Computer mini-tools and Fathom

Bakker (2002) makes the distinction between two broad categories of statistics educational software: route-type and landscape-type. This is a distinction similar to specialised and general tools made by Olds, Schwartz, and Willie (1998, cited in Kaput, 1992), or black-box and white-box software (Buchberger, 1989), or black-box, glass-

55 box and open-box software (Hosein, Aczel, Clow, & Richardson, 2008). Route-type, specialised, or black-box software, all have in common that the operation of the software is concealed and cannot be modified readily by the user. In contrast for landscape-type, general and white-box software the internal operation of the software is, to a degree, visible and the user has a level of control over how the software operates. The two broad categories of software have both advantages and disadvantages when learning statistics and are now discussed in turn.

Route-type software, occasionally known as mini-tools or applelets, is software specifically designed for a specific task or technique or for the development of a specific skill. This focussed approach allows the teacher to set clearly defined boundaries for the work-space, which is an advantage if classroom management and discipline is a significant issue. The disadvantage is that it constrains the students to examine only a limited number of often tightly controlled alternatives, which is at odds with current research and thinking on statistics education. Cobb (1999) used route-type software with features not generally available with commercially available software tools, but which were based on current statistical educational research. In particular, the software fitted the thinking of the user, rather than the user fitting the software. It also fitted the student’s learning trajectory by providing a familiar entry point but continued to support the student as more sophisticated statistical thinking emerged. Cobb and McClain (2004) selected software mini-tools using two criteria. They argued that software should be developmentally appropriate, which would ensure that the cognitive load and the entry time to develop a basic proficiency were relatively short, and the software should also support more sophisticated thinking as students’ thinking developed. They noted that the inherent small scale of mini-tools may also limit the scope for students’ development.

Landscape-type software is characterised by an open construction environment where students are not guided, or constrained, to take a particular route to a solution, and this allows students to follow their own individual learning trajectory. Students are also able to create their own, often unconventional, representations of data. This structure also gives teachers greater flexibility to develop their own learning sequences. Route-type software has provided software developers with the foundations, insights, and prototypes to develop landscape-type products such as Fathom. The major disadvantage with landscape tools, according to Bakker (2002), is that students are given too many

56 options, the learning environment is too complex and students may become confused. The main objective of the lesson can be lost, and there is the potential for off-task behaviour.

Fathom (Key Curriculum Press, 2005) has been favourably received by the education research community (Hammerman & Rubin, 2004; Lane-Getaz, 2006; Lock, 2002; Maxara & Biehler, 2006). Fathom has simulation features and offers multiple ways of presenting data. It encourages students to manipulate data and transform information. The software allows samples and sampling distributions – the subject of this study – to be examined for structure, shape, and other characteristics that single statistics can only present numerically. The graphical representations encourage the development of an intuitive sense of the shape of the data distribution, and a view of the data aggregate. New technology tools, such as Fathom, do not change the complexity or the quantity of data, but give people more options for presenting information and this may aid interpretation, and ultimately, decision making. Novel presentations of data are a catalyst or boundary object (Hoyles et al., 2004) for classroom discussion. Students need to be comfortable with letting complex ideas simmer (Hammerman & Rubin, 2004) and to learn that there is rarely one clear way to make a decision when dealing with complexity and variability.

The most significant, and on-going, research into the use of Fathom in schools is by Rolf Biehler and co-workers in Germany. Having identified the need for statistics education software (Biehler, 1997) Biehler selected Fathom because, as he put it, it had tools for exploring data, tools for elementary simulations, tools for studying mathematical functions, and at the same time served as a meta-tool and meta-medium where teachers and learners could adapt working and learning environments (Biehler, 2003). Working with 17-19 year old students and with undergraduate teachers – students at least three years older than the students in this study – Biehler examined exploratory data analysis tasks (Biehler, 2003, 2006), computer based simulation of statistics and probability (Biehler, 2006; Maxara & Biehler, 2007), and more recently computer based modelling of sample size (Biehler & Prommel, 2010). The research was conducted in classroom environments with follow-up interviews with selected participants to provide more detailed information using instructional guidebooks and modifiable Fathom worksheets. Task complexity ranged from simple introductory tasks utilising data graphical display features to more complex tasks that included formal

57 theoretical probability calculations. This research allowed the researchers to identify and categorise students’ working styles (Maxara & Biehler) and further refine the teaching resources used. The most recent work (e.g., Biehler & Prommel) suggests that Fathom has become progressively institutionalised in the German senior high school system. The studies now include a preliminary fifteen lesson pre-course designed to improve the efficiency by which the software is learnt through building intuitions, learning the basic steps of simulation, and acquiring basic and stable Fathom competencies. Given the age and maturity of the research cohort the level of sophistication of formal mathematics is considerably higher than that expected of a high school student. The use of Fathom based simulation techniques is discussed subsequently (Section 2.5.7).

Lane-Getaz’s (2006) progressive integration of Fathom into a senior high school statistics program provided an example of a teaching professional’s use of the software. In the second year of the program, two of five topics, bi-variate data and inference, were delivered in Fathom. In the third year of the study, Fathom was used throughout the course including in the final assessment research project. Lane-Getaz concluded that students, as part of this course, demonstrated improved statistical thinking, used statistics more appropriately and accurately, and their interpretations and conclusions showed measurable improvement. The improved performance of the students was attributed to a number of contributing factors that included Fathom, the use of investigative projects, process orientated software, engaging activities employing the big statistical ideas, formative assessment and the teacher’s ability to interweave topics into a conceptual whole.

The statistics education research literature identified two broad classifications of software as route-type or landscape-type software. Route-type software are task specific tools that provide highly guided learning experiences, but are potentially inflexible; landscape-type software offers greater flexibility and a variety of learning pathways, but has the risk that the students becomes confused. Fathom, a landscape-type software, offers many of the features identified as desirable by statistics education researchers. The software is also used in senior schools and tertiary institutions, and it is the subject of current research, in both Germany and the USA, and consequently it is the software used in the study presented here.

58 2.5.5 The role of computers and software in the classroom

Statistical education researchers recommend that software support learning, rather than occupying a central role. The software is invariably used as part of a classroom culture that promotes enquiry, discussion and active learning, exposes students to the big ideas of statistics, and uses authentic tasks and authentic assessment (Bakker, 2004; Ben-Zvi & Arcavi, 2001; Ben-Zvi, Garfield, & Zieffler, 2006; McClain & Cobb, 2001). Cobb (1999), for example, in a study of a Year 7 group, did not introduce software until lesson 5 in a sequence of 34, and Lane-Getaz (2006), in an extended classroom study, concluded that Fathom was only a contributing factor to improved student outcomes. Ben-Zvi (2000) viewed computers specifically as cognitive tools that have the potential to improve learning. All of the studies presented by Ben-Zvi and Garfield (2004) involved the development of students’ conceptual models and thinking that were largely independent of the type of technology used. Kaput (1994) noted the importance to learning of computer-based representations that facilitated a connection between human experience and mathematics.

Insights may also be gained from the recent introduction of other electronic technology, such as Interactive Whiteboards [IWB] (e.g., Moss, Jewitt, Levacic, Armstrong, Cardini, et al., 2007). One large-scale study (Glover & Miller, 2001) examined teachers’ use and integration of IWB in 25 UK schools. Through analysis of 100 video- taped lessons the researchers identified three developmental stages: (a) supported didactic, where the IWB is used as a visual aid only; (b) interactive, where the IWB is used to stimulate response from the students; and (c) enhanced interactive, marked by teachers’ thinking that “seeks to use the technology as an integrative part of teaching [...] to integrate concept and cognitive development in a way that exploits the interactive capacity of the technology” (cited in Goos, Dole, & Makar, 2007, p. 326). In this study Fathom was used in an “enhanced interactive” way to support learning. Vale and Leder (2004), in a study of study of middle school students’ views of computer-based learning in mathematics, reported that girls viewed computer-based learning less favourably than boys. Girls were less inclined to consider software relevant to their mathematics learning, with a tendency to see computer-based study as providing skills in computer use, not necessarily mathematics. Boys were more positive about computer-based learning, including that they found using computers pleasurable

59 and that it supported their mathematics learning. Success with computers and students’ interest were linked positively with high achieving girls and boys more positive about computer use. This research has implications for this study, which includes providing computer-based learning opportunities that gives a priority to mathematics learning over computer use and utilising boys’ interest in computers.

The software Fathom has features thought desirable by statistics education features, must be incorporated into the pedagogy thoughtfully to be effective. The literature recommends that software should be used to support learning rather than play a central role and that mathematics learning must have a priority over computer use. The attitudes of girls and boys to computer-based learning may differ substantially.