The research was conducted in one of partner schools of the larger scale EdQual project. The school is a mixed secondary school with a population of about 950 students, 43 teachers in 2008, located in the capital city and receiving among the best primary school leavers. The average class size in this school is 30. They have about 40 computers all linked together forming a network in the same room as a smart board, projector, scanner and Internet; all donated by the New Partnership for African Development (NEPAD) e-school initiative in November 2005. Compared to other Rwandan schools of this size, this is not an unusual number of computers to have, but there is a significantly lower class size than in most schools, as a result of school’s intention to excel. The computer room is open for teachers but not for unsupervised learners.
Two mathematics teachers (Isaie and Laetitia) were already involved in EdQual and participated in this particular study which focuses on the use of Geometers’ Sketchpad (GSP) from May 2008 until June 2009. They teach mathematics in senior two/grade 8 and senior three/grade 9 at ordinary level. The teachers are certified from two different institutions of higher learning for teaching Mathematics and Physics at secondary level and have 2 and 3 years experience, respectively.
Initially, to avoid interfering with the school calendar, the teaching experiment was included in the usual teaching process but as time elapsed, the computer room was used by either the administration for community development of professionals or for learners’ computer skills lessons. We needed to organise class activities in extra ordinary periods; that is in hours not normally reserved for teaching. During our sessions, learners had to sit in pairs at a computer and were called upon to share and discuss their views. This is an unusual situation in Rwandan classrooms where teachers usually stand in front of the class and talk and learners talk only to answer teacher’s questions or ask clarifications.
The overall collaborative action was based on teacher–researcher meetings during teachers’ break time and teaching in the school. Prior to the classroom activities, three meetings of about 1 h each were organised. These meetings were devoted to explaining the purpose of the research and the role of each participant since the process is quite different from the usual ones in the larger project. We discussed topics to be taught, considering the national mathematics curriculum and generally we jointly planned and co-taught as much as conditions allowed. (Sometimes one teacher was busy teaching his/her class). There were often two or three teachers in the same classroom. Each teacher had previously used a spreadsheet at least once in his/her class.
Over the period of the research, there were meetings or telephone conversations. Initially, we had planned to have 1 h a week for classroom activities but the improvised school agenda was not in our favour. We had in total ten meetings and eight classes. Telephone conversations were to confirm whether a given class is due or not while other meetings were to preparing classroom activities. Short meetings, organised after each classroom session, were aimed at sharing interpretations of particular events and to develop innovative approaches for future lessons.
During the academic year 2008, we delivered lessons in one of the senior two classes (there were three classes of this grade) and in one class of senior three in 2009. This was deliberately done for two main reasons. First, after introducing the software and objectives of our study to 2008 learners, we wanted to work with the same learners in 2009. Second, we wanted teachers to practise in their usual classes. Unfortunately our first aim was not achieved since the school administration mixed learners at the beginning of the 2009 academic year. Thus some of the 2009 class were not previously introduced to the software. Taking into account this situation we grouped learners (generally in two’s or three’s) by familiar with non familiar with the Geometer’s Sketchpad (GSP).
Given this school description and overall activity organisation, I present the process of change using some classroom stories. This is within collaborative action research already implemented by the EdQual project. Data from this research are composed of researcher–teachers discussions during pre-observation and post-observation phases of lessons, and researchers’ notes during the lesson. In this case study, there were three significant episodes contributing to the teachers’ change.
Findings from students’ answers
One classroom observation provoked the awareness that what we, as teachers, take as obvious is not so for learners. During the academic year 2007, Isaie had a lesson on multiplication of integers with senior one learners. During the pre-observation discussions on what strategies to adopt for enhancing critical thinking, problem solving and argumentation skills, we opted for grouping learners in fours. (The normal seating is that two learners share one desk, so two adjacent desks formed one group making a total of eight groups). The task was to discuss properties of multiplication and fifteen minutes were given for this task. Then group representatives simultaneously and silently wrote results on the chalkboard which was divided proportional to the number of groups. Examining properties one after another, the teacher and learners had to decide which group had done the task correctly. Identifying the unit element, three groups were convinced that ‘0 is the identity’ while five had said ‘1 is the identity’. Asked to justify their answers, the three groups said ‘because 0 is the only identity we have learnt in our last lesson’. (They were referring to addition since the topic came after addition of integers and its properties). At the end of the lesson the teacher expressed surprise how large a number of learners could not easily recognize 1 as unit element.
The teacher was surprised at learners’ justifications. He revealed ‘since I am teaching I had never given learners opportunity to discuss in their groups. So listening to learners gave me opportunity to know that what I was taking as the easiest thing might be difficult’. Nevertheless at the end of our discussions, the teacher noticed that organizing such group discussions was time consuming even though beneficial. These observations constitute the beginning of teachers’ awareness (Mason 1998) and therefore a desire to change their practices aiming to make learners more participative.
Drawing on Lerman (2002) quoting Mason (1998), we hope to work with teachers to educate their awareness so they are sensitised mathematically to work with their students in a mathematically informed and appropriate fashion. Since the Geometer’s Sketchpad (GSP) was new tool, I had to introduce its main features to my two colleagues during preparation hours; the challenges of new technologies in general were previously discussed in workshops organised by EdQual. In order to improve collegiality and community learning, I taught the two first classes in the presence of the two teachers. I did this partly to familiarise my colleagues with the self-reflection phase to be held after each lesson in this case study.
Before we started our class activities in the GSP environment, I asked learners to construct a square on their sheets. Their construction was mainly based on counting the ‘small squares’ in their squared sheets. Challenged to explain what they can do if they do not have a squared sheet but only compass and ruler, learners affirmed never having used instruments in constructing geometrical figures.
After the lesson, I asked Isaie how he draws a square on the chalkboard and he confirmed the students’ answer: ‘I just draw with hands the four sides and assume that it is square, I do not use compass neither ruler’. Concerning geometry, the methodological approach suggested by the national curriculum is that teachers invite learners to construct all geometrical figures using suitable instruments. While constructing a square using geometrical instruments, there is the opportunity to internalise physical properties that include equal and perpendicular sides and right angles. One can observe here that teachers either give little time to explore the curriculum guide or under estimate the importance of the usefulness of the methodological column. (The curriculum starts with general objectives and the rest is presented with three columns of specific objectives, subject content and methodological remarks). GSP was challenging us as teachers to reflect on our practice by using appropriate instruments to construct geometric figures which we have the habit to neglect.
Findings from collaborative action
A powerful experience as repeated by teachers was when I led discussions with their learners. Lave (1996) argues for the social nature of learning based on the premise that human beings are relational, social beings, who of necessity participate and engage with one another. Therefore, learning is a community practice, not an individual enterprise. During a GSP assisted lesson taught by Isaie, I had discussions with learners about whether or not a square is a rectangle. Seventeen learners said no; one said yes and five abstained. The ones who said yes argued that a square is a rectangle because both are polygons.
Drawing on the Aristotle’s famous deduction, I asked learners whether or not a human being is an animal in order to bring attention to their conclusion. All learners responded affirmatively.
- Researcher::
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Is a human being an animal?
- Learners (all)::
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Yes
- R::
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Is any animal a human being?
- L (all)::
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No
- R::
-
So drawing on this, your home work is to justify whether or not a square is rectangle
What is important here is not to say that this was the best way of directing learners to the right answer; rather this episode is a simple illustration of the use of our environment and the need for giving time to reflection for our learners instead of staying restricted to mathematical deduction and induction.
Findings from teacher–learner interactions
A pair of learners is inscribing a square in a circle in a lesson taught by the teacher Laetitia. A pair of learners (Christine and Maseka) starts drawing a circle and, using the segment menu, constructs a diameter as a segment joining two arbitrary points on the circle but trying to pass through the centre of the circle. They draw another segment intersecting the preceding one approximately at the centre. Finally, they join endpoints of these segments to obtain the following figure (French words mean ‘the square constructed by’; learners were asked to write their names on the drawings in order to match with their worksheets during data analysis):
- The teacher (L)::
-
Is this a square?
- Learners::
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We need to check if the sides are equal?
(Using the “measurement menu” they find sides are not equal)
- L::
-
What else can you check?
- Learners::
-
Angles
[Measuring angles they find them not all equal and conclude it is not a square]
- L::
-
So what can you do to make a square?
(No answer)
- L::
-
How do you call these two segments (pointing diameters on the screen)?
- Learners::
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They are diagonals
- L::
-
How must be diagonals of a square?
- Learners::
-
Oh…. they must be equal
- L::
-
Only equal?
- Learners::
-
Also perpendicular
- L::
-
So now you need to draw perpendicular diameters?
- Learners::
-
Yes, but how to do it?
(The teacher explains the process of constructing perpendicular lines and learners carry on.)
After this experience, learners were able to write down in their own words the process used in constructing not only a square but also perpendicular lines.
Teacher–student interactions were facilitated by the dynamic geometry which not only gave learners the opportunity to express themselves and manipulate objects but also helped teachers acquire the habit of asking learners to explain their thinking/doing. In the post-observation discussion, the teacher said: ‘if I was using chalk and board I could not have a particular attention to any one learner but within the GSP all learners being simultaneously working get direct feedback’. This reveals the awareness of the teacher in relation to the advantages of a learner-centred methodology but the traditional way of teaching does not encourage them to practise it.