Learning Trajectories

At the heart of the Learning and Teaching with Learning Trajectories [LT]2 website are mathematical Learning Trajectories. Learning Trajectories have three parts - a learning goal, a developmental path along which children develop to reach that goal, and a set of activities matched to each of the levels of thinking in that path. Together, these help children develop to higher levels of mathematical thinking.

Children follow natural progressions in development and learning. While no child's path is exactly the same, understanding common paths helps us know what to do and when to support early math learning.



Frequently Asked Questions (FAQs)

What is a Learning Trajectory [LT]2?

Children follow natural developmental progressions in learning. Curriculum research has revealed sequences of activities that are effective in guiding children through these levels of thinking. These developmental paths are the basis for the learning trajectories.

Why are trajectories important?

Research shows that when teachers understand how children develop mathematics understanding, they are more effective in questioning, analyzing, and providing activities that further children’s development than teachers who are unaware of the development process. Consequently, children have a much richer and more successful math experience in the primary grades.

Why use learning trajectories?

Learning trajectories allow teachers to build the mathematics of children – the thinking of children as it develops naturally. So, we know that all the goals and activities are within the developmental capacities of children. We know that each level provides a natural developmental building block to the next level. Finally, we know that the activities provide the mathematical building blocks for school success.

When are children “at” a level?

Children are at a certain level when most of their behaviors reflect the thinking – ideas and skills – of that level. Often, they show a few behaviors from the next (and previous) levels as they learn. Most levels are levels of thinking. However, some are merely “levels of attainment” and indicate a child has gained knowledge. For example, children must learn to name or write more numerals, but knowing more numerals does not require deeper or more complex thinking.

Can children work at more than one level at the same time?

Yes, although most children work mainly at one level or in transition between two levels (naturally, if they are tired or distracted, they may operate at a much lower level). Levels are not “absolute stages.” They are “benchmarks” of complex growth that represent distinct ways of thinking.

Can children jump ahead?

Yes, especially if there are separate “sub-topics.” For example, we have combined many counting competencies into one “Counting” sequence with sub-topics, such as verbal counting skills. Some children learn to count to 100 at age 6 after learning to count objects to 10 or more, some may learn that verbal skill earlier. The sub-topic of verbal counting skills would still be followed.

How do these developmental levels support teaching and learning?

The levels help teachers, as well as curriculum developers, assess, teach, and sequence activities. Through planned teaching and also encouraging informal, incidental mathematics, teachers help children learn at an appropriate and deep level.

Should I plan to help children develop just the levels that correspond to my children’s ages?

We should support children's learning and development based on their needs. While we list common ages associated with levels, there are many cases in which children will need support at lower or higher levels. 

What is a sub-trajectory?

A subtrajectory is a group of levels that are loosely coupled with a larger trajectory. While present within the development of the larger trajectory, the dependence of building iteratively from one level to the next may not be essential. Thus, subtrajectory levels may appear 'out of order' in terms of the child's age and development.


The Story of Learning & Teaching with Learning Trajectories

Special Thanks To

Institute of Education Sciences
The research reported here was supported by the Institute of Education Sciences, U.S. Department of Education, through grant numbers R305K050157, R305A120813, R305A110188, and R305A150243. to the University of Denver. The opinions expressed are those of the authors and do not represent views of the Institute or the U.S. Department of Education.