education primary' in 2009 academ ic year. The students w ere in year 1 o f their three- year diplom a program m e.
O nly sixteen participants cam e from the highlands districts. O ut o f tw o hundred and tw elve participants only sixteen obtained a credit in m athem atics and over sixty p er cent o f them failed the m athem atics exam ination in Form E. This suggests that the m ajority o f the participants in this study did n o t m aster m athem atical skills and
techniques necessary for achieving either a pass or a credit. This evidence confirm s m y general observation over the years that the m ajority o f students w ho register for DEP program m e h av e failed m athem atics in Form E, and as such, they h av e w eak understanding and know ledge o f m athem atics. A typical C O S C m athem atics exam ination p aper set at the end o f Form E is characterised b y tasks, w hich call for procedures and recall o f and use o f algorithm s in a range o f topics. See A ppendix 7 for one sam ple page each, from m athem atics exam ination papers 1 & 2, C O SC 2009, as an indication o f th e assessm ent standards w hich students are expected to meet. It was im portant to obtain inform ation relating to participants perform ance in Form E as m athem atical proficiency is often considered to be a problem for a large percentage o f DEP candidates. T he inform ation also helped m e to have a better picture o f the level o f p articip an ts’ relationships w ith representations and m athem atics in general.
I w as also interested in establishing th e age o f all participants in m y sam ple because it w ould b e easy to infer from such inform ation w h eth er each participant jo in ed th e college (LCE) straight from high school or had been teaching for som e years before enrolling for D EP program m e. U sually students w ho are in their tw enties com e from high school w hile those in the late thirties and forties w ould b e in
possession o f a P rim ary Teachers C ertificate (PTC), w h ich used to b e offered by the
104
LCE in the past. So these older participants are in m ost cases experienced teachers who have com e back to the college to upgrade their teaching qualification w ith the diplom a (D EP) program m e.
T he F requency Table 4.1 below presents p articip ants’ ages in this study and show s that they range from eighteen to fifty. T he table show s that over 57% o f the participants are betw een ages tw enty-one and tw enty-five. This m eans a m ajority o f participants are in their early tw enties, w hich then suggests that m o st DEP candidates com e straight from high school.
T able 4,1: F requ en cy T able show ing A ges o f Participants
A g e F re q u e n c y P e r ce n t 20 and under 33 15.6 21 - 2 5 121 57.1 2 6 - 3 0 28 13.2 3 1 - 3 5 14 6.6 3 6 - 4 0 9 4.2 41 - 4 5 6 2.8 4 6 - 5 0 1 .5 Total 212 100.0
T hus p articip ants’ responses in Section 2 o f the survey instrum ent largely depend on their high school m athem atics. On the other hand, participants aged thirty-six and above are few and are likely to possess a teach er’s certificate. F o r th ese participants, responses to th e tasks in Section 2 o f the survey instrum ent arise from b o th their know ledge o f school m athem atics and their experience o f teaching prim ary school m athem atics.
In order to gather additional background inform ation about the participants in T ier 1 , 1 q ualitatively analysed their responses to tasks 8 and 9 o f Section 1 o f the
survey instrum ent. I read all tw o hundred and tw elve p articipants’ responses and as I w as reading and studying these I m ade note o f the com m on issues that participants w ere m entioning. I w rote a list o f em erging them es for each question. I counted the n u m ber o f occurrences o f each o f the themes. In order for a com m on issue to qualify as an acceptable them e it had to have been m entioned by at least tw en ty participants. T he counting w as n o t only necessary for identifying acceptable them es but also to enable m e to develop a deeper understanding o f participants’ experiences o f being m athem atics learners in Lesotho schools.
T able 4.2 show s that in general participants experienced m athem atics as a challenging venture (n = 92) and also as a school subject that requires effort and thinking from the learner (n - 8 5). These two them es are dom inant and therefore it could b e argued that in general, participants p erceive m athem atics as a difficult and co gnitively challenging subject requiring effort and a lot o f thinking from learners. It is w orth noting, b u t not surprising, that a considerable numberf?? = 56) o f participants hold a v iew that m athem atics as a body o f k now ledge involves learning operations, num bers, figures, sym bols, place value, and form ulae. T his finding is characteristic o f m any school learners in the U nited K ingdom and the U nited States o f A m erica
(Boaler, 2009).
T able 4.2 suggests that participants did n o t only experience difficu lty in
learning m athem atics because o f the dem ands o f m athem atics p e r se, b u t also because they experienced unpleasant relations with th e ir m athem atics teachers (n = 5 5 ).
C onsider for exam ple the follow ing story told by one o f the participants (P articipant No. 143) in th e follow ing figure 4.1.
Table 4.2: Themes emerging from data and their frequencies
Em erging Them es from Data F requency
M athem atics as enjoyable subject at prim ary but n o t in secondary schools
48
M athem atics as a challenging venture 92
M athem atics as problem solving activity 26
M athem atics as a subject that requires effort and thinking from learners 85 M athem atics as know ledge o f operations, num bers, figures, sym bols,
place value, and form ulae
56
R eferring positively to m athem atics teacher(s) 38
R eferring negatively to m athem atics teacher(s) 55
P ositive and negative experiences 34
Figure 4.1: A m elia ’s response to Q 8 (Section 1)
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M athem atics results o f this participant for all three national exam inations (Standard 7, F orm C, and Form E) bear witness to this story. A m elia, a fem ale aged 45, got a third class pass in m aths in Standard 7, obtained an E sym bol w hich is close to a fail in Form C and failed m athem atics in Form E. She m akes reference to her teacher w ho used to sm ack learners for failing to com prehend w hat rules and calculations m eant. A s a result, learners developed negative attitudes tow ards m athem atics. C orporal punishm ent is still rife in m any schools in Lesotho (de W et, 2007). D e W et (ibid.) conducted a study in L esotho w here she explored perceptions and experiences o f school learners w ith regard to school violence. To indicate the seriousness o f corporal punishm ent in schools in Lesotho, I m ake reference to some com m ents m ade by participating learners in de W et’s study. She cites the follow ing: “ Som e students are w hipped as if they are not hum an beings. T eachers do not like the students . .. ” (p. 681). It is unfortunate that this sort o f situation is not uncom m on despite the fact that the Lesotho Education A ct (2010) declares it illegal. The A ct states th at “ a learner shall not be subjected to cruel, inhum an and degrading pu n ishm ent” (p. 164). Perhaps teachers continue to hit learners because o f the B asotho culture that beating a m inor by an adult is view ed culturally as part o f
rein fo rcin g discipline in the hearts o f young ones. I am afraid it follow s that i f learners continue to be beaten in m athem atics lessons, it rem ains an em pty dream to think that B asotho learners w ill b y chance love and enjoy learning m athem atics and hence obtain good results in m athem atics.
In analysing item 9 (w hat kind o f training do y o u think w o u ld help y o u in order to teach m athem atics w ell at p rim a ry sc h o o ll), I follow ed the sam e p rocedure as in item 8. In response to this item, participants stated that they expect the program m e
(D EP) to equip them w ith know ledge and skills as follows. T able 4.3 below represents the issues they raised:
T able 4.3: P a rticip an ts’ expectations and frequencies
E m erging Them es from Data Frequency
Use teaching m aterials (e.g. pictures, figures, diagram s, exam ples, experim ents) to help pupils leam m ath
55
Explain m athem atics w ell to learners 30
D evelop u n derstanding o f m athem atics 68
G uide/teach pu pils how to pass m athem atics 23
L eam how pupils leam m athem atics easily 62
A ccording to T able 4:3, it seems reasonable to conclude that in general participants w ere expecting the program m e to em pow er them w ith skills and know ledge that w ould enable them to have a better understanding o f m athem atics (n = 68 ). T hey also h ave an am bition to leam how to m ake m athem atics teaching and learning easy for pupils (n = 6 2 ) . I f participants view m athem atics as a difficult subject, then learning how to teach it effectively and give m eaning to m athem atical ideas seem s to b e a key aim for th e ir courses. T hat is, the m athem atics courses offered at the college w ould have to devise m eans through w hich student teach ers’ attitudes could b e changed so that th ey could b e innovative and resourceful in learning effective w ays o f m aking m athem atics accessible to prim ary school learners. This is a v ery b ig challenge. I hope this study can help m athem atics teacher training. In w hat follow s, I focus on the analysis o f particip an ts’ responses to tasks in Section 2 o f th e survey instrum ent.
The seven m athem atics tasks in Section 2 can b e p u t into four categories. Tasks 1 - 3 are sim ilar. Each one is accom panied b y a potential iconic representation. Participants are expected to link the representation w ith their w orking out o f solutions. Task 4 (a - c) belongs to another category becau se here participants are offered som e iconic representations and expected to identify and m ention operations w hich th e y m ig h t represent. In the third category, task 5 involves a statem ent in sym bols, m ost often solved b y use o f the standard algorithm for addition. Fourth category tasks 6 and 7 are such that participants w ere asked to arrive at solutions to word problem s and show how they did so. Item 6 asks specifically for a
representation the student teacher m ight use to explain to a child how th e solution was achieved. In w hat follow s, I present analysis o f particip an ts’ responses to tasks in term s o f highest and low est facility and then I focus on each o f the above-m entioned categories.
T able 4.4: R ank ing T ier 1 tasks according to the high est and low est facility
Overall Responses to Tasks in Section 2
H ighest facility T ask description L ow est facility T ask description
T ask 6 (n — 204) 9 6 .2 % correct B uying exercise books (division) T ask 7 (n = 8) 3 .8 % correct H andshake problem T ask 5 ( n = 197) 9 2 .9 % correct A ddition o f a 2 and a 3 digit num ber
T ask 4(b)(n = 78) 3 6 .7 % correct L inear m ultiplication {string o f beads) T ask 2(a) (n = 192) 9 0 .6 % correct A ddition on a num ber line
T ask 1(a) (n = 127) 5 9 .9 % correct Subtraction w ith counters T ask 1(b) < n = 188) 8 8 .7 % correct A ddition with counters T ask 4(a) ( n = 156) 7 3 .5 % correct M ultiplication array {no statem en t in sym bols)
T ask 2(b) <n - 179) 8 4 .4 % correct
Subtraction on a num ber line
T ask 3 (n = 164) 7 7 .3 % correct
M ultiplication array {statem ent in sym bols) T ask 4(c) (n = 177) 8 3 .5% correct M ultiplication as a collection o f discrete sets
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T he above tab le provides a summary o f item s and inform ation on how
participants perform ed on each. The ranking o f th e tasks is according to th e num ber o f participants w ho gave com pletely correct answ ers to each item. It appears that the great m ajority o f participants found m ost o f the tasks quite easy w hile alm ost all students found T ask 7 quite challenging. The table show s that participants found Task 6 easiest o f all. 96.2% o f participants gave a fully correct answ er to this task. Item 6 involves the context o f buying exercise books and it is a routine textbook type task w ith Tow level cognitive dem and’ (Stein, G rover, and H enningsen, 1996).
P articipants w ere at liberty to suggest the representation o f th eir choice that w ould help them to explain to a child how the answ er w as obtained. T he task m ight be perform ed b y division (4 )4 4 to obtain the answ er o f 11 exercise books) or, by repeated subtraction ( 4 4 - 4 - 4 - 4 - 4 - 4 — 4 - 4 - 4 - 4 - 4 - 4 ) o r , as m any participants opted to represent the problem , b y repeated addition (4+4+ 4+4+ 4+4 +4+ 4+4+ 4+4 = 44). It should not be a surprise that respondents perform ed well on the sym bolic p art o f this w ord problem because it is very sim ilar to a question that is found in th e tex tbook for Standard 4 (N C D C , 1997, p. 23). H ow ever, the p art o f task 6 that asks for use o f a representation to explain the operation to a child requires PCK and m ight b e expected to have been challenging for respondents. Y et they nearly all could do it correctly.
T ask 5 w as found to be the task w ith the second highest facility. 92.9% o f participants gave fully correct answers. It involves addition o f a 2 and a 3-digit num ber, an op eration that calls for autom atic recall o f facts and th e u se o f a standard algorithm . In this item participants w ere only asked to show how th ey w orked out the answ er and m o st used the predictable, vertical approach.
T he task w ith low est facility (task 7) yet having a ‘high level o f cognitive d em an d 5 (Stein et al, 2000) appeared to be inaccessible to the great m ajority o f participants. O nly 3.8% o f respondents offered a correct answ er to this task
(handshake problem ). It is a non-routine m athem atical task that participants w ould not be expected to have m et before and it could n o t be solved w ith a standard algorithm .
T ask 4(b) is the task w ith second low est facility w here 36.7% o f participants gave a fully correct answer. The task involves linear m ultiplication represented by m eans o f a string o f beads. In this task participants w ere challenged to think o f a suitable sym bolic representation. This diagram could b e said to represent
m ultiplication (6 x 6 = 36) or division (36 6). Som e participants offered repeated addition as an interpretation, w hich was acceptable.
There w as an interesting disparity betw een the num bers o f participants who gave fully correct answ ers to item s 1(a) and (b) and 2(a) and (b). This w arrants closer study. T ask 2(a) is the item w ith the third highest facility w ith 90.6% o f respondents offering correct solutions. This task involves addition as counting all o r counting on from the first addend on a num ber line, a task w ith w hich m ost respondents appeared fam iliar. T h e item w ith fourth highest facility is task 1(b) w ith 88.7% correct
solutions. T his task involves an addition sum to be linked w ith a representation o f counters arranged in a static structure o f two groups o f 5 b y 2 (see figure 3.2, p. 84). W hile all participants m ight be expected to b e fam iliar w ith the use o f counters such as stones and sticks, it is noted here that they found it som ew hat easier to dem onstrate