Conservation of Quantity: An Overlooked Construct?

Theoretical Perspective

Developing number sense requires the acquisition of many different critical concepts that, if not mastered, may impact development and achievement (e.g., see Fosnot & Dolk, 2001; Sarama & Clements, 2009). Of these, we focus on conservation of quantity: "a collection's quantity does not change when the objects are moved around" (Krasa & Shunkwiler, 2009).

While debate occurs about when a child conserves number (e.g., Halford & Boyle, 1985; Sarama & Clements, 2009), children as young at 10 months demonstrate an ability to compare quantities, By approximately age 7, however, researchers have concluded that most children consistently conserve a number and have abstracted a "rule" (i.e., if nothing is added or subtracted matching sets remain equivalent) without being distracted by perceptual cues such as length or the spreading out of objects (Baroody, 1987; Sarama & Clements, 2009). However, such conclusions have been drawn using a limited range of tasks. In our work with struggling students we found that students continue to have difficulty with conservation when provided a range of differing tasks. Thus, we sought to answer the research question: What are the differences and similarities in student understanding for struggling first and second grade students for the three task levels for conservation of quantity?

We separated conservation tasks into three levels that corresponded roughly with student performance. Figure 1 characterizes these various types of tasks. The levels differ by the number of groups of objects that exist.

Figure 1: Task Levels for Conservation of Quantity.

Level	Description	Sample Task
1	A group of objects is rearranged.	The child is shown 15 counters. After the child recognizes that there are 15 counters, the counters are arranged in a "line" and the child is asked to determine the number of counters.
2	Two groups of objects are provided and the same number of objects are removed from one group and added to the other group.	Thirteen counters are placed into two groups in opaque cups. The child is asked how many counters are in the two cups. Two counters are moved from one cup into the other and the child is asked how many counters are in the two cups altogether.
3	Multiple groups of objects are provided and the same number of objects are removed from one group and added to another group.	Objects are placed into 2 groups of 10 and one group of six. The total number of objects is determined. One object is moved from one of the groups of 10 into the group of six and the child is asked to determine how many objects.

Method

We worked individually with 3 students (who were identified as struggling) for two years with 45-minute sessions twice per week for 16 weeks per year. Basic demographic data along with their scores on a standardized mathematics measure, KeyMath3, are provided in Table 1.

Instruction was provided to pairs of students, following the teaching experiment model of Steffe and Thompson (2000). All instructional sessions were video recorded.

Table 1. Participant Demographic Data

Variable	Amanda	Becca	Caitlan
Race	White	Black	Hispanic
Age at beginning of study	6 years	6 years	7 years
Primary language	English	Arabic	Spanish
KeyMath3: grade equivalent score Beginning of 1st year      Operations      Total test End of 1st year      Operations      Total test End of 2nd year      Operations      Total test	n/a n/a k.8 k.4 1.9 1.9	k.6 k.2 1.1 k.8 2.6 2.1	k.8 k.4 k.6 k.8 1.4 1.2

To analyze student performance, we created individual descriptions of student strategies for the various tasks the students completed from each of the three levels of conservation tasks. We then compared these strategies across the three students.

Results

Due to space constraints, we provide brief findings from Levels 1 and 2 to exemplify the differences and similarities in student performance for the three struggling case study students.

Level 1: A single group of objects are rearranged

Amanda, November Grade 1

Amanda counts 15 counters in a pile and says that there are "15 total." The teacher arranges 15 counters in a long line and asks, "How many counters are there?" Amanda is unsure how many there are. The teachers reminds her taht there were 15, and Amanda says that she is still unsure. When the teacher pushes the pile back together, she says there are 11 counters.

When the other two case students completed these tasks near the beginning of grade 1, they responded similarly to Amanda. However, by the beginning of Grade 2, all three students recognized that the quantity did not change when one group of objects was rearranged.

Level 2: Two groups of objects are provided and the same number of objects are removed from one group and added to the other group.

Becca, November Grade 2

The teacher says and demonstrates: There are some counters in the cups. If I move three counters from this cup to the other cup will the number of counters all together in both cups be more, less, or the same as what we started with?"

Becca says she thinks there would be more "because in this cup there are more [touches cup where counters were added] and this one is less [touches cup where counters were taken away]." The teacher asks about both cups and Becca again says she still there are more counters.

The other students demonstrated similar difficulties with tasks that involved moving objects between two groups. They were unsure about how moving objects between groups impacted the total number of objects. Students demonstrated difficulty maintaining a focus on the total number of objects when the number of objects in each group changed and had difficulty applying these ideas to tasks that involve compensation (e.g., 3 + 5 = 2 + 6).

Importance of the Research

This study extends our understanding of magnitude by providing a framework for magnitude tasks and descriptions of student strategies. This study demonstrates the complexity of understanding magnitude and the challenges that students may face with this construct.