3.G 1: Geometry Squares But Not Rhombuses (#209)


A Year in the Life: Ambient Math Wins the Race to the Top!
Day 209

For one year, 365 days, this blog will address the Common Core Standards from the perspective of creating an alternate, ambient learning environment for math. Ambient is defined as “existing or present on all sides, an all-encompassing atmosphere.”

And ambient music is defined as: “Quiet and relaxing with melodies that repeat many times. Why ambient? A math teaching style that’s whole and all encompassing, with themes that repeat many times through the years, is most likely to be effective and successful. CCSS math standards are listed here in blue, followed by ambient math suggestions.

Geometry 3.G
Reason with shapes and their attributes.
1. Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals).  Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.

Form drawing nicely covers this standard by focusing on squares that can be fitted within a perfectly-drawn circle.  Hand-drawn forms are key, until formal geometry is taught in Grade 6.  That’s time enough to learn the formal names and terms.  No need now to be so explicit that each shape is identified by name.  More effective, much deeper learning is possible if the first interface with a concept is intuitive and broad, not specific.

Confidence and solid knowledge are built on just such a foundation.  Waldorf education parallels the stages of child development with the stages of human development.  For this reason, perspective drawing is not taught until Grade 7, paralleling the cultural context of the Renaissance when art images first reflected the leap into 3-dimensional seeing.

Plane forms are appropriate at this time, but not yet dimensional forms.  So the Rhombus needs to wait till a later grade, because its complex angles make it a dimensional rather than flat, plane figure.  Many variations on the square are experienced with form drawing, and this variety engenders a flexibility of thinking which spills over into all other areas.

Another invaluable quality of form drawing is that the forms are experienced on a much deeper level because they are created from virtually nothing.  Each form is described either verbally or with a picture story first, then moved, large and with whole-body movement.  The form is then “drawn” in the air before committing it to paper.

All of the square forms are accurately drawn because they’re based on the circle.  A yellow crayon or pencil is used to draw many circles, filling up the whole page, until the perfect circle is “found” then darkened.  After the corners of the square are located on the circle, the square is drawn with exploratory lines until the “right” lines are found.  This sort of freehand drawing is so much more empowering than tight, technical drawing because the forms are artfully built.  See below for contrasting examples of finding the “square spiral” within nested squares.


As always, knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of knowledge as a worthy goal. Tune in tomorrow for more Grade 3 math CCSS and their ambient counterparts.

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