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Fractions
Fractions in Elementary Grades (Pedagogical Content Knowledge Project)
 by Julio Flores, Melissa Franklin, TuanAnh Huynh, Janelle Sprague (UC Irvine, August 2008)
Contents 
Cognition and Learning Background: Elementary Algebra
California State Curriculum Standards for Fractions
2nd Grade
 Number Sense 4.0 Students understand that fractions and decimals may refer to parts of a set and parts of a whole
3rd Grade
 Number Sense 3.0 Students understand the relationship between whole numbers, simple fractions, and decimals
4th Grade
 Number Sense 1.0 Students understand the place value of whole numbers and decimals to two decimal places and how whole numbers and decimals relate to simple fractions.Students use the concepts of negative numbers
5th Grade
 Number Sense 1.0 Students compute with very large and very small numbers, positive integers, decimals, and fractions and understand the relationship between decimals, fractions, and percents. They understand the relative magnitudes of numbers
 Number Sense 2.0 Students perform calculations and solve problems involving addition, subtraction, and simple multiplication and division of fractions and decimals
6th Grade
 Number Sense 1.0 Students compare and order positive and negative fractions, decimals, and mixed numbers. Students solve problems involving fractions, ratios, proportions, and percentages
 Number Sense 2.0 Students calculate and solve problems involving addition, subtraction, multiplication, and division
 Even though order of fractions is introduced in the third grade and equivalence is taught in the fourth grade, and even after two or three years of learning fractions, students still struggle with many concepts surrounding fractions such as order and equivalence (Yoshida & Sawano, 2002).
Students’ Developing Understanding of Fractions
 Studies have suggested that building on students’ informal knowledge, which is knowledge that has been constructed by the individual in response to their reallife experience, is an evident way to develop their understandings of mathematics (Mack, 2001)
 Research has found that in elementary school children develop a conceptual transitional shift in their ability to coordinate relations between parts and wholes. This became present in students performance on the following task (Saxe, G.B., Gearhart, M., & Seltzer, M., 1999)
Example
 When students were asked by an interviewer, "Write the fraction that shows the amount of the shape that is gray," younger children tend to interpret the goal in terms of whole numbers rather than in terms of partwhole relations. For the discontinuous quantity items in Figure la, children count the grays and then indicate the product of their count with a numeral, writing, "2" for the first item and "3" for the second item. For the four continuous quantity items in Figure Ib, children may write "3," "1," "5," and "2," respectively. (Saxe, G.B., Gearhart, M., & Seltzer, M.1999).
Cognitive Obstacles and Common Misconceptions
Why are basic facts of fractions so difficult for students to understand?
 Students need a deep conceptual knowledge base of fractions to build on before learning rules of operations on fractions (Aksu, 1997 and Peck & Jencks, 1981).
 Students initial understandings may assist the development of their understanding of complex content domain (Mack, 2001).
 However, no matter how limited student understanding may be, is important because it provides a link to the more complex ideas students are building (Mack, 2001).
 Students have trouble with fractions because they don’t see them as real numbers (Niemi, 1996).
 Students need to see a fraction as a relation between two numbers rather than just the number of parts selected (the top number) over the number of parts in all (the bottom number). (Niemi, 1996).
 Division of fractions has been noted as one of the most complicated and least understood content domains (Aksu, 1997 and Tirosh, 2000).
EqualPartitioning
 The first cognitive obstacle in learning fractions is equal partitioning, for example half of a whole, was found to be knowledge acquired in Kindergarten according to Nunes and Bryant (1996). Nunes and Bryant also found that this early acquisition of equal partitioning before being introduced to fractions was easier for students than having learned equal partitioning after being introduced to fractions. This is partly because equalpartitioning is not a teaching goal in the unit of fractions.
 Many researchers have stressed the importance of equal partitioning for understanding fractions (Niemi, 1996, Peck & Jencks, 1981, and Yoshida & Sawano, 2002).
 To increase understanding of fractions teachers should incorporate equal partitioning in addition to the regular fraction content (Yoshida & Sawano, 2002).
EqualWhole
 The second cognitive obstacle is the concept of equalwhole which therefore makes it hard for students to understand the order of fractions as quantities, but not hard for them to order fractions as symbols (Yoshida & Sawano, 2002).
 A typical error made by children was drawing a fraction in which the size of the whole that each fraction represented was in direct proportion to the size of the denominator; that is, each representation of one was a different size (Yoshida & Sawano, 2002).
Comparing Fractions
 Students with poor conceptual understanding are able to compare fractions only by using rules they are unable to correctly draw or explain their answers (Peck & Jencks, 1981).
 Some students think factions are equal if they have the same number of pieces left over (Peck & Jencks, 1981).
 Example: ¾ is the same as 2/3 because both have one part left over.
Mixed Numbers
 Students are unsure what mixed numbers mean.
 When asked what does 5¼ equal? A) 5 + ¼ B) 5  ¼ C) 5 x ¼ D) 5/ ¼, less than half of the students were able to identify A as the correct answer (Niemi 1996).
Adding Fractions
 When adding fractions, students sometimes add numerators together and then add denominations together (Peck & Jencks, 1981).
 Example: ¾ + ½ = 4/6
 Underdeveloped conceptual understanding can also lead to student errors when adding fractions (Peck & Jencks, 1981).
 Students may also apply rules incorrectly (Peck & Jencks, 1981).
How to Reveal Student Misconceptions
 Have students explain their answers or give justifications for their methods (Niemi, 1996).
 This must be more than just stating the rule or algorithm used and stating the steps.
Pedagogical Tools and Strategies
Dividing Fractions
 The most commonly taught algorithm for division of fractions is “invert and multiply.” *The problem is that this algorithm is taught procedurally with no explanation and students often simply memorize it, while finding little sense in the procedure (Sharp, 2002). This causes some students problems when they apply the procedure. Many students forget whether to invert the divisor or the dividend or what to do with the numbers after they invert.
 Even if they get the answer, many students have no idea why the quotient is correct. The reason that so little explanation is given by teacher for this topic is that it requires algebraic thinking. Now this would not be a problem, however, if the procedure is typically taught in the upper elementary grades (4th or 5th) because students do not deal with algebra until middle school. In addition, elementary teacher knowledge of algebra may not be strong enough to teach students why the algorithm works (Weinberg, 2001). *By teaching students the concept behind division of fractions, the algorithm will make more sense to them.
 There are many ways to teach this topic without appearing to make students randomly invert and multiply fractions.
 Work on fostering the inverse relationship between multiplication and division. Students see the invert and multiply procedure as an isolated activity because they do not see this relationship (Sharp, 2002).
 Example: “Sally takes a trolley ride that costs $0.25 per mile. She rides the trolley for 20 miles. How much does she pay?”
 A student could solve this by multiplying 20 by 0.25
 Or it can be solved by dividing 20 by 4 because 4 miles costs $1.00.
 This will get some students to see the inverse relationship.
Common Denominators
 Another algorithm to teach division of fractions is the common denominator method. This should make sense to students because it is similar to addition and subtraction of fractions (Kwarteng & Ahia, 2006).
 Convert the denominators of any two fractions to common denominators. Then divide the numerators. For example: ½ ÷ ⅛ becomes 4/8 ÷ 1/8 = 4. This problem can be related to the repeated subtraction method of division.
Adding Fractions
 To add two fractions using a drawing as a model (Peck & Jencks, 1981)
 Have students draw 3 rectangles.
 Slice the rectangles into first fraction one way (vertically)
 Slice the rectangles into the second fraction the other way (horizontally)
 Shade the first rectangle to show the first fraction.
 Shade the second rectangle to show the second fraction.
 Add the two shaded regions together on the third rectangle.

 Note: This can also be done using just one rectangle, but for greater clarity, it helps to use three.
Other Fraction Problems
 Make up problems from students everyday lives instead of using synthetic materials such as pattern blocks or fraction circles. Use things such as candy bars and cups of water. To stimulate student interest, show a photograph (samples below) of the information given in the problem such as a photo of a candy bar or pizza (Sharp, 2002).
 Other contexts that can be used include: gum (fifths), candy bars (halves and fourths), pizzas (eighths), pies (sixths), and juice (various portions).
 Select fractions carefully. Unit fractions are the easiest to work with. In addition, fractions that have denominators with powers of two are easier to work with at first such as ½, ¼, ⅛. The ones that should be introduced next are those with even denominators such as 1/6, 1/10, 1/12. Fractions with odd denominators will make more sense to students once they understand fractions.
 Start the problems with whole number answers. Then move to problems with larger dividends. Quotients including remainders can be given to students who are ready to move on.
 A sample problem with meaningful context called “Holiday Bows” (taken from Bulgar, 2002).
 Students were given real ribbon, scissors, a meter stick, and string to solve this problem: How many bows, each onethird meter in length, could be made from a piece of ribbon that is six meters in length?
 When students used their knowledge of fractions in this example, they recognized that each meter contains an equal number of fractional pieces and they multiplied this number by the number of meters. This will allow them to make sense of the algorithm.
 Students were given real ribbon, scissors, a meter stick, and string to solve this problem: How many bows, each onethird meter in length, could be made from a piece of ribbon that is six meters in length?
 Here are some sample problems (taken from Sharp, 2002):
 1. Here I have 2 ¼ cups of orange juice. I take medicine each day and my doctor wants me to limit the amount of orange juice I drink when I take my medicine. I can have ¾ cups of orange juice each day with my breakfast. For how many days can I have orange juice? (Whole number answer)
 2. When I got home last night, I found my dog not feeling very well. Our vet said to give our dog some medicine. She gave us 15 tablets. Because our dog is very large (100 pounds), the vet said to five the dog 1 ⅔ tablets each day. For how many days will the medicine last? (Large dividend)
 3. My mom made this teddy bear. She is thinking about making a whole bunch of them. She used ribbon for the bow. This ribbon is 11 feet long. She wants to use 1 ½ feet of ribbon for each bear. For how many bears can she make a bow? How much of a bow does she have left over? (Remainder)
 4. This pizza is one of the pizzas from Barb’s birthday party. At the end of the party, 7/8 of the pizza is left. Barb wants to store it in the refrigerator. She uses several plastic containers. Each container holds 3/8 of a pizza. How many containers will she use? How many containers will be completely full? How full will the last container be?
 Note: By using context questions such as this, students may not question why the quotient is larger than the dividend. When students simply use algorithms to solve problems, they will believe that division always makes something smaller.
Curricula and Technological Resources
 These curricula and resources were selected because they offer a variety of ways for students to be able to create meaning of fractions. Many times students learn fractions as isolated pieces of information but they don’t know how to make sense of it. “Providing many kinds of representations can help students with this problem, as long as teachers help students connect their understanding of concepts to the different representations” (Millsaps and Reed, 1998). These resources would allow for continuous and diversified exposure to the content and therefore would make the material comprehensible to the students.
Additional Information
 Elementary and Middle School Mathematics: Teaching Developmentally by John A.Van De Walle (2007)
 Chapters 16 and 17 focus on Fractions and offer great information and fantastic activities for use and implementation in the classroom.
Technology Resources
Online:
 National Council of Teachers of Mathematics (NCTM) website:
 Fraction Track
 http://standards.nctm.org/document/eexamples/chap5/5.1/index.htm
 game about equivalent fraction concepts using number lines.
 National Library of Virtual Manipulatives (NLVM) website:
 Virtual Manipulatives
 http://matti.usu.edu/nlvm/nav/vlibrary.html
 Fraction Bars
 http://nlvm.usu.edu/en/nav/frames_asid_203_g_l_t_l.html
 Fraction Pieces
 http://nlvm.usu.edu/en/nav/frames_asid_274_g_2_t_l.html?open=activities
 Fractions Comparing
 http://nlvm.usu.edu/en/nav/frames_asid_159_g_2_t_l.html
 Fractions Visualizing
 http://nlvm.usu.edu/en/nav/frames_asid_103_g_l_t_l.html
 Fractions Adding
 http://nlvm.usu.edu/en/nav/frames_asid_106_g_2_t_l.html
 Fractions—Rectangle Multiplication
 http://nlvm.usu.edu/en/nav/frames_asid_194_g_2_t_l.html
 Fractions Bar Applet
 www.arcytech.org/java/fractions/fractions.html
 applet for connecting fractions, decimals, and percents
 Fraction Model—Version 3
 http://illuminations.nctm.org/tools/tool_detail.aspx?id=45
 equivalence of fraction, decimal, and percents are presented
 Fraction Pointer
 www.shodor.org/interactive/activities/fracfinder1/index.html
 good for connecting area model and number line
 Who Wants Pizza? A Fun Way to Learn Fractions
 http://math.rice.edu/%7Elanius/fractions/index.html
 fun and interactive lessons from meaning to multiplication of fractions.
For Purchase:
 Fraction Attraction by Sunburst
 Four multimedia activities cover ordering, equivalencies, relative size, and multiple representations of fractions. Diagnostic feature helps students correct conceptual errors.
 http://store.sunburst.com/ProductInfo.aspx?itemid=176675
 Games of Math 4 – Division and Fractions
 Students begin at their appropriate level and practice division and fractions.
 http://store.sunburst.com/ProductInfo.aspx?itemid=196771
Literary Resources
 The Doorbell Rang, Hutchins (1986)
 This story demonstrates division and fractions of a whole as children gather and share some cookies.
 Gator Pie, Mathews (1979)
 Story about a group of alligators attempting to split a pie so that everyone receives a piece.
 The Man Who Counted: A Collection of Mathematical Adventures, Tahan (1993)
 Story about a wise man who uses his mathematical skills to solve problems.
 Apple Fractions, Jerry Pallota
 Playful elves demonstrate how to divide apples into different fractions.
 Eating Fractions, Bruce McMillan
 Simple concept book about fractions. Uses food to demonstrate simple fractions and even includes recipes.
 The Hershey’s Milk Chocolate Bar Fractions Book, Jerry Pallota
 This book presents fractions through the use of a Hershey’s chocolate bar.
 Fraction Fun, David A. Adler
 This book introduces fractions through the use of pizza to teach numerators and denominators.
 Jump, Kangaroo, Jump! : Fractions, Stuart J. Murphy
 A kangaroo and his friends divide into groups.
 The Wishing Club: A Story about Fractions, Donna Jo Napoli
 The characters in the story need to combine their fraction pieces to make one whole.
Annotated References
Aksu, Meral. (1997). Student Performance in Dealing with Fractions. Journal of Educational Research; 90 (6) 375380.
Bulgar, S. (2003). 'Using research to inform practice: Children make sense of division of fractions. Paper presented at the 27th International Group for the Psychology of Mathematics Education Conference, Honolulu, HI.
FreduaKwarteng, E., & Ahia, F. (2006). Understanding division of fractions: An alternative view. Unpublished manuscript, University of Toronto, Toronto, ON, Canada.
Mack, N.K. (2001). Building on informal knowledge through instruction in a complex content domain: Partitioning, units, and understanding multiplication of fractions. Journal for Reference in Mathematics Education. 32, 267295.
Niemi, D. (1996). A Fraction Is Not a Piece of Pie: Assessing Exceptional Performance and Deep Understanding in Elementary School Mathematics. The Gifted child quarterly. 40 (2) 70 80.
Peck, D.M., Jencks, S.M. (1981). Conceptual Issues in the Teaching and Learning of Fractions. Journal for Research in Mathematics Education. 12 (5) 339348.
Saxe, G.B., Gearhart, M., & Seltzer, M. (1999). Relations between classroom practices and student learning in the domain of fractions. Cognition and Instruction. 17, 124.
Sharp, J., & Adams, B. (2002). Children's constructions of knowledge for fraction division after solving realistic problems. Journal of Educational Research, 95, (6), 333347.
Tirosh, D. (2000).Enhancing prospective teachers' knowledge of children's conceptions: The case of division of fractions. Journal for Research in Mathematics Education. 31, 525.
Van de Walle, J.A. (2004). Elementary and Middle School Mathematics: Teaching Developmentally. Boston: Pearson Education, Inc.
Weinberg, S.L. (2001, April). Is there a connection between fractions and division? Students' inconsistent responses. Paper presented at the Annual Meeting of the American Educational Research Association, Seattle, WA.
Yoshida, H, & Sawano, K (2002). Overcoming cognitive obstacles in learning fractions: Equal partitioning and equal whole. Japanese Psychology Research. 44, 183195.