Florida international University
Assessing Conceptual Understanding of Rational Numbers and Constructing a Model of the Interrelated Skills and Concepts Students continue to struggle to understand rational numbers. We need a system for identifying students’ strengths and weaknesses dealing with rational numbers in order to jump the hurdles that impede instruction. We need a model for describing learning behaviors related to rational numbers – prerequisite skills and the development of a sense of rational numbers – that is dynamic and allows for continuous growth and change. It would inform us of the important background knowledge that students bring with them and the prior experiences that influence their level of understanding. It would further enable us to assess students’ current levels of understanding in order to prescribe the necessary instruction to continue their progress. Designing a method for assessing students’ conceptual understanding of rational numbers that has this potential is a challenge. In this paper, I will discuss where the call for conceptual understanding stems from in the recent past, what has already been done involving assessment of conceptual understanding, what this reach has revealed about acquiring skills and a number sense with rational numbers, and I will describe a plan using this information for developing a continuum of rational number skills and concepts. Background on Reform in Mathematics as it Relates to Conceptual Understanding National assessments and reports often act as a jumpstart for research agendas, curriculum development, and professional development training. Analysis and assessment of student learning weaves its way into all three categories as the message of current reform in mathematics becomes clear. Assessment is not something done to the students but for the students and therefore must inform instruction. It is important to recognize that the research I have proposed has not come out of nowhere and is not a new idea. What is being suggested is to bring it all together in a practical way. Briefly highlighting various assessments and reports that have identified the importance of conceptual understanding enables one to trace back to the “hatching” of the idea. In addition, outlining the course that got us to where we are today, trying to determine what it means to understand something, and investigating how understanding can be assessed assists us in continuing that course of action in the right direction. In 1980, recommendations were made by the National Council of Teachers of Mathematics for reforming mathematics instruction in “An Agenda for Action.” These recommendations were based on results of the second National Assessment of Educational Progress (NAEP) and on data collected by the National Science Foundation (NSF) largely from a study called “Priorities in School Mathematics” (PRISM). Specifically in the area of fractions, NAEP contended that students’ inabilities to compute with fractions was the result of dependence on rote memorization of algorithms and a focus strictly on routine problems. Among eight recommendations, “An Agenda for Action” called for problem solving to be the focus of school mathematics in the 80’s and basic skills in mathematics to include more than computational fluency. The fourth NAEP showed improvement but indicated that mathematics instruction still lacked depth, particularly in the area of conceptual development of fraction and decimal skills. In 1983 the National Commission on Excellence in Education released a report, “A Nation at Risk,” which identified, analyzed and made recommendations for addressing problems and deficiencies in American education. The recommendations where made in an...