In order to identify, understand, and assess children’s mathematical thinking, we have created two continua of numeracy development over time. Wherever possible, we drew on the existing theory and research to identify, infer, and label children’s behaviour and thinking.

In particular, we identified and described the mathematical strategies—the actions, words, or written work children use to calculate or solve problems. We also labelled the key ideas we inferred they had, or were building, as they used a given strategy. We combined our field observations and inferences to create the Student Continuum of Numeracy Development: Addition and Subtraction and the Student Continuum of Numeracy Development: Multiplication and Division, which document children’s mathematical growth over time. The numeracy continua provide a graphic outline of this general progression, with the earliest strategies at the left, and the more efficient and complex ones at the right, while the inferred key ideas are shown across the bottom. Although there is an overall progression forward, the continua also document what we saw and what all observant teachers know: children’s mathematical progress is neither linear nor strictly developmental; instead, it jumps forward and backward and sometimes shifts laterally. The continua, therefore, allow for a variation in paths that the children in the videos took as they progressed mathematically. Finally, the continua are lenses through which teachers can view the mathematical development of children and can be used in classrooms to assess and plan for instruction.

We make use of two overall terms in the continua of children’s mathematical development in numeracy:

- strategies
- key ideas

A strategy is what you hear and see as children work to solve a problem or to do a calculation. For example, suppose a child solves 5 + 7 by reversing the numbers, then raising 5 fingers as a physical model to be counted, putting each finger down while counting 8, 9, 10, 11, 12. The child is using the strategy of counting on from the larger number. The child is also demonstrating the use of knowledge you cannot see: a mathematical key idea.

While **key ideas** can have different meanings, we use the term narrowly, rather than broadly, to describe the important mathematical properties or ideas that children construct as they work with different strategies. The key ideas are not physically visible to us as teachers; instead we must infer them from the strategies we see children using. In this example, the child is making use of the commutative property: knowing that the order in which the numbers are added does not change the sum. In this case, the child has determined that 5 + 7 = 7 + 5. We can see that this child would have the mathematical power to add more disparate numbers, such as 5 + 11, by counting on. This child would only need to count on from 11 rather than from 5. Key ideas are therefore important. They can be roadblocks for children who have not yet constructed them (in this instance, labouriously counting on from 5 through 16) and mathematically powerful for those who have. While one might be tempted to try to teach key ideas directly, Kamii reminds us that, given effective instruction, children can “reinvent” them in order to do a calculation or to solve a problem.

The key ideas appear as bands across the bottom of each continuum. Each band spans a range of strategies, indicating when children first begin to develop the key idea through to when they have a solid understanding of it. For example, children may make use of the commutative property to count on from the larger number, to use a known fact, or to add up over 10. Children begin to construct the key ideas at somewhat different times in the two continua because multiplication and division key ideas are more complex. With multiplication and division, the child is dealing with relationships of multiple rather than single objects. These two terms provide the organizational backbone of this book.

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.

The continuum captures the strategies in use and the underlying key ideas over the two grades we examined. Precursors (e.g., basic components of counting or the key idea of magnitude) could appear to the left of the figure if we had included kindergarten. Subitizing would also typically appear even before counting. We have nonetheless included it here because of its repeated use by children. You will notice several strategies that are delineated by broken lines. These broken lines represent strategies that we have yet to see in the videos but know will typically begin to appear in Grade 3. They have been included to show the full continuum of students’ early numeracy development.