Learning Trajectory

Learning Trajectory

The learning trajectory is intended to help teachers identify different stages of understanding uncovered during students' work in the diagnostic tasks [39, 45, 70]. The indicators match a prior synthesis of math education research [13, 20, 30, 54, 62, 72, 85, 86]. The questions in the Noticing Guide directly align to the stages shown in the trajectory to help teachers begin to plan instruction. Students' evidenced understandings may fall between two of the stages in the trajectory. If this is the case, Task Set One (see "Instruction") will help teachers further understand their view of students' reasoning.

Overview

We have found that there are two main indicators that provide the most robust insight into students' conceptions of fractions across the trajectory stages:

Partitioning Plan: describes the student’s evolving understanding of how to link their partitioning of the whole(s) to the number of sharers to create a fractional quantity.
Coordination of Equal Parts With Wholes: describes a student’s evolving understanding of the importance of equal shares while exhausting the referent quantity shared.

Both indicators assume that a student already conceives as the whole as being divisible. The interaction and interplay between these two indicators yield a clear picture of a student’s initial conceptions of fractions. Over time, the indicators combine to produce a more robust understanding of fractions as coordinated quantities with referent wholes. Examples of how a student might think within the diagnostic fraction tasks at each of the stages of the trajectory are included.

Note: We strongly recommend downloading the below PDF files over opening as a Google Doc in order to maintain original content and formatting

Stage Summaries - Text

Learning Trajectory OVERVIEW (1).pdf

Learning Trajectory

Stage Summaries - Table