Well-prepared beginning teachers of mathematics at the early childhood level are conversant in the developmental progressions that are the core components of learning trajectories and strive to see mathematical situations through children’s eyes. [Elaboration of C.3.1]

The younger the child the more important teachers' uses of children’s thinking and learning are as starting points. Further, the younger the child, the more difficult *decentering* and seeing mathematical situations through children’s eyes will be. Fortunately, we know in great detail how children think and learn mathematics in the early years, and for candidates to become conversant in these *developmental progressions* and utilize them is important when they plan for and interact with children in preschool and primary settings.

These developmental progressions are paths most children follow in learning a mathematics topic. These paths are children’s natural ways of learning. For example, consider that children first learn to crawl, then walk, then run, skip, and jump with increasing speed and dexterity. These are the levels in that developmental progression of movement. Children similarly follow natural developmental progressions in learning the concepts and skills within a certain domain or topic of mathematics. When teachers understand these developmental progressions, and select, sequence, and modify activities on the bases of them, they create mathematics learning environments that are particularly developmentally appropriate and effective. These developmental progressions, then, are the core of a learning trajectory (which includes, as described previously, the mathematical goal, or content, and instructional activities and strategies corresponding to each level of the developmental progression).

Developmental progressions begin when life begins. Young children have certain mathematical-like competencies in number, spatial sense, and patterns from birth. While they develop and learn, they progress through identifiable *levels of thinking—*periods of time of qualitatively distinct patterns of thinking about mathematics. As an example, children develop increasingly sophisticated counting strategies to solve increasingly difficult types of arithmetic problems. For example, even very young children, shown one chip on a plate covered then shown another chip placed under the cover, can make their plates "look just like mine”—an early* visual addition.* Later, these children use a counting-all procedure. Given a situation such as combining six red apples and two green apples, children count out objects to form a set of six items, then count out two more items, and finally count all those items and say, “Eight.” After children develop such methods, they eventually phase them out, in favor of other methods. On their own, children as young as 4 or 5 years may start *counting on,* solving the previous problem by counting, "Siiiiix, … seven, eight. Eight!" The elongated pronunciation of the first addend substitutes for counting the initial set one-by-one. That approach is used as if they first counted a set of six items.

Thus, counting skills—especially sophisticated counting skills—play an important role in developing competence with computation. Counting-on when increasing collections and the corresponding *counting-back-from* when decreasing collections are critical numerical strategies for children. However, they are only beginning strategies. If the amount of increase is unknown, children use *counting-up* to find the unknown amount. If five items are added to so that one now has nine items, children may find the amount of increase by counting and keeping track of the number of counts, as in (with drawn-out pronunciation of “five” and later, “nine”): “Fiiiive; 6, 7, 8, 9. Four (as in four counts, the amount of increase)!” And if items are removed from nine items so that five remain, children may count back from nine to five to find the unknown decrease as follows: “Niiiine; 8, 7, 6, 5, 4. Four!!” However, counting backward, especially more than two or three counts, is difficult for most children unless they have consistent instruction. Instead, children might learn counting-up to the total to solve a subtraction situation. For example, “I took away 5 from 9, so 6, 7, 8, 9 (raising a finger with each count)— that’s 4 more left in the 9.” Learning this way, including the complementary use of number composition and decomposition strategies (e.g., break apart to make a 10) is more developmentally appropriate and more effective at achieving fluency than jumping immediately to verbal memorization (Baroody, 1999; Henry & Brown, 2008).