Goals:

Cardinal equivalence is the ability to understand that apples and oranges are equivalent numerically when there is three of each. Kelly Mix (1999a; 1999b) studied the developmental emergence of cardinal equivalence in 2- to 4-year olds In her experiments, Mix (1999a) manipulated three different set types: a) similar matching set (black disks to black dots choice cards), b) homogeneous set (pasta shells to black dots choice cards), and c) heterogeneous set (random objects to black dots choice cards). In the second experiment (1999b) she used sets of objects presented in different contexts: a) simultaneously versus sequentially, and b) varied events (puppet jumps and light flashes). Taking together, her findings indicated the following developmental progression: at 3-years of age, children judged only sets with highly similar looking items as equivalent. At 3½-years, however, children were able to judge as numerically equivalent sets containing items that were homogeneous among themselves but different from the choice cards and presented either simultaneously or sequentially. At 4½-years of age children were able to judge as numerically equivalent sets made up of random objects (heterogeneous sets). This age group also had a more proficient counting ability. Mix concluded that young children rely, at first, on the surface similarity of the sets they are comparing to judge equivalence. Only later, with increased counting skills and numerical understanding, does this focus on surface similarity give way to numerical comparisons of two sets.

While a lot is known about the development of counting skills and equivalence understanding in young typically developing children, the same is not true for children with disabilities. The few studies conducted with this population have children who are older than the typically developing children as participants, which do not accurately tell us whether the behavior documented results from developmental delays or from years of ineffective schooling.

In order to investigate the emergence of numerical concepts in children developing atypically, it is imperative to focus the research endeavor on the skills and understandings of young children with disabilities  Because the development of counting and equivalence judgment seems important to support further mathematical understanding, and because there is a strong correlation in the development of both skills, the present study aimed to investigate the ability to recognize equivalence on homogeneous and heterogeneous sets in 4 ½ -year-olds with atypical development, and it also examined the relationship between participants’ counting abilities and cardinal equivalence.

Methodology:

Participants:

Eleven children with atypical development participated in the study. Four children with Pervasive Developmental Disorder (PDD); five with general Developmental Delays (DD); and two with Motor Disabilities (MD, i.e., cerebral palsy and infantile stroke). The average age for the group of participants was 54 months (ranging from 46 to 60 months). All the participants were enrolled in an inclusion preschool program for children with and without special needs in the Boston area.

Procedures:

The experimental session with each child was video taped. For the equivalence task there were 11 training trials and 11 test trials. Each trial was composed of a triad. Each child was presented with a target card and two choice cards. The two choice cards included one with the target numerosity (2, 3, 4, and 5) and another with the wrong numerosity. Choice cards differed from the target card either in length or density. There were two conditions to this task: 1) High similarity, and 2) Low similarity. The task was presented in counterbalanced order of conditions. Additionally, each child was requested to count sets of 2, 3, 4, 6, 8 and 10 items and to produce cardinalities of 2, 3, 4, 5, and 6 items.

Results:

A foreigner to the study coded 20% of the data and 92% of reliability was obtained. One-sample t-test was conducted on the group total scores of Training, Test, High and Low Similarities. The results (in Table 1) indicated that participants scored better than chance in Training and they also scored better than chance on both High and Low Similarity sets. In addition, the paired t-test conducted to evaluate whether the mean difference between training and test for groups 1 and 2 were significant showed no significant differences indicating that the order of the training did not affected performance on test. The frequency of correct judgments occured more on sets with small quantities.  Correlation coefficients were computed for counting assessment, cardinality, and equivalence test. It showed significant correlations between counting and equivalence, r (9) = .61, p< 0.05, and between cardinality and equivalence, r (9) = .66, p< 0.05.  Although counting and cardinality correlated highly with equivalence, a difference of the strength of the correlation is detected. While counting accounts for 37% of the success in equivalence judgment, cardinality accounts for 43%. That explains why for four participants in the study the correlation between counting and equivalence was not straightforward. Although these children had above chance level scores in counting, they obtained low scores in equivalence. This suggests that the ability to count in children with disabilities might be a procedural isolated competency that needs to be connected with other conceptual numerical understandings.

Implications for practice:

This study showed that 4½-year-old children with disabilities have a strong procedural knowledge of count and an emergent and powerful conceptual knowledge of cardinality and equivalence. Therefore, it seems important that the teaching of math focus on the integration of procedures and concepts involved in this domain.

Recent studies claim that the learning of numerical abilities requires the interconnection of conceptual and procedural knowledge. Although the interaction between conceptual and procedural knowledge seems important, the majority of educational practices designed to assist children with special needs still prioritize one single dimension of knowledge and ignore the other. Instruction that focuses on the separation of concepts and procedures may put at risk atypically developing children’s ability to use their numerical concepts and skills meaningfully and flexibly. The meaningful learning of mathematics requires the integration of procedural and conceptual knowledge in order to create preschool programs that improve the mathematical learning of young children with special needs.