This case study focuses on how a high school student learned the meaning of the velocity sign (by using MBL). As a framework Nemirovsky highlights the difference between "symbol system" (the analysis of mathematical representations in terms of rules) and "symbol-use" (some rules of the system that is used, embedded in personal intentions, in specific history and in a situation). Traditionally new rules are introduced by a teacher and students have to practice it in many contexts. However, the rules are abstract descriptions of dynamic processes, the rules are a fiction. Nemirovsky stresses the importance of ways of symbolzing, that involves developing meanings of graphical shapes and about the situation being represented by the graph, from which a style of symbol-use emerges. Communication is an important aspect in symbol-use, because symbol-use is a dynamic game among different perspectives. Focusing on symbol-use, the velocity sign emerges as a particular way of symbolizing.
Nemirovsky expects that his learning of the learning processes of students will enable us to illuminate new approaches to introducing calculus notions: what it takes to learn a particular way of symbolizing or how MBL tools may take part in students' making sense of the connection between symbols and physical behavior.
One of the main observations is that Laura's difficulties began when she met signed graphs and negative velocity using the motion detector. Her struggle was not with the general distinction between speed, position and direction, but with the interpretation of the qualities of the toy-car motion with signed graphs. There were some basic connotations that Laura brought to her thinking with graphs, such as "up" was more and "down" was less, as well as negative being "below" positive. (MBL-argument 1) She had to refine her idea in order to develop a new approach to graphing motion in which the computer representation made sense. Without the motion detector, Laura could have produced signed graphs of velocity by following the explicit specification of new symbolic rules, so that she could plot data according to those rules. Laura used the motion detector to experiment with a symbolic behaviour that was meaningful for others (experts), but that was dissonant with her own ways to meaning. Her hand motion, her expactations about graphical shapes, her kinesthetic sense, and her shifts of perspective all became part of her wanting to be meaningful in a new territory. The motion detector did not describe how things are, it offered Laura a symbolic behaviour that she had to interpret according to the multiple frameworks available for her. It was this interpretative effort that moved her to revise her own understanding.
(MBL-argument 2) The experience with the motion detector mediated the conversation in a way that empowered the users to explain their thinking, puzzlement, and suggestions.
From the point of view in which mathematics learning is identified with the pratice of rules - with the acquisition of a symbol system - understanding the velocity sign amounts to little more than the acceptance of a convention (the direction of linear motion). However, the exploration of symbol-use in all its richness helps us become aware that rules are only a part of a broader perspective encompassing many aspects of the individual's experience.
Nemirovsky concludes that learning is not a matter of showing how it works, but of offering opportunities of symbol-use which encourage the learner to revise what he/she knows and expects. This case study suggests that learning graphing is learning how to do things with graphs.
Nemirovsky's alternatief voor MBL is twijfelachtig: Without the motion detector, Laura could have produced signed graphs of velocity by following the explicit specification of new symbolic rules, so that she could plot data according to those rules. Vooral dat following is natuurlijk maar de vraag.