Co-planning, implementing and observing lessons

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• Research and Supports
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Co-planning: Each team developed one or two research lessons which addressed specific needs of the students as identified through the diagnostic assessment data. To inform the co-planning, teams were provided with research articles that focused on student learning of fractions and encouraged to read those of interest to them. Diana Chang, a Masters of Education student, presented her thesis work on students' use of the number line to one group of teachers. Finally, the board leaders in math, the project lead, and the Trent research team provided additional information from current research as well as provincial perspectives during meetings.

The exploratory lessons developed by the teams included precise decisions such as:

• selecting fractions to challenge students' choice of representation (such as 25);
• including benchmark fractions (e.g., 12, 23, 34) and non-benchmark fractions to determine the extent to which students' possess generalized knowledge and understanding;
• providing multiple opportunities for student choice, including their selection of the task, the representation tools (e.g., hand constructed vs. commercially produced), and the strategies used (e.g., ordering fractions by considering the relative closeness to benchmark fractions and 0 or 1 rather than determining a common denominator);
• introducing and/or supporting the use of appropriate mathematical language, including reading a fraction as a number (e.g., 14 is "one-fourth" rather than "one over four");
• providing multiple opportunities for students to engage in math talk with their classmates and teacher throughout a three-part math lesson.

Implementing & observing lessons: Teams selected one classroom as the site to meet for an exploratory lesson. Prior to that lesson, all teachers were encouraged to teach the lesson in their own classrooms. Their student work and insights informed the planning of the exploratory lesson. This allowed for a richer observation experience as well as identification of trends in student thinking and collaborative consideration of teacher moves (both in the moment and over time) to resolve misconceptions.

Following the initial exploratory lesson, teams summarized a number of challenges and misconceptions with fractions. A selection is included below:

 Challenges with content Use of imprecise fraction language reinforces student misconceptions of fractions (e.g., 4⁄10 is "four tenths" but reading it as "four over ten" leads some students to understand it as "four tens", represented by four sets of ten, or as literally "four over ten" represented as a ratio – see picture on right). These multiple naming strategies appear to add to student difficulty in constructing a meaningful understanding of fractions. Over-reliance on circle representations for teaching and learning leads to limited success in representing fraction amounts that do not easily lend themselves to partitioning in a circle (such as 2⁄5 and 4⁄10 – consider the difficulty of partitioning a circle into tenths or fifths compared to fourths or eighths). Students' limited understanding of the meaning of fractions results in inappropriate strategies for comparing fractions (e.g., 2⁄5 is equal to 1⁄10 because 2 fives are 10 and 1 ten is 10). Students who understand that a part-whole area model requires the pieces to be of equal area can struggle with how much precision is required in representations. When comparing fractions, sometimes approximate drawings are sufficient, but other times exact drawings are required, depending upon the fractions' relative equivalency. There appears to be a tendency to move quickly to symbolic notation of fractions and procedures related to fractions rather than to make informed choices about the best representation for sense-making. Some common misconceptions about fractions: Size of the partitioned areas doesn't matter even when using an area model. When we are thinking about parts of a whole in an area model, all the parts have to be the same size. The numerator and denominator in a fraction are not related (little-to-no understanding of the relationship between the numerator and denominator, i.e., that the fraction has two numbers that together represent a value only because the numbers have a relationship). Fractions cannot represent 'parts-of-a-set' relationships. All representations of fractions must always show the 'parts' as attached or touching, including set representations. Equivalent fractions always involve doubling. Number lines are a non-changing whole (where 1 is always the whole) as well as a flexible whole (where the entire length is the whole, regardless of the whole number labels which extend beyond 1). Some students used both definitions simultaneously when ordering fractions (absolute value of 1⁄2 on a number line vs. a relative value of 1⁄2 of the number line length).

Issues around planning and teaching were also identified through the teaching and observing of multiple lessons.

• Some classrooms engaged students who would normally work on a modified program in the exploratory lesson with great success. The teachers wondered how and when they could blend the programming for these students.
• The teachers identified a tension with wanting to correct student misconceptions and allowing time for the students to resolve these notions themselves. Which misconceptions will likely resolve in an appropriate time period and which ones would require intervention?
• There was discussion around lesson refinements which would increase the alignment and connections between the lesson goals, the three parts of the lesson and the success criteria for students.

This discussion led to a list of considerations for subsequent teaching and learning:

1. Expose students to a range of representations.
2. Get students to connect representations with stories in context to make better decisions about which representation(s) to use and when.
3. Increase exposure/discussion/class math talk to enhance the language of fractions.
4. Enable students to make precise drawings when they want/need to (e.g., provide grid paper).
5. Think more about how to teach equivalent fractions.
6. Think more about the use of the number line, including having students revisit their number line over the unit to revise the location of fractions and also to place additional numbers (such as percent and decimal numbers where appropriate).
7. Increase student understanding of the similarities and differences of set and measurement models. For example, students may sort representations as set, part-whole, or both (could use epractice, document pictures from lesson, or student-generated representations).
8. Think more about the precise selection of fractions that will push student thinking with a specific concept (e.g., compare 319 and 389 to see if they are looking at the fraction as a number).
9. Use a task for the summative that is open enough to allow for the achievement of levels (such as ordering fractions on a number line with a variety of specific criteria).
10. Include success criteria to allow students to better distinguish the learning (as identified through the learning goal) from the task.
11. Implement low-cost, low-prep strategies for increasing student understanding, including having students edit and revise their work over time, using sticky notes for ordering and comparing fractions to allow for easy modification of work, and skip counting by fractions to emphasize fractions as numbers and the role of the numerator and denominator.

Subsequent lessons in the classes focused on the area of need for the specific class. However, many of these lessons shared common elements. Where possible, teams met over a period of time to share their student work and to collaborate about what types of tasks and probes would support student learning. Teams had opportunities to discuss problems of practice, which included strategies for collecting data for reporting, sequencing and timing of content, and sharing of resources which supported both their classroom practice and their professional learning.