Math Relearning/EngageNY/GPK/Module 2: Shapes

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These are my comments on EngageNY's Prekindergarten Module 2.


New propositional concepts from this module:

  • Objects have shape as one of their attributes.
  • Shapes consist of their outlines and not their interiors.
  • Shapes have attributes, such as their number of sides and corners.
  • Shapes can be classified and named by their numbers of sides and corners and by the straightness or roundness of their lines.
  • Objects have positions in relation to each other that can be described using terms such as above, behind, below, and between.
  • Three-dimensional shapes have two-dimensional faces of particular shapes.
  • Three-dimensional objects have functional properties particular to their shapes, such as the ability to stack, roll, or slide.
  • When a quantity is separated into groups, each group is smaller than the original quantity.
  • Real-world objects have shapes that can be identified with mathematical shapes.
  • Objects in the real world can be modeled by simpler objects that represent them.

I might not include term definitions in these lists of propositional concepts, since they're defined in the curriculum material.

Connections to earlier concepts:

  • Shapes can be matched and sorted into groups by their attributes.
  • Certain attributes of a shape can be counted, such as its sides and corners.
  • Shapes in a group can be counted.

Before deciding on Common Core to learn math, my approach to geometry was going to be somewhat different. I was going to start with measurement, which would include counting, and I would've used measurement to introduce the concept of length, which would introduce the notion of continuous quantities. Counting would introduce the idea of discrete quantities. After measurement I'd have covered geometry and introduced the idea of a shape by describing an angle as a line with a discrete change in direction and a curve as a line with a continuous change in direction. Then I'd have described common shapes, and that's as far as I got. Common Core takes a more holistic approach and starts with the knowledge students already have, guiding them to analyze that knowledge so they begin learning the formal properties of familiar shapes.

"the whole triangle consists only of its outline" - I've wondered that. This curriculum is achieving my purpose for it, filling in gaps in my knowledge.

Topic A: Two-Dimensional Shapes

Lesson 1: Find and describe circles, rectangles, squares, and triangles using informal language without naming

We learned how to sort and count everyday objects in Module 1. Now we're sorting mathematical objects--shapes--and doing it by counting their features--their sides and corners.

I notice the teacher does the sorting at first, which ends up prioritizing the number of sides and corners as a criterion. Is there any mathematical reason to sort shapes by some other attribute?

Topic B: Constructing Two-Dimensional Shapes

Lesson 7: Construct a rectangle and a square

"Which balls are bigger, when we made two balls or when we made four balls?" - Ah, slipping in a little 2.MD.2. Smaller units means more units in the measurement.

Topic C: Three-Dimensional Shapes

Whoa, 3D shapes already in preschool, slow down! Okay, not really. But a third dimension does add potentially a lot of complexity. For example, consider 3D ambigrams. There's one on the cover of the book Gödel, Escher, Bach. Each block is shaped so that it looks like a different letter from the angle of each axis, as shown by the shadows on the walls and floor. It takes some time for me to picture the shape of a block in my mind so that it forms each specific shadow from each angle. Fortunately, we're only working with familiar shapes in preschool, so the students won't have to construct brand new shapes in their minds.

In my pre-Common Core contemplations, I hadn't thought about how to approach 3D shapes, so this is new to me. Somehow their very physical, experimental approach to it is striking to me, even though they've been taking that approach for everything else. The kids learn about the shapes by using them and watching the results, especially by noticing their behavior while building with them, observing their fitness for different building purposes (e.g., flat faces are good for stacking; pointed ones aren't). The stamp activity is another good one, helping the kids isolate the shapes' features by observing their "footprints."

Lesson 11: Identify, analyze, sort, compare, and build with solid shapes

The curriculum takes the physical behavior of shapes seriously enough to devote a lesson to it, as if it were a class in engineering, though transformations analogous to rolling and sliding are part of geometry.

Lesson 12: Position solid shapes to create a model of a familiar place

This modeling exercise fits into K.G.5, so I don't know why the authors didn't list a corresponding pre-K standard.