ASSESSMENT PRACTICES
6.2 Description of teacher educators' instruction
6.2.3 Description of Wilson's instruction
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with an aspect. He concluded the lesson by telling them to try the questions again on their own, and assured them that he would return their scripts in due course.
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Drawing on the previous knowledge, Wilson began the day's lesson on a 4 x 4 magic square. He worked out two examples on the board and also gave students some questions as a self-test of their understanding (for example, draw a 4 x 4 magic square with the numbers 5, 6, 7, 8, …, 20).
Students worked for about 10 minutes, after which Wilson called on a student to record her answer on the board. She recorded the result as shown in Table 6.2
Table 6.2: 4 x 4 magic square
After the student presented her work on the board, Wilson called on the class to confirm whether her solution was right:
Wilson: Is she correct?
Student: Sir, her answer is wrong. She has missed the steps up the numbers.
Wilson: Which part of her solution?
Another student: I think almost all the entries. For example, column two has a total sum of 48, but if you check that of the first column, it is 50, which is not the same.
The explanation the student gave, hinted how his colleague and others who might have made a similar mistake could rectify their mistakes. Wilson therefore calls on the student to record his answer for the 4 x 4 grid on the marker board. After he was done with this, Wilson invited questions from the students, but none of them asked a question. After waiting for about 3 minutes with no questions, Wilson continued with the lecture, moving on to another concept, figurative numbers.
Wilson talks of figurative numbers as numbers that can be shown by dots or counters, and arranging them to form a geometric shape. He mentioned, among others, triangular numbers, square numbers, and pentagonal numbers as examples of figurative numbers. To show how dots
5 18 19 8 12 10 11 9 16 14 15 13 17 6 7 20
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are arranged to form the various polygons, he began his activity with triangular numbers, using bottle tops. He used the bottle tops to illustrate the formation of the first three triangular numbers and modelled it on the marker board using dots (see Figure 6.7).
Figure 6.7: Wilson's modelling of triangular numbers.
After Wilson was done with the modelling of triangular numbers on the board, students started talking to each other, which led to noise in the lecture hall. As he tried to ask what the noise was all about, a student asked "Sir, how can one be considered as a triangle?” Another added that one gave us a dot, it did not form a triangle, so how can it be considered as a triangular number? Wilson answered: “Yes, one did not show a triangle, but you have to picture the one dot as a triangle in your mind.”
He noticed that students were not satisfied with the answer that was given and therefore promised to show a video on the concept to illuminate the process and how one is considered as the first figurative number at his next meeting. Wilson asked his students to go into their groups and used think points to encourage them to think of an emerging pattern and generate the next four triangular numbers. The group activities ensured collaboration among students and facilitated learning.
Students remained in their group for the rest of the lesson as the instruction process continued with square numbers and pentagonal numbers. Wilson ended the lesson by asking students questions to summarise the day's activity.
The researcher did not observe the follow-up lesson where Wilson promised to show the video;
instead, the researcher was taken to a different group where the lesson on matrices was taught. The lesson started by recapping the previous work and providing feedback on previous topics. About 20 minutes was spent solving questions on how to find the sum to infinity of a given series or sequence; for example, given the series 2 + 1
2+ 1
8+ ⋯ find the sum to infinity.
1 3 6
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After students were done solving the question, Wilson said "I hope the groups are ready for the presentations”, and the students replied "Yes, sir, we are ready" (chorus answer). The teacher asked
"Which group is to present first?”. “Group four”, the students answered.
Wilson took his seat and called on group four members to start the presentation. The group began their presentation on the matrix by giving the definition and how the names of a matrix are derived.
Members of the group took turns to explain an aspect of the concept through activities. Different groups followed one after the other to do their presentations on areas concerning the topic. After each group presentation Wilson required students to ask questions and comment on the presentation. He was always the last to comment before the next presentation.
In some situations, two groups presented on the same area, for example, how to solve simultaneous linear equations in matrix. One of the group discussions focused on writing the equation in matrix form and multiplying by the inverse to find the unknowns, while another group adopted Cramer's rule for solving systems of equations. There were six groups in all, with an average of six members each. In his closing remarks, Wilson asked his students to self-test themselves by solving more questions on their own, since the presentations brought to finality the discussion on the topic. He also asked them to read on binomial theorem, the topic for the week to come.
In his third observed lesson, Wilson was teaching binomial theorem, as he had mentioned in the closing remarks of his second observed lesson. Wilson used reflective questions to established students' knowledge on the topic. He then introduced his students to Pascal's triangle and led the students to generate it. Activities were used to enhance students' ability to recognise and use Pascal’s triangle for solving binomial expansions (see example below).
Example
(3𝑥 − 2)5 = 1(3𝑥)5+ 5(3𝑥)4(−2) + 10(3𝑥)3(−2)2+ 10(3𝑥)2(−2)3+ 5(3𝑥)(−2)4 + 1(−2)5
= 243𝑥5− 810𝑥4+ 1080𝑥3− 720𝑥2+ 240𝑥 − 32.
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He followed this up with a series of examples. Self-test questions were used by Wilson to assess students' understanding of the concept. In addition, he explained and introduced the rule for the binomial theorem.