Part of this is because of the way we were taught fractions but other parts have been because fractions is no longer a whole number system but a rational number system. You have to go back to your grade 9 mathematics class to remember your classification of numbers. Basically, fractions now deal with parts of wholes instead of just the whole number. However, for four years of our students lives we only deal in whole numbers.

The research suggests four underlying reasons that may result in students struggling with fraction concepts. The first is a natural progression of understanding which teachers tend to forget, moving too quickly through the curriculum for their students (Fosnot & Dolk, 2000; Lamon, 1996; Pothier & Sawada, 1983). Second is the instructional practices that result in poor understanding of part-whole relations (Cramer, 2002; Fosnot & Dolk, 2000; Kamii, 1999). Third is the teaching of algorithms and procedures without developing conceptual understanding (Battista, 1999; Cramer, 2002; Fosnot & Dolk, 2000; Kamii, 1999; Mack, 1999). Finally, emphasizing the teaching of and memorization of computational skills, resulting in poor problem-solving skills with fractions (Asku, 1995).

Students go through five stages of developing learning in fractions (__ Pothier and Swanda__, 1983). The first stage is a sharing stage. At this stage, students learn the basic language of fraction sharing along with a natural procedure for halving (Pothier & Sawada, 1983). Students understand that when there are two people, each person gets a piece of the whole but that piece does not necessarily equal a half. It is interesting to note that this is developed at a social level, which suggests that students do come to school with some fraction concepts and schemas from which to build their knowledge; this is in contrast to the traditional argument that the teacher imparts all knowledge to their students. The second stage is a mastery of the halving process that students created in the earlier stage. This is a critical stage in developing fractionconcepts because it is here that students learn equivalency (1/2 = 2/4) at a basic level when they start to share pieces equally and realize what happens to the whole piece as they share with more people, or that child can double the number of parts to obtain fractional parts whose denominators are half the size (Pothier & Sawada, 1983). Pothier and Sawada‟s third stage is a development of fair sharing. At this stage, students realize that partitions are classified as “fair” or “not fair.” In addition, students also learn addition and subtraction of fractions (1/4+1/4= 2/4), when they give the pieces they create to each other. This critical stage was confirmed by Reyes (1999) who reported that students who struggled with creating benchmarks and equivalent fractions needed more practice representing equal fair shares. According to Reyes (1999) and the Ontario Curriculum benchmarks are critical fraction placements where students see other fractions are in comparison to those placements. Those critical fractions are 0, 1/2, and 1; sometimes, it also includes, 1/4, and 3/4.1 When I refer to Benchmark or Benchmark model it is this definition to which I refer. In the fourth stage, students recognize the inefficiency in the doubling strategy when dealing with odd fraction denominators. At this stage, students use a counting strategy to calculate thirds, fifths, ninths, and so on. Pothier and Sawada‟s developmental stages were replicated in later research by Lamon (1996) who followed students from grades 4 to 8 suggesting that student‟ progress from using inefficient calculation strategies to more efficient strategies. Lamon noted that although Pothier and Sawada‟s fifth stage, using multiplication, was described as theoretical in their research, her findings confirm that students can reach this fifth stage. In addition, Lamon also pointed out that students used their social sharing strategies to solve their particular problems. Ignoring the natural progression of student development is one aspect of teaching that often leads to further struggles in fractions (Pothier and Sawada,1983).

So the question becomes how do we help students develop fractional understanding in the classroom?

1) This takes time and I think we need to do this a lot earlier than the curriculum tells us too. In fact, it starts in Kindergarten. As the research shows the first two stages of development happen at a young age as students understand the concept of a half. They can also build an understanding of what a half is and its relationship to a whole.

2) Fractions are a relationship to a whole and we have to treat it this way. In my research, I found that many of the problems centered around fractions being known as shading in sections (i.e: please show 3/4 by shading in 3). Rarely did we say 3 of 4 or that the piece we shaded were 1/4 and we have 3 1/4's. This understanding of part-whole relationships is fundamental to all student's future learning of fractions.

3) We teach algorithms first before conceptual learning. Often we fall to the algorithm (invert and multiple and common denominators) because that is what we remember. However, this is very confusing for students success of understanding fractions. Students and adults need more work with the concept of fractions in order to help them build real understanding of the relationships. The more concrete work we can do in early grades the better for older grades.

2015 Fractions Learning Pathways. www.edugains.ca for full resource.

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