Differentiating with Trajectories

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smith
Nina Smith

Math Interventionist K-8
Marcy Open School
Minneapolis Public Schools

and

smith
Sarah Moffett

Math Teacher grades 3-5 and Math Specialist grade K-8
Marcy Open School, Minneapolis Public Schools
MCTM Region 5 Director

A learning trajectory is a common developmental path to understanding a concept. Douglas H. Clements describes a simple example: “children first learn to crawl, which is followed by walking, running, skipping, and jumping with increased speed and dexterity.” Many respected researchers have studied how children develop mathematical ideas and have identified stages of understanding. Some have developed assessments to identify where children fall on trajectories. Others have collected lessons and activities that are designed to move students thinking along these trajectories.

While traveling, to a particular destination, you must have some way of knowing how to get there. As teachers we are expected to teach the state standards and hold our students to high expectations. These two things are the magic formula for closing the vast achievement gap in Minnesota.  Simply following this formula without a path for getting our students there is at best wishful thinking and at worst abusive.

As we taught, we could see the gaps in students’ understanding. The issue was we couldn’t name what ideas students were missing and what they really understood other than, “They just don’t have number sense.”  So, we stuck to the formula, and tried to help students as best we could; mainly by teaching procedures and re-teaching the same way.  Deep down we knew our students needed something different but we just didn’t know what that might be.

Then, in a summer class taught by Aneesa Parks (Minneapolis Public Schools Math Specialist), we realized we were asking children to jump before they were strong walkers. More importantly, we discovered what the something different that we needed was. It was Math Learning Trajectories for Conceptual Place Value.  We finally had a way to discover and name the things our students do understand about place value, and what the next steps were for their level of understanding.

What we did

Our first step was to use a formative assessment developed by another MPS Math Specialist to interview 2nd grade students and collect some information around their thinking with place value.  This was a quick assessment that took about 5 minutes per student.

Next, during our planning meeting, we laid out Aneesa’s place value trajectory and asked ourselves:

  • How do the questions in the formative assessment relate to the trajectory?
  • Where do our students fall on the trajectory?  How do we know?
  • What does this tell us that our students understand? – Name It.  What do they need? – Name It.
  • How can we use this information to form guided groups?
  • What resources will we use to address the needs of each group?

In our next post we will share how we answered these questions, how we planned differentiated instruction and how our work affected our students.

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The following resources were used in developing our plan:

Place Value Trajectory developed by Aneesa Parks

Conceptual Place Value formative assessment by Jeanne Lawless’ (Minneapolis Public Schools Math Specialist)

Developing Number Concepts books (1-3) by Kathy Richardson

How Children Learn Number Concepts by Kathy Richardson

Learning and Teaching Early Math the Learning Trajectories Approach by Douglas Clements

NZMaths – Numeracy Projectshttps://nzmaths.co.nz/numeracy-projects

K-5 Math Teaching Resourceshttps://nzmaths.co.nz/numeracy-projects