Characterizing and assessing number sense in third through eighth grade students

One goal of mathematics curricular and instructional reforms in the United States is to help students build “mature number sense,” which involves making sense of numbers and operations, using reasoning to notice patterns, and flexibly selecting the most effective and efficient problem-solving strategies. In support of this goal, mathematics educators have developed a variety of instructional practices designed to move students beyond seeing mathematics as a set of disconnected procedures and facts to appreciating it as a coherent set of ideas and tools.

These practices have been growing in popularity across mathematics classrooms, but it is unclear if students’ mature number sense is improving. One reason for slow progress is that the field currently lacks a rigorously validated measure of mature number sense that is both widely accepted and easy to use. Moreover, even though mathematics educators generally agree on the importance of mature number sense, there is little evidence demonstrating that it is a measurably distinct characteristic of mathematical cognition.

Lack of such knowledge is a critical problem because, without it, researchers cannot explain, predict, and study mature number sense, and teachers are left to rely on their own intuitions about whether and how to focus on it in their classrooms. In this project, researchers will advance fundamental knowledge of mathematical cognition through the development of innovative metrics and methods to assess mature number sense with valid and reliable scores for use with students in grades 3-8.

The central hypothesis is that mature number sense is a measurable characteristic of mathematical cognition, separable from both domain-specific knowledge (e.g., arithmetic fluency, fraction knowledge) and domain-general constructs (e.g., creativity, cognitive flexibility), that predicts math-related beliefs and achievement. The research team will work with a diverse sample of students in grades 3-8 to refine, extend, and validate a brief assessment of mature number sense.

They will start with a measure that they iteratively developed and validated in a large pilot study with middle and high school students, and they will optimize it as the “middle school form” (for grades 6-8). They will collect additional evidence of validity for this form by comparing it to other, more lengthy measures of mature number sense and by conducting a longitudinal study to examine if scores on the measure in fall predict grade-level mathematics learning and math-related beliefs in spring.

Next, they will extend the measure downward to include upper elementary grades 3-5. Items will be iteratively developed and examined for construct relevance using expert ratings, classical and IRT item statistics, and retrospective think-aloud strategy reports of students during problem solving. The researchers will examine if the factor structure of the elementary form is the same as that found with the middle school form, testing measurement invariance across the two grade-level groupings.

They also will develop a technical manual for researchers and teachers. By the end of the project, there will be a practical, validated assessment of mature number sense for students in grades 3-8 that is easy to administer in schools, after-school programs, research laboratories, and homes. The target audiences for this measure include basic researchers interested in studying mathematical cognition, intervention researchers interested in studying the effects of innovative instructional practices on students’ number sense, and teachers who wish to monitor their students’ progress toward the “processes and proficiencies” with longstanding importance in mathematics education.

This project is funded by the EHR Core Research (ECR) program, which supports work that advances fundamental research on STEM learning and learning environments, broadening participation in STEM, and STEM workforce development.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.