Defining Representation
 Introduction | Concrete Models | Written Representations | The Teacher's Role | Summary | Your Journal

As young students learn to do mathematics, it is important for them to have time to develop their own strategies and processes for solving problems in order to make sense of what they are doing. Models serve as a link between a concept and the symbols that can be used to express that concept. Concrete models can help children represent their thinking about a mathematical concept. By "concrete models", we mean something that exists physically in the world and that generally the student can manipulate. Think back to the children who were estimating and counting beans in Part A of this session. The children began by using the models of specific numbers of blocks as a reference for their estimate. In this case, the models were provided by the teacher; however, children should certainly be able to use their own models as an estimation strategy. The students also used concrete models to show their counting strategies. By grouping the beans into sets of 10, they were able to demonstrate the counting process they had used.

Models can be used as representations in all of the Content Standards. Examples of concrete materials that can be used to model, for instance geometric concepts, include pattern blocks, and geo-blocks. Graphs using real objects, introduce children to organizing data. Linking cubes, base-10 blocks, and other counters can physically represent certain number concepts. Everyday items such as paper clips or file cards are often used to represent early measurement ideas. Whenever possible, children should be given their own choice of materials to model mathematical procedures and represent their thinking. Models imposed by the teacher may not provide the same meaning for students -- they make sense of what they are doing by modeling with their own concrete materials.

We know that children begin to develop understanding of mathematical ideas on a concrete level. Physical materials give students an opportunity to express their ideas before they are able to record them with pencil and paper; they also give students the flexibility to make, test and refine their conjectures. This does not imply that, once students have developed writing skills, physical models should be eliminated. Many mathematicians and scientists rely on concrete models to test and refine their conjectures!

Watch the video segment (duration 0:28) in the viewer box on the upper left to hear a reflection from Delia Hakim, an elementary school mathematics teacher.

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