Young children base their quantitative comparison of “more” or “less” on perception or appearance (Copeland, 1970) before developing a thorough understanding of number and reliance on counting or matching procedures (Steffe, Thompson, & Richards, 1982). Speed and accuracy of perception may be affected by the arrangement of items to be compared. While canonical patterns and familiar arrangements tend to be easier for young children to enumerate (Mandler & Shebo, 1982), some patterns or random arrangements actually make it more difficult for children to determine the number of objects present (Clements, 1999) and may inhibit subitizing.

I conducted an individual structured interview with 104 preschool-age children (mean age four years, nine months). I asked children to respond to four quantitative comparison items presented using one of four randomly assigned arrangements of counters: (AO) action-on-objects subitizing (a unique combination of the spatial-relative elements of geometry with number composition where counters are arranged three-in-a-row, left-to-right, top-to-bottom arrangement); (HL) horizontal-linear array (counters placed in a straight horizontal line); (DD) dice/domino configuration (mimicking the arrangement of dots on a die for numerosities one through six and the standard domino configuration for numerosities greater than six); and (TF) ten-frame format (a two-by-five rectangular frame where the top row of the ten frame was filled first, starting at the left, followed by the bottom row).

I sat across the table from the child and placed a chicken stuffed animal on the child’s left and an elephant stuffed animal on the child’s right, 12-16” apart. I told the child: “Here are two friends, a chicken and an elephant. We are going to play a game with them. We are going to give the chicken and the elephant some toys to play with, and you get to tell me if they have the same number of toys or if one friend has more toys.” I placed a foam board upon which counters were glued in the assigned physical arrangement in front of each stuffed animal and asked: “Do the friends have the same number of toys to play with?” If the child answered yes, I moved on to the next item. If the child answered no, I asked: “Which friend has more?”

Sixty-six percent of the children answered all four of the comparison items correctly, and a further 17% of students made only one error. The overall mean score for all children was 3.47 on a 4-point scale. The total mean score for girls was 3.51, slightly exceeding the boys’ mean score of 3.43. However, a two-tailed t test indicated no statistical difference between boys’ and girls’ scores (p =. 710). For arrangement of counters, this statistical procedure yielded an F ratio of .654 and a probability value of p = .582. For gender, the ANCOVA yielded an F ratio of .356 and p value of .552. Finally, for the interaction between arrangement of counters and gender, the ANCOVA yielded an F ratio of .146 and a p value of .932. Inasmuch as the p value for each element of the ANCOVA well exceeded .05, there is no statistical evidence that spatial arrangement of counters, gender, or the interaction between arrangement and gender affected children’s success in making quantitative comparisons. It appears that the spatial arrangement of counters did not affect children’s accuracy in making quantitative comparisons.

I reviewed the interviews to explore the observable strategies and coded them as: (a) subitizing, (b) explicit counting (one-to-one correspondence with gesturing), (c) looking first followed by counting, or (d) unidentifiable. Subitizing appeared to be the preferred method of comparison in 74% of AO group responses, 60% of HL group responses, 72% of DD responses and 74% of responses made by students in the TF group.

I recorded students’ response times and coded response times as: (a) less than or equal to one second (deemed indicative of subitizing), (b) two seconds (possibly indicating subitizing or counting), (c) greater than or equal to three seconds (deemed indicative of counting), or (d) unclear or impossible to judge. Seventy-three percent of responses were offered within one second, the range deemed indicative of subitizing. This finding converges with the direct perception or subitizing strategy exhibited by many children. The AO group offered more subitizing-range responses than the other physical arrangements offered (80%). The HL array and TF format provided nearly identical percentages of responses within the subitizing range, with 72% and 73% respectively. The HL array gave the highest percentage of responses for the three or more second range (21%) with the TF format giving 16% of responses in this range, perhaps indicating a reliance on counting or other time-intensive approaches. The DD configuration group gave the lowest percentage of responses in the subitizing range (68%).

This study has implications for mathematics educators, preschool teachers, and researchers who are interested in children’s strategies for making quantitative comparisons and the effect of spatial arrangements of counters on children’s speed and efficacy in making comparisons. First, this study demonstrates that preschoolers are capable of making quantitative comparisons, a stepping stone to fluency with number. Preschool mathematics curricula that do not currently emphasize quantitative comparisons may be expanded to do so. Second, the overwhelming number of responses that fell within the subitizing range indicates that preschoolers prefer to use subitizing to enumerate and/or compare small sets of counters. Games, activities, and mathematics tasks that involve subitizing range numerosities may serve to strengthen and provide useful practice with this already innate ability in children. Third, regular experience with the unique spatial arrangements highlighted in the study may help students recognize the common mathematical nature of different configurations. Teachers could expose children to a variety of spatial arrangements for identical comparison situations thereby expanding children’s repertoire of number representations. Finally, the relatively recent model for early enumeration, action-on-objects subitizing, may or may not influence children’s strategies or success in other mathematical contexts and should be investigated further to determine other useful applications for the configuration.