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Examining educational staff’s expansive learning process, to
Proceedings of the 10th ERME Topic Conference MEDA 2020 - ISBN 978-3-9504630-5-7 54
access to materials and technology to use in their teaching. It is not just important that the teachers are offering professional development courses, but also, as reported by Black et al. (2013), there is a need to establish communities of practice, to provide ongoing support and sharing of resources. However, there is an increased need to focus on how to establish professional development in order to support the educational staff (the participating classes’ maths teachers and teacher assistants) to embed the students’
CT in mathematics. There is therefore a need to explore how the educational staff can be supported in using robots as a manipulative artefact in mathematics and how the built-in computational thinking in the robot can help to support the students' mathematical understanding.
THEORETICAL BACKGROUND
In adopting the term ‘learning environment’, I consider the teaching and learning situation as a whole (Bottino & Chiappini, 2002). This means that I am interested in analysing teaching and learning processes to use in the educational staff’s professional development. As a part of this, robots have an important role as a manipulative artefact in mediating teaching and learning processes.
The expansive learning process (Engeström, 2001) gives me a framework that is useful for describing and developing the educational staff’s collective professional development process. The expansive learning process is concerned with the historical and cultural development of activity, and I am specifically interested in the mediation role of the digital manipulative artefact, and how the educational staff are using it to develop the students’ mathematical understanding. For Engeström (2001), it is important to learn from new forms of activity which are not yet available. In this case, the knowledge and skills are learned as they are being created by the educational staff.
“An expansive learning activity will produce culturally new patterns of activity, and expansive learning at work will produce new forms of work activity at the workplace”
(Engeström, 2001 p. 139). The typical sequence of learning action in an expansive cycle is the following: Questioning is the first action, where a question will be asked, criticising or rejecting some of the already accepted practices and existing knowledge.
Analysing is the second action. This analysis involves a mental, discursive, or practical transformation of the situation to discover reasons or explanatory mechanisms.
Analysis evokes the “why” questions and exposes the principles. Modelling is the third action. Here, the participants construct an explicit and simplified model of the new idea that explains and offers a new resolution to the problematic situation. Examining the model is the fourth action. Here, the participants are running, operating, and experimenting with the new model in order to understand its dynamics, potentials, and limitations. Implementing the new model is the fifth action. Here, the participants are using and testing the new model. Reflecting is the sixth action. Here, the participants are evaluating the new model and perhaps adjusting it. Consolidating the model and its outcome into a new stable form of practice is the final action (Engeström, 2001). These seven steps for increased understanding should be seen as an outwardly expanding cycle but with many different kinds of action that can take place at any time. Mapping
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the educational staff’s learning action can provide a collective mirror for the educational staff and help them to identify problems in the activity. The seven steps also allow the researcher to identify and analyse which types of learning action is most dominant in a particular period of time.
In order to support the educational staff’s expansive learning processes, a teaching sequence, centred on the use of robots as a manipulative artefact in geometry, was analysed together with the educational staff. I then focused on the moment of the teaching sequence where the use of the robots was expected to unfold mathematical meanings. The teaching sequence was considered using three second-grade classes. I will present the analysis of selected episodes drawn from the teaching sequences that have been used in the educational staff’s professional development to support their expansive learning processes. The concept of teaching sequences emerged from the Theory of Semiotic Mediation (Bartolini Bussi & Mariotti, 2008), and offers a framework to design teaching sequences embedding digital manipulative artefacts, and to analyse the collected data in order to gain insight into the use of robots to support students’ CT and mathematical understanding. The structure of a teaching sequence can be defined as a didactical cycle that consists of an iteration where different typology of activities consist such as; activities with an artefact, individual/small group production of signs, collective production of signs, and mathematical discussion (Bartolini Bussi & Mariotti, 2008).
The teaching sequences have been developed by the educational staff, and the tasks and activities were the same for all three classes. This allowed me to look at the same teaching sequence concerning geometry in three different classes, and explore the potential for using robots to gain a deeper understanding of geometry for the students.
This leads to the following research question: How can the use of a teaching sequences support the educational staff’s expansive learning process in using robots as a digital manipulative artefact to aid the students’ computational thinking and mathematical understanding?
METHOD
The study has been conducted with the participation of three second-grade classes (A, B and C), and the classes’ educational staff. The teaching sequences were developed throughout two lessons, each of them lasting one and a half hours in each class.
Ethnographic data in the form of observations and video have been used to analyse the teaching sequences, and unfold the students’ use of the robots to gain a better understanding of how the educational staff can create tasks and activities that support the students’ CT and mathematical understanding. Through professional development sessions, the educational staff was presented with quotes from the teaching sequences, and were then guided by the researcher to analyse and gain a better understanding of the tasks and activities. This analysis was used to gain new insight, and to develop the teaching sequence further. There have been two professional development sessions with the educational staff, one after the second didactical cycle, and one after the last
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didactical cycle. These sessions were videotaped, which allowed for mapping and analysing the educational staff’s expansive learning process (Engeström, 2001).
Overview of the teaching sequence
The teaching sequences that the educational staff has formulated for the students were structured in order to support the students' geometric thinking. According to Goldenberg, Dougherty, Zbiek and Clements (2014), is it important to help students develop a precise language about geometric figures, to give them words and language as well as the ability to participate in discussions about the categorisation of the figures.
Attention will therefore be paid to tasks and activities that encourage the students with help from the manipulative artefact to categorise the geometric figures, and to use the appropriate words. As outlined, there was a need for the educational staff to develop and model new solutions such as tasks and materials that were to be used in connection with the robot and the geometric field of study.
As stated above, the teaching sequences are followed by an iterative process of didactic cycles.
The first didactic cycle involved the robots. The task was that the students should become familiar with the robot, and figure out how it works.
The second didactical cycle focused on how to get the robot to make a square, and then they had to investigate how small and how large a square the robot could make.
According to CT, the students were asked to create an algorithm for making the square.
The third didactical cycle; here the students were asked to work with the geometric properties of polygons. The task (Fig. 1) was to make the robot land on, for example, all squares, triangles, etc., and describe the characteristics of the individual polygon.
Figure 1. Task geometric properties
The fourth didactical cycle had a problem-solving approach, and the students had to figure out which type of polygon the robot could make. The students were required to investigate the type – from one-sided to ten-sided polygon – the robot could make, and see if they could make any generalisations from it. In the next section, I will focus on the first and the third didactical cycles to unfold how the teaching sequences were used as a part of the educational staff’s expansive learning process.
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Focus on the first didactical cycle
The students worked together in pairs, and each pair had their own robot. During the first task, the students were required to become familiar with the robot. The teacher provided a short overview of the different buttons on the robots, and after that the students was asked to figured out what the different buttons were for. Subsequently, the teacher followed-up on it in the class discussion. The following quote demonstrates how the students are trying to understand how the robots work. The quote is taken from class A, and the discussion section at the didactical cycle.
Student 1: We clicked a whole lot and but it didn’t do what we wanted it to.
Student 2: It was because we forgot to press delete. Then we clicked two forwards and two to the side, two backwards and two to the side.
Teacher: What did you think it was doing? A square?
Student 1: Yes, but it didn't. It just started driving around and going backwards (the student moves his body to show what the robot had done). Because we forgot to press delete.
Teacher: If you had clicked on delete, would it then have made a square? Did you try it?
Student: You must press two forward, one left, two forward, one left, two forward, one left, two forward, one left.
Teacher: This is something we should actually try out in a moment. I actually saw it when I was with you when you said that you were going to try to make it do a square and it just didn't.
Student 1: (The student turns around to demonstrate the ability to make a square.) This was a good example of what the student was struggling with in the first task. Many had trouble remembering to clear the robot after ending an activity, and thereby trouble with creating a new activity. Student 1 also used his body to demonstrate the movement of the robot, which helped him to make the movement more understandable. The fact that the students were initially given time to investigate the robot made them more focused on the mathematics in the later exercises.
Professional development
After the second didactical cycle, the educational staff were given a training session to help them develop the teaching sequences further. The quote above was shown to them to let them know how the students had worked with the robots. The quotes also showed that, even when the task did not include mathematics, the group attempted to get the robot to make a square. The quote showed the potential for using the robots as a digital manipulative in mathematics, but also the need to develop tasks and activities to support this. Here, the educational staff started to ask questions on the previous practices, as part of their expansive learning processes. They became aware of the fact that there was a need to design tasks and additional material that could be combined
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with the robot if it were to be used to support the students' mathematical concepts of understanding. It appeared that the robot could not be regarded as a manipulative artefact for mathematics itself, but that it was the didactical framework and the task which created the possibilities for supporting the mathematical concept of understanding. For this reason, the educational staff had to act to ensure that the developed materials could be used in connection with the robot if it were to support the geometric subject area.
Focus on the third didactical cycle
In the third didactical cycle, the students worked on two tasks. In the first, they were asked to work with the geometric properties of polygons, see Fig. 1. The students had to make the robot land on, for example, all squares, triangles, etc., and describe the characteristics of the individual polygons. When the students sorted the polygons, it helped to initiate a process in which they focused their attention on the characteristics of the polygon. This would allow them to increase their knowledge of the individual polygons. The students must both relate to the robot through CT, where they must first get an overview of, for example, the triangles on the worksheet, and then create an algorithm that moves the robot from one triangle to another. The students use the robots as a manipulative object by describing the characteristics that lie behind the polygon the robot lands on.
In the second task, the students had to categorise different polygons on the basis of different criteria such as a right angle, acute angle, etc. This helped to support the students' study of various properties such as a 'right angle' and 'equally long sides’.
Categorising the polygons from Fig. 1 on the basis of new criteria helps to support students' reasoning at a higher level of abstraction (Goldenberg et al., 2014). This also gave the students the opportunity to distinguish between polygons which resembled each other and to become aware of the common characteristics of polygons which did not appear to have the same characteristics. Through their work on categorising the polygons, the students developed an understanding of the fact that the different polygons could have the same characteristics. When the students worked with the robots as a manipulative artefact, they are working with CT when they are programming the robots. Through CT, the students work to get the robots to move in different sequences, for example when the robot has to move around all the triangles (Fig. 1). During the task, they were continually debugging and correcting their codes if the robot did not land on the desired polygon. In this way, the students were trained in their CT when they introduced what they had worked with in the classroom as well as the way in which they had solved the rewarding task.
Professional development
After the fourth didactical cycle, the educational staff were given another session. The session focused on the third and fourth didactical cycles, and the teaching sequences as a whole. The educational staff was asked to mention which part of CT the students had been working with in the two types of tasks in the third didactical cycle. However,
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this proved difficult for them: “I actually don’t know. I know that we were working computational with the robot, but which part of CT we used, is hard for me to say”
(Teacher, Class C). This showed that the educational staff were using the digital manipulative artefacts as a tool to let the students solve the task. It is thereby suggested that the educational staff gain a better understanding of CT, so they can support and distinguish signs of CT that can support the students’ mathematical understanding.
Together with the researcher the educational staff were questioning and analysing the teaching sequence to gain a better understanding of the possibility to work with the robots as a digital manipulative tool in mathematics. Through the teaching sequence the educational staff modelled the didactical cycles, and examined the teaching sequence by using the robots to develop the students’ CT and mathematical understanding. “This form of development has given me the courage to do more, and use the technology in small steps” (Teacher, class A). The educational staff stated that working with the digital manipulative artefacts through a collective professional development process had an impact on them, based on the fact that they had to develop new teaching sequences together. In other words, it has given the educational staff the courage to work with technologies in teaching. The educational staff felt that they were gradually being supported in their own expansive learning process and were getting help to change their activities through joint collective actions that helped to develop their practice regarding the use of digital manipulatives to support of the students’
mathematical and computational understanding.
CONCLUSION
Expansive learning should be seen as an iterative process, where the educational staff together with the researcher examine the current practices, along with the teaching sequences. Considering the educational staff’s expansive learning processes, they are still in the beginning of the processes. By analysing the didactical cycles, it helps the educational staff to gain a greater understanding of how the robots could be used as a manipulative artefact to support the students’ mathematical understanding.
From an educational perspective, working with a robot as a digitally manipulative artefact helps the students to reason, problem-solve, generalise, and predict, which may lead to a deeper mathematical understanding. The possibilities for supporting the students' mathematical learning are present with digital manipulative artefacts under the right pedagogical and didactical prerequisites. During the study, it was found that the robot itself could not be regarded as a mathematical manipulative artefact, but that the didactical cycles and the work on the tasks, which meant that the robot, and the built-in CT helped to support the students' development of their mathematical and computational understanding.
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REFERENCES
Bartolini Bussi, M. G. &Mariotti, M. A. (2008): Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective. In: Handbook of international research in mathematics education, New York. 746-783.
Back, J., Brodie, J., Curzon, P., Myketiak, C., McOwan, P.W., & Meagher, L.R.
(2013). Making computing interesting to school students: teachers' perspectives.
ITiCSE '13.
Bottino & Chiappini (2002): Advanced Technology and Learning Environments: Their Relationships Within the Arithmetic Problem-Solving Domain, in Handbook of International Research in Mathematics Education, red. English, L.D. chapter 29.
Engeström, Y. (2001): Expansive Learning at Work: Toward an activity theoretical reconceptualization, Journal of Education and Work, 14:1, 133-156
Goldenberg, P. Dougherty, B., Zbiek, R.M. & Clements, D. (2014): Developing Essential Understanding of Geometry and Measurement for Teaching Mathematics in Pre-K-Grade 2, National Council of Teachers
Grover, S. & Pea, R. (2018). Computational Thinking: A competency whose time has come. In Computer Science Education: Perspectives on teaching and learning, Sentance, S., Carsten, S., & Barendsen, E. (Eds). Bloomsbury.
Lee, I., Martin, F., Denner, J., Coulter, B., Allan, W., Erickson, J., Malyn-Smith, J., &
Werner, L.L. (2011). Computational thinking for youth in practice. Inroads, 2, 32-37.
Nugent, G., Barker, B.T., Grandgenett, N., & Adamchuk, V. (2009). The use of digital manipulatives in k-12: robotics, GPS/GIS and programming. 2009 39th IEEE Frontiers in Education Conference, 1-6.
Wing, J. M. (2006): Computational thinking, in Communications of the ACM, 49(3), p. 33–35.
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