Cluster Concepts

When students learn concepts and relationships in isolation, they often forget these ideas or are slow in making the connections among them.28 A thinking process called "clustering," used to group, unify, integrate, and/or make connections among concepts, is something that students and adults use routinely, often without even realizing it.29

A teacher can use this approach to cluster mathematical ideas, concepts, relationships, and objects in order to reveal common characteristics. This presentation in turn helps students to categorize and classify those ideas or objects and to remember properties and attributes, which makes the learning more meaningful.

Clustering quadrilaterals

Some students can best understand quadrilaterals when the teacher presents them as a unifying concept. In the process of clustering, a teacher guides students in studying, comparing, and classifying all quadrilaterals according to their attributes-not only the square and the rectangle, but the plain quadrilateral, the rhombus, the parallelogram, the trapezoid, and the isosceles trapezoid. This enables students to have a better sense of four-sided figures, their properties, and characteristics, so they can eventually internalize the idea and even develop a hierarchy of quadrilaterals based on properties.

Other ideas for clustering in geometry and measurement

special lines in triangles such as medians, perpendicular bisectors, altitudes, and bisectors of angles
special lines of circles
the classification of Platonic solids
the relationship of various types of tessellations
proportionality
Archimedean solids
inscribed and circumscribed geometric figures
the classification of angles
nets of specific solids, such as cubes, pyramids, etc.
the relationship between area and perimeter of quadrilaterals, triangles, circles, etc.
the symmetry of geometric figures in two and three dimensions
the similarity and congruency of polygons
the relationship between surface area and volume.

Robert Reys et al., Helping Children Learn Mathematics, 7th ed. (New York: John Wiley & Sons, 2001), p. 33-34.
Ashish Ranpura, "How We Remember, and Why We Forget," Brain Connection (June 2000). Available only online at http://www.brainconnection.com/topics/?main=fa/memory-formation3.