Students struggle with writing a sum in sigma notation once the sum has gotten slightly more complicated than the original examples that they have worked with. In addition for the area formula the first term of the sum does not follow the pattern so it must be kept outside the sigma form. Once they have figured this out though the main confusion comes from the very unnatural conclusion in their mind that an infinite process can yield a finite number. Although when they consider the size of the snowflake physically they can see that the area is not going to be infinite, they still have a hard time connecting that conclusion to the series form of the area. Then they are even more confused by the notion that the perimeter could indeed be infinite despite the fact that it is contained inside a finite space. I think these are great ideas for students to have to confront immediately in this class as there are a lot of concepts both in series and improper integrals that deal with the notion of infinity and it is an area that students have many misconceptions. I like to have a quick discussion after students have turned in the synthesis questions to highlight some of these findings to the whole class as sometimes it seems that the students can have a reasonably good understanding of the concepts worked on but still might not have put them together to think about these bigger ideas. |