Operations with Fractions

What I Knew and What I Wanted to Know What I Knew Fractions represent part of a whole. Students are taught to add and subtract fractions by finding a common denominator but not for multiplying and dividing fractions. Many teachers do not have a good conceptual understanding of fractions. Students come to 6th grade without a strong foundation in rational numbers. What I Wanted to Know How fractions are explained and taught in younger grades and why many students struggle with this concept. Is it because of the concept itself or is it the way it is taught that causes confusion? Ways to represent adding, subtracting, multiplying, and dividing fractions to increase student understanding. The most confusing aspects of operations with fractions for students. Why do we teach students to invert and multiply fractions in order to divide? Concrete and abstract representations (and how to explain them).

The Big Idea Fractions are a difficult concept to learn and teach due to the many layers of thinking within the concept (researchers name 5 layers). Students are taught algorithms for fraction operations without understanding why or how the algorithms work, thus creating problems for students. Lamon Brown & Quinn

Key Conceptual Ideas Categories and Representations Research shows that students learn fractions through multiple representations (symbols both written and spoken, manipulatives, real-world situations, and pictures). o Lamon pg. 360 Fractions are made up of 5 different sub-categories (not just part/whole). These 5 different categories are 5 different ways of thinking about fractions. This creates complexity within the concept. They are just a SMALL subset of rational numbers. o Charalambos, Charalambos, & Pitta-Pantazi pg. 293 o Brown & Quinn pg. 28 o Lamon pg. 365

Key Conceptual Ideas Curriculum and Textbooks Textbooks are lacking multiple representations of fractions - the most typical representation is symbolic representations. Teachers who use only a textbook for instruction do not provide students with all the representations that are beneficial for students. o Hodges pg. 79 Curriculum is lacking in the instruction of fractions; only teach one sub-category or teach sub-categories as different entities. Rules and algorithms that teachers teach students do nothing for rational number sense. o Lamon pg.642 & 643

Key Conceptual Ideas Algorithms Students learn algorithms without understanding them. Then, they have difficulty applying the algorithms to the correct problems. Students have no idea how or why the algorithms work. o Brown & Quinn pg.30

Curricular Connections Playing instruments and singing (band and chorus) - counting beats and notes Slope and other algebraic concepts (understanding slope and proportionality of slope) Transformations in a coordinate plane (stretching and shrinking); scale

Real World Applications Language of fractions ("fraction" can mean multiple things) Smart shopping (rates)

Works Cited Brown, G, & Quinn, R. (2006). Algebra students' difficulty with fractions. amt, 62, Charalambous, C, & Pitta-Pantazi, D. (2007). Drawing on a theorhetical model to study students' understandings of fractions. Educational Studies in Mathematics, 64, Hodges, T, Cady, J, & Collins, R. (2008). Fraction representation: the not so common denominator among textbooks. Mathematics Teaching in the Middle School, 14(2), Lamon, S. (2007). Rational numbers and proportional reasoning. Students and Learning,