Students’ design of inquiry with didactic tools: A Japanese case Tatsuya Mizoguchi, Berta Barquero, Marianna Bosch To cite this version: Tatsuya Mizoguchi, Berta Barquero, Marianna Bosch. Students’ design of inquiry with didactic tools: A Japanese case. Thirteenth Congress of the European Society for Research in Mathematics Education (CERME13), Alfréd Rényi Institute of Mathematics; Eötvös Loránd University of Budapest, Jul 2023, Budapest, Hungary. ฀hal-04421401฀ HAL Id: hal-04421401 https://hal.science/hal-04421401 Submitted on 27 Jan 2024 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Students’ design of inquiry with didactic tools: A Japanese case Tatsuya Mizoguchi1, Berta Barquero2 and Marianna Bosch2 1 Tottori University, Japan; mizoguci@tottori-u.ac.jp 2 Universitat de Barcelona, Spain This paper presents activities of Japanese pre-service teachers designing an inquiry learning activity in the context of study and research paths for teacher education, an instructional format proposed by the Anthropological Theory of the Didactic. It discusses the conditions and constraints of these activities from the perspective of the didactic tools proposed to teacher-students for the design of teaching and learning sequences. Japanese pre-service teachers are instructed on how to design a structured problem-solving (SPS) lesson in previous courses and, in many cases, lessons are designed and implemented in this format during teaching practice in schools. In the teacher education activity considered in this paper, some inquiry tools designed in didactics research appear useful for the students’ analysis of mathematical activities. The activity also indicates that specific design tools for SPS lessons do not always find their correspondence when designing inquiry learning. Keywords: Teacher education, task design, study and research path, structured problem-solving, anthropological theory of the didactic. Introduction Inquiry has been recently introduced in the Course of Study in Japan as a complement to the dominant structured problem-solving (SPS) approach (Stigler & Hiebert, 1999). Teacher education programmes and teaching practices at many universities cover SPS knowledge and skills with a variety of tools for the design and analysis of mathematical activities. What they have in common is a practical approach based on the question of how to better teach the content defined by the national curriculum. To approach inquiry-based activities in teacher education, the first author of this contribution has implemented a sequence of courses that combine SPS with teacher educational proposals emerging from the Anthropological Theory of the Didactic (ATD) that fall in the so-called “paradigm of questioning the world” (Chevallard, 2019). However, whereas the notion of inquiry used in the ATD is closer to “a way of teaching in which students are invited to work in ways similar to how mathematicians and scientists work” (Artigue & Blomhøj, 2013, p.797), this is not the case of the “inquiry” that appears in the Japanese Course of Study. There, inquiry is closer to “appropriation” or “application”, that is, an activity carried out after the main knowledge tools are acquired – through SPS. Our research interest here is not in the possibility of an immediate shift of all educational activities into inquiry-based ones but in the development of teacher education programmes that enable the teaching-learning process of such inquiry to be put into practice. Given the teachers’ limited experience with inquiry in Japan, we consider important identifying how their own practice of an in-depth inquiry can be effective in enhancing their mathematical analysis attitudes and task-design strategies. The research question we address in this contribution is how to combine the teacher education tools provided by the SPS approach with those proposed by the ATD to help pre-service teachers design inquiry-oriented teaching-learning processes and what are the specificities of each approach in this combination. Theoretical framework, context, and study design This paper analyses the activities of pre-service teachers through the proposal of a study and research path for teacher education (SRP-TE) process (Barquero et al., 2018). Study and research paths (SRPs) are proposed in the ATD as a model of inquiry towards a shift from the traditional paradigm of “visiting works” where curricula are proposed as a set of knowledge or tools to be learnt (or “to be visited”), to the new one of “questioning the world” where learning processes start from the consideration of open socially relevant questions to address. SRPs are represented through the following schema, called the Herbartian schema: [𝑆(𝑋; 𝑌; 𝑄0 )➦{𝐴♢𝑚 , 𝑊𝑛 , 𝑄𝑝 , 𝐷𝑞 }]➥ 𝐴♡ (Chevallard, 2019). In this schema, 𝑆(𝑋; 𝑌; 𝑄0 ) is the didactic system formed by a group of students 𝑋 and a group of teachers 𝑌 addressing an open question 𝑄0 . 𝑄0 is at the origin of the inquiry process, and the main aim is to collectively elaborate a final answer 𝐴♡ . During the inquiry process, 𝑋 and 𝑌 raise derived questions 𝑄𝑝 , search and use bodies of knowledge or “labelled answers” 𝐴♢𝑚 they make available, together with empirical data 𝐷𝑞 and other works 𝑊𝑛 . A summary of the inquiry process followed in an SRP can be described only focusing on the questions and answers that appear, thus generating a questions-answers map (QA map) of the inquiry process (Winsløw et al., 2013). SRP-TE applies SRP to the context of teacher education. It starts with an initial question 𝑄0𝑇 related to the teaching profession, for example: “How to teach inquiry?” (Module 0). In the teacher education process to address 𝑄0𝑇 , teachers experience an SRP (Module 1), then analyse it with different didactic and mathematical tools (Module 2), before designing a possible implementation and analysing it (Modules 3 and 4) as shown in Figure 1. Module 1. Let teachers experience a teaching project –based on an SRP – designed by the educators. Role of student Module 0. Start with a professional questions as starting point of the SRP-TE and first look for existing/available answers Module 4. Collective a posteriori analysis of the implementation. Role of teacher, analysts and designer Module 2. Collective analysis of the SRP that comes to be experienced. Role of mathematical and didactic analysts Module 3. Design and implementation of an adapted version of the SRP for a specific group of students Role of designers Figure 1: Module’s structure of an SRP-TE The teacher education programmes at Tottori University (TU) are organised as shown in Figure 2 for primary and secondary school teachers. The classes under consideration in this paper are ‘Lesson Design in Mathematics II’ (for secondary school teacher) and ‘Psychology and Epistemology of Mathematics Education’ (for primary school teacher). They are placed as final classes in the curriculum (‘Teacher training courses’) for both teacher education programmes. Therefore, the didactic tools used by the students in these classes have been learnt in previous courses. Figure 2: Structure of mathematics teacher education programme at TU The didactic tools learnt and available in each class are shown in Table 1. Table 1: Didactic tools learned in the classes Class Didactic tool Introduction to Didactic Studies QA map to describe problem solving processes Mathematics Kyozai-kenkyu: Epistemological and didactic analysis of for Primary school Teachers mathematical knowledge at stake Didactic Theories in Mathematics Lesson plan according to the traditional SPS format Activity-assistance chart Psychology and epistemology of mathematics education All above didactic tools Design of mathematics lessons II Table 1 contains a number of Japan-specific didactic tools. In particular, kyozai-kenkyu strategy (cf. Shinno & Mizoguchi, 2021) is considered the most important teacher activity in traditional Japanese lesson study (Watanabe et al., 2008). It corresponds to the teacher’s preparatory work on all aspects of teaching, including the analysis of mathematical knowledge, task design, conceptualisation of student activities, etc. The SPS format is currently known internationally: Japanese mathematics lessons are basically implemented in this format (Figure 3, left side), so students are demanded to learn this style of lesson. Although details are omitted here, it has been practically reported in Japan that many problem-solving lessons are not successfully implemented due to insufficient design of jiriki-kaiketsu (individual problem solving) and neriage (discussion in a whole class) parts. Therefore, in the class of TU, the structure of these parts is characterised by mathematical activities to help students better understand and design this style of lesson, as shown in Figure 3 (right side). The activity-assistance chart is a template for the design of jiriki-kaiketsu and neriage parts. Before making the lesson plan, students use it to organise the proposed mathematical activities and foresee the kind of assistances teacher might provide to help students progress (Figure 4). Our research focuses on how the above-mentioned didactic tools related to SPS can or cannot work successfully to design inquiry-based learning activities, using the implementation of the SRP-TE as a “case study”. It also aims to analyse it qualitatively to understand the functioning of the SRP-TE modules in this institutional setting. This is done by analysing pre-service teachers’ activities in the above two courses that were conducted jointly in the 2022-23 academic year during 13 sessions of 90 minutes. The participants were one student for primary school teaching and four students for secondary school teaching. Figure 3: Standard SPS model (left) with embedded mathematical activity (MA) (right) Figure 4: An example of the activity-assistance chart (from Mizoguchi, 2013) The object of analysis includes the SRP carried out by the students and the SRP lesson(s) designed by them. The data consisted of in-class discussions taken as field notes and students’ reports submitted at the end of the course. We conducted an exploratory study based on the identification of the elements of the Herbartian schema in the inquiry and design processes of pre-service teachers during the SRP-TE, to analyse their role first as tools for the progress of the inquiry and, afterwards, as tools for the design, paying special attention to the relationships between both. SRP-TE in practice at TU The course was implemented following the modules’ structure of an SRP-TE (Figure 1), but it only included from Module 0 to halfway through Module 3, with only the design phase, since no practice was carried out. Table 2 shows a brief description (excerpts) of each session. Module 0: Introducing the professional question to address. The purpose of implementing an SRP-TE in this course is to enhance the pre-service teachers’ mathematical analysis techniques. In previous courses, there were around five groups of three to five students per group each year. The main objective was to design a teaching sequence about the relevant knowledge of a specific mathematical topic. Students were proposed concrete questions to address related to the topic and carry out the inquiry in teams. In the 2022-23 class, as the number of students was lower, it was decided to let the pre-service teachers pose the questions for inquiry in an open way and start from the teaching question 𝑄0𝑇 about how to teach or design an inquiry activity. Table 2: Description of each session Phase 1: Task design and implementing the SRP [Module 0-1] Designing a task involving socially valuable mathematical knowledge and the potential to expand the questions. Trying to solve this task as a group of students. Phase 2: Analysis of the implementation of the SRP [Module 2] What are the different phases or steps structuring the activity? What have been the questions you faced during the different phases of the activity? What new questions can you raise now when considering the work done and thinking about other possible paths or strategies? What mathematical knowledge and tools have you used? What other mathematical tools can be used as an alternative or as a new development? What questions or doubts related to mathematics can you raise after the analysis? Phase 3: Design the lesson plan(s) [Module 3a] Represent the questions listed in Situation 2 on a QA map for the lesson(s). Based on the QA map produced, compile an activity-assistance chart for the teaching-learning process. Prepare a lesson(s) plan. There were two potential questions raised by pre-service teachers in session 3 [25 0ct 2022]. One was “How much more forest area does the world need to increase to achieve carbon neutrality?”, the other was “What equipment would be needed to replace all the energy currently obtained from Japanese fossil fuels with renewable energy?”. The questions were examined from the perspective of what mathematics is expected to be used in the process of inquiry. The former question was finally adopted on grounds such as the possibility to find data on population projections and the integrals expected in relation to the CO2 absorption of forests per unit area. Module 1 and 2: Experiencing and analysing an SRP. Students have already experienced SRPs in previous courses, as well as used QA maps for their analysis. Therefore, they proceeded from the beginning by representing their inquiry process on a QA map. From Session 4 [25 Nov 2022] to Session 9 [13 Dec 2022], the discussions followed the progress of the students’ inquiry by way of seminars in each session. Students developed their work outside of class time in groups and presented and discussed it in class. The first problem raised was the use of the QA map as main resource. Obviously, students raised derived questions, but no answers were provided for each of them, so the path leading to the final answer (𝐴♡ ) was not available at this point (Sessions 5 [15 Nov 2022] and 6 [22 Nov 2022]). After obtaining an approximate overview, the class moved on to an activity to generate an answer to each question. During the sessions, the class discussed the addition of subquestions that seemed to be missing and the way the questions were organised. Finally, students modelled the situation with a linear function 𝑦 = 𝑎𝑥 + 𝑏, where 𝑦: CO2 emissions in 2050 (in tonnes 𝑡); 𝑎: CO2 absorption per unit area of forest (𝑡/𝑘𝑚2 ); 𝑥: increased forest area (𝑘𝑚2 ); 𝑏: current CO2 absorption (𝑡). When applying this mathematical model, students noted several points: differences in CO2 absorption by tree species are ignored; it was assumed that the current vegetation and forest proportion remains the same; differences in CO2 absorption by age of trees are not considered. The values obtained were the following. For 𝑦: 𝑦 = 44836569834 ≒ 44.8 × 109 ( 𝑡 ). For 𝑎: 𝑎 = (current global CO2 absorption per year ( 𝑡 )) / (current forest area ( 𝑘𝑚2 )) = (15.6 × 109 ) / (40.6 × 106 ) = 384 (𝑡/𝑘𝑚2 ). For 𝑏: Assuming that the current CO2 absorption consists of two types of CO2 absorption, one by forests for one year and another by the sea surface for one year, 𝑏 = 15.6 billion (forest, 𝑡) +9.53 billion (sea, 𝑡) = 25.13 × 109 (𝑡). The result calculated under these conditions was 𝑥 = 0.051 × 109 , i.e. approximately 51 million (𝑘𝑚2 ). To elaborate the final answer 𝐴♡ , new data was necessary. The current world land area is 148.94 million (𝑘𝑚2 ). As 40.6 million (𝑘𝑚2 ) are forest area, the remaining area is 108.34 million (𝑘𝑚2 ). 63.4 million (𝑘𝑚2 ) of this area is classified as desert climate, and the remainder is 44.9 million (𝑘𝑚2 ). Assuming that forestation is not possible in desert climates and defining the ‘forestable area’ as the land that is not currently forested and not classified as desert climate: forestable area = 44.9 million (𝑘𝑚2 ) < 51 million (𝑘𝑚2 ) = forest area that should be increased to achieve carbon neutrality in 2050 which was found in another QA path. Therefore, based on the assumptions made here, the students concluded that “it is not possible to increase CO2 absorption and achieve carbon neutrality in 2050 solely by increasing forest area” (𝐴♡ ). Figure 5: Students’ completed QA map Module 3: Design stages (only). Discussions to prepare the design began in session 10 [10 Jan 2023]. First, each teaching hour was organised based on the completed QA map (Figure 5). The group members were to allocate lessons to their respective responsibilities and design their own one. Students already had experience in designing SPS lessons in other classes and used the activityassistance chart as a didactic tool. Therefore, each lesson was provisionally designed using it. The activity-assistance chart succinctly summarised the items needed to design an SPS lesson, and students have experienced the chart’s effectiveness in practice. However, in translating their SRP into a learning process, they became aware that this chart was difficult to use. What they had written so far in the ‘activity’ column of the chart was the final state of the (expected) activity. In contrast, in their actual SRP, questions and answers were always expressed as pairs, as represented in the QA map. This means that the way students generate such questions is critical in the teaching and learning process, something that is not always consistent with their previous experience. However, it may actually be largely due to how they have used the chart, rather than to the functionality of the chart itself. In addition, they encountered aspects of the ‘assistance’ column of the chart that did not necessarily correspond to their previous experience of what to assist with. All these aspects were discussed in class with the educator, particularly regarding the potential assistance to provide, the (lesson) teacher’s questions that could encourage the generation of student questions, and the suggestion of mathematical tools and resources to address these questions. Discussion and conclusion Here we consider two didactic tools in particular: the QA map and the activity-assistance chart. The students initially envisaged teaching under a traditional Japanese teacher education culture, which is inserted in the paradigm of visiting works. They asked themselves: what mathematics are to be taught, and how such mathematics appear in the process of inquiry? However, the progression of the inquiry while making the QA map naturally led them to think differently, more in line with the paradigm of questioning the world. QA maps are useful for tracing the process of inquiry (Florensa, et al., 2021). They are usually generated in a way that looks back over the whole activities (Mizoguchi et al., 2022). Therefore, regarding the transition of the modules in Figure 1, the SRP performed in Module 1 is often analysed in Module 2 mainly using QA maps. In the SRP-TE described in this paper, instead of a linear progression from Module 1 to Module 2, pre-service teachers’ activities moved back and forth between both. The QA map was developed in response to the progression of the inquiry at each moment. This enabled students to reflect on each stage of the inquiry activity and anticipate the postinquiry design. This suggests that the QA map works as another tool for kyozai-kenkyu in helping students design an inquiry-based learning activity. For its part, the activity-assistance chart is a tool involved in the design of an SPS lesson, and its suitability for inquiry-based learning needs to be considered. The existent didactic contract (Brousseau, 1997) on the use of this chart by pre-service teachers in designing SPS lessons may not be totally effective and it could act as an obstacle when the purpose is the design of an inquiry-based activity (e.g., through the design of an SRP). It could be suggested that this is more than just a difference between SPS and SRP. Rather, it points at a difference in the didactic paradigm underlying these tools. In SPS lesson study, the assumed paradigm is the one of visiting works, in which each lesson, even if based on the approach of an open question or problem, is associated to a previously determined piece of knowledge. In SPS, answering the initial question is a means for students to (meaningfully) construct, elaborate or retake the mathematical knowledge at stake. For its part, an SRP based on the paradigm of questioning the world gives total priority to the generating question and leaves the inquiry process open towards the elaboration of a final answer 𝐴♡ . Nor the answer nor the tools required for its elaboration are preconceived by the teacher. 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