Math Relearning/EngageNY/GPK/Module 3: Counting to 10

These are my comments on EngageNY's Prekindergarten Module 3.

Overview
This module is mostly an application of Module 1 to a wider range of numbers, so there aren't many new concepts.

New propositional concepts from this module:


 * The concept of "none" is symbolized by the numeral 0.

Connections to earlier concepts:


 * In the standard sequence for counting, each number is one greater than the previous number and one less than the following number. Numbers have smaller numbers embedded in them. So any counting number can be decomposed into two parts. It's helpful to view a counting number as a combination of the previous number in the counting sequence and 1 more. It's helpful to view a counting number between 5 and 10 as 5 and a certain number more.

People can subitize (recognize without counting) quantities of 2 or 3. I think this is why the curriculum emphasizes the relationship of other numbers to 5: 5 is both a factor of 10, which is the central number in our place-value system, and a combination of two numbers we can subitize. But the students also learn how to decompose numbers into other pairs that don't involve 5, such as in the Concept Development of Lesson 26, where they identify all the number pairs that make up 9, including 9 and 0.

It would be good to experiment with ways of quickly counting larger groups of objects when they're in convenient arrangements. This module does some of this. Ten-frames are one example. They'll show up in Kindergarten. The dot arrangements on dice are another, though that only goes up to 6, unless you combine dice, which takes you up to at least 12. Then there are ingredient arrangements on crafting tables in Minecraft, which take you to 9. You could also use the corners of geometric figures. I don't know how high you could comfortably take that. I'd probably recognize up to 9 because of the Enneagram, though looking at Lesson 27, I see it's harder without the interior lines. With more complicated shapes like the pentagram (for 10), which let you visually group the items, you could go higher.

Overview
Lesson 5 introduces the array visual model, which "is an arrangement of a set of objects organized into equal groups in rows and columns. Arrays help make counting easy. Counting by equal groups is more efficient than counting objects one by one." {How to Implement 28} I want to keep track of models like this, since I'm paying attention to mental aids for doing math, but I don't know how exactly I want to record them. I could put them in the propositional concepts lists, but I'm not sure they belong there. Plus they're already defined in the material, though there's a benefit in restating the definition in the context in which the model's used.