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TRANSITIONS- Understanding the Learning Progressions

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This current VADOE project is called TRANSITIONS


Article In Press

Using a coach-facilitated professional development model, we used Lesson Study to focus on the teaching and learning of algebra in the transitions grades. Using this approach, teachers in grades 5-9 focused on problem solving across the learning progression of algebra. Results from the analysis of teacher reflections, video analysis, and lesson study debriefs revealed that the design of the coach-facilitated professional development and Lesson Study offered opportunities for coaches and teachers to mutually develop in their content and pedagogy while deepening their understanding of students’ learning progressions in algebraic thinking.


Keywords: Teacher Education-Inservice/Professional Development, Instructional Activities and Practices, Learning Trajectories (or Progressions), Algebra and Algebraic Thinking

Theoretical Framework

Transitional years for students moving from elementary to middle to high school are critical junctures where we as educators want to ensure a smooth transition in terms of their academic learning progression. Understanding the learning progression is important for teachers for it serves as the guidepost for analyzing student learning and tailoring their teaching sequence. Learning progressions, according to the National Research Council, “are descriptions of the successively more sophisticated ways of thinking about a topic that can follow one another as children learn about and investigate a topic” (2007, p. 214). The importance of learning progressions is found in the research on development of algebraic reasoning through problem solving in earlier grades. Blanton and Kaput (2005, 2008) reported that teachers become better at teaching algebraic reasoning when their own mathematical knowledge and understanding is increased and their algebra “eyes and ears” allow them to bring out algebraic reasoning while looking at student work and carefully listening to student discourse and questions. It takes a teacher who has a deep and profound understanding of fundamental algebra to explore the foundational concepts of algebraic reasoning through patterning, relations, functions, and representations using algebraic symbols and utilizing mathematical models to represent relationships (NCTM, 2000). For our research and professional development design, we focused on helping teachers anticipate students’ hypothetical learning trajectory (HLT) of algebraic thinking by posing problems that represented patterns, function and algebra during Lesson Study to validate their understanding of the learning progression in algebra. HLT is “a prediction of how the students’ thinking and understanding will evolve in the context of the learning activities” (Simon, 1995, p. 136). Through the coach-facilitated lesson study, we wanted our teachers to witness the “actual learning trajectories which can be specified only during and after a student has progressed through such a learning path” (p. 135).

Context for our Study

Forty-six teachers from grades 5 – 9 and 11 math coaches met for a one-week summer institute and continued in school-based Lesson Study teams during the academic year. The content focused on algebraic thinking across the transition grades. For this research report, we focused our analysis on the summer content institute and the follow-up lesson study data sources to demonstrate the progression of development in teachers’ vertical articulation of the algebraic strand and how coach-facilitated Lesson Study and professional development supported teachers development. We challenged teachers across the transition grades to see beyond the boundaries of elementary, middle, and high school and to see beyond their own grade level to enhance their content knowledge and deepen their understanding of students’ learning progression in algebra.

Research Questions

This study explored the following research questions:

1) How did the focus on the learning trajectories of algebra across the transition grades and transforming instructional practices help teachers go beyond the borders of their grade level standards and their traditional instructional practices?

2) What were the affordances of using the coach-facilitated Lesson Study model in the school-based professional development model?

Data Analysis

To begin analyzing the themes, we used the document analysis technique using teachers’ individual reflections and exit passes, transcripts of the Lesson Study debriefs and symposium presentations and the researcher memos. We systematically analyzed the data by developing initial codes and used the method of axial coding to find categories in such a way that drew emerging themes (Miles & Huberman, 1994). To verify and compare recurring themes and categories, the research team worked individually on coding the documents before comparing preliminary codes in order to agree upon recurring themes from the reflections. Dedoose, a data management tool (Dedoose Version 6.2.7) was used to code and analyze the data.


For the first research question, regarding learning trajectories of algebra across grade bands and transforming teachers’ instructional practices we found the following recurring themes:

Going beyond borders of grade bands through vertical articulation of learning trajectories

The majority of our Lesson Study teams were composed of multi-grade teams that ranged from 3rd to 9th grade. Each team chose a rich task that had multiple access points with teachers working together to modify one problem to elicit an important mathematical standard for each grade level. As teachers negotiated and discussed the ways they would modify the problem, their knowledge of the learning progression of algebra deepened. Such vertical articulation not only helps to combine curriculum content across grade levels but also helps to relate knowledge (or skills) to a wider range of content, to provide a deeper understanding for the same content (cognitive process), and helps to design and develop new life-long competencies. In one group, teachers worked on a task called “The Age Problem”. In this problem requiring discovery and exploitation of patterns of multiples, the teachers used the same prompt for grades 5 thru 9 by changing the expected student outcomes. The 5th graders were expected to be able to find the pattern to get some answers, the 7-8th graders were expected to be able to write at least a recursive formula and expand the pattern, and the algebra students were expected to write an explicit formula for the arithmetic sequence and then generalize to other similar problem situations. In this way, the teachers took into account the learning progression of algebraic thinking using a single problem across grade levels. After defining expectations, the teachers anticipated the hypothetical learning trajectories based on what they thought their students would do with the given task. One of the teachers stated, “Working in a cross-grade level team enabled us to plan for… extensions and remediation opportunities to provide all students an access point. It was very helpful to hear new strategies/ideas and see where students need to take the knowledge (for future grades).”

Crossing the borders from a traditional instructional approach to experimenting with a problem-based approach to teaching

One of the recurring themes in teachers’ reflection was teachers taking initiatives and experimenting with teaching through the problem-based approach. In order to make this paradigm shift, teachers realized that they needed to provide more time and space for students to grapple with the mathematics. “The main thing that I have learned is allowing the students the necessary time to try to figure out the problems on their own. It is extremely hard to stand back and watch the students struggle and not jump in and point the in the exactly correct directions.” Another catalyst for this shift was when teachers observed that students who experienced productive struggle obtained success. This served as the “proof of concept” that they needed to continue with this problem-based approach, “Students are getting better at explaining their thinking and showing their thought process using different representations…The more they are exposed to problem solving strategies and different ways to predict patterns, the more depth there is to their work.”


For the second research question regarding the affordances of using the coach-facilitated professional development model, we found the following themes:

Teachers moving beyond traditional instructional practices

The coaches were recruited with the school teams to facilitate the professional learning with teachers. Through their differentiated PD, we wanted coaches to focus more on their role as facilitators who could support teachers as they implemented problem-based learning and examining student thinking by examining their work. One coach noted the shift in her teachers’ interactions and her ability to bring more focus on learning trajectory into their meetings:

The math teachers on each grade level meet twice weekly as a team. The majority of their time [was] spent writing lesson plans together.… Now,… the focus of their weekly meetings has shifted to evaluating student work… [and providing] an opportunity to revise the assignments given to students so that questions require higher order thinking which require students to engage in algebraic habits of mind such as generalizing patterns to reach the abstract representation.

Coaches saw this willingness by their teachers to cross the boundary to reform-based teaching practices as an opportunity to support change for positive student outcomes aligned with school and district initiatives.

Teachers not used to teaching using groups and more open-ended tasks saw the difference in how students reacted. If teachers can transfer this knowledge into building lessons from the current curriculum rather than thinking of the lessons they did as “special” we will be on the right track. I see many opportunities to be able to link the lesson study experiences with other site-based efforts.

Teachers changing ideas promote student thinking and practices

Where teachers were convinced to begin to change their practice after seeing their students engaged in the mathematics learning, the coaches had a parallel experience watching their teachers take on the instructional practices that promote students thinking through questioning during a problem-based lesson.

I’ve noticed a change in the teachers as well – instead of “is that right or wrong,” they are looking for more than just a correct answer.… Just like the students are learning that math isn’t black and white and there are many ways to get to the same conclusion, the teachers are as well.


School-based professional development and Lesson Study created a community of practice that provided opportunity for educators not only to co-create but to bring together all their expertise and strength. This collective knowledge yields more than the sum of their individual knowledge which in turn enhances one’s individual instructional practice and also provides opportunities for school teams to develop collective teaching agency. The coach-facilitated PD provided teachers with the immediate trust in the work they were doing because they came with the school-based coach. Knowing that their coach endorsed the instructional practices and the problem-based approach validated our methods and gave them more impetus to put forth effort in their professional learning. Through our coach-facilitated Lesson Study we are able to provide the unusual and powerful opportunity for in-service teachers at multiple grade levels to collaboratively plan a learning task and observe students engage in the task. Through our analysis of this activity, we found several revealing outcomes. Teachers with the various expertise were able to mutually offer and benefit from the vertical articulation and in essence provided a collaborative coaching environment. In addition, highlighting what they learned about how students responded to the task offered teachers the time and space to discuss the learning trajectories in algebraic thinking. The work with learning trajectories supported vertical teaming by teachers, for it allowed a “chance for teachers to discuss and plan their instruction based on how student learning progresses. An added strength of a learning trajectories approach is that it emphasizes why each teacher, at each grade level along the way, has a critical role to play in each student’s mathematical development” (Confrey, 2012, p. 3). As we continue our work, we will continue to explore how coach-facilitated PD and Lesson Study can help sustain the PD efforts and make best practices “stick” with teachers.


Blanton, M., & Kaput, J. J. (2008). Building district capacity for teacher development in algebraic reasoning. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the Early Grades (pp. 361-388). Mahwah, NJ: Lawrence Erlbaum Associates/Taylor & Francis Group.

Confrey, J. (2012). Articulating a learning science foundation for learning trajectories in the CCSS-M. In Van Zoerst, L. R. Lo, J. J. & Kratky, J.L. (Eds.). Proceedings of the 34th annual meeting of the North American Chapter of the International Group for the Psychology Mathematics Education (pp. 2-20). Kalamazoo, MI. Western Michigan University.

Lewis, C. (2002). Lesson Study: A handbook of teacher-led instructional change. Philadelphia: Research for Better Schools.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

Simon, M. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114-145.



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