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Investigating Lebanese teachers’ mathematical, pedagogical and self—efficacy profiles: A case study Iman Chahine*, Heather King Georgia State University, University of Johannesburg *Email: [email protected]

ABSTRACT We examined the mathematical, pedagogical and self-efficacy profiles of 13 grade 4 to 6 teachers sampled from five randomly selected schools in Lebanon. The purpose of this study is to explore the impact of providing appropriate training,effective practices and resources for teaching a well-designed curricular intervention. This intervention focused on teachers’ sense of self- efficacy and their expectations of students’ academic success. Participating teachers were randomly assigned to experimental (TREAT) and Control (COMP) groups. The study employed a quasi-experimental and survey design. We used three instruments to collect data: Pre and Post Mathematics Teaching Efficacy Belief Instrument (MTEBI); Teacher Mathematical Profile Questionnaire (TMPQ); and Teacher Pedagogical Profile Questionnaire (TPPQ). TREAT and COMP teachers were provided with independent training before the implementation of the intervention. Data was analyzed using measures of central tendencies and multivariate techniques. Results of the mathematical and pedagogical profile analysis showed that both groups of teachers used more procedural problem solving techniques rather than conceptual ones. Additionally, results of MANOVA indicated that there were no statistically significant differences between TREAT and COMP teachers’ responses on pre and post Personal Mathematics Teaching Efficacy (PMTE) and Mathematics Teaching Outcome Expectancy (MTOE). However, TREAT teachers had significantly higher academic expectations of their students than did the COMP teachers.

http://dx.doi.org/ 10.5339/nmejre.2012.2 Submitted: 3 December 2011 Accepted: 8 May 2012 ª 2013 Chahine, King, licensee Bloomsbury Qatar Foundation Journals. This is an open access article distributed under the terms of the Creative Commons Attribution License CC BY 3.0, which permits unrestricted use, distribution and reproduction in any medium, provided the original work is properly cited.

Cite this article as: Chahine I, King H. Investigating Lebanese teachers’ mathematical, pedagogical and self—efficacy profiles: A case study, Near and Middle Eastern Journal of Research in Education 2012:2 http://dx.doi.org/10.5339/nmejre.2012.2

Page 2 of 17 Chahine and King. Near and Middle Eastern Journal of Research in Education 2012:2

INTRODUCTION Calls for increased productivity and enhanced accountability in schools worldwide have been very audible. In an accountability-oriented era, teachers in many countries find themselves, more than ever, in need of support. Many critics globally agree that most of the currently fashionable methods of accountability advocated by educational policy makers have damaged the professional performance of many teachers by destroying the image they hold as key participants in the teaching and learning process.1 By the same token, the compounded focus on developing systems of educator effectiveness – cultivating highly-effective teachers and leaders by re-examining and realigning a range of policies and practices for recruiting, developing, retaining, and rewarding educators has created an unprecedented pressure on schools. Predictably, a climate of compounded tension combined with heavy teaching loads, large classes, a tight daily schedule, and rules that discourage mutual interactions among colleagues have made teachers’ work more isolated and ultimately isolating.2 Notwithstanding the state of affairs currently prevalent in many educational systems in more affluent countries, the climate is far more dismal in countries struggling with economic, social, and political crises. Lebanon is one such country that has had its share of violence and political and economic instability, which impacted the country’s vital infrastructure and compromised its human and material capitals.3 The regional political struggles juxtaposed to internal social and economic fluctuations have made it hard for the Lebanese people to sustain economic stability and to secure essential and equitable services in education. Notably, more research aimed at studying teacher efficacy as a component of teaching effectiveness.4 Additionally,5 acknowledges the relationship between teachers’ sense of efficacy and the reform vision of mathematics teaching. The author explains: “failure to explore, identify, and build new foundations of efficacy in teaching mathematics may seriously limit the impact of reform” (p. 387). After years of civil war the return to more stable times and the arrival of a new government in late 1998 promised an era of political stability, economic development, and educational advancement for Lebanon. At the outset, the government initiated a number of reforms which were encouraging. The policies of reform included prioritizing poverty alleviation plans6; advocating social justice through equitable distribution of essential services7; pledging a commitment to greater accountability in public administration8; and designating education as a central pillar of its policy.9 However, while a consensus-building process has been proposed from which a long-term national education strategy will result, the general profile of the public education system is perceived to be low, with a disproportionately skewed distribution of qualified teachers and educational resources, the poor taking the lowest shares.10 Since 1994, the school education system in Lebanon has been organized as follows: (a) six-year elementary cycle; (b) three-year intermediate cycle; and (c) three-year secondary cycle. Together elementary and intermediate cycles constitute the nine-year basic education cycle in addition to two-year pre-school cycle that is instituted in the majority of schools. The Lebanese public education sector is relatively centralized with decentralization mostly focused on the day-to-day logistics involved in the functional operation of schools.1 As a result, there is an extensive flow of information and instructions from the government to the schools particularly with limited efforts on behalf of the government to enhance national capacities in education planning and management. As documented in a series of reports published by the national govenemnt, UN agencies and the World Bank,11 effectiveness, equity, and quality are the most cited challenges facing the Lebanese education sector. To reduce the gap left by the weakened public education system, schools subsidized by active NGOs, diverse religious communities, and individual entrepreneurs flourished, all with various objectives and differing performance. Aside from the large religious organizations (e.g., Al-Mabarrat, Makassed, Catholic, etc.) that run their own schools, the private sector always played a pivotal role in education provision and financing in Lebanon. Overall, three main school systems exist namely, public, semi-private subsidized by religious NGOs, and private schools supported by secular organizations or private companies.12 While nearly 70 percent of student enrollment is reported in non-government schools,10 parents’ alignment with religious and political affiliations in semi-private education seems to determine to a great extent the academic quality criteria by which schools are selected.13 Overall, the quality of instruction in public and semi-private schools, particularly for teaching mathematics, is below the average expected threshold.14 With few research studies reporting potential


monitoring systems of educator effectiveness in Lebanon, there is unprecedented need to examine teachers’ pedagogical content knowledge and their perception of self-efficacy in order to design reform initiatives that will enhance and support their instructional effectiveness15 contend that, remarkably, teachers’ positive perceptions of their self-efficacy remarkably drive their willingness to enact change in their teaching practices. Needless to say, teacher performance has been inextricably linked to student achievement16 – 19; we argue that surveying teachers’ challenges and opportunities for progress is essential to their professional growth. By the same token, examining the impact of curricular support efforts on teachers’ perceived self-efficacy could help inform and thereby direct the design of rigorous professional development initiatives focused on building a sustainable cadre of proficient teachers responsive to students’ growth toward improved academic outcomes. RESEARCH ON TEACHER SELF-EFFICACY Albert Bandura, one of the pioneers in investigating the construct of self-efficacy, argues that selfefficacy is a regulatory mechanism that influences behavior through four processes: cognitive, motivational, affective, and selection.20 Contrary to other cognitive theories, which focused on the effectiveness of explicit responses, Bandura’s main interest is studying the effect of beliefs in personal abilities on inducing particular responses in observed, overt performances. In this respect, Bandura distinguishes between two forms of expectations: efficacy expectations and outcome expectations, p. 193.20 While some researchers view teaching efficacy as a “unitary” construct, others distinguish two types of teaching efficacy: personal and general.5 From the latter perspective, teachers’ sense of efficacy is defined as a construct in which the generalized behavior of an individual is based on two factors: a belief about action and outcome and a personal belief about one’s ability to cope with a task.21 A handful of research has emphasized the role that teachers’ perceptions of self- efficacy play in their teaching and student learning.5,22 While5,22 have both considered teacher efficacy as a type of self-efficacy,5 defines efficacy as: “the belief that [teachers] can affect student learning” (p. 387). Empirical evidence from research supports the claim that teachers’ sense of self-efficacy correlates with students’ cognitive development as well as with their own affective growth.23 To this end, several calls have been raised to investigate interventions which potentially increase teachers’ sense of selfefficacy.24 Such attempts range from professional development workshops to in-service teachers’ training programs and extend further to building collaborative school communities.25 In a study involving 1213 elementary teachers,26 found that teachers who interacted more frequently with peer coaches from their own schools and with expert teachers from other schools had higher general teacher efficacy than those who did not. The argument forwarded was that collaboration could influence teachers’ perceptions of how effective they are by initiating and maintaining a sense of shared judgment. In a similar vein,5 cites four major components of teaching that represent “promising sites” for building and maintaining efficacy beliefs namely the choice of problem; anticipating student reasoning; initiating and guiding discourse; and judicious telling. These components align with the recent recommendations set forth by the National Council of Teachers’ of Mathematics reform vision.27 On the other hand, studies indicate that increased collaboration could decrease teachers’ confidence if feedback from peers is negative.22 This confirms further the view that teaching efficacy is a complex construct that involves a multiplicity of variables which are closely linked to the cultural norms that thrive in the social contexts where teachers are practicing their daily teaching. THE STUDY PURPOSE The study examines the mathematical and pedagogical profiles of 13 mathematics teachers teaching in five semi-private schools subsidized by a charitable, religiously affiliated organization in Lebanon. The study further explores the impact of providing appropriate training in effective practices and resources for teaching a well-designed curricular intervention on teachers’ sense of self-efficacy in delivering high quality instruction and teachers’ expectations of students’ potential for academic success. HYPOTHESIS The study investigated the hypothesis that there are statistically significant differences in measures of self-efficacy between teachers trained in using effective practices and are supported by appropriate resources for teaching rational number concepts and those who did not take the training. The


underlying assumption of the study is that with proper professional development supplemented by appropriate resources for instruction, teachers will have an “enhanced” sense of self-efficacy in teaching fraction related concepts.

METHOD Sample We randomly selected 5 schools from a population of 14 schools to participate in the study.3 At the time of the study, 13 teachers were selected and were randomly assigned to treatment (hereafter TREAT) and comparison (hereafter COMP) groups. Two of the five selected schools had two teachers each teaching two grade five classes, two other schools had one teacher each teaching one class, and the final school had one teacher teaching all four classes. The distribution of grade five teachers and students in each of the five participating schools between the TREAT (TR) and COMP (CO) groups is given in Table 1. Table 1. Distribution of grade five classrooms, teachers, and students in each of the five participating schools between the TREAT and COMP groups. School 1 TR

Teachers Students

1 25

School 2

CO

1 26

TRE

1 52

School 3

CO

1 46

TR

1 52

School 4

CO

1 37

TR

3 87

School 5

CO

3 114

TRE

1 69

CO

– –

Teacher Demographics Out of the 13 teachers participating in this study, one was male and 12 were females teaching grades 4–6. Only one teacher had received a Bachelor’s Degree in Mathematics, two had a Bachelor’s Degree in Education, another two teachers had a National Teacher Certification, two had a high school degree, and six held other degrees including Political Sciences, Business Administration, Social Studies, and Engineering. Participants teaching experience ranged from 2 years to16 years with a mean of 5.86 years and standard deviation of 3.5 years for the TREAT group, and a mean of 9.5 years and standard deviation of 5.5 years for the COMP group. Instruments We used three instruments to collect quantitative data: a) Mathematics Teaching Efficacy Belief Instrument (hereafter MTEBI) for assessing teachers’ mathematics teaching efficacy belief; b) Teacher Mathematical Profile Questionnaire (hereafter TMPQ) for assessing teachers’ fractional content knowledge; and c) Teacher Pedagogical Profile Questionnaire (hereafter TPPQ), which examines the didactical and instructional decisions and techniques that teachers employ when teaching rational concepts. All instruments were translated to Arabic the language undertaken in the study. MTEBI questionnaire MTEBI is a modified version of the Science Teaching Efficacy Belief Instrument (STEBI) that has been adapted to teaching mathematics. Numerous validation studies have shown that MTEBI is a valid and reliable instrument for assessing mathematics teaching self- efficacy and outcome expectancy.28,29 MTEBI consists of 21 items, 13 items designed to measure the Personal Mathematics Teaching Efficacy (hereafter PMTE) subscale and eight items measuring the Mathematics Teaching Outcome Expectancy (hereafter MTOE) subscale. Each item is scored on a 5-point Likert scale with 5 ¼ Strongly Agree and 1 ¼ Strongly Disagree (See Appendix A). Pre-MTEBI was administered to the TREAT and COMP groups at the beginning of teacher training and before the curricular intervention. The rationale behind using this instrument is to obtain initial rating of TREAT and COMP teachers’ self-efficacy before engaging in the study. At the end of the curricular intervention, a post MTEBI was administered to both TREAT and COMP teachers to examine whether experiences encountered during the curricular intervention have influenced TREAT teachers’ beliefs in their ability to teach mathematics to their students. The post MTEBI questionnaire consisted of the original 21 pre-MTEBI items supplemented by a set of six items that measured teacher expectations of students’ potential for academic success as developed by Oh


et al.30 (see appendix B). This instrument used a 5-point Likert scale with responses ranging from 1 ¼ Strongly Agree to 5 ¼ Strongly Disagree and was administered after the curriculum intervention. Surveying teachers’ expectations There has been extensive research on the differential attitudes that teachers may unconsciously exhibit towards their students and its consequent impact on students’ achievement.31 – 33 According to,30 teachers discriminate between students based on many attributes including gender, race/ethnicity, socioeconomic status, language, ability tracking, and negative perceptions regarding students’ performance. Additionally, a handful of research has documented that ethnic minorities and poverty stricken children are the most direct target of low expectations and that teachers tend to expect better performance from middle and upper class students.34,35 While measuring teachers’ expectations was not an initial purpose of this research study, our field observations during data collection along with informal conversations with teachers and other school personnel motivated this addendum to the study. The rationale behind surveying teachers’ expectations is to explore their perceptions of students’ potentials for academic success and to explore how these perceptions impinge on the quality of their instructional strategies. A significantly high percentage of the student sampled in this study comes from low socioeconomic backgrounds. In all the schools subsidized by the charitable organization, approximately 20% of students are orphans of war and receive food, shelter, and social counseling in addition to education.36 One of the participating schools has a dormitory for boys and a teacher from the TREAT group acts as an afterschool student counselor and tutor for resident students. TMPQ & TPPQ instruments The TMPQ assessment profile includes 21 items designed to examine teachers’ content knowledge of rational numbers. At the beginning of this questionnaire, teachers’ comfort level for teaching rational number concepts was assessed using a 5-point Likert scale ranging from 1 ¼ extremely uncomfortable to 5 ¼ extremely comfortable in responses to the question: “How would you rate your comfort level in teaching mathematics?” On the other hand, the TPPQ comprises 10 question items designed to assess teachers’ pedagogical profile. Items on the TPPQ profile include problem situations involving fraction related concepts and teachers were asked to explain what instructional strategies they employ for teaching such concepts to their students. A listing of problem types and number of items on TMPQ and TPPQ assessment profiles is given in Table 2. Table 2. Number of Items on TMPQ and TPPQ by Problem Type. Problem type

Comparison Ordering fractions Concept of unit Multiplication: one-step Quotative division Partitive division Concept of equal parts Equivalence a) by representation b) by procedure Estimating sum/difference a) numerically b) representation Qualitative thinking Exact differences/sums: Numerical & representational Missing value problems Division: one-step decimals Effect problem Ratio problem

Number of question items

8 2 5 2 5 2 1 1 4 1 2 2 2 1 1 1 1

Items on both assessment profiles were adopted from a study by Post et al.37 and were translated to Arabic the language of instruction in the participating schools. In the translation process, we made numerous adjustments on the adopted instruments to adapt and appropriate the questions to the


Lebanese cultural and educational contexts. The modifications we incorporated include using Arabic names, Lebanese currency units, and converting measurement scales to the metric system. The three aforementioned instruments were administered at the beginning of the training sessions and before the curricular intervention. THE INTERVENTION We designed the intervention to bridge teachers’ intuitive understandings of fractions with formal ways of representing fractional quantities and operations. In planning the intervention, we not only aimed to increase teachers’ conceptual knowledge of fractions, but also to foster their ability to see the conceptual underpinnings of mathematical symbols and procedures particularly in relation to fraction comparison and operations on fractions. The tasks were constructed purposefully to support teachers’ abilities to provide conceptually-based explanations for mathematical procedures and to choose effective representations of mathematical concepts. During the training sessions, teachers’ deep misunderstanding of basic fraction concepts were revealed and then clarified by comments and questions from other peers. Through negotiations and open dialogue, teachers were afforded a space to practice as autonomous learners, collectively engaged in exploring new landscapes where they were able to construct their own understanding of rational number concepts. We postulated that this new environment would help reduce the anxiety level that teachers experience consistently when confronted with problem solving tasks that involve unfamiliar problem situations. Procedure Following the initial contact with the five selected schools, we met with the principal in each of the five schools and provided a general background on the nature and duration of the research activities as well as the process of teachers’ selection. Upon the random assignment of teachers into TREAT and COMP groups, a timetable was set for carrying out the study and curricular training for teachers was scheduled in each of the five randomly selected schools. Each school principal received an Arabic version of the intervention modules and a sample resource kit that was distributed to students and teachers during the curricular intervention. Training for Teachers As mentioned in Chahine et al.,3 independent training sessions were held to prepare teachers prior to the curricular intervention for TREAT teachers, i.e., teachers using curricular intervention and COMP groups, i.e., those using Lebanese curriculum. Three two-hour training sessions were administered for TREAT teachers and another three two- hour seminars were given to COMP teachers during the course of the study. The first session was held the first week of intervention, the second was administered two weeks during the intervention, and the third was carried out two weeks prior to the completion of intervention. The sessions were administered in Arabic and the topics discussed were related to the teaching and learning of fractions for elementary students. At the beginning of the training sessions and prior to the curricular intervention, both TREAT and COMP groups completed TMPQ and TPPQ assessments and pre MTEBI questionnaire. At the end of the study, we administered the post MTEBI questionnaire to measure the impact of teaching the curricular intervention on teachers’ mathematics teaching self-efficacy. Training for TREAT teachers During the orientation sessions, TREAT teachers were introduced to the teaching philosophy and the fractional content necessary to teach the curricular intervention to students. Teachers were engaged with research-based instructional strategies that have shown to contribute to improved student learning and, consequently, facilitate the implementation of curricular intervention. By examining related research on how best to teach the new content, teachers were exposed to novel techniques to reinforce their instructional capacity thereby broadening their content and pedagogical knowledge and skills for teaching rational number concepts. Training techniques included explaining learning theories, modeling of a skill under simulated conditions, feedback about peers’ enacted performance, and peer-coaching. Horsley et al.38 argue that “through an effective introduction to a high-quality set of curriculum materials, teachers can become clear about the goals for student learning so that they can make sound judgments about the use of materials and strategies in the classroom” (p. 64).


For example, teachers were engaged in developing teaching strategies to help students solve the following problem: “Car A can travel a greater distance in 3 hours than car B can travel in 2 hours. If possible, find which car will travel a greater distance: car A in 5 hours or car B in 6 hours”. Throughout training and as teachers actively experimented with the materials as learners and enacted the various teaching strategies they could implement in their classes, they became more familiar with the content and pedagogy of the curricular intervention. At the end of the first two-hour session of orientation, TREAT teachers were provided with sample lesson plans and resources to incorporate while teaching. Prior to the study completion, two two-hour training followed to reinforce and support teachers’ implementation of the intervention during the sixweek period of the study. Training for COMP teachers Training for teachers in the control group, i.e., teachers of the Lebanese curriculum involved working in study groups in which they discussed and reflected on problem situations related to fraction concepts. We organized groups of teachers around teaching different situations which involved fraction concepts such as equivalent fractions and operations on fractions. For example, teachers were asked to develop short lesson plans to teach fraction concepts associated with a problem situation such as the following “Hani drank 3/5 times as many liters of milk as Waleed. Hani drank 8/10 of a liter of milk. How many liters of milk did Waleed drink?” Two weeks after implementing the intervention, a second training followed. COMP teachers undertook training related to their regular curriculum for the same period as the TREAT group. DATA ANALYSIS To test the group mean differences between COMP and TREAT groups on a linear combination of pre and post MTEBI scores, a one-way MANOVA was conducted. The mean scores for pre MTEBI and post MTEBI were designated by GRADE and GRAPOST respectively. The data was examined to ensure its compliance with assumptions for MANOVA i.e., independence of observations, multivariate normality, and equality of variance- covariance matrices. Analysis on Subscales for Pre and Post MTEBI We adopted Enochs et al.29 classification of MTEBI (Mathematics Teaching Efficacy Belief Instrument) into two scales: PMTE (Personal Mathematics Teaching Efficacy) and MTOE (Mathematics Teaching Outcome Expectancy) to test whether there were statistically significant differences in TREAT and COMP teachers’ responses particularly on these scales. An initial analysis of teachers’ responses indicating their degree of agreement with each item by responding on a Likert measure from 1 ¼ Strongly Agree to 5 ¼ Strongly Disagree revealed sharp inconsistencies in the reliability scores between the pre and post testing scales. We suspected that the efficacy scale influenced responses by the wording of the question items (either positive or negative). Oh et al.30 found a similar result when examining efficacy using a4 efficacy scale. Notwithstanding the relatively large sample size of 87 participants included in the study, the authors noted low reliability scores (.62) for some items on the scale attributing this low reliability to “awkwardness in wording” of the items, p. 67.30 To overcome this caveat, we analyzed the content and recoded the 5-point Likert scale for positive and negative items in the questionnaire to off-set possible differences as a result of wording in the questionnaire. For example, items q3, q6, q8, q15, q17, q18, q19, q21 were recoded such that the scale runs from 1 ¼ Strongly Disagree to 5 ¼ Strongly Agree in order to reflect the appropriate efficacy level. To examine the main effect of curriculum on teachers’ responses on Pre and Post MTEBI tests and that of the interaction between groups and efficacy scales for each test, a mixed-model ANOVA was run. This model entailed a 2 £ 2 £ 2 mixed between-within-subjects design involving one between – subjects independent variable, i.e., GROUP with two levels (TREAT and COMP), the within-subjects independent variable EFFICACY with two levels (PERSONAL, OUTCOME), and the within-subjects independent variable MTEBI with two levels (PRE and POST). The rationale behind using such a design was to examine whether there were differences in the mean scores on each of two efficacy scales, i.e., PERSONAL and OUTCOME between TREAT and COMP teachers on PRE and POST MTEBI tests. Additionally, we investigated the effect of interactions between the performance on PRE and


POST MTEBI across the two efficacy subscales between the two groups. Results of the mixed-model ANOVA analysis showed that only the main effect of EFFICACY test (Wilk’s Lambda ¼ .071, F (1, 11) ¼ 144.914, p ,.001) was statistically significant. This indicated that there were differences in the mean scores averaging across both groups (TREAT and COMP teachers) on the two efficacy scales (PERSONAL and OUTCOME). The multivariate j2 ¼ .929 signified a large effect size and eventually confirmed that around 93% of the multivariate variance in responses was associated with the difference between teachers’ responses on the two EFFICACY scales. Additionally, examining the average mean of scores for PMTE and OMTE scales across pre and post-tests, it was evident that on average, teachers had higher scores on the Personal Scale (M ¼ 58.64 i.e., around 76%) than the Outcome Scale (M ¼ 30.46 i.e., around 90%). Moreover, the main effect of MTEBI (Wilk’s Lambda ¼ .998, F (1, 11) ¼ .026, p ¼ .874 was not statistically significant. This indicates that there were no differences between the PRE and POST MTEBI mean scores within either the TREAT or the COMP groups. Furthermore, none of the effects of two and three-way interactions were statistically significant (MTEBI by GROUP: Wilk’s Lambda ¼ .980, F (1, 11) ¼ .225, p ¼ .644; EFFICACY by GROUP: Wilk’s Lambda ¼ .951, F (1, 11) ¼ .563, p ¼ .469; MTEBI by EFFICACY: Wilk’sLambda ¼ .997, F(1, 11) ¼ .035, p ¼ .856; and MTEBI* EFFICACY *GROUP: Wilk’s Lambda ¼ .962, F (1, 11) ¼ 429, p ¼ .526. The means and standard deviations for TREAT and COMP teachers’ responses on pre/post PMTE and pre/post MTOE are given in Table 3. Table 3. Means and Standard Deviations of TREAT and COMP Teachers’ Scores on Efficacy scales. Efficacy Scales

Group

Pre PMTE

TREAT COMP TREAT COMP TREAT COMP TREAT COMP

Post PMTE Pre MTOE Post MTOE

Mean

56.48 61.79 58.24 58.46 27.71 33.67 27.42 34.00

Standard deviation

6.39 4.50 8.58 6.38 4.23 13.47 12.52 6.32

Analysis of Teachers’ Mathematical Profile Analysis of central tendency measures of teachers’ confidence levels showed that teachers in both groups were confident in their ability to teach rational number concepts (mean ¼ 4.77, SD ¼ .44). While 23% of the teachers indicated that they were comfortable teaching fractions, more than 77% noted that they were extremely comfortable. To examine teachers’ mathematical profile, we investigated TREAT and COMP teachers’ content knowledge by calculating the percentages of correct answers on each of the problem types covered in the TMPQ and TPPQ assessments. To assess teachers’ responses, we adopt three scales: correct ¼ 1 point; partially correct ¼ .5 point; and incorrect ¼ 0 point. The lowest percentage of correct answers reported was on effect problems (46%). We attributed the low success rate to teachers’ unfamiliarity with such types of rate problems, which involve qualitative change in the numerator and denominator examining the ‘effect’ in the fraction as a rate. Heller et al.39 identified nine qualitative rate- change problems and capitalized on mastering the problem solving of such problem types as a prerequisite for acquiring proportional reasoning skills. The second two lowest percentages correct were on one-step division (around 54%) and partitive division (61.5%). Furthermore, the highest percentages of correct answers were detected for questions involving concept of equal parts (100%); numerical estimation of sums and differences (100%); and ratio problems (100%). The percentages of correct answers per problem type as well as the means and standard deviation of responses for TREAT and COMP teachers are given in Table 4. In general, and upon examining mathematical content knowledge across TREAT and COMP groups (see Table 4), it was evident that COMP teachers showed higher percentages of correct answers than TREAT teachers on 10 of the 17 problem types covered in the TMPQ and TPPQ assessments. Surprisingly, this was in contrast with findings of the students’ data where TREAT students outperformed COMP students on both TREAT and school tests.3


Table 4. Percentage of Correct Answers, Means, and Standard Deviations of TREAT and COMP Teachers’ Responses Per problem Type in TMPQ and TPPQ. TREAT Teachers Problem type

Comparison Ordering fractions Concept of unit One-step multiplication Quotative division Partitive division Concept of equal parts Equivalence a) by representation b) by procedure Estimating sum/ difference: a) numerically b) using a representation Qualitative thinking Exact difference/sum: Numerical & representational Missing value problem One-step division Effect problem Ratio problem

Total points

Percent correct

Mean

8 2 8 3 5 2 1

98 78.6 80.9 90.5 85.7 64 100

7.86 1.57 4.86 2.71 4.28 1.28 1

1 4 1

92.8 96 100

2

COMP Teachers

Standard deviation

Percent correct

Mean

Standard deviation

.378 .79 1.67 .49 .95 .49 0

100 91.6 97 94 95 58 100

8 1.83 5.83 2.83 4.75 1.17 1

0

.93 3.85 1

.19 .38 0

100 100 100

1 1 1

0 0 0

85.7

1.43

.79

95.8

1.92

.20

2

85.7

1.71

.49

83

1.67

.52

2 2 1 1 1

82 85.7 57 28.6 100

1.64 1.71 .57 .28 1

.63 .49 .53 .49 0

100 100 50 66.6 100

2 2 .50 .67 1

.41 .41 .41 .42 .41 0

0 0 .55 .52 0

Analysis of Teachers’ Pedagogical Profile Teachers’ responses on the TPPQ assessment were evaluated according to three classifications: a) accuracy of answers; b) type of knowledge involved in the problem solving strategy i.e., conceptual/procedural; and c) problem representation. In specifying the type of knowledge that prevailed in teachers’ responses when solving items on the TPPQ, we adopt Rittle-Johnson et al.40 definitions of procedural and conceptual knowledge. Rittle-Johnson et al.40 define procedural knowledge as “the ability to execute action sequences to solve problems” (p. 346). They also argue that procedural knowledge involved mainly the use of previously learned step-by-step techniques and algorithms to solve specific types of problems. Furthermore, the authors explain that conceptual knowledge entails “implicit or explicit understanding of the principles that govern a domain and of the interrelations between units of knowledge in a domain” (p. 346). By the same token,40 define problem representation as “the internal depiction or re-creation of a problem in working memory during problem solving” (p. 348). A handful of research studies support the hypothesis that forming correct problem representations is one mechanism linking improved conceptual knowledge to improved procedural knowledge. Furthermore, a number of studies investigated the role that problem representation played in changing performance either positively or negatively, depending on the circumstances.41 Following the above definitions, we coded data from the TPPQ assessment according to the following scheme: 1)

Procedural knowledge. Explanations that involve the use of a formula or the execution of any previously learned algorithm were indexed as procedural. Correct procedural explanation received 1 point and percentage correct was computed per test item. Partially accurate explanations received 0.5 point and no explanation was scored as 0. 2) Conceptual knowledge. Solutions that involved generating novel, unfamiliar explanations to problems were labeled as conceptual. Accurate conceptual explanations received 1 point; partially accurate explanations received 0.5 point; and no explanation was scored as 0. Conceptual and procedural explanations employed by the teachers when solving the problems on TPPQ assessment were identified based on their novelty. We categorized an explanation that relied on the use of familiar and routine procedures, algorithms or formulas (e.g., executing actions) already


known for solving a problem as procedural. For example, using the cross product approach to answer a missing value problem was labeled as procedural (see Figure 1). Problem solutions that involved generating a new procedure or an explanation rooted in conceptual understanding of fraction related concepts were labeled as conceptual. For instance, explanations which included connections between symbols and concrete models using different representations were coded as conceptual (See Figure 2).

Figure 1. A problem solved using a procedural problem solving strategy.

The percent correct of conceptual and procedural explanations employed by TREAT and COMP teachers when responding to TPPQ assessment are given in Table 5. Problem representations We also investigated teachers’ competence to represent correctly the problems when using either procedural or conceptual knowledge by examining the succession of accurate explanations including the use of drawings, diagrams, or visual models. Partially accurate explanations received .5 points and no explanation was scored as 0. We further categorized the problem representations that the teachers used in their problem solving strategies and in their description of how they explained the concept to their students according to whether they were pictorial (e.g., use of pictures, drawings, etc.), symbolic (e.g., use of a mathematical formula), a combination of pictorial and symbolic, or others (e.g., use of graphs, texts, diagrams, etc.). The percentages of occurrence of problem representation for TREAT and COMP teachers’ explanations when responding to the TPPQ assessment are given in Table 6. These percentages are projected graphically for TREAT and COMP teachers in Figures 3 and 4 respectively. As noted from Table 5, TREAT teachers provided more procedural than conceptual explanations for almost all problem types excluding the partitive division problems where only 14% of the explanations were procedural. As regards the partitive division problems, 86 % of the explanations were conceptual and 71% of which were accurate. Moreover, while all of the procedural explanations provided for missing value problems and ratio problems were accurate, none were correct for effect problems. Similarly, COMP teachers used more procedural than conceptual explanations on six of the nine problem types; however, all their explanations were correct for four of these problem types. Additionally, all conceptual explanations for seven of the nine problem types were correct.


Figure 2. A problem solved using a conceptual problem solving strategy.

Overall, TREAT teachers used more procedural explanations in solving problems than COMP teachers on six of the nine problem types measured by TPPQ assessment. These problem types include comparison, concept of unit, one-step multiplication, missing value, one-step division, effect problems, and ratio problems. However, the COMP teachers provided more accurate explanations on four of these problem types: comparison; concept of unit; one-step division; and effect problems. In terms of the problem representation dimension, we inspected thoroughly the explanations provided by the teachers and sorted it according to problem representation used. Percentages of occurrence and correct explanations per problem representation were also calculated. Overall, on all the nine problem types covered in the TPPQ assessment, TREAT and COMP teachers mainly used symbolic or a combination of symbolic and pictorial representations to explain their answers to the problems. However, it was noted that the highest percentage of employing symbolic representations in solving the problems was detected among TREAT teachers on six of the nine problem types, i.e., comparison, concept of unit, one-step multiplication, missing value problems, effect problems, and ratio problems. Results of Surveying Teachers’ Expectations While TREAT and COMP teachers showed high comfort level and confidence in their ability to teach mathematics (as reported earlier), both groups were not sure (mean ¼ 3.01, SD ¼ 52) about what to Table 5. Percentage of Correct Answers on the concept structure of TREAT and COMP Teachers’ Responses per Problem Type on TPPQ assessment. TREAT Teachers Problem type

Comparison Concept of unit One-step multiplication Quotative division Partitive division Missing value problem One-step division Effect problem Ratio problem

COMP Teachers

Procedural/conceptual

Percent correct

Procedural/conceptual

Percent correct

86/0 86/0 71/14 57/29 14/86 86/14 100/0 43/14 100/0

86/0 86/0 86/100 71/14 86/71 100/100 71/0 0/0 100/0

50/50 67/33 33/67 83/0 17/83 67/33 67/33 33/17 83/17

100/100 100/100 83/83 83/0 100/100 100/100 83/100 17/100 100/100


Table 6. Percentage of problem representation of TREAT and COMP teachers’ responses per problem type in TPPQ. TREAT Teachers Problem type

Compare Concept of unit One-step multiplication Quotative division Partitive division Missing value One-step division Effect problem Ratio problem

COMP Teachers

Picture

Symbol

Picture þ symbol

Other

Picture

symbol

Picture þ symbol

Other

– – 14 – 43 – – – –

86 71 57 57 – 86 86 43 100

– 14 29 – 14 – – – –

– – – 29 43 14 14 14 –

– – 17 – 33 17

83 50 17 83 – 50 100 17 67

– 50 67 – 67 33 – – 33

17 – – – – – – 33 –

– –

expect from their students. Oh et al.30 employed an average expectation level of 3.61, a standardized threshold for measuring expectations. In comparison to the standardized threshold, the reported mean expectation level of teachers (3.01) in the current study is below the average level suggested by Oh et al.30 and thus was considered low. The means and standard deviations for TREAT and COMP teachers’ responses on each of the expectations’ questions are given in Table 7. While there were no statistically significant differences detected between the responses of TREAT and COMP teachers on five of the six questions, TREAT teachers disagreed (mean ¼ 4.28, SD ¼ 1.49) that most students in their schools will perform below the national average, while COMP teachers were not sure (mean ¼ 3.66, SD ¼ 1.36). We inferred from this result that TREAT teachers exhibited more confidence in their students’ potentials for achievement after the curricular intervention. Additionally, comparing teachers’ expectations of their own students’ academic performance to their expectations of students in general, it was evident that, on average, TREAT teachers agreed that nearly all of their students were at or above the national average. However, they were not sure (mean ¼ 3.00, SD ¼ 1.41) that students, in general, could perform at the national average. On the other hand, COMP teachers were not sure of their students’ projected performance (mean ¼ 3.00, SD ¼ .89) and that of others (mean ¼ 3.33, SD ¼ 1.03) in their schools. Therefore, we conclude that TREAT teachers had higher academic expectations of their students than those of others, while COMP teachers were not sure.

100 90 Percentage of Occurrence

80 70 60 Others Pictorial + symbolic symbolic Pictorial

50 40 30 20 10

pr R ob ati le o m

pr Ef ob fec le t m

Co m

pa

ris

on Co nc ep t un of m i ul On t tip e lic -st at ep io Qu n ot di at vi iv sio e n Pa r di tit M visi ive iss on in pr g v ob al le ue m On s e di -st vi ep sio n

0

Problem Types Figure 3. The percentages of occurrence per problem representation for TREAT teachers’ explanations on TPPQ assessment.


100 90

Percentage of Occurrence

80 70 60 Others Pictorial + symbolic symbolic Pictorial

50 40 30 20 10

pr R ob ati le o m

pr Ef ob fec le t m

ce pt un of m ul On it tip e lic -st at ep io Qu n o di tat vi iv sio e n Pa di rtit M visi ive iss on in pr g v ob al le ue m On s di e-st vi ep sio n

Co n

Co m

pa

ris

on

0

Problem Types Figure 4. The percentages of occurrence of problem representation for COMP teachers’ explanations on TPPQ assessment.

Table 7. The means and standard deviation of TREAT and COMP teachers’ responses per question item on a 5-point Likert scale. TREAT Teachers Questions

Q22. Most of the students in my school will be at about the national average in academic achievement. Q23.Most students in my school are capable of mastering grade level academic objectives. Q24. Teachers in my school generally believe most students are able to master the basic reading/math skills. Q25. I expect that most students in my school will perform at about the national average in academic achievement. Q26. Nearly all my students will be at or above grade level by the end of this year. Q27. I expect most students in my school will perform below the national average in academic achievement.

Mean

Standard deviation

COMP teachers Mean

Standard deviation

3.14

1.21

3.66

1.03

2.00

0

2.83

1.33

2.28

.75

2.83

.75

3.00

1.41

3.33

1.03

2.28

1.25

3.00

.89

4.28

1.49

3.66

1.36

POSSIBLE LIMITATIONS OF THE STUDY In studying the effect of supported intervention on teachers’ mathematics self-efficacy when teaching fraction related concepts several constraints are anticipated. First, although there was a random selection of schools, the choice of teachers teaching intact rather that randomized classes may constitute a threat to the internal validity of the present study. Selecting intact groups, which may not be equivalent in characteristics, could result in uncontrolled selection threats.42 Second, the wording in the efficacy instrument might have hampered teachers’ understanding of some statements thereby contributing to measurement errors. For example, question eight on the MTEBI states “I will generally teach mathematics ineffectively”; the choice of the words may have influenced teachers’ understanding of the intended connotations and hence affected the response they used to indicate their agreement or disagreement. Third, the very small sample size which was limited to the number of teachers teaching only grade five in the five selected schools might have jeopardized the statistical power of the analysis model. Fourth, generalizability beyond the population selected might also be an


issue to consider as the schools selected serve a particular population subsidized by a charitable organization that caters for orphans of war. Furthermore, the use of MTEBI as a measure of rather complex construct such as teachers’ sense of self-efficacy when teaching mathematics may be restraining. Teachers’ perceptions of their efficacy are continuously in flux and are closely influenced by their prior teaching experiences and the educational capital they bring to their practice. DISCUSSION AND IMPLICATIONS In this study, we envisaged that providing a pedagogical space where teachers can engage in facilitating purposive mathematics content will reasonably enhance their mathematics self-efficacy and possibly foster more confidence in their potential to enact reform in their classrooms. We postulated that immersing teachers in a well-structured curriculum for teaching rational number concepts accompanied by the necessary training support and relevant resources will assist teachers in recognizing their strengths to impact effectively students’ performance. Notwithstanding the positive association between teachers’ self-efficacy, reform teaching, and student learning, findings of this study show that teachers’ sense of self-efficacy did not change significantly as a result of improved pedagogical and technological support. We conclude that perhaps for the sample of semi-private schools included in this study, teaching efficacy is not necessarily correlated with teachers’ immediate teaching environments, but rather the entire school culture might have influenced the perceived collective efficacy beliefs that teachers hold as a group. 15 argue that “the sense of collective efficacy in a school can affect teachers’ self-referent thoughts and, hence, their teaching performance and student learning” (p. 8). Perhaps until teachers are afforded opportunities to reflect on their beliefs about self and group capabilities to pledge commitment to capitalize on their expertise as action researchers posing and responding to their own challenges, a shift in their sense of self-efficacy is less likely to happen. Particularly in the Lebanese context, and with the logistic challenges facing the semi-private and public educational sectors, there is a heightened demand to reinforce the role of teachers as decision-makers in promoting students’ understanding, in pushing forward professional development planning, as well as maintaining teachers’ image as agents of change.43 With the rising concerns for cultivating highly effective teachers by endorsing full-fledged teacher evaluation systems that are performance-based, more efforts are encouraged to locate factors that impact teachers’ beliefs and how these beliefs impact student learning. A good starting point would be to foster collegiate collaborations between teachers in their daily practice and to reinforce collective efficacy beliefs within the school cultures that will lessen the state of isolation that teachers are experiencing and empower them to be more proactive in finding new avenues to enhance students’ potentials for academic excellence. REFERENCES [1] Rallis SF, MacMullen MM. Inquiry-minded schools: Opening doors for accountability. Phi Delta Kappan. 2000;81(10):766–773. [2] Huffman D, Kalnin J. Collaborative inquiry to make data-based decisions in schools. Teach Teach Educ. 2003;19(6):569–580. [3] Chahine IC, Post T, del Mas R. The effect of using a research-based curriculum on learning basic rational number concepts by Lebanese students. NMEJRE. 2011;3:1–9, doi:10.5339/nmejre.2011.3 [4] Gibson S, Dembo MH. Teacher efficacy: a construct validation. J Educ Psychol. 1984;76(4):569 –582. [5] Smith J. Efficacy and teaching mathematics by telling: a challenge for reform. J Res Math Educ. 1996;27(4):387–402. [6] Shaar N. Work on poverty in the economic and social commission in Western Asia. 2004, November. Paper presented at the Fourth Regional Workshop on Poverty Statistics in the ESCWA Region. Amman. [7] Bahous R, Nabhani M. Improving schools for social justice in Lebanon. Improving Schools. 2008;11(2):127–141, doi:10.1177/1365480208091105 [8] Iskandar A. Public service accountability in Lebanon. 2000. Retrieved from: http://ddc.aub.edu.lb/projects/pspa/ PSAccount/PSAccount-0.html [9] Zakaria Z. The role of education in peace building: a case study. Atlanta, Georgia, USA: United Nations Children’s Fund (UNICEF); 2011. [10] World Bank. Project appraisal document for a general education project. Report number: 20152-LE, 2000. [11] Chahine IC. The role of translations between and within representations on the conceptual understanding of fraction knowledge: a trans-cultural study. J Math Educ. 2011;4(1):47–59. [12] Ayyash-Abdo H, Alamuddin R, Mukallid S. School counseling in Lebanon: Past, present, and future. J Counsel Dev. 2010;88:13–17. [13] Lynch S. A is for Al-Mahdi. 2010. Retrieved from http://www.nowlebanon.com/NewsArchiveDetails.aspx?ID¼180177 [14] Henningsen M, Zebian S. High-level thinking, reasoning, and communication in Lebanese elementary mathematics classrooms: a preliminary Technical Report on the Fall 2002–2003 Classroom Observations of the MAthematics


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Appendix A. Mathematics Teaching Efficacy Beliefs Instrument (MTEBI). 1. When a student does better than usual in mathematics, it is often because the teacher exerted a little extra effort 2. I will continually find better ways to teach mathematics 3. Even if I try very hard, I will not teach mathematics as well as I will most subjects 4. When the mathematics grades of students improve, it is often due to their teacher having found a more effective teaching approach. 5. I know how to teach mathematics concepts effectively 6. I will not be very effective in monitoring mathematics activities. 7. If students are underachieving in mathematics, it is most likely due to ineffective mathematics teaching 8. I will generally teach mathematics ineffectively. 9. The inadequacy of a student’s mathematics background can be overcome by good teaching. 10. When a low-achieving child progresses in mathematics, it is usually due to extra attention given by the teacher. 11. I understand mathematics concepts well enough to be effective in teaching elementary mathematics 12. The teacher is generally responsible for the achievement of students in mathematics. 13. Students’ achievement in mathematics is directly related to their teacher’s effectiveness in mathematics teaching 14. If parents comment that their child is showing more interest in mathematics at school, it is probably due the performance of the child’s teacher. 15. I will find it difficult to use manipulatives to explain to students why mathematics works. 16. I will typically be able to answer students’ questions. 17. I wonder if I will have the necessary skills to teach mathematics. 18. Given a choice, I will not invite the principal to evaluate my mathematics teaching. 19. When a student has difficulty understanding a mathematics concept, I will usually be at a loss as to how to help the student understand it better. 20. When teaching mathematics, I will usually welcome student questions 21. I do not know what to do to turn students on to mathematics.

SA

A

UN

D

SD

SA SA

A A

UN UN

D D

SD SD

SA

A

UN

D

SD

SA SA SA

A A A

UN UN UN

D D D

SD SD SD

SA SA

A A

UN UN

D D

SD SD

SA

A

UN

D

SD

SA

A

UN

D

SD

SA

A

UN

D

SD

SA

A

UN

D

SD

SA

A

UN

D

SD

SA

A

UN

D

SD

SA SA SA

A A A

UN UN UN

D D D

SD SD SD

SA

A

UN

D

SD

SA SA

A A

UN UN

D D

SD SD

Adopted from Enochs et al.29 Please indicate the degree to which you agree or disagree with each statement below by circling the appropriate letters to the right of each statement. SA ¼ Strongly Agree; A ¼ Agree; UN ¼ Uncertain; D ¼ Disagree; SD ¼ Strongly Disagree.


Appendix B. Teachers’ Expectations Questionnaire. 1. Most of the students in my school will be at about the national average in academic achievement. 2. Most students in my school are capable of mastering grade level academic objectives. 3. Teachers in my school generally believe most students are able to master the basic reading/math skills. 4. I expect that most students in my school will perform at about the national average in academic achievement. 5. Nearly all my students will be at or above grade level by the end of this year. 6. I expect most students in my school will perform below the national average in academic achievement.

SA

A

UN

D

SD

SA

A

UN

D

SD

SA

A

UN

D

SD

SA

A

UN

D

SD

SA

A

UN

D

SD

SA

A

UN

D

SD

Adopted from Oh et al.30, “Inner City Teachers’ Sense of Efficacy towards Minority Students”. SA ¼ Strongly Agree; A ¼ Agree; UN ¼ Uncertain; D ¼ Disagree; SD ¼ Strongly Disagree.