Limits

Limit is a core concept in mathematics. In teaching calculus it plays the role of a foundational concepts which further concepts build on.

Decoding work on teaching limits has been done by Harald Hofberger and Peter Riegler with the support of members of the BayZiel Decoding group.

Decoding work done

Decoding work originated from a Decoding interview in which two physicists by training helped a mathematician to uncover tacit mental moves when dealing with limits.

Identification of bottleneck

An analysis of the transcribed interview revealed 13 bottlenecks related to limits, which had been addressed at least shortly in the course of the interview. These bottlenecks are not strictly independent from each other. Some of them can also be generalized beyond the context of limits, such as

Students to not recognize what they need to do in order to solve a (limit-)problem.

The major part of the Decoding interview, however, focused on bottlenecks related to the definition of limit and related to dealing with mathematical definitions in general. In hindsight, these bottlenecks could be described collectively via

Students view the mathematical definition of limit as unduly complex.

In their published analysis, Hofberger and Riegler focus on the following aspects of perceived complexity:

1. Students often fail to realize that the mathematical definition of limit (e.g. of a sequence) is nonconstructive, i.e. it does not allow to calculate the limit. It only allows to verify or falsify that a given test value in fact equals the actual value of the limit. In contrast, a constructive definition would allow to determine the actual value of the limit.
2. Student often fail to align or contrast the definition of limit with their idiosyncratic, intuitive notion of approaching a limit. They tend to view the definition as an over-complex statement of the obvious.
3. Students often fail in grasping the meaning of the various symbols used in the definition of limit. This particularly applies to the symbol epsilon and its role as an upper bound distance measure.

One could phrase this cluster of bottlenecks in terms of the following intended learning outcome:

Students align their intuitive notions of approaching with the formal notion of limit in mathematics. They can explain how the seeming complexity of this definition is needed for an unambiguous and contradiction-free formalization of the limit concept.

Description of mental tasks needed to overcome the bottleneck

One of the central issues in the Decoding interview has been the notion of "approaching", often colloquially used in the context of limits (such as in "the sequence approaches the limit 42"). Used in the mathematical context of limits "approaching" has a very precise meaning, eventually defined via the concept of limit, which differs slightly, but significantly from its colloquial meaning. The mathematical notion of "approaching" does not rule out that distance increases intermittently, while in its colloquial usage "approaching" is usually associated with continuously getting closer. The mathematical definition of limits takes great pains to include the case of an intermittent increase in distance.

Take the following two statements as examples:

(1) Anna is approaching her 50th birthday.

(2) While square dancing Anna is approaching Ben.

Statement (1) does not allow the possibility that the (temporal) distance between Anna and her Birthday increases. While Statement (2) does allow for the possibility that the distance between Anna and Ben increases temporarily, the mental image of most readers presumably excludes this possibility.

Modelling the tasks

The authors suggest an activity to contrast the colloquial and mathematical meanings of "approaching". With respect to the bottlenecks identified in the post-hoc analysis of the interview, they also suggest activities, for instance to help students recognize the difference between constructive and nonconstructive definitions and to recognize the nonconstructive nature in the case of limit.

Practice and Feedback

The authors suggest a diagnostic question to be used for eliciting students' understanding of "approaching."

Sharing

The results of this investigation have been published in a book chapter. In more details the findings are described in an unpublished manuscript.

Researchers involved

Harald Hofberger, Peter Riegler

Available resources

These resources are available for registered users only:

• video recordings of the interview
• corresponding transcripts: part1, part2, part3, part4, part5