Virginia Mathematics Teacher Fall 2016

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, (Figure 6). 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 linear addition, working backwards, partitioning, and comparison strategies.

proportional 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 17 th century to 2002. reasoning through

Figure 1. Solution by Guess and Check

Virginia Mathematics Teacher vol. 43, no. 1

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