2 01 +
79
w hich m akes all o f this to b e 201. Researcher: H ow w ould you do it now? Tseli: I think I w ould use a num ber line.
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W hen asked to explain the m eaning o f the arrays that she had draw n, Tseli provided a clear explanation saying “actually I get it and then getting th e answ er, after that I ju s t confirm by; I ju st confirm by ... w hether I w rote it correctly or som ething,
Tseli suggested the use o f a num ber line an alternative representation. In order to dem onstrate to m e how she could use a n u m ber line to do this operation (201 + 79), she drew the follow ing num ber line:
F ig u r e 5 .1 5 : T S e li’s r e s p o n s e to T a s k 5 d u r i n g t h e i n t e r v i e w
E xplaining h er use o f the num ber line appeared to challenge T seli and the follow ing excerpt provides evidence o f that:
Tseli: U h . . , h a . .. h a ... h a ... {laughs), O K, I ’m ... now , now I w o n ’t do it; I w o n ’t do a num ber line. I think it is giving m e a little ... or som ething, really I cannot dem onstrate it to young children. T seli had com fortably used the algorithm to add a tw o-digit nu m b er ( 7 9 ) to a three- digit num ber (201) and obtained the correct answ er (280). S he w as then able to dem onstrate th e operation by m eans o f an array, draw ing 280 individual circles w hich m ade full sense but w as a bit unwieldy. W hen Tseli resorted to the use o f a num ber line she h ad d ifficu lty in convincingly explaining how her diagram w as show ing the operation and the answ er correctly. H ow ever, it is interesting th at T seli could reason successfully and flexibly on th e num ber line but had difficulty in articulating her
160
ac? a A •" 1 -
thinking. She appeared to add 80 in jum ps o f tw enty starting at tw o hundred, to w hich she had also added a single hop, labelled “+1” . H er total w as th en correctly, two hundred and eighty, but o f course she had only needed to add seventy-nine so she also show ed one hop to the left from 280 and labelled it 1” . T his w as a fair attem pt to reconcile (200+1 )+(80 - 1 ) but Tseli appeared to lack confidence in articulating it fully and had learners in m ind as she explained that she w ould not feel com petent to use a n um ber line in a m athem atics lesson. It could be concluded from h er w ork that although Tseli drew a variety o f diagram s successfully, she appeared to lack
confidence in explaining how she used them to confirm th e answ er w hich she was already certain of, through follow ing the algorithm .
T se li’s attem pt to get a solution for Task 7 (hand-shake problem ) was less than successful. She m ultiplied 9 by 8 and did not attem pt an iconic representation. In contrast to this, w hen asked about the use o f representations w hen teaching
m athem atics Tseli said that it is im portant to use them because learners can see and touch them . The follow ing excerpt presents the conversation held w ith Tseli:
T seli: I think, I think I love them, I think they are very interesting and they are very good for the young brains because th e fact that they can be seen and be touched ... W e m ake m athem atics real to children ... R esearcher: O K and w e m ake them learn m athem atics from things they know ? Tseli: W e teach them w hat they d o n ’t know through w h at they know , I
th in k it’s interesting.
Tseli: So far alm ost, I think it’s a m ultiplication square, especially because m ultiplication can be ...
It is w orth noting th at Tseli argues that through fam iliar representations learners could be taught new m athem atics concepts. She m entions th e use o f a m u ltiplication square. This show s that at the end o f Y ear 1 o f study, participants h ad learned som e other representations that could be used to enhance teaching and learning o f m athem atics other than stones frequently m entioned by people in Lesotho w ho h av e no t undertaken a teacher training course. However, know ing o f these representations is n o t enough, the teacher needs to b e able to be able to use them w ith confidence and ease and be able to com m unicate their thinking processes as they do so.
T h r e e C re d it S t u d e n t s