The manners in which the history of mathematics may be used in the teaching and learning of mathematics can be classified into three major categories of approaches.
Within each of these categories history may be included in different grain sizes, i.e.
the use may be scaled from very small to very comprehensive. I shall refer to these categories as theilluminationapproaches, themodulesapproaches, and thehistory-based approaches.
The Illumination Approaches
In the illumination approaches the teaching and learning of mathematics, whether it is the actual classroom teaching or the textbooks used, is supplemented by historical information. As indicated above, these supplements may be of varying size and scope.
The smaller of these are what Tzanakis and Arcavi (2000, pp. 208, 214) refer to as
“isolated factual information” or “historical snippets” which may cover names, dates, famous works and events, time charts, biographies, famous problems and questions (e.g.
Swetz, 1995; Swetz, 2000), attribution of priority, facsimiles and so forth. Also the telling of anecdotes and stories belongs to this category (e.g. Siu, 2000a). One way to think of these smaller supplements in the illumination approaches is to think of them as
‘spices’ added to the mathematics education casserole.
At the other end of the scale of the illumination approaches we find what might be referred to as “historical epilogues” (or prologues). This term is inspired by Lindstrøm (1995), who at the end of every chapter in his calculus textbook has an historical epilogue:
a section where the history of the mathematics presented in the chapter is outlined along with names, dates, motivating problems, references to original works, anecdotes, discussions of attributions of priority as well as the developmental progress. If original sources are to be used within the illumination approaches, the historical epilogues would be the place to do this, although it would probably only be in the form of small extracts.
The Modules Approaches
The modules approaches are instruction units devoted to history and quite often they are based on cases. The term “modules” is taken from Katz and Michalowicz (2004).
Just as in the previous category of hows, the modules approaches may vary in both size and scope. The smallest of these would be what Tzanakis and Arcavi (2000, pp.
217-219) refer to as “historical packages” which are collections of “materials narrowly focused on a small topic, with strong ties to the curriculum, suitable for two or three class periods, ready for use by teachers in their classroom”.
2.4 Three Categories of Approaches 27
In the middle of the scale we find modules of perhaps ten to twenty class periods.
Such modules need not be tied to the mathematical topics in the curriculum, due to which they also provide the opportunity to study branches of mathematics that are not usually part of the curriculum at a given school level. The ways to implement both the historical packages and the larger modules are numerous. They may be introduced through textbook studies, through readings of original sources, or through student projects. Other ways may be through historical plays, the Internet, worksheets, historical problems, mechanical instruments, etc. (Fauvel and van Maanen, 2000, pp.
214-232).
At the upper end of the scale we find the full courses (or books) on the history of mathematics within a mathematics program. These may include an account of historical data, a history of conceptual developments, or something in between (Tzanakis and Arcavi, 2000, p. 208). Such courses can rely on original or secondary sources (or both) depending on the level of history studies intended. Of course, these approaches can also be implemented in many other ways than just through an actual course or reading of a book. One example could be that of extensive student research projects as described in Fauvel and van Maanen (2000, pp. 215-216) and practiced at Roskilde University.
The History-Based Approaches
The last category of approaches covers the ones directly inspired by or based on the development and history of mathematics. Unlike the modules approaches, these approaches do not deal with the study of history of mathematics in a direct manner, but rather in an indirect one. The historical development is not necessarily discussed in the open. On the other hand, it often sets the agenda for the order and way in which the mathematical topics are presented. Turning to the number sets again for a concrete example, the evolution of these would mean that the natural numbers (N) would be the first taught, then the positive rationals (Q+) and some of the positive irrationals (R+) before turning to zero and the negatives (Z), the remaining reals (R), and finally the complex numbers (C). In this respect, the history becomes a fully integrated part of the approaches themselves – one might think of them as ‘historical approaches’. A frequently mentioned and debated example of this is the so-called genetic approach (principle), which I shall return to.
This completes the presentation of the three categories of how to use history in mathematics education. In the following I shall turn to the ICMI Study’s characterization of the hows.
Categorizing the ICMI Hows
The three ways in which history may be used in mathematics teaching listed by the ICMI Study are: (a) learninghistory, by the provision of direct historical information, (b) learningmathematical topics, by following a teaching and learning approach inspired by history, and (c) developingdeeper awareness, both of mathematics itself and of the social and cultural contexts in which mathematics has been done (Tzanakis and Arcavi, 2000, p. 208). In the following I shall recapitulate each of the three ways and discuss them according to the previously proposed categories of hows.
The direct historical information way in the ICMI Study consists of two different cases; (1) the isolated factual information and (2) the full courses or books on the history
28 The ‘Whys’ and ‘Hows’
of mathematics. The first of these is, as indicated earlier, similar to the illumination approaches in the lower end of the scale. The second is the upper end of the scale of the modules approaches. Generally Tzanakis and Arcavi (2000, p. 208) describe the emphasis of the direct historical information way as being “more on resourcing history than on learning mathematics”.
About the second way of using history, Tzanakis and Arcavi (2000, p. 208) state that
“this is essentially what may be called the genetic approach to teaching and learning”, which places it in the category of history-based approaches. As mentioned earlier, the genetic approach is a frequently debated way of involving history and for this reason I shall elaborate a bit on the topic. The word genetic comes from the Greek wordgenesis, which in English translates to either creation or development. Actually there are several variations of the genetic approach or thegenetic principle as it is also called where the word ‘principle’ refers to a principle of education (Mosvold, 2001, p. 31). Schubring (1978), in his extensive study hereof, distinguishes between two genetic principles: (1) thehistorical-genetic principle, which aims at leading students from basic to complex knowledge in the same way that mankind has progressed in the history of mathematics, and (2) thepsychological-genetic principle, which is based on the idea to let the students rediscover or reinvent mathematics by using their own talent and experiences from the surrounding environment. As examples of each of these I shall briefly present the ideas of Toeplitz’genetic method and Freudenthal’sguided reinvention.
Otto Toeplitz named and addressed the genetic method in a lecture dated 1926 (Toeplitz, 1927). Burn (1999, p. 8) explains (or interprets) Toeplitz’ method by saying that “the question which Toeplitz was addressing was the question of how to remain rigorous in one’s mathematical exposition and teaching structure while at the same time unpacking a deductive presentation far enough to let a learner meet the ideas in a developmental sequence and not just a logical sequence.” Tzanakis (2000, p. 112) describes the genetic method as a method in which there is no uniquely specified way of presenting a given subject, which means that the approach is not a method in the strict sense of an algorithm, but rather a general attitude towards the presentation of a scientific subject. In such a presentation the motivation behind introducing new concepts, theories, or key ideas of proofs is based on the evolution of the subject. Problems and questions which were stimulating for the historical evolution are then reconstructed in a modern context and notation, so that they become more accessible to the students. For samples on the genetic method put to practice see e.g. Toeplitz (1949), Toeplitz (1963), Edwards (1977), Stillwell (1989), Tzanakis (2000), Burn (1999), and Burn (2005). Worth mentioning is that Toeplitz in his 1926-lecture distinguished between two different types of the genetic method, thedirect and theindirect:
[A]ll these requisites [...] must at some time have been objects of a thrilling investigation, an agitating act, in fact at the time when they were created.
If one were to go back to the roots of the concepts the dust of times [...]
would fall from them and they would again appear to us as living creatures.
And from then on there would have been offered a double road into practice:
Either one could directly present the students with the discovery in all of its drama and in this way let the problems, the questions, and the facts rise in front of their eyes – and this I shall call thedirect genetic method – or one could by oneself learn such an historical analysis, what the actual meaning and the real core in every concept is, and from there be able to
2.4 Three Categories of Approaches 29
draw conclusions for the teaching of this concept which as such is no longer related to history – theindirect genetic method. (Toeplitz, 1927, pp. 92-93, my own translation from German)
Toeplitz was not the first – nor the last – mathematician to believe that history could be a guide for teaching and learning mathematics. Prominent mathematicians as, for instance, Poincaré (1899), Klein (1908), Polya (1962), and Weil (1978) have expressed their positive beliefs regarding this matter. Also Hans Freudenthal was inspired by the evolution of mathematics, although putting it to a slightly different use than Toeplitz.
Freudenthal (1991, p. 48) says: “Children should repeat the learning process of mankind, not as it factually took place but rather as it would have done if people in the past had known a bit more of what we know now.” Freudenthal dislikes the term genetic method since “it does not allude to any activity of the learner”, instead he stresses that the students should be guided to “an activity” by their teacher, and that “the learner should reinvent mathematising, not mathematics” (Freudenthal, 1991, pp. 46, 49). ‘Mathematising’ may be described very shortly as “the entire organising activity of the mathematician” (Freudenthal, 1991, p. 31). The question as to what the learner is to mathematise is also answered by Freudenthal, his answer being “reality”, by which he means the “learner’s own reality as laid open to him by his guide” (Freudenthal, 1991, p. 50). (See also e.g. Freudenthal, 1973; Gravemeijer and Doorman, 1999; van Amerom, 2002; Fung, 2004.)
The third way of the ICMI Study, the ‘mathematical awareness’ way, is divided into the study of the ‘intrinsic nature’ and the study of the ‘extrinsic nature’ of mathematical activity. The intrinsic nature involves important aspects of doing mathematics such as the part of general conceptual frameworks and of associated motivations; the evolving nature of mathematics exemplified by notation, terminology, methods, representations as well as notions of proof, rigor, and evidence; and the role of paradoxes, contradictions, intuitions, and the motivation for generalizing, abstracting, and formalizing (Tzanakis and Arcavi, 2000, pp. 211-212). The extrinsic nature of mathematical activity deals with the fact that mathematics is often regarded as a discipline which is largely disconnected from social and cultural concerns and influences. Tzanakis and Arcavi (2000, p. 212) state that the history of mathematics may illustrate the superficiality of such a view in the way that it can: show how a vast number of mathematical aspects are related to other sciences and humanities; show how the social and cultural milieu may influence the development, or the delay of the development of certain mathematical domains; show that mathematics is an integral part of the cultural heritage and practices of different civilizations, nations, or ethnic groups; and show that currents in mathematics education through time have reflected trends and concerns in culture and society.
In the context of the three categories of approaches, the modules approaches seem the most suitable for illustrating the intrinsic and extrinsic natures of mathematics.
However, in my view the mathematical awareness way is not as clear an approach as the two other ICMI ways. In fact, it seems more concerned with the whys than with the hows. A point which will be subject to discussion in the following section.
30 The ‘Whys’ and ‘Hows’