- Student Learning

Student engagement with rich tasks

As student needs became more apparent, teachers made precise and purposeful task selections. Sources for these tasks included professional journals and research articles, textbooks, resource books, as well as lessons co-planned by the team.

Three main questions emerged related to the teaching and learning of fractions:

- Which representations support students in acquiring a deep understanding of fractions?
- When is accuracy in representations important?
- When should misconceptions be permitted to stand?

*1) Which representations support students in acquiring a deep understanding of fractions?*

Students relied heavily on hand-constructed circle models to represent their thinking. This representation proved limiting when trying to compare fractions such as 2/5 and 4/10, for the simple fact that it is spatially difficult to partition a circle into tenths and fifths. Some representations also reinforce the common student misconception that a fraction represents two numbers (the two and the five in two fifths, for example) and not one singular quantity or number as is the case. This misconception is reinforced, for example, when students count the sections of a circle without understanding that the circle represents the whole. The use of linear models and rectangular models avoided some of these challenges and supported an increased understanding of fraction concepts.

*2) When is accuracy in representations important?*

When hand-drawing a representation, it was important for students to understand how much accuracy is required. Some students found it difficult to articulate the situations that required less or more accuracy. For example, when using a hand-drawn representation to compare ^{2}⁄_{5} and ^{8}⁄_{9}, a high degree of accuracy may not be necessary for students who easily see that ^{2}⁄_{5} is closer to 0 and ^{8}⁄_{9} is closer to 1, so will conclude that ^{2}⁄_{5} is less than ^{8}⁄_{9}. However, when comparing ^{2}⁄_{5} and ^{1}⁄_{3}, more precise representations may be required to ensure that an accurate comparison is made.

*3) When should misconceptions be permitted to stand?*

In this study, teams found that letting student misconceptions stand, even overnight, did not solidify the misconception but rather gave the student time to think it through. Careful selection of rich tasks challenged students to revisit their fragile understandings and through discussion/exploration both in class and beyond class time, cemented more precise and accurate conceptions of fractions.

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