The Theory:
Knowing that I need to work on my own understanding of how math
concepts develop (see last blog), I try as much as possible to ‘consult with
the experts’ when planning. We are just swinging back into Geometry, so I went
to Van De Walle’s “Teaching
Student-Centred Mathematics” (Grades K-3) and the K-3 Guide to Effective Instruction
in Mathematics – Geometry and Spatial Sense (see link below) for support. Kindergarteners
are learning to “explore, sort, and compare” as well as “identify and
describe, using common geometric terms” two-dimensional shapes and
three-dimensional figures. The “identify” is only one small part of that, the
bigger part is becoming aware of geometric properties. “We know now that rich
experiences with shape and spatial relationships, when provided consistently
over time, can and do develop spatial sense” (Van De Walle, p. 187).
Kindergarteners are generally working at Level 0 of van Heile’s Levels of
Geometric Thought (see Guide, p. 12) where students identify identify
two-dimensional shapes and three-dimensional figures by their appearance as a
whole.
Students do not describe properties
(defining characteristics) of shapes and figures. We are supporting them to
work towards understanding that all shapes or figures within a class share
common properties (e.g., all rectangles have four sides, with opposite sides
parallel and congruent). Progression
from one level to the next is less dependent on students’ age or maturation than on instruction that
promotes reasoning about geometric ideas.
Van de Walle and the Guide recommend
that students have a lot of experience with sorting and classifying shapes.
The Practical:
Following ideas from the Guide about
triangles, after a quick “Minds On” (telling a partner what shapes they could
see in a piece of art), we began the “real triangles” problem. I gave pairs a
baggie with about 10 different triangles (traditional and non-traditional forms) printed on paper. I did not say that the
bag contained triangles, just that it had pictures of shapes. Their
problem to solve was to find and bring back to the circle the “real triangles.”
While they worked on sorting with their partners, I took anecdotal notes on the
language that I heard. For our
consolidation, we sat in a circle with the “real triangles” on the floor in
front of us to show others. A few students shared why they had picked these.
The idea of three sides, and three “points” (vertices) was shared. The idea that
they were actually all triangles was not. Hmmmm. Right in line with what I had
learned about Level 0!
Our next step… Sort again, only this
time, pose the question, “What are the other shapes?”
The Guide also has a great idea on
page 18, to continue with shapes that are ‘trianglish’ (e.g. open three sides;
soft corners…) and to ask why they aren’t triangles, and how they could be made
into triangles!
We’ll see how that goes, then move to
the “real rectangle!”
My key learning here continues to be
the need for my own continued learning about the development of mathematical concepts,
paired with effective problems that invite the students to engage with
meaningful mathematics.
References:
K-3 Guide to Effective Instruction in Mathematics – Geometry and
Spatial Sense
http://etfo-ot.net/Site/wp-content/uploads/2009/05/geomspatsense-guideeffectinstrmathgrk-3.pdf
Teaching Student-Centred
Mathematics, K-3, John Van de Walle and LouAnn Lovin.
