38 The ‘Whys’ and ‘Hows’
given by Karl.-H. Schwager in 1956 points out that history is always made by adults and that it therefore may be difficult for a child to fully understand the problems related to history (quoted in Schubring, 1978, p. 189). Also Lietzmann, Klafki, and Freudenthal have, according to Schubring (1978, pp. 189-197), criticized the historical-genetic principle. They were of the opinion that history should not make up the base for the education, but that the students themselves should be the starting point. Both the pedagogic and the didactic perspective should be the students’ own experiences from the environment surrounding them every day. Recent criticism of Toeplitz’ genetic method does not seem easy to find (Mosvold, 2001, p. 32), however one discussion of the indirect genetic method by Kronfellner may be found in Fauvel and van Maanen (2000, pp. 71-73).
The sixteenth and last objection in Siu’s list is in fact more of a critical question than an objection to the use of history, since it asks: “Is there any empirical evidence that students learn better when history of mathematics is made use of in the classroom?”
(Siu, 2007, p. 269). With this ‘objection’ in mind, let us take a look at the list of empirical research studies which I have come across in my survey of the literature.
2.8 A Survey of Empirical Studies on Using History 39
Jahnke et al. (1996) Bartolini Bussi and Pergola (1996) Calinger (1996) Bero (1996)
Katz (2000) Bruckheimer and Arcavi (2000); Dorier (2000); Isaacs et al. (2000);
Winicki (2000)
Horng and Lin (2000) Hsiao and Chang (2000); Hsieh (2000); Hsieh and Hsieh (2000); Lakoma (2000); Lin (2000); Liu (2000); Ming (2000); Ohara (2000); Prabhu and Czarnocha (2000); Su (2000); Troy (2000); Tsukahara (2000); Winicki-Landman (2000)
MJRME Special Issue Barabash and Guberman-Glebov (2004); Fung (2004); Waldegg (2004);
Zormbala and Tzanakis (2004)
Furinghetti et al. (2007) Demattè (2007); Horng (2007); Isoda (2007); Liu (2007); Smestad (2007);
Su (2007); Tzanakis and Kourkoulos (2007); Winicki-Landman (2007) ESM (1998-2007) Philippou and Christou (1998); Radford (2000b); van Amerom (2003);
Durand-Guerrier and Arsac (2005); Bakker and Gravemeijer (2006);
Farmaki and Paschos (2007); Arcavi and Isoda (2007); Furinghetti (2007); Radford and Puig (2007); Thomaidis and Tzanakis (2007) Barbin et al. (2008) Dimitriadou (2008); Glaubitz (2008); González-Martín and Correia de
Sá (2008); Liu (2008); Morey (2008); Paschos and Farmaki (2008);
Thomaidis and Tzanakis (2008) TSG23, ICME11 Lawrence (2008)
Cantoral et al. (2008) Ceylan Alibeyoglu (2008); Gonulates (2008); Jankvist (2008c); Kourk-oulos and Tzanakis (2008); Nataraj and Thomas (2008); Peard (2008);
Reed (2008); Smestad (2008)
WG15, CERME6 Blanco and Giovart (2009); Jankvist (2009d); Kjeldsen (2009b); Lawrence (2009); Tardy and Durand-Guerrier (2009); Thomaidis and Tzanakis (2009)
Other samples McBride and Rollins (1977); Fraser and Koop (1978); Arcavi et al.
(1982); Arcavi et al. (1987); Harper (1987); Sfard (1995); Demattè and Furinghetti (1999); Lit et al. (2001); Jankvist (2008j); Jankvist (2008b);
Charalambous et al. (2009); Jankvist (2009b); Jankvist (2009c); Kjeldsen and Blomhøj (2009)
Ph.D. dissertations van Amerom (2002); Bakker (2004); van Gulik-Gulikers (2005); Goodwin (2007)
Table 2.2 A list of the 81 empirical studies found in the samples in table 2.1 as well as others, which I have come across in my survey. Remarks: Samples which was not found to include empirical studies are not listed in the table. The surveys of proceedings edited by Horng and Lin (2000), Barbin et al. (2008), and Cantoral et al. (2008) do not include the papers written in Taiwanese, French, and Spanish. Nor does the survey of contributions in Barbin et al. (2008) take into account abstracts, since these do not display any empirical data.
Overarching Studies
Fraser and Koop (1978); Smestad (2007); Smestad (2008); Gonulates (2008)
The studies concerning mostly the use of history as an affective and motivational tool are:
Studies Using History as an Affective and Motivational Tool
Charalambous et al. (2009); Hsiao and Chang (2000); Hsieh (2000); Hsieh and Hsieh (2000); Lawrence (2008); Lin (2000); Liu (2007); McBride and Rollins (1977); Philippou and Christou (1998); Troy (2000)
Studies focusing on history as a cognitive tool (e.g. epistemological obstacles) make up the largest group of empirical studies:
40 The ‘Whys’ and ‘Hows’
Studies Using History as a Cognitive Tool
Arcavi et al. (1982); Arcavi et al. (1987); Arcavi and Isoda (2007); Bakker (2004); Bakker and Grave-meijer (2006); Barabash and Guberman-Glebov (2004); Blanco and Giovart (2009); Bruckheimer and Arcavi (2000); Bartolini Bussi and Pergola (1996); Ceylan Alibeyoglu (2008); Dimitriadou (2008);
Dorier (2000); Fung (2004); Furinghetti (2007); Glaubitz (2008); González-Martín and Correia de Sá (2008); Goodwin (2007); van Gulik-Gulikers (2005); Horng (2007); Isoda (2007); Kjeldsen (2009b);
Kourkoulos and Tzanakis (2008); Lakoma (2000); Lawrence (2009); Lit et al. (2001); Liu (2000);
Liu (2008); Ming (2000); Morey (2008); Nataraj and Thomas (2008); Ohara (2000); Paschos and Farmaki (2008); Peard (2008); Prabhu and Czarnocha (2000); Radford (2000b); Reed (2008); Su (2000); Su (2007); Tardy and Durand-Guerrier (2009); Thomaidis and Tzanakis (2008); Thomaidis and Tzanakis (2009); Tsukahara (2000); van Amerom (2002); van Amerom (2003); Winicki (2000);
Winicki-Landman (2000); Winicki-Landman (2007)
Of studies somehow referring to the evolutionary arguments for using history, historical parallelism, etc., we find:
Studies on Historical Parallelism Etc.
Bero (1996); Durand-Guerrier and Arsac (2005); Farmaki and Paschos (2007); Harper (1987);
Radford and Puig (2007); Sfard (1995); Thomaidis and Tzanakis (2007); Tzanakis and Kourkoulos (2007); Waldegg (2004); Zormbala and Tzanakis (2004)
Of course, some studies recognize history as a cognitive tool and at the same time as a motivational or affective tool (or some other combination of the roles), but I have tried to classify them according to what appears to be their main objective.
It should be mentioned that some of the studies concerning history as a tool also acknowledge the role of history as a goal, although this is not their main objective (e.g.
Hsieh and Hsieh, 2000; Liu, 2000; Su, 2007). Other studies focus on history as a goal, but mention the possible side effects in terms of history as a tool (e.g. Kjeldsen and Blomhøj, 2009; Jankvist, 2009b). Of studies mainly concerned with using history as a goal we find:
Studies Using History as a Goal
Demattè (2007); Demattè and Furinghetti (1999); Isaacs et al. (2000); Kjeldsen and Blomhøj (2009) Jankvist (2008b); Jankvist (2008j); Jankvist (2008c); Jankvist (2009b); Jankvist (2009c); Jankvist (2009d)
Thus, only four studies in the above list, not counting my own, focus on the use of history as a goal.
Categorizing the above empirical studies was not always an easy task. Some of the classifications are results of interpreting what seemed to be the authors’ underlying motivation. In particular, this was the case when having to separate some of the tool studies into those using history as a motivational tool and those using it as a cognitive tool. This being said, it was fairly easy to separate the studies focusing mainly on history as a tool from those focusing mainly on history as a goal.
I shall not go into individual descriptions of the 81 studies in table 2.2, arguing for each of their classifications above, since this would be a rather lengthy process and more importantly, it would not necessarily add much to the research study of this dissertation, the reason being that it first and foremost is concerned with the use of history as a goal.
For the same reason I shall, however, consider the few empirical studies on history as a goal listed above.
2.8 A Survey of Empirical Studies on Using History 41
The Empirical Studies on History as a Goal
Demattè and Furinghetti (1999) discuss a questionnaire of 39 questions used to study 13 year old students’ beliefs about mathematics as a socio-cultural process. The questionnaire was given to a total of 288 students. Several of the 39 questions relate directly to matters of history as a goal as discussed in the sections above. An example of a finding related to these questions is that students on the one hand have doubts about the existence of mathematics independent of the Western culture, but on the other hand believe students all over the world to study the same mathematics (Demattè and Furinghetti, 1999). Based on the (history as a goal) questions, the authors conclude that the typical student has only primitive ideas on the birth and development of mathematics. The line of thought from this study is continued in Demattè (2007), a paper displaying 35 different questionnaire questions relating to students’ images of mathematics and discussing briefly some of the students’ answers to them.
Isaacs et al. (2000, p. 123) describe a “cultural origins of mathematics” unit introduced to first year pre-service primary mathematics teachers with the goal of modifying their “world view of mathematics”. Through historical examples of geometry the authors tried to show the students that geometry is a practical science used to solve problems; a constructive and aesthetic medium where patterns, transformations, and geometrical relationships predominate; the interplay between geometry and religion in history and culture; geometrical measurements as an introduction to numbers which are not rational; and matters of logical justifications in geometry (Isaacs et al., 2000, pp. 124-126). 57% of the students found that the unit had helped them change their attitude towards mathematics, 25% were undecided and 18% disagreed. In particular, the authors mention the deepening answers of one student, who pointed to the fact that the unit had helped dispel some of the myths associated with mathematics: that it is based on genius mentality, it has nothing to do with reality (life), that mathematics must be done mentally and fast (if you cannot, you are not mathematical), and that mathematics is purely a Western phenomenon.
In their paper, Kjeldsen and Blomhøj (2009) discuss how history (and philosophy) of mathematics have been integrated at university level mathematics education through problem-oriented project work (based on examples from Roskilde University natural science basic studies). For two out of three examples of students’ projects, the authors illustrate how the students gained insights into elements of the history of mathematics.
The first example is a project on a mathematical model as an explanation of cell division proposed by Rashevsky in 1934. Kjeldsen and Blomhøj (2009, p. 94) describe some of the (history as a goal) outcomes of this project as: “the students learned about some of the problems in the history of the development of a new scientific discipline of mathematical-biology in the 1930s. Through this project work, the students managed to put themselves in a position where they could understand the discussion that took place in the 1930s, and they were able to uncover (some) of the causes behind the conflict.
This gave the students a consciousness about history of mathematics, tolerance and insights into problems of communication that can be observed today between scientists from different sciences about ‘proper’ methods to produce scientific knowledge.” The other example concerned whether linear programming, as developed by Dantzig and Kantorovich, respectively, was in fact a multiple discovery. The student’s (history as a goal) profits from the project were: “how mathematics has developed historically in
42 The ‘Whys’ and ‘Hows’
space and time, in culture and society. This case especially demonstrated that despite the universal character of mathematics, its historical development is not unaffected by the surrounding society” (Kjeldsen and Blomhøj, 2009, p. 100). The framework used by Kjeldsen and Blomhøj in discussing these student outcomes is the so-called KOM-report (Niss and Jensen, 2002), which I shall rely on myself and therefore discuss in some detail in chapter 3.