In this section I describe and analyse the data that I collected from two interviews with Mr.
Ken and the five lessons observed during the data collection process. I support the results by discussing these in relation to the relevant literature I also use tables to present pertinent results of the analysis of this case study. An analysis of the data is presented by first giving an overview of all five lessons observed. This is then followed by an analysis of the teacher’s interviews and the teacher’s responses to learner’s questions regarding the task.
The tasks were administered to 37 of Mr. Ken’s learners from a grade 10 mathematics class.
The five lessons totalled to 2½ hours of viewing, each period’s duration was 30 minutes long.
What is of interest is the content of these lessons and how Mr. Ken explained why he chose these tasks. Since he has taught this geometry section before, the teacher now wants to do something different that would push the learners thinking (i.e. what is going on in these lessons is not what he would normally teach). Therefore it took the teacher 5 lessons to reach the aims of these two tasks so as to complete these two tasks with his class. By and large from the summary below, we see that the teacher works largely with conjectures to reach the
solutions that I discussed earlier in this chapter. I demonstrated that these lie inside Van Hiele level 3 and level 4.
4.4.1. Overview of the lessons:
This summary and the table below shows an overview of the five lessons observed:
Lesson One: During the first lesson the teacher gave the first task of “How many diagonals are there in a 700 –sided polygon?” to the learners without any particular instruction besides to use any method to solve the problem. Groups of various learners found it difficult to get started. After some time and with the teacher refusing to give suggestions or directions i.e.
Mr. Ken insisted that they make a start on their own, the groups then got started by attempting various methods and approaches. Mr. Ken observed for 5 - 8 minutes and then he started to interact, prompt and build on what the learners had done. The learners then started deducing, testing and justifying solving the problem. They used examples and non–examples that involved mathematical concepts like: sides, diagonals, vertex, pentagon and polygons.
Lesson Two: The same task continued with more testing, investigating and again with prompting, questioning etc from the teacher. New mathematical concepts were dealt with like:
odd number of sides and even number of sides, hexagon and the number of diagonals from a vertex of a polygon.
Lesson Three: The conclusion of the first task was arrived at after more justifying and conjecturing with the number of diagonals in a 21 – sided and 24 –sided polygon and eventually a 700 – sided polygon. The teacher prompted the learners to look for a pattern by using the smaller polygon that lead to the solving of the 700 – sided polygon.
Lesson Four: The second task of “Where 4 rectangular properties has access to the road but not to the lake must retaining their surface area and have access to both the road and lake.” was given to the learners. The learners found it difficult so the teacher sketched the scenario and eventually demonstrated the solution, while still encouraging learners to work towards their own solutions. The teacher then moved the learners to prove the required theorem which forced the learners to start to define, justify, hypothesis, investigate and conjecture, so that they could see that the rectangles had to be changed to parallelograms.
Lesson Five: In the concluding lesson, still working on the second task, learners displayed misconceptions about the vertical height, horizontal height and diagonal length. The teacher dealt with these misconceptions as they occurred so that the learners could move to proving the theorem. Learners nevertheless, concluded that the two conditions for this theorem must hold for the theorem to be true. This was consolidated after more defining, justifying, investigating, testing and conjecturing.
Date Duration Topic Concepts Discussed Comments
Tues
Conceptual and a bit of strategic competence. Justify and
Table 10 - 4.4: An overview of the five lessons' observed
The above table provides an overview of the five lessons, where the first three lessons focus on one task on polygons and diagonals that deals informally with conjecturing and proofs.
The fourth and fifth lessons focus on the second task dealing with the formal proof of quadrilaterals.
I began by making an assessment of what I consider as the teacher’s mathematical work, by referring to specific sections involving the teacher’s input and the learners’ responses, which the teacher guides throughout the lessons. Although the teacher’s practice is in the foreground for this study, the learners’ responses are important because engaging these is what constitutes the mathematical work that the teacher has to do.
4.4.2. Constitution of the five lessons:
What I am going to present now is a table that summarises the analysis of the lesson of event by event (see Appendix D for the full table). I identified: 3 events (notions) and 8 sub-events (sub-notions) in lesson one. Most of these entailed working with learners on justifying their mathematical thinking, to the teacher and the class as a whole. In lesson two, there were 2 main events (notions) and 10 sub-events (sub-notions), and most of them were focused on testing conjectures that were being made. There were 4 events (notions) and 6 sub-events (sub-notions) in lesson three and most of them were conjecturing. Similarly there were 3 events (notions) and 7 sub-events (sub-notions) in lesson four and most of them were justifying, 4 events (notions) and 6 sub-events (sub-notions) in lesson five and most of them were conjecturing.
So what then is different across the lessons and the mathematical work the teacher did? What geometry work did the teacher try to do, in terms of the above explanation of the five strands in relation to the condensed six problem solving types and what kinds of appeals did the teacher make as he worked to have his learners come to know and be able to do the tasks he presented to them.