Durmus and Karakirik (2006) posited that inspirational educational activities and cognitive tools might develop students’ active involvement in the teaching-learning process and promote their reflection on the concepts and relations under investigation…
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117). Physical or real-world features do not define a concrete experience in a mathematical context; it is by how significant the connection is to the mathematical ideas and situations. For example, a student might create the meaning of the concept "four" by building a representation of the number and connecting it with either real or pictured blocks. Virtual manipulatives, also called computer manipulatives, appear to offer interactive environments where students can manipulate computer objects to create and solve problems. Furthermore, perhaps because they are receiving instant feedback about their actions, students then form connections between mathematical concepts and operations. However, whether using physical or virtual manipulatives, it is necessary to connect the use of a specific manipulative to the mathematical concepts or procedures that are being studied (p. 119). Some researchers have observed that some of the constraints inherent to physical manipulatives do not bind virtual manipulatives. Use of models and/or manipulatives gives assessment of mathematical learning a cohesive connection to mathematical instruction (Kelly, 2006). Kelly’s study examines the relationship between mathematical assessment and the use of manipulatives. ...
The use of such assessments in combination with the use of manipulatives should build strong student investment in the teaching-learning process while developing deeper mathematical learning. Physical Manipulatives Relative to the teaching and learning of mathematics, physical, or concrete, manipulatives are three-dimensional objects used to help students bridge their understanding of the concrete environment with the symbolic representations of mathematics (Clements, 1999; Hynes, 1986; Moyer, 2001; Terry, 1996). There has been historical documentation of the use of manipulatives such as the abacus, counting sticks, and of course fingers, prior to the Roman Empire (Fuys & Tischler, 1979). Examples of teacher-made manipulatives include those that use materials such as beans, buttons, popsicle-sticks, and straws (Fuys & Tischler). Today’s teachers have access to a wide variety of commercially available manipulatives designed to aid in the teaching of most elementary mathematical concepts. Examples include Algebra tiles, attribute blocks, Base-10 materials, color tiles, Cuisenaire rods, fraction strips, geoboards, geometric solids, pattern blocks and Unifix cubes. The appearance of commercially made manipulatives in the United States increased during the 1960s after the work of Zolten Dienes and Jerome Bruner was published (Thompson & Lambdin, 1994). Many educators continue to view manipulatives as teaching tools that involve physical objects that teachers use to engage their students in practical and hands-on learning of mathematics. These manipulatives continue to be instrumental to introduce, practice, or remediate mathematical concepts and procedures. Concrete manipulatives come in a variety of physical forms, ranging from grains of rice to
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