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Teacher Professional Learning Instructional Decisions Student Learning Fractions(Content)

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Teams spent a lot of time thinking about each lesson. They focused not only on the content, but on the instructional strategies that best aligned with that content. They needed to understand how the concept of the lesson connected across other strands and grades. Planning lessons based on student understanding meant that there was variation from class to class based on student need, even within similar grades.

Key learnings from co-planning and implementation of lessons were:

  • Context: when fractions were represented using a set model that involved a food context, students wanted all items to be the same size, which is not necessary for set models. Teams also noted that manipulatives were an excellent example of a real context in-and-of themselves.
  • Task design: Teams were able to see how design of the tasks influenced student responses and success. For example, when students were asked to place numbers, including fractions and decimals, on a number line they had more success than when asked to place fractions alone. Inclusion of improper fractions and mixed numbers along with proper fractions allowed students to build stronger understandings of the related nature of these numbers. Teachers selected tasks that allowed a significant amount of time for students to deeply engage and reason.
  • Common difficulties: Teachers designed lessons to build robust understanding and address common misconceptions. Understanding some consistent errors in student reasoning helped the teams to intentionally bring these to the surface so they could be addressed. See examples in chart below:

    Common difficulties

    Lesson implications

    Fragile understanding of the meaning of numerator and denominator

    Encourage students to select their own tools and representations in order to develop a sense of the whole as well as to consider the role of partitioning to explore the relationship between the numerator and denominator

    Limited procedural understanding for generating equivalent fractions

    Engage students in constructing equivalent fractions through their own reasoning using manipulatives rather than learning a single algorithm such as doubling both the numerator and denominator

    Conflation of characteristics of "parts of a set" and area models (e.g., parts of a set must always be the same size)

    Engage students in lessons that expose them to i) area models partitioned in non-congruent yet equal sized segments; ii) sets (collections of objects of varied sizes) (See resources: Math for Teaching: Fractions document.)

    Rather than focusing on eradication of these difficulties, the lessons were designed to allow students to explore their own understanding and grapple with concepts to realign, deepen and consolidate: "Mistakes are an important part of mathematical learning, what Borasi (1994) called 'springboards for inquiry.' Eggleton and Moldavan (2001) asserted that mistakes are an inevitable part of problem solving and indeed 'if no mistakes are made, then almost certainly no problem solving is taking place'" (Bruce & Flynn, 2011).
  • Learning goals: With the increased use of more open tasks, the teams found it sometimes difficult to know just which pieces to have students discuss and deconstruct during the debrief and consolidation phase of the lesson. This difficulty highlighted the need to carefully consider the learning goals and anticipate student responses to the task in order to ease the selection of responses to highlight and to do so in an appropriate sequence. The teams also found it helpful to have some pre-planned questions that were focused on the learning goals for use during the consolidation.

Teachers built a comprehensive resource-bank which, along with their increased pedagogical content knowledge, allowed them to make more precise decisions about subsequent lessons. Moreover, they were able to validate their thinking through discussions with colleagues over time and by referencing the research materials available to them.

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