Challenging but Achievable Math for Young Children

This white paper explores how learning trajectories (LTs) help teachers challenge students at a developmentally appropriate level. LTs describe structured pathways for understanding how children develop mathematical thinking and how instruction can align with those developmental levels. Each trajectory includes three components: a mathematical goal, a developmental progression of children’s thinking, and teaching practices that support learning at progressively challenging levels.

The approach emphasizes observing children’s thinking and using formative assessment to match instruction with their current level of understanding. Teachers who understand standards, children’s mathematical thinking, and effective instructional strategies can design environments and activities that support meaningful engagement in mathematics.

Developmental progressions illustrate how children move through stages of understanding. In counting, early stages include verbal counting where children may initially say number words together (“one twothreefour”) and later recite them distinctly. Object counting introduces additional complexity, requiring one to one correspondence between number words, pointing actions, and counted objects, as well as spatial awareness when objects are arranged randomly.

A key milestone in counting development is the cardinality principle: recognizing that the final number word in a count represents the total number of objects in a set. Children may correctly count objects yet still recount the set when asked “how many,” indicating that this understanding is still emerging.

Instructional strategies aligned with learning trajectories include structured activities and interactive tasks that highlight correspondence, counting, and cardinality. Examples include digital or hands on counting activities and classroom tasks such as revealing objects one at a time while counting to emphasize the relationship between enumeration and total quantity.

Research synthesizing multiple longitudinal investigations involving hundreds of teachers and thousands of students reports stronger mathematics learning outcomes when instruction aligns with learning trajectories compared with other curricular approaches. Studies also report benefits across diverse contexts and highlight the importance of equitable expectations and culturally responsive teaching practices.

Clements, D. H., Guss, S. S., & Sarama, J. (2022). Challenging but achievable math for young children: Learning trajectories help teachers challenge children at just the right level for their best learning. Mathematics Teacher: Learning and Teaching PK–12, 115(7), 468–475.