developing deeper insights and sharpening up of ideas.

•

  Gijsbers & Pepin: Context based tasks on differential equations to improve students' beliefs about the relevance of mathematics • Nagari Haddif: Principles of redesigning an e-task based on a paper-and-pencil task: The case of parametric functions • Borys & Choppin: Tensions between resource perspectives and trends in the design and dissemination of digital resources • Brocardo, et al.: Tasks to develop flexible multiplicative reasoning • Seidouvy & Eckert: Designing for Responsibility and Authority in experiment based instruction in mathematics • Cohen-Eliyahu: The role of design in conceptual change: The case of proportional reasoning

  Noll, et al.: How to design educational material for inclusive classes -Jukic Matic: Teachers' pedagogical design capacity and mobilisation of textbook -Wynne & Harbison: Task design within the Universal Design for Learning Framework to support inclusion in the mathematics classroom -Llanos & Otero: Changes in the images and arguing from mathematics textbooks for the secondary school in Argentina along 67 years -Dooley & Aysel: Using variation theory to explore the re-teaching phase of lesson study -Cizmesija et al.: Asymptotes and asymptotic behaviour in graphing functions and curves: an analysis of the Croation upper secondary education within the anthropological theory of didactics Session 4 Theme: Issues linked to prospective teachers' work with tasks and resources • Dempsey & O'Shea: Critical Evaluation and Design of Mathematics Tasks: Pre-Service Teachers • Kilic, et al.: Pre-service teachers' reflections on task design and implementation

•

  Stylianides & Stylianides: Promoting prospective elementary teachers' knowledge about the role of assumptions in mathematical activityThe issues identified in this session related to four main themes. First, although not directly addressed in the three papers, the issue of design capacity building was raised: what teacher design might mean (see also sessions 1 & 2), who is designing what (see also sessions 1 & 2), and how design capacity building relates to teacher education. Linked to latter, a second point was discussed: the relationship between teacher knowledge (see Stylianides & Stylianides) and design capacity. As a third point for discussion, participants reflected on the implementation process in terms of preservice teacher learning/knowledge development (seeKilic, et al.). Under a fourth issue it was discussed how teacher knowledge (in teacher education) could be enhanced through task design (Dempsey & O'Shea).

	Session 5
	Theme: Issues linked to design and use of resources in professional development and collective
	work

• Eckert: Agency as a tool in design research collaborations • Essonnier, et al.: Factors impacting on the collaborative design of digital resources • Gueudet & Parra: Teachers' collective documentation work: a case study on tolerance intervals • De Moraes Rocha & Trouche: Documentational trajectory: a tool for analysing the genesis of a teacher's resource system across her collective work The issues identified in this session related to three main themes. First, it was emphasised that teachers' learning trajectories of professional learning are typically not short-term, and hence that there is a need to research them longitudinally (see De Moraes Rocha & Trouche). Second, the dynamics in collaborative task design of teachers and teachers working/designing collaboratively with curriculum resources, were highlighted (see Eckert; Gueudet & Parra): in terms of lesson preparation, design of learning trajectories, and/or teacher re-design in class. Linked to that a third point was raised: the cultural aspects of teachers working with curriculum resources (see Essonier et al.).

Proceedings of CERME10


text:
Eugenio Chandia
email: eugenio.chandia@ciae.uchile.cl2sip

Maria Luz Montes

Incorporating mathematical problem solving in the Chilean school curriculum

Keywords: Problem solving, math curriculum, professional development

Problem solving is one of the skills that is expected to develop in children who learn mathematics. To do this, school mathematics curriculum worldwide has incorporated this skill for teachers to promote it in classrooms. Though research results show what should be made, these have been parceled and make difficult an effective incorporation of the skill, its promotion and students' learning in schools. Therefore, the following work presents a professional development strategy that incorporate diverse research results related to the promotion of problem solving in math class. This allows to achieve a professional development complex system whose main axis is the school teacher.

Introduction

In Chile, the General Law of Education states that "students develop knowledge, skills and attitudes that allow them to understand and use basic mathematical concepts and procedures, related to numbers and geometric forms, in everyday problems solving, and to value the contribution of mathematics to understand and act in the world", (article 29, section f). However, in the Curricular Study Programs of Chilean Mathematical Education there is no concrete proposal for teachers to promote such skills and attitudes in the classroom. Then, this poster shows one way to develop and include the problem solving in the Chilean Curriculum National of Math Education. To do this, is necessary to know what is needed for a successful curricular proposal implementation in math. The NCTM (2014) suggests the following points to implement a successful curricular development proposal: a) Teachers need to have a professional development of high quality to maximize the efficacy of the materials and activities they propose in the curriculum, since even the best textbooks and resources can be misinterpreted or misused. b) Collaboration among teachers throughout the school year may result in appropriate adjustments and activities adaptations for dosing topics to address the strengths and weaknesses of each student. For this reason, along with thinking about the curricular development proposal, a professional development proposal that supports teachers to incorporate in a better form the new requirements that are demanded need to be thought. On the other hand, [START_REF] Marrongelle | Scaling up professional development in an era of common state standards[END_REF] indicate that the characteristics that make more effective a professional development program are: to be performed regularly and connected to practice; to be focused on students learning; to be directed to the teaching of a content; to be aligned with objectives of the school; and to build strong links among teachers. Considering the perspective mentioned above, in the following parts will be described the process for the implementation and the incorporation the PS in the curriculum.

Proceedings of CERME10

Methodology

The project aims to impact the: instructional practices of participant teachers related to the delivery of opportunities to develop skills in the students, focused in problem solving; teachers' skills to solve problems; teachers' skills to modify and adjust activities to promote the development of skills in their students; students' skills to solve problems; students' skills to represent, report, discuss, argue and explain mathematically. For this purpose, a proposal for curricular development that was adapted to the reality of the Chilean school system in Mathematics Education was designed. The project includes two main stages. In the first one a proposal for curricular development based on a literature review was designed, based on the Chilean curricular framework. The proposal included classroom activities for students, instructional methodologies for teachers and an annual curriculum integration plan for each participating school. In the second one, there was implemented a pedagogical technical support model in order to teachers appropriate the activities and methodology to be performed later in the classroom. The curricular development proposal will be implemented in 4 schools, which compromised their participation of teachers and students from 1st to 4th grade and the time necessary to do so. It had a population of 24 teachers and 600 students from 1st to 4to grade in total.

Discussion and conclusions

The project is still being conducted, but there are preliminary results on its implementation. In the stage 1 of the proposal design was observed that while the curricular proposals from countries such as Singapore or Finland have among their curricular objectives to promotion of mathematical skills as the problems solving, the curricular resources available for teachers in those countries do not guarantee by their own, the skill promotion and not the acquisition of teachers' instructional practices. Again, research results corroborate the lack of professional development strategies that allow the skills acquisition by teachers [START_REF] Bunyi | The quest for quality education: the case of curriculum innovations in Kenya[END_REF]. Given this lack and the need of the Chilean teachers in terms of mathematical knowledge, as well as in pedagogical skills to promote mathematical skills and attitudes towards it in classrooms, it is that the strategy of weekly and monthly support was crucial, both for the acquisition of pedagogical skills and mathematical knowledge.