To identify which design elements could be relevant to students when learning basic logical competencies in a technology-enhanced and real-world object based learning environment, we conducted an educational case study.
Using case study principles to reach our research goal
Since using case studies has a long tradition in mathematics education research in examining students’ solution processes and methods (e.g. Cobb 1986), this research method should also be appropriate for our study. Furthermore, as case studies can be used not only to investigate solution processes and methods in problem-solving but also to explore students’ emotions when solving problems in mathematics classrooms (Eynde and Hannula 2006), using case study principles should provide valuable results for our research aims. Our study focused on students solving a particular problem (developing an MS Excel program to solve the Logifaces problem) and our research goal was to explore which design elements could be relevant for students in technology-enhanced and real-world object based learning environments.
According to Cohen et al. (2007), case studies require a clearly defined limited system of real people in real situations experiencing a specified intervention. This limited system of real people in real situations should extend the understanding of concrete ideas and interventions beyond abstract theories. In this study, the limited system was defined as three groups of students throughout four teaching units. The situation to be investigated was defined as the students of these three groups, or more precisely students’ needs and requirements concerning a technology-enhanced learning environment based on real-world objects when learning basic logical competencies. The intervention included basic logical competencies were learned by students using Logifaces-stones and MS Excel.
According to the work of Yin (1984), our educational case study can be characterised as an explorative case study. The explorative character of our educational case study is because our study aims to develop hypotheses regarding design elements of learning environments in which real-world objects and technologies are linked. According to Cohen et al. (2007), among others, participatory observations or post-observation recordings are data collection methods that generally apply to case studies and are specifically appropriate for our case. Participatory observations were selected as the data collection instrument because a researcher was present in all teaching units and also interacted with the students when needed. The interactions of the researcher with the students also resulted in ongoing mini-interviews (Bakker and van Eerde 2015). The mini-interviews always lasted less than 3 min and were intended to help clarify why students encountered difficulties. The researcher made observational recordings immediately after the occurrence of any phenomena or after conducting mini-interviews. The data collected during the lessons were supplemented with final written feedback from students after each double lesson. According to Kane and Staiger (2012), collecting supplementary observation data through student feedback should lead to an increase in educational quality. Written feedback was chosen as a data collection tool to gather feedback from all students and to make it clear that their feedback could not be traced back. By making written feedback untraceable, it could be expected that the honesty of student’s feedback was possibly increased.
Using grounded theory approaches when collecting and evaluating research data
When collecting and evaluating research data, we applied techniques and principles of grounded theory approaches (GTA). In our study, we followed the constructivist interpretation of GTA (Charmaz 2006) and a GTA interpretation according to Strauss and Corbin (Khan 2014). A constructivist interpretation of GTA and a GTA interpretation according to Strauss and Corbin means, on the one hand, that the previous knowledge of researchers and the current scientific body of knowledge should be included in the development of theories and hypotheses. On the other hand, this interpretation of a GTA follows that any hypothesis or theory developed in the course of research depends on the perspectives of researchers and cases under investigation.
This constructivist interpretation of GTA was particularly relevant to our exploratory educational case study, as, on the one hand, the researchers could not be described as neutral, as they sometimes took on participating and supporting roles. On the other hand, it must be assumed that theories and hypotheses on design elements of technology-enhanced and real-world object based learning environments developed in our exploratory education case study would have been different if our study had been conducted with other classes, at a different time, or at other schools. According to Cohen et al. (2007), results or hypotheses that depend on the framework conditions of a study are a specific feature of case studies. However, if the conditions and frameworks of the study or case are described in detail, theories or hypotheses developed in a case study can be applied to similar cases, phenomena or situations.
Coding techniques of grounded theory approaches
In analysing the research data and developing theories and hypotheses, we have followed a four-part approach, namely: 1) screening of new data, 2) open coding, 3) axial coding, and 4) selective coding.
We followed Ritchie’s (2012) approach to initially view the new data. Initially viewing new data means that, in a first step, all researchers read the newly collected raw data. This repeated reading of the raw data was intended to give all researchers an overview of the current status of our educational case study and to be able to derive initial topics from the raw data. In the next step, the newly collected raw data were transcribed and then coded using a QDA software. Our approach to coding is based on the theoretical guidelines and practical applications of Breuer et al. (2009), Charmaz (2006) and Mey and Mruck (2011).
In the first phase of coding, we applied the techniques of open coding. The goal of open coding is to break up the collected data. To break up the collected data, we asked the questions what, how and why. The resulting first open codes were then grouped according to similar characteristics and definitions and provided with new keywords. This grouping of first open codes resulted in open codes of a higher degree of abstraction (see Table 1, columns 1 and 2).
The open codes of a higher degree of abstraction were then used for axial coding. By coding open codes axially, a synthesis of the research data should be achieved again. In axial coding, open codes were grouped around a central open code (phenomenon) according to causes, activities and consequences (see Fig. 2). The open codes of the area of activities were then grouped and used as categories for selective coding. For selective coding, these categories were linked, and dependencies were identified. Identifying dependencies allowed us to develop the design elements relevant for students in terms of technology-enhanced and real-world object based learning environments for learning basic logical competencies core categories: (A) Using open tasks with multiple solutions, (B) Just-in-time feedback and (C) Novelty effects in the learning process.
Student quotes given in the section Results have been translated from German to English by us. Individual student quotes were accompanied by information whether this feedback was collected via written feedback [F] or mini-interview [I]. If feedback was collected via mini-interviews, the composition of the student group in terms of gender is also given. Here, g stands for girl and b for boy.