The Pancake Story and the Epsilon–Delta Definition

The complexity in understanding the ϵ–δ definition has motivated research into the teaching and learning of the topic. In this paper I share my design of an instructional analogy called the Pancake Story and four different questions to explore the logical relationship between ϵ and δ that structures the definition. I designed the story from an anti-deficit perspective on students’ mathematical thinking. I leverage students’ intuitions about quality control, and explicitly build on students’ prior knowledge that I gathered from student interviews. I illustrate the utility of the story in supporting students’ sense-making of the formal definition by sharing changes in students’ claims and justifications about the relationship between ϵ and δ. Students' justifications also provide examples of the kind of thinking an instructor might expect to see in the classroom when using the story. I end the paper with some classroom implications and a teaching episode illustrating how students might use the story to make sense of limit more broadly.