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Enhancing Pedagogical Practices Through Professional Development in Proportional Reasoning

Padmanabhan Seshaiyer and Jennifer Suh

Abstract: There is a continuous need to develop more content-focused professional development (PD) programs for teachers that can help lead to improvements in teacher content knowledge in proportional reasoning. This work presents how the pedagogical practices of a group of 85 elementary and middle grades teachers that participated in a summer institute through a Mathematics and Science Partnership (MSP) program, were impacted in proportional reasoning through problem solving activities. The observations on their understanding of proportional reasoning through poster artifacts, the reflections on their work through journals as well as misconceptions in their problem solving found in this work are presented. Impact on student learning through follow-up lesson study are also discussed.

Project Overview: Over the last seven years, the Center for Outreach in Mathematics Professional Learning and Educational Technology (COMPLETE) at George Mason University (GMU) has been supported by the Virginia Department of Education MSP Program to coordinate projects that has helped to provide PD to teachers from various school districts in the Northern Virginia area.  Topics in these PD have included Building numbers and number sense for elementary grades; Rational numbers and Proportional reasoning in middle grades and; Expeditions in Science Technology Engineering Education through Mathematics at high school level. The projects were coordinated by a mathematician and a mathematics educator from GMU in collaboration with partnering districts that included Alexandria, Arlington, Fairfax, Fall Church, Frederick, Loudoun, Manassas City, Manassas Park, Prince William and Virginia Council for Private Education. The project website provides more details: http://completecenter.gmu.edu.

Introduction: Proportional Reasoning is fundamental to many important mathematical concepts and is often regarded as the pathway to performing well in higher level mathematics. Teachers have also been urged to focus students’ attention on the meaning of problems and to help students value different mathematically correct solutions to a single problem (NCTM, 2000). There is a great need for research in evaluating the effect of solving one proportional situation via multiple solution strategies for example using unit rate strategy; repeated subtraction strategy; equivalent fractions strategy; size-change strategy; cross multiplication using equal rates or ratios strategy, relative and absolute thinking strategy and; reasoning up and down strategy (Lamon, 2007). Teaching proportional reasoning through problem solving therefore requires depth of mathematical knowledge for teaching that not only includes understanding of general content but also having domain specific knowledge of students. Research has shown that a content-focused PD leads to improvements in teacher content knowledge with a focus on student learning goals, highlighting the concepts being addressed, how they are developed over time, difficulties students may encounter, and how to monitor student understanding (Suh and Seshaiyer 2013, 2014a, 2014b). To evaluate the collaborative nature of designing PD, Suh, Seshaiyer, Freeman and Jamieson (2011) used the collective self-study method to examine how purposively designed experiences such as the content-focused institute in the summer with school-based follow-up Lesson Study cycles in the fall encouraged vertical articulation of algebraic connections. In this work, we present one such PD program that systematically introduced the concept of proportional reasoning through problem based learning activities. Specifically, the paper presents how the pedagogical practices of a group of 85 elementary and middle grades teachers that participated in a summer PD institute were impacted in proportional reasoning. The study explored the following research question: How can professional development for mathematics teachers be designed and implemented so that the teachers develop deep understanding of proportional reasoning?

The goals for our project were for elementary and middle grades teachers to relearn proportional reasoning through multiple representations in problem solving and linking related problems and concepts. This project was designed based on the current research and needs in mathematics education in the state. The program included a content-focused summer PD institute and a follow-up Lesson Study (Suh and Seshaiyer, 2014c) throughout the academic year focused on engaging teachers in active learning through rational numbers and proportional reasoning tasks, exploring pedagogical strategies, utilizing mathematics tools and technology, and promoting connections aligned and coherent to the elementary and middle school curricula. Daily activities in the summer institute included modeled lessons using a variety of mathematics tools and technology and in-depth conversation about the proportional reasoning, pedagogical strategies such as using problem solving and multiple representations. The purpose of the PD was the development of mathematical teaching knowledge through a collaborative network of pre-service and in-service teachers who collaboratively plan lessons and exchange best instructional practices and effective uses of technology tools to design instructional tasks which promote algebraic conceptual thinking.  Teacher collaboration enhances their professional practice which then affects students' learning. The teachers were engaged in content-focused activities to help them become aware of the specific math content topics in rational numbers and proportional reasoning (Lamon, 1999): Relative and Absolute Thinking; Measurement; Quantities and Co-Variation; Reasoning up and down; Unitizing; Sharing and Comparing; Proportional Reasoning; Equivalence; Reasoning with fractions; Part-whole comparisons with unitizing; Partitioning and Quotients; Rational numbers as operators; Rational numbers as measures; Ratios and Rates; Distance-rate-time relationships; Similarity and percents; Changing fractions. Each of these content topics were motivated through sample benchmark problems that aligned with the changes in the 2009 VA Standards of Learning. While we considered several during the institute, we include here one specific problem that was provided to the teachers as an opening problem for the day, the cathedral problem that is adapted from Burns, S. (2003): While building a medieval cathedral, it cost 37 guilders to hire 4 artists and 3 stonemasons, or 33 guilders for 3 artists and 4 stonemasons. What would be the expense of just 1 of each worker?  Note that guilders here refers to currency used in the Netherlands from the 17th century to 2002.

Data Analysis through Poster Artifacts and Reflections: The teachers worked in groups attempting to solve the problem multiple ways and were asked to create a poster representing their solution. Each teacher was also asked to reflect on how they participated in the problem solving process and how they would take this problem back to their classrooms to present to their students. Photographs of all the posters created by the teacher groups of the cathedral problem were taken and the data from the posters were analyzed for content, connections between concepts, and any possible differences related to the time already spent in the seminar.   Data was also analyzed from the teacher reflections for common themes as well as individual perspectives. Some of the solutions from the various poster artifacts created by teacher groups are indicated in Figures 1-4. The analysis and discussion of these poster artifacts indicated strategies employed by the teachers including guess and check, tabular and pictorial representation, linear addition, working backwards, partitioning, and comparison strategies.




As these poster illustrations clearly indicate the teachers exhibited a wide variation in their thinking. They generated a lot of interesting conversation that helped the instructors to bring a nice closure to this problem using proportional reasoning.  Teachers reported that the reasoning up and down helped them to break problems into chunks and build on those chunks.  They saw how building on known concepts or known quantities gave them a sense of control as opposed to the lost feeling we sometimes experience during the introduction of a completely new idea.  The teachers realize that the latter is a source of concern, frustration, and fear in their students.  One teacher commented that she never realized how emotional the process could be and that she was gaining a new perspective on her students and how she interacts with them.  Another teacher wrote that she would use reasoning up and down to help her students focus on what they already know and then guide them in building on that knowledge.  Several teachers remarked on the importance of labeling processes so that students have a clear picture of how the concepts tie together; this leads to the development of conceptual understanding and the internalization of concepts and processes for the students. Another teacher reflected, I am also starting to think differently about analyzing student work.  When problems have the opportunity of yielding a variety of correct answers, it is important to consider what the student is doing and what math they can do and understand."

Along with in-class activities, the teachers had the opportunity to reflect on all the problems that they worked on each day. The teacher reflections enabled us to focus on the understanding, reactions, and feelings of the individual teachers. While the posters showed how people in a group approached problem solutions in a variety of ways, the reflections gave us insight into how the individual teachers were feeling about the sessions, about their own competence, and about their classroom practices.  Several themes emerged including the value of the struggle, the joy of using conceptual thinking, the importance of clarity, the advantage of building, the benefit of collaboration, and recognizing that there are multiple valid ways in which to approach problem solving, which leads to viewing student work with new eyes. In the institute, there was also a focus on formative and performance based assessment and error analysis to improve student learning. Following this, in the fall semester the teachers worked on Lesson Study as a team with K-12 math teachers, special educators, LEP teachers led by a mathematician, a mathematics educator and a math coach (Suh and Seshaiyer, 2014a, 2014b). The follow-up lesson study was one mechanism used to sustain the learning and keep the focus on the content, helping teachers understand how the tasks they are using in their classrooms are intended to contribute to student understanding.

Program Evaluation and Assessment: Data collected from the MSP programs included baseline data on participating schools, teachers, and students including the numbers served, qualification levels of teachers, proficiency levels of students; program results for teachers related to changes in content knowledge and highly qualified status and for students related to changes in academic achievement. For the teachers’ content knowledge, a pre- and post-assessment of teacher content knowledge was administered before and after the summer institute and following the final follow-up meeting. Special efforts were made to create these assessments to reflect VA SOLs at various grade levels. A pre-survey solicited information regarding their opinions, their preparation, their teaching practice, and the quality and impacts of their professional development experiences. To assess the project’s impact on classroom instruction and student achievement, data such as student assessment and artifacts of student work was collected during the academic year at the follow-up Lesson Study sessions. Examples of student sample work from a Lesson study is illustrated in Figures 7 and 8 where students demonstrate their thinking of solving problems using techniques in proportional reasoning such as unitizing. These included having teachers develop and implement assessments of their students learning. These assessments were aggregated as part of the report of impact on student achievement. To assess impact on classroom instruction, teacher reports and activities related to their dissemination plans was aggregated during the follow-up sessions throughout the school year. The primary indicator of student achievement was the change in students’ strategic competence in math. Students learnt to think about situations in relative rather than absolute terms. For example, the students in a particular lesson study were asked to compare two situations of increase in numbers from 500 to 800 versus 300 to 600.  Students thinking in absolute terms may answer that these situations have the same amount of increase, however, a relative multiplicative thinking helped the students to think proportionally to help them understand and justify the differences in the two situations. Reports were prepared each year that identified the major findings of the evaluation documenting evidence of increased knowledge and skills and improved classroom practice as a result of teachers’ participation in the project. A written evaluation was conducted at the end of the program year to determine how well the overall goals and objectives have been achieved. The summative evaluation measured the extent to which the project goals were met, the degree to which the participants gained the desired skills and knowledge, and the project’s effectiveness in improving classroom practice and student outcomes. We were fortunate to have support for doctoral students and lead specialists from the MSP award that allowed them to serve as knowledgeable others to sustain a collaborative coaching effort to continue to engage the teachers after the completion of the summer institute.

Impact and Outcomes:  Our results have revealed that the design of the PD and Lesson Study offered opportunities for teachers to become inspired by what their students were capable of doing. Despite encountering challenges as they transformed their teaching approach, the MSP program has greatly impacted the teachers to find ways to rethink their mathematics instructional and pedagogical practices in order to promote higher ordered thinking skills in their students. The successful completion of the summer PD followed by collaborative coaching & Lesson study also gave the opportunity for the 85 participating teachers to earn 3 credits of graduate coursework at George Mason University. Struggling through problem solutions with colleagues, analyzing their approaches, questioning their reasoning, understanding how to develop open tasks, comparing their work with others and, contributing to group efforts were noted by the teachers as being very beneficial to their discoveries in the Summer PD institute and the follow-up Lesson study.


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

Lamon, S. J. (1999). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers. London: Lawrence Erlbaum Assoc.

Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning. (pp. 629-668). Charlotte, NC: Information Age Publishing.

Suh, J. & Seshaiyer, P. (2014a). Developing Strategic Competence by Teaching Using the Common Core Mathematical Practices, Annual Perspectives in Math Education, Chapter 8, pp 77-87.

Suh, J. & Seshaiyer, P. (2014b). Examining teachers’ understanding of the mathematical learning progression through vertical articulation during Lesson Study, Journal of Mathematics Teacher Education, pp: 1-23.

Suh, J. & Seshaiyer, P. (2014c). Mapping teachers’ understanding of the mathematical learning progression through vertical articulation during Lesson Study. American Educational Research Association Online Repository, Philadelphia, PA.

Suh, J.M., Seshaiyer. P., Freeman, P. & Jamieson, T.S. (2011). Developing teachers’ representational fluency and algebraic connections. In Wiest, L. R., & Lamberg, T. (Eds.). Proceedings of the 33rd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. 738-746.

Suh, J. & Seshaiyer, P. (2014b). Examining teachers’ understanding of the mathematical learning progression through vertical articulation during Lesson Study, Journal of Mathematics Teacher Education, pp: 1-23 (2014).

Suh, J. & Seshaiyer, P. (2013). Mathematical Practices That Promote 21st Century Skills, Mathematics Teaching in the Middle School, NCTM, 19(3), pp 132 - 137.

Seshaiyer, P., Suh, J., & Freeman, P. (2011). Unlocking the locker problem with technology”, Teaching Children Mathematics, NCTM, Vol 18(5), pp 322-325.

Padmanabhan Seshaiyer

Professor of Mathematical Sciences

George Mason University



Jennifer Suh

Associate Professor of Mathematics

George Mason University



Virginia Council of Teachers of Mathematics
PO Box 73593
Richmond, VA 23235

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