Long-term

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There were several dilemmas that the teams grappled with related to long-term planning. The teams found that allowing students to interact with the fractions content in smaller chunks, punctuated throughout the school year, benefitted many students. Students were able to make connections across number systems such as decimals and percents because this approach to teaching decreased the compartmentalization of fractions concepts. For example, students were able to connect their work with number lines in fractions to measurement work. Teachers were able to make more precise decisions about next steps for instruction with the increased time between the lesson chunks. However, this type of planning was demanding on teachers, both in terms of their time and their pedagogical content knowledge. It also presented challenges with alignment with board directions about sequencing and timing, data collection for reporting requirements, and alignment with colleagues teaching the same grade.

Teams developed and tested short punctuated lesson sequences (bundles of 3 to 4 lessons) and observed the impact on students' ability to connect new learning to prior knowledge and build flexible thinking. Typically, students treat each math unit as separate, and have difficulty making connections to and drawing on what they already know. For example, in one classroom, students who had just finished a unit on geometry weren't able to think flexibly about using pattern blocks as a tool to explore fractions. In this case, students focused on vertices and sides of pattern blocks, rather than other attributes such as area that would support fractional thinking. Conversely, when teams purposefully integrated fractions and decimals, students were able to identify, and take advantage of, the connections between the two number systems.

In order to further students' flexible thinking, teachers built a community of learners comfortable with explaining, discussing, and defending their thinking as a critical first step. In addition, the increased precision of teacher language for fractions (e.g., 27 read as 'two-sevenths' rather than 'two out of seven' or 'two over seven') reinforced the understanding that a fraction is a number (rather than two numbers separated by a line). Similarly, reading the decimal 0.25 as '25 hundredths' rather than 'zero point two five') reinforced place value.

One particularly interesting finding was that when students used manipulatives as the primary site of problem solving, rather than just a communication tool after the fact, they were not only more engaged in the learning, but also developed deeper understanding of fractions concepts. The use of a variety of representations also allowed teachers to uncover misconceptions that students held, even when they did the symbolic (numerical) manipulation correctly. This reinforced the importance of using manipulatives consistently as an integral part of learning throughout the year.