Confrey, Berger and Wilson propose to develop a software diagnostic tool for integrating diagnostic interviews, group administered assessments, and student data in real-time so that teachers can enter and view student status information. This project would concentrate on rational number learning in grades 3-8. The design is based on a model of learning trajectories developed from existing research studies.
The diagnostic system to be developed for teachers would be used in assessing their students' knowledge and would identify difficulties in understanding five key clusters of concepts and skills in rational number reasoning. It would also investigate the diagnostic system's effects on student and teacher learning in relation to state standards, assessments, and curricular programs. The five areas include understanding: (1) multiplicative and division space; (2) fractions, ratio, proportion and rates; (3) rectangular area and volume; (4) decimals and percents; and (5) similarity and scaling.
The diagnostic measures will include diagnostic interviews collecting data using a handheld computer, two types of group-administered assessments of student progress, one set along learning trajectories for each of the five sub-constructs and one composite measurement per grade. The diagnostic system will produce computer-based progress maps, summarizing individual student and class performance and linking to state assessments.
Mathematics understanding is a critical educational endeavor. Beginning even before children start school, mathematics learning is cumulative, becoming more sophisticated, over years. Various sources of information show that U.S. students overall show major weakness in multiplicative reasoning, that is, understanding fractions, ratio, as well as multiplication and division. These weaknesses, signaling weak conceptual preparation in mathematics, are distressingly commonly even in high school, and prevent students from succeeding in algebra and higher math, and achieving scientific and technical careers that are so important today. Learning trajectories are research-based descriptions of student mathematics learning. Our research group has been interested in developing learning trajectories, and building instructional materials and assessment tools from them, with which teachers could carry out more conceptually-based instruction and gain ongoing insight into their students’ conceptual progress and challenges. The focus of such learning trajectories is how students’ conceptual understanding of mathematics grows and can be supported. The Delta Project set out to develop learning trajectories for these multiplicative reasoning topics. Our vision was that learning trajectories and electronic diagnostic assessment systems, hand in hand, would get rapid feedback to teachers and students on what students are learning. We hoped that eventually, a system we developed could be used routinely in classrooms. The rapid feedback from an integrated curriculum and assessment tool would provide teachers and students immediate feedback (data internal to the classroom) to help them strengthen conceptual learning and reasoning, and help teachers recognize student conceptual strengths and weaknesses and quickly adjust their teaching on a daily basis. Further, the learning trajectories and diagnostic assessments would be linked to state educational standards. We focused initially on math topics that provide very early foundation for multiplication, division, ratio, and fractions in the early years of school (grades K-3). Bringing together previously disparate research literature, re-interpreting and extending some of the research, we constructed a learning trajectory for equipartitioning, which is based on fair sharing actions and construction of equal groups that are a hallmark of multiplicative reasoning. This work demonstrated that the early roots, for children, of multiplication and division are different from those of counting and addition/subtraction, offering an avenue for teachers to help students build multiplication, division, and fraction concepts early and robustly. We then created a digital (tablet-based) digital assessment system and new curriculum materials around this equipartitioning learning trajectory. To find out whether students would accept and how they would learn with the system, we conducted several in-class teaching sessions with 3rd through 6th graders in actual class situations. The students enjoyed using the tablet based system, demonstrated growth in their mathematical reasoning, and provided many suggestions to improving the usability of the assessment system—all critical landmarks in the development of such innovations. During the final two years of the project, the Common Core State Standards (CCSS) were adopted by most of the States. To support teachers in interpreting and implementing these standards with learning trajectories, we constructed the other learning trajectories of this project and embedded them in a website we created, TurnOnCCMath.net, (freely available for use by any interested persons). We pursued this effort in order to support educators in interpreting the Standards, and strengthen their own conceptual instruction. Thousands of educators have repeatedly used the website; we have consistently received feedback from teachers across the county expressing the value of the learning trajectories in helping make sense of the standards. Therefore we believe this work is having a growing practical impact in teaching and instructional planning. One of NSF’s purposes in funding research is to promote development of promising innovations that can serve public needs on a large scale. Universities sometimes partner with external non-profit or commercial organizations, to build out promising innovations at a large scale (research groups in universities are not typically set up for this purpose, so next steps are typically carried out by private or other firms). Our project was identified by an education company committed to next-generation digital curriculum and assessment systems; the company licensed the learning trajectories and the diagnostic assessment system, and is currently building a fully interactive digital curriculum for classroom use, based on our project’s outcomes. This is a success reached only by a few projects, and promises to make new types of conceptually based curriculum and rapid instructional feedback systems available to large numbers of children and teachers. Our group is a leader in the U.S. in systematically pursuing construction and dissemination of new learning trajectories for K-8 mathematics. This work could not have been accomplished without the support of the National Science Foundation: we wish to express our gratitude to the Foundation, and the taxpayer support this represents, for the opportunity to make contributions to educational research and development.
