Abstract
This paper aims at contributing to remedy the narrow treatment of functions at upper secondary level. Assuming that students make sense of functions by working on functional situations in distinctive settings, we propose to consider functional working spaces inspired by geometrical working spaces. We analyse a classroom situation based on a geometric optimization problem pointing out that no working space has been prepared by the teacher for students’ tasks outside algebra. We specify a dynamic geometry space, a measure space and an algebra space, with artefacts in each space and means for connecting these provided by Casyopée. The question at stake is then the functionality of this framework for implementing and analyzing classroom situations and for analyzing students’ and teachers’ evolution concerning functions, in terms of geneses relative to each space.
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Notes
We are aware that ‘genesis’ is used for denoting, in MWS, processes connecting the epistemological and the cognitive planes in a space. However, the way we use this word is consistent with the theoretical construct of ‘instrumental genesis’ (Lagrange 1999), a basis of the work we carried out for more than 15 years. Confronting these two conceptions is a promising perspective.
Studying variations of a continuous function is for instance proving that it is decreasing on some interval and increasing on another adjacent interval, and thus that it has a minimum. See examples in the teaching situations above.
The current programs of study and accompanying documents in France can be found at (http://eduscol.education.fr/).
We submitted a dynamic geometry figure with A a fixed point, C a free point, and B and D constructed in order that ABCD is a rectangle with sides parallel to the axes, to 34 students in a 10th grade class. The students had basic knowledge in dynamic geometry allowing them to understand how the rectangle is constructed and what happens when C is dragged. We asked them whether B and D are free points. Half of the class answered positively and explained that these points “are able to move”. See also Laborde, Kynigos, Hollebrands & Strasser (2006, p. 285).
About quantification and its importance in students’ understanding of functions, see Thompson (2011).
In addition, considering functions in “real world situations” would suppose a space where students could work in a non-mathematical space, for instance around a physical device involving a mechanical dependency. See Lagrange (2013) for an example. Here, for the sake of simplicity, we will keep the type of task of geometric optimization as an object for reflection, and then restrict ourselves to three spaces.
“Much like British A-levels or European Matura, the baccalauréat allows French students to obtain a standardised qualification, typically at the age of 18. This then qualifies holders to (…) go on to tertiary education”. (https://en.wikipedia.org/wiki/Baccalauréat). The students observed here pass the scientific baccalauréat involving a set of subjects among which mathematics has the heavier weight.
More data about this study is provided by Minh (2012b).
Some evidence supporting this assumption is provided by Lagrange and Caliskan (2009) from a study of textbooks and classroom practices in France.
There is a special functionality, based on algebraic theorems dealing with functions, that helps building proofs.
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This research was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED), under Grant Number VI1.99-2012.16.
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Minh, T.K., Lagrange, JB. Connected functional working spaces: a framework for the teaching and learning of functions at upper secondary level. ZDM Mathematics Education 48, 793–807 (2016). https://doi.org/10.1007/s11858-016-0774-z
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DOI: https://doi.org/10.1007/s11858-016-0774-z