The field of mathematics education research has well documented elementary students' and teachers' difficulty with rational numbers and fractions (e.g. Armstrong & Bezuk, 1995; Mack, 1998; Meagher, 2002; Moss & Case, 1999; Tirosh, 2000).   Several studies have examined elementary children's understanding of fractions and fair sharing.  For example, Empson, Junk, Dominguez, and Turner (2006) examined students' sharing strategies in first, third, fourth, and fifth grades and found two major categories of spontaneous strategies.  Charles & Nason (2000) interviewed a small group of third grade students and described four classes of strategies.  Kouba (1989) examined first grade students' strategies when solving two types of partitive division tasks as part of a larger study of multiplication and division.  In all, studies demonstrate a greater level of abstraction in dealing with fractional amounts with increasing age.  In contrast to this body of work, only a handful of studies have examined how preschool and early elementary school children solve fraction and fair sharing problems (Empson, 1999; 2003; Hunting & Davis, 1991; Miller, 1984; Pothier & Sawada, 1983; Wing & Beal, 2004).   Theoretical Framework & Research Questions

We examined young children's naïve understanding of the division of quantities, and differences in strategies for solving fractional problems between pre-K3, pre-K4 and K children prior to formal instruction.  We investigated the following research questions:  (1) In what ways do young children demonstrate their fractional understanding, specifically mixed fractions, by means of verbally presented fair sharing tasks?  (2) How and in what ways do children's visual representations and oral descriptions differ across pre-school and Kindergarten age groups?

Although previous studies have attempted to elucidate individual students' processes in learning to comprehend fraction tasks at young ages, these studies have tended to involve small numbers of students Kindergarten age and up, and approach the problem from a nonsystematic framework (Empson, 1999, 2003; Hunting & Davis, 1991; Pothier & Sawada, 1983).  Other work has begun to examine the social activities of sharing as these build on young children's informal knowledge (Empson, 1995; Kieren, 1992; Lamon, 1996; Mack, 1993; Piaget et al, 1960; Smith 2002; Streefland, 1993; Vergnaud, 1983).   However, we are concerned with the way in which very young students', ages three to six, and their informal understandings of fair sharing can support and lead students' thinking about fractional concepts and partitioning.  Data Collection & Analyses

Thirty-six pre-school and Kindergarten children ranging from 3 years, 8 months to 6 years, 6 months participated in this study (17 females and 19 boys).  All of the students attended a private Montessori school.  Each student was interviewed, quantitatively and qualitatively assessed, and video recorded by the principal author.  All seven of the fraction items in this study were framed socially; that is in the context of sharing items with friends.  The first two items presented the possibility for one-to-one correspondence, with 1) three objects and three friends, and 2) six objects and three friends.  The next four items presented situations containing more objects than friends (e.g. 5 oranges and 3 friends), resulting in a mixed number or improper fraction as the correct answer.  The final question presented the students with two objects to be shared fairly among six friends (See Table 1).

 

	Fair Sharing Task Read Aloud	Concept Investigated
1	Amanda wanted to share 3 muffins with her 3 friends.  How can she do this fairly?	1-to-1 correspondence
2	Chris wanted to share 6 crackers with his 3 friends.  How can he do this fairly?	Distributing 2 wholes ·            count by 2s ·            count by 1
3	Jade wanted to share 6 carrot sticks with her 4 friends. How can she do this fairly?	Distribution of wholes ·            Dividing into half ·            Dividing into fourths
4	Eric wanted to share 7 cookies with his 3 friends.  How can he do this fairly?	Distribution of wholes ·            Distribute 2 each ·            Dividing into thirds
5	Sam wanted to share 5 oranges with his 3 friends.  How can he do this fairly?	Distribution of wholes ·            2 remain ·            Dividing into halfs ·            Dividing into thirds
6	Matthew wanted to share 7 pretzel rods with his 4 friends.  How can he do this fairly?	Distribution of wholes ·            3 remain ·            Dividing half, fourths ·            Dividing into fourths ·            Dividing into ¾
7	Emily wanted to share 2 granola bars with her 6 friends.  How can she do this fairly?	Less items than friends ·            Dividing into thirds ·            Dividing into sixths
Table 1:  Core of oral assessment protocol.

            Using grounded theory (Glaser & Strauss, 1967), we examined the entire data set to determine emergent themes and strategies across students and items.  Both students' verbal and written responses were considered together when coding each response.  Each item was coded as correct or incorrect, in other words, as a fair or unfair distribution of the snack.  And in addition, the type of strategy the student used was also described.  Examples of each strategy including a student's written work and the accompanying transcript that exemplifies each strategy are provided in Table 2.  Table 3 presents the number of children exhibiting each of the strategies described for each of the tasks presented in this study, as a function of age group. Table 4 depicts a progression of children's problem solving skills with task 3 as they mature.

            In order to determine whether age-related differences in performance were significant, a number of chi-square cross-tabulation tests were performed on the number of children (or frequency of strategies used) in each age group, separately for each task. Only the strategies for which there were cell counts for at least one age group for that task were included in the analysis for that particular task. Although the results must be interpreted with caution given the small data set, and the small frequencies within some cells, these tests are the most appropriate for providing statistical verification of what can be seen visually when examining Table 3. For all tasks, except for task 1, there were significant differences in strategies between Pre-K3, Pre- K4 and K children. Task 1, χ²  (4) = 5.79, p = .22.; Task 2, χ²  (6) = 32.06, p <.001; Task 3, χ²  (10) = 18.42, p = .05; Task 4, χ²  (10) = 29.52, p = .001; Task 5, χ²  (10) = 35.42, p <.001 Task 6, χ²  (10) = 32.41, p <.001 and Task 7, χ²  (6) = 33.02, p < .001.

 

Incorrect/Unfair Strategy	Student Work	Age: Years, Months	Transcript
Distribute All Wholes but not Fairly		3,11	E: Chris wanted to share 6 crackers with his 3 friends.  How can he do this fairly? C: (Drawing lines) E: How many crackers did each friend get? C: Wee (hitting paper with pencil) E: How many did this friend get? C: One E: How about this friend? C: Two E: How about this friend” C: Three E: Is this fair? C: Yeah.
Distribute Wholes but Partition Incorrectly		5,5	E: Sam wanted to share 5 oranges with his 3 friends.  How can he do this fairly? C: Like this . . . (Drawing a line from each face to orange) This one's left, so I can like cut it up (partitions far left orange and then second from left and lines connecting to faces).   . . . I'm not sure I can fix that. E:  Can you tell me how many oranges each of the friends get? C: I'm not sure. . . . All the lines are all mixed up.
Distribute Pieces Only		4,6	E: Emily wanted to share 2 granola bars with her 6 friends.  How can she do this fairly? C: (Draws partitions) E:  How much does each friend get? C: Then you just cut some more in half and then . . . then they can get some more (drawing lines connecting faces and items).  Then they each get three.
Distribute Wholes Only		4,6	E: Jade wanted to share 6 carrot sticks with her 4 friends.  How can she do this fairly? C: She could just give five to the kids. E:  Can you show me how you would do that. C: (Drawing) This one could have this one, this one …. E: So how many did each friend get? C: Ten E: How many are leftover? C: Five
Change Task (add items or people to avoid partitioning)		5,4	E:  Matthew wanted to share 7 pretzel rods with his 4 friends.  How can he do this fairly? C: We can give one to each person.  (Adding circles for friends)  Now this is enough for everyone.
Correct/Fair Strategy	Student Work		Transcript
Distribute  Wholes then Economical Partition		5,7	E: Sam wanted to share 5 oranges with his 3 friends.  How can he do this fairly? C: Five oranges (drawing 3 lines from friend to orange) and we have two more left.  So then we cut this one.  And then they would get one whole and two . . . two halves of a third.
Distribute Wholes then Extra Partitions		5,6	E: Matthew wanted to share 7 pretzel rods with his 4 friends.  How can he do this fairly? C: Ok first I'm gonna give them one whole pretzel rod.  There's three left.  So I'm gonna cut these into fourths.  One, two, three. (drawing partitions). E:  So how much does each friend get? C: Three little pieces E:  And how much all together? C: Four E: Four what? C: One big pretzel rod and three little pieces.

Table 2: Examples of each student strategy.