Synthesizing Rational Number Reasoning for Urban Schools

Principal Investigator: 
Project Overview
Background & Purpose: 

This study’s purposes are 1) to collect the (English-language) research literature on rational number reasoning; 2) review all this literature, provide descriptions of many aspects of each study, and generate a new synopsis of each article; 3) assemble all the citations and descriptions of the literature into an online searchable database; and 4) write syntheses of each content topic area. For the purposes of this study, Rational Number Reasoning (RNR) is defined to encompass the following topic areas: equipartitioning; multiplication and division; ratio, proportion, and rate; fractions; area and volume (in relation to multiplication, division, ratio and fractions); similarity and scaling; and decimals and percents.


All research literature in the form of refereed publications, book chapters, and conference presentations that could be identified by our team’s searches, as well as via solicitation of contributions from members of the mathematics education and cognitive psychology research community.

Research Design: 

This is a longitudinal synthesis project that will identify learning trajectories. Studies are summarized and results are synthesized by theoretical analysis, comparison of constructs and empirical findings using a concept map.


This research project has generated a number of insights, including:

  • Rational Number Reasoning is complex, and yields to a Learning Trajectories strand analysis;
  • Equipartitioning/Splitting is a foundation for Rational Number Reasoning;
  • Division and multiplication should be derived from equipartitioning/splitting, and coordinated with counting, addition, and subtraction;
  • Three dominant meanings for a/b capture most of RNR reasoning; and
  • Fundamental revisions are needed in the sequencing of RNR construct learning.
Publications & Presentations: 

Confrey, J., Maloney, A., Nguyen, K., Wilson, P. H., and Mojica, G. (2008) Synthesizing Research on Rational Number Reasoning. NCTM Research Pre-Session, Salt Lake City, UT.

Confrey, J., (2008) A Synthesis of the Research On Rational Number Reasoning: A Learning Progressions Approach to Synthesis. Invited Lecture to the International Conference on Mathematics Instruction XI, Monterrey, Mexico.

Confrey, J., (2008) Engaging in Reasoning about 21st Century Mathematics: A New Approach to Rational Numbers. Plenary Address to Annual Meeting of the North Carolina Council of Teachers of Mathematics, Greensboro, NC.

Confrey, J., (2008) Learning Trajectories and Rational Number Reasoning. Workshop on Higher Cognition in Adolescents and Young Adults: Social, Behavioral, and Biological Influences on Learning National Science Foundation, Washington, DC.

Confrey, J., (2008) From Learning Trajectories to Diagnostic Assessments for K-8 Rational Number Reasoning: Defining Learning Progressions in K-8 Mathematics. Council of Chief State School Officers National Conference, Orlando, FL.

Other Products: 

Online searchable bibliographic database of summaries and relevant catalogue of characteristics of research literature. Open-source database application code. Syntheses of research literature, by content area and identification of levels of consensus on learning trajectories.