Figure 5.19: L itha’s response to Task

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D espite h is good perform ance in other tasks, L itha had not yet got the correct answ er to this task. W hen probed further, L itha w orked out the answ er successfully.

1. Litha: Therefore; it starts ... it’s going to be e i g h t ...

2. R esearcher: The person is the 8th one, bu t the rem aining people w ill b e how many?

3. Litha: 7 ... OK, it is going to be ... is going to be 8, 7, 6, 5, 4, 3, 2, 1 and then w e add.

T alking about his thinking for the handshake problem appeared to b e a ZPD for Litha w here m y single prom pt in turn 2 appears to have been enough to lead him to a correct solution. D uring the interview he could not think o f constructing any representation that could have possibly assisted him to arrive at the correct answer. H is initial thinking that a person has two hands is irrelevant in this context because a person uses one hand and not both in the process o f shaking hands. H ow ever, i f he had halved 72 he w ould have got the correct answer. B y talking and thinking tow ards a solution he arrived at the sum o f 8+7+6+5+4+3+2+1 handshakes.

M atoti’s C ase

M atoti is a young w om an aged tw enty-three years. She lives in B otha-B uthe district. M ato ti’s m athem atical background appears quite good for the reason that in Standard 7, she obtained a 2 Qd class in m athem atics, a sym bol C in F orm C, and a credit in Form E. W h en asked to w rite a story about h er experiences o f learning school m athem atics, M atoti m entioned that it is crucial to know the four m athem atical operations (addition, subtraction, m ultiplication, and division) at th e prim ary school level. S he said the follow ing w ith regard to high school m athem atics:

Figure 5.20: M a to ti’s account o f learning school m athem atics

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A ccording M atoti, h ig h school m athem atics is about know ing “ form ulas” for various problem solving tasks w here the know ledge o f form ula is im portant in enabling one to get answ ers easily. T his m ay explain w hy M atoti obtained good results in school

m athem atics exam inations. It m ight be she w as good at m em orising and using form ulae fo r calculations.

D uring th e interview , M atoti w as asked to m ake com m ents on her answ ers to tasks 1, 2, 5, and 7. Responses to tasks 1 and 2 w ere m arked incom plete because M atoti had. not w ritten sym bolic answers and yet she had dem onstrated the operations on the diagram s accom panying the tasks. She w as asked to explain w hy she had not w ritten the answ er. C onsider the following excerpt:

M atoti: I ’m not sure b u t ... here w e are subtracting and by shading these circles it seem s that all the circles w ere 17. N o w I ’m going to subtract 5, then these 5 , 1 coloured differently from these ones because they have ...

R esearcher: D id you double shade? M atoti: Y es, I shaded them twice. R esearcher: So w hat is the answer?

M atoti: The answ er is that one w hich has the straight lines. R esearcher: Oh! W hat is the answer?

M atoti: T he answ er is 12.

D raw ing from the excerpt above it can b e said that M atoti knew the answ er although she did not w rite it dow n at the time. It m ight b e a m ere om ission in not w riting answ ers. She repeated th e sam e error on T ask 2. Like other participants, M atoti got a correct answ er for Task 5 using the algorithm . She then drew a full 280 circles. D uring the interview , M atoti was asked i f she could think o f another w ay o f

doing this task. She suggested a num ber line. She w as then asked to draw it. Consider F igure 5.22 below :

F ig u r e 5 .2 1 : M a t o t i ’s u s e o f n u m b e r lin e d u r i n g t h e i n t e r v i e w

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T his diagram presents an exam ple o f an em pty num ber line. M atoti m arked 200. She then added 70, + 9, + 1 to th e 200 dem onstrated by arrow s p o in tin g to the right. W hile th e answ er is obviously 280 M atoti did not w rite it on th e num ber line. She w rote other n um bers (0, 50, 1 0 0 ,1 5 0 , 250, and 300) on the n um ber line as indicators. This tim e around M atoti used the num ber line correctly for this operation (201 + 79), but still she did not w rite the answer. A fter taking som e m athem atics courses at the college, M atoti has deepened her understanding o f connections b etw een sym bolic representation (201 + 79) and m ore than one iconic representation (the array o f dots and a n um b er line). H er explanation o f w hat she does on the num ber line also confirm s M ato ti’s m athem atical understanding o f addition. C onsider the follow ing excerpt:

M atoti: It w ill be 200 + 70, and 270 + 9 is 279, plus 1 gives m e 280. R esearcher: W hy d id n ’t you say that 9 + 1 is ju s t 10?

M atoti: Y es, I can m ake it, 9 plus 1 is 10, then 1 will ...

R esearcher: Tell m e w hy d id n ’t you use num ber line instead o f these dots? M atoti: I did not know ...

R e s e a r c h e r : D i d y o n k n o w h o w t o u s e a n u m b e r l i n e t h e n ? M atoti: Before?

Researcher: Y es like, w hen y o u ’re doing this, did you know how to use a num ber line?

Matoti: No!

It follow s from th e excerpt above that M atoti in T ier 1 w as not fam iliar w ith how a num ber line w orks. This evidence shows that th ere is the po ssibility for som e learners to com plete high school m athem atics w ithout know ing how to u se a n u m ber line for addition an d subtraction o f w hole num bers. T his is surprising considering that the n um ber line has been a recom m ended representation for arithm etic operations in the Lesotho prim ary m athem atics syllabus, books 2 and 3 (1998).

W hen asked to com m ent o f her answ er to T ask 7, M atoti said she w as not sure about h e r answ er - “there are 18 handshakes” . She drew 9 people in circle holding hands, w hich m ay h ave lead to the answ er o f 18. D esp ite the fact th at M atoti had perform ed w ell in school m athem atics, she m ade a draw ing that could not lead her to finding the correct answ er to this Task. Perhaps this is because this task could not be solved follow ing an algorithm already know n to her. P articipants had to b e creative enough to b e able to construct self representations w ith m athem atical properties, w hich allow ed them to reason tow ards a solution. A t th e end o f year 1, w h en asked about m athem atical representations that she w ould com fortably use in h er class w hen on teaching practice (TP), M atoti m entioned th at she w ould u se gam es and other teaching aids that she could not m ention by nam e, despite m athem atics education

lectures in th e use o f m ultiple representations. T his indicates to m e that she m ay still understand m athem atics prim arily in term s o f procedures and form ulae.

T h a n d i’s C ase

A t th e beginning o f this study, Thandi is tw enty three years old. T handi is a w om an w ho resides in a tow nship situated about 20 kilom etres south o f M aseru. T h an d i’s background in m athem atics appears relatively strong. In Standard 7 term inal exam ination she obtained third class (3) in m athem atics. In Ju n io r C ertificate (JC) national exam inations she got an E symbol in m athem atics. In F orm E exam inations T handi obtained an E sym bol in her first sitting and a C sym bol in her second sittin g ', w hich is a credit. This suggests her am bition to im prove. In the follow ing extract Thandi tells a story about her experience o f learning school m athem atics: