This paper reports preliminary results of the Year 1 of NSF DR K-12 exploratory project addressing an innovative approach to preK students’ development of algebraic and quantitative reasoning through measurement. This approach builds on measurement concepts and algebraic design of the pre-numeric stage of instruction found in the successful Elkonin-Davydov elementary mathematics curriculum from Russia. The purpose of the study is to adapt and refocus the conceptual framework of the E-D pre-numeric stage with respect to preK and kindergarten students.

 

The current reform efforts in mathematics education reinforce and provide new emphasis on the role and importance of algebra and algebraic reasoning throughout the entire preK-12 curriculum (National Mathematics Advisory Panel, 2008; NCTM, 2000, 2006; RAND Mathematics Study Panel, 2003). In particular, the preK-2 Principles and Standards for School Mathematics (NCTM, 2000) expect that preK-2 students should be able to “represent and analyze mathematical situations and structures using algebraic symbols” and “use mathematical models to represent and understand quantitative relationships” (p. 92). As a result, the last decade has been characterized by increased attention of educational researchers and policy makers to the ideas and concepts of early algebra (EA).

The proposed paper focuses on the quantitative approach to EA which provides an entry point into EA based on the visual interpretations of arithmetic operations on physical quantities through measures, magnitudes, and number lines. In particular, one of the comprehensive measurement-based approaches to EA was developed by Davydov in an elementary curriculum known in Russia as Elkonin-Davydov (or E-D) curriculum. The US implementation of the E-D measurement ideas is known as the Measure Up project. While the E-D curriculum with its proven track record focuses on the development of quantitative and measurement reasoning among elementary-aged children in grades 1–6, it is feasible that much younger children, even four-year-olds, can access the pre-numeric ideas. This is supported by research by Baillargeon (2001) and Wynn (1997) who showed that infants as young as two-months old demonstrate the development of number and measurement concepts.

The study addressed the following research questions:

(1) To what extent can the pre-numeric stage of the E-D measurement approach be applied to the preK students' development of quantitative reasoning?

(2) How early can algebraic thinking be developed when quantitative reasoning is used as a foundation for PreK students' mathematics?

(3) What are major issues related to the implementation the pre-numeric stage of the E-D measurement approach in terms of the US preK mathematics education and professional development of preK teachers?

Qualitative data were collected from the lessons and interviews during the first year of the NSF DR K-12 exploratory project starting Fall 2010. All the individual teaching experiments and interviews were videotaped, and demographic data and artifacts of student work were collected. The participants included six four-year old preK students (three girls and three boys) enrolled in a preK-K lab school at a large Mid-West public university. The sample was randomly selected in order to represent different populations of preK students in terms of gender, SES, and basic counting experiences.

The initial tasks developed in this PreKEA project were focused on the identification of a hypothetical learning progression based on the Elkonin-Davydov approach to early mathematics. Of primary interest was the early development of number through quantitative reasoning and relationships. To determine the hypothetical learning progression, the materials adapted from the Elkonin-Davydov curriculum (Measure Up) were carefully analyzed with regard to quantitative relationships with continuous quantities (including length, area, volume, and mass).

The mathematics in each of the lessons in the Measure Up curriculum from Grade 1 was documented in three ways: the learning activity that children carry out, the materials that embody the mathematics, the representation used to describe the result of the learning activity, and the evidence by which children's understandings are determined. Using a review of literature on early childhood mathematics, a preliminary set of tasks was created.

The tasks were focused on the pre-numeric aspects of the mathematics. These aspects included 1) identification of attributes of objects that can be compared; 2) direct comparison of area, length, volume and mass; 3) indirect comparison using intermediate measures; and 4) prediction of quantitative comparisons. The representations used to show the results of the comparisons and predictions were identified and incorporated in the research tasks.

Once the tasks were identified, the protocols for each task were created. These protocols included the necessary materials, actions on the part of the interviewer, the anticipated actions of the child, and questions to elicit responses from the child. The tasks and protocols were vetted across the two co-PIs.

A combination of small group, pair and individual interviews were conducted in Spring 2011. While an initial set of 18 tasks had been developed, children reached a cognitive threshold earlier than had been expected. When children were no longer successful with the tasks, the interviews ceased at Task 6. Additionally, because the children appeared to have some issues in completing the tasks, adaptations that can be incorporated in revised tasks for the 2011–12 academic year were discussed. The data analysis will continue in Fall 2011, and it will focus on establishing proof-of-concept that young children can access and attain quantitative and algebraic reasonings through the use of tasks that employ a measurement approach.

The project described in this paper will identify key concepts of the E-D pre-numeric stage relevant to four-year-olds, and it will develop and explore lesson units which can be integrated into US preK settings. This study can benefit disadvantaged students by using an innovative approach to EA instruction that has the potential to broaden access and at an early stage change the situation when disproportionately many disadvantaged students are not prepared adequately for learning quantitative reasoning and algebra. With research in preK narrowly focused on particular topics, the results of this project have the potential to inform a broader field including mathematics education and early childhood education with evidence that young children can access and interact with more complex mathematics, extending beyond counting.