Proof blocks: autogradable scaffolding activities for learning to write proofs

This dissertation introduces \tool, a tool which enables students to construct mathematical proofs by dragging and dropping prewritten proof lines into the correct order instead of needing to write them from scratch. First we present implementation details of the tool, as well as a rich reflection on our experiences using the tool in courses with hundreds of students. When writing a problem, the instructor specifies the dependency graph of the lines of the proof, so that any correct arrangement of the lines can receive full credit. We develop a novel algorithm which enables assigning students' partial credit on \tool{} problems based on the number of edits that their submission is from a correct solution, and give benchmarking results showing that this performance gives significant performance gains over the baseline algorithm, enabling large scale classroom deployment. Finally, we will present evaluations of \tool{} both in learning and assessment contexts. For assessment, we provide statistical evidence that \tool{} are easier than written proofs, which are typically very difficult. We also show that \tool{} problems provide about as much information about student knowledge as written proofs. Survey results show that students believe that the Proof Blocks user interface is easy to use, and that the questions accurately represent their ability to write proofs. Through our experiments targeted at measuring student learning, we provide evidence about various aspects of the process of students learning proof by induction. We give evidence that using Proof Blocks alone is not as effective for learning as using Proof Blocks after reading educational materials first. We also show that students who read a book chapter and complete a Proof Blocks activity perform marginally better than students who only read the book chapter, but it is not clear if the source of this improvement is the Proof Blocks or just exposure to more examples. Together with the evidence from out study showing that students who write proofs from scratch also learn no more than students just reading the book chapter, we conclude that though students can have clear learning gains very quickly when they have low knowledge of proof by induction, subsequent incremental learning gains are very difficult, and it is not well understood how to design a learning activity to help students experience these learning gains in a short period of time.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Computer Science

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2023

Author dc:creator

• Poulsen, Seth

Contributors dc:contributor

• Herman, Geoffrey L
• West, Matthew
• Zilles, Craig
• Viswanathan, Mahesh
• Koedinger, Kenneth

• Educational Software Mathematical Proofs Graph Algorithms

Statement dc:rights

• Copyright 2023 Seth Poulsen

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/121414

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F121414