Limits
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- Last edited 86 days ago by Peter Riegler
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The notion 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 Hofberger and Riegler with the support of members of the BayZiel Decoding group.[1]
Contents
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:
- 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.
- 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.
- 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
Modelling the tasks
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Practice and Feedback
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Anticipate and lessen resistance
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Assessment of student mastery
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Sharing
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Researchers involved
Harald Hofberger
Available resources
These resources are available for registered users only:
See also
Notes
References
External Links
- ↑ Harald Hofberger, Peter Riegler (2022): Studentische Schwierigkeiten mit dem Grenzwertbegriff und mögliche Implikationen für die Lehre, in P. Riegler & Ch. Maas (Hrsg.): Scholarship of Teaching and Learning in der Mathematik. Mathematik-Lehre forschend betrachten, DUZ open