Elementos del Tanque Cilíndrico de Techo Flotante.

knowledge of instructional processes for students who learn atypically. Co-teaching is a means of providing the desired learning and teaching outcomes that can benefit both students and teachers. Sileo and van Garderen (2010) conclude that, although co-teaching structures can enhance student learning, it is also important to consider the subject matter. General and specialist educators can work together to blend their knowledge bases. This relationship is invaluable because it weds content and strategy specialists and allows teachers an opportunity to meet all students’ mathematical learning needs. “The greatest premise of co-teaching is the teacher’s ability to provide academic and behavioural support for all students” (Sileo and van Garderen, 2010: 15). Unfortunately in developing countries such as Zimbabwe, the economy cannot support a multiplicity of teachers in the form of general and special teachers. So the students’ opportunity to learn becomes compromised.

Bass (1993) points out that the curriculum was still organised in ways that prevented many students from gaining access to the Mathematics that they need. Bass brought in an element of what students need rather than what is given to them, irrespective of the relevance. The curriculum should consist of what the learners find useful in their lives. This does not mean that the sole intention of the curriculum is to provide students with only what they need, since this would make it narrow, but what they need can help to motivate them to want to learn. It can spark an interest in students to pursue Mathematics further than the classroom. It is important to consider what the learners with VI need so that they can find learning interesting and worthwhile, and hence contribute to an improvement in performance..

2.3.3 Other Studies on OTL

Several researches reveal that OTL can be used to aid the understanding of the teaching and learning process. It is clear that most researchers connect students’ achievement with their opportunity to learn the content. Some researchers point out that the study of students’ opportunities to learn provides great insights into variation in student achievement.

Researchers have described the opportunity to learn in a variety of ways. Schwartz (1995) and Scherff and Piazza (2005) maintain that OTL include the provision of curricula, learning materials, facilities, teachers and instructional experiences that enable students to achieve high standards. It also includes a teacher’s reported content coverage, time allocated for instruction or instructional time that is actually used to deliver instruction. The achievement of students is not a simple issue, for learners perform differently even under the same conditions. Some learners may not recognise the opportunities offered to them to learn because of other reasons that impact on them socially or otherwise. Scherff and Piazza (2005) bemoan the fact that the indicator of whether a school is considered successful is usually student achievement scores, yet a single score can mask the complexities of teaching and learning, as well as the factors that impact on test results. The concept of opportunity to learn (OTL) can, therefore, guide the assessment of schools and place them in proper perspective, especially when striving not to leave any child behind.

The USAID paper (2008) identifies eight crucial elements that create what they refer to as a basic opportunity to learn. These elements are; total instructional time, the hours in a school year and days that the school is open, teacher attendance and punctuality, student attendance and punctuality, the teacher-student ratio, instructional materials per student, and the classroom time spent on the tasks and skills taught per grade. Hence, without a strategy to monitor these elements closely, and directing funding to ensure that a minimum level is attained, children cannot be provided with a basic opportunity to learn. The USAID paper further argues that the

basic OTL index starts from a relatively simple premise that learning is to some degree a function of time and effort. Without adequate time on task, no learning is possible. Investments in teachers, materials, curricula, and classrooms are wasted if they are not used for a reasonable period of time. Therefore there is a direct relationship between learning and OTL.

Cueto, Ramirez and Leon (2005) note that the difference between what is intended and what is implemented is due to many factors. These factors include the fact that the curriculum is too long, the students do not master some of the competencies and so the teachers do not have sufficient time to cover them all, the teacher may have different priorities regarding what should be taught, or the teachers do not have the educational material needed to teach some competencies (Cueto et al., 2005). Our understanding of the causes of poor performance by students with VI may become clearer if student achievement is related to the opportunity to learn, regardless of family background. It becomes a more relevant issue to consider what happens in the classroom, that is, the teacher’s teaching practices and the quality of the delivery of lessons, for these can be changed or improved (Stevens, 1993).

2.4 THEORETICAL FRAMEWORK

It is generally understood that students with VI do not perform as well as sighted students especially in Mathematics. Students with VI do not seem to get the same OTL as their sighted peers due to a number of reasons. According to the CIET (1999: 219), some of the reasons for this discrepancy are that:

To begin with, most children with VI are still kept in homes away from the public eye such that when they are finally taken on board they are well above the age of their classmates

due to lack of knowledge of Braille. Belay (2005) confirms this assertation about some African and Asian nations whereby disabilities are consigered as some form of a curse or punishment and so they hide their disabled children. hideWhere some have had the opportunity to attend school at an early age, the resource units established at their schools are not fully equipped due to financial challenges faced by the government.

Secondly,There are not many teachers professionally qualified to teach students with VI. Those that have specialised are deployed (after in-service training)to their original schools after qualifying even if there is no need for a specialist teacher. The newly qualified diploma holders were not effective in their areas of specialisation.

Then there are not enough resources (textbooks, braille writers, computers, etc) to go round and finally, students with VI are given the same examination as the sighted regardless of their circumstances (e.g. diagrammatic images for Mathematics).

These factors impact negatively on the education of children with disabilities and specifically on children with VI who are learning mathematics. The researcher believes these students encounter more problems in Mathematics than in other subjects. They have to learn extra braille, to develop good listening skills and good memory in order to get by in Mathematics lessons.

It is a fact that ‘doing Mathematics’ involves working out problems on paper. Because there are so many symbols and formulae to deal with in Mathematics, merely listening to the teacher teaching does not help the student to acquire Mathematics concepts. Mathematics requires exactness, definiteness, totality and comprehensibility of presentation, so oral communication becomes very arduous (Kohanova, 2006). Even the sighted students face challenges when they have to just listen to the teacher, they have to write something. Oral communication has to be

supported by text or pictures. These issues call for the teacher to be well versed in supporting student engagement in a Mathematics lesson. The study agrees with Kohanova (2006) who believes that the process of acquisition of knowledge by non-sighted students is different from that of sighted students. Kohanova describes the sort of interactions which take place in an integrated setting between

• teacher and the knowledge to be taught, • teacher and non-sighted student,

• non-sighted student and knowledge to be learnt,

• Tutor and non-sighted student (involving individual help),

• Non-sighted student and sighted student (which involves cooperation).

All these interactions have a bearing on how the student with VI can learn Mathematics.

The main theory that guides this research emphasises the interrelationship between learners and contexts, and specifically posits that learning is a function of what people do given what they have opportunities to do (Greeno & MMAP, 1998; Gresalfi, 2009), in Gresalfi, Barnes & Cross (2011). Grasalfi et al. base their theory on Ecological psychology by Gibson (1979). Ecological psychology claims that what happens in any particular moment is based on a constructed set of actions defined by the affordances of the environment for a particular action, the intention of the agent to take up the affordances and effectivities of the agent to actually realise these affordances. Affordances refer to the set of actions that are made possible by a particular object, what, in the context of this research could be refered to as opportunity to learn. They further explain that the extent to which an affordance can be acted on has to do with one’s effectivities, that is, an individual’s ability to realise those affordances. In other

words, there is a possibility of missing out on opportunities that may be made available to learners.

Gresalfi et al. (2011) describe how particular moves by the teacher might become affordances for particular forms of engagement for students. The forms of engagement considered were procedural, conceptual, consequential and critical engagements. They analysed an ecological framework together with the conceptualisations of engagements by considering how the kinds of instructions teachers give, the structure of whole class discussions and more importantly, what teachers emphasises in those discussions which might make the particular forms of engagement more or less likely for students. Their analysis has to do with everything that goes on in a lesson, which is what this study shall be looking at during lesson observations. The objective will be to assess how the OTL or affordances impact on student understanding and ultimately on performance in Mathematics. It is important to note that an affordance can only be afforded if it can be recognized and acted on. Greeno and Grasalfi (2008) posit that what makes an affordance actionable is inherently a dynamic relation between the environment and the person. The focus here is on interaction, which presupposes that a particular task might make something possible but does not make it obligatory.

This study will take the same line of enquiry as was done by Gresalfi et al. (2011) since part of the methodology being used here involves observing interactions between the teacher and the students with VI in a classroom situation. Hebert et al. (1997), in Gresalfi et al. (2011) describes five critical features of classrooms, namely, the design of tasks, the role of the teacher, the social culture that develops, the kinds of tools available to support learning and the importance

of creating equitable opportunities to learn. This study focuses on all five features since they are seen to contribute to OTL and to influence the performance of students.

There is no doubt that task design is a key affordance for supporting student engagement. Stein, Smith, Henningsen and Silver (2000) in Gresalfi et al. (2011: 251) observe that tasks can support particular forms of cognitive engagement ranging from low-level engagement, for instance, those that require rote learning or memorisation, to those that support more sophisticated forms of engagement (referred to as ‘doing math’). Hence they conclude that for students to engage deeply with Mathematics content, they must be given tasks that create opportunity for such engagement. They add that teacher practices such as the kinds of questions they ask, the ways they organise student work and how they frame activities help to shape the ways that tasks are implemented, resulting in a ultimate engagement. The type and quality of classroom discourse shape the kinds of learning and the depth of understanding that is realised in a classroom (Ball, Lubienski & Mewborn 2001; Carpenter, Franke & Levi, 2003). Literally what it boils down to is that the design of a task together with the way its implementation is supported can constitute opportunity to learn mathematics.

Besides the ecological theory, the researcher also refered to the works of Raymond Duval and that of van Hiele. Having realised the problems that students with VI encounter in dealing with questions on geometry, the researcher feels it is prudent to consider the plight of these children with VI in light of the van Hiele model of the development of geometrical thought; and Duval’s model of geometrical reasoning. Geometry makes up about half of the ‘O’ level syllabus content and for student to pass they need to have a good grasp of both the algebraic and

geometrical concepts. Jones (1998) says the van Hiele model of thinking in geometry is a teaching strategy based on levels of thinking commonly referred to as the van Hiele levels. This model fosters the idea that students’ initial curricular encounter with geometry should be of the intuitive, explanatory kind (van Hiele 1986 : 117). The learner then progresses through a series of levels characterised by increasing abstraction. The levels are

• Visualisation (the student recognises individual shapes), • Analysis (shapes become bearers of their properties),

• informal deduction or abstraction (the student develops relationships between and among properties, properties are ordered) ,

• deduction (students prove theorems and are able to work with abstract statements) and • rigour (students understand the relationships between various systems of geometry, can

compute, analyse and create proofs). (Gujarati, 2014).

Gujarati (2014) explains that the van Hiele levels do not explicitly tell teachers how to teach geometry, but can help teachers to assess the level their students are working at by looking at some of the characteristics of that level. The geometrical levels are sequential and students must progress though them in order. Progress through the levels is more dependent on the instruction received rather than age or maturation.

On the other hand, Duval (2006) says that to understand the difficulties that many students have with comprehension of mathematics, we must determine the cognitive functioning underlying the diversity of mathematical processes. We need to understand the cognitive systems that are required to give access to mathematical objects, and whether these are common to all processes of knowledge or they are just specific to mathematics.Duval 2006: 104) further says that

“research about learning of mathematics and its difficulties must be based on what students do by themselves, on their productions, on their voices”.

A close analysis of the works of van Hiele and of Duval reveals that there are lots of similarities in the way students learn geometry. The two models together with the main framework, the ecological framework, are considered in this research as providing a strong basis on how students with VI can successfully learn Mathematics. The ideas presented in the models were used during he collection and analysis of data to gain an insight into how OTL impacts on performance by students with VI in Mathematics. In the next section I describe how students with visual impairment are exposed to educational opportunities in different parts of the world.

Gauvain and Cole (1997), in describing Vygotsky’s idea of interaction between learning and development comment that when we determine a child’s mental age by using tests, we are dealing with the the actual development level. When the child is subjected to some scaffolding such as repetition, offering leading questions or initiating a solution and having the child complete it, then the child gets to a higher level of operation. Vygotskian principles on the zone of proximal development is best understood as the difference between what a learner can do without help and what he or she can do with help. In Vygotsky’s perspective, the role of the teacher is mediatingthe child’s learning activity as they share knowledge through social interaction. Scaffolding refers to the way the adult guides the child’s learning through focused questions and positive interactions. This approach is a particularly useful approach when dealing with students who are visually impaired.

2.5 EDUCATION FOR STUDENTS WITH VISUAL IMPAIRMENT