How using dynamic geometry changes the way teachers and students talk about shape and space An NCETM research study module

This module is based on Sinclair, N. and Yurita, Y (2008) To be or to become: how dynamic geometry changes discourse Research in Mathematics Education 10(2)

About this module This study module explores a research paper. It is designed to be self-explanatory. It begins by outlining the area of interest, explaining how it could benefit you to read it. It explains how the study was conducted. It summarises the findings. It encourages you to relate the research to your own teaching. In most sections you can read an extract from the relevant section of the paper. You can download the full paper here. We recommend that you read the paper once through before you do this module.here

Why you should read this paper Reading this paper could help teachers to better support students to move between static and dynamic geometry. It explains that: Teachers and students may perceive, and refer to, geometrical objects differently when they use dynamic geometry software There need to be accepted ways to handle situations that did not occur in the static case Teachers and students might see and reason about the dynamic shapes in different ways (when students have not constructed the geometrical objects themselves)

Questions addressed in this paper What changes occur in the transition from static to dynamic representations in terms of the teacher’s way of communicating about geometric objects, relationships, and claims? What is the impact of dynamic representations on students’ and teachers’ thinking? Background – what do we already know? The research questions

using measurements Discussion At this stage it may be a good idea to reflect on your own experience of using dynamic geometry software. Think about how you talk about geometric shapes when you use dynamic geometry software? Do you talk about shapes ‘moving’, for example? How do you think your students’ conceptions could be affected by their perception that geometric shapes ‘move’ when using dynamic geometry software? Investigate some of the dynamic geometry constructions referred to in this paper (note: works with Firefox). parallelogramrhombus

Methods – what did the researchers do? Four weeks observing a grade ten geometry course (students aged fifteen to sixteen). Two teachers and 23 students in one class. Classroom observations, videos, and questionnaires. The transition from non-technology to Sketchpad-based lessons with a projected display The lessons before, during and after the introduction of Sketchpad. The methods

What happened FIRST - without Sketchpad A worksheet with diagrams of quadrilaterals A worksheet with diagrams of quadrilaterals - similar and different Identifying quadrilaterals from definitions AND THEN - with Sketchpad Exploring pre-constructed parallelogram in a Sketchpad file - dragging Making the parallelogram into a rhombus Using measuring tools Using measuring tools to confirm identification of quadrilaterals Acting on a rhombus - can properties be broken? The accounts

Framing – theoretical background Learning geometry is the process of changing ways of talking about geometric objects and relationships Tools enable, shape and mediate a teacher’s mathematising. The tools have a long term effect on mathematical thinking which can be observed in the patterns of the teacher’s communication The teacher’s communication has four basic features: vocabularies – mathematics uses some words in a specific way (e.g. ‘regular’ polygon) visual mediators – such as the symbols used to show that two lines are parallel routines – such as the ways in which mathematical forms are categorised or grouped narratives – in mathematics these might include definitions, theorems and proofs The framing

Finding – talking about the geometrical shape WITHOUT THE SOFTWARE The shape possesses properties The geometric diagram is already made Reference to the parallelogram Is this parallelogram a rhombus WITH THE SOFTWARE How was the shape ‘made’? The dynamic diagram is a result of human intention-and-purpose Represents all parallelograms Can we make the parallelogram into a rhombus? The findings

Discussion – talking about geometric shape WITHOUT THE SOFTWARE The research found that a geometric diagram on paper is seen as a fixed thing, which someone else has made. It is possesses properties which the teacher and students are not able to change. How do you and your students refer to geometric diagrams on paper? WITH THE SOFTWARE The research found that the teachers and students talked about the diagrams as something they could make and change. How do you and your students refer to geometric diagrams within the software environment?

Findings – acting on the shape WITHOUT THE SOFTWARE Visual mediators (e.g. marks to show that lines are parallel or equal in length) are obvious and are used in identifying shapes It is difficult to use visual mediators to make arguments Using new routines on static shapes (in the imagination) Visual mediators are static WITH THE SOFTWARE Introduction of measurements as mediators (e.g. length of a line) Introduction of transformation routines (acting on a shape, trying to break it) Dragging leading to pathological cases The image as a source of knowledge The findings

Discussion – acting on geometric shape WITHOUT THE SOFTWARE Markers such a parallel lines are used to identify shapes but not very often to make arguments. After using the software the students imagine using it. How can you help students to use markers to make arguments? WITH THE SOFTWARE Measurements such as line length are used to identify a shape. A shape can be transformed or ‘broken’ using dragging. You or your students drag a parallelogram so that it collapses into a line. How can this provide an opportunity for fruitful mathematical discussion?

Finishing off – what could this mean for your teaching? The research found that teachers and students refer to, and act upon, geometric shapes in different ways depending on whether they are using the dynamic geometry software or not. Can you find ways of bringing dynamic-geometry- thinking into static-geometry lessons?

End of module The module links to the slides that follow. They are not part of the main presentation.

Background In dynamic geometry environments: objects can be constructed using tools analagous to straightedges and compasses on-screen objects can be transformed and measured elements of objects can be dragged Research on dynamic geometry environments: has focused on student learning when students interact directly with the computer points to the ways in which using the software can support students’ reasoning and proof demonstrates that students begin to think about dynamic constructions as a set of drawings (which is close to a ‘figure’) shows how students’ conceptions related to constructions within dynamic geometry environments tend to differ from the conceptions they form when they use ruler and compass constructions We know less about: the use of dynamic geometry environments in the context of whole class teaching using interactive whiteboards the back- and-forth between static and dynamic geometry Back to the module

Except 1 – the research questions In this article, we attempt to address some questions arising from two considerations described above. We focus primarily on the role of the teacher in mediating dynamic representations, which leads us to the following more specific question: What changes occur in the transition from static to dynamic representations in terms of the teacher’s way of communicating about geometric objects, relationships, and claims? We will be especially interested in how the teacher’s way of thinking about dynamic representations differs from their thinking about the static counterparts. Our research question is examined in the specific context of reasoning about quadrilaterals and, in particular, working with the properties of various quadrilaterals and the relationships between these properties. Back to the module

Except 2 – the methods We spent four weeks observing a grade ten geometry course (students aged fifteen to sixteen). Our research participants were two teachers and 23 students in one class. …The research setting was a high school (students aged 15-18) attracting mostly middleclass students in the American midwest. We video-recorded four weeks of daily one-hour geometry lessons, corresponding to a teaching unit on quadrilaterals (following a previous unit on triangles). This period of time included the transition from non-technology to Sketchpad-based lessons. The teacher used the software and projected a display on a screen at the front of the room. In this article, we focus on the lessons before, during and after the introduction of Sketchpad, during which we see the emergence of a distinctive mathematical discourse of dynamic geometry use. Back to the module

Except 3 – what happened? On the first day, students were given a worksheet entitled “Similar and Different” … on which ten quadrilaterals were drawn, each having markers to indicate equal lengths, equal angles, and right angles. The students were asked to discuss the similarities and differences they observed. On the second day… students were asked to use [a] new worksheet to help identify the shapes given on the “Similar and Different” worksheet. Thomas launched the third lesson with the following two questions: “Can a parallelogram be a kite? Is H really a parallelogram?” He then asked Sarah to open a file in Sketchpad that contained a pre-constructed parallelogram (with several construction lines hidden, so that only the line segments were visible). …The students had a quick response to Thomas’ question about what Sarah would have to do to make the parallelogram into a rhombus (refer to Figure 3): ‘’Make all sides equal.’ … Sarah measured the four sides of the parallelogram and began dragging its vertices so as to make the measurements equal. Back to the module

Except 4 – the framing …thinking can be conceptualised as a special case of the activity of communication, that is, as a type of discursive activity… changes in thinking result in learning, and learning a school subject such as geometry is defined as the process through which a learner changes her ways of talking about geometric objects and relationships in a certain, well-defined manner. Further, teaching a school subject involves changing students’ discourse. … the discursive features of teachers’ communication in the classroom will be highly influential in student learning since their own ways of thinking and communicating will also change. … the tool cannot be seen as a disposable element of the teacher’s own mathematising; rather, it enables, shapes and mediates it... the use of these tools has long-term effects on mathematical thinking, which can be observed in the characteristic discursive patterns produced through their use. Back to the module

Except 5 – the findings In describing the dynamic parallelogram, Thomas talks about the way that the shape was “made”: that is, the parallel properties that went into its construction. Compare this with the way he talked about geometric shapes before, which was in terms of the properties that the shape possesses or in terms of its identity (as exemplified by the presence of the verb ‘to be’ in the following phrases: What is this? Is this a square? Is the parallelogram a rectangle?) …. the term ‘a parallelogram’ does not seem to signify a specific static drawing anymore, but rather becomes a name for a dynamic shape that preserves its identity (that of ‘being a parallelogram’) through the shape-altering movement on the screen Back to the module

Worksheet 1 Back to the module

Measurements to confirm equal lengths Back to the module