I recently read a paper by Bent Flyvbjerg in which he discusses and justifies the usefulness of case study/ case methodology in social science. I am wondering whether and how is assumptions can be applied to mathematics. I am not summarizing the paper and am not providing its application to math; but simply sketch my thoughts.
Old-fashioned definition: “Case Study. The detailed examination of a single example of a class of phenomena, a case study cannot provide reliable information about the broader class, but it may be useful in the preliminary stages of an investigation since it provides hypotheses, which may be tested systematically with a larger number of cases. (Abercrombie, Hill, & Turner, 1984, p. 34)”
Cases in mathematics:
Claim: “General, theoretical (context-independent) knowledge is not more valuable than concrete, practical (context-dependent) knowledge”.
Bent Flyvbjerg discusses the role of cases and theory in human learning and emphasises that case study (e.g. carefully chosen experiments, experiences, cases) produces the type of (concrete, practical) context-dependent knowledge that research on learning shows to be necessary to allow people to develop from rule-based beginners to virtuoso experts. In contrast, textbooks provide (general, theoretical) context-independent theories (focus on universals) and bring the students just to a beginner’s level.
(This supports our work towards context-dependent mathematical learning objects; although we have so far defined context by the logical/ narrative/ social relation of mathematical knowledge — but maybe by”social” we actually mean concrete/ individualized/ practical aspects)
We make use of cases in mathematics (and maybe should adapt mathematical learning respectively rather than solely presenting abstract/ universal theories): William Farmer one’s told me about a colleague who had to give a lecture on algebra unprepared. Consequently, when presenting a proof, he had to continuously revise his steps, wipe the board, and start over again. This incremental approach actually helped the students to really understand how mathematics is practised, i.e. that the universal and abstract theories are not invented from scratch but have to be iteratively developed based on many cases — examples, counterexamples, etc. And also Cristian Calude has recently pointed out once more that the way of doing math cannot at all be compared to the way of presenting the final results!
Generalization and Force of Example
Claim: “One can often generalize on the basis of a single case (i.e. does not necessarily need statistics/ quantitative studies), and the case study may be central to scientific development via generalization as supplement or alternative to other methods. But formal generalization is overvalued as a source of scientific development, whereas the force of example is underestimated.”
In “Mathematical Naturalism” Philip Kitcher illustrates that the development of mathematics can be seen as a stepwise process from
So what is the value of generalization and examples in mathematics? Although finding mathematical results is based on a case-based generalization process, mathematicians are particular interested in discussing the final general, abstract, and universal structures rather than looking at the concrete objects. However, particularly in teaching examples and exercises are essential, they help teachers to guide students from theory to theory (Michael Kohlhase says examples are theory morphism, i.e. “structure-preserving mappings between two mathematical structures.”).
Claim: “The case study is useful for both generating and testing of hypotheses but is not limited to these research activities alone.”
Bent Flyvbjerg cites Eckstein, John Walton, Karl Popper, who underline that case study is a mean for testing theories (Eckstein), produces the best theories (Walton), and is ideal for generalizing using falsification (critical reflexivity) (Popper). In the paper, theory is defined in two ways …
“[..] theory in its “hard” sense, that is, comprising explanation and prediction [..] theory in the “soft” sense, that is, testing propositions or hypotheses” [..]
See wikipedia definition of theory, mathematical theory and list of first-order theories. See also “A shorter model theory” by Wilfrid Hodges (1997).
“In mathematical logic, a theory is a set of sentences in a formal language. One way to specify a theory is to define a set of axioms. The theory can be taken to include just those axioms, or their logical or provable consequences, as desired.”
“A syntactically consistent theory is a theory from which not every sentence in the underlying language can be proved.”
“A satisfiable theory is a theory that has a model. This means there is a structure M that satisfies every sentence in the theory. “
Falsification is widely used to test and, if needed, refute mathematical theories: “If just one observation does not fit with the proposition, it is considered not valid generally and must therefore be either revised or rejected.” (cf. Flyvbjerg).
Claim: “The generalizability of case studies can be increased by the strategic selection of cases. When the objective is to achieve the greatest possible amount of information on a given problem or phenomenon, a representative case or a random sample may not be the most appropriate strategy. This is because the typical or average case is often not the richest in information. Atypical or extreme cases often reveal more information because they activate more actors and more basic mechanisms in the situation studied.”
This takes on the discussion by Kerber et. al., which have addressed the typicality of examples. However, in mathematics (in particular teaching) we also make us of atypical examples, counter examples, and near-miss examples. (Note, we might want to use our study of typical examples to eventually identify atypical examples. Moreover, we might want to apply the types of cases by Flyvbjerg to mathematical examples and exercises)
Flyvberg presents several strategies for selection different cases (see figure below): Among others …
- extreme cases: Getting a point across in an especially dramatic way, see e.g. Normen’s motivative example for management of change — Adriane 5
- critical cases: Require experience, looking for “most likely” and “least likely” cases, i.e. cases to either clearly confirm or irrefutably falsify propositions and hypothesis
- paradigmatic cases: Cases that highlight more general characteristics of the societies in questions; Kuhn showed that scientific paradigms cannot be expressed as rules and theories as there exists no predictive theories how predictive theory comes out; discovering paradigmatic cases requires intuition and taken-for-granted procedures
Strategies for the Selection of Samples and Cases
Do Case Studies Contain a Subjective Bias?
According to Flyvbjerg, it is falsification, not verification, that characterizes the case study. Moreover, the question of subjectivism and bias toward verification applies to all methods, not just to the case study.
This question also applies to mathematics. Rarely any human being is able to provide fully objective illustration, thus, also mathematical results have an individual touch and can include subjective, context-dependent parts influenced by the type of problem or personal views. In mathematics, single subjective cases do not lead to accepted and verified results. Instead we can observe a communal and peer-reviewed process within which a given proof is falsified and tested (see discussion with Cristian Calude).
Case studies often contain a substantial element of narrative. Good narratives typically approach the complexities and contradictions of real life. Accordingly, such narratives may be difficult or impossible to summarize into neat scientific formulae, general propositions, and theories.
I recommend to read pages 238-239, as Flyvberg’s illustration open a very new perspective on our work on mathematical examples (see below): “The opposite of summing up and “closing” a case study is to keep it open. [..] I tell the story in its diversity [..] I avoid linking the case with the theories [..] Instead, I relate the case to broader philosophical positions that cut across specializations. In this way, I try to leave scope for readers of different backgrounds to make different interpretations and draw diverse conclusions regarding the question of what the case is a case of. [..] Case study is a “virtual reality” [..] Students can safely be let loose in this kind of reality, which provides a useful training ground with insights into real-life practices that academic teaching often does not provide. [..]“ … maybe mathematical examples are more than theory morphisms (i.e. clear mappings between theories), maybe that have to leave room for imagination and interpretation. However, math is different to social science such as political studies or philosophy and mathematical knowledge is of a very different kind then social-science knowledge: it is abstract, well-structured, extraordinary interlinked, has a precise syntax and semantics. Well, but these characteristics make access to mathematical knowledge also so hard for novice: Maybe before being abstracted into clear structures in the human mind it has to be of a different form to be more easily understood by novice.
“One might say that the rule formulation that takes place when researchers summarize their work into theories is characteristic of the culture of research, of researchers, and of theoretical activity, but [..] something essential may be lost by this summarizing [..] “
This is indeed a problem in mathematics: When getting reading to write down their mathematical concepts, mathematicians do not fully articulate their thoughts but leave out parts that well-experienced mathematicians can fill in. However, students are lacking the observations/ experiences/ examples/ cases that experts have acquired through-out the years and have difficulties in understanding theoretic and abstract writings.
Further Reading
- Robert Stake’s (1995): The Art of Case Study Research
- Charles Ragin and Howard Becker’s (1992): What Is a Case?
- Scenario-based techniques