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Cognitive & Developmental Issues
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Concepts such as symmetry, translations, reflections, area, and volume can be very abstract and are often not connected to the student's world. When a new concept is introduced, students need tactile and visual experiences to assist them in understanding the concept. For example, area and volume become more real when students cover a rectangle with squares or fill a rectangular solid with cubes. Using manipulatives allows the students to relate new ideas to things they already know and understand. After students feel comfortable with the concepts using manipulatives; students are ready to move from the concrete to the visual.
For example, a figure drawn on paper could be folded along its line of symmetry to verify that it is in fact symmetric. The best use of instructional time is to wait to introduce some abstraction until students have moved into van Hiele's Level 2 (analysis).
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A Suggestion for Using Concrete Representations
Use a mirror or Mira™ to illustrate that a pattern block is symmetric about a line. Have students trace a flat plastic bear (or other animal) in one position, and then move or translate the bear to a new position and trace it again. Have them place the bear on one side of a line drawn on a sheet of paper, and then place a Mira™ on the line. Have students draw the bear they see reflected in the Mira™ on the other side of the line. Students now have a reflection of the bear across the line of reflection. They can look at properties of the reflection and compare these properties with what happens when they look at their own reflection, wave, or wink in a mirror.
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