Instrumental orchestration in Chinese mathematics lessons with dynamic geometry Fangchun Zhu fangchun.zhu@ens-lyon.fr Supervisor: Sophie Soury-Lavergne Binyan Xu

Outlines Context Research questions Instrumental orchestration 30/10/13 Outlines Context Research questions Instrumental orchestration Types of instrumental orchestration Methods Analysis based on one Chinese mathematics lesson Conclusion Examples of students’ operating dynamic geometry Workshop

Context There are three important areas in mathematics educational research on technology (Sinclair et al., 2016): (1) the introduction and design of new technology, both hardware and software, (2) theory and methodology for a better understanding of the role of existing and emerging technology, and (3) empirical studies on the use of technology in teaching and learning . But how technology affects mathematics teaching and learning, is far from being explained clearly (Bretscher, 2014). What is difficult for teacher to integrate technology into his or her mathematics teaching? Lagrange et al (2003) has shown that there are too many variables to manage in the classroom: one of these variables is that teachers need to make transformation between old and new didactical practices (Assude & Gelis, 2002). In order to help teachers integrate technology into mathematics didactical process, we need to make deep understanding of teachers’ managing mathematics lessons with technological tools. To account for this topic, Trouche (Trouche, 2004) introduces instrumental orchestration to describe this process. In spite of technology’s recognized potential for teaching and learning, its integration into mathematics education lags behind the high expectations that many researchers and educators had some decades ago (Drijvers, Doorman, Boon, Reed, & Gravemeijer, 2010) Cayton, Hollebrands, Okumuş, & Boehm, 2017) (Assude, 2007). It is critical that if one teacher does not perceive that integrating technology especially dynamic geometry software into his class is helpful to his didactic process, he will not use these technologies any more unless he faces some additional requirements such as curriculum. It means that the use of technologies in teaching often involves changes in the working environment of lessons: change of room location and physical layout, change of class organization and classroom procedures (Jenson & Rose, 2006) and the resources they need to use in the class such as mathematics tasks (Hegedus et al., 2017). Specially in the case of dynamic geometry systems, we can see the shift in considering mathematical activity and teacher profession caused by the introduction of dynamic geometry into mathematics classroom (Trgalova, Jahn, & Soury-Lavergne, 2009).

Research question This study wants to describe teachers’ real practices with dynamic geometry software and give a deep understanding of how mathematics teachers integrate dynamic geometry software in their lessons.

Instrumental orchestration An instrumental orchestration is defined as how teachers want to organize the class teaching with different kinds of artefacts available in learning environment in order to help students learn mathematics (Trouche, 2004). We can distinguish instrumental orchestration into three different elements: a didactic configuration, an exploitation mode and a didactical performance (Drijvers, Doorman, Boon, Reed, & Gravemeijer, 2010). Instrumental orchestration tries to answer questions about what technological artifacts mathematics educators should introduce to learners and what guidance they should provide so learners can appropriate and use artifacts as instruments to mediate their activity with various artifacts (Trouche, 2004; Drijvers, Doorman, Boon, Reed, & Gravemeijer, 2010).

Instrumental orchestration Focus in this research Didactic configuration An arrangement of artefacts in the environment, or in other words, a configuration of the teaching setting and the artefacts involved in it. The process of teacher’s choice and use of the mathematical tasks or the geometry software in the classroom and how to configure them in the classroom. Exploitation mode The way the teacher decides to exploit a didactical configuration for the benefit of his or her didactical intentions. The sequence of the tasks teachers showed to the students and the role mathematical software played in the tasks. It is one way of decision teacher takes. Didactical performance The ad hoc decisions taken while teaching on how to actually perform in the chosen didactic configuration and exploitation mode. The interaction between teachers and students: the questions teachers posed, the feedback teachers gave and the purpose of each questions.

Types of instrumental orchestration Type of instrumental orchestration Description Didactical configuration Exploitation modes Character Technical-demo It concerns the demonstration of tool. Digital mathematics environment, including computer, internet, screen, software is necessary and also one projector to show tasks on the screen is also critical. The classroom arrangement needs allow all the students can follow the teachers’ demonstration One of the possible way to exploit is that teachers can introduce one technology in a situation or with a task, also students’ work can also be the departure of the teaching process The teaching process controlled by teacher Students watch and apply what teacher said The main issue is about the problems of technology Topic of this process comes from students or difficulties identified by teacher Explain-the-screen Teachers explain what happened on the screen to the whole class by operating the software. It is similar to the Technical- demo ones Teachers can take students’ work or their own solution of the task as the topic of the teaching process The teaching process controlled by teacher or student Other students just listen to the teacher The issue is not just about technology in relation with the problems of mathematics The topics come from teacher or students’ work

Types of instrumental orchestration Type of instrumental orchestration Description Didactical configuration Exploitation modes Character Guide-and-explain Students begin to explain the mathematics contents shown on the screen guided by the questions posed by teachers Similar to explain-the-screen Teachers can take students’ work or their own solution of the task as the topic of the teaching process and pose questions while teaching The teaching process controlled by teacher Students answer questions The issue is not just about technology in relation with the problems of mathematics The topics come from teacher or students’ work Link-screen-board This type begins to concern the relationship between two main resources in classroom: screen and board. Teachers begin to think how to represent what happens in the technological environment only with paper, book and blackboard The configuration is also similar to the above two types, digital mathematics environment with a projector is critical. In this type, some traditional resource like blackboard is needed also. And we need to let both blackboard and screen are visible to the students. Similar to “explain-the-screen”, students’ work or teachers’ own solution to a task can be the departure of the teaching process The teaching process controlled by teachers Students just listen to the teacher use more than one resources in class such as technology, board, chalk and so on, issues related to the two types of representations The topic comes from students’ work or a task

Types of instrumental orchestration Type of instrumental orchestration Description Didactical configuration Exploitation modes Character Discuss-the-screen Students participate into the discussion. Teachers may just help or talk with the students if they need help. Digital mathematics environment and projector which can show tasks or students’ work are necessary. And the classroom arrangement need to make students discuss easily Students’ work, a task, or a problem or approach set by the teacher can be the topic of discussion Students began to participate in the process; teachers are not the center of the class Teacher or other students give feedbacks The topic is about mathematics The issue comes from students’ work or a task Spot-and-show one or more students explain their work to the whole class and others or teacher give some feedback or ask some questions. Including digital mathematics environment Teachers choose some students whose work are shown on the screen to explain their solutions, other students or themselves give some feedbacks Students themselves make the explanations not the teacher Teacher and other students give feedback Students’ work was shown on the screen as a topic of the following class

Types of instrumental orchestration Type of instrumental orchestration Description Didactical configuration Exploitation modes Character Sherpa-at-work Teachers can allow students to share his/her construction steps to the whole class. We can describe a student in this role as a Sherpa-student (Trouche, 2004). This so called Sherpa-student uses the technology himself to present his or her work, or do what teachers tell him or her to do with software. The facilities are similar to “Discuss-the-screen”. The classroom need to allow the so-called Sherpa-student control the software and other students can follow his operation. Teachers can let Sherpa-student explain his own work or ask him to carry out the activities of other students. One student (Sherpa student) control the software Teacher or other students give instructions Topic is about mathematics or technology Present the work or do what teacher asked to do Board-instruction It is a traditional teaching orchestration in classroom. In this type, technology is not critical. Teachers use traditional resources like board to explain what they need their students learn during this lesson. Didactical configuration is more similar to the types above, but technology is not essential in this type Teachers may stand in front of the board and teach Teacher controls the teaching process Students listen to the teacher and answer questions Teaching content is about mathematics The content may come from students’ work or teacher’s preparation

Method Lessons observed in this research Two lessons were analyzed based on the framework of instrumental orchestration. Teacher included in this research Teacher: ZH (pseudonym) who have about 20 years of teaching experiences and is very familiar with using dynamic geometry software in mathematics teaching. The first lesson we call it “open lesson” in China, which means other teachers or experts could attend and listen. After the lessons, the experts would make some comments or suggestions to this teacher. We choose this kind of “open lesson” because teachers who make these kinds of lessons always prepared adequately and try to show their best teaching process to others. So teachers always use as many resources as they can even these resources may not be often used in their regular lessons. These lessons can help us to see more properties of Chinese mathematics lessons and they can also reflect Chinese mathematics education tendency in these years. The second lesson was held in regular mathematics classroom in Grade 8 (one year after). By contrasting these two lessons we can see the transformation of the teacher’s integration of dynamic geometry software in mathematics teaching and find if the practice in open lesson can affect teacher’s regular didactical process. In his school, one private junior high school in Shanghai, teachers are encouraged to use new technologies in their class such as iPad or smart phone. The school committee bought many software to support teachers develop their didactical methods such as the Geometry Sketchpad, which is used almost in every school in Shanghai China. Before the lessons, ZH often let students learn the new contents by themselves and finish some exercises with the help of some videos prepared by ZH. Then during the lessons, he can begin with the difficulties students faced when finishing the exercises. From ZH’s practices, we can find out the possible integration of dynamic geometry into mathematics lessons.

Tasks in China Task Description ZH designed one diagram of triangle whose length of side AC and BC is determined (AC = 4, BC = 6). Point A could be moved while points B and C were stable. From the diagram, the trace of point A is like a circle. The first task contained one moving point P. At first, point P is coincide with point A then it moves from A to B to C to D. Students need to find out the mathematical relation between the area of triangle APD and the length of moving trace. The second task contained two moving points, point A and C, but these points do not move according to line but like a circle. There is a right triangle ABC, ∠C=90°, BC=4, AC=3. ZH makes triangle ABC turns around point B to make point A on the line BC (point A’), students need to find the length of segment AA’ The third task, ZH designed one rectangle ABCD. There is a point E which moves along segment BC, from point B to C. Point B’ is a symmetry point of point B according to segment AE like the figure shown. Students need to find what kind of trace point B’ moves when point E is moving along BC

Results: type of instrumental orchestration in China First lesson Duration Second lesson Technical-demo   Explain-the-screen 5 7 minutes 7 8 minutes Guide-and-explain 3 11 minutes 5 minutes Link-screen-board 1 30 seconds Discuss-the-screen 2 2 minutes Spot-and-show Sherpa-at-work Board-instruction

Example episodes Discuss-the-screen All the students in this lesson were divided into small groups (two or three students) ZH shown the task and diagram students need to discuss and let the diagram move automatically. The discussion process was not controlled by ZH, he just decided the questions to talk about and the total time of discussion. The topic of this discussion is a mathematics task prepared by ZH before the lesson. ZH walked around and listened the students’ talking. There is no interaction between ZH and his students. Dynamic geometry software played as a projector to show the task, Students did not operate software any more.

Example episodes Explain-the-screen and guide-and-explain Before the discussion episode in the first lesson, ZH presented the mathematics task with the help of dynamic geometry software. There is no interaction between ZH and his students Only when ZH needed to present or explain the requirements of the tasks, he chose to control the explaining process. After that, ZH changing his organization into letting students make explanation by posing several questions. He chose some of the students to answer these questions in order to let others know what happened on the screen and the geometry features. ZH put the diagram designed based on the task situation on the screen with the help of dynamic geometry. One student was asked to explain what happened on the screen, during that time, we could find point P began to move from point A to B then, from B to C. In the first lesson, ZH put the task on the screen but now the diagram did not move at beginning. Only when he noticed the students made some misunderstanding, ZH began to operate the software and drag the point in order to show why they were wrong. For example, one student tried to answer the second sub-questions of this task (the minimum of side AB). He first gave a wrong answer: 2.2. ZH dragged point A to let the student notice the answer is not correct. During this episode, the interaction happened between two persons: the student chosen to answer the questions and ZH. Other students just listen their talking and did not give any feedbacks. The topic in this episode is about the same mathematics task in discussion. ZH posed a series of questions and let students explain the screen by answering them. So we can see this kind of process as “guide-and-explain”. It concerns whole-class explanation guided by what happens on the screen and the questions prepared by ZH. The explanation goes beyond techniques, and involves mathematical content. After all the students finish their explanations, ZH began to make conclusion or explain the solution if the students’ answers were not correct.

Example episodes Spot-and-show ZH put students’ work on the screen and let her explain her solving strategy ZH gave the students some feedbacks or posed some questions to let them make deep thinking. The topics came from the students’ work based on the task situation. In this lesson, ZH did not chose the students’ work before the class, he chose them when students solved the problems and at the same time he put them on the screen and made them as a topic of following discussion.

Conclusion The model of teaching process of this Chinese teacher The following model could be used to present the process of ZH’s didactical practice: explain-the-screen (teacher, if needed) -----whole-class-discussion ----- spot-and-show (if possible) ------ guide-and-explain (one or more students) ----- explain-the-screen (teacher). Main features of the instrumental orchestration in Chinese mathematics classes Teacher makes more control in teaching process Students have less time to interact with technology ZH also tried to organized the teaching process in order to let students have a chance to participate Tendency of Chinese mathematics education ZH began to notice students could explain mathematics contents by themselves Letting students have more time to participate mathematics learning process is one of the important tendency of Chinese mathematics education.

Questions What is the relation between teacher’s document work and teacher’s knowledge?

Examples of students’ operate dynamic geometry from other lessons

Examples of students’ operate dynamic geometry from other lessons

Examples of students’ operate dynamic geometry from other lessons

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Thank you for listening

Workshop Task Description The first task contained one moving point P. At first, point P is coincide with point A then it moves from A to B to C to D. Students need to find out the mathematical relation between the area of triangle APD and the length of moving trace. The second task contained two moving points, point A and C, but these points do not move according to line but like a circle. There is a right triangle ABC, ∠C=90°, BC=4, AC=3. ZH makes triangle ABC turns around point B to make point A on the line BC (point A’), students need to find the length of segment AA’ The third task, ZH designed one rectangle ABCD. There is a point E which moves along segment BC, from point B to C. Point B’ is a symmetry point of point B according to segment AE like the figure shown. Students need to find what kind of trace point B’ moves when point E is moving along BC If you will use these tasks in mathematics lessons, how do you modify them to meet your own teaching objectives?