Back-mapping and Learning Trajectories

Basically, we want to help kids think mathematically! It is why we, the adults in mathematics education, do what we do. But, there are so many variables, factors, causes, element, and it is complex.

To equip us the best, to help students, I have listed two nationally recognized tools below.  They are available for use at no charge due to the funding from the Hiesing-Simons Foundation and Bill and  Melinda Gates Foundation. They are aligned to the standards, from credible sources, based on current research, and have been presented and shared with state supervisors of mathematics nationwide.

Before we jump to the tools, consider another question that I am often asked: Do we focus on core instruction or intervention? The answer is both. Prevention and early detection are the best intervention and we will not intervene our way out of good core instruction. We want to continue to improve instruction each year with the goal that we are decreasing the number of students who will need intervention. At the same time, we need to intervene with those students who need it at the earliest possible time to maximize impact on the student. The ultimate goal is to have as little as a need for intervention as possible.

The use of both tools below would be greatly enhanced by user knowledge and understanding of Number Sense. Otherwise, educators may make decision to ignore or skip necessary experiences for students.  For example, students who are struggling to count (including verbal, object, cardinality) may need experiences to develop subitizing.  Not sure what this is. It has been around since the 1940’s in the research and is foundational and critical if we want students to add and multiply.   A common error educators make is to ignore or not spend enough time on developing student subitizing.  Another common error is to focus on a strategy itself and not the cognitive development.  Focusing on strategies for the sake of strategies will not develop the cognitive subitizing our students need. Teaching should be focused on cognitive development that leads to fluency, NOT teaching strategies to get right answers.  This will fall off for the students in 4th and 5th grade when the numbers get bigger and more complex and those strategies are no longer applicable or efficient.

My favorite feature of these tools is that most of the resources can be used with minimal prep time, are turn-key, and can be used with students the next day.

Step 1 – Use to back map: The Coherence Map – Developed by Student Achievement Partners (Jason Zimba lead writer of the standards) and available on the websiteScreen Shot 2018-10-28 at 10.56.50 AM.png

  • a digital map of the standards that can be used to see the coherence of the standards
  • it can be used to back map and identify at what point in the standards the students struggle or have gaps
  • contains aligned tasks and assessment items to use with students
  • from the authors of the standards
  • understanding of the progression documents will strengthen use of the Coherence Map (be sure to read the Front Matter to understand increase understanding)
  • do not use the Coherence Map to forward map as there are multiple forward paths

Step 2 – Use to forward map: Learning and Teaching with Learning Trajectories– Developed by Dr. Clements and Dr. Sarama.  Doug Clements worked on the progression documents, with the standards writing team, and is nationally recognized as a lead researcher in early childhood mathematics.   (About Clements and Sarama)

Screen Shot 2018-10-28 at 11.02.28 AM Screen Shot 2018-10-28 at 10.58.15 AM

  • a digital resource of the learning progressions that can be used to move students forward in their cognitive development of mathematical thinking conceptually, conceptual understanding and skills for fluency
  • a Learning Trajectory is a content-specific learning path or a developmental progression that can be used to build conceptual understanding.
  • contains
    • 17 learning trajectories (birth to 3rd grade) for students to access the standards
    • short developmental video examples
    • resources for 1 on 1, small group, whole group task and computer
      • only 4 computer games because there is no computer substitution for student experiences with teachers
  • has a document that shows the Gold Standards alignment
  • can set up a classes of students
  • be used to observe and record individual child’s progress on the in each trajectory
  • parents can Access
  • User Guide – screen shots of setting up account






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