1. Enrique de la Torre Fernández (Universidade da Coruña, Spain) Adelina Silva Muslera (Universidad Autónoma de Querétaro, Mexico) GeoGebra Conference 2009 1

2. Introduction Our concern is about the use of technologies in the mathematics classroom. We questioned ourselves mainly as the students use technology to conceptualize the differential and integral calculus. We need a software that will adjust to that concern and a methodological frame that fit in our plans of research. This it is a first result of that concern. 2

3. Instrumental approach The instrumental genesis is described like the process by means of which a device is transformed into an instrument by the subject. A device is a material or abstract object, given to the subject. An instrument is a psychological construction, building from the device; it is any object that the subject associates with its action to make a task. The instrumental genesis has two components, the instrumentalization: is relative to the discovery, personalization and transformation of device by the subject. and the instrumentation: is relative to the emergency and evolution of the schemes of students for the accomplishment of a given task. 3

4. Instrumental approach The instrumental approach has been developed and applied in the context of algebra software (CAS), where different types of restrictions have been identified, what often are learning opportunities and which provide data for the professor and for the researcher. This idea supposes that technology is a mean where abstract relations and structures are expressed. The dynamic geometry software (DGS) is also a mean where it is possible to express the geometric relations by the way of to drag with the mouse or the facility to redefine, as the algebra software does with the algebraic expressions. 4

5. Context and methodology 28 first course students of Chemistry in the Faculty of Sciences (University of Corunna, Spain). Time destined for practices, one hour for week. 10 sessions Subjects were: derivatives, applications of derivatives, integral and applications of integral. 5

6. Context and methodology 16 computers in the classroom. Students had access to paper and pencil. Some of them made use of their personal calculators. Software: GeoGebra. CamStudio (to record the work the students develop in computers). 6

7. A interpretative qualitative research (ethnographic methods ). We use the participant observation like a strategy for collecting data. We want to understand the reality within a context given, from the meaning of people implied in that context. Context and methodology 7

8. Example of derivatives Second day: two students manipulate a computer and generate a video, also each one of them wrote their answers with paper and pencil. Problems proposed : 1. Is the function derivable at x=2? Justifies your answer. 2. To find the equation of a tangent to the function f(x)= 3x 3 +14x 2 +3x+8 what go through the coordinates origin. 8

9. Example of derivatives The solution to the first exercise it could be obtained directly when they analyze the function and they see that function is not defined at x=2. Nevertheless, as they have the computer on, the first thing they make is to design the function. They use the commando Function: Function [x+1/2x^2-8] The software indicates a mistake. 9

10. Example of derivatives For many minutes, the students rewrite the sentence, placing parenthesis in the numerator and denominator, and trying the menu for to create a slider. This is the so called “fishing behavior”: they test without affecting its organization or her verification, hoping that in a reasonable time something interesting will be. 10

11. Example of derivatives Techniques of paper and pencil for the first exercise: 11

12. Example of derivatives Techniques of paper and pencil for the first exercise: 12

13. Techniques of paper and pencil for the second exercise: Example of derivatives 13

14. Techniques of paper and pencil for the second exercise: Example of derivatives 14

15. Conclusions One of the potentialities of GeoGebra, the functions graphs, becomes a restriction for the students. When the students make derivatives, usually they do not see the graph of the derivative function, and when the derivative function is showed in the screen, it is something new for them. This can be denominated a obstacle in the sense that it is a lack of balance in the conceptual aspects. 15

16. Thank you! 16