Understanding Fractions

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In this fractions action research project, gaining a deeper understanding of the various conceptions of fractions allowed teachers to better understand and respond to student thinking.

The part-whole relationship of simple fractions is highly emphasized across North American mathematics programs. Most students involved in this project demonstrated a consistently solid understanding of a half of an item, and when presented in the context of sharing food, also identified the need for the two pieces to be of equal size or area. This comfort level was evident when working with students to explore common benchmark fractions, such as 13 and 34. However, when asked to represent less common fractions, such as 25 or to interpret the fraction 66, many grade 4 through 7 students struggled. Through this research project it was evident that students switched indiscriminately between the use of:

• fractions as part of a whole using a set representation,
• fractions as part of a whole using an area representation, and
• fractions as a ratio (part-part).

These meanings and representations of fractions occur throughout the elementary and secondary Ontario curriculum.

Researchers have identified multiple ways of understanding, perceiving and representing fractions (Lamon, 1999; Marshall, 1993; Mosely & Okamoto, 2007). This research informed the development of the Math for Teaching: Ways We Use Fractions document, which highlights multiple interpretations and representations of a fraction depending on the context or situation.

These include interconnected understandings of fractions as:

• Linear Measures: situations in which a rational number's distance from zero is important
• Part-Whole Relationships: situations in which a part is compared to the total amount
• Part-Part Relationships: ratio situations in which separate quantities are compared
• Quotient perceptions: situations that highlight the process or result of a division
• Operator perceptions: situations in which the role of a rational number in enlarging or shrinking a quantity is highlighted

The junior grades represent a key learning period for students with respect to fractions as they move beyond part-whole with continuous models to work with sets as well as part-part relationships. When educators have a deep understanding of fractions, they are more capable of responding to student needs, with an emphasis on conceptual understanding.

Within the junior curriculum, students are expected to be able to represent, compare, and order fractions. The development of these actions K-12 is shown and further explored below.