Chapter 6 Teacher Strategies and Learners' Performance
6.2. Analysis of data
6.2.1. Test question 2 (a)
The learners scored the 1least number of marks in the test for the question that was displayed on the c:hallkboard as:
Question 2
(a) Give the additive inverses of the following:
1. +4 2. -3 3. -2
The ,teacher often mentioned "additive inverses" during lessons. The table below indicates the number of times the term "addit,ive inverses" was used during the three .lessons.
Lesson Number of times Number of times mentiioned by teacher echoed by learners
,8 4
2 5 1
3 13 3
Table 11 Table showmg number of times term "additive mverse" was used by . . teacher and learners in various lessons
The learners thus heard the term "addit,ive inverse'' 26 times and chorused the term 8 times during the thr,ee lesson presentations but 13% of the learners did
not even attempt th1is problem. The learners were introduced to the term
"additive inverses" in ,lesson 1 1in the following manner:
T: ... So we have to calculate using what ... for example, what is positive one plus negative one? Hands up. What is the answer here? Yes? Yes?
L 1: Zero.
T:
Yes, zero. What is negative four plus positive four? Class?L: Zero.
T:
Zero. Whal is negative 100 plus positive 100? All of you it is .. ? L: Zero.T: Zero. So, let's say here, let's say here that given this four and that given this six. You want to remove this positive four before in order to get what?
Zero. You see what we calf an additive inverse. We call what?
(On the board tihe teacher demonstrated using the example x + 4 = 6.)
L: Additive inverse.
T: Additive inverse of, for example, of positive one is negative one. What goes with x and positive one and negative one is · ... you get what?
L: Zero.
T: Which means that xis in the opposite one is negative one and again opposite one is ...
L:
Negative one.T: Because they give you what?
L:
Zero.T: So, if you use the additive inverse, you get what? ... Zero.
Here the teacher emphasised that the sum of a number and its additive inverse is zero. The teacher did not explaiin why the additive inverses are useful i.e. the functional understanding was not considered. The teacher placed much emphasis on the instrumenta'I understand:ing required for the technique to be demonstrated.
On the cha1lkboard the teacher wrote the following ,examples, together with the term "additive inverse":
+1 + -1 = 0 -4 + + 4 = 0
·100 + +100 = 0 additive inverse
The teacher used these examples to demonstrate the theory that was later to be applied in solving the equation. In the example that fol:lowed, the teacher
demonstrated how to solve for x using the equation x + 4 = 6. She said:
. . .Right, so let's write the unknown x, so the additive inverse of positive
four is negative four . . . .
The teacher thus renamed the "plus four" as "positive four". The teacher then used the example, "x min1us two equals to four", but asked for the additive inverse of "negative two". This is what the teacher said:
Let's say you are given this. Let's say you
are
given x minus two equals to four. Find the value of x. Again you use whatever? You use the additi�e inverse of? Negative two. 1What is the additive inverse of negative two? Hands up. Hands up so I can see. Yes.Once aga1in the teacher renamed the "minus two" as "negative two".
The teacher often renamed the "minus" as "negative and "plus" as "positive"
during presentations. The teacher did not, however, us,e the notation using the superscript, i.e. + 4 or -2, again aft,er the three examples were written on the chalkboard when the teacher introduced the learners to this term in lesson 1.
The learners always made use of additive inverses whilst dealing with equations.
The"+" and u_" signs seen in equations did not represent "positive" or "negative"
but were perhaps interpreted as symbols indicating an operation.
In the test question 2 (a) the learners were asked to find the additive inverses of integers that were not pa:rt of an equation. The I1earners, other than simply observing the three examples written by the teacher on the chalkboard, did not use the notation used 1in the test question. The additive inverses of the integers given in the test were not wnitten with a superscript but as +4, -3 and -2.
Perhaps this is one of the rreasons why only 87% of the ,Jearners attempted this test question and only 24% of these learners gave the correct solution. The notation used in th,e test may well have been u nfamiliiar to the learners.
During the observed !lessons the learners were reminded to "remove" terms as well as "bring" terms to the other side using additive inverses. The teacher treated the terms as objects, or the unknown as "the thing that we don't know", and used addiitive inverses in order to isolate the unkn,own. The teacher did not stress the importance of ensuring that equivalence is ma:intained when the
ba1lance algorithm is employed. The teacher emphasised the fact that the sum of
· the number and its additive inverse is zero and thus the unknown becomes isolated on one side of the equation. During 11esson presentations the additive inverses of terms were not dealt with as representing integers but as objects that had to be removed. The learners always found the additiive inverses of terms in an equation by dealing with the unknown as if it were an object.
The seUing out of the question may a:lso have confused the 33% of the learners who attempted this test problem. The subdivision of the question (a), using
numbers, may have been an unfamiliar format. This particular setting out of subsections was on'ly seen in the test and was not used by the teacher to set out problems dur,ing lesson presentations.
6.2.2. Test question 1 (b)