The three linked lessons taught during practicum formed a short unit of work called ‘What’s the Time?’ (See Appendix B for Brief Lesson Plans). This unit was concentrated on the Measurement strand of the Stage 1 Mathematics syllabus, specifically the skills involved in reading analogue and digital time (MS1.5; NSW BOS, 2002).
When planning this unit, I considered the research of Van de Walle (2001) which advocates an alternative approach to the teaching of time. Reflecting this approach, my first lesson involved exposing students to the hour hand of a clock, in isolation, as a means of estimating time in approximate terms, for example, just after one o’clock. Supplementary to this, students were asked to interpret approximate descriptions to represent a suitable positioning of the hour hand.
The lesson was structured to commence with a whole class discussion of prior knowledge about analogue time, specifically student knowledge of what time is represented when the hour hand points to a given point on the clock face. A model clock was used as stimulus for this discussion. Shared terminology was introduced for describing time, in terms of the hour hand position, in approximate terms. As an additional extra, brief discussion occurred about predicting the position of the minute hand, given the hour hand position.
Although I was informed that all students had some experience with time, in this first lesson I chose to have students work independently on a double-sided worksheet, in order to allow informal diagnostic assessment of students’ abilities, which would form the basis of differentiation of learning in subsequent lessons where necessary. While students were completing the tasks, I circulated to respond to student questions and monitor progress.
Whilst some difficulties arose due to confusion between the terms ‘almost’ and ‘about’, most students were able to successfully use approximate terms to describe the position of the hour hand (See Appendix C for annotated work sample). During the tasks that required students to draw their estimations of the hour hand position given an approximate description, some students were prompted to also attempt to draw the minute hand position. These tasks revealed that some students in the class were able to accurately represent the hour hand, and predict the positioning of the minute hand also (See Appendix D for annotated work sample), demonstrating skills well beyond what is expected at a Stage One level.
Taking into consideration the overall success demonstrated by students in the first lesson, in the subsequent lesson I moved on to addressing the representational value of each ‘hand’ on an analogue clock. Continuing to apply Van de Walle’s approach (2001, p. 301), this second lesson introduced the concept of counting by fives to determine how far after a given hour the minutes have reached. After an initial explanation of this process by the teacher, students broke into groups of three (groups were selected by students, generally on the basis of friendship or by convenience) and were allocated one question to complete collaboratively. This gave students an opportunity to clarify whether or not they were confident in applying the count by fives strategy. Supported by Plowden (1967, cited in Ogden, 2000, p. 212), this use of group work provides “opportunities for pupils to explain to and teach each other”.
On this basis, some students moved off to work independently on the set tasks whilst a small group of students remained that I worked with more intensively, using additional scaffolding and examples of applying the strategy. This mode of voluntary grouping worked well as it allowed students who were confident to remain actively engaged whilst providing additional support for those who needed it. This is supported by Groundwater-Smith, Ewing and Le Cornu (2003, p. 93) who assert that it is “important to allow students to work at their own level and pace so that they do not become restless or frustrated”.
To summarise the lesson, students were paired up, on the basis of who finished the set task first, whilst continuing to apply knowledge of students who would not form a productive partnership. In these pairs, students worked to set questions for each other to answer. This involved one partner positioning the hour and the minute hands, and the other partner applying the count by fives strategy to tell their partner what time was shown. The aim of this task was to develop students’ skills in ‘questioning’, which is a Working Mathematically outcome for Stage One (NSW BOS, 2002, p. 19). Additionally, the choice to use groups of two, rather than larger groups, was made on the basis that the questions were not problematic or open, that is, only one correct answer was possible. In groups of three or more, only one student would be able to contribute to each question, after which additional responses would be surplus. This cooperative task was not very successful, probably as time was not allocated for the explicit development of cooperative skills (Bobis, Mulligan & Lowrie, 2004, p. 322). This resulted in many pairs simply writing and answering questions for themselves.
In this lesson, addressing the position of the hour and minute hands simultaneously appeared to cause confusion for some students. In some cases this was merely due to confusion between which hand was the hour and which the minute. In other questions, where students previously recognised that the hour hand does not remain fixed in the same position for the hour’s duration, students appeared to be confused by which hour was represented when the hour hand was approaching the next hour. Additionally, in some instances, students simply treated both hands in the same manner, either both as hour hands (9:55 interpreted as 9:11), or both as minute hands (11:05 interpreted as 55:05) (See Appendix E for annotated work sample).
Taking these difficulties into consideration, in the final lesson, students were organised into small abilities groups (4-6 students per group). These groups were formed by the teacher on the basis of formative assessment of students’ work samples. In this lesson, students were converting analogue times to digital times and vice versa. Whilst some groups of students, who had demonstrated a strong grasp of the concepts, continued to address times with minute values of each multiple of five, other groups’ activities were simplified to reflect more stage appropriate tasks. Two groups addressed only times on the hour and half hour, whilst three groups addressed times for each quarter hour also. Each small “working group” (Galton & Williamson, 1992, p. 9) was seated together, rather than working at their usual desks, to allow cooperative learning and peer clarification of answers. Although this approach required slightly more preparation, less teacher assistance was required during on-task time which allowed me to monitor more students rather than focusing on assisting students with difficulties.
Again the final section of this lesson required students to work in pairs, setting and answering questions of each other. One student would show an analogue time and the other student would need to find the corresponding digital time. However, in this lesson, pairs were allocated by the teacher, with the small ability groups breaking into pairs. Explicit instruction about the process of working cooperatively was also provided (Groundwater-Smith, Ewing & Le Cornu, 2003, p. 94). Selection of the pairs prior to commencing the lesson, on the basis of ability and knowledge of which students cooperate productively, worked well. Students were engaged with this task and worked well to assist each other when encountering difficulties.
In summary, throughout the course of the three lessons, continual reflection allowed for the adjustment of task difficulty and grouping strategies. Tasks were differentiated to accommodate different levels of ability apparent in the class. Lesson structure was also altered to allow opportunities for students to make decisions about the pace and level at which they wished to work. Small groups were employed primarily to allow cooperative clarification of ideas between peers. A particular focus was also made upon cooperative pair work, with the adoption of explicit skills instruction improving student outcomes. As knowledge of student abilities, both on tasks and as cooperative partners, was improved, the lessons were able to be better tailored to suit the specific needs of the students in the practicum class.