Entering into dialogue about the mathematical value of ... - Springer Link

Math Ed Res J (2018) 30:21–37 DOI 10.1007/s13394-017-0218-2 O R I G I N A L A RT I C L E

Entering into dialogue about the mathematical value of contextual mathematising tasks Caroline Yoon 1 & Sze Looi Chin 1 & John Griffith Moala 1 & Ban Heng Choy 2

Received: 6 February 2017 / Revised: 5 June 2017 / Accepted: 5 June 2017 / Published online: 24 June 2017 # Mathematics Education Research Group of Australasia, Inc. 2017

Abstract Our project seeks to draw attention to the rich mathematical thinking that is generated when students work on contextual mathematising tasks. We use a design-based research approach to create ways of reporting that raise the visibility of this rich mathematical thinking while retaining and respecting its complexity. These reports will be aimed for three classroom stakeholders: (1) students, who wish to reflect on and enhance their mathematical learning; (2) teachers, who wish to integrate contextual mathematising tasks into their teaching practice and (3) researchers, who seek rich tasks for generating observable instances of mathematical thinking and learning. We anticipate that these reports and the underlying theoretical framework for creating them will contribute to greater awareness of and appreciation for the mathematical value of contextual mathematising tasks in learning, teaching and research. Keywords Mathematising . Reporting . Design-based research

* Caroline Yoon [email protected] Sze Looi Chin [email protected] John Griffith Moala [email protected] Ban Heng Choy [email protected]

1

The University of Auckland, Auckland, New Zealand

2

National Institute of Education, Nanyang Technological University, Singapore, Singapore

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The context of modelling activities and mathematising tasks in New Zealand Create an unbiased draw for an ultimate frisbee tournament Divide a catch of fish fairly between two families Choose the best route to the airport during rush hour The above scenarios are examples of everyday concerns of sports officials, recreational fishers and taxi drivers. They are also the kinds of scenarios that can be featured in mathematical modelling activities: tasks that require students to get their hands dirty in the messy reality of a context while devising a mathematical description (a model) of the scenario. Modelling activities are distinct from word problems, which also attempt to connect mathematics to the real world, but do so in a more contrived manner. For example: &

&

A modelling activity might give students a city road map and information about bus and train services (see Fig. 1a) and ask them to generate a method for choosing the ‘best route’ from any part of the city to the airport, where ‘best’ involves weighing up different factors such as time, route directness, cost of tickets and overall distance. A word problem might give students the diagram in Fig. 1b and ask: What is the shortest distance from Sally’s house to the airport? Answering this word problem requires simply calculating four sums and identifying the smallest.

Word problems are standard mathematics problems, ‘dressed up’ (Niss et al. 2007) in a sanitised version of the real world, where the mathematics required to solve the problem is carefully controlled and predictable. In contrast, modelling activities embrace the messiness of a real-world context and encourage students to mathematise this information in the form of a model, while drawing on whatever mathematics is appropriate. Over the past decade, employers, researchers and curriculum designers have promoted mathematical modelling activities as resources for developing twenty-first century competencies such as problem-solving, teamwork and communication (e.g. English and Gainsburg 2016; Ministry of Education 2007). In most uppersecondary and tertiary mathematics classrooms in New Zealand; however, modelling activities are limited to one-off rainy-day events, or not implemented at all. Their low implementation rate is partly an ironic reflection of their rich instructional value:

a

b

Fig. 1 a Material provided in modelling activity. b Diagram for a word problem

The mathematical value of contextual mathematising tasks

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modelling activities generate mathematics that is so diverse and unexpected that student performance is difficult to document and assess. This contrasts with word problems: standard measures such as percentages and grades can describe student performance on word problems but fall short when attempting to capture the diversity and complexity of mathematical modelling solutions. As a consequence, classroom and research reports of student performance on modelling activities often emphasise student engagement and real-world competencies but underreport the mathematical content of student solutions. This contributes to scepticism among students, researchers and classroom decisionmakers (such as teachers, lecturers, heads of departments and course coordinators) about the mathematical value of modelling activities: ‘These are fun problems, but where’s the maths?’ This gap between calls for teaching with modelling activities and their limited use in practice may also be explained by differences in the reasons for valuing modelling. Many researchers advocate for teaching modelling for its own sake, so that students learn modelling-specific competencies such as making assumptions, mathematising, critiquing and refining a model (Maaß 2006). This typically involves project work done over large amounts of time (a full day or more) in which students experience, reflect on and develop modelling-specific competencies. In contrast, teachers who are conscious of curricular and time constraints often regard modelling as a ‘vehicle’ (Julie and Mudalay 2007) for teaching some specific mathematical content that is aligned to the curriculum and are unlikely to allocate the time needed to teach modelling for its own sake. We perceive a need for tasks that represent some middle ground between modelling activities and word problems, and that can serve as a way of encouraging dialogue between researchers and teachers about the role of modelling activities in mathematics teaching and learning. For this project, we designed a suite of what we call contextual mathematising tasks that invite students to mathematise a meaningful context situation. These tasks have been designed to be implemented within 50 min—the length of a typical class period, lecture or tutorial in New Zealand schools and universities. The task design was influenced by principles for designing ‘model-eliciting activities’ (Lesh et al. 2000), and while all of the tasks invite students to mathematise, they vary in the degree to which they invite other modelling competencies. We now describe an example of one of these contextual mathematising tasks. The food court seating task The food court seating task begins with students viewing and trying to identify images of recognisable dining locations. These images are chosen so they are suitable for each audience. For example, when we implement the task with students in Auckland, one of the images comes from the top of the Sky City Tower. With Wellington audiences, we include an image from the Wellington airport. Afterwards, students are introduced to Taki, a young man who works at a local mall in South Auckland. Taki is frustrated that the food court in the mall is so busy during his lunch break that he cannot sit down to eat his lunch, and he plans to ask management to redesign the seating area so more people can sit and eat their meals. Students are given diagrams showing the existing food court seating arrangement (Fig. 2) and are presented three alternative arrangements proposed by Taki and his friend, Donelle (Fig. 3a–c).

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Fig. 2 Existing food court seating arrangement

Students are asked to critique these latter three proposals and to describe some common sense assumptions for designing seating arrangements. Typical student responses include all seats must be accessible, there should be a variety of seating options so that people do not have to sit next to strangers and there must be enough space for each chair. Next, students are told that the managers of the mall have received Taki’s request favourably but have developed some new rules that must be abided when designing new seating arrangements (see Fig. 4). Usually by this time, about 15 min have passed since the beginning of the task. Finally, students are invited to design a new seating arrangement that maximises the number of people that can be seated within the given area, while following the rules, and to generalise their arrangement for any rectangular area (see Fig. 5). They are encouraged to spend about 35 min on this.

Research questions and scope Our project seeks to enter into dialogue with various stakeholders about the mathematical value of contextual mathematising tasks. We will design new ways of reporting that raise the visibility of the mathematics students engage with as they work on contextual mathematising tasks. These reports will be used to initiate conversations with students and decision-makers from upper-secondary and tertiary classrooms, as well as researchers of mathematical thinking and learning about: & & &

Enhancing students’ mathematical solutions in contextual mathematising tasks Integrating contextual mathematising tasks into mathematics content teaching Analysing the mathematical thinking generated by contextual mathematising tasks

Our reports will go beyond the current practice of labelling contextual mathematising tasks with lists of mathematics topics selected from existing curriculum documents: ‘This task involves proportional reasoning and geometry’. We seek to develop more nuanced ways of reporting that are based on theoretically grounded


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Fig. 3 a Taki’s proposed seating arrangement. b Donelle’s counterproposal to Taki’s proposed seating arrangement. c Donelle’s revised proposed seating arrangement

analyses of students’ mathematical work and are guided by one overarching research question: How can we capture and describe student mathematical thinking in a way that raises its visibility while retaining and respecting its complex nature?

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Fig. 4 The list of rules for the food court seating task

This question can be operationalised into two sub-questions: 1. What kinds of reports will document, measure and describe the mathematical thinking of students working on contextual mathematising tasks in a way that makes this complex mathematical thinking visible to students, teachers and decision-makers?

Fig. 5 The problem statement in the food court seating task


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2. What theoretically grounded analytical framework can be used to capture and describe rigorously the complex mathematical thinking of students working on contextual mathematising tasks, in order to inform the design of practical reports? We will develop two products that align with each of our sub-questions: 1. A practical set of reports for engaging students, teachers and researchers in conversation about the mathematical value of contextual mathematising tasks 2. A theoretically grounded framework for creating the reports This project builds on New Zealand research from two other nationally funded projects. The first project created principles for designing modelling activities for secondary school that can be integrated into instructional sequences (see the LEMMA series, for example, Yoon et al. 2016). These principles were then extended to design conceptual readiness tasks that prepare students for lectures at the undergraduate level (Barton et al. 2017). The principles that emerged from both projects informed the design of contextual mathematising tasks used in the current project. We now look beyond questions about the design of these tasks and initiate conversation with students, teachers and researchers about how they might use contextual mathematising tasks in their practices. Our project builds on a large international research base that investigates the use of modelling activities in mathematics instruction at university and secondary school (e.g. Stillman et al. 2007). It is also situated within a smaller, emerging research area that designs diagram-based ways of describing the mathematics students engage with as they work on rich, open-ended tasks (e.g., Williams 2007; Yoon 2015). In combining these two research influences, this project seeks to raise the visibility of modelling activities as research tasks for investigating mathematical thinking and learning, as well as their more familiar role as instructional resources.

Description of research design The project’s dual focus on practice and theory lends itself to a design-based research approach (Cobb et al. 2003), which coordinates theory and practice to design instructional artefacts and the means of supporting them. Since our practical reports need to be accountable to a range of stakeholders, we adopt a multitiered component in our design (Lesh et al. 2008), in which three tiers of participants (students, teachers and researchers) co-design (with researcher-designers) the reports that address their tier’s needs. Research will be conducted in two phases, during which reports for each tier will be designed, tested and revised over 46 cycles in total (see Table 1). Phase 1 will involve rapid prototyping (Joseph 2004), whereby student and teacher reports are designed in quick succession to provide timely workable solutions that can be rapidly tested and refined. Phase 1 emphasises the design of low fidelity prototype-reports for students and teachers through long chains of design cycles. At the student tier, 12 reports (indicated in Table 1 by S1.1, S1.2, … S1.12) will be designed and tested in quick succession, with results from each test influencing the design of the subsequent report.

S1.2

S1.3

S1.4

S2.2a S2.3a

S2.2b S2.3b

S2.2c S2.3c

Design S2.1a

S2.1b

S2.1c

Theory Ongoing interactions between theory Theoretical and design framework analysis

S1.1

S1.6

S1.7

R2

S2.4c S2.5c S2.6c

S2.4b S2.5b S2.6b

S2.4a S2.5a S2.6a

Ongoing interactions between theory and design

S1.5

S1.9

Theoretical framework analysis

S1.8


R3

T2.1

T2.2

T2.3


S1.10 S1.11 S1.12


R1

R4

T2.4 T2.5 T2.6


T1.1 T1.2 T1.3 T1.4 T1.5 T1.6 Theoretical framework analysis


Ri Design cycle for the ith researcher-oriented report

Tn.i Design cycle for the ith teacher-oriented report in phase n

Sn.i Design cycle for the ith student-oriented report in phase n. Each student group in phase 2 is involved in three design cycles, which is denoted by a, b, c in subscripts

Phase 2

Phase Theory Identifying theoretical 1 Design influences

Table 1 Project activity occurring at theory and design levels in phases 1 and 2

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At the teacher tier, six reports (indicated in Table 1 by T1.1, T1.2, … T1.6) will be similarly designed, tested and revised. The researcher-oriented reports will only be tested on one occasion during this initial phase (R1). Throughout these design cycles, we will record memos of the theory we engage during the design, testing and revisions. We will then pool these records at the end of phase 1 to articulate a theoretical framework that guides our design. Phase 2 will be characterised by more frequent shifts between the worlds of theory and practice as we seek to refine our theoretical framework and fine tune the design of our practical reports. Opportunities to analyse and revise the theoretical framework will be interspersed between shorter sequences of design cycles. We will conduct three more cycles of the researcher-oriented reports (R2, R3, R4) as we move our focus towards becoming explicit about the theoretical groundings of our analytical framework. We will now describe the research setting and participants involved in both phases before providing more details about how we will design and test the reports and articulate and revise the theoretical framework. Research setting Research will take place at a diverse range of instructional sites in New Zealand (see Table 2), which are purposefully chosen to enhance the generalisability of our results. Six mathematics courses are chosen from three educational sectors: secondary school, foundation studies (transitional programmes to prepare students for tertiary study) and undergraduate mathematics. The courses within each educational sector are further differentiated: the two secondary schools have different decile ratings 1 (decile 3 vs. decile 7); the two foundation studies programmes prepare students for different universities; and the two undergraduate mathematics courses cater for students with different majors (non-maths majors vs. maths majors). Participants Our research involves three types of participants: teachers, students and ‘outside’ researchers. Teacher participants Table 3 provides details of the six teacher participants. We use the term ‘teacher participant’ rather than their official job titles, as their relevant role in this project is as teachers of mathematics. With the aim of enhancing the generalisability of our research results, we select teacher participants with a wide range of experience in using mathematics education research to inform their teaching. We recruit some teachers with limited experience in mathematics education research alongside others with more experience to increase the representativeness of our sample. These teachers also have a wide range of reasons for choosing to participate in our study. Given the diversity of our teacher participants, it is more likely to get divergent opinions and robust debate during our dialogues with them. This is a hallmark of sound 1 School decile ratings in New Zealand describe the socioeconomic position of a school’s student community relative to other schools, with a decile 10 being the highest socioeconomic rating, and decile 1 being the lowest socioeconomic rating.

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Table 2 Instructional sites and teacher and student participants Site Institution

Educational sector

Secondary

Description of mathematics course (approx. class size)

1

Public school X (decile 7) Year 13 mathematics with calculus class (25)

2

Public school Y (decile 3) Year 12 mathematics class (25)

3

University Z

Pre-degree bridging course in algebra (100)

4

University W

Pre-degree bridging course in algebra (40)

Undergraduate 5

University W

First year undergraduate service mathematics course for non-mathematics majors (1000)

6

University W

Second year undergraduate discrete mathematics course for mathematics and computer science majors (150)

Foundation Studies

design-based research because it encourages researchers to test, examine and revise the credibility of the artefacts they design. Student participants Table 4 provides details about the 54 students who will participate in the project. During Phase 1, six students from each course will work in groups of three, resulting in 12 groups of three (36 students in total). When the same six courses are offered again during Phase 2, three new students from each course will Table 3 Teacher participants Site Teacher participant pseudonym Mathematics education Motivation for participation and professional role research experience 1

Ed: secondary school mathematics teacher and HOD

Conducted MSc research in mathematics education

Keen to use modelling activities as internal assessment projects

2

Dee: secondary school mathematics teacher and acting/assistant HOD

Graduate diploma in secondary teaching, limited experience in education research

Keen to experiment with activities to engage students and to help students gain the credits they need

3

Kat: foundation studies mathematics lecturer, course coordinator and programme leader

Conducting PhD research in mathematics education

Keen to see how project can contribute to her teaching and research

4

Hal: foundation studies mathematics lecturer and course coordinator

Conducted MPhil research in Open-minded about value of education contextual mathematising tasks for raising student engagement

5

Ash: university lecturer, course coordinator and undergraduate coordinator in mathematics

Limited experience in education research

6

Gus: university senior lecturer in No experience in education mathematics, research research mathematician

Keen to reflect on pedagogical practice and develop education research experience Open-minded about relevance of mathematics education research to his teaching


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Table 4 Student participants Site Description of students (student age)

No. of students in No. of students in phase 1 phase 2

1

Year 13 secondary school students in mathematics with calculus Two groups of class three students (17–18 years)

One group of three students

2

Year 12 secondary school students in mathematics course (16–17 years)

Two groups of three students


3

Post-secondary school students in mathematics course for pre-degree qualification (18+ years)



4

Post-secondary school students in mathematics course for pre-degree qualification (18+ years)



5

First year undergraduate students in general mathematics course Two groups of for non-mathematics majors three students (18+ years)


6

Second year undergraduate students in discrete mathematics course for mathematical sciences majors (19+ years)



n = 36

n = 18

Total number of students in each phase:

work in groups, resulting in six groups of three (18 students in total). Students will participate in the research in two ways. First, they will work on three to four contextual mathematising tasks in their groups, thereby providing data on mathematical thinking which we will use to create reports for students, teachers and researchers. Second, they will evaluate reports of their own mathematical thinking; these evaluations will be used to inform further cycles of design. We will work with students from a diversity of educational sectors to gain a range of perspectives on what constitutes a useful report for them so as to generate a range of mathematical thinking for reporting. ‘Outside’ researcher participants We will consult with 6–12 mathematics education researchers (referred to here as ‘outside’ researchers) who are not named as investigators on this project. These will involve members of and academic visitors to the Mathematics Education Unit in the University of Auckland, as well as researchers at regional and international conferences. Outside researchers will evaluate reports of student mathematical thinking through interactive seminars.

Design cycles for practical reports In order to design practical ways of reporting on students’ mathematical thinking, we first need to engineer observable instances of that mathematical thinking. We do this by implementing mathematising sessions in which we collect data on students’ mathematical thinking. Data from these mathematising sessions will be analysed and transformed into reports of students’ thinking. The reports will then be tested out with the relevant

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tier of participants, and data from the testing at each site will be used to inform the next cycle of report design. We will now describe each of the four stages of this process. Stage 1: implementation of mathematising sessions Each group of student participants will meet over four sessions during phase 1 and three sessions in phase 2, yielding 66 one-hour mathematising sessions in total. In each session, student groups will work on a contextual mathematising task in the presence of a researcher, who will present and clarify the task, but will not offer mathematical hints. The tasks will feature concepts from discrete mathematics that are likely to be unfamiliar to the students, yet accessible through the real-world contexts. Students will be video and audio recorded, and transcripts will be generated of their work. The researcher will take observation notes and collect all student inscriptions produced on paper. These mathematising sessions will be conducted outside of class time, and outside of course requirements. This is to allow (1) high-quality data to be collected in semi-clinical settings without classroom noise or interruptions to produce transcripts that can be analysed in depth and (2) students to engage in the mathematising sessions unaffected by course assessment concerns. Stage 2: design of reports Once data have been collected from each mathematising session, three researcherdesigners will work as a team to design reports of the mathematical thinking from each session. Reports aimed at students will be designed with the objective of encouraging students to reflect on their mathematical thinking and enhance their performance on future contextual mathematising tasks. These reports will be generated soon after the students have finished their mathematising sessions so that the students can recall their mathematising experiences. In phase 1, reports will be created within 1 week of completion, which will allow researchers to have sufficient time to analyse the data on student thinking and identify narratives that will help achieve the reporting objective. Once analytical procedures are established in phase 2, the time interval will be reduced to a mere 10 min post-completion, allowing researchers to create formative feedback reports under similar time pressures that teachers encounter in classrooms. Researcher’s observation notes and students’ inscriptions will be used to create these reports, which will then be presented verbally to the students. If relevant, de-identified reports of other students’ work on the same contextual mathematising tasks may also form part of the report. Reports that are aimed for teachers will be designed with the objective of encouraging teachers to plan how they might integrate the contextual mathematising tasks into their future teaching. These reports will be generated from more intensive analyses of transcripts of students’ work over a longer period of time (between six and 12 weeks). Detailed annotated transcripts of student thinking will be created from data collected on students’ multimodal forms of expression that include speech, inscriptions (including words, symbols, graphs, drawings) and gestures (pointing/deictic, iconic/imagistic and metaphorical/symbolic). Research suggests that evidence from such a variety of semiotic resources can contribute to a comprehensive picture of student thinking (Arzarello et al. 2009). These transcripts will be analysed to identify the mathematical structures to


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which students attend, including mathematical objects, relations, attributes and operations, according to the spot diagram methodological approach (Yoon 2015). A combination of data mining and epistemological analysis will be used to construct coding schemes for the transcripts emerging from each task. These can range from less than 20 codes in relatively simple initial mathematising tasks to 200+ in later tasks, where a wider range of concepts are drawn on with deeper levels of complexity. Coding will be conducted separately by two researchers, who will compare and revise the coding schemes to achieve a satisfactory level of inter-rater reliability. Codes will then be used to construct spot diagrams, which will subsequently be interpreted to identify key instances of conceptual development. Reports that are aimed for outside researchers will be designed with the objective of encouraging researchers to consider contextual mathematising tasks as research tools for generating observable evidence of rich mathematical thinking. These will be generated using a combination of the techniques used in designing reports for students and teachers. Stage 3: evaluation of reports Table 5 describes the data that are collected during the three tiers of practitioner testing. Students and teachers will give feedback on reports via questionnaires and semistructured interviews. Two versions of a questionnaire will be created (one for teachers; one for students), using a combination of closed and open items to seek their evaluation of the reports. Most questionnaire items will be the same across both versions. Closed items will invite participants to record their judgements of the reports they have read, covering such features as the range of mathematical concepts covered, the depth of the mathematical concepts covered, the readability of the reports and the usefulness of the reports. The open items will invite participants to describe how they might use the information from the reports, either to prepare for future encounters with contextual mathematising tasks (in the case of students), or for future lesson planning (in the case of teachers). Interviews with students and teachers will follow a semi-structured interview protocol in which project researchers will probe their evaluations of the reports. Participants will be asked how they will use the information either for future lesson planning or study and to expand on what other kinds of information would be helpful in achieving these goals. Outside researchers will give verbal feedback on reports through seminar participation. Project researchers will present reports of student thinking to outside researchers in interactive seminars and will record outside researchers’ verbal feedback through field notes. Stage 4: data analysis Data collected during the three tiers of report evaluations will be analysed to inform further revisions to the design of the reports and underlying theoretical frameworks. The 54 student responses to closed items on the questionnaire will be analysed as time series data, to identify trends via simple moving averages. Trend graphs will be plotted and analysed to identify changes, if any, in student responses to the reports. The 12

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Table 5 Data collection and data analysis for evaluating reports Activity

Participant action

Teacher evaluation or reports

Teacher views the reports, One questionnaire responds to questionnaire response individually

12 teacher questionnaire responses

Questionnaire responses analysed using summary statistics to inform semi-structured interview questions

Teacher engages in a one 1-hour semi-structured interview

12 h of teacher interview video recordings

Transcripts of teacher interviews created and analysed using FOCUS framework

54 student questionnaire responses

Questionnaire responses analysed using summary statistics to inform semi-structured interview questions

Student evaluation of reports

Data collected in To t a l d a t a Data analysis each test cycle collected

One 1-hour video and audio recording of teacher interview

Each group views report on Three individual questionnaire their group’s responses mathematical thinking, and students answer questionnaire individually Students engage in one 30-minute semi-structured group reflection

One 30-minute 30 student video and audio group recording of a interviews group interview

Transcripts of student interviews created and analysed using learner self-regulation framework

Outside Project researchers present rereports of student searcher thinking to outside evaluaresearchers, who give tion of verbal feedback in reports seminar setting

Four sets of Field notes of field notes discussion and questions raised by seminar participants

Summaries of outside researcher feedback created

teacher responses to the questionnaires will be used to inform questions in follow-up interviews, as will student responses on the open items. Transcripts of teacher and student interviews will be created from the video and audio recordings. These will incorporate important non-verbal data that are relevant to the interview, such as pointing and inscriptions, but at a less detailed level than the transcripts of student mathematical thinking. Transcripts of teacher interviews will be coded using Choy’s (2015) FOCUS framework, which distinguishes between three components of teacher noticing: attending to, making sense of and deciding to respond. Transcripts of student interview responses will be analysed using a self-regulation framework (Zimmerman and Schunk 2011) to identify students’ forethought, performance control and self-reflection. Field notes of outside researcher feedback will be summarised and used to create criteria for subsequent design cycles. Theoretical development Design-based research seeks to create design solutions that transcend the local context through theory building (Barab and Squire 2004). Accordingly, this project will develop a theoretical framework that guides the way we analyse and create reports on student mathematical thinking. This framework will be a bricolage (Lester 2005) of


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relevant theories about the nature of mathematics, the nature of mathematical thinking, what counts as evidence of such thinking and how such thinking can be analysed and reported. Far from being closely guarded theories, they are articulated and ‘put to work’ in the 46 design cycles, undergoing scrutiny through the practical tools whose design they inform. At the outset of our project, we have identified two main theoretical frameworks influencing the design of our reports. The first was Mason’s (2004) structures of attention framework, which comprises both what one attends to, as in the objects (e.g. functions, graphs, equations); and how these objects of one’s attention are attended to. Mason (2004) proposed that when something is learned, what changes cognitively is the structure of attention. Our reports focus on what students attended to and how they attended to these things, with the goal of raising students’, teachers’ and researchers’ awareness of the structure of students’ mathematical attention. We focus on what students do rather than what they did not do, what they are lacking or what they could have done. At the same time, we wish to include in our reports suggestions of how certain ways of attending both help and hinder students, in accordance with Brousseau’s (1997) theory of ‘obstacles’. Brousseau claimed that students experience difficulty when some aspect of their knowledge that previously gave them success now functions as an obstacle to success. Brousseau suggested that the process of learning mathematics can be construed as a process of overcoming obstacles, and that this can be achieved by drawing students’ attention to the obstacles they encounter during a task, highlighting both the advantages and the disadvantages of this particular piece of knowledge. We expect our set of theoretical influences to change and expand as we progress through the project and work to articulate and refine a theoretical framework for designing reports.

Anticipated outcomes We anticipate being able to share three types of outputs with the research and practitioner communities at the end of this project. First, we will produce over 50 reports of mathematical thinking generated by students working on contextual mathematising tasks. These can function as case studies that give students, teachers, decision-makers and researchers detailed information about the kinds of thinking that can accompany specific contextual mathematising tasks. Such case studies can help teachers anticipate the mathematical thinking that might occur when they implement contextual mathematising tasks, provide students with detailed formative feedback that can inform subsequent mathematical study and help decisionmakers justify the use of contextual mathematising tasks in classrooms to students, parents and administrators. Second, our project will generate several new contextual mathematising tasks on discrete mathematics topics that will be made available at the conclusion of the study. These may be useful as exemplars for internal assessments in upper secondary school in New Zealand or as collaborative-based tutorials in tertiary foundation study courses. At the undergraduate level, they may encourage more group-based teaching modes to complement lecturing practices.

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Finally, our theoretical framework will provide a generalizable and sharable conceptual tool that we will share with the research community as a way of creating further similar reports for further contextual mathematising tasks implemented in different settings. Ultimately, we hope that these three outputs will encourage students, teachers, decision-makers and researchers to share stories that raise awareness of and appreciation for the mathematical value of contextual mathematising tasks and modelling activities in learning, teaching and research. Acknowledgements This project was supported with funding from the Teaching and Learning Research Initiative, administered by the New Zealand Council for Educational Research.

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