Primary Math Continuum

Learning Trajectories for Primary Grades Mathematics Developmental Levels 

https://www.mheonline.com/assets/pdf/program/building_blocks_learning_trajectories.pdf

Learning Trajectories Children follow natural developmental progressions in learning, developing mathematical ideas in their own way. 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 Building Blocks learning trajectories.

 Learning trajectories have three parts: a mathematical goal, a developmental path through which children develop to reach that goal, and a set of activities matched to each of those levels that help children develop the next level. Thus, each learning trajectory has levels of understanding, each more sophisticated than the last, with tasks that promote growth from one level to the next.

The Building Blocks Learning Trajectories give simple labels, descriptions, and examples of each level. Complete learning trajectories describe the goals of learning, the thinking and learning processes of children at various levels, and the learning activities in which they might engage. This document provides only the developmental levels.

Frequently Asked Questions (FAQ)

1. 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, because the research on which they are based typically involves higher-income children.

2. 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.

3. 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. So, another way to think of them is as a sequence of different patterns of thinking. Children are continually learning, within levels and moving between them.

4. 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.

5. How do these developmental levels support teaching and learning?

The levels help teachers, as well as curriculum developers, assess, teach, and sequence activities. Teachers who understand learning trajectories and the developmental levels that are at their foundation are more effective and efficient. Through planned teaching and also encouraging informal, incidental mathematics, teachers help children learn at an appropriate and deep level.

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

No! The ages in the table are typical ages children develop these ideas. But these are rough guides only—children differ widely. Furthermore, the ages below are lower bounds of what children achieve without instruction. So, these are “starting levels” not goals. We have found that children who are provided high-quality mathematics experiences are capable of developing to levels one or more years beyond their peers. Each column in the table below, such as “Counting,” represents a main developmental progression that underlies the learning trajectory for that topic. For some topics, there are “subtrajectories”—strands within the topic. In most cases, the names make this clear. For example, in Comparing and Ordering, some levels are about the “Comparer” levels, and others about building a “Mental Number Line.” Similarly, the related subtrajectories of “Composition” and “Decomposition” are easy to distinguish. Sometimes, for clarification, subtrajectories are indicated with a note in italics after the title. For example, in Shapes, Parts and Representing are subtrajectories within the Shapes trajectory. Clements, D. H., Sarama, J., & DiBiase, A.-M. (Eds.). (2004). Engaging Young Children in Mathematics: Standards for Early Childhood Mathematics Education. Mahwah, NJ: Lawrence Erlbaum Associates. Clements, D. H., & Sarama, J. (in press). “Early Childhood Mathematics Learning.” In F. K. Lester, Jr. (Ed.), Second Handbook of Research on Mathematics Teaching and Learning. New York: Information Age Publishing.