PARASITOSIS GASTROINTESTINALES EN PASTURAS TROPICALES

The manipulation of concrete objects is not, in itself, enough to give learners the opportunity to understand abstract, symbolic representations of mathematical ideas (Blair, Blair & Schwartz, 2012). It is critically important that learners understand these symbolic representations as they advance through school (Uttal, O’Doherty, Newland, Hand & DeLoache, 2009). Manipulating concrete objects in order to understand mathematical concepts is certainly important, particularly in the early stages of learning, but learners must be able to connect concrete and symbolic representations. Thus, the essential duty for mathematics teachers is to help learners to understand, and to manipulate, symbolic representations.

Learners need repeated experiences and a wide variety of concrete materials to make these connections strong and stable. Teachers often compound difficulties at this stage of learning by asking learners to match pictured groups with number sentences before they acquire sufficient experience of relating varieties of physical representations with the various ways of stringing mathematics symbols together, and the different ways we refer to these things in words. The fact that concrete materials can be moved, held, and physically grouped and separated makes them much more vivid teaching tools than pictorial representations.

Because pictures are semi-abstract symbols, if introduced too early, they may confuse the delicate connections being formed between existing concepts and the new language of mathematics. Similarly, Marshall and Paul (2008) note that structured concrete materials are beneficial at the conceptual development stage for mathematics topics at all grade levels. Concrete objects provide a way around the opaqueness of written mathematical symbols. Evidence from research indicates that learners who use concrete materials actually develop more precise and more comprehensive mental representations often

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show more motivation and on-task behaviour, better understanding of mathematical ideas, and are able to apply these to real-life situations (Hiebert & Grouws, 2007).

According to DeLoache (2004) the concept of dual representation can shed light on this fundamental problem. The central tenet of this concept is that all symbolic objects have a dual nature; they are simultaneously objects in their own right and representations of something else. To use a symbolic object effectively, one must focus more on what the symbol is intended to represent and less on its physical properties. Symbols may be difficult to teach to learners who have not yet grasped the concepts that they represent (Ball, Thames & Phelps, 2008). At the same time, the concepts may be difficult to teach to learners who have not yet mastered the symbols. This scenario presents teachers with a dilemma of how to sequence concepts and symbols during teaching.

Hiebert (1988) proposes a theory that may help to explain learners' “overly mechanical behaviour” of learning. The theory is based on how learners develop competence in dealing with the written symbol systems of mathematics. Hiebert (1988) suggests a series of cognitive processes whose cumulative effect yield competence with written mathematical symbols. He identified five major types of processes:(1) connecting individual symbols with referents; (2) developing symbol manipulation procedures; (3) elaborating procedures for symbols; (4) routinizing the procedures for manipulating symbols; and (5) using the symbols and rules as referents for building more abstract symbol systems.

Connecting symbols with referents: In school mathematics, written marks in textbooks

represent quantities or operations (processes) on quantities. To connect written symbols with appropriate referents, learners must be familiar with the relevant quantities and actions on the quantities, and they must be familiar with the written characters that will be used to stand for the quantities and actions. Then they must create a correspondence between the written characters and the quantities or actions to which they refer. Familiarity with quantities that can be used as referents is part of many learners' informal knowledge. Learners often engage in activities with materials and ideas to find how many, how much and when. These everyday experiences generate knowledge of quantities and actions on quantities that can provide the initial referents for written

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mathematical symbols (Nunes, Bryant & Watson, 2007). Learners competence with written symbols develops as construct connections between individual symbols and familiar referents. Meanings for individual symbols are created as connections are established between the written marks on paper and the quantities or actions that they represent (Pape & Tchoshanov, 2001). The process involves building bridges between symbols and referents and crossing over them mentally many times.

The significance of the connections between numeric symbols and quantities is that they provide mental paths from the symbol to the referent. Learners can recall the mental image of related quantities and reason directly about the quantity to solve the problem if it is presented to learners in the form of written symbols (as in ordinary classroom lessons). The advantage is that the quantities serve as "conceptual entities" (Greeno, 1983), as cognitive objects that the problem solving procedures take as arguments. For learners who are new to the domain, such conceptual entities are likely to support the problem solving process.

Developing symbol manipulation procedures: The second cognitive process required to

continue the development of competence with symbols is directed towards the development of symbol procedures. The procedures are formulated by manipulating the referents of the individual symbols, observing the result, and then paralleling the action on referents with an action on symbols.

Routinizing symbol procedures: The symbol system is used more efficiently if the

procedures are well practiced. When procedures are practiced so often, they can be executed automatically, with little conscious thought, and then the user achieves maximal efficiency.

Building more abstract symbol systems: Symbol systems themselves develop by

building on one another (Goldin & Kaput, 1987). Learners' competence with symbols continues to develop as more abstract systems are encountered, and the ways in which they build on earlier familiar systems are recognised. One way in which later systems can build on earlier ones is through the transfer of meaning directly from the early symbols and rules to the later system. A second way is through the recognition of a correspondence between two different symbol systems. Learners can transfer meaning from a familiar symbol system to a new, more abstract system if they have established

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meanings for the familiar symbols (the first two processes have been thoroughly engaged), and if they recognise a mapping between the systems so that the familiar symbols and rules can serve as referents for the new system.