Proof-of-Concept Learning LLC

Founder: Edith Aurora Graf, Ph.D.

Guide mathematics learning by making connections between cognition, assessment, and artificial intelligence.

I offer research consulting services to teams interested in supporting mathematics learning. Whether your goal is to design an assessment, develop educational technology, or if you are considering whether and how to apply artificial intelligence to education, I can provide a research-based perspective. My work has spanned mathematics assessment design, automated item generation (AIG), learning progressions/trajectories, and more recently, applications of AI to mathematics learning and assessment.

About

Who I Am

I specialize in mathematics cognition, learning, and assessment design, primarily at the middle- and high-school levels. I founded Proof-of-Concept Learning LLC because I believe all learners can benefit from engaging with interesting mathematics. What do I mean by interesting mathematics? I do not necessarily mean the study of advanced topics, since there are many foundational ideas that can be explored in great depth. Interesting mathematics might involve application of learned concepts and procedures to a real-world problem, or it might entail using different representations and different strategies to solve the same problem, or even posing conjectures and proving them. And this is by no means an exhaustive list!

How can we support learners as they engage with interesting mathematics, which is by definition challenging? First, cultivating persistence is important⏤and as with many endeavors, community can play a central role here. Whether in a work, school, or online context, discussing mathematical ideas with others can inspire thought and provide constructive critique. But, there does need to be a willingness to contribute ideas, make mistakes, and learn from them. Educators often make the case that we need to create spaces for learners to share their ideas without fear of being wrong, and I agree, but I think this is most likely to be successful if the behavior is modeled. In other words, I think we all stand to benefit from putting our ideas out there. My favorite block eraser bears the quote fragment, "To err is human," from An Essay on Criticism by Alexander Pope. Inspired by my block eraser, I read the essay, which also includes this quote: "But you, with pleasure, own your errors past, and make each day a critique on the last." In addition to emphasizing the importance of learning from our own mistakes, the value of supportive critique from others is pointed out.

You may wonder where I stand on AI in education. I have mixed feelings⏤I think there is both great potential and great risk. While humans are expected to err (in mathematics and in life), we generally don't expect computers to miscalculate or provide false information. And I think it is this expectation, as much as anything else, that leads learners to trust the output when maybe they shouldn't. And that's risky, because it has the potential to solidify misconceptions. However, when guided and reviewed by humans, or co-produced with humans in an iterative way, AI-supported approaches stand to provide great gains in creative output and learning. For example, AI-supported AIG, when guided by a sound theoretical framework and carefully checked, has great value, particularly as we learn how to generate more open-ended tasks and score them using automated methods. I do think in the future the technology may do better in mathematical and scientific domains, though this improvement may require fundamental change rather than incremental modifications.

"Consider the role of a mentor, guide, or reviewer. If it is not to assist the recipient in doing their best work, then what is it?"

— Edith Aurora Graf

Curriculum Vitae

CVs & Documents

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Full Academic CV

Complete CV

The comprehensive record of my academic and professional history — including all publications, presentations, grants, teaching experience, service, and professional affiliations.

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Summary

Short CV

A concise two-page overview highlighting key positions, selected publications, and core areas of expertise — ideal for a quick professional introduction.

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Scholarship

Research

This page provides abstracts and summaries for featured publications and presentations. For a complete list of work, see my full CV.

Publications

Presentations

Graf, E. A., Shin, H. J., Yan, D., & Von Davier, A. A. (2026). Automated item generation: The promises and the challenges. In Reference module in social sciences. Elsevier. doi:10.1016/B978-0-443-26629-4.00236-7

This article briefly reviews traditional approaches to automated item generation (AIG) and discusses an iterative workflow for these approaches. It then examines the promises and challenges of traditional approaches — now rapidly being supplanted by AI-assisted methods — before introducing AIG terminology and discussing the promises and challenges of AI-assisted AIG. A revised workflow contrasts the two approaches, arguing that while the overall structure has not changed with mainstream AI adoption, how each step is performed has changed substantially. The article concludes with an illustrative example of AI-assisted AIG, a psychometric comparison of AI- vs. human-authored items, and a discussion of agentic AI for AIG.

Graf, E. A., van Rijn, P. W., & Eames, C. L. (2021). A cycle for validating a learning progression illustrated with an example from the concept of function. The Journal of Mathematical Behavior, 62, 100836. doi:10.1016/j.jmathb.2020.100836

A learning progression (or learning trajectory) describes the evolution of student thinking from early conceptions to the target understanding within a domain. As a complex theory of development, it requires both conceptual and empirical support. Building on earlier work proposing a four-step validation cycle — Theory Development, Examination of Empirical Recovery, Comparison to Competing Models, and Evaluation of Instructional Efficacy — this paper presents revisions suggested by a group of experts focused on learning sciences and classroom assessment. The adapted cycle is described and its first stages are illustrated through the validation of a learning progression for the concept of function.

Graf, E. A., & van Rijn, P. W. (2015). Learning progressions as a guide for design: Recommendations based on observations from a mathematics assessment. In Lane, S., Raymond, M. R., & Haladyna, T. M. (Eds.), Handbook of test development (2nd ed.). Routledge. taylorfrancis.com

A learning progression models the development over time of student understanding about particular content. Although the term was recently introduced in the context of science education, the idea of characterizing the development of student learning has a much longer history. Many learning progressions have roots in Piagetian theories of cognitive development, but beyond maturation they are assumed to reflect the influence of instruction — characterizing how student understanding develops in situ, in the classroom context. They are appealing because they may afford opportunities to report current levels of student understanding and suggest how to guide further learning.

Graf, E. A., & Fife, J. H. (2012). Difficulty modeling and automatic generation of quantitative items: Recent advances and possible next steps. In Gierl, M. J. & Haladyna, T. M. (Eds.), Automatic item generation (pp. 157–179). Routledge. taylorfrancis.com

This chapter discusses recent research and future directions for difficulty modeling and automatic item generation (AIG) of quantitative items. It defines background terminology, examines the relationship between cognitive and difficulty modeling, and reviews research identifying features of quantitative items that affect difficulty. The authors argue that an iterative approach to item design, evaluation, and revision can guide the transition from weak to strong cognitive theory, and that AIG can facilitate such an approach. The chapter explores difficulty modeling from the perspective of transfer of learning, reviews psychometric methods that support it, and discusses AIG advances that can assist difficulty modeling, instructional diagnosis, and automatic scoring.

Graf, E. A., van Rijn, P. W., Lizano, C. L., Andrews-Todd, J., Jiang, Y., & Lee, J. (2025, December). The role of rich mathematical conversations in learning trajectory research [Online keynote presentation]. Conference on Learning Trajectories and its Implications for Curriculum and Instructional Reform, Hangzhou Normal University, Hangzhou, China.

This keynote presentation is focused on whether chat-based discussion among students solving learning progression-based mathematics tasks can lead to more advanced performance and understanding.

Graf, E. A., Forsyth, C., Ruiz Diaz, S., Yan, D., & Jiang, Y. (2025, April). Mathematical explorations in an LLM. In E. A. Graf (Chair), Applications of generative AI to mathematics education: Opportunities and challenges. Annual meeting of the National Council on Measurement in Education (NCME), Denver, CO.

In this preliminary work, mathematics problems in arithmetic, algebra, and number theory were posed to an LLM which provided answers, solution steps, and confidence judgments.

Graf, E. A., Lizano, C. L., van Rijn, P. W., & Crombie, W. O. (2024, April). Designing learning progressions to advance equity in assessment and learning. Training session at the annual meeting of the National Council on Measurement in Education, Philadelphia, PA.

This training session spans lessons learned over almost 10 years of research on learning progressions, and covers learning progression design, task design, rubric design, empirical recovery, and using learning progressions as a benchmark to measure growth as a result of chat-based collaboration.

Graf, E. A., & van Rijn, P. W. (2019, April). Cycle for validating a learning progression. In L. Ketterlin Geller (Chair), Validating theories of learning for classroom assessment design: Sources of evidence. Annual meeting of the American Educational Research Association, Toronto, ON.

This presentation presents a draft cycle for validating a learning progression. Steps evaluating both empirical recovery and practical utility are included.

What I Offer

Services

I work with researchers, psychometricians, software engineers, and educators to support learning and assessment in mathematics.

Guidance and Review

If you are working on a project or developing an educational technology that would benefit from a research perspective, I can help. I provide guidance and review and can serve on advisory boards or review materials developed for mathematics learning or assessment.

Preparation of Reports

Whether you are in the planning stages or analyzing results, I have experience preparing both literature reviews and technical reports, and bring perspectives from cognitive science, mathematics education, assessment, and measurement to both.

Speaking Engagements

I am available to give keynote addresses or to speak on expert panels, and always look forward to exchanging ideas on the future of mathematics learning, assessment, measurement, and technology, especially in the age of AI.

Interested in working together?

Get in Touch

Contact

✉️

Email ea.graf@proofofconceptlearning.com

🔗

LinkedIn linkedin.com/in/edithauroragraf