PRUEBA DE ECONOMÍA SEGUNDO TRIMESTRE – GC-LM

of how the Fathom software acted upon the user to support learning and students’ development of use of the Fathom tool as a mathematical instrument. Central to instrumental genesis is the notion of schemes, which are the mental processes needed to use the software tool effectively (Section 2.5.6.1) To support examination of students’ development of schemes this study borrowed from situated abstraction the three-step epistemic approach of Hershkowitz et al. (2001). The analysis was not intended to provide a comprehensive use of instrumental genesis framework, but rather to provide additional insights and evidence of students’ learning trajectories.

As the use of instrumental genesis is relatively novel, an example is given from the study. In this study a student was presented with the artefact of a blank Fathom workspace and the student assembled a Fathom die simulation, but the Fathom die became an effective instrument only when the student accepted the virtual die as a fair representation of a die. When instrumental genesis is applied to the die simulation this allows the diagram presented earlier as Figure 2.3 to be refined and re-presented as Figure 3.3. Schemes are largely invisible and elusive, and their identification is somewhat problematic and subjective, but in this instance the two schemes associated with the development of the instrument are first, the procedural process of assembling the die simulation and second, the acceptance of the die as fair.

Figure 3.3. An example of instrumental genesis as applied in this study.

Blank Fathom + Assembled Fathom die, Fathom virtual workspace Fathom die is fair die

[ARTEFACT] + [SCHEMES] [INSTRUMENT]

136 Of principal interest in this study is instrumentation, which is where the software acts upon the user. Students began the study with no knowledge of the Fathom die simulation; they could initially either consider the simulation as fair, unfair, or defer forming an opinion until having the opportunity to examine the simulation. The process of testing the Fathom die and comparing the random nature of its behaviour with other die provided the opportunity for the process of instrumentation and the software to act upon the student to occur. The evolutionary and personal nature of instrumental genesis seemed self-evident: students’ development of use of the software and students’ acceptance of the simulation (and the extent of that acceptance) occurs within an individual student’s own time-frame and learning pathway.

The framework was used to identify specific instances of how the software appeared to support students’ transition from (a) frequency to proportional data analysis, (b) language use of “tossing a coin” to “sample size,” (c) small to large sample sizes, (d) interpreting a graph to choosing a graph to “tell a story” and (e) elementary to more sophisticated interpretation of a graphical representation of the cumulative proportion of heads.

The terminology of situated abstraction was incorporated into instrumental genesis to examine how students abstracted mathematical meaning. Hershkowitz et al. (2001) three step epistemic process of constructing new knowledge, recognising an existing mathematical structure as relevant, and building-with to apply knowledge to a new context provided by the terminology used. This structure was used to describe specific instances of students’ mathematical learning of the fairness measure, students’ development of understanding of the re-sampling terminology of “sample size used to calculate a measure” and “the number of measures collected,” and students in the detailed study’s interpretation of graphical representations of the cumulative proportions of heads.

Instrumental genesis was also used in this study to examine students’ development of procedural use of Fathom. Procedural use was defined as the basic skills required to complete a Fathom task, such as construct a graph or summary, and procedural use of Fathom to construct graphs and summaries are examples of schemes. The software was used to support learning and to allow students to attend to the mathematical concepts – developing skills in the use of the software was incidental to the mathematics, but

137 Fathom could be a potential constraint on learning if students found the software difficult to use or it was not introduced effectively.

The study focussed on students’ use of Fathom at four points in the classroom study at Lessons 1, 3, 5, and on the post-study assessment, and these four points provide a developmental path for analysis (Section 4.5.4.1). Lesson 1, the exploratory data analysis of the New York marathon data set, produced students’ response to their first use of Fathom, and Lesson 3 gave students’ response to their first use of a Fathom simulation. Lesson 5 compared male and female students’ responses to two different versions of the one Fathom worksheet, and the post-study assessment assessed students’ independent use of Fathom under traditional examination conditions. In all four instances students’ responses were assessed as the proportion of students able to use Fathom effectively such that Fathom was not likely to be a barrier to learning. These data were supported further in the detailed study by an examination of students’ ability to demonstrate their recall of the four basic operations of taking a sample, creating a summary, changing sample size, and creating a graph. In the detailed study students were assessed as the proportion of students able to complete “by self,” “after demonstration,” and “after demonstration and support.”