Mathematics Teaching Via the Lesson Study Model

Anass Bayaga

School of General and Continuing Education, Faculty of Education,University of Fort Hare, South Africa

E-mail: abayaga@ufh.ac.za

KEYWORDS Assessment. Instructional Strategies. Introductory Activities. Pedagogy. Teaching Method

ABSTRACT The paper presented empirical work related to effective teaching of mathematics in order to determine majorissues of importance for future research and to understand the issues in relation to theory and application of Lesson StudyModel (LSM) in South Africa context. The study applied two- phased sequential mixed methods. In the first phase, analysis ofMANOVA and repeated-measures ANOVA were done to investigate whether there was a significant difference or not betweengroups in respect of experimental and control groups. The two main findings included (a) LSM was a better predictor ofimproving mathematics teaching and (b) distinct views on LSM could be identified by the mathematics teachers in the processof using LSM. One of the implications from the study was that LSM could be accepted as a turning point in developing themetacognitive skills, emphasising the reflective teaching and learning and providing internal consistency of instructional planning.

INTRODUCTION

Both past and present research (Ross andBruce 2005; Takahashi et al. 2006; Takahashi2007) in mathematics have lamented over ef-fective knowledge of mathematics both in teach-ing and learning. The authors suggest that math-ematics education research have be taken forgranted that effective mathematics teaching re-quires sound content knowledge as well asknowledge of the process of pedagogy; suggest-ing that teaching and learning of mathematicsare confined to both transmission (teaching) andacquisition (learning) of mathematical concepts.The view that lack of understanding of trans-mission and acquisition of mathematical con-cepts was further noted by writers (Kim andBaylor 2007; Takahashi 2007), who implied thatthe argument about the interphase betweentransmission and acquisition of mathematicalconcepts is a major problem in that most poornational results in mathematics emanates fromlack of attention from above. This concern isparticular in South African mathematics edu-cation (Takahashi 2007).

The concern points to one main issue, thus,the understanding that effective pedagogy re-quires both extensive content knowledge as wellas pedagogical knowledge from teachers. Con-sistent with the above, review of South Africanliterature suggests that there is still a disjunc-ture between extensive content knowledge aswell as pedagogical knowledge (Takahashi2007). Other international literature (Kajanderand Lovric 2005) has raised similar concerns,

suggesting that the concern appears to be a glo-bal concern.

It was in this vein that the Japanese devel-oped a remedial model to better understandtransmission and acquisition of mathematicalconcepts, thus improving mathematics teachingand learning. This model was called the lessonstudy model. As suggested by the lesson studymodel literature (Chokshi and Fernandez 2004)the model has been a success story both in Ja-pan and United States of America. The authorssuggested that LSM is a teacher directed pro-fessional development model developed in Ja-pan. It is grounded in research and not like tra-ditional lesson planning. Hence, the lesson pro-cess begins by teachers and other stakeholderslooking critically at their mathematics curricu-lum in a manner that alludes to continues pro-fessional development in mathematics.

Thus teachers and stakeholders need to lookat their curriculum in their schools in relationto what they know about their student’s learn-ing. This suggests a two- dimensional view in-volving: (a) looking at both what learners arehaving troubles learning and at (b) how theseconcepts are being taught; implying that pro-spective teachers, as well as teachers, collabo-ratively engage in meaningful dialogues aboutanswers to such views. It is in this connectionthat this study adopts and adapts LSM to applyin South African context.

Development of Lesson Study Model

It is imperative to note that in applying thismodel of lesson study in South African context,

© Kamla-Raj 2013 Int J Edu Sci, 5(1): 11-18 (2013)

the study retains essential features of the Japa-nese model, making necessary changes to adaptto the contexts and purposes of South Africahigher education. Thus, whether in Japan or theSouth Africa, lesson study involves a small teamof instructors (researcher and the candidateteachers), working together to design, teach,study, and refine a single class lesson. This workculminates in at least four tangible products; thefirst including a detailed usable lesson plan,second is an in-depth study of the lesson thatinvestigates teaching and learning interactions,third is explaining how students respond to in-struction, and fourth is how instruction mightbe further modified based on the evidence col-lected. In this direction, the process adopted andadapted LSM by (a) formulating learning goalsor objectives (b) designing the research lesson(c) designing the study (d) teaching and observ-ing the research lesson (e) analysing the evi-dence (f) repeating the process(g) documentingthe lesson study.

Noting that aspects of lesson study resembleother teaching improvement strategies such asbackward design and classroom assessment, acloser look at how the lesson study process playsout in higher education, however, reveals im-portant differences with other teaching improve-ment activities in the South Africa. The differ-ence suggests that to implement LSM, the re-searcher(s) ought to defining the problem, planthe lesson, teaching and evaluating the lessonand reflect on its effects and finally revise thelesson. This process implies that lesson study isone component of a system designed for con-tinual professional development.

In Japan, lesson study is either done by teach-ers across a district, or by teachers within aschool. But in this study, both the researcherand the candidate teachers collaboratively con-ducted the study. The reason was to have imme-diate discursive views from participants who arerequired to be implementing the model aftercompletion of the course (cf. Method). However,the topic for the lesson study was unanimouslychosen by both the teachers and the candidateteacher, but was linked to larger national goals(National Curriculum Statement-NCS). The les-son study model focuses on one of the area, thusmathematics professional development of teach-ers. This is one version of an “inquiry groupstudies” as a way to improve mathematics teach-ing. In this specific study, as part of a goal toimprove candidate teacher’s problem-solving,

the study worked on a lesson study topic of sub-traction with. Teachers, (intermediate phase),met weekly (block sessions) to design, teach andevaluate research lesson. The next step involvedto revise the lesson, re-teach it, evaluate, reflecton the lesson again, and share results, this pro-cess took up to a year for each group involvedin the study. Thus, the first group (experiemen-tal-2008 cohort) used LSM process for one aca-demic year as part of the mathematics lectures.The 2009 cohort (controlled) used the normallesson plan without applying any aspect of LSM.

Takahashi (2007) noted that lesson studyempowers individual teachers and leads tosteady incremental improvement in teaching,rather than fast reform which is often the un-achieved goal of South African approaches tochange. Takahashi (2007) summarize lessonstudy through an eight step problem-solvingprocess, although others divide this process dif-ferently.

The next section of the paper describes themotivation for the propositions (hypotheses) inrespect of the theory underlying the LSM.

Motivation for Propositions

Until recently, the LSM perspective has beenpromoted largely within disciplinary boundariesand in isolation from each other, though re-searchers have seriously addressed the scope formore integration (Takahashi 2007). Some re-search (Kajander et al. 2005) showed that math-ematics lecturers do not use the same defini-tions of content knowledge as well as knowl-edge of the process of pedagogy. When teach-ing, in particular mathematics, South Africaneducators focus on a structured lesson plan,whereas the Japanese concern themselves withLSM which incorporated a number of additionalsuch (a) brief description of classroom context(b) materials/resources, (c) instructional objec-tives (d) introductory activities (e) instructionalstrategies/student activities (f) closure (g) assess-ment (h) duration and (i) alignment/consistency,noting that most of these are components of alesson plan.

But LSM have been very influential not onlyin Japan but also in the USA. The reason beingthat the model has made an important contri-bution to the understanding of mathematicsteaching across the Asian pacific and the West(Chokshi and Fernandez 2004; Fernandez andYoshida 2004). Thus, the hypothesis is that the

ANASS BAYAGA12

most efficient and effective student learning ofmathematics would result when classroom in-struction and materials are aligned with lessonstudy model. Following the above hypothesis,research on curriculum alignment in Japan andthe US lately tends to favor LSM as a positiveinfluence on achievement of Mathematics(Fernandez and Yoshida 2004).

When the literature was examined in SA,there were a limited number of publicationsabout the LSM, although there were severalpublications about the curriculum studies.While, a few of these curriculum studies are re-search (Reddy et al. 2007), most of them areabout the description and discussions about thecurriculum development, teacher education andmultigrade teaching. Although, the introductionof the LSM to Japanese education was quiteparallel to its development and application inthe United States, it is taking some time for it tobe implemented in practice and in theory inSouth Africa.

Following the lack of attention given to LSMin South African schools, this study used meth-odologies derived from LSM to study the effec-tive teaching of mathematics in a historicallyBlack South African University students pursu-ing a Postgraduate Certificate of Education(PGCE) Programme. The paper presents em-pirical work related to the scholarly search foreffective teaching of mathematics in order todetermine major issues of importance for futureresearch and to understand the issues in rela-tion to theory and application of LSM in SouthAfrica. This study could therefore contribute tothe South African literature on mathematicseducation, since there are few experimental stud-ies related to the LSM. Moreover, the findingsof this research might start to guide the attemptsto develop teaching at pre/in-service teachereducation by using the LSM.

Research Hypotheses

Following the assertion that application ofLSM improves effective teaching and learningof mathematics. The following have beenhypothesised:

Hypothesis 1

Ho=There is no significant difference in us-ing LSM in the teaching of mathematics

Ha= There is significant difference in usingLSM in the teaching of mathematics

Hypothesis 2

Ho = LSM is not a better predictor of im-proving mathematics teaching

Ha= LSM is a better predictor of improvingmathematics teaching

Hypothesis 3

Ho= Usage of LSM by teachers is not evalu-ated differently by different teachers

Ha = Usage of LSM by teachers is evaluateddifferently by different teachers

Hypothesis 4

Ho = Distinct views on LSM could not beidentified by teachers

Ha = Distinct views on LSM could be identi-fied by teachers

METHODOLOGY

The purpose of this two- phased sequentialmixed methods (Creswell 2003) study was toobtain statistical results (phase I) from a sample(43) of Post Graduate Certificate of Education(PGCE) students over a period of two years. Inthe second phase, interviews were done in or-der to explore different aspects of planning viathe LSM. The sample consisted of 43 cohorts ofthe PGCE students in a mathematics class (412Eand 422E) over four semesters in University ofFort Hare in the greater Eastern Cape Provinceof South Africa. Thus, the sample was made ofcandidate teachers preparing to teach the inter-mediate phase (4-6 grades) of the South Africaschooling system. Suggesting that a replicationof this study should take cognisance of the co-hort of students in terms of (a) duration ofprogramme and (b) purpose of the programmeprescribed by the South African National Cur-riculum Statement. This was because PGCEprogramme was designed for graduates ofprogrammes other than a degree in the field ofeducation.

These were then followed up with few re-spondents (13) to explore those results in depth.Thus, in the first phase, inferential research hy-potheses (cf. research hypotheses) was con-

MATHEMATICS TEACHING VIA THE LESSON STUDY MODEL 13

structed in order to compare the effects of usingthe LSM constructs on a teachers ability to im-prove effective mathematics teaching.

In this study, the experimental group was the2008 cohort (22 candidates), while the 2009 (21candidates) were the controlled group. In theexperiential group, the candidates were exposedto teaching using LSM, while the controlledwere those using the lesson plan (LP) method.The mean scores of exams of the groups werecompared by the researcher. It was hypothesisedthat there would be significant difference be-tween those candidate teachers using the LSMand those using LP. Another recommendationthereof is that candidate researchers may usethe same cohort as experimental and controlledas opposed to that used in this study.

Procedure of LSM

After getting information about the LSM toapply to the experimental group, next, the fea-tures that incorporated LSM and LP were used.Thus examples in the literature were searchedand examined. A set of criteria that was used toassess teaching of mathematics was determined.The LSM characteristics had nine components:(a) brief description of classroom context, (b)materials/resources (c) instructional objectives(d) introductory activities (e) instructional strat-egies/student activities (f) closure (g) assess-ment/re-evaluation (h) duration and (i) align-ment/consistency were explained in the infer-ential analysis.

Thus, after giving some exercises contain-ing fifteen (15) questions, candidate teacherswere asked to match objectives and the cogni-tive categories. They were also asked to write atleast one question related to every sub-categoryin the cognitive process dimension. Then, theknowledge categories were explained, they wereasked to give two factual, conceptual, and pro-cedural knowledge examples from their topicarea. Since, metacognitive knowledge was newfor them, the instruction continued at a slowpace and every single sub-category was ex-plained thoroughly. After explanations and dis-cussions about these sub-categories, various ac-tivities and discussion were undertaken.

In the control group, the course was in-structed in a traditional way with the methodsof lecture, question-answer, and discussion andcandidate teachers were given traditional infor-

mation about how to prepare lesson plans toteach. In this regard, the research (lecturer) firstasked them to examine lesson plans and gavethem feedback about the lesson plans they hadprepared.

Details of Sampling

The study assigned one of the year groups(2008 cohort) as the experimental group (n =22) and the other (2009) as the control group (n= 21). The mean ages of the PGCE candidateteachers in the experimental and the controlgroup were 21 years and 22 years respectively.Ethnically, while 4% of them made up Whites,the rest (96%) was Black students (Coloured andBlack Africans). The analysis of repeatedMANOVA and univariate repeated ANOVAwere done to investigate whether there was asignificant difference or not between groups inrespect to components in the LSM. In order toexamine the lesson of the experimental groupfrequency and percentage were calculated. Inorder to test inter-rater reliability of the scoresobtained from the experts who assessed the les-son plans, ANOVA test, which was done hadintraclass correlation coefficient based onSpearman-Brown formula as 0.94, suggest areasonably strong reliability.

RESULTS AND DISCUSSION

This section addresses the results of the studyin light of the hypotheses posed. It consists ofsix sub-sections. The first includes test of as-sumptions of repeated-measures ANOVA andMANOVA (Tabachnick et al. 2001; Hosmer andLemeshow 2000) and preliminary analysis ofthe groups. Although, the test of Mauchly’ssphericity was as explained it is imperative tonote that subsequent test have been conductedand explained as the entire sections developed.

Test of Assumptions: Mauchly’s Sphericity

Mauchly’s sphericity1 test for first hypoth-esis was conducted to examine the form of thecommon covariance matrix. The results revealedthat sphericity assumption was not violated (X2

= 32.5; df 3; p=0.45). Thus, the chi-square ap-proximation for this test was 32.5 with 3df andan associated probability greater than 0.05.Since, this was greater than the alpha level of

ANASS BAYAGA14

0.05, the study was confident that the data didmeet the sphericity assumption.

Preliminary Analysis

Firstly, a preliminary analysis to determineif there was any statistically significant differ-ence between two groups was conducted in termsof their performance in their exams (cf. proce-dure of LSM). The results revealed that, themean scores of the experimental group(M=238.10, SD = 7.00) and that of control groupis 238.18, SD = 9.90) was not significant. Thusfindings of the independent sample t-test [t(46)= 0.06, p>0.05] was not statistically significant.The results suggested that the candidate teach-ers in the experimental and control groups aresimilar. Secondly, the researcher administereda pretest of instructional planning and evalua-tion course content via LSM to the groups todetermine comparable levels of understandingof the content prior to the experiment. Notingthat KR-20 reliability coefficient of the pretestwas 0.86. The scores that were obtained fromthe pretest were examined by using independentsamples t-test in order to determine if there wasany statistically significant difference betweenthe two groups. Again the results suggested thatthe mean of the pretest scores of the experimen-tal group was (M=39.6, SD = 5.34) and that ofthe control group (40.24, SD = 6.00) was insig-nificant. The t-test, which was done with themeans of the pretest scores [t(46) = 0.56, p>0.05]was not statistically significant. Suggesting thatthe experimental and control groups are not dif-ferent in respect to the pretest that was appliedat the beginning of the semester of each year.

Hypothesis 1

This section sought to examine the null hy-pothesis that there was no significant differencein using LSM in the teaching of mathematics.A comparison of planning skills of the groupswas conducted in order to investigate any sig-nificant difference in the LSM of the teachersin the control and experimental group in respectto components of LSM.

Multivariate analyses of variance(MANOVA) were done. The F ratio forMANOVA indicated that the differences be-tween the two groups mean scores were statisti-cally significant at the 0.05 level, [F (9, 44) =

5.62, p<0.05]. That is, the experimental andcontrol groups had statistically significant meanscores on the collective dependent variables (cfprocedure of LSM). The multivariate eta squaredof 0.65 (based on Wilks lambda) implied thatthe magnitude of the difference between thegroups was not small (Cohen 1988). That valueindicated that 65% of multivariate variance ofthe dependent variables was associated with thetreatment (LSM). Because a statistically signifi-cant MANOVA F was obtained for the collec-tive dependent variables, univariate ANOVAwas conducted to further understand how thetwo groups would be affected by the interven-tions regarding each of the dependent variables.

The results showed that a statistically sig-nificant mean difference existed between thegroups with respect to closure, assessment, re-teaching and alignment/ consistency (p< 0.05).Additionally, a one-way between-groups analy-sis of variance was conducted to explore the im-pact of age on levels of LSM. Subjects were di-vided into three groups according to their age(group 1:20 or less; Group 2:21 to 25; Group3:26 and above). There was a statistically sig-nificant difference at the p<0.05 level for thethree age groups [F (3,432) =4.53, p < 0.05].Despite reaching statistical significance, theactual difference in mean scores between thegroups was small. The effect size, calculatedusing eta square, was 0.2. Post-hoc comparisonsusing the Tukey HSD test indicated that themean score for group 1 (M=20.14, SD=4.67)was significantly different from Group 3(M=21.36, SD=4.35). Group 2 (M=21.26,SD=4.30) did not differ significantly from ei-ther group 3 as evidenced in the indexes.

Candidate teachers’ opinions about the plan-ning with the LSM stated that the LSM hadpositive effects on their study of mathematics.While only two teachers pointed out that theLSM created some difficulties while teaching,others mentioned the contributions of the LSM:The researcher on the other hand notes that theLSM was very useful in planning even thoughusing the LSM seems to be complex and diffi-cult. A respondent2 (mama3) noted that:

…while writing the objectives by using theLP was easier; I am having difficulty placingobjectives by using the LSM.

When teachers were asked the tasks they en-joyed or had most difficulty with the LSM, theystated that the most enjoyed task was to fill in

MATHEMATICS TEACHING VIA THE LESSON STUDY MODEL 15

the LSM. Even though they pointed out that theyhad some difficulty in filling the LSM as part ofthe requirement, they stressed that they reallyenjoyed filling in the LSM collaboratively andthey got excited as if they were curiously solv-ing a puzzle. Buyambo (a respondent) capturethis as saying:

It was delightful and stimulating. Complet-ing the LSM by thinking about and analysingour own knowledge each time was more amus-ing than making something by using memory.Placing every single objective into the LSM andwriting activities and assessments for them werelike solving a puzzle, I enjoyed that a lot.

Later, participants come back to objectivesand showed three elements all together, whichenable them to think more deeply. The evidencesuggested most difficult task in planning was toseparate factual knowledge (fk) from concep-tual knowledge (ck) and to understand meta-cognitive knowledge (mk). Metacognitive know-ledge was a little harder to understand, since itwas a new concept that was not taught in anyother course before. The lecturer (researcher)stated that since they were studying in the fieldof mathematics education, lessons needed tocontain objectives about applying proceduralknowledge. Moreover, they pointed out that they(participants) rarely used those objectives con-taining metacognitive knowledge, since theywere having some trouble understanding thistype of knowledge.

When the teachers’ answers are examined,it can be said that all of them had difficulty infilling in the LSM. However, it could not beconcluded this process had negative inferences,since this process was interesting and enjoyableat the same time. According to teachers, the mosttime consuming task was to determine the placeof objectives in the dimension of knowledge andcognitive process.

Conclusively, as evidenced by both the in-ferential analysis together with the empiricalopinions from the respondents, the study confi-dently rejects the null hypothesis and concludesthat there was significant difference in usingLSM in the teaching of mathematics. The nextsection examined the second hypothesis.

Hypothesis 2

This second sub-section addressed the nullhypothesis that LSM was not a better predictorof improving mathematics teaching.

A one-way between-groups MANOVA wasperformed to investigate LSM differences in de-pendent variables. Three dependent variableswere used: (a) brief description of classroomcontext-BDC (b) materials/resources-MR (c)instructional objectives IO. The independentvariable was LSM. Thus, if there is a predictionbetween the dependent variables and LSM thenLSM influence effective mathematics teaching.Preliminary assumption testing was conductedto check for normality, linearity, univariate andmultivariate outliers, homogeneity of variance-covariance matrices, and multicollinearity, withno violations noted.

There was a statistically significant differ-ence between experimental and control on thecombined dependent variables: F(3, 413) =2.02,p=0.02; Wilk’s Lamda = 0.87; partial etasquared = 0.3. When the results for the depen-dent variables were considered separately, theonly difference to reach statistical significance,using a Bonferroni adjusted alpha level of 0.014,was BDC: F(1, 421)= 6.51, p=0.004, partial etasquared = 0.4. An inspection of the mean scoresindicated that experimental reported higher lev-els of BDC (M=34, SD=7.9) than control(M=21, SD= 5.2).

Because the study was conducted with twodifferent groups additional analysis was con-ducted to determine the difference with regardsto time of the treatment (Intervention). A one-way repeated measures ANOVA was conductedto compare scores on the confidence in copingwith mathematics test at Time 1 (prior to theintervention) (M=17.00, SD=4.21), Time 2 (fol-lowing the intervention) (M=23.01, SD=4.42)and Time 3(three-month follow-up) (29.13,SD=4.20). The means and standard deviationssuggest significant results. There was a signifi-cant effect for time [Wilks’ Lamda=0.33, F (2,24) =38.10, p<0.05, multivariate partial etasquare = 0.85].

As evidenced above, the inferential analysissupports the alternate hypothesis while reject-ing the null. The study therefore concludes thatLSM is a better predictor of improving math-ematics teaching. The next section examined thethird hypothesis.

Hypothesis 3

The third hypothesis to be addressed washypothesis 3. The section on hypothesis three

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(3) suggested that the usage of LSM by teacherswas not evaluated differently by different teach-ers. The first multivariate test of a within-sub-jects effect was the within-subjects main effecttest. It examined changes in evaluation rate ofLSM as a function of mathematics teaching. Thenull hypothesis was that the mean evaluationrate does change across respondents.

The results was significant, since the F ratiofor this hypothesis4 was very large [F (2, 141) =2431.1, p = .0001], the study confidently rejectedthe null hypothesis and conclude that the evalu-ation rate changes in the population from whichthe sample was drawn.

Hypothesis 4

The main hypothesis (Ho) tested was thatdistinct views on LSM could not be identifiedby teachers using LSM.

The mean view of the one-way repeated-mea-sures ANOVA was 3 strongly support on the firsttest; 5 strongly support on the second test; and14 strongly support on the third and final test(for experimental group-22 sampled). TheANOVA shows that these views are significantlydifferent, F (2, 17) = 32.11 p<0.0005. Repeated-measures t-tests showed that subjects were sig-nificantly slower on the first test than they wereon the second (test 1 versus test 2: t (6) = 5.32,p<0.001; but there was further increase incompletion mean-view between the second andthird tests (test 2 versus test 3: t (6) = 5.51, p =.001, significant).

It appears that practice produces an initialrapid improvement in subjects’ view of perform-ing a task, slows it, but then additional practiceleads to increase and or further improvement.

Note that before the hypothesis 4 was con-ducted, an assumption was tested. The follow-ing section gave the brief results of a test to sat-isfy one of the requirements for doing repeated-measures ANOVA-called “Sphericity Assump-tion” in this particular case.

Recall, the sphericity assumption is that thevariances of variables are equal – it is the equiva-lent of the homogeneity of variance assumptionapplied in the “between subjects case” (cf.Tabachnick et al. 2001 for details), noting thatif the test produces a significant result, the sphe-ricity assumption has been violated. This meansthe p-value for the test of the within-subjectsfactor needs to be adjusted, thus the p associ-ated with the Huyn-Feldt correction. In this par-ticular case, the Mauchly Sphericity test was notsignificant (p = 0.169, which is greater than .05),so no violation of assumption.

As seen from Table 1, the sphericity assump-tion was satisfied for these data. In this case,there was a highly significant effect of the“meanview” variable; in other words, there is asignificant difference between the three tests interms of the average views taken to completethe task (p<0.0005).

According to the responses, although theyfound LSM a little confusing at the beginning,they started to enjoy preparing their teaching(re-teach) because of the examples given at thecourse and the discussions. This, as a respon-dent (Job) captured, stated that:

When I first started LSM, I was afraid of be-ing unsuccessful since it seemed confusing.However, I understood that the task was not thatdifficult. Moreover, I realised that preparingLSM would not take much effort, since deter-mining activities and assessments were mucheasier after completing the objectives. Prepar-ing LSM gave me opportunity. Thus, to preparemore detailed, re-teach and to think deeply. Ibelieve that I can prepare clearer, more under-standable, and more validly.

On the other hand and consistent with therespondents, participants opinions about plan-ning with the LSM was conclusive on one thing.The participants pointed out that teachers werecurious and interested in preparing LSM. Thestudy observed high motivation towards mak-ing teaching suggesting that this motivation was

Table 1: Test of assumption

Within subjects Mauchly’s Approx. χ2 df Sig. vEpsiloa

Green house Huynh- Lower-e-Geisser Feldt bound

Test-mean views 0.601 4.131 2 0.168 0.662 0.701 0.519

MATHEMATICS TEACHING VIA THE LESSON STUDY MODEL 17

a= may be used to adjust the degrees of freedom for the average tests of significance. Corrected are displayed in the tests of within-subjects effects table, which was explained.

caused by the structure of the LSM since they(respondent) knew that they were studyingsomething model, which was not observed inthe controlled group.

In general the findings of the study suggestthat planning with the LSM is effective and joy-ful even though it is more time consuming andrequires more effort. It could be said that bothresearcher and respondents had a positive atti-tude towards planning mathematics teachingwith the LSM.

CONCLUSION

Based on the analyses in section three (3)together with the empirical evidence from re-spondents, the results have been consistent withthe success of previous writers.

Two main findings were distinct and em-erged, thus; (a) LSM would be a better predic-tor of improving mathematics teaching and (b)distinct views on LSM could be identified bythe mathematics teachers in the process of us-ing LSM. The results of this study suggestedthat it is harmonious with and confirms thosestudies discussing potential benefits of planningof mathematics teaching with the LSM. It couldbe said that there would be several improvementsin mathematics curricular development by theapplication of the LSM in South African math-ematics education.

RECOMMENDATIONS

The implication from the study was thatfirstly; the LSM could be accepted as a turningpoint in developing the metacognitive skills,emphasising the reflective teaching and learn-ing, and providing internal consistency of in-structional planning. The LSM may provide aframework within which prospective teachers

as well as teachers could model not only theway they teach, but also the way they examineand analyse their teaching.

NOTES

1 For practical purposes, this is important only in helpingone to decide which output to use, and if the output shouldbe adjusted. If one can use the univariate output, onemay have more power to reject the null hypothesis infavour of the alternative hypothesis. However, theunivariate approach is appropriate only when thesphericity assumption is not violated. If the sphericityassumption is violated (where p<0.05), then in mostsituations its better off staying with the multivariateoutput.

2 Names are pseudonym3 Pen name4 Most calculation report a separate multivariate test

statistic (Pillais’, Hotelling’s, Wilks’, and Roy’s); but theWilk’s test is commonly used.

REFERENCES

Chokshi S, Fernandez C 2004. Challenges to importingJapanese lesson study: Concerns, misconceptions, andnuances. Phi Delta Kappan, 4(3): 520-525.

Hosmer DW, Lemeshow S 2000. Applied Logistic Regression.US: Wiley-Interscience.

Kajander A, Lovric M 2005. Transition from secondary totertiary mathematics, McMaster University experience.International Journal of Mathematics Education inScience and Technology, 6(2-3): 149-160.

Kim Y, Baylor AL 2007. Pedagogical agents as social modelsto influence learner attitudes. Educational Technology,47(1): 23-28.

Reddy V, Kanjee A, Diedericks G, Winnaar L 2007.Mathematics and Science Achievement at South AfricanSchools in TIMSS 2003. Pretoria: Human SciencesResearch Council.

Tabachnick BG, Fidell LS, Osterlind SJ 2001. UsingMultivariate Statistics. US: Allyn and Bacon Boston.

Takahashi A 2007. Lesson study in North America. In: M Isoda,M Stephens, Y Ohara, T Miyazaki (Eds.): JapaneseLesson Study in Mathematics. Singapore: WorldScientific Publishing, pp. 175-181.

Takahashi A, Watanabe T, Yoshida M 2006. Developing goodmathematics teaching practice through lesson study: AU.S. perspective. Tsukuba Journal of Education Studyin Mathematics. 25: 197-204.

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