3.5 The Case of Mathematics

Five mathematics lessons were observed, one large group lecture and four small group lectures. The large group lecture was of one-hour duration and small groups extended for 2 hours. The large group lecture was taught by Mrs. Q and the four small groups by Mrs. M. The large group lecture was on the teaching of directed numbers and the small group lectures addressed the following topics: the primary school mathematics syllabus, the teaching of subtraction and addition, transformation and construction.

3.5.1 Aims and Objectives

The mathematics course consists of twenty (20) modules whose content is clearly spelt out in thirty-six (36) objectives. The modules and objectives appear in two distinct categories in the documented curriculum, namely, subject matter and methods. Twenty-two (22) of the thirty-six objectives cover the subject matter and only fourteen concentrate on the teaching methods. The aim of the mathematics course in all the three years is to stimulate interest in mathematics and to produce teachers who “can create an appropriate learning atmosphere and impart knowledge of subject so pupils ... are aware of different learning/teaching methods... have sound knowledge of organizing maths lessons... are familiar with the current school maths syllabus”. All the five lessons, which were observed, were related to the aims and objectives of the course as spelt out in the curriculum document.

The practice of making objectives explicit in class is not common at the college and the five mathematics lessons observed were no exception.

3.5.2 The Large Group Lecture by Mrs. Q

The large group lecture was on directed numbers - the adding and subtracting of one-digit numbers. Mrs. Q wanted the students to develop a lesson plan on directed numbers. She started by introducing parts of a lesson plan and wrote the elements of the lesson plan on the chalkboard as follows: subject, topic, class, size, lesson objectives and presentation. Student teachers grumbled upon the mentioning of the words lesson plan - “lesson plan again” implying that they have already been taught how to draw up a lesson plan either by the same lecturer or others in the same or even in another department. In her introduction she wanted students to elaborate on the parts of a lesson plan. She asked questions such as “What do you think the objective of this lesson should be?” To which one student responded: “At the end of the lesson students should be able to add numbers.”

After the introduction Mrs. Q demonstrated to student teachers how to construct a teaching aid - the Number Line - for teaching directed numbers. In her demonstration she illustrated on the chalkboard and followed the illustration by constructing the Number Line using paper. She then gave the students time to construct their own teaching aid. As they were busy with the activity she was moving around and talking to individual students.

The lesson was conducted in a typical auditorium whose fixed furniture makes students sit in rows with the lecturer in front of the class. As the students were constructing the teaching aid Mrs. Q could only interact with those at the extreme ends of the row and those in the front row.

The lesson's focus was on pedagogic content knowledge, that is, a combination of content on directed numbers and how to teach directed numbers to primary school children. The interaction was minimal and the students tended to respond in chorus to questions asked. In her introduction Mrs. Q indicated that the lesson plan was for a Standard Seven class - the final year of primary school. The topic of the lesson - addition and subtraction of one-digit numbers - is, however, a Standard One topic in the primary school syllabus. Most of the class time was used to demonstrate construction of a Number Line.

Classroom management was problematic during this lesson. Throughout the one hour of teaching the students streamed in and out of the lecture hall with plates of food in their hands. This class overlapped with the serving of lunch and the students did not want to miss both the lunch and the lesson, which caused unnecessary disruption to Mrs. Q's class. Although she started her class on time, that is 14.00 hrs, after an hour she had not quite finished with her lesson. At 15.00 hrs another lecturer was already in the lecture hall to start her class with the same group of students. Mrs. Q ended her lesson abruptly with students having indicated that they had not understood what the lesson was about. She could not address the students' concern because the other lecturer had started distributing handouts to students and Mrs. Q had to leave the lecture hall unceremoniously.

3.5.3 Small Group Lectures by Mrs. M.

All of Mrs. M's lessons were of two-hour duration. Of the four lessons she taught, two focused on subject matter and two on methodology. Her first lesson was on the primary school mathematics syllabus. Students were required to work in groups to study the Standard 1 maths syllabus and decide how many periods were to be allocated to each topic and how many periods of mathematics were to be scheduled per week. These are the instructions the lecturer gave;

In your groups, I want you to look at the scope of the standard 1 syllabus and find out if it is allocated enough time. I would like you to refer to the green book which is just the syllabus without the Teacher's Guide and find out how many hours are scheduled for this class. I want you to break that topic down into the lessons that are required. Tell me what you think you should do in each lesson - if you think the topic should be chopped down into a number of periods you should say so.

After working in their groups for 20 minutes the student teachers were asked to present their group findings. Each group had a spokesperson presenting their deliberations and Mrs. M asked questions for clarification. Discussions were limited to the group representative and the lecturer as exemplified below:

After the presentations by two groups it was time up and the lesson was brought to an abrupt ending with neither summing up nor any evaluation of the lesson.

Addition and subtraction The second of Mrs. M's observed lessons was based on homework she had given to students in a previous class. She had assigned students to prepare a lesson plan on the teaching of addition and subtraction and to develop accompanying teaching/learning materials. The lesson started 30 minutes late because of Mrs. M's late arrival.

Mrs. M instructed the students to present the homework by simulating a primary school classroom and assuming the roles of school inspector, schoolteacher and pupils. Two role-plays were presented by different student teachers: one on the teaching of addition and the other on the teaching of subtraction. Student teachers used improvised teaching materials such as sticks and stones. The following is an extract of one simulation:

Mrs. M's lesson ended with her thanking the students for playing the roles of teacher very well but she did not debrief the simulations. She did not bring closure to her lesson and it ended as abruptly as the previous one.

Transformation. The next observation of Mrs. M started with her making reference to the previous lesson and reminding students that the lesson was on transformation. The previous lesson had been on the enlargement of objects and had laid a foundation for the current one. Mrs. M went on to explain the concept of enlargement, emphasizing the change in size. In her delivery of content she clarified the following concepts: reflection, translation, rotation. In her illustrations she emphasized the importance of the centre of enlargement.

She used mathematical instruments and the chalkboard for her illustrations and talked the students through each step of the illustration. The lesson was highly interactive with Mrs. M giving students exercises to work out ratios on the chalkboard and guiding them with questions. Mrs. M reinforced the correct responses by repeating the students' answers as illustrated below:

The lesson proceeded with Mrs. M assigning exercises. Students worked on the task individually while she moved around to see what they were doing. The lesson ended with student teachers being at different levels; some had completed the task, had been checked by the lecturer and had been released. The other group had not finished their task and she advised them to complete their task as homework.

Construction The last lesson of Mrs. M to be observed was on construction. The two-hour lesson started late because she arrived fifteen minutes into the lecture. She introduced the topic by making illustrations on board. The lesson was highly interactive with Mrs. M asking questions while doing illustrations on the chalkboard, and students responding, mostly in chorus.

She demonstrated how to bisect a line and an angle. She gave a detailed explanation of the steps to follow in finding the centre of rotation and the angle of rotation. Besides posing questions and giving explanations, Mrs. M required students to use the chalkboard to demonstrate what they had learned. She also gave them individual work and assisted those who were experiencing problems. At one point she had to provide an explanation to the entire class because of a common problem most students were experiencing. The students who had a better grasp of the concepts were given an opportunity to explain to the others how they had gone about solving the problem.

The lesson ended with Mrs. M emphasizing the importance of ensuring that the lines meet at a particular point of the centre. She then asked students who had finished to leave while those who hadn't finished were asked to remain behind and finish the assignment. The lesson ended 15 minutes early with Mrs. M leaving some students working on the assignment.

3.5.4 Summary

Both lecturers who were observed in the teaching of mathematics involved students in their learning. They used various strategies to ensure that student learning takes place. The use of questions, demonstrations and practice was very effective in promoting students' involvement. Demonstration is particularly a potent strategy in large classes and Mrs. Q used this strategy successfully in her large group lecture thus modeling to prospective primary school teachers how learning can be promoted in big primary school classes. Classroom management in Mrs. Q's class was very poor, with students moving in and out throughout the lesson. The disruption did not provide a conducive learning environment. It is not surprising that at the end of the lesson some students did not know what the lesson was about. Mrs. M did not model good time management skills. She arrived late in several lessons and left early in one instance. Her late coming led to an abrupt ending of almost all of her observed lessons. Whereas in two lessons the lecturers dealt with the lesson plan, their own lessons were not exemplary in modeling closure.

Although the mathematics course, as presented in the documented curriculum is biased toward subject matter - fourteen modules out of twenty, and twenty-two objectives out of thirty-six - the lessons which were observed were balanced between subject matter and methodology. The large group was an integration of the two, focusing on pedagogic content knowledge, and the four small groups were divided equally into subject matter and methodology. Both lecturers presented mathematics as a practical subject, and even the subject matter topics emphasized the “knowing how” instead of the “knowing that” (Eraut, 1994).

The most commonly used teaching/learning materials were the chalkboard and the mathematical instruments. In more than one situation, however, the lecturers asked students to improvise. In the large group lecture paper was used to construct a teaching aid and in one small group lecture students had to bring to class teaching/learning materials they had constructed themselves. This is another way of communicating the practical nature of mathematics at this level and the lecturers acted as good models for driving this point across. Textbooks and handouts were not used in the observed lessons.

Students' assessment was carried out throughout the lessons and homework assignments were also given as part of the assessment. After every demonstration students were given an opportunity to practise what they had learnt by either using the chalkboard themselves or engaging in individual exercises. The lecturers went round to check individual student's work and assessed their understanding of the concepts, and where necessary they gave additional clarification.

The observed lecturers used a variety of teaching styles, tilted more towards student-centred methodologies. Some of the teaching styles which were employed were demonstration, group work, simulations, and role-play, questioning technique and individual work. Most of the teaching styles were appropriate for facilitating the learning process. In one of the lessons students were required to analyze the primary school syllabus and relate the activity to the school situation as they knew it. In their task of analyzing the Standard 1 mathematics syllabus students engaged in interesting discussions. Issues such as “you have to think about a learner - some are slow learners and standard 1 pupils are young and likely to forget” were comments that came from the student teachers themselves. This reminder came constantly from lecturers and it seems to have successfully engendered the required awareness among the prospective teachers. However the opportunity for learning which could have been obtained from discussing the role-play was missed. The lecturer did not debrief the students' role-plays and the lesson did not, therefore, promote the learning that was expected. The questioning technique provided an interactive environment between the students and the lecturers although the tendency for students was to answer in chorus, a practice which the lecturers discouraged as can be seen in Mrs. Q's remark: “I won't listen to your music. I expect you to respond individually - you must raise up your hands.”

The content which was, presented in the entire five lessons links very well with the primary school syllabus and the teaching at this level. Although the linkage is strong between the observed lessons and the primary school syllabus some worrying observations were made.

Mrs. Q spent too much time demonstrating the construction of the Number Line, firstly on the chalkboard and then using paper. The teaching aid is simple and should not have used too much time. The large group is meant to introduce a topic to the entire class and engaging the class in the construction of a teaching aid seems inappropriate. However, looking at this classroom practice from a different perspective, especially in the context of primary school classrooms for which Mrs. Q could was preparing student teachers, the activity might be regarded as indicating creativity on her part.

The lesson that Mrs. Q selected for the lesson plan on directed numbers is a Standard One topic although she used it for a Standard Seven. One of the major objectives of the maths course is to enable students to read the primary syllabus and to relate topics to the relevant standard. Failure by the lecturer to make this relationship sends a wrong message to prospective teachers - that a teacher is at liberty to teach any topic to any class. It also sends a wrong message about the application of learning theory to subject content and creates an undesirable disjunction between foundation courses and curriculum studies courses.