1. THE CASE OF CALCULATORS IN ELEMENTARY MATHEMATICS Jean-Francois Maheux Université du Québec à Montréal (CANADA) maheux.jean-francois@uqam.ca http://www.math.uqam.ca/maheuxjf

3. The importance of technology in education keeps increasing On the other hand, even when made available, technology is not necessarily used in the schools (Cuban, 2001) Preservice teachers also resist learning technology for their teaching (Laffey, 2004) Technology in education

4. The case of calculators presents the situation in an interesting light Cheap, known for its great possibilities for at least 30 years (Groves, 1995, Suydam, 1978) Missing in the litterature is the epistemological question, which I found to be of primary importance to understand “what counts” in mathematics education (Maheux, 2010; Roth & Maheux, 2011) Mathematics can be considered in different ways, each connecting with different approaches to teaching and learning (e.g. Ernest, 1994) … and technology may impact our understanding of the nature of mathematics (Raju, 2001; Leung, 2006) An epistemological issue?

5. I question the educational use of technology with the distinctions made by Heidegger (1954) : We shall be questioning concerning technology. Questioning builds a way […] a way of thinking. All ways of thinking, more or less perceptibly, lead through language in a manner that is extraordinary. [… In so doing we] prepare a free relationship to [technology, one that] opens our human existence to the essence of technology. When we can respond to this essence, we shall be able to experience the technological within its own bounds I take the case of simple calculator in elementary mathematics as a focal point to read Heidegger while reviewing some of the abundant literature on the topic. Reading Heidegger

7. In the way we generally think, Heidegger suggests, technology is a means to an end, and it is a human activity. As early as 1976, Texas Instrument introduces the “Little Professor” their first “educational toy” calculator. Nowadays the company makes educational technology the core of their business (TI, 2002). From about the same time, math educators began asking how or when the calculator could be best used and to what ends (Roberts, 1980) Heidegger is not satisfied with this “instrumental definition” and wants to distinguish between questioning technology and asking technological questions. In his view, these are technological questions that take us away from reflecting on our relationship to technology itself. Questioning technology

8. We think technology as a contrivance, worried about how to manipulate it to reach our ends. It is not technology that we question, but our ability to master it. In doing so, we conceptualise technology as something neutral : “calculators may be used in an innovative or in a conservative way … in every classroom, teachers use the … calculator in a different way” (Romano & da Ponte, 2009, p. 4). We ask valuable questions and progress, improving technology and the way we use it. But we remain in the dominance of technology, blind to its essence and our assumptions about it. A neutral contrivance

9. We ask technological questions from within a technological way of thinking about technology, and thus asking questions that are produced by technology itself. This is correct, but does not help in understaning our orientation to technology Reviews around educational use of calculator confirms Heidegger’s impression. Arnold (2004) essentially find in the 1990 they same key questions raised by Suydam (1978) around teachers’ and students (a) use, (b) attitude and (c) required/developed skills For Heidegger, we miss the “essence” of technology, which is what technology really is to us: we shall never experience our relationship to the essence of technology so long as we merely conceive and push forward the technological, put up with it, or evade it. (p.4) What we do, what we miss

11. Heidegger reflects about our instrumental conceptualisation, and it basic principle: causality. He reframes causality to a relation of responsibility for revealing something, in the sense of “unleashing” it A stark contrast with the ‘neutral contrivance’: Technology is no mere mean, it changes our ends and impose it own ‘truth’ Technology is something we are responsible for, while at the same time, there is also a bringing forth into existence that comes from technology itself. This bringing forth escapes our control, but, in the end, we are nevertheless responsible for it. Technology is no mere mean

12. Literature shows technology requires changes on the part of the teachers and the students (among others) E.g.: Ruthven (2009), like Duffin (1989), mention the need for new approaches to teaching, meaning that students must be supported in a very different way once the calculator enters the classroom. It is not only about abandoning hand-written calculations (Shuard, 1991) but event changing expectations of children's mathematical activity (Groves & Cheeseman, 1993) what must be taken on is not the technology itself, but the responsibility for it Technology is change, and what must be taken on is not the technology itself, but the responsibility for it. Responsibility for educational techonologies

13. What gives technology such an ascendant? Heidegger finds: From earliest times until Plato the word techne is linked with the word episteme. Both words are names for knowing in the widest sense... Such knowing provides an opening up. As an opening up it is a revealing. (p.13) Education technology summons us to change the way we do things, but also calls upon the way we conceive of what we do. Calculators not only require teachers to teach mathematics differently, but also impose another way to see mathematics, and mathematical activity Also a change in what is understood to be “knowing mathematics”: an epistemological shift difficult for mathematician themselves! The Meaning of Technology

14. A similar situation appeared when we moved from abacus to handwriting calculation, which is itself a technology: it seems that mathematics was “denaturize”, and calculations reduced to getting a result following a receipt, rather than “an art” in itself Enframing: The True Danger of Technology Heiddeger calls this “enframing”, the true danger of technology. It comes downs to reducing nature (mathematics) to a “standing reserve” And this actually does NOT come from ‘technology’: Technology only embodies this general attitude to order, measure, control…

15. Enframing is something that we do and must be attentive to Precisely because technology is something we are somehow obviously responsible for, it reminds us of this, and that there is another way to “reveal” the world/mathematics This other way, Heidegger affirms, is art. Art is similar to technology because it is, too, a way of revealing, a way to known, to understand. But contrarily to technology, art does not strive to make the world into a standing-reserve. Art upholds the way things reveal themselves to us, and celebrate our "taking part" in this revealing It is easy to think of mathematics as an art (e.g. Lockhart 2009) I’m sure most people use a calculator for everyday arithmetic. And why not? … my point is [that] Mathematics should be taught as art for art’s sake (p. 34) The saving power of technology

16. Shifting (mathematics) education to the realm of art will not be an easy task. Epistemological issues are in the coming, but technology itself is able to support such a change, although it needs to thought as such with teachers, students, and all those involved. It is only, when we become aware of our technological orientation that we can develop a free relationship to technology. Only when we constantly question it, what we do with it, what it does to us. There might still be a lack of confidence or knowledge in how to use calculator, etc. but that is not the most important aspect. What most matters is the upholding of non-technological epistemology of mathematics and mathematical activity, one in which we make central the artistic way of revealing: playful explorations, aesthetic appreciations, empathic meaning making, astounding encounters, and so on. Shifting education to art?