Discussion at the JEM Symposium on “math tutoring: tools and feedback” in Heerlen, Netherlands.
I have been reading a two papers by Manfred Kerber, Erica Melis, and Jörg Siekman on the typicality of examples which raised my interest and which I would like to further elaborate on. See below my notes on both papers. During the JEM Symposium I tried to get other perspectives on this topic:
Johan Jeuring mentioned that their worked out examples (in their exercise web service) are actually (hopefully/ ideally/ potentially) typical examples to illustrate a problem. The study of Gemma Corbalan showed that this kind of feedback was more helpful then only providing feedback to a concrete solution.
However, Johan was wondering whether the context-aware handling of typical exercises would provide a significant variance, that is, if the users’ subject views on what is typical really varies enough that a context-aware exercise framework and respective markup makes sense (in particular for mathematics; analogously to our notation framework). However, he also mentioned that in particular in computer science there are a lot of changes going on e.g. wrt. to which programming language is used for illustrating specific programming concepts.
George Goguadze said that he would not base the selection of examples on the typicality. First of all, teachers have a subjective view on which examples/ exercise are typical for a particular audience. And secondly, students have a subjective understanding what is typical (e.g. base don their prior education). But his comment actually illustrates that their are different understandings of typicality, so explicating these might potentially be helpful. For example, teachers can provide typical examples in their lecture, but receive feedback on the student’s view on these. Based on the exercise material (and a pool of further exercises) of the teacher, a tools could select the typical exercises from a student’s perspective (based on the interrelation and annotation of exercises).
Notes on the two papers by Manfred Kerber etAl:
- Typical examples are examples that are representative for a particular situation or concept; they play an important role in human knowledge representation and reasoning (see also reasoning by analogy, case based reasoning)
- Typical examples are easy to remember and are sufficient to catch important aspects of the general case.
- We can distinguish different types of typical examples (depending of the type of concepts or situation they help to explain): procedural examples (e.g. illustrating an induction proof), typical examples for concepts (e.g. a violin, a chair), typical examples of a situation (e.g. opening a door); typical examples for a proof or a theorem, typical examples for a plan
- Typicality of examples is a well-known research area in empirical psychology (see references below)
- Representation of examples by a data structure like a semantic net, frame structure, or within neural net.
- Typicality ratings distinguishes typical examples from atypical examples: direct ratings (Is this a typical example?); reaction time (Answer true or false: [example] is a [category-name]); reproduction of examples (list or draw an example of a category member); asymmetry of similarity rating (less typical examples are often considered to be more similar to typical example than the converse); generalization (Humans generalize more likely from typical examples than from arbitrary examples)
- CoPs and typicality: Can typicality be influenced by context parameters? Do interpretation (preference) vary among different communities of practice? Can context-aware handling of typical example help to find and select typical example for a CoP or an individual? Do the typical examples (identified for a specific context/ CoP) improve understanding (in contrast to the default typical example)?
- CoPs and typicality: Are some notations typical for a CoP (e.g. if their are most frequently used by this CoP or preferred by most members of the Cop)?
Further References:
- Afterthoughts on analogical representation (1975) by A. Sloman
- Mathematics and Plausible Reasoning (1954) by Pólya (BOOK)
- G. Lakoff. Women, Fire, and Dangerous Things What Categories Reveal about the Mind. The University of Chicago Press, 1987. (BOOK); see slides
- C. Mervis and E. Rosch. Categorization of natural objects. Annual Review of Psychology, 32:89 115, 1981.
- L. Rips. Inductive judgements about natural categories. Journal of Verbal Learning and Verbal Behavior, 14:665 681, 1975.
- How to Solve It: A New Aspect of Mathematical Method by Pólya; 1973 (BOOK)
[...] takes on the discussion by Kerber et. al., which have addressed the typicality of examples. However, in mathematics (in particular teaching) [...]