Description:
This is a worksheet I made for a Calculus 1 discussion session. (There is a link to the worksheet on the bottom of the page). Its goal is to familiarize students with the epsilon-delta definition of limits in calculus:
Epsilon-delta definition of the limit
The limit of a function f(x) at a point a is L if for every ϵ>0 there exists a δ>0 such that we have |f(x)-f(a)|< ϵ whenever |x-a|< δ.
It is probably not a surprise that most students are puzzled when they see this the first time. In fact, students usually struggle with making sense of this definition throughout the semester. My attempt to help students understand it was creating a game that can be played by two players.
Reflection:
All teams got to the point where they understood the rules of the game well enough so they could play according to the rules. This is important, because the way the players think when playing the game should be approximately the same way one should think about the epsilon-delta definition. Most students also figured out which player has the winning strategy, which means that they could apply the definition to these functions.
The real question is: did they understand that they were applying the definition? I do not think they really did, but I hope that they at least gained a very vague understanding of how the game relates to the epsilon-delta definition. It is very likely that most students remained confused about the definition, and they would not have been able to formulate it without memorizing it word by word.
Understanding the epsilon-delta definition takes lots of time and practice, so it is unlikely that there is a way to effectively explain it to calculus students in a short time. But I do think this game is an improvement over most attempts to explain the definition to students in lecture-style.
DELTA pillars:
learning communities and diversity: students with different thoughts on the definition and the game were able to share these with each other to obtain better understanding.
teaching-as-research: I didn't think about it that time, but next time I will give students follow-up problems in order to assess their understanding.