From the NCSM President
By Valerie L. Mills


Recognizing and Leveraging Mathematical Trajectories in Teaching and Learning

The Common Core State Standards (CCSS) for Mathematics opens with the words: "Toward greater focus and coherence." The mathematics education research community has repeatedly suggested that if our instruction had both greater focus and greater coherence, we would see gains in student achievement. The question that this opening text raises is, "Can a new list of standards move us toward greater focus and coherence?" I believe the answer is yes, these standards can support the community to move toward greater focus and coherence if we can learn to recognize and then leverage the intentional mathematical content trajectories that live under the surface of the CCSS. The embedded trajectories were informed by both research on children's cognitive development and by the logical structure of mathematics, and they suggest the next big steps we might take toward making mathematics accessible to all of our students.

In this and the next issue of the NCSM Newsletter, I will explore how leaders can work to recognize and then leverage these trajectories to better serve our students. To support this transition I have also developed a brief set of slides that can be downloaded from the NCSM Website-Members Only side-and used with educators. These slides are intended to help NCSM leaders facilitate discussions that will help teachers grow in their own understanding of concept and skill progressions. I hope you find this resource useful!

See the Powerpoint Mathematical Trajectories Part 1

Recognizing Trajectories

For many educators, learning to use a new set of standards means digging into the list of standards that were written specifically for their grade or course. This makes sense if the question that needs to be answered is, "Have all the standards been taught?" However, a 2011 report from the Consortium for Policy and Research Education (CPRE) titled, Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction, suggests this approach to understanding and implementing the CCSS is not sufficient if we are to fully understand the intent of the standards and achieve the added focus and coherence as intended. Using the role of the number line as an example, the report notes that, "A teacher or test designer seeing exclusively within the grade level will miss the point [of the fraction standards associated with the number line in grade 3]. Multi-grade progression views of standards can avoid many misuses of standards," (Daro et. al, p. 43). Unfortunately, despite the CCSS authors' attention to these trajectories while writing the standards, the design of the document does not make them immediately apparent and leaders need to work to help teachers recognize and use these connections to support teaching and learning. The trajectories are in some ways an evolution of what has been known as vertical alignment. Vertical alignment has focused on what topics are taught at each grade. The trajectories are different in that they suggest how deeply and in what ways students explore a topic across multiple grades. This shift is designed to help teachers attend to the development of mathematical connections among enduring mathematical ideas in ways that may have been missed previously and can now be emphasized.

Resources for More Information

Two extraordinary resources offer a place to begin working with teachers to shift their attention from a single grade/course toward a broader understanding of the development and connections among mathematical ideas across grades and courses. The first is Turn On CC Math led by Jere Confery (member of CCSS validation committee) and Alan Maloney at North Carolina State University. Their team has created a terrific website that organizes the standards into strings of content laid out in color-coded, tessellated hexagons that make visible the mathematical connections within and across grades. In addition to the organization of the standards, the team has written elaborations of the standards in teacher-friendly language. A second set of important resources comes from Bill McCallum's (co-author of the CCSS) project at the University of Arizona, Progressions Documents for the Common Core Math Standards For the Progressions project, teams of CCSS writers, mathematics educators, and mathematicians are compiling a set of narrative documents that each describes the progression of a topic across a number of grade levels. Both resources are well researched and written.

What We Know and Don't Know Yet

Before we leave this topic, I need to offer a word of caution. While the use of mathematical trajectories is likely to be productive for educators and students, the trajectories represented in the CCSS are not all fully described and some are more extensively studied than others. Both of the websites refer to gaps and areas that need more detail and/or research. In many ways our community is just beginning to generalize our thinking about the sequences and ways students might best develop an understanding of mathematics and so there is surely much more for us to learn together.

Textbooks and Trajectories

In addition to recognizing the content trajectories in our standards, a second part of this work is learning to identify how instructional materials treat the development of these ideas. The challenge here is to draw teachers' attention to the match or mismatch between their instructional resources' (e.g., textbooks, software, or websites) presentation of the ideas and the development called for in the CCSS. In short, not only do teachers need to look carefully to see that mathematical content is present in their textbooks, they also need to consider how it is presented.

Professional Learning Communities (PLCs)

Conversations among teachers to explore the concept and skill progressions written into the standards and the alignment or misalignment of these with textbooks are well suited to PLCs. Trajectory explorations can be approached within a wide range of professional learning settings and time frames. They can be broken into a few 60-minute sessions or during a full-day release. Exploring concept and skill trajectories can be a powerful strategy to include in a school improvement plan focused on most any mathematical topic. The work can be approached with teachers in a variety of configurations (e.g., within a grade, across grade bands, or across K-12). Please visit the NCSM Website-Members Only side-to download a brief slide show for some ideas about how to frame PLC conversations around these ideas and the websites listed above. The notes sections contain facilitation tips.

Recognizing and understanding teaching and learning trajectories is the first step toward using them to improve teaching and learning experiences.
I hope you will look for opportunities to begin working with your colleagues on teaching and learning trajectories this fall!

Valerie L. Mills, NCSM President, is a Supervisor, Mathematics Education Consultant for the Oakland Intermediate Schools, a resource center serving 28 school districts in southeast Michigan. She can be reached at

^ Top