Learning Trajectories Approach to Help All Students Progress in Math
Who does this approach work with? All categories of learners who struggle to learn and understand math, including
- Learners with and without disabilities, including mild, moderate, and significant disabilities
- Learners with complex communication needs
- Learners in Pre-K through adult/transition classrooms
Why Try Something New?
Educators have access to scopes and sequences for grade-level learning. This includes textbooks, standards, pacing guides, and benchmarks.
If some learners have not yet mastered knowledge needed at their grade-level those resources may not be enough. Educators may need more help knowing where to start or how to navigate instruction.
The learning trajectories use a developmental approach.
Step 1: Get to Know the Trajectories
The Learning Trajectories come from researchers Doug Clements and Julie Sarama. Download the Learning Trajectories and Learning Trajectories Handouts and familiarize yourself with them.
There are three components of the trajectories including
- Math goals
- Developmental sequences through which learners naturally progress
- Activities matched to each of those levels
These three components put together can be used to assess, set goals, and plan for instruction.
Watch the video for a short introduction to this module, including a description of the approach by Doug Clements.
Step 2: Assess
Use the Learning Trajectories Handouts, or scripted trajectories, to plan how you will assess your student. Watch the video to hear more about how the handouts are meant to be used.
Keep in mind:
- Avoid using the age equivalents noted on the trajectories when interpreting and reporting progress. Because learners progress at different rates, age equivalents are not needed and can cause confusion, frustration, or embarrassment.
- Be flexible when establishing a basal and a ceiling. As much as possible, start at the beginning of the trajectory and ask questions even after the student may begin answering consecutive items incorrectly. Often, students are able to demonstrate understanding of items all over the trajectory, which would go unnoticed with a more traditional ceiling.
- Ask enough questions to get the two key pieces of information. Overwhelmingly, when educators use this trajectory they learn two important things:
- Learners understand more math than the educators thought they did.
- Learners do not know math that the educators assumed the students knew, which helps explain difficulties with instruction and learning.
Step 3: Plan for Instruction
After you have assessed your learner, visit Doug Clements and Julie Sarama’s website LearningTrajectories.org to plan for instruction. There is no cost to use the resources, but you will need to set up an account to access them.
The complete set of trajectories, along with instructional activities matched to developmental levels within those trajectories, are located at the website.
Watch the video to learn how to set up an account and use the Learning Trajectories resources.
Step 4 (Bonus Step): Set and Monitor Goals using the Common Core Essential Elements, IEP, and Learning Trajectories
If you work with learners who study the Common Core Essential Elements, you can use the K-12 Essential Elements/Trajectories spreadsheet (co-constructed by Michigan educators after attending Foundations of Math: Teaching Students with Significant Disabilities) to use the Common Core Essential Elements to set IEP goals and then use Learning Trajectories to establish a starting point for instruction and document progress toward annual goals.
Watch the video to learn how the spreadsheet was designed to align Essential Elements with Learning Trajectories.
FAQ #1: Is this for learners beyond elementary level?
The Learning Trajectories are only for early elementary topics. What about learners who are older than elementary age or are in upper elementary or secondary classrooms?
Often times difficulties with math learning are not related to the current content being taught, but due to a lack of foundational skills or understanding that are a prerequisite for understanding the current content. For example, a learner struggling with algebra may not be confused by algebra, but is not secure in concepts related to numeracy or composition of numbers. A learner struggling with time or money may not be secure in decomposing and composing numbers or visualizing quantities that are unseen.
While the Learning Trajectories may not address all math learning challenges for older learners, using them as a first step in diagnosing learning difficulties can be critical. This is particularly true for learners with persistent difficulties in math. The trajectories can reveal important information about strengths and difficulties that can be addressed, help learners to progress through developmental levels, and begin to achieve at grade level.
FAQ #2: Is this applicable to learners with disabilities?
I teach learners with disabilities. How do I know the Learning Trajectories, researched with students without disabilities, are appropriate for them?
Research tells us that all learners go through the same levels of understanding when learning math, regardless of disability, even if some learners may require extra time, assistive technology, or different materials.
Hundreds of Michigan educators who work with learners with moderate, severe, and multiple impairments, as well as hundreds who work with learners with mild disabilities and students who struggle but do not have disabilities, have found that the Learning Trajectory approach is an effective way to learn more about their learners, plan appropriate instructional activities, and help their learners deepen math understanding.