Exploration Approach

This section discusses the exploration approach which is used to examine how

and when students use software for solving tasks under their own volition, that is,

without a teacher intervention or not following a pre-existing procedure. By

interviewing teachers, Ruthven, Hennessy and Brindley (2004) noted that software

packages used by secondary school students in the subjects of mathematics and English

were used for:

supporting processes of checking, trialling and refinement, notably with respect to checking and correcting basic elements of work, testing and improving problem strategies and solutions, and editing and redrafting written texts. (p.271)

The key point to note from their interviews was that students used the software

for testing and checking processes which implies that students were using software for

exploring processes or scenarios. Similarly, Pierce and Stacey (2001) reported from a

as an ‘independent expert’. The students used the CAS to explore properties of

functions for testing or making their own conjectures. For example, the students

changed the values of coefficients and exponentials in mathematical functions to see the

change in its corresponding graphs.

Trouche (2000) developed a classification of how students use technology (CAS

or graphical calculators) when solving tasks. His category of ‘overall calculator use’

indicated how often students were using these tools under their own volition. Hence, his

classification of students is dependent on how and when students explore with the CAS

or graphical calculators. He provided evidence for his categories by observing a senior

level high school class undertaking a design engineering project that covered calculus

and elementary analysis. He classified students into five extreme categories: theoretical,

rational, thinker, experimenter and scholar (see Table 4).

Table 4: Types of students based on their use of technology from Trouche (2000) (my translation)

Student Information Source Meta-cognitive activity Privileged method of proof Overall calculator use Usefulness of calculator

Theorist Notes Interpretation Analogy Average High

Rationalist Pen/Paper Inferences Demonstration Low Low

Tinkerer Calculator Investigation Accumulation High Low

Experimentalist All Comparison Confrontation Average High

Scholastic None Investigation Copy/ Paste Average High

Trouche indicated that when solving tasks, each type of student privileges

specific information sources and calculator uses. For example, the theorists use

references (notes, paper), work towards interpretation for understanding, use analogies

for proof, spend about average time exploring on the calculator overall but their

exploration time spent is usually fruitful. Trouche suggested that some students may

have a predisposition as to how they used the software, that is, students had a particular

However, these particular styles of exploring with software by Trouche may be

rather strategies that students employ depending on the topic. Coupland (2004) in her

study investigated how students appropriated the use of Mathematica (a CAS). She

issued both the ALMQ and a Mathematica Experience Questionnaire to her students.

The former measured students’ processing levels and the latter questionnaire measured

the students’ mathematical engagement with the software and their computing

experience. Through examining the responses from 113 students, she noted that

students’ uses of the software were dependent on their processing levels. Her analysis

showed that students with a deep processing level and a low computing background

reported that they were still able to appropriate the tool to allow for mathematical

engagement. In an earlier study, Laurillard (1979) found that processing levels may be

dependent on the learning context. As Coupland showed that students’ appropriation of

software is influenced by their processing level, then it is possible that Trouche's student

categories are not stable, that is, students may opt for any of these strategies depending

on the subject or task.

Coupland also found that students’ exploration with mathematical software

using their own initiative was quite poor. Students were requested to mark on a visual

analogue scale, a position on the line anchored by ‘disagree’ on the left and ‘agree’ on

the right. The line was approximately 41 millimetres. With 113 students returning

completed questionnaires, she noted for one item, “I often used Mathematica to explore

my own questions about mathematics”, that the students scored poorly. The students’

mean score for this item was 10.3 out of 41 which indicated a high disagreement with

this statement.

2.7.1 Exploration: Performance, Tasks and Processing Levels

might impact on students’ performance on mechanical and constructive tasks. If

students choose not to use the software boxes then they would more likely have

procedural or arithmetic errors for both the mechanical and constructive tasks. Further,

through exploration, students can test scenarios and via self-explanations, can build

their conceptual understanding. This could potentially impact on their performance for

not only the constructive tasks but perhaps also the interpretive tasks.

Moreover, Coupland found that students with a deep level of processing were

more likely to choose to use the software for mathematical engagement than students

with lower levels of processing. This might suggest that students with a deep level of

processing would be more likely to explore with the software boxes for a purpose such

as confirming answers or testing hypotheses.

2.7.2 Exploration and the Software Boxes

The frequency of exploration may also be dependent on the software box itself.

Both glass-box and black-box software are able to solve procedural tasks easily as the

student is only required to click the buttons to get the answer. However, the open-box

software requires students to determine what they would do at each step and this may

mean that students might be more reluctant to use this software for solving procedural

tasks. This behaviour may thus impact on how students explore using the software

boxes when solving the mechanical and constructive tasks. Students using the black-box

and the glass-box software may then explore more for the mechanical and constructive

tasks compared to students using the open-box software as there is a sense of more

immediacy. As the mechanical tasks are relatively simple, that is, requiring only the

inputting of values and executing a command, students may choose to always solve