Phases of Strategy Development
The strategies in each chapter are grouped into four general, increasingly sophisticated, phases of development on the continua:
- Direct Modelling & Counting
- Counting More Efficiently & Tracking
- Working with the Numbers
- Proficiency
When children use direct modelling strategies, they fully represent the problem with objects, then count the objects to find a solution. For example, to find 5 + 7, children would represent both numbers with objects, such as their fingers or marks on a page. As children shift to the next phase of development, they transition from direct modelling to tracking. With direct modelling, the objects children count are visible. When children shift to counting more efficiently & tracking, they may still use their fingers or marks on a page, but these are used to track their mental count. For example, to add 5 + = 12, children would track the missing addend as they count on from 5 to 12. Their 7 raised fingers would be a physical track of their mental count. When children shift to working with the numbers, they are no longer counting or tracking, but instead, operating on or with the numbers. For example, to add 5 + 7, they might decompose the 7 into 5 + 2 so they can add 5 + 5 to get 10, then add 2 more to get 12. When children achieve proficiency, they are demonstrating some or all of these strands of mathematical proficiency identified by the National Research Council.5 To paraphrase these strands, we can say that children:
- understand the mathematics they are using;
- are skillful, accurate, and fluent, choosing appropriate and efficient strategies;
- are able to communicate and justify their thinking and reasoning; and
- have a productive disposition, viewing mathematics as useful and believing in their own capacity to use it.
For example, children who have achieved proficiency not only know 5 + 7 = 12 as a “fact,” but they also understand the other relationships within this equation. For example, they know that 5 + 7 could be adjusted to get 6 + 6 = 12, or that the sum of 5 + 7 is 2 less than the sum of 7 + 7, or 2 more than the sum of 5 + 5, and so on. These children are able to communicate their thinking and justify why these other relationships are true. Furthermore, they have confidence in their ability to find the sum of 5 + 7 in another way if they momentarily forget their fact.
The phases of strategy development found on the continua are also displayed in The Interviews section of each chapter, on the side tabs, for quick access for assessment and teaching purposes.