Faculty Advisor or Committee Member

Erin Ottmar, Advisor

Faculty Advisor or Committee Member

Ivon Arroyo, Reader

Faculty Advisor or Committee Member

Neil Heffernan, Department Head




This thesis grounds the design of learning technologies in cognitive learning theory to explore deeper formative measurement of the learning process. This work implements Graspable Math (GM; Ottmar, Landy, Weitnauer, Goldstone, 2015), a dynamic learning technology that has been designed using perceptual-motor learning theory, which has been shown to have a strong connection to mathematical reasoning (Kirshner, 1989; Kellman, Massey, & Son, 2010; Goldstone, Landy, & Son, 2010). With this dynamic mathematics learning technology, we can measure the algebraic problem solving process in ways that are not possible with pencil and paper or other more traditional learning technologies. By collecting this data, this research will explore how to move beyond traditional correctness-based assessment and design more formative measures of the learning process. This work provides a rich perspective on the evolution of research on mathematical proficiency, how this research is applied in practice, and an in-depth example of how one technology-based learning environment has been developed to measure mathematical proficiency. This work has three main objectives: 1) develop a theoretical framework to assess mathematical proficiency within GM, 2) explore GM-based measures of mathematical proficiency across K-12 populations, and 3) design GM-based tools that are grounded in theory on mathematical proficiency. This work first presents a conceptual model that maps student behavior data measured through GM onto the five theoretical strands of mathematical proficiency as defined by the National Research Council’s 2001 publication, Adding it Up. The first study reveals underlying constructs in Elementary student data and suggests there is an added benefit of including these formative measures within predictive models. Above and beyond background characteristics and summative measures of knowledge, formative measures of the learning process revealed subtle interactions based on student behaviors and prior knowledge. These constructs also show potential in mapping onto certain strands of mathematical proficiency. The second study compares underlying constructs within Elementary data to High School data using exploratory factor analysis and finds similar factors across both populations. These results suggest that certain constructs may underlie different age groups and have the potential to be used as measures of mathematical proficiency. While the first two sections describe the definition and measurement of mathematical proficiency within GM, the final section explores the implementation of these measures within the design process of new GM-based activities for students and tools for teachers. Ultimately, the goal of this work is to serve as an example method for other researchers, educators, and designers to move beyond summative measures of assessment and enhance the formative assessment capabilities of learning technologies by grounding measures in theories of learning.


Worcester Polytechnic Institute

Degree Name



Learning Sciences and Technologies

Project Type


Date Accepted



Restricted-WPI community only


formative assessment, learning technologies, mathematics education

Available for download on Friday, April 22, 2022