Matematicas_en_SAT_Informe_Ford_Sarah_Richards_Agosto_25 by Gerardo Montoya

Improving Maths Education in Rural Honduras A research study on maths teaching and learning in the Sistema de Aprendizaje Tutorial (SAT)

Sarah Richards, Consultant

August 2010

This report is dedicated to the Bahaâ&#x20AC;&#x2122;i youth of Iran. They have been denied access to higher education for decades, and are beacons of hope, compassion and fortitude.

Contents Acknowledgements .......................................................................................................................................................... 5 Executive Summary ....................................................................................................................................................... 6 Background ...................................................................................................................................................................... 11 Objectives.......................................................................................................................................................................... 13 Methodology .................................................................................................................................................................... 13 Limitations of the study .............................................................................................................................................. 13 Creating an atmosphere of consultation .............................................................................................................. 15 In the classroom ........................................................................................................................................................ 15 Improving the educational value of classroom talk ................................................................................... 17 General Training and consultation skills ........................................................................................................ 19 The use of questions and answers.......................................................................................................................... 20 Allowing time for reflection ................................................................................................................................. 20 Questions: increasing participation .................................................................................................................. 20 Questioning: a formative assessment tool ..................................................................................................... 21 Tutor/student questions ....................................................................................................................................... 22 Three ways of working................................................................................................................................................ 22 The use of tutor/student presentations of work ......................................................................................... 23 Working in small groups........................................................................................................................................ 25 Peer teaching and ‘social loafing’ .................................................................................................................. 25 A forgotten method? .......................................................................................................................................... 26 Individual work: the third force ......................................................................................................................... 27 Thinking time ........................................................................................................................................................ 27 Formative Assessment: fine tuning acompanimiento..................................................................................... 28 Taking responsibility for learning .......................................................................................................................... 28 Unsupervised working ........................................................................................................................................... 29 Homework ................................................................................................................................................................... 29 Formative assessment helps students take responsibility for learning ............................................ 29 Tools and Representations ........................................................................................................................................ 30 The importance of the texts.................................................................................................................................. 30 Multiple representations ....................................................................................................................................... 31 Updating and revision of texts ............................................................................................................................ 33 Guías de Aprendizaje y Evaluación (Ciclo Común) ..................................................................................... 35 Practical application of maths ............................................................................................................................. 37 3

Statistics ....................................................................................................................................................................... 37 Linking SAT and community maths .................................................................................................................. 38 Evaluation ......................................................................................................................................................................... 38 Community Evaluation ........................................................................................................................................... 39 Evaluation – testing and retesting..................................................................................................................... 40 Maths: a problematic part of the program ..................................................................................................... 40 Percentages of retakes for 2008 ......................................................................................................................... 40 Comparing SAT performance with the government system .................................................................. 41 Tutors and mathematics: knowledge, attitudes and beliefs about teaching and learning ..................... 42 Tutor attitudes to maths: the positive results of training ........................................................................ 42 Building on an apprenticeship of observation.............................................................................................. 44 Theory- based practice ........................................................................................................................................... 45 Maintaining positivity: getting unstuck........................................................................................................... 45 Tutor knowledge ....................................................................................................................................................... 46 Basic mathematical operations ......................................................................................................................... 47 The transition to Bachillerato ......................................................................................................................... 48 Preparing for training .................................................................................................................................................. 50 To study or not to study texts .............................................................................................................................. 50 Maintaining the quality of maths training ........................................................................................................... 50 The Training................................................................................................................................................................ 50 Supporting tutors’ development of practice in the field........................................................................... 51 The role of the field assessor .................................................................................................................................. 51 Institutional links .......................................................................................................................................................... 52 Conclusions ....................................................................................................................................................................... 55 References ........................................................................................................................................................................ 60 Appendix 1: Observations: maths trainings .................................................................................................. 62 Appendix 2 – Summary of the field assessors’ session observations ................................................. 63 Appendix 3: Summary of field assessors comments on classroom observations (not monthly reports) ......................................................................................................................................................................... 66 Appendix 4: Questionnaire Survey .................................................................................................................... 69 Appendix 5: Additional comments of the tutors on the questionnaire .............................................. 77

Acknowledgements This report was made possible through grant number 1070-0932 from the Ford Foundation. I would also like to acknowledge the assistance of all those currently involved in the implementation of SAT-Honduras who gave their time and professional assistance during this investigation, including the Executive Director of Bayan Association, Ing Soheil Dooki, for his sustained encouragement; the mathematics coordinator, Ing Ruhollah Sayyรกh, for his fine example and patient, good humoured support; and the many other people, tutors, field assistants, coordinators and administrative staff who all gave generously of their time, inspired me by their dedication, and treated me with great kindness. This report would have been impossible without them and I am very grateful. In addition, I am very grateful to Dr Chris Martin (ex Ford Foundation) and Prof Diana Coben for helping identify the need for this research, and for their useful ideas on how to undertake it, to Prof Malcolm Swan for his valuable comments, to my colleagues at Abingdon & Witney College for their warm support and the granting of extended leave and finally, a very heartfelt thank you to Dr Michael Richards for his editing and wise advice. The views and opinions expressed in the report, as well as any errors, are those of the consultant1 and do not necessarily represent those of the Asociaciรณn Bayรกn or the Ford Foundation.

Consultant qualifications include: BSc in Psychology (Open University, UK), Post Graduate Certificate in Education (Oxford University), MA in Education (Oxford Brookes University) and SAT Tutor (Impulsor level). Experience in education includes 20 years teaching and teacher training/mentoring in England, Malawi, Ghana, Mexico and Honduras. The consultant has delivered workshops/papers at various international conferences/training events in England, Ireland, Scotland, USA and Mexico. In 2002, she was on the final evaluation team of the UK Department for International Development (DFID) project to establish SAT on the north coast of Honduras (1997-2002). sejrichards@gmail.com 1

Executive Summary The Sistema de Aprendizaje Tutorial (SAT) is an interdisciplinary secondary school program designed to provide a high quality and appropriate education in rural areas. The quality of the Honduran SAT program has been nationally and internationally acknowledged. In Honduras, SAT is implemented in a partnership of the Ministry of Education and four NGOs. Asociación Bayán’s responsibilities include: philosophical guidance; the adaption and updating of educational materials; implementation of the curriculum; all training; and monitoring and evaluation of the educational processes. Asociación Bayán and the Ford Foundation commissioned this report to:   

Investigate tutor training for maths and the impact that this has on the mathematical understanding and pedagogy of the tutors. Assess other aspects of SAT affecting the development of the students’ mathematical capacity in the classroom. Make recommendation to improve the effectiveness of the maths component in SAT, with particular reference to tutor training.

It is further stated that the results of the study will be made available to the educational community so that they can be of use to other educational programs, both national and international. It would be difficult and counterproductive to attempt to completely separate the maths component, especially as regards the pedagogic aspects, from the overall SAT program, in view of some of the generic pedagogic challenges faced. Therefore some more general analysis of the learning process and challenges in SAT is in order. In a little over a decade SAT has expanded dramatically from about a hundred students to nearly eight thousand. This rapid expansion is a remarkable achievement made possible by the energy and sacrifice of a dedicated team of people, and has involved the formidable challenge of training about 700 tutors (there is a high turnover rate) and 80 field advisors and coordinators. One of the key challenges to maintaining the quality of SAT is that it is based on several assumptions that are notoriously difficult to change, about the purpose of education and learning processes which underpin the whole SAT methodology, but which the trainees, despite a rigorous selection process, often do not initially share. The most critical assumptions are that: 

tutors and students are co-workers committed to individually and collaboratively developing their spiritual, practical and intellectual capacities and service-orientated attitudes; 6

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the learning process is most effective when it is active, participatory and applied to solve real life problems; the role of the tutor is to guide and accompany students on their developmental journey - this is very different from the role of a traditional ‘transmission’ orientated teacher. A new type of educator is required, ‘con una amplia visión, no solo de los aspectos educativos, sino también de lo relacionado a la gestión comunitaria y desarrollo social.’2

The tutors, who are often without a sound maths education themselves3, face the daunting task of enabling poorly prepared students4 to access, and succeed with, a curriculum that is much more conceptually challenging (and necessarily so) than they have themselves experienced. It is therefore unsurprising that maths has been the most problematic area of SAT since the beginning of the program5. It is an indication of the fundamentally sound nature of the training that almost all the questioned tutors reported that their own maths had benefitted significantly, and that they had a more positive attitude towards maths as a result of their training with SAT. The vast majority of the tutors reported enjoying teaching maths, and felt they understood the material well enough to explain it to their students. A major contributing factor to this success is the inclusive and collaborative learning environment created in most of the trainings, and, very significantly, field reports and observations indicate this is almost always replicated in the SAT classrooms. The strong emphasis on the practical applications of maths, conceptual understanding and the integrated nature of the curriculum appears to be profoundly motivating for both tutors 2

‘A broad vision of not only of education but also of the promotion of social and community development’ (taken from the introduction to Aprender Enseñando, the SAT teacher training text). . 3 Honduras was reported to have the worst maths education in Central America in the Informe de Progreso Educativo de Honduras. (2005). Programa de Promoción de la Reforma Educativa en América Latina y Caribe (PREAL). Washington, DC/Chile: Inter-American Dialogue/Corporación de Investigación y Desarrollo 4 Rural schools are more likely to have several grades in one classroom and experience much more frequent teacher absences than urban schools. 5 If students score less than 60% in an end of module test they have to retake it. In 2009, the percentage retaking maths was 45% compared to a 16% average for the other four areas of SAT.

and students, but during observations of trainings it was apparent that some key aspects of the SAT methodology are not being implemented, or are only being partially implemented. These are the aspects of the methodology that are most associated with effective maths pedagogy according to international research. The neglected activities include ones that are strongly associated with conceptual development, increasing classroom participation, taking ownership of the learning process, and assisting teachers in assessing and responding to student learning. For example, activities explicitly recommended in the SAT texts, but not seen in 195 hours of training/classroom observations, are: ď&#x201A;ˇ

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Individual work followed by small group work, in which each student in turn should explain/justify their method of resolving a problem or explaining a concept to the other members of the group Use of the SAT Learning and Evaluation Guides

The SAT texts are innovative, engaging and generally very supportive to the learner, but they sometimes assume a knowledge and understanding of maths, and its associated pedagogy, which tutors have not yet developed. It may be that many tutors are satisfied with their delivery of the curriculum because they are unaware of the more challenging aspects of the SAT methodology, and the concomitant need for greater conceptual understanding. Many key teaching staff have therefore identified as an urgent need the development of additional materials and activities which can enable the tutors and students to overcome the deficiencies of their previous education. There is a strong commitment by SAT coordinators to do this. A more complete utilisation of the SAT methodology may reduce the need for additional materials, although some important (according to recent research and international good practice for effective maths teaching) tools/activities should be considered in terms of their potential to significantly enhance the learning process in SAT. Teaching staff are very aware of the need to focus more on conceptual development and pedagogy, but find that this is very difficult due to the severe time constraints of the current training schedules. The more careful planning of training could help with this, but there is also a need to increase (carefully selected) field support staff capacity to deliver more specialist maths training in small, local trainings, and to mentor tutors. The process of training maths specialists has begun and two trainings have taken place. 8

Interviews revealed that most tutors are also studying for undergraduate degrees, and are taking classes on education. But no linking of university studies and SAT, and very limited reference to SAT educational theory, was observed during the trainings, apart from during the initial training. A greater understanding of the theoretical underpinnings of the SAT methodology, its relationship to wider educational theory, and how these relate to their own beliefs and practices would significantly help tutors to more fully implement the SAT methodology, and to become more reflective and effective practitioners (this finding is supported by educational research literature). The rapid expansion of SAT has placed Asociación Bayán under considerable stress, and all the interviewed coordinators expressed serious concerns about maintaining quality. Inconsistencies in data collection on field activities (although there are plans to produce a manual), serious archiving problems,6 and the lack of effective fora for discussing concerns, are (in the consultant’s view) are impacting Asociación Bayán’s capacity to address these issues. It is vital that such constraints are urgently addressed. It is clear that the quality of the tutors as effective teachers is the key to the success of the SAT program. The high turnover rate of tutors7 means that the SAT trainings have to be very effective. It is vital that tutors experience the full SAT methodology while in training, understand the reasons for it, and are supported in their adoption of participatory teaching practices. Therefore the most important recommendations of this consultancy report are that: 

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Activities which are a part of the SAT methodology, but which have only partially been adopted, or have not been adopted, are fully implemented, unless consultation with FUNDAEC identifies them as being no longer appropriate. Priority is given to building the capacity of: o trainers to use the SAT methodology fully; o the field assessors to be able to ‘accompany’ the tutors in the development of their general pedagogic skills; o carefully selected maths field support staff to be able to effectively support the tutors in the development of their maths pedagogy/knowledge. Effective fora for the consultation of pedagogical and other educational issues are established. Tutors are mentored in order to help develop their capacities to reflect on their practice, and to be able to systematically experiment with small changes (action research). Relevant insights, techniques and tools originating outside SAT, particularly those associated with increased classroom participation, conceptual development and ‘formative assessment’, are identified and considered for adoption and adaptation,

See Limitations: page 13/14 See Background: paragraph 5

ď&#x201A;ˇ

since (according to the consultantâ&#x20AC;&#x2122;s experience and relevant research literature) these have the potential to significantly enhance key aspects of the SAT pedagogy/curriculum. The collection and analysis of field data and feedback is systematised and standardised (where appropriate), and archiving systems are improved.

Introduction Background The Sistema de Aprendizaje Tutorial (SAT) was developed in Colombia in the 1970s to provide a good quality and appropriate secondary education in rural areas. The curriculum is interdisciplinary and focused on developing students’ spiritual, moral, practical and intellectual potential in order that they can, along with other community members, take charge of their own autonomous and sustainable development (Box 1). The quality of SAT has been nationally and internationally acknowledged8. Box 1 The search for a valid rural education implies changes in the relationships between many of the elements of the educational system - time, space, students, teachers, school, and the community. For example, the relationship between the students and the professors in the programs of FUNDAEC is one of co-workers embarked on an enterprise of great importance - the search for the path of development of their people. The student is not considered an empty container to be filled drop by drop but a mine of hidden talents and potential that need to be discovered, perfected, and directed toward the service of others. Source: Arbab el al (1988)

In Honduras, SAT is implemented in a partnership of the Ministry of Education and four NGOs, including Asociación Bayán. Asociación Bayán, among other services, provides: philosophical and pedagogical guidance, including curriculum development; all the training; monitoring and evaluation; and adapts or updates SAT texts in consultation with the founding NGO of SAT, the Foundation for the Application and Teaching of Sciences (FUNDAEC), Colombia. SAT’s dramatic expansion from about a hundred students to about eight thousand in a little over a decade has presented the formidable challenge of inducting and training over 700 new tutors, as well as about 80 field advisors and coordinators. The SAT philosophy and pedagogy is a new and challenging paradigm for the majority, because most Hondurans, especially those from poorer backgrounds, have mainly experienced the ‘transmission’ or ‘empty container’ method of teaching (UNDP, 2009a). It is widely recognised that teachers tend to adopt the style of teaching

A 2005 Ministry of Education report on education in Honduras noted that: "SAT .... through the development of contents relevant to the aspirations and real needs of the rural population, has shown itself to be adapted to the special conditions of rural people." In 2002, SAT-Colombia received a Club of Budapest “Change the World – Best Practice Award”; and in 2008 Asociación Bayán was invited to present SAT-Honduras as a ‘best practice’ sustainable development case study at the UN Commission for Sustainable Development (CSD-16) meeting.

that they have experienced as students (Schwille & Dembele, 2007). However SAT has been able to achieve some significant positive changes in teaching style. This is probably because of its strong philosophical base, and the fact that the training already has some elements known to increase active teaching methods, such as systematic training over a long period and support by peers in the field (Wiliam, 2006). On the other hand SAT and Asociación Bayán have also faced the challenge of a high turnover rate of tutors, which is to be expected in a rapidly expanding program. From 2000 to 2008 just over 500 tutors were trained, and about 20% left after an average 2.4 years. The high turnover rate is partly due to the understandable tendency for tutors, most of whom are undergraduates, to look for a secure job after completing their degree. Those still in SAT average about 2 years of service9. This is problematic in that research shows that experience is a key factor in improved teacher performance (Chingos & Peterson, 2010; Wiliam, 2007); the short average length of tutor service also increases the pressure on tutor training to be particularly effective. The SAT methodology (Box 2) is largely inspired by the Baha’i Writings, in which consultation is conceptualised as the collaborative search for truth and the process by which decisions are made for community action. The criteria for consultation are demanding and include (italicised phrases are from the Bahá’í writings10): ‘purity of motive‘; an atmosphere of ‘love and harmony’; objectivity or detachment; ‘courtesy’; ‘absolute freedom’ of expression; and the responsibility of all to give an opinion supported by logical arguments rather than personal emotions.

Box 2 The method of teaching, reflected in the design of textbooks, is one of raising questions and trying to find answers in an atmosphere of consultation between teachers and students. Source: Arbab et al (1988)

In SAT there is great emphasis on the quality of the textbooks and tutors. The textbooks are fundamental to the methodology, and are designed to be worked through systematically by the students, accompanied and helped, where necessary, by their tutor. This is very different to normal Honduran practice in which the teacher stands at the front and teaches, and the students listen. The SAT texts, through a series of thought provoking questions, explanations and graded activities, develop key concepts, skills (intellectual and practical) and attitudes so that students can become independent and rigorous mathematical thinkers, and accurate, efficient practitioners.

Tutor data was only available up to 2008 so average length of service could have changed. For example, ‘Should any one oppose, he must on no account feel hurt for not until matters are fully discussed can the right way be revealed. The shining spark of truth cometh forth only after the clash of differing opinions.’ (Abdu’l Baha, cited in Shoghi Effendi, 1922). 10

There is substantial research evidence that teachers have a major influence on student achievement; students with the best teachers learn at twice the rate of those with average teachers (Rivkin, 2005). Asociación Bayán, in consultation with the Ford Foundation, has identified or hypothesised that improving the level of maths teaching would have a substantial impact both on overall academic progress and the running of small businesses, agricultural and community projects. For these reasons Asociación Bayán and The Ford Foundation commissioned this study and report.

Objectives   

To investigate tutor training for maths, and the impact that this has on the mathematical understanding and pedagogy of the tutors. To report on other aspects of SAT affecting development of the students’ mathematical capacity in the classroom. To make recommendations to improve the effectiveness of the maths component in SAT, with particular reference to tutor training.

Methodology  

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Semi-structured interviews with coordinators, field advisors and tutors (Appendix 1) Observations of maths and general training (156 hours), SAT lessons (15 attended in the Departments of Atlantida, Colon and Santa Barbara) and a range of meetings, including coordinator meetings (Appendix 1) Participatory observation, including that derived from conducting tutor training in maths11 and English, and providing support to other trainers/coordinators Questionnaire surveys, focusing on tutor experience and attitudes to maths learning and teaching (applied to 190 tutors of whom 133 responded (70%), adapted from proven questionnaires by Swan (2006) and a seminal study by Fennema & Sherman (1976) as translated by Ignacio et al (2006) (Appendix 4) Review of documents: SAT Texts; Tutors’ worked Texts; Field Assessor Observation Reports and Monthly Reports; Asociación Bayán Nacional Profiles; data on tutor length of service; one Asociación Bayán internal report; and reports to donors (see below).

Limitations of the study 

The paucity of reports: after repeated requests for any report which contained references of any sort to education, I was advised that due to hard disc failures and archiving problems only the following reports were available: 2010 January Informe de Actividades; the Perfiles Nacionales for 2010 (Block 1), 2008 (Blocks 2 & 3) and

Maths: 4 hours, English: 12 hours

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2009 (incomplete due to the coup); and three reports to donors12, two for 2007 (Jan – June and July – Dec) and one for 2004-2006.13 SAT students were not consulted in this study, partly since the three year Hewlett study by the Universities of Berkely, New York, Wellesley College (USA) and the National Pedagogic University (Honduras) will contain an in-depth impact analysis of student achievement and experience for the first two years of study14. I decided that it was beyond the scope of this report to analyse the interactions between mathematics and other curriculum areas, economic activities and community development (in SAT these process are all integrated). It is however hoped that this report will contribute to a holistic reflection on the SAT program. The Asociación Bayán plans to review working conditions and practices and therefore these are not addressed in this report.

Reports: 

Scaling up the System of Tutorial Learning (SAT) as an alternative education program tackling rural poverty and gender inequity in Honduras:01:07:04-30:06:06 (No:1045-0688)

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Building Sustainable Livelihoods through Agriculture 01.01.07:31.07.07 (CIDA)

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Fortaleciendo la Candidación del Programa Educativa Rural SAT en Intibucá, La Paz y Comayagua. Junior 07 – Dec 07 (IBIS)

There is very little documentation of SAT’s development in Honduras; this, combined with the loss of many key reports means that the opportunity for Association Bayán to reflect on its experience and for other organisations to benefit from them is seriously diminished. When the innovative nature and success of the program is taken into account this is a significant loss. 14 2007 (Draft proposal) 4 mains goals were: 1) To develop and apply appropriate measures of secondary school quality; 2) To evaluate the impact of SAT on treated communities, particularly student participants; 3) To explain the impact (or lack thereof) of the SAT program; 4) To extract lessons for the scaling-up of SAT in Latin America and expansion to new countries.

Findings Box 3 En SAT la experiencia en matemática es fascinante porque aprendes a desarrollar tu propio concepto y te pones a utilizar tu lógica matemática que a veces esta muy dormida en problemas de la vida diaria y te llevas a resolver esos problemas de la vida diaria. Source: SAT tutor comment on questionnaire

SAT is implemented by a dedicated team that believes strongly in its value and is committed to its sustainable development. Although many senior staff said that SAT’s rapid expansion has made it difficult to maintain the high standard of training achieved in earlier years, the vast majority of tutors report that the maths training has had a significant positive impact on them (Box 3) and that it prepares them well for teaching. This report finds that the maths tutor training is basically sound, but that senior staff concerns are valid and there is an urgent need to address them for quality to be maintained and improved.

Creating an atmosphere of consultation In the classroom In the major recent Human Development report of Honduras by UNDP (2009a), which focused on youth issues and contains a major survey of Honduran youth attitudes (UNDP, 2009b), Honduran youth responded that, after poverty, their greatest problem was violence, principally feuds and fights. The SAT centres are located in communities where family feuds are common and therefore can be a difficult environment in which to develop an Box 4 ‘atmosphere of consultation’. The class worked in small interacting groups for two hours and then, during the recess, most of the centre’s seventy students played football together with vigour and good humour. The tutors were unable to explain how the students formed their teams - age, sex and community did not appear to be factors. Observation: March 2010

15 16

In interviews with first year tutors and other tutors recalling their first year, frequent mention was made of the difficulties and sustained effort required to establish and maintain classroom norms of respect and collaboration. The majority appear to achieve it; in the Field Assessor reports, the boxes for: ‘Se observa un ambiente alegre y de motivación en el grupo’15 and ‘Se observa respeto hacia las opiniones de los compañeros’16 were ticked 96% of the time (Appendix 2). This is a

‘A happy and motivated atmosphere was observed.’ ‘Students were respectful of one another’s opinions.’

Box 5 When a student in one group was undergoing chemotherapy and not robust enough to leave the house so his SAT group met in his home during his treatment and convalescence. Two years later he is studying Bachillerato with the same group. A first year tutor realised at the beginning of the year that the low literacy levels of some of his students would impede their progress so he gave extra classes every Saturday for the first block. A tutor accompanied a group of his students on the two hour walk to their community and stayed for the night in order to get to know their families and experience their daily journey. Parents, students and tutor worked together, bringing sand from the river so that they could build a new classroom.

yes/no Box, but judging by additional comments (Appendix 4), they appear to be using fairly similar indicators17 and, in most of the sixteen classes observed, there was a collaborative, purposeful and cheerful atmosphere (Box 4). Tutors remain with the same students for up to six years, and are responsible for academic progress and pastoral support. It seems, from interviews and conversations with tutors and coordinators that most tutors demonstrate commitment to, and develop strong bonds of affection for, their students (Box 5). The fact that they often go to considerable trouble to get to know their home circumstances, provide extra help and are active with their students in community projects (Appendix 3) also contributes to the generally united classroom atmosphere. It was noticeable that although some assessor monthly reports were fuller and franker than others, most contained observations about the learning environment that could usefully be reflected on.

Source: observation Oct 2008 & April 2010, coordinator interview May 2010 tutor interview with video evidence May 2010.

It is recommended that: 1. Asociación Bayán continues to build on these good foundations by continuing to reflect and experiment with ways of further developing ‘an atmosphere of consultation.’

Example of monthly reports were received from 8 out of 12 regions and observation reports on 36 tutors completed by 8 assessors. This represents less than 1% of the tutors it and cannot be considered representative.

Improving the educational value of classroom talk Asking probing questions in order to consult on them is at the heart of SAT methodology (Box 6). This approach is supported by wide body of research showing the value of discussion in the learning process (Anthony & Walshaw, 2009). Given the constant talking observed in SAT

Box 7 Mercer (1995,2000) has described in some depth the types of interaction that promote effective and ineffective learning. In particular he demonstrates the superiority of exploratory talk over disputational and cumulative talk. Exploratory talk consists of critical and constructive exchanges, where challenges are justified and alternative ideas are offered. Disputational talk consists of disagreement and individualised decision making. This is characterised by short exchanges consisting of assertions and counter-assertions. In cumulative talk speakers build positively but uncritically on what each of the others has said. This is most typically characterised by repetitions, confirmations and elaborations. In short the most helpful talk appears to be that where the participants work on and elaborate each other’s reasoning in a collaborative rather than a competitive atmosphere. Exploratory talk enables reasoning to become audible and knowledge becomes publicly accountable. -my

Box 6 Un proceso en el cual se tiene como referencia el método científico: el grupo explora, identifica problemas, formula interrogantes e hipótesis, consigue información, experimenta, descubre, crea modelos, aplica, explica y plantea nuevas preguntas … Source: Valcarcel & Correa (1995)

classrooms, and the vital role that tutors think this plays in learning,18 it is essential to note that the evidence indicates that some types of discussion are much more productive than others (Box 7). The concept of ‘consultation’ is far more complex than that of ‘exploratory talk’ (Box 7), but they share common characteristics, and developing tutor/student capacity for ‘exploratory talk’ can be considered as contributing to the creation of ‘an atmosphere of consultation’.

The whole-group discussions observed in the trainings and classes were almost exclusively of the ‘cumulative talk’ type. The better trainers sometimes tried to facilitate ‘exploratory’ discussions, with some success, but generally it seemed as italicsthough the trainers needed further Source: Swan (2006:87) guidance in how to promote ‘exploratory talk.’19 For example, it was observed that in the tutor trainings, tutors rarely built on previous contributions by themselves or others and this was never observed in a classroom. 18

86% of tutors reported (in the questionnaire survey) that their students usually learnt through discussion. It may be that assessors (and tutors) have different conceptions of what it means to take part in a discussion. For example in the Informes de Visita (Appendix 2) in response to ‘Durante su estadia en el aula cuantos participantes vio opinar’ the reported percentage of students ranged from 100% to 6%. It was noticeable that assessors tended to give a similar estimate for all their observations. It may be that some assessors count giving an answer as ‘opinar’ while others require justification or elaboration as well. 19

The texts offer many opportunities for activities associated with conceptual development, but most tutors seemed unaware of them. This is indicated by the fact that, although discussions which involved sustained reasoning (photograph) were rarely observed.20 96% of tutors reported that usually ‘mis estudiantes realizan todos los ejercicios del texto de matematicas’.21 Most exercises are diligently done, but it was also observed that opportunities to probe and extend conceptual understanding and analytical capacity are lost (Box 8). Developing student conceptual understandin Box 8 g and capacity to think analytically is internationally It was clearly explained how to record acknowledged to be challenging. Trainers are a quantity in different bases. The aware of the difficulties (Box 9) and keen to tutors then constructed tables to develop this key aspect of SAT methodology. show how quantities were

Box 9 Entonces en general SAT tiene un desafío y es justamente ayudar a sus participantes a pensar más. Este es el problema. El Doctor Gustavo Correa decía si tan solo logramos a ayudar a los jóvenes pensar es un gran logro (¿). Porque están acostumbrados que le profesor les digan todo y ellos toman notas y cuando toman examen es muy literal y entonces en SAT les ponen difícil porque les está preguntando a cado rato ‘¿Y Usted que piense y porque es falso, y porque es verdadero, cual es la justificación?’

represented in bases 2 – 10 (some used beans to model the bases). Students worked in small groups and, with the help of the trainer, nearly all had completed the table by the end of the session. No time was given at the end of the session for observations, questions and the forming generalisations related to the activity Source: Observation May 2010

It is recommended that:

2. Tutors are explicitly taught the characteristics of ‘exploratory talk’, provided Source: coordinator interview Abril 2010 opportunities to develop these skills, and mentored in facilitating appropriate kinds of ‘exploratory talk’ in the classroom. 3. Topics where tutors regularly experience difficulties are identified before training so that questions, reflection, activities and discussion can be carefully planned. 4. Opportunities for developing mathematical thinking are clearly identified during training, and appropriate activities are clearly demonstrated.

Trainers frequently asked students a question in order to probe understanding but discussions based around these answers were very rare. 21 ‘My students do all the exercises in the maths’ textbook’. 20

5. Tutors should have a written record (e.g., training handouts) to support classroom practice and/or training in the taking of pedagogical notes, since inspection of many tutor texts revealed that tutors very rarely take pedagogical notes22.

General Training and consultation skills The process of developing an ‘atmosphere of consultation’ (Arbab, 1988) clearly transcends subject areas; it is the foundation of the SAT methodology. The thrice yearly, two week trainings are seen by coordinators as the key to raising tutor awareness of the qualities and attitudes necessary to create a transformative learning environment. Five different trainers were observed delivering Aprender Enseñando. It is clearly desirable that trainers contribute creatively to the course content and method of delivery, but the style of facilitation and content of the orientation varied to such a degree that tutors cannot be considered to have received the same training. In some of the trainings there was an open and accepting atmosphere, and tutors appeared to speak freely, but in others there was little opportunity for the tutors to express their thoughts, and on one occasion an opinion (valid in my view) was dismissed as incorrect. It is important that tutors have substantial and varied experience of an ‘atmosphere of consultation’, particularly during their early training; otherwise it is harder for them to develop such an atmosphere, in far more challenging circumstances, in their own classes. It is recommended that: 6. Aprender Enseñando is used to develop consultation skills and its delivery is collaboratively planned so that tutors have a comparable experience.

A quick inspection of approximately 60 tutor texts revealed that some tutors never take pedagogical notes, even when it is recommended that they do. For example on one occasion after a very clear analogy and diagram was used by a trainer to clarify a concept it was suggested that a note was made of this. Subsequent inspection of the books revealed that only 15/21 tutors had done this and that the notes, when taken, were often very brief. When note taking is not suggested pedagogical notes are very rare. On one occasion tutors were provided with paper to make additional notes and glue to stick this into their books. This increased the quantity and legibility of notes.

The use of questions and answers Allowing time for reflection

Box 10 No se trata de entregar conceptos totalmente elaborados, sino de plantear preguntas y situaciones que ayudan al estudiante a precisar ideas en las cuales posiblemente no se había detenido a pensar. ¿Cómo se llego a nociones como punto, recta, plano? ¿ Qué relaciones hay entre ellas? ¿Para qué sirven esas ideas?

The questions in the texts have many purposes, among them bringing to the student’s attention something that s/he may not have considered before (Box 10). The student may not be expected to say or write anything immediately after this reflection, but the act of doing it aids learning, encourages independent thinking, and enriches subsequent discussion. There is evidence that a slight pause between asking a Source: Bosquejo General: El Espacio y Sus question, even the simplest one, and a Elementos. student/tutor being required to answer, increases participation (Hodgen & Wiliam, 2006). It was observed that some trainers allowed time for reflection while others did not. It is recommended that: 7. Trainers allow time between questions and responses, and discuss the reasons for doing this with the tutors.

Questions: increasing participation It was observed that a large number of questions are asked by the trainers/tutors. Almost always, in both training and classes, either a particular person is nominated to answer (sometimes before the question is asked) or it is addressed to the whole group, and someone shouts out the answer. This means that, although most tutors/students contribute in most sessions, a vocal minority tends to dominate.

It is recommended that: 8. Trainers/tutors are taught a range of techniques for increasing tutor/student participation when answering questions.

Questioning: a formative assessment tool Box 11 Practice in classrooms is formative to the extent that evidence about student achievement is elicited, interpreted and used by teachers, learners or their peers to make decisions about the next steps in instruction that are likely to be better, or better founded, than decisions made in the absence of the evidence that was elicited. Source: Black & Wiliam (2009)

There is strong research evidence that improving formative assessment (Box 11) is the most effective way of raising student performance (Hattie & Timberley, 2007). Questioning can be a good formative assessment tool if the teacher asks appropriate questions, listens to the answers, and builds on the information gained. Student answers sometimes require probing and reflecting on in order to understand what is meant and to expose underlying assumptions which may be impeding progress. This can take time, but is essential for building up clear

communication. It was observed that some trainers are much more skilled than others in this process. The SAT tutors seemed unaware of the power of using student errors to explore and develop conceptual understanding, as is shown by the fact that 90% of the tutors in the questionnaire reported that they tried to teach so that the students did not make mistakes. Many of the questions in the SAT texts are designed to develop understanding; exploring mathematical ideas, making mistakes and subsequently analysing and rectifying them is an important part of this process (Box 12).

Box 12 Es necesario tener lápiz, papel y borrador en mano, no para copiar, sino para resolver personalmente todos y cada uno de los ejercicios, corriendo el riesgo de cometer errores, de tomar caminos esquivados y asumiendo también la responsabilidad de revisar, corregir y de proponer soluciones originales y creativas. Source: Bosquejo General: Manejo de Variables, Unidad 1

It is recommended that: 9. Tutors are supported and mentored in the didactic use of questions and ‘error analysis’ as a formative assessment tool; this includes being informed of the utility of this approach;, experiencing it in training; knowing where it is most appropriate; and being mentored in developing these skills by field assessor accompanimiento. 10. Students are encouraged to experiment, take risks, make mistakes, identify and correct them. This would facilitate exploratory talk.

Tutor/student questions Developing student capacity to form probing questions is an important part of the SAT methodology23. There are two ways this happens: Firstly, students are specifically asked to make up a problem for other students to solve. 92% of the tutors reported that ‘aprendo mucho inventando nuevos problemas’24. Creating problems or questions often requires more complex processing of the concepts than responding to a readymade question. This type of activity assists in the internalisation and development of concepts (Swan, 2006); the form and content of a student-made question gives insights into understanding, and so can be used for formative assessment. Also being able to create mathematical problems is a useful skill for tutors to acquire so that they can provide students with additional examples, and support them in forming their own problems. Secondly, there are questions that originate from the students’ own engagement with the topic and/or development processes. In the trainings for maths specialists25 it was observed that participants asked questions about mathematical reasoning and representations, but in the normal trainings and classes almost all the questions made by tutors/students referred to algorithmic procedures as opposed to more substantial conceptual or community development issues. It is recommended that: 11. Activities where question formation is explicitly asked for or which lend themselves to this are clearly identified in the trainings to help tutors more fully utilise the texts. 12. Trainers experiment with using small group work as a way of encouraging the formation of more reflective student questions, and encourage tutors to do the same.

Three ways of working The tutor guide Aprender Enseñando (Dooki & Arias, 2007) is studied by tutors in their first training. It is a basic introduction that divides SAT pedagogy into three parts: Lectura comentada; Estudio en pequeños grupos; y Estudio individual.26 In the field assessor report form (Appendix 3) it is made explicit that lessons are expected to be a combination of these 23

El estudiante aprenda a plantearse el mismo los problemas, buscar los datos y encontrar las respuestas (Bosquejo General, Suma y Resta) 24

I learn a lot inventing new problems The maths coordinator is building up a team of maths specialist who will support other tutors, run the micro centres (local, short trainings given when the need arises) and the best will deliver training. So far about 24 have attended a training, eight either have a maths degree or are near completing one 26 ‘Reading and discussion’, ‘Small group work’ and ‘Individual work’ 25

three groups: ‘Usa tanto la lectura comentada como trabajo en grupo y lectura individual?’27 is one of the criteria of a good class. All except one of the tutors (out of 29)28 were reported by field assessors to be doing this. The three methods are explained quite generally in the tutor guide, so that it is likely that each assessor/tutor/trainer has a different understanding of what these terms mean. In interviews and meetings it was clearly demonstrated that the SAT coordinators are deeply committed to improving the pedagogy of SAT, and would like to have more time to reflect on experience and initiate change. Each of the three methods above has an important contribution to make towards creating an atmosphere of consultation but it is possible that, as they are only described very generally, that staff members have very different conceptualisations as to their principal characteristics. So it would be fruitful if assessors, coordinators and tutors were given time to discuss what each one of the methods means to them and a richer conceptualisation is attained. If this does not happen it could be damaging, particularly in the case of trainers, if individual assumptions about the methods cause a departure from the core SAT methodology. It is recommended that: 13. Sufficient time is allowed for this at coordinator meetings for consultation on pedagogy.

The use of tutor/student presentations of work In about a third of the observed classes, one or two students were invited to the board to demonstrate their calculations. The student would, usually silently, write up the procedure and answer. The tutor or another student would comment if there was a mistake in the working out and/or the final answer. Use of the board followed a similar pattern in the trainings. Some trainers had more tutor demonstrations than others, but the main purpose of the activity was almost always to check the correct use of an algorithm. In smaller groups the whole group would, sometimes animatedly, join in with the calculation, but with larger groups there was less participation. Quite often it would become an

‘Equal amounts of whole group reading and commenting on text (combined doing related activities), small group work and individual work.’ 28 6 observations received no comment for this category 27

exchange between the trainer and tutor (both with their backs to the group), the trainer trying to plug a hole in the tutorâ&#x20AC;&#x2122;s understanding or of the algorithm. Checking algorithms in this way is time consuming. Also 23% of tutors reported finding it stressful. This might not be too concerning as long as it is not humiliating, which it appeared to be sometimes. Tutor whiteboard presentations should be used as a stimulus for discussion when a variety of responses are valid, and these can also be compared and evaluated. Presentations, such as that illustrated in the photograph (a tutor is telling colleagues how he explains factorisation), can help Box 13 weaker members of the group because peers often Effective teachers encourage their use more accessible language than the teacher. But in students to explain and justify their order to be effective, the tutor presentations need to solutions. They ask them to take and defend positions against the be clearly delivered and allow sufficient time to contrary mathematical claims of develop discussions and follow-up questions. It would other students. They scaffold also be better to have fewer presentations, but for student attempts to examine them to be well thought out and closely analysed. The conjectures, disagreements and aim should be to move towards analysing counterargument ... as attention mathematical methods and approaches and their shifts from procedural rules to practical applications rather than practising making sense of maths ... students algorithms (Box 13). become less preoccupied with It is recommended that:

finding the answers and more with the thinking that leads to the answers.

14. More efficient methods to check answers and Source: Anthony & Walshaw (2009):19 correct procedures (e.g., use of small individual white boards, peer checking, etc.) be used in the trainings and classes instead of whole class presentations by individual students/tutors. 15. Student/tutor presentations of work are used as the basis for discussions. 16. Appropriate activities are identified beforehand and adequate time allocated for them.

Working in small groups Peer teaching and ‘social loafing’

Peer teaching is encouraged in SAT, for example, 78%29 of tutors reported other tutors asking them for help with maths. There is strong research evidence to support peer teaching. In an internationally acclaimed meta-study (Hattie, 1999) it was found to have an above average impact on student achievement, and to be more effective than homework in improving student performance. With a peer, as opposed to a teacher, people often feel more relaxed, more able to ask questions, request clarification, try out something new and risk making mistakes. This complements the SAT philosophy that the whole group should move forward together, and no one should be left behind. In small groups those who are more advanced gain by explaining and supporting their peers as this helps them consolidate their own knowledge. This contrasts with whole group work when those who are already proficient can be bored by trainers/tutors revisiting material, as was seen during the whole class checking of algorithms. It was observed that in the trainings the tutors almost always work in unchanging friendship groups of two to four people, and on almost every occasion the exercises are done collaboratively. However it was also observed that occasionally more ‘social loafing’ than peer teaching or collaborative group work took place. It is also well known that constant group work can encourage passivity and dependency.

Another observed problem of constant collaboration was that struggling tutors were not always identified and given the help they needed. For example, in the training for El Sistema de Numeracion Decimal, a tutor was observed after the lesson copying out a table First year tutors (who had not yet taught maths/completed training) were asked not to respond to this question. Of the other tutors 28% failed to respond. 29

of numbers made with different bases. When asked about the table, she could not express any of the given numbers using a binary or any other base. Her textbook was correct, but it is probable that her teaching of this section would have been unsatisfactory (appropriate support, was of course, given to the tutor). The observed small group working was usually informal and unstructured; on only a few occasions was a group given a task that they were expected to provide feedback on as a group. Therefore the focus of the group/pair activity tended to be on each person filling in their textbook, sometimes with one member of the group playing the role of a transmissionbased teacher, and occasionally with the weaker tutors just copying out the answers. It is recommended that: 17. Tutors are given clear criteria of how to assess their own and others’ work, and guidelines are written on how to peer teach/accompany a tutor/student without creating dependency (perhaps using the Guias de Aprendizaje y Evaluaciόn) 18. Tutors/students are encouraged to work in different groups. 19. Tutors/students capacity to work in groups is developed by using different types of group work and providing opportunities for analysis of the effectiveness and inclusivity of the group work. 20. Tutors experience and reflect on a range of techniques for improving group work (see below) 21. Tutors are made more accountable for their progress in the trainings (e.g., a test at the beginning and end) and/or tutors’ texts are formally reviewed before the end of the training (this was a goal in the October 2008 Perfíl Nacional but not observed in the maths trainings). A forgotten method?

The following explicit instruction appears in the text Fracciones y Porcentajes (6.4): ‘para revisar los ejercicios, formen grupos de 4 estudiantes y cado uno explique a los compañeros de grupo como resolvió uno de los ejercicios’30. Similar instructions appear in many other places in Box 14 this and various other texts. The instruction implies ‘los momentos más importantes de that the students have done the exercise on their cada lesión son los dedicados a own, have had time to reflect on it, and then that explicar a los compañeros la forma each student should verbalise their understanding cada uno resolvió los ejercicios’ of the exercise and explain it in a small group. It Source Destrezas Algebraicas En la Vida Rural:303 appears to be vital part of the methodology (Box 14) and the effectiveness of this technique is also strongly recognised in the literature, for example:

“Revise your answers in groups of four, each person explaining how s/he did one of the exercises.”

‘In small supportive groups students learn how to make conjectures and engage in mathematical argumentation and validation’ (Anthony & Walshaw, 2009). In spite of the explicit guidance to use it, this method was never observed. In the Teacher’s Guide for this text there is no mention of a different review method having being introduced, and therefore it can be assumed that it remains a key SAT method as developed by FUNDAEC but it appears not to have been adopted in SAT-Honduras. It is recommended that: 22. SAT Text instructions, and specifically this one, are implemented, unless consultation with FUNDAEC reveals a method is no longer regarded as appropriate. 23. Recommended activities/methods, if not explicit in the text, are clearly explained to the tutors.

Individual work: the third force Thinking time The SAT texts are full of questions. Some of the questions are to draw the students’ attention to a concept or aspect of a concept of which he/she might be unaware and which is about to be developed in the text. The student is expected to reflect (briefly) on the question before reading on. Students are also sometimes explicitly told to work through an exercise independently. Independent working is an essential part of learning, as explained in a recent overview of international research on effective maths pedagogy: ‘Teachers should ensure that all students are given opportunities to think and work quietly by themselves when they are not required to process the varied and sometimes conflicting perspectives of others’ (Anthony & Walshaw, 2009).

In the tutor questionnaire, 88%31 reported that ‘mis estudiantes realizan los ejercicios por sí mismo y consultan a sus compañeros de vez en cuando’. It was observed that there were short periods of spontaneous independent working (2–5 minutes) in most SAT classes, but that this rarely happened in the trainings. The limited time for training was given as the reason. This is however a false economy since (according to research) it impedes learning, creativity and independent thinking, as well as making it harder to assess tutor progress. Not using this method in training gives tutors the impression that independent working is of less value than group work. It is recommended that: 24. Sufficient time should be allocated for ‘independent working’ in the trainings. 25. The reasons why independent working is important should be presented and discussed in the trainings.

Formative Assessment: fine tuning acompanimiento32 In most trainings and classrooms there is a clearly visible interest and concern about the progress of each tutor/student, and teaching is adapted in response to this. Interviews with tutors resulted in many examples of the considerable effort expended in acompanimiento, or accompanying each student on their developmental journey. Improving formative assessment skills would greatly assist trainers/tutors in their acompanimiento because the systematic use of a wider range of techniques would improve the quality of the information obtained about teaching/learning, and thus make more attuned adjustments possible. Many of these techniques are suggested in the texts but are not, as yet, being fully used. It is recommended that: 26. A range of formative assessment techniques is used; those in the texts are identified and, where necessary, additional ones are introduced and trainers/tutors/students are supported in developing their capacity to use them.

Taking responsibility for learning It is expected that SAT students will ‘dejan de ser observadores pasivos de hechos y circunstancias para convertirse en personas activas en la exploración de caminos de mejoramiento personal y comunitario’33(Valcarcel & Lizcano, 1995:6). This is achieved slowly by the acquisition of new habits and attitudes. One of the key attitudes/habits that 31

99% responded to question An analogy that is used for both the tutors’ and assessors’ role is that of a person who ‘accompanies’ another on their educational journey. 33 Stop being passive observers of events and become active in the search for new ways of changing oneself and the community for the better. 32

need to be developed in order to achieve this paradigm shift is taking responsibility for one’s own learning. Tutors identified this as their second most important priority when working with the students34 (Appendix 3).

Unsupervised working During the tutor training periods SAT groups are often left unsupervised. Twelve tutors were asked what their students were doing during this two week period. Only one replied that they were not doing a SAT related activity. The others either had special tasks (e.g. reading a book together and doing a report) or were working on the group agricultural plot. No independent maths work was reported, apart from revising for tests. The tutors were confident that the unsupervised work would be done. On two occasions35 the tutor was absent and the class was observed working collaboratively with their maths textbooks. And just as impressively, in all the maths classes observed student discussions Box 15 focused on the maths. Once or twice a tutor urged Yo prefería que hagamos las tareas en students to work faster, but it was never necessary to el aula porque así yo los miro que ask a student to return to the task.

Homework

trabajan realmente. Si les doy un trabajo y se encuentra a un amigo que ya está en secundo y lo ayuda? Hacen la mayoría del trabajo dentro del aula. Yo les doy cosas para hacer en la casa que yo se que pueden hacer.

Coordinators and assessors reported that in some groups the assigned homework was not done or if it was done, copying was widespread, while in other Source: Interview with first year tutor groups homework was well done and appeared to be whose group had no retakes in first having an impact on learning. The texts emphasise the module. May 2010 importance of fluency, speed and accuracy when calculating as well as conceptual understanding. Homework can be very useful in developing the later (Box 15) It is recommended that: 27. Tutors are given opportunities to share good practice on homework (and other aspects of practice) e.g., trainings, micro-centres, tutor meetings, bulletin, website.

Formative assessment helps students take responsibility for learning It is easier for students to be responsible for their own learning when they know what they are supposed to be learning and the criteria for success, and when they are given clear guidance on what they have to do to improve their performance. It is suggested in Aprender Enseñando that the Guías de Aprendizaje y Evaluación should be studied closely by 34

When asked to allocate 100% between six possible priorities (appendix 4) the two highest scoring ones were: ‘Motivarse y tomar responsabilidad sobre su propio aprendizaje’ (mean: 24%) and ‘Utilizar la matemática para explorar y describir el mundo y resolver problemas de su vida cotidiana’ (mean: 32%). 35 Observations made during a previous visit, October 2008

tutors and students together as a way of evaluating learning, but at no time was this observed in either the trainings or classes. It is recommended that: 28. The use of the Guias de Apprendizaje y Evaluación as a formative assessment tool is investigated, or students are given other means to help them monitor their progress.

Tools and Representations If the metaphor for the learner is a ‘mine rich in gems to be discovered and polished’ rather than ‘an empty pot to be filled’, then appropriate ‘tools’ (Box 16) are needed.

The importance of the texts

Box 16 Effective teachers draw on a range of representations to support their students’ mathematical development. These include the number system itself, algebraic symbolism, graphs, diagrams, models, equations, notations, images, analogies, metaphors, stories, textbooks and technology. Such tools provide vehicles for representation, communication, reflection and argumentation. They are most effective when they cease to be external aids, instead becoming integral parts of students’ mathematical reasoning. As tools become increasingly invested with meaning, they become increasingly useful for furthering learning.

In her seminal study Ma (1999) found that the most effective Chinese primary school teachers Source: Summary of research findings developed a ‘profound understanding of Anthony & Walshaw (2009) fundamental mathematics’ (PUFM) after becoming teachers, and that this understanding continued to develop throughout their careers (in contrast to US teachers who usually do not develop PUFM). When interviewed, the teachers with PUFM attributed this to: teaching more than one grade and intensive study of the teaching materials both independently and with colleagues (Ma, 1999). This is good news for SAT in that tutors teach different grades, and the texts are studied as part of the training and so there is a good basis on which to build.

Box 17 I now offer a new perspective from which to expand our understanding of the concrete. The more connections we make between an object and other objects the more concrete it becomes for us. The richer the set of representations of the object, the more ways we have of interacting with it, the more concrete it is for us. Concreteness, then, is that property which measures the degree of our relatedness to the object, (the richness of our representations, interactions, connections with the object), how close we are to it, or, if you will, the quality of our relationship with the object. Concreteness is not a property of an object but rather a property of a person’s relationship to an object. Concepts that were hopelessly abstract at one time can become concrete for us if we get into the ‘right relationship’ with them.

Multiple representations It is international good practice, as in SAT, to view the learner as an active participant in the learning process. It is widely agreed that each person constructs his/her own understanding by interpreting and internalising new information and experiences. An implication of this is that the more complex and varied is the experience of a concept, the more ‘concrete’ or real it is to the learner (Box 17). Some of the texts have rich and varied representations of concepts (e.g., El Sistema Decimal), while others (e.g. Manejo de Variables Unidad 1) assume a familiarity or ‘internalisation’ of algebraic symbolism which tutors appear not to have developed (as shown by the errors made). The better trainers use analogies, examples and sometimes objects (e.g., a pair of scales when explaining equations), but it is likely that many of the tutors do not subsequently use these as very few take notes about pedagogy.

Source: Wilensky (1991)

In other texts, there are several situations in which tutor/student knowledge or understanding is assumed which they do not yet have (Box 18). The act of constructing a representation and experimenting with it makes a concept more ‘concrete’. These sorts of activities are recommended in the text but were rarely observed in the

Box 18 On turning to page 232 El Espacio y sus Elementos at a micro centro with four students. There was a murmur of recognition when Pythagorus was mentioned. With a little prompting of one another they recited, in unison: ‘the square of the hypotenuse is equal to ..’ When asked what that meant, someone said: ‘a squared plus b squared ..’ And when asked what that meant, no one could offer an explanation. The trainer drew a diagram drawn on the whiteboard showing the sides squared and demonstrated how the squares of the two sides were equal to the square on the hypotenuse. The tutors expressed pleasure at the insight that this diagram gave them and said that now they could understand the diagram in the text. Source: Observation March 2010

trainings. The reason given for this was lack of time. Perhaps some of these issues stem from the fact that the texts were written for Columbia where the standard of maths education is higher. In Honduras is it necessary to provide varied representations at secondary level to make up for educational deficiencies at the primary level (in the photograph students are doing an exercise on multiples representations with algebra36) Many coordinators and tutors identified extra materials as an urgent necessity. It is recommended that: 29. Tutors are informed of the utility/theory of multiple representations and related activities in the text are clearly identified. 30. The texts are analysed with tutors to identify where extra representations and tools are needed, and these developed in collaboration with the tutors. (The Maths Coordinator has for many years wanted to address this concern and in June 2010 Asociaci贸n Bay谩n submitted a proposal to Irish Aid for funding). 31. When the representations are physical they should, as far as possible be made with locally available and sustainable/recycled materials (e.g. photographs: representing the difference between (3 + 2 )2 and 32 + 22 using bottle tops and seeds to represent the decimal system)

Activity taken from The Standard Units developed by Malcolm Swan, Susan Wall & others

Updating and revision of texts The texts are innovative and engaging. The maths is integrated with the other core SAT areas, communication, technology, science and most importantly, service to the community. The constant application of the maths to worthwhile practical applications is motivating. This is demonstrated by the vast majority of tutors37 reporting that studying with SAT had made them feel much more positively about maths and generally giving the practical nature of the texts as one of the reasons. However there are some concerns: 



Box 19 Lo mismo que las otras unidades hay un énfasis en el desarrollo de actitudes positivas, como las siguientes: compromiso personal en el aprendizaje proprio y del grupo; satisfacción por enfrentar y vencer retos; reconocimiento del valor de lo que se aprende para la formación personal y para la solución de problemas; honestidad en la búsqueda de soluciones, convicción en las respuestas; reconocimiento de los propios errores y interés por corregirlos

Most coordinators interviewed spontaneously complained about the Source: Bosquejo General, Manejo de number of errors in the texts because they Variables, Unidad 4 undermine tutor/student confidence in the texts38. Some of the information is out of date, and therefore now incorrect. For example, in Matematicas 1 Conjuntos y Numeros, (revised by Asociación Bayán in 2008), the number of species is given as about a million. Current estimates are between two and one hundred million (an interesting fact in itself) and no mention is made of the rate at which species are judged to be disappearing. Other opportunities to engage tutors and students with current issues are missed. Examples of this are: the balance between using animals for economic gain and humane treatment, and the limitations of double entry book keeping e.g. it does not record the social or environmental cost/benefits. In Para Salvar las Barreras it is suggested that additional material is introduced where appropriate but this was rarely observed. The study and practice of maths offers many opportunities for the development of a range of attitudes/qualities/skills, for example patience, perseverance, courage, attention to detail, etc. Trainers/tutors rarely referred to this. These are sometimes mentioned in the Guides to Learning and Evaluation and in the introductions to the texts, as in Box 19. This example is from the second year of Bachillerato but it would be beneficial to give more emphasis to attitudinal development from the first text/guide.

Percentage For example in Manejo de Variables: Unidad 2 (adapted and contextualised by Asociación Bayán 2004) there are at least 14 typographic errors, while some are minor (e.g. omission of question marks) six could impede understanding (e.g. incorrect answer) 38



It is regrettable that the standard of illustration is not higher, for example, many of the drawings of animals in Conjuntos y Numeros look as though they have been rather carelessly traced. In one instance the illustrations cause confusion and need additional material or redrawing.39 It was reported by the maths coordinator that some of the longer exercises consistently fail to engage the students (e.g., the final exercise for the community water supply and study of house plans). The exercises were developed in Columbia where the original students were older than the majority of the present ones in Honduras, perhaps this is the reason why they are less successful in Honduras than Columbia. It is desirable that the students are fully conversant with the metric system but Honduras still mainly uses the Imperial system and more opportunities should be given for them to become competent in both systems and conversions. The texts produced in Honduras do not always follow FUNDAEC referencing conventions (and international good practice). In Aprender Enseñando there is reference in the introduction to material from the Ruhi Institute being used but it is not clear where. Substantial passages40 from Para Salvar Las Barreras (FUNDAEC, 1995) have been used without identifying the source.

It is recommended that: 32. Errors be corrected (the most urgent task), new illustrations commissioned in order increase the utility of some representations (e.g. Suma y Resta), and the general attractiveness of the texts improved. 33. The texts are analysed to identify where exercises need to be modified/changed to make them more engaging to Honduran students and where extra information or discussion topics could be introduced. This could be put the Teaching and Evaluation Guides. 34. Activities are developed to enable students to become competent with metric and imperial systems. 35. Texts follow referencing conventions with regard to acknowledging the origin of material taken from other sources.

The tutors in the Maths Specialist Course reported that students find the diagrams confusing in the first exercises in Suma and Resta, La Union de Conjuntos:1. Tutors said that that these exercises are demoralising since addition is one of the few aspects of maths that students feel confident about, and they get very dispirited when they find they cannot do the first exercises in the book. The Maths Specialists suggested developing a range of activities such as stories the students could act out in order to clarify the concept of combining sets with identical elements. It might be clearer to use intersecting Venn diagrams to represent this concept rather than the current diagrams. 40 e.g. pages 40 -44 in Aprender Enseñando are very similar to pages 28 – 32 in Para Salvar Las Barreras but not referenced

Guías de Aprendizaje y Evaluación (Ciclo Común)41 Informing learners of the content, purpose and criteria for success of the curriculum is a key factor in enabling students to become independent learners (Anthony & Walshaw, 2009); this is emphasised in Varcarcel and Correa (1995) and they identify the important role of the Guias de Aprendizaje y Evaluaciόn42 (GAE) in this. The GAE are booklets of 4 to 10 pages that accompany each text. They contain an outline of the curriculum, indicators for assessing student progress, and two to five short paragraphs of ‘observaciones’ or guidance. This guidance tends to be fairly general.43 Box 21 Ser una organización comprometida con el bienestar individual y colectivo, la excelencia, la innovación y la transparencia; con una sólida base ética, moral y espiritual, coherente en su accionar; con una actitud y práctica de aprendizaje. Source: Bayán website

Box 20 En realidad, más que guías de la evaluación, son guías de estudio para ser analizadas por el tutor y los estudiantes antes de iniciar cada unidad. Se buscar con esto promover una verdadero participación de tutores y estudiantes en el proceso de evaluación, mayor conciencia sobre los propósitos que se pretenden y hacer realidad la evaluación permanente y participativa Source: Aprender Enseñando:41

The role of the GAEs in evaluation is discussed in the Evaluation section below but evaluation is not the key purpose of the GAE, as is explained in Box 20. The GAEs are designed to help tutors and students become active, analytical and collaborative participants in the learning process by informing them of the curriculum. Asociación Bayán’s vision (Box 21) has ‘transparency’ as a key value. One of the ways this is demonstrated in an educational institution is by making the curriculum and evaluation criteria clear to the teachers. Therefore for both philosophical and practical reasons it is important

that tutors are trained to use the GAEs. Unfortunately this does not appear to be happening: in 186 hours of observation, the GAEs were never used or referred to. The GAE for Book 4 (multiplication and Source: Asociación Asociación Bayán division) was not usable because it had not website been updated for the revised text. Tutors cannot use the GAEs without training because they assume a knowledge and

Box 22 The secret of what anything means to us depends on how we've connected it to all the other things we know. That's why it's almost always wrong to seek the "real meaning" of anything. A thing with just one meaning has scarcely any meaning at all Source: Minsky, 1987 p. 64

The Bachillerato GEA were not made available until 08/07/10 and so are not included in this report. Learning and Evaluation Guides 43 ag., ‘representar fracciones en la recta numérica implica comprender muy bien los conceptos que se están manejando’ and ‘insistimos en la necesidad de seguir muy cerca el trabajo de cada estudiante para verificar el desarrollo de las capacidades …’ 42

Box 23 Teachers’ manuals provide the background for the maths in the corresponding textbook and suggestions of how to teach it. The introduction of a typical teacher’s manual gives an overview of the textbook, its main topics, the rationale for the text book’s organisation, the relationship between the topics in the textbook and the topics of the preceding and succeeding volumes. The main body is a section by section discussion of each topic and subtopic of the textbook. The discussion of each topic focuses on these questions: What is the concept connected with the topic What are the difficult points of teaching the topic What are the important points of teaching the topic What are the errors and confusions that students tend to have when learning this topic.

methodology which the tutors do not necessarily have. Another important role that the GAEs could play is in enabling tutors and students to create links between concepts and topics. The formation of links is believed to be important in the process of building up rich conceptualisations (Box 22). The GAEs, with their brief, clear descriptions of concepts, skills and attitudes could help tutors and students do this. When the tutors are in the communities they often have very limited access to support. The field assessors may visit a SAT centre once every two weeks but they, and other tutor colleagues, often do not have the knowledge to help (it is anyway unrealistic to expect field assessors to know the specifics of 70 texts). Almost every coordinator and tutor interviewed spontaneously requested additional materials so that (often isolated) tutors could have something to refer to for guidance and extra information.

Ma (1999) in an internationally acclaimed study found that having guidance such as that in Box 23 Source: Ma (1999) played an important role in developing teacher capacity. The GAEs, Aprender Enseñando and 44 texts comprise the tutors’ written orientation/instructions on mathematics and pedagogy. These contain a wealth of guidance on different topics but it is often difficult to locate it because there are no indexes and tables of contents (if present) tend to be very general. There is a need to identify what support the tutors need and, if present in the texts, make it easier to locate and, where necessary, develop new materials. This would assist with micro centro45 trainings as well as individual study. It is recommended that: 36. The Guias de Aprendizaje y Evaluación are studied in the training and tutors are mentored in their use in the communities. 37. There is increased reflection and use of the introductions to the texts. 44

The introductions to the texts are an under-utilised resource. They are read through at the beginning of a training and were not observed to be referred to subsequently. 45 Short training organised locally for a small group of tutors to address a specific training need.

38. Appropriate additional materials, are developed in collaboration with tutors, to support the tutors in their mathematical development and teaching practice (Irish Aid funding has been requested for this). The Guias de Aprendizaje y Evaluación are adapted if necessary, studied in the trainings and tutors are mentored in their use in the communities. 39. Appropriate additional materials, are developed in collaboration with tutors, to support the tutors in their mathematical development and teaching practice (Irish Aid funding has been requested for this).

Practical application of maths Linking the maths being learnt in the classroom to real life applications is widely acknowledged to be motivating and conceptually enriching. The tutors expressed great enthusiasm for the practical nature of SAT maths in interviews and questionnaire responses. Interestingly, assisting with their own university studies was the most common example given of how studying maths had benefitted them46 (Box 23). 78%47 said that they usually related the maths they taught in class to maths in daily life, and 96%48 said that maths was useful in community development projects. When asked for examples of how students used their maths in the community, the most Box 24 frequently mentioned activities were ‘shopping’, Quien trabaja por el mejoramiento de ‘measuring out land’ and ‘planting’. Surprisingly las condiciones de vida en el campo debe ser capaz de analizar indicadores there was no mention of statistics. Statistics

Coordinators report that statistical projects (for example, on chicken production, community health, etc.) are sometimes done in a perfunctory

sociales, entender como se construyen, aplicarlos para poblaciones especificas, determinar su utilidad y también sus limitaciones e incluso hacer propuestas para mejorar los indicadores existentes. Source: Bosquejo General: Destrezas Algebraicas en la Vida Rural

way because these projects are not as important as the end of module tests in determining if a student passes the module. The data in the texts is often out of date (e.g., in El Estado de Salud de La Comunidad the data is for 1997), and

There were many references to ‘daily life’ but ‘university studies’ was the only specific example. 98% of respondents answered this question 48 95% of respondents answered this question 47

sometimes statistical concepts are introduced but not elaborated sufficiently in the text (e.g., the concept of the mean in the text Aritmética en la Investigación Científica). In the training for Destrezas Algebraicas en la Vida Real very little discussion or analysis of social indicators was observed which was surprising as it is a fundamental tool in community development (Box 24)

Linking SAT and community maths 53% of tutors reported that ‘mis alumnus comparten en la clase los métodos matematicas que sus padres utilizan en su vida cotidiana’. When asked for examples, after some head scratching, tutors gave examples of simple agricultural situations, e.g., spacing between plants. The educative process should generate ‘una armoniosa fusión de saberes, del saber tradicional con el conocimiento moderno (Valcarcel & Correa, 1995). It appears that this process and potential is under-developed in Honduras-SAT as yet. It is recommended that: 40. More up to date data is used in the trainings. 41. Tutors are given copies of this data to use in the communities. 42. Statistical concepts inadequately explained in the Box 25 texts should be identified, and En SAT pretendemos evaluar procesos, más extra exercises developed. que resultados. No se trata de registrar si el 43. Trainings are carefully planned estudiante adquiero tal concepto o so that key statistical concepts desarrollo tal capacidad o actitud, sino de are adequately developed analizar sus progresos en caso de que sean 44. Ways of increasing the value mínimos, tratar de detectar las causas y that students place on encontrar la manera de superar esa statistical projects are situación; lo mismo en casa de buenos avances, promover el ir más allá en la investigated (and the copying búsqueda de aplicaciones o de ampliaciones. of previous years’ assignments Source: Aprender Enseñando:40 stopped). 45. Ways of integrating traditional representations and methods (estimation, measurement, etc.) are investigated.

Evaluation In SAT, evaluation is more concerned with continuous reflection on the process rather than making judgements on a final result (Box 25). Effective evaluation requires acting and observing simultaneously; this is a power of the mind which has to be developed (Box 26).

The main purpose of evaluation in SAT, as stated in Aprender Enseñando, is ‘assessment for learning’. Aprender Enseñando states that the GEA should be used to capture the learning process; and make adjustments based on the information. It is difficult to see how tutors can use the GEAs to do this because the GEAs are mainly lists of the concepts, capacities and information that students should develop or acquire, (and their related indicators), but there is no mention of how the process should be captured. It is assumed that the tutor will be able to interpret the GEAs, but as they have not studied them in the trainings this is not possible. At the moment they are more a guide for ‘summative’ or end of course assessment than for ‘assessment for learning’. It is recommended that:

Box 26 The image of two birds, one eating and the other watching pervades Eastern and Western art. There are multiple interpretations ... an educational interpretation is that the two birds are forms of attention ... the eater is the part of us that gets caught up in doing while the watcher is the internal monitor-witness that observes without judging. The watcher has been likened to conscience which needs to be awakened and developed...When applied to teaching it means it means having a part of you separately witnessing the lesson, able to observe and suggest alternative actions Mason & Johnston-Wilder (2004)

46. Tutors are systematically supported in the use of GEAs for assessment for learning and, if necessary the GEAs are adapted or additional material provided.

Community Evaluation All of the parent groups are expected to be active in the evaluation of learning and identification of goals. Some groups are very active (Box 27).

Box 27 En algunas comunidades de X además de la sociedad de Padres y Madres de familia existe un asociación a la que se le llama la Junta del Colegio. Esta junta está conformada por 5 a 7 miembros, electos en una reunión donde todos los habitantes de la comunidad participan (tengan o no hijos estudiando en el centro educativo). Esta junta vela por el desarrollo de actividad para todo el centro ... Su labor es apoyar al máximo el bienestar de la comunidad educativa y se encargan de las actividades macro, es decir actividades en las cuales toda la comunidad debe estar involucrada. Es importante mencionar que muchos de los miembros de la junta son jóvenes, que tiene niños pequeños que no van ni al Kínder o están en la escuela y algunos no tienen hijo. Source: coordinator email junio 2010

Evaluation â&#x20AC;&#x201C; testing and retesting Percentage of students retaking test by area 50 40 30 20 10 0

Maths: a problematic part of the program

Of the five SAT curriculum areas, students generally have least success 2008 with the mathsâ&#x20AC;&#x2122; tests. Data on test 2009 performances are available from 2008, 2010 and shows that maths has almost invariably been the area with the highest number of test retakes (Figure Figure 1 1). It was reported orally that maths has been problematic since the beginning of the program. This can be largely explained by the fact that tutors come from a low quality educational system (Box 28) and that the tutorsâ&#x20AC;&#x2122; literacy levels affects maths understanding (Box 29). It is widely believed by tutors that the maths in the SAT

percentage of students retaking test

2008 Block 2: Impulsor II 100 80 60 40 20 0 A B C D E F G H I J K K L M N Region

Photograph:Ruth Moulton

curriculum is at a higher level than other Honduran Bachilleratos; one reason for this could be that it demands more conceptual understanding and for this to be articulated. It is therefore unsurprising that there are significant problems.

Percentages of retakes for 2008 Students take a test at the end of each module and if they score less that 60% they are required to retake it. Data on retake rates for different regions was available for 2008 (Block 2 & 3). 2008. 2008, Block2, Impulsor 2 was selected as an example because it was the most complete. The wide range is remarkable and sometimes difficult to explain. The two regions with the highest retake rates border on the region with the lowest rate (I, M & K). The tutors would have attended the same training and the student population was similar.

Some centres had lower retake rates at each level. It has been suggested that one reason for this could be that different criteria are used to define a retake. In 2008 a student could retake the test various times, and some centres might have counted each attempt as a retake while others only counted the student once. Also, it was reported to the consultant, some centres allow students to redo a couple of answers in a test if they only are a few marks short of 60, and do not count this as a retake, while other centres do not allow this. It is also possible that good practice is affecting retake rates. If so, this should be identified and widely shared. A standardised system of data collection and analysis is necessary to be able to make any judgement. Data on summative assessment (if valid and reliable) can be valuable in the evaluation process. Currently the test data are of limited value as the retake rate is a blunt instrument for measuring performance. All students gaining 0-59 marks have to retake. There is no way of knowing if a student scored 59 or 0, and whether those passing scored 60 or 100. Data on maths performance is held in the Institutes to which each SAT centre is affiliated, but is difficult to obtain. It would greatly assist Asociación Bayán in identifying trends and assessing the effectiveness of educational interventions, as well as to monitor the progress of individual students, if data on test scores were collected. It is recommended that: 47. Criteria for defining a ‘retake’ are standardised (Asociación Bayán is developing a data entry system to improve accuracy of entry and calculations, and there are plans to produce a data collection manual for staff) and reports are archived. 48. Data on student and tutor test scores are collated and analysed.

Comparing SAT performance with the government system The Universidad Pedagogica Nacional Francisco Morazan is in the process of carrying out a study for the Secretaria de Educación Pública (SEP) comparing the ‘rendimiento academico y factores asociados’49 of SAT over 2008-2010 with a control. The control is the Secondary School or Centro de Referencia (CER) that the SAT group is associated with for administrative purposes. There are 53 SAT Centres (853 students) and 25 CERs (1,234 students) in the study. There was not a statistically significant difference between the two groups but as the questions are based on the CER and not the SAT curriculum, the validity of the comparison is questionable50. It was interesting to note that the SAT students who had not been taught 49

Academic achievement and associated factors. However, due to frequent teacher strikes/absences, the CER students may not have been taught the material that the test was based on either! 50

statistics, probability and geometry did about as well as the CER students, who supposedly had been taught these subjects! The Hewlett study, which has a more rigorously defined comparison group, plans to apply questions which have been designed for an international comparison study. Tutors and mathematics: knowledge, attitudes and beliefs about teaching and learning51 It is widely recognised that obtaining information on attitudes and beliefs is problematic, and the results should therefore be treated with caution. However, as SAT is concerned with the development of attitudes and beliefs and the impact they have on action, it is something that must be attempted. Based on empirical evidence, it is essential to ‘start where the learner is’ and this requires knowing something of the underpinning premises that the learner is using to construct his/her understanding of mathematics.

Tutor attitudes to maths: the positive results of training The tutors reported generally positive attitudes towards maths in the questionnaire survey (Appendix 3). Almost all the tutors said that being good at maths was something that was accessible to all, not just the ‘brainy’. They appeared to enjoy learning maths, and feel ‘un gran satisfación con exíto con un problema matematico’52. All53 SAT tutors thought ‘las matematicas son utiles y necessarias en todos los ambitos de la vida’54. But, interestingly, only 76% thought that maths was necessary for understanding the sciences. There is a tendency in attitudinal questionnaires to report what is considered to be the ‘correct’ attitude, and so the answers may be aspirational or compliant, but, even taking this into account, there is a pleasing positivity about maths. A remarkable 98% of tutors55 reported feeling more positive about maths after studying the SAT texts, and 66%56 of tutors felt they were ‘capáz y hábil’ at maths. 83% reported that 51

A rigorous selection process, interviews and two week initial training, assist Asociación Bayán in selecting tutors who appear to have an attitude of learning and are orientated towards service to the community. Co coordinators expressed the expectation that as the number of SAT graduates working as tutors increases, the training process will become easier because they will already be familiar with the texts and core values/assumptions. 52 ‘Very satisfied with when a maths’ problem is solved’. 53 Over 99% of respondents replied to this question/statement 54 ‘Maths is useful and necessary for all aspects of life’. 55 All tutors who had studied with SAT for more than a year reported improved attitude to maths 56 95% of respondents replied to this question/statement

after training they felt they understood all the material and could do all the exercises. This is supported by interviews and casual conversations in which tutors often spontaneously said how much they had learnt with SAT. It is an outstanding achievement to have enabled the vast majority of tutors to feel more positive about maths (Box 28).

Tutors particularly mentioned the practical application of maths, the emphasis on ‘porque?’57, and how easy it was to ask questions in SAT trainings. However there appears to be a minority who struggle in each training and, are either left behind or slow the progress of the whole group. If the standard of maths in SAT is to be raised it is vital that these tutors are identified and systematically supported.

Box 28 A mí me encanta las matemáticas y yo siento que lo que he aprendido en SAT me ha ayudado mucho en mis estudios universitarios y en mi vida diaria. Siempre me he sentido que me gusta las matemáticas pero desde que estoy en la sistema he descubierto que se resuelven muchos problemas de la vida con mejor facilitad. Estudiar los textos de SAT y es muy importante ya que ayudan en gran manera en nuestra vida como profesional también para tener alguna destreza conocimiento al momento de usar el estudio superior en la universidad. Source: comments on tutor questionnaires

It is recommended that: 49. Tutors who find maths difficult are identified at the beginning of their training so that they can be given extra support 57

‘why?’

Building on an apprenticeship of observation Almost all tutors have had a long apprenticeship of observation in the transmission approach during their twelve or so years of schooling and the questionnaire survey indicated that many hold beliefs consistent with this approach (Appendix 3). Research shows that this deep identification with often respected teachers58 and their mode of delivery means that ‘superficial attempts’ to change teachers to a more active style will either be ‘neglected’ or ‘ignored’ (Schwille & Dembélé, 2007:30). The skills that tutors have acquired in the transmission method (e.g., clear explanations, see photograph) should be recognised and developed and their beliefs about teaching and learning explored (Box 29). One coordinator said that he noticed that tutors who were less good at maths tended to have more participatory lessons, and that the most knowledgeable ones tended to take over and teach in a more formal way. This implies that tutors may have a tendency to adopt more transmission style if their knowledge increases and trainers should be aware of this. Changes in beliefs59 and practice are complex but it is possible that changes in practice precede changes in belief (Swan, 2008). It is therefore important to use participatory and active Box 29 methods when training to demonstrate alternative Any attempt to encourage teachers methods and tutors should be given specific and to modify their beliefs about supported opportunities to develop their practice teaching and learning must towards active methods which have been shown to therefore entail the explicit be effective. It is very probable that most tutors are recognition of existing beliefs and disposed towards doing this as they reported their situated nature, attend to the developing student capacity for independent constraints and difficulties under learning as a priority (Appendix 3). Research with which teachers work and allow teachers indicates that sustainable changes in teachers time to reflect on the practice are achieved by making a few, small contradictions between ‘good changes in consultation and collaboration with other practice’ and ‘existing practice’. teachers (Hodgen & Wiliam, 2006). Field assessors Source: Swan, 2006:173 could play a very important role in facilitating this.

Surprisingly (considering reports of the standard of maths teaching in Honduras) 79% of tutors thought that they had had good teachers at school (questionnaire). This is at variance to a discussion on previous maths’ experiences in the maths specialist training when several tutors said how ‘traumatised’ they had been. 59 For example, a change from seeing the student as ‘a mine rich in gems’ rather than ‘an empty pot’.

Theory- based practice Research indicates that the ability to relate practice to theory is an important tool in the process of reflecting on experience. Most tutors are studying at the university and it would be enriching for them to form links between the educational theory that they are learning there and SAT theory and practice. It is possible that the failure to fully implement the SAT methodology is due to a lack of understanding of the reasons for some of the practices. The importance in SAT of relating conceptual understanding to the practical application of knowledge was mentioned during trainings, but few other instances of linking practice to other aspects of the SAT theoretical framework or wider educational theory were observed. Also developing tutors’ theoretical knowledge would assist tutors to modify their practice in a systematic way as they could research and modify their practice using a theory of change model. 60 It is recommended that: 50. The tutors’ pedagogic/theoretical learning in SAT is linked to the educational theory and pedagogy studied in the university. 51. Tutors are systematically informed about SAT methodology and theory. 52. Tutors form groups to conduct simple collaborative action research on SMALL changes in practice based on a theory of change model. 53. Assessors are trained to support tutors in developing more participatory practices and to promote and support action research.

Maintaining positivity: getting unstuck ‘Being stuck is an honourable state’ (Mason & Johnston Wilder, 2004:287). Being stuck helps learners develop powers of reflection, perseverance and courage. It is often a necessary stage when studying and applying maths and learners should realise this. The most stressful thing for tutors (25% of responses61) was ‘cuando me atasco o bloqueo en la resolución de un problema’. When asked how they overcame ‘maths blockages’, tutors did not have any defined strategies apart from ‘not giving up’.

a cost effective and perhaps most efficient way of conducting educational research. The second most stressful thing was being unexpectedly asked to do a presentation at the whiteboard (19%). 61

Box 31

It is recommended that: 54. Tutors are taught techniques for overcoming ‘bloqueos’, and that this is normal and even useful for the learning process.

Box 30 The knowledge gap between the US and Chinese teachers parallels the learning gap between Chinese and US students ......the quality of teacher subject matter knowledge directly affects student learning – and it can be immediately addressed Source: Ma (1999)

Tutor knowledge Tutors are not expected to be ‘teachers’. They guide their students through each of the five components of the program and it would be unrealistic to expect all tutors to have the knowledge of a specialist maths teacher but a sound basic knowledge is necessary (Box 30). All coordinators interviewed expressed concern about the level of tutors’ mathematical knowledge although 66% of the tutors reported themselves as ‘capáz y hábil’. It should be remembered that judgements about one’s own performance are made via comparisons within a reference group.62 Many of the tutors may feel that they are relatively good at maths63, but this does not necessarily mean their maths is good enough to fully

El sistema educativo de Honduras presenta un retraso de 100 años respecto a las naciones vecinas de Costa Rica y Panamá, según un informe divulgado este lunes por la Organización de Naciones Unidas para la Ciencia, la Cultura y la Educación (UNESCO) y el Banco Centroamericano de Integración Económica (BCIE). En materias como matemáticas o lenguaje el porcentaje de aprobados es inferior al 47 por ciento, una tasa muy lejana al 70 por ciento exigido en los Objetivos de Desarrollo del Milenio (ODM) establecidos en 2000 por Naciones Unidas. A pesar de que la tasa de alfabetización de Honduras ascendió del 68 al 80 por ciento en la década de los noventa, el documento de UNESCO señala la necesidad urgente de mejorar el sistema educativo hondureño. 'Estamos 100 años atrás de lo que están Panamá y Costa Rica, sin compararnos con países como Canadá, Estados Unidos o Japón, sino a nuestros vecinos más cercanos', señaló el coordinador de la Red de Educación de Honduras, Elmer Villeda, durante la publicación del informe. En los últimos tres años apenas se han impartido un cincuenta por ciento de las clases establecidas en el calendario escolar hondureño, según otro informe publicado por la Federación Iberoamericana del Ombudsman (FIO), una red que sirve de apoyo a todas las defensorías del pueblo de América Latina, informó el diario 'El Heraldo'. Source: 20 de abril http://noticias.terra.es

Students at the university report knowing more maths in their peers and on at least one occasion being asked to assist the lecturer in teaching the other students (email correspondence with coordinator) 63 Tutors often reported how studying maths with SAT had meant they were one of the better students on their university maths courses.

Box 32 Hay un ejercicio que hacemos con ellos por ejemplo. ¿4 x 6 es lo mismo de 6 x 4? y la mejoría, el 99% dice que sí, es el mismo. Pero si se presentan situaciones de la vida cuando no es la misma … claro tiene razón, porque ellos han aprendido las tablas aritmética y allá no más. No han aprendido a relacionarlo con algo cuando les viene a SAT les cuesta. Source: coordinator interview, May, 2010

access the SAT curriculum (Box 31 and Box 32). 64 The students entering SAT come from rural primary schools. Frequent teacher absence and small schools, in which five grades are sometimes taught together, means that the students have usually received a woefully inadequate primary education. Teaching quality is important for all students, but it has most impact on the weakest (Hodgen & Wiliam, 2006). This increases the pressure on SAT tutors to be good at their job.

Box 33 Una cosa curiosa también es que nos hemos encontrado que los tutores tienen muchos dificultades en la parte de comunicación, de leer, de escribir, comprehension de lectura. Tienen muchísima dificultad en matemáticas. ... el lenguaje que se utiliza en los textos supone que la persona, en la parte de comunicación, comprende lo que está pidiendo. Por ejemplo dice agrégale el doble del producto de tal cosa por tal cosa…y allí nos detenemos. Source: coordinator interview March 2010

Basic mathematical operations Tutors sometimes had difficulties with the vocabulary (Box 33), underlying concepts and procedures for basic mathematical operations. For example, long division. In Book 5, Aritmetica Box 34 en la Investigación Científica, it was observed that When doing an exercise requiring the many tutors had problems with the long division calculation of the average of food consumed and weight of 25 chickens algorithm, and none used the recommended six tutors came to the board to vocabulary at the beginning of the session. One demonstrate how they did this. Each tutor who came to the board could not even begin one painstakingly used the standard to do the calculation, and two admitted not long division algorithm even when knowing what to do with the decimal point. The the number to be divided was 75000. Trainer however was excellent at probing and No one questioned this or suggested checking tutor understanding, and by the end of an alternative. the session all the tutors appeared to be Source: Observation February, 2010 proficient. It often seemed that the tutors did not 64

Tutors take the end of module test at the same time as their students and this information is used by coordinators to identify where more training is needed. This information does not appear to be collected centrally.

have a sense of the numbers that they were dealing with; increasing the use of estimation would help develop this. It was very rare to see different methods of working out a problem discussed (Box 34) although it is a part of the SAT methodology.

It is recommended that: Box 35 Nobody has ever been able to reach the concrete. The concrete is so abstract that nobody can reach it. We can only function because of abstraction. Abstraction makes life easy, makes it possible ...The essential point is this algebra is an attribute, a fundamental power of the mind. Not of mathematics only. Source: Gattegno, 1970, cited in Mason & Johnston-Wilder, 2004

55. Estimation techniques are taught and practised. 56. Different procedures for working out a calculation are compared (when appropriate) particularly in respect to their efficiency. The transition to Bachillerato Just as students have difficulties adapting to the SAT system when they enter Ciclo Comun so to the transition to Bachillerato is problematic and these difficulties often stem from inadequately understood concepts in Ciclo Comun.

There is rightly a strong emphasis on developing algebraic competency in SAT (Box 35). Tutors and coordinators identified algebra generally, and in particular the expression of everyday problems algebraically, as the most Box 36 problematic and disliked topic. Difficulties are Micro centro de polinomios con apparent from the beginning. For example, at a tutores de II Bach, A & B. El micro centro se realizo con el objetivo de course for maths specialists sustained help was reforzar tutores ya que durante la needed to define the elements in the set ‘people visita pedagógica realizada al grupo attending this course’. Other basic ideas from the se observo debilidades en el manejo first maths module, such as ‘an element can de contenido, se reforzaron conceptos belong to both a set and a subset’ had to be como los siguientes, propiedades de la clarified. If the most competent tutors were multiplicación, terminos semejantes, unsure, it is to be expected that the average tutor productos notables, propiedades de la will struggle, and the average assessor will be potenciación, grado de un polinomio. unable to offer much support in the classroom. Source: assessor monthly report, March 2010

In Bachillerato, weaknesses present in Ciclo Comun become more apparent, e.g., in all six classes observed of Polynomios II, tutors taught a mechanical method for expanding quadratic equations. Understanding of this method was not probed, and a substantial number of student errors went uncorrected. Twenty 48

Bachillerato tutors were asked if they could suggest an alternative way of representing a quadratic equation in order to make it more meaningful to students (e.g., a diagram, graph or real life example). Only one tutor was able to do this, relating it to an area of land. In three SAT classes, students were asked if they could make a diagram of what they were doing. None could at the beginning of the lesson, but following an example, most students were able to do this and expressed relief at having gained a new level of understanding (as did the tutors). It appears from monthly reports that some assessors are very effective at identifying when Box 37 support is required and providing it (Box 36) Destrezas Algebraicas:327: A water supply problem 26 tutors in the group working in friendship groups of two to four. The text book gives 9 variables (no of inhabitants, average number of animals per family etc) to make into a formula for calculating the daily water requirement. All groups were flummoxed and unable to start task. Many requested trainerâ&#x20AC;&#x2122;s assistance. Two groups, with trainer help, developed formulas which they explained coherently at the board, correcting errors after trainer input. Most groups began to make inroads into the task after significant assistance from trainer and other tutors, but one group of three was unable to make any independent progress on the task. After lunch, in the next session a new topic was started. Source: Observation May 2010

Finally it was noted that opportunities for developing algebraic skills in the Ciclo Comun are sometimes not used. For example, in a lively and practical lesson on triangles and parallelograms, insufficient time was allowed for students to explore and develop their own formulae.

With the help of a skilled trainer many tutors make substantial progress but limited time in training means that additional support is needed by some tutors (Box 37 and photograph).

It is recommended that: 57. Activities in the texts that are particularly suitable for developing algebraic skills in the Ciclo Comun texts should be identified before training and carefully planned. 58. Work with algebra should include multiple representations.

Preparing for training To study or not to study texts Tutors do not usually work through the texts before trainings although they are requested to do this65. When interviewed a tutor who had done the exercises reported finding the training boring as it mainly consisted in redoing the exercises. If all tutors were expected to read the texts, but not actually do the exercises, before the training and note where they had questions or comments it would assist trainers in allocating training time efficiently. It is recommended that: 59. Tutors read the texts before training and note where they have difficulties. At the end of the training review, this list of difficulties can be reviewed, progress assessed and further support needs identified. 60. Tutors read texts before training and note down ideas for additional teaching materials, activities etc. so that these can be shared at the training.

Maintaining the quality of maths training The Training Some coordinators are seriously concerned that the rapid expansion of SAT and loss of some key personnel has diminished the quality of training, not only from a technical point of view but, more vitally, there has been a loss of ‘la mística del trabajo’ or ethos. In the great majority of maths trainings there was a purposeful and collaborative atmosphere and the most tutors appeared to make good progress. It was noticeable that the trainer’s style had a large impact on the learning environment; in some observed sessions there was a rather subdued atmosphere, questions were rarely asked by tutors, and a mainly ‘transmission’ teaching style was used. It also appeared that some sessions were not carefully planned. This resulted in key concepts/skills/attitudes not being clearly identified and developed. This probably significantly affected tutor progress and their capacity to adequately accompany their students. Although most tutors engaged wholeheartedly with the training process, there was a (sometimes sizeable) minority in some sessions that did not. Concern was also expressed that the incidence of unprofessional conduct during trainings was increasing. Some tutors also complained about feeling pressured by other members of the group not to ask questions or discuss issues so that the text could be completed quickly. It was reported by coordinators that, worryingly, it tended to be tutors who have been longer in SAT who were less motivated during the trainings. 65

There is evidence that a few tutors (perhaps ones with the weakest maths) copy out other tutor’s answers if asked to fill in the texts before the trainings

It is recommended that: 61. The training of trainers is made a priority. 62. All training sessions are planned. 63. Maths trainers have relevant teaching experience. 64. The most capable and enthusiastic tutors who are studying, or have studied, maths and pedagogy at the university are trained as trainers (this is already happening)

Supporting tutors’ development of practice in the field The importance of providing teaching manuals, additional teaching tools and videos are mentioned in other sections of this report. In some areas the provision of micro centros is well developed and plays an important role in supporting tutors. In other areas micro centros cannot be as responsive to the tutor needs because of a lack of trained staff. The maths specialists could play an important role in the provision of micro-centros but it is necessary to train them. This is happening, in the past seven months 24 tutors/assessors received a five day maths specialist training. The role of the field assessor The field assessors are another set of SAT heroes; on motorbikes, buses or foot they travel on bad roads or paths to visit remote communities. It was not unusual to meet an assessor (or coordinator) in bandages from a recent motorbike accident (Box 38) or, as happened recently, shocked by an assault in which his motorbike was stolen and he was tied up and thrown down a hillside.

Box 38 Two hours from home and an hour away from Comayagua an assessor collided with a motorbike that was going the wrong way up a one way street, both motorbikes lost balance and fell and both riders were taken to see the community nurse; the assessor was bandaged up and given something for the pain. He carried on his journey to the Institute in order to hand in paperwork so his tutors would be paid. When he arrived at the Institute the director was shocked by his injuries and took him to the hospital. He was bandaged up again and given an injection for the pain. A mechanic mended and adapted his motor bike so he could ride it with injured leg. He arrived in Comayagua about an hour later, soon after 7.00pm. He went to the SAT centre to collect text books. It was a wet, cold and windy night. The coordinator, who had just got home after a 12 hour day travelling to the communities, returned to the centre to hand over the texts, check that he could get to his sister’s house in Comayagua and that he would attend properly to his injuries. When he arrived at his sister’s house he had to be helped off the bike and carried in. Three months later he is ’90%’ recovered. Observation and tutor interview March and August 2010

According to the coordinators, because of lack of training, some assessors have a limited understanding of the SAT methodology, and this affects the quality of the acompanimiento they provide the tutors. Field assessors, although they sometimes attend general trainings, 51

appear to receive very little specialist training; the one assessor training organised (to the consultant’s knowledge) during the first half of 2010 was cancelled. Field assessors who have not been SAT tutors often do not know the texts well. It would of course be difficult for any field assessor to become an expert in all 70 texts, but they could specialise in one area (as with the maths specialists) as well as general pedagogy.

It is recommended that: 65. Field assessors’ training needs are reviewed and met.

Institutional links In the Final Report to the donor 1045-0688 (1 July 2004 – 30 June 2006) it says that one of the lessons learned was: ‘La participación de los actores como las universidades en el desarrollo del programa incrementa la credibilidad del programa, facilita la consecución de becas de estudio para los estudiantes, y reduce Box 40 los costos relacionados a consultorías y asesoramientos’.66 This was surprising to read Tenemos un Convenio Marco con given that, while Asociación Bayán has excellent UNAH, links with various government departments, Don X los contactó a Curla de la Ceiba there was no contact related to pedagogy or tambien, para otro convenio, no educational theory with a Honduran university prosperó or other educational establishment in the first half of 2010 as far as the consultant was aware. La universidad pedagogica ha dado Bayan has had an agreement with the Universidad Autonomia de Honduras (UNAH) since 2001 which states that they will work together to ‘mejorar la calidad professional de los docentes y superar la calidad de vida de los pobladores de la Mosquitia67 (this was later extended to all regions in which SAT was working). However despite many attempts and considerable effort on Bayan’s part (Box 40) this relationship has not developed.

algo de cursos, como "Preparación de Instrumentos de Evaluación" para Bayan, donde participaron varios coordinadores La universidad pedagogica fue abordado en multiples ocaciones para un reconocimiento oficial a las capacitaciones de Bayan, no aceptaron, etc. Source: email from Executive Director, August 2010

The participation of universities in the program’s development has raised the status of the program, helped obtain scholarships for students and reduced consultancy and evaluation costs. 67 Improve teaching quality and the quality of life for the inhabitants of the Mosquitia. 66

This is a loss for UNAH because Asociación Bayán, with 14 years of experience implementing SAT, has much to contribute to Honduran educational discourse.

Box 39 It could be claimed that a people were in charge of their own development only if they were learning systematically about the changes that occurred in their society, and were consciously incorporating in their continuous learning process appropriate elements from the universe of knowledge: their own, the modern knowledge system, as well as the experiences of other people and groups in the world who are facing similar challenges. Source: Arbab et al, 1988

This relative isolation, national and international, means that Asociación Bayán is not, in the view of the consultant, doing enough to draw on recent research findings around international best practice in maths teaching, and to see if it contains elements that are appropriate for adoption and adaption.

Although the quote in Box 39 is referring to community development it could be argued that it is these activities which maintain the vigour and development of any process. Some coordinators have expressed the concern that the SATHonduras risks stagnating if it does not become part of a wider dialogue, more closely linked with the Honduran educational system68, and to rigorous national and international educational debate. Therefore, even though there are huge time pressures on key staff, time should be found for this. The comparative evaluations being undertaken by the Universidad Pedagógica Nacional Francisco Morazan, and the Hewlett study have not, as yet, contributed to any developments in practice or theoretical understanding. To date there has been no substantive progress report of the research being undertaken under the Hewlett study but it is expected that it will make a substantial contribution to understanding the taught and learned69 SAT curriculum and its impact on students. Finally, it is surprising to the consultant that there are not closer links with FUNDAEC. Networking or developing a forum with other SAT Programs for informal discussion of methodological/pedagogic/ philosophical questions would be invaluable for Asociación Bayán, and FUNDAEC might also benefit from such a link (for example, when the Ciclo Comun texts were revised, it could have been useful to have sought Asociación Bayán’s reflections on its considerable experience with SAT).

Recent changes in the Honduran curriculum may require adjustments in the SAT curriculum e.g., the teaching of the division algorithm, 69 The ‘official’ curriculum is what tutors are expected to teach, as set out in the GEA and texts but what is ‘taught’ curriculum (what actually happens) will depend on tutor knowledge, interests, and skills. Students also affect the curriculum by their previous knowledge, attitudes towards learning etc. 68

It is recommended that: 66. Asociación Bayán’s teaching staff form links, both formal and informal with other educational institutions, and engage more actively with national and international discourses on education. 67. An inter SAT Program learning network be created.

Conclusion The maths component of the tutor training program has achieved many notable successes:  



The training and supporting in the communities of about 700 tutors over the last decade. The vast majority of tutors questioned reported that: SAT had made them feel more positively about maths; after the trainings they could do the exercises in the text book; they felt confident explaining the exercises and enjoyed teaching maths; those studying maths at the university said that SAT had significantly helped them with their studies, that they understood more and were able to use maths more in practical applications. It was observed that the inclusive and relaxed, but purposeful, atmosphere of the trainings is usually replicated in the SAT classrooms.

There is therefore a sound foundation to build on. The rapid expansion of SAT has however put the training process under considerable strain; all interviewed coordinators expressed concerns about maintaining the quality of SAT. It was also found that some key elements of the SAT methodology were not being fully implemented, and that there were other factors which, if addressed, could significantly improve the program. The most important recommendations of this report are therefore that: 



Asociación Bayán continues to build on these good foundations by continuing to reflect and experiment with ways of further developing ‘an atmosphere of consultation’. Activities which are a part of the SAT methodology, but which have only partially been adopted, or not been adopted by Asociación Bayán, are fully implemented, unless consultation with FUNDAEC identifies them as being no longer appropriate. Priority is given to building the capacity of: o trainers to use the SAT methodology fully rather than partially o the field assessors to ‘accompany’ the tutors in the development of their general pedagogic skills; o carefully selected maths field support staff to effectively support the tutors develop their (more specific) maths pedagogy/knowledge. Relevant insights, techniques and tools originating outside SAT, particularly those associated with increased classroom participation, conceptual development and ‘formative assessment’, are identified and considered for adoption and adaptation, since (according to the consultant’s experience and relevant research literature) these have the potential to significantly enhance some key aspect of the SAT pedagogy/curriculum. 55

 

Effective fora for the consultation of pedagogic and other educational issues are established. The collection and analysis of field data and feedback is systematised and standardised (where appropriate) and archiving systems are improved.

Other recommendations are that:  



 



Aprender Enseñando is used to develop consultation skills, and its delivery is collaboratively planned so that all tutors have a comparable experience. Tutors are explicitly taught the characteristics of ‘exploratory talk’, provided opportunities to develop these skills, and mentored in facilitating appropriate kinds of ‘exploratory talk’ in the classroom. Training sessions are planned for developing a wide range of pedagogic skills, particularly those associated with assessment for learning, conceptual development, and increased participation. Recommended activities/methods, if not explicit in the text, are clearly explained to the tutors. Topics in which tutors regularly experience difficulties are identified before training so that sufficient time can be allocated to them, and related questions, reflection, activities and discussion adequately planned. Sufficient time is allocated for ‘independent working’ in the trainings. Tutors have a written record (e.g., training handouts) to support classroom practice and/or training in the taking of pedagogic notes, since inspection of many tutor texts revealed that tutors rarely take pedagogical notes. Trainers allow time between questions and responses, and discuss the reasons for doing this with the tutors. Activities in which question formation is explicitly requested, or which lend themselves to this, are clearly identified in the trainings in order to help tutors more effectively utilise the texts. Trainers experiment with small group work as a way of encouraging the formation of more reflective student questions, and encourage the tutors to do the same. Tutors are supported and mentored in the didactic use of questions and ‘error analysis’ as a ‘formative assessment’ tool; this includes being informed of the utility of this approach; experiencing it in their training; highlighting appropriate SAT text examples; being made aware of the key relationship between errors and ‘misconceptions’; and being mentored in developing their own questions in field assessor accompaniment. Trainers/tutors are taught a range of techniques for increasing tutor/student participation when answering questions.

    



  



Trainings are planned so that key statistical concepts are developed and understood. More up to date data is used in the trainings, and tutors are given copies of these data to use in the communities. The reasons why independent working is important are presented and discussed in the trainings. Maths trainers have relevant experience. The most capable and enthusiastic tutors who are studying, or have studied, maths and pedagogy at the university are trained as trainers (this is already happening to some extent). Field assessors’ training needs, as regards supporting tutors’ pedagogic development, are reviewed and met. Statistical concepts inadequately explained in the texts are identified, and additional exercises developed. Ways of increasing the value that students place on statistical projects are investigated (and the copying of previous annual assignments is stopped). Ways of integrating traditional representations and methods (estimation, measurement, etc.) are investigated. Tutors are systematically supported in the use of the Learning and Assessment Guides for assessment for learning and, if necessary, additional materials are provided. More efficient methods to check answers and correct procedures (e.g., use of small individual white boards, peer checking, etc.) are used in the trainings and classes instead of whole class presentations by individual students/tutors. Student/tutor presentations of work are used as the basis for discussions. Tutors/students experience and reflect on a range of techniques for improving group work A range of formative assessment techniques is used; those in the texts are identified and, where necessary, additional ones are introduced; and trainers/tutors/students are supported in developing their capacity to use them. The use of the Guias de Apprendizaje y Evaluación as a formative assessment tool is investigated, or students are given other means to help them monitor their own progress. Tutors/students are given clear criteria of how to assess their own and others’ work, and guidelines are written on how to peer teach/accompany a tutor/student without creating dependency (perhaps using the Guias de Aprendizaje y Evaluaciόn). The texts are analysed to identify which exercises need to be modified/changed to make them more engaging to Honduran students, and where additional



 

  

 



     

information or discussion topics need to be introduced (these could be put into the Teaching and Evaluation Guides). Errors in the texts are amended and new illustrations are commissioned where the current ones are unclear (e.g. Suma y Resta) or unattractive (e.g., Conjuntos y Numeros) When the representations are physical, they are made, as far as possible, with locally available and sustainable/recycled materials. Texts produced in Honduras follow FUNDAEC (and normal practice) referencing conventions with regard to identification of the origin of material taken from other sources. Activities are developed to enable students to become competent in both metric and imperial systems. Tutors are taught techniques for overcoming ‘bloqueos’, and that the process of meeting and overcoming maths learning blockages is perceived as normal and often key to the learning process. Estimation techniques are taught and practised. Work with algebra includes multiple representations. Tutors read the texts before training and note where they have difficulties. At the end of the training review, this list of difficulties can be reviewed, progress assessed, and further support needs identified. Tutors who find maths difficult are identified at the beginning of their training so that they can be given extra support. Sufficient time is allowed at coordinator meetings for consultation on pedagogy, particularly with respect to the development of an enriched and common understanding of the SAT methodology. Asociación Bayán’s teaching staff form links, both formal and informal, with other educational institutions, and engage more actively with national and international discourses on education. An inter SAT/PSA Program learning network is created. The tutors’ pedagogic/theoretical learning in SAT is linked more closely to the educational theory and pedagogy studied in the university. Tutors are given opportunities to share good practice e.g., via intranet, trainings, micro-centres, tutor meetings, the SAT Bulletin, etc. Tutors read texts before training and note down ideas for additional teaching materials, activities, etc., so that these can be shared at the training. Tutors/students are given opportunities to work in different groups during training. Tutor/student capacity to work in groups is developed by using different types of group work, and by providing opportunities for analysis of the effectiveness and inclusivity of the group work. 58



  

Tutors form groups to conduct simple collaborative action research on small changes in practice based on a ‘theory of change’ model, and field assessors are trained to promote and support this action research. Criteria for defining ‘retake’, ‘opinar’ and other terms on the feedback forms that are subject to very different interpretations are standardised (it can be noted that Asociación Bayán is developing a data entry system to improve accuracy of entry and calculations, and there are plans to produce a data collection manual for staff). The indicators and procedures used in the Hewlett study for classroom observations are investigated for adoption/adaptation. Data on student and tutor test scores are collated and analysed. Reports are archived so that reflection on, and documentation of, processes are facilitated.

References Abdu’l Baha. Cited in a letter dated 5 March 1922 written by Shoghi Effendi to the Bahá'ís of the United States and Canada, published in "Bahá'í Administration: Selected Messages 19221932", pp. 21-22 Anthony, G. & Walshaw, M. (2007) Effective Pedagogy in Mathematics. Educational Practices Series – 19. Geneva. International Bureau of Education. FUNDAEC: its principles and its activities Arbab, F. Correa, G. & de Valcarcel, F. (1988). FUNDAEC: its principles and its activities. Cali: CELATER Atherton, J S (2010) Learning and Teaching; What works and what doesn't [On-line] UK: Available: http://www.learningandteaching.info/teaching/what_works.htm Chingos, M. & Peterson, P. (2010) May:draft. Predicting Teacher Effectiveness by college selectivity, experience etc. Paper prepared for conference on: Merit Pay: Will it work? Is it Politically Viable? Sponsored by the Harvard Program on Educational Policy and Governance. Hattie, J. (1999) Influences on Student Learning: Inaugural Lecture, Professor of Education, University of Auckland. Hattie, J., Timperley, H. (2007) The Power of Feedback. Review of Educational Research. Vol 77 (1): 81-112 Hodgen, J. & Wiliam, D. (2006) Mathematics inside the Black Box: Assessment for learning in the mathematics classroom. London: inferNelson Ignacio, G. I., Barona, E. G., & Nieto, L. B. (2006) El Dominio Afectivo en el Aprendizaje de las matemáticas. Revista Electronica de Investigacion Psicoeducativa 8, Vol 4 (1): 44-72 www.investigacion-psicopedagogica.org/revista/ Informe de Progreso Educativo de Honduras. (2005). Programa de Promoción de la Reforma Educativa en America Latina y Caribe (PREAL). Washington, DC/Chile: Inter-American Dialogue/Corporación de Investigación y Desarollo Ma, L. (1999, 2010ed). Knowing and Teaching Elementary Mathematics. New York: Routledge. Mason, J. & Johnston-Wilder, S. (Eds) (2004) Fundamental Constructs in Mathematics Education. Abingdon:Routledge

República de Honduras. (2010). Visión de País 2010-2038 y Plan de Nación 2010-2022. Tegucigalpa. www.visiondepais2010-2038.com Rivkin, S. G., Hanushek, E. & Kain, J.F. (2005) Teachers, Schools and Achievement, Econometrica 73(2): 417-458 Schwille, J. & Dembele, M. (2007). Global perspectives on teacher learning: improving policy and practice. Paris:UNESCO Swan. M. (2006). Collaborative Learning in Mathematics: a challenge to our beliefs and practices. London/Leicester. NRDC/NIACE Wilensky, U. (1991). Abstract Meditations on the Concrete and Concrete Implications for Mathematics Education in in I. Harel & S. Papert (Eds.) Constructionism. Norwood N.J.: Ablex Publishing Corp. UNDP. (2009ª). Informe Sobre Desarrollo Desarrollo Humano. Honduras 2008/2008. De la exclusión social a la ciudadania juvenil. Programa de las Naciones Unidas para el Desarrollo (PNUD). Tegucigalpa, Honduras UNDP. (2009b). Encuesta Nacional de Percepción sobre Desarrollo Humano 2008: Juventud, Desarrollo Humano y Ciudadanía. Programa de las Naciones Unidas para el Desarrollo (PNUD). Tegucigalpa, Honduras Wiliam, D. (2006). Assessment: learning communities can use it to engineer a bridge connecting teaching and learning. Journal of Staff Development Vol 27(1), 16-20 William, D. (2007). Assessment for Learning: why, what and how? Professorial lecture, Institute of Education, University of London, 24 th April. Reports to donors: 

Scaling up the System of Tutorial Learning (SAT) as an alternative education program tackling rural poverty and gender inequity in Honduras: 01:07:04 to 30:06:06 (No:1045-0688)



Building Sustainable Livelihoods through Agriculture 01.01.07 to31.07.07 (CIDA)



Fortaleciendo la Consolidación del Programa Educativa Rural SAT en Intibucá, La Paz y Comayagua. 06.07 to 12.07 (IBIS)

Internal Reports: 2010 January Informe de Actividades; Perfil Nacional for 2010 (Block 1) Perfil Nacional 2009 (incomplete due to the coup) Perfil Nacional 2008 (Blocks 2 & 3)

Appendix 1: Observations: maths trainings70 Date 25.01-29.01 8.02-9.02 18.03 26.03 26.04 27.04 28.04 29.04 30.04 6.05 13.05 14.05 17.05 18.05 19.05 25.05 26.05 27.05

Area 1 2 3 3 1 4 1 1 1 4 1 1 1 1 1 5 5 5

Text Impulsor texts – maths specialists Matematics 5 Micro capacitación – Polinomios II Micro capacitación – Medidas del Espacio Matematicas 1 & 6 Matematicas 1 & 6 Matematicas 3 & 1 Matematicas 3 Matematicas 3 Matematicas 6 Polinomios 1 Polinomios 1 Destrezas Algebraicas Destrezas Algebraicas Destrezas Algebraicas Matematicas 3 Matematicas 3 & Matematicas 6 Matematicas 3

Hours 45 8 4 4 7 4 7 4 2 2 7 7 4 7 4 3 4 2

Observaciones: maths lessons fecha 22.02 1.03 2.03 17.03

Región 6 6 6 7

22.03

Textos Matematicas 5, Matematicas 5, Matematicas 4 Matematicas 5 Manejo de Variables, Polinomios II, Matematicas 5 Manejo de Variables, Polinomios II, El Espacio y Sus Elementos Matematicas 5, Polinomios II, Polinomios II,

horas 3 1 4 4 3

Observaciones: reuniones fecha 2.03 8.03-9.03 31.05

Lugar 6 3 9

grupo Asesores Coordinaros Asesores

horas 2 12 2

Summary of Interviews (semi-structured) grupo

coordinaros

Assessors campo Tutores

del

Notes

duración (approx) 0.5 – 3.0 horas 0.3 – 0.5

0.3 – 0.5

During visits to centres and during breaks in training sessions a

5 coordinators were interviewed on more than one occasion.

31 hours of general training were observed: Aspirantes: Filosofía de SAT, Aprender Enseñando, Tecnología 1, Second Year: 1 Descripciones,Third Year: Alfabetización, Lecturas 5

hours

large number of conversations were held with tutors; these gave valuable insights and informed the framing of interview questions.

Appendix 2 – Summary of the field assessors’ session observations71 ¿El tutor utiliza metodología del enfoque analítico y participativo (no magistral)? ¿Los estudiantes se sientan en circulo? ¿El tutor formula preguntas adicionales a las que hay en el texto? ¿Las preguntas del tutor son simples, precisas, entendibles y necesarias? ¿Antes de avanzar en otro tema existe una conclusión del punto anterior? ¿Se estimula la participación de todos? ¿Durante su estadía en el aula cuantos participantes vio opinar

¿Usa tanto la lectura comentada como trabajo en grupo y lectura individual? ¿Asegura que los participantes utilicen las puntuaciones de lectura correctamente? ¿Da la oportunidad de lectura a todos los participantes? ¿Hay investigación del conocimiento local (tradicional) suficientemente? ¿Se asegura la comprensión del grupo antes de pasar a otros temas? ¿Verifica los textos para asegurar la precisión de las respuestas escritas allí? ¿Verifica la ortografía y gramática de los participantes en sus textos ¿Avance del contenido está de acuerdo al tiempo transcurrido y previsto? ¿Estima que el grupo podrá finalizar los textos hasta la fecha de evaluaciones?

Si 30

86%

No 0 0%

na 5

14%

26 27

74% 77%

4 3

11% 9%

5 5

14% 14%

74%

17%

74%

14%

2 10

23%

25 74% © 8/11, 10/16,18/21,14/15 12/12,6/9,20/23, 22/24, 13/17, 8/12,10/15,11/11, 13/17 (Y) 4/23, 8/16, 6/18, 7/17,3/19, 6/24, 4/12, 9/20 27/27 20/20 23/23 80%

17%

28 26

74%

23%

86%

14%

69%

29%

71%

20%

57%

20% 8

23%

54%

20% 9

26%

69%

11% 7

20%

71%

20%

3 assesores (Area 5) (18 tutors) C1- 8, C2- 8, C3 – 2, 2- 6 hours, Oct 09, June 09, April 09, 3 assessors (Area 10) (11 tutors) C1 -4, C2 -4, C3-1, B2-2, 2 – 5 hours, (abril 2010) 1 assessor (Area 11) (3 tutors) C1, B1, B1 + 3 tutors 71

¿La asistencia diaria esta al día? ¿Comienza puntualmente las sesiones (incluyendo luego de los recreos)?

29 27

83% 77%

1 2

3% 6%

5 6

14% 17%

¿Se observa un ambiente limpio y organizado? ¿Se observa un ambiente ordenado y disciplinado en general? ¿Los participantes cumplen con las responsabilidades asignadas y son obedientes? ¿Se observa respeto hacia las opiniones de los compañeros? ¿Se observa respeto hacia el tutor/a? ¿La practicas agropecuarias y de Servicio cuentan con un plan de fechas ¿Todos los estudiantes participan en las actividades ¿Los recursos para las practicas están disponibles ¿La comunidad participa en la ejecución de las practicas y proyectos ¿Se busca la participación de organizaciones para los proyectos ¿El/la tutor/a monitorea el avance de las practicas con la frecuencia requerido ¿El registro de las experiencias y practicas se lleva con el rigor debido ¿Se lleva asistencia en cuanto a las practicas ¿Las practicas se llevan en forma paralelas al avance de contenidos ¿Se observa proyecto de servicio a la comunidad (si es positivo descríbalo) ¿Se implementan los experimentos en el aula durante la lección? ¿Tienen disponible los instrumentos e insumos oportunamente ¿Demuestra seguridad en la realización de las experiencias? ¿Hubo algún experimento durante la visita? (si si explique cuál) ¿Hubo algún experimento se concluyo con éxito? (si no explicar las causas) ¿Elabora su plan de clase y lo utiliza para el mejor desarrollo de su clase? ¿Están al día sus informes administrativos para la Dirección Departamental ¿Es muy puntual y es un ejemplo positivo en cuanto a su conducta

86%

11%

80%

11%

86%

11%

83%

14%

30 8

86% 23%

0 2

0% 6%

5 25

14% 71%

23%

77%

29%

71%

23%

74%

14%

86%

26%

74%

23%

69%

23%

74%

23%

77%

11%

14%

74%

94%

63%

17%

20%

77%

14%

89%

11%

¿Planea actividades académicas pertinentes para mejor comprehension del grupo? ¿Promueve actividades de recaudación de fondos con sus estudiantes ¿Decora apropiadamente su aula para apoyar la educación del grupo ¿Se observa un ambiente alegre y de motivación en el grupo

77%

23%

80%

14%

69%

86%

23%

86%

14%

Appendix 3: Summary of field assessors comments on classroom observations72 (not monthly reports) M E T O L O G I A

Se sientan en grupos ya que el mobiliario ahorita no da para que estén en circulo también la cantidad de alumnos Recomendación: elaborar preguntas para repaso de la lección y verificar los textos de los estudiantes Durante esta visita los estudiantes hicieron exposición No están en circulo por que el espacio es reducido están sentado en forma de U Que pregunta a los estudiantes que expliquen lo que entienden de la que ¿ Explore a los estudiantes ¿ qué va a estudiar antes de entrar al contenido de texto Cuando use ‘b’ no dijo ‘b’ grande o ‘v’ pequeña dijo a los estudiantes ‘b’ labial o ‘v’ dental. Fue una jornada muy interesante Se recomienda que pregunta a cada estudiante que lee lo que entiende do lo que leyó x Se debe realizar preguntas …cada párrafo Se debe enfocar a las temas Bien participativos (2) Esta cumpliendo con la metodología (3) Constructivista y participativa (2) Expositiva, analítica, Constructivista, dinámica y participativa Material didáctico (3) Material didáctico bonito

E V A L U A C I O N P R O C E S S O D I S 72

siempre ratifique la ortografía de las palabras que utilizan los estudiantes al escribir las respuestas a las preguntas del texto No estaban dentro del aula de clases Hacer un repaso del tema anterior antes de seguir otro Se dieron algunas sugerencias a la tutora para prepares(¿) Cada vez que termine una lección revise las respuestas escritos por los estudiantes y rectifique la ortografía x (y no solamente verbal) (2) Revisar los textos para verificar se esten trabajando Avanzar con las (¿) para poder (¿) con el plan de evaluación En el texto de la matamatica no hay revisión de las respuestas Algunas estudiantes presentaron debilidad al escribir algunas palabres El en texto de conjuntos y números en el lección ……no lo hay …… respuestas Hasta cierta tematico si hay pero en mayoría no le hay. No en todos los textos esta revisado toda la temática (2)

Debe d ser puntual en la hora de entrada Ensenar a los estudiantes a respetar las ideas de los demás En conjuntos de 5 estudiantes una exposición Number of comments per tutor 1 – 15.

C I P L I N A

Les (¿) fuera del aula porque la maestra de primeria tenían reunión a pesar de esto estaban bien organizados bajo la sombra de un árbol Algunos alumnos no trabajan en los textos inspectados por lo menos dos.

P R A C T I C A S

En este bloque no llevan prácticas agrícolas x Hacer un cronograma de actividades antes de empezar el proyectos entre todos los padres de familia ¿ en terreno. En las practicas apoyan los padres de familia El día de hoy no estuvo en el aula, estaban trabajando en el campo con las prácticas de tecnología. Se sugiere elaborar un plan para las practicas agropecuarias

E X P E R I

No presencia ningún experimento (11) Insectario se concluye con éxito muy presentable Experimento sobre la temperatura Agua caliente, tibia y fría

Plan for ‘un archivo del la conducta de cada estudiante’ Cumplen con sus obligaciones asignadas (5) Trabajan bien en sus textos Bien aplicados Cumplen con sus normas Todas se integran al trabajo Bien disciplinados Son educados Portan uniformes completos Plan for ‘un archivo del la conducta de cada estudiante’ Buena cooperación entre alumnos Respeto entre compañeros Trabajan mutualmente Estudiantes respetosos al tutor Buena relación (5) Excelente relación Muy respetosas Muy buena entrega y compromiso ante el grupo Excelente relaciones con las personas de la comunidad (4)

- proyectos de la comunidad (2) Estimule a los estudiantes a realizar proyectos de servicio a la comunidad Hay un proyecto planificado pero aun no ha sido ejecutado Cría de pollos campaña de alfabetización no lo hubo x campaña de limpieza en la comunidad (2) Agua caliente, – Parcela bien cultivada

M E N T O S T U T O R

Tiene que hacer su horario de clase y pegarlo en un lugar visible dentro de aula. (3) No presento el plan de clase por olvido No andaba su plan de clases y que estaban trabajando en el campo Se sugirió laborar puntualmente el plan de clase y ¿ del mismo ¿ evitar (¿) y (¿) mejor su clase Cada día que se imparta las clases debe tener sus planes de clase para cada capacidad Tratar de mantener el entusiasmo en los estudiantes No decora el aula porque es aula prestada y los (¿) de la escuela no les gusta que peguen cosas en las paredes No tienen paredes están recibiendo clases en el galpón (porque no hay nada puesto en los paredes) No existe material (¿) en el aula de clase. So se observo algunos aspectos .. que la visita fue en la parcela … mayor tiempo Planificación esta al día

Appendix 4: Questionnaire Survey The percentages on the left hand side are the percentage of students responding to each question.73 The numbers on the left are the percentage of responses in each category. (Questionnaire on following page)

If a question does not have a percentage it is because there appeared to be confusion amongst the first year students as to whether or not they should answer the question. Taking this into account the response rate was about 95% for all questions. 73

El objetivo de este cuestionario es recolectar información que nos ayude a mejorar la capacitación y el apoyo que le damos. Esperamos que los resultados de estos puedan ser compartidos y utilizados por capacitadores de otras partes del mundo. No estamos colectando información que revele su identidad, por lo que sus respuestas permanecerán como anónimas. Utilizaremos esta información para conocer su opinión sobre las matemáticas y el proceso de enseñanza de las mismas. Por favor, responda a las preguntas en cuanto las lea – nos gustaría conocer su primera opción, y siendo éste un cuestionario anónimo, siéntase libre de responder de la manera más honesta posible. Este cuestionario ha sido elaborado para conocer su valiosa opinión COMO TUTOR/A de matemáticas. Para contestar el cuestionario, marque con una X la respuesta que considere más oportuna. Las escalas de valores son las siguientes:

Casi siempre, Generalmente, 50% del tiempo, Rara vez, Muy rara vez.

1 93% 2 97% 3 98% 4 96% 5 97% 6 95% 7 98% 8 96%

9 94%

Mis estudiantes realizan todos los ejercicios del texto de matemáticas. Mis estudiantes resuelven los ejercicios por sí mismos y consultan a sus compañeros de vez en cuando. En la resolución de ejercicios matemáticos, mis estudiantes solo utilizan los métodos que yo les enseño. Al resolver los ejercicios matemáticos, mis estudiantes comienzan con los más “fáciles” y dejan para el final los “difíciles”. Yo motivo a mis estudiantes para que terminen los ejercicios lo más rápido posible. Mis estudiantes comparan diferentes métodos de resolución de problemas matemáticos. Cuando doy una explicación, lo hago a toda la clase de una vez. Relaciono las matemáticas que mis estudiantes han aprendido en textos anteriores con lo que enseño en el texto actual. Mis estudiantes me sorprenden con respuestas, comentarios o preguntas que encuentro originales.

Casi siempre

Generalmente

50% el tiempo

Rara vez

Muy rara vez

10 Evito que mis estudiantes cometan errores 96% razón por lo cual explico las cosas con cuidado desde el principio.

11 Todos los ejercicios que mis estudiantes 95% trabajan en clases pueden ser resueltos con explicaciones que se encuentran en el texto. 12 Mis estudiantes aprenden discutiendo sus 97% ideas. 13 Mis estudiantes aprenden trabajando en 98% pares o en grupos pequeños. 14 Mis estudiantes inventan sus propios 95% métodos para resolver ejercicios matemáticos. 15 Yo les indico a mis estudiantes cuales 94% ejercicios deben resolver. 16 Yo disfruto enseñando matemáticas. 96% 17 Mis estudiantes aplican en su vida cotidiana 95% lo aprendido en mis clases. 18 Yo relaciono lo avanzado en mis clases de 95% matemáticas con lo que mis estudiantes aprenden en los otros textos. 19 Mis estudiantes justifican sus respuestas en 95% los ejercicios de Verdadero/Falso. 20 Yo motivo a mis estudiantes a que razonen 96% la resolución de los ejercicios matemáticos, aunque les tome más tiempo. 21 Yo relaciono lo avanzado en matemáticas 97% con la afirmación de valores espirituales. 22 Las matemáticas son útiles en los proyectos 95% de servicio a la comunidad. 23 Me siento segura/o cuando explico a mis 95% estudiantes cómo realizar los ejercicios matemáticos. 24 Me siento segura/o cuando ayudo a mis 98% estudiantes a entender cómo funcionan las operaciones y formulas matemáticas. 25 Mis alumnos comparten en la clase los 96% métodos matemáticos que sus padres utilizan en su vida cotidiana. 26 Tomo tiempo de mis clases para 95% contextualizar lo aprendido en la realidad de la comunidad en la que trabajo.

Casi siempre

Generalmente

50% el tiempo

Rara vez

Muy rara vez

Los temas de matemáticas que encuentro más fáciles de enseñar son:

Los temas de matemáticas que encuentro más complicados de enseñar son:

Cada sección presentada a continuación, excepto la última, cuenta con 4 afirmaciones. Lea cada una y valórelas de modo que el total sume 100%. Estas valoraciones deben reflejar su opinión con respecto a ellas. Si lo desea, puede escribir comentarios personales. Las Matemáticas son: Un conjunto de verdades universales, reglas y procedimientos estandarizados que deben ser enseñados a los estudiantes Una materia creativa en la que cada individuo crea sus propios conceptos y métodos. Una de las herramientas fundamentales para el avance del desarrollo autosustentable de la comunidad. Un conjunto interactivo de ideas creado en forma colectiva a través del intercambio de perspectivas e interpretaciones. Si considera que no hemos presentado una definición que es importante, por favor compártala con nosotros en este espacio. Asegúrese de que este concepto sea parte de la suma del 100%

32 20 28 19

Aprender es: Un proceso individual basado en observar, escuchar y hacer ejercicios similares hasta que la nueva conducta se afiance. Un proceso individual basado en la exploración y la reflexión. Un proceso interactivo en el que los estudiantes son desafiados a desarrollar su entendimiento a través de actividades desarrolladas y analizadas en grupos. Un proceso motivado por el deseo de contribuir a la sociedad y en el que cada persona desarrolla sus potenciales. Si considera que no hemos presentado una definición que es importante, por favor compártala con nosotros en este espacio. Asegúrese de que este concepto sea parte de la suma del 100%.

20 21 31 28

Enseñar es: Introducir el aprendizaje con conceptos sencillos, yendo paso a paso hacia conceptos más difíciles. Dar instrucciones verbales y verificar, a través de ejercicios, que éstas hayan sido comprendidas, para corregir vacios de comprensión a través de los errores. Facilitar discusiones en las que el estudiante conjuntamente con su tutor profundice el conocimiento de las matemáticas y exploren la interconexión e interdependencia de lo aprendido para el desarrollo personal y social. Saber reconocer cuándo un estudiante está listo para aprender; proveer un ambiente estimulante que facilite la exploración y que, a través de experiencias cuidadosamente estructuradas, evite aprendizajes errados. Establecer una serie de diálogos no estructurados y estimulantes entre el tutor y los estudiantes, en los que los conceptos y su interrelación son explorados verbalmente. Si se generan conceptos errados, estos son expuestos para su análisis y corrección. Si considera que no hemos presentado una definición que es importante, por favor compártala con nosotros en este espacio. Asegúrese de que este concepto sea parte de la suma del 100%.

Cuando enseño, mis prioridades son dirigidas a que mis estudiantes logren: Motivarse y tomar responsabilidad sobre su propio aprendizaje.

Utilizar la matemática para explorar y describir el mundo y resolver problemas de su vida cotidiana.

Completar los textos y estar preparados para los exámenes.

Explicar los conceptos y formulas matemáticos.

Comportarse correctamente, participar en actividades y ser solidarios con sus pares.

Reconocer la calidad de su trabajo y la de otros y saber qué hacer para mejorar. 11 Si considera que no hemos presentado una definición que es importante, por favor compártala con nosotros en este espacio. Asegúrese de que este concepto sea parte de la suma del 100%.

Este cuestionario ha sido elaborado para conocer su valiosa opinión COMO ESTUDIANTE de las matemáticas.

Para contestar el cuestionario, marque con una X la respuesta que considere más oportuna. Las escalas de valores son las siguientes: Muy de acuerdo, De acuerdo, No he decido, En desacuerdo, Muy en desacuerdo Muy de acuerdo

De acuerdo

No he decido

En des acuerdo

Muy en des acuerdo

Las matemáticas son útiles y necesarias en todos los ámbitos de la vida. Las matemáticas son difíciles, aburridas y alejadas de la realidad. En matemáticas es fundamental aprenderse de memoria los conceptos, formulas y reglas. Las únicas matemáticas que me interesan son las que tengo que enseñar, porque las tengo que conocer. 6 Las destrezas o habilidades que utilizo en 98% clase para resolver problemas no tienen nada que ver con las que utilizo para resolver problemas en la vida cotidiana. 7 Busco distintas maneras y métodos para 98% resolver un problema.

8 Aprendo mucho inventando nuevos 99% problemas.

9 Si no comprendo las matemáticas 98% difícilmente podré asimilar y dominar otras asignaturas relacionadas con ella (como física, química etc.) 10 Mi rendimiento en matemáticas depende en 98% gran medida de la actitud del/a capacitador/a hacia mí.

11 Cuando dedico más tiempo de estudio a las 99% matemáticas, obtengo mejores resultados en la resolución de problemas

12 95% 13 94%

Cuando resuelvo un problema, suelo dudar de si el resultado es correcto. Me considero muy capaz y hábil en las matemáticas.

14 95% 15 99%

Estoy calmado/a y tranquilo cuando resuelvo problemas de matemáticas. Los/as buenos/as profesores/as que explican con bastante claridad y entusiasmo

1 98% 2 96% 3 93% 4 97%

y son agradables hacen que gusten las matemáticas.

67 Muy de acuerdo

De acuerdo

No he decido

En des acuerdo

Muy en des acuerdo

18 Las matemáticas son para cabezas 96% inteligentes y creativas.

19 La gente que es buena en matemáticas no 95% tiene que gastar tiempo pensando cómo resolver un problema. 20 Las clases de matemáticas se me hacen 95% eternas, son muy pesadas, no estoy a gusto y siento deseos de salir corriendo. 21 Ante un problema complicado suelo darme 96% por vencido/a fácilmente. 22 Cuando me enfrento a un problema, 93% experimento mucha curiosidad por conocer la solución. 23 Me provoca gran satisfacción resolver con 95% éxito un problema matemático. 24 Si no encuentro la solución de un problema, 94% tengo la sensación de haber fracasado y de haber perdido el tiempo.

25 En mis estudios matemáticos muestro 95% perseverancia y paciencia.

16 96% 17 96%

27 28

Cuando estaba en la escuela y en el colegio tuve bueno/as maestro/as de matemáticas. La gente a la que le gustan las matemáticas suele ser un poco rara.

Si tiene más de un año trabajando con SAT, por favor conteste el siguiente grupo de preguntas: Mi percepción de las matemáticas es más positiva desde que recibo las capitaciones de SAT. Otros tutores me buscan para ayudarles con las matemáticas. Después de una capacitación siento que no entiendo toda la materia y que no puedo resolver todos los ejercicios. Si está estudiando o ha estudiado matemáticas en la universidad o en la Pedagógica, por favor conteste la siguiente pregunta: Mis estudios de matemáticas con SAT me han ayudado con mis estudios de matemáticas en otras instituciones.

Lea las afirmaciones sugeridas a continuación y marque con una X el número que mejor represente cómo se siente usted cuando realiza cada una. La escala es de 1 a 5, siendo 1 – calmado y totalmente sin estrés, y 5- muy nervioso y/o muy estresado.

1 Muy calma do/a 1

Revisando el texto de matemáticas.

Planeando la clase que voy a enseñar al día siguiente. Viendo el capacitador explicar álgebra en el pizarrón. Empezando a estudiar el texto antes de ir a la capacitación. Interpretando gráficos matemáticos.

3 4 5 6 7 8

9 10

5 Muy Estresa do/a

66 62

Llegando a la capacitación y pensando en la capacitación de matimáticas. Cuando voy a trabajar en el pizarrón durante la capacitación. Cuando el/la capacitador/a me propone ‘por sorpresa’ que resuelva un problema.

Cuando me atasco o bloqueo en la resolución de un problema. Si hay otra acitividad matemática que le de estrés anótela aquí:

Si tiene algún comentario sobre su experiencia estudiando matemáticas con SAT, o si desea hacer alguna sugerencia, por favor compártala con nosotros en este espacio.

Appendix 5: Additional comments of the tutors on the questionnaire 1. 2. Pues, bueno yo he aprendido mucho de las matemáticas en SAT estoy bien agradecido por las capacitaciones. Y le ayuda a la preparación tanto persona como ensenar al alumnos unos no la ha ¿a la clase y preparándolo. Se siente con confianza y eso le ¿ a alumno. 2. 7. Las matemáticas vista en SAT han despertados interés en mi persona yo ya no tengo miedo a matemáticas porque he aprendido como poder a desarrollo a cómo enfrentarle. También he aprendido comprender situaciones de la vida porque (¿) miedo de la matemáticas. 3. 10. A mí me encanta las matemáticas y yo siento que lo que he aprendido en SAT me ha ayudado mucho en mis estudios universitarios y en mi vida diaria. 4. 12. Hay textos de mi matemática que se hacen fácil desarrollarlo de igual manera que hay textos que se debe dar más tiempo para desarrollarlos ya que tiene un nivel de complejidad y se vuelven un poco más difícil su realización en los textos de Bachillerato se le debe dar mas ¿??? en estos textos en las capacitaciones. Ya que se necesita ir con los conceptos darlos a impartirlos a los estudiantes. 5. 15. SAT nos ayuda mucho en las clases de matemáticas porque debemos contenidos muy amplios y bien definidos 6. 18. En mi comentario lo que les puedo decir es que en SAT es donde yo he aprendido mucho de matemática muy bonito la moralidad. Se aprende mucho en capacitaciones ya que los capacitadores son personas muy preparados. 7. 19. Mi experiencia como tutora de SAT ha sido muy importante porque aprendemos la clave las operaciones de matemática de diferentes maneras y las comprábamos con situaciones de la vida diaria. 8. 20. Las matemáticas estudiados con SAT me ha servido mucho en mis estudios universitario también a encontrado el porqué de los ejercicios. 9. 24 Los textos de matemáticas son muy interesantes pero se le debe dar más tiempo ya que esto texto de matemática necesita más practica que los de lectura (Matemática 3) 10. 27 Siempre me he sentido que me gusta las matemáticas pero desde que estoy en la sistema he descubierto que se resuelven muchos problemas de la vida con mejor facilitad. 11. 31. todo lo que he aprendido hasta hoy en las matemáticas de SAT me parece muy interesante porque el alumno y yo comprendemos lo que hacemos. 12. 34. Por los momentos pues tengo un mes de trabajo y me siento bien al saber que tengo la capacidad de ayuda a la comunidad y a su persona y se obtiene diferentes expresiones a vivir en otro parte y compartir unos conocimientos diferentes. 13. 37. Soy ingresada de SAT 100% y para mí las matemáticas aprendidas en SAT ya que el tutor se detiene y se explica detalladamente tratando dar la explicación más clara y dar un mejor resultado en la evaluaciones 14. 39. En SAT los experiencias en matemáticas es fascínate porque aprende a desarrollar su propia concepto y te pone a utilizar tu lógica que ha a veces esta muy dormida y puedes aplicar esta lógica matemáticas en problemas de la vida diaria y te lleva resolver estos problemas de la vida diaria.

15. 43. Todavía no tengo un ano pero siento que ya me ha ayudado bastante las capacitaciones de matemática 16. 44. Desde del momento que comencé a trabajar siento que he aprendido muchas conceptos que en el colegio no aprende. 17. Sugerencias que el capacitador nos trate con paciencia y confianza 18. 45. Es un proceso muy bonito y importante porque son ejercicios mucho mas explicativos que un educación de nuestro mismo nivel y estos ejercicios los podemos utilizar para resolver ejercicios de la vida cotidiana 19. 52. Las matemáticas de SAT me ha ayudado a ser mas analítico y a ponerlo en práctica en la vida real 20. 59 Es la primero vez que recibo una capacitación de SAT de matemáticas y fue muy productivo. Me fue muy bien y creo con dedicación estaré preparado a dar esto texto con el ayuda de Dios 21. 65. Mi experiencia es que encuentro mucho más fáciles los textos. Creo que es una forma muy efectiva de enseñanza 22. 75. Pienso que para las matemáticas es necesario usar diferentes materiales el tutor o capacitador tiene que tener paciencia para que los estudiantes le comprended y captan mejor 23. 76. Soy egresado de SAT y me siento totalmente satisfecho porque educación que he recibido soy tutor del mismo programa y me siento feliz de poder compartir mis conocimientos a mis alumnos y a otras personas, Gracias. He tenido la fortuna de ser una persona humilde y agradable ante la sociedad en SAT aprende muchas cosas que me ha gustado en la vida. Agradezco a la Asociación Bayan y a SAT por el gran apoyo que me ha brindado. Espero seguir adelante, seguir adelante y siempre resaltado al nombre de SAT donde quiero que yo vaya 24. 77. Hasta el momento los textos estudiados con SAT me ha gustado mucho los mejores capacitadores de matemáticas que he tenido son X y Z 25. 78. Estudiar los textos de SAT y es muy importante ya que ayudan en gran manera en nuestra vida como profesional también para tener alguna destreza conocimiento al momento de usar el estudio superior en la universidad. Sugerencia que los textos de SAT tenga suficiente espacio como para desarrollar ejercicios con mejor orden también que haya una secuencia en algunos temas que están muy similares. 26. 83. La experiencia de trabajo con SAT es muy bueno ya que he mejorado en el área de matemáticas porque durante estuve en la escuela y en el colegio no había aprendida mucho sobre la materia. En SAT encontramos con una matemática bien desarrollada donde retomarnos la matemática estudiado anteriormente y cosas de nuestra vida diaria 27. 92 Estudiando matemáticas en SAT me ha mejorando miro la matemática en manera diferente, y me gusta más y me siento más confianza y miro que la matemática no es como de niño pensé que era. Que SAT ha desarrollado en me el interés para seguir aprendiendo mucho más sobre matemáticas. 28. 99. Hay algunos temas que son más complicados que otros por eso sugiera que para algunos textos siempre haya micro centros aun que hayamos terminado el texto erg expresiones y ¿ Algebraicas 29. 101. Mas que un comentario tengo un profunda agradecimiento ya que todos las capacitaciones de matemáticas que he recibido me ha ayudado mucho porque antes me forma parte de SAT les tenía miedo a las matemáticas y ahora es una de las clases que 78

30.

31.

32. 33. 34. 35.

más me gusta y lo más importante que a mis estudiantes les gusta y disfrutan las clases de matemáticas. 105. Experiencia la manera de ver las matemáticas con la metodología me ha ayudado sobre la manera y he aprendido a razonar mejor los problemas presentados. Sugerencia la capacitación de matemáticas debería ser al principio de la primera semana 106. Respeto al texto que sea siempre de ver en los primeros días porque uno tiene más relajado en cambio cuando los dan al final de la semana uno se desespera o se estresa mas por el cansancio o porque nos desesperamos por unos que no ponen atención. 121. En SAT se obtiene la herramienta necesaria para poder entrar diferentes ejercicios y poder resolverlos 123. En SAT la matemática se explique de una manera muy intangible porque se aplica a la realidad y se explica con modelos sencillos pero muy importantes. 128. Hacer preguntas, contestar respuestas breves que el alumno afirmaciones se presentan para analizarlos en sentido positivo y negativo 132 Las matemáticas así como el trabajo de SAT no ha sido solo una fuente de empleo sino que ha cambiado positivamente mi vida. Gracias a Dios por el SAT.