ASSESSMENT PRACTICES
6.2 Description of teacher educators' instruction
6.2.2 Description of Sekyi's instruction
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The process continued until they arrived at the answer: one bundle of 5 bundles of 5 sticks and 3 bundles of 5 sticks. This was finally written mathematically as 1305. She followed it up with the following examples for the students to work out:
a) 234 + 114 b) 1012 + 1102 c) 278 + 158
Her third lesson was on the properties of operation and was presented by students via group presentations. Groups took turns to present aspects of the subject matter. The groups presented on the commutative property, associative property, and distributive property. Two groups presented on commutative property. One group presentation was on the commutative property of addition, while the other group's presentation was on the commutative property of multiplication. The same went for the associative property, while the last group presented on the distributive property. After each presentation, group members were questioned by other students. Emily was always the last person to comment on a presentation. After all the groups had finished presenting Emily said “This ends the lesson on properties of operation. I am not treating it again.” She concluded the lesson by asking students a few questions to highlight key concepts learned on the day.
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Sekyi: How can we find the zeros of the x2+ 5x + 2 = 0 using the method of completing squares?
Students: First we have to transpose the 2 to the other side of the equation
Another student: Then find half of the coefficient of x, square it and add to both sides of the equation.
The review continued to the end by arriving at the zeros for the equation. He then began the lesson for the day by sharing the learning intentions. Sekyi was teaching graphing of linear inequalities.
He started by exploring students' existing knowledge on how truth set of inequality is represented on a number line, as per the transcribed excerpt below:
Sekyi: If you have the inequality x > 1 and represented on a number as shown [see figure 6.3a], what can be said about the representation?
Student: It means 1 is not part of the solution.
Sekyi: What if the line is represented like this? [See figure 6.3b]
Student: Sir, for that, 1 is part of the solution.
6.3: Sekyi number line representation of 𝒙 > 𝟏 𝒂𝒏𝒅 𝒙 ≥ 𝟏
Based on the response, Sekyi introduced his students to how to graph inequalities on the Cartesian plane, by taking them through the strategies required in drawing inequalities and for shading the required region representing the solution for the given inequality. Some of the key things mentioned were:
1. Using broken lines when the giving inequality has one of the symbols less than or greater than (< 𝑜𝑟 >). He explained that the points on this line do not form part of the solution set.
2. Using a continuous line for either less than or equal to symbol or greater than or equal to the symbol (≤ 𝑜𝑟 ≥) and the points on the line forms part of the solution set.
1
-1 0 -1 0 1
Figure 6.3a Figure 6.3b
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After taking the students through the strategy with illustrations, Sekyi read out the question: Show the required region represented by the linear inequality 𝑥 + 𝑦 > -2. He worked out the example with his students and plotted the graph (Figure 6.4) on a graph board placed at the right corner of the lecture hall. After the plotting and guiding the students to shade the region which represents the solution for the inequalities, Sekyi asked his students to do the plotting in their graph books.
Figure 6.4: Sekyi's graph for the inequality 𝒙 + 𝒚 > −𝟐
After illustrating the procedures involved through the example, Sekyi engaged his students in a series of hands-on activities with other inequalities symbols to ensure better conceptualisation of the topic. Sekyi also used self-test activities to test student understanding; for example, ‘Plot the graph of the inequality x + y ≤ 4 and shade the required region’. While students were working on the task Sekyi was moving around the lecture hall, observing as students worked on the task.
After working for about 15 minutes, students voluntarily presented their results on the board for discussion. With just 2 minutes to end the lesson, Sekyi announced an impending quiz to be written the following week:
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Sekyi: … the quiz will be based on units 4 and 5. (Field note).
Sekyi's second observed lesson was with level 200, where he was handling Methods of Teaching Basic School Mathematics modules. He was teaching the subtraction of fractions but began the lesson by reviewing the previous lesson on additions of fractions. After greeting the class, he wrote a fractional sum: 1
2
+
13 on the board and asked his students to solve it using the paper-folding technique.
Sekyi: Show and explain how you will guide Grade 6 learners to find the sum of 1
2
+
13
usinga paper-folding technique.
Student: Sir, you first pick a strip of paper and fold it into two equal parts horizontally and shade one part representing 1
2.
Sekyi: After shading the one part, what will be the next thing to do?
Another student: You then refold the paper vertically into three equal parts and shade one part for 1
3. So, the answer is 5
6 .
A further student responded to his colleague by drawing her attention to the fact that it should be 4 out of 6, not 5.
Student: No, the answer is 5 out of 6 not 4. It can be observed that after the folding we had 6 equal divisions and when you check, one part is shaded twice, so we count that part 2 which gives 5 not 4.
Sekyi confirms the answer and asks her to show this by drawing on the marker board.
Figure 6.5: Diagram depicting 𝟏
𝟐
+
𝟏𝟑 using paper folding.
1 3 1
2
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The lesson continued with the day's topic, subtraction of fractions. After the review on the addition of fractions, Sekyi worked out an example (see Example 2) with his students after explaining the subtraction concept.
Sekyi: Describe how you will guide a Grade 6 student to solve 5
7− 2
7 using concrete material.
Although the premise of his question is to describe "how you will", the question was not directed at his students to do it. He used this question to demonstrate the teaching of the concept of subtraction of fractions.
Figure 6.6: Diagram depicting 𝟓
𝟕− 𝟐
𝟕 .
Although the question did not specify the exact material to be used, Sekyi utilised the paper-folding technique to explain. He noted after the student had folded the paper into 7 equal parts, assisting him or her to shade 5 parts, representing 5
7. He then assisted the student in cancelling 2 of the 5
7
of the shaded portion; the remaining shaded out of the 7 becomes the answer.
The explanation was not seen as convincing for some of the students. A student was observed whispering to her friend “I do not understand what he did”. The friend told Sekyi, but the said student denied having said it, and Sekyi then proceeded as if nothing had happened. He further used a number line and Cuisenaire rod to illustrate the procedure to his students. Realising that time for the lesson was almost up, he concluded by asking them to read on multiplication and division of fractions before the next lesson. He also wrote the following problems on the board for them to try them on their own using the Cuisenaire rod technique:
3 7
5 7 2
7
146 1. 1
2 − 1
4 2. 1
2− 1
5
In his third observed lesson, Sekyi led his students (level 100) to discuss the questions for the midterm quiz, which they had written in the previous week. His third lesson therefore saw him giving feedback to students on their midterm quiz, as illustrated in the transcribed excerpt below.
Student is asked to read out the question and reads as follows: Determine whether or not the indicator y = −2 is a solution to equation 3(y + 1) = 4y − 5.
Sekyi: Good, so what were you supposed to do?
Student: Sir, the question asked that we indicate if y = −2…, so you can solve and compare the answer.
Sekyi: You solve for what?
Student: Sir you solve for y from the equation, and when you get y, let’s say y = 8, then you compare it with y = −2.
Sekyi: Is there another way of solving it?
Another student: When you check, the equation involves a variable which is y, and you have been given y, you just have to substitute y [into the equation], and after that, you compare the right-hand side to the left to see if the values are the same.
Sekyi recommended and explained that the two methods could be used. However, he cautions his students to stick to instructions whenever they are writing a paper:
The question stated that determine whether or not …., so just solving and not concluding is not enough. What I have seen in the script I am marking is, most of you did not conclude, so be careful else you will lose marks in the final examination. (Field note)
The discussion continued with the other questions up to the end of the instruction, with Sekyi activating his students as resources of their own learning. Students took charge of the instructional process, while Sekyi provided assistance when he saw that students were encountering difficulty
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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.