Theoretical Framework

2. ICT tool use

In this chapter I focus on the use of ICT¹ tools in (mathematics) education.
A. Problem statement

It is important to study the way in which tools can be used to facilitate learning. How are tools used and what characteristics do they have to have.

B. Theoretical overview

First I will sketch a general overview, ending in a description of the instrumental approach of tool use. Here I will use the construct of figure 1, looking at how tools affect aspects of the teaching and learning of mathematics.

Tool Student Mathematical Activity Teacher Curriculum Content

From the Second Handbook of Research on Mathematics Teaching and Learning (Lester jr., 2007)

Not only mathematical activity, students, teachers and curriculum are affected, also the relationship between these aspects.

Technology and Mathematical activity

Many people use tools all the time. The Vygotskian notions on tool use (Vygotsky, 1978) sees a tool as a mediator, a " new intermediary element between the object and the psychic operation, directed at it" . In mathematics, tools have to have certain characteristics to be beneficial. The Handbook on Research mentions three important issues:
Externalization of representations

Heid (1997) also mentions this. The important question remains: how is mathematical activity influenced or changed by tool use? Feedback is mentioned. Otherwise time-consuming "production work" as well. Unlike the physical tool a cognitive tool provides a "constraint-support system" (Kaput, 1992) for mathematical activity.

Mathematical fidelity

A representation must be faithful to the underlying mathematical properties; this is mathematical fidelity (Dick, 2007). In essence this means that a tool can represent maths incorrectly. This also has to do, in my opinion, with the difference between "use to learn" and "learn to use" (cTwo, 2007), as the latter means one has to know the shortcomings of a tool, but this knowledge could also lead to a better understanding of a concept, thus a tool is used to learn.

Another aspect is the underlying machine code for a certain tool. It often is the case that certain extreme values yield strange results. So an important question is: "is the mathematical fidelity of a math system good enough to support maths at secondary school?". It will almost surely be a trade-off between this and the amount of time needed to improve the system.

Cognitive fidelity

This is the "degree to which the computer's method of solution resembles a person's method of solution.". To make sure that transfer of knowledge or skill takes place

Technology and students

An important distinction in type of activity is between exploratory and expressive tools and activities (Bliss & J., 1989). They reside on a continuum. So when a procedure is described it´s exploratory but choosing one’s own procedure is expressive (albeit somewhat limited). Initial play with a technological tool is often beneficial: it stimulates expression but also builds a purposeful relationship with the tool , and thus instrumental genesis (Guin & Trouche, 1999) can take place. However, structured guidance is often necessary, as to avoid the "play paradox" (Hoyles & Noss, 1992). This means that " playing" with a tool sometimes enables students to accomplish an activity without learning the intended concepts. To solve the paradox "reflection" on the task at hand is advised.

When studying student use of a tool the construct of a 'work method' could work: Guin and Trouche (1999) see five work methods: random work method, mechanical work method, a resourceful work method, a rational work method and a theoretical work method.

The combination of the type of activity (exploratory, expressive) and work method should enable us to deduct what students are thinking.

Technology and practice

In it important that there is pedagogical fidelity in tool use. This means that students actually learn what the teacher has intended. So here we consider the match between technology and practice..

An interesting choice is whether sometimes "privileging" is appropriate: using tools when basics are known and rules are "internalized" (in a sense trivialized). Before that, tool use is prohibited. The concept of privileging can also apply to certain mental activities, like proofs etc. This coincides with the white box versus black box discussion (Buchberger, 1989), which states that “privileging” with tools –meaning that tools may only be used when a concept is understood- is necessary. This prevents students from just “executing an algorithm" without knowing what they are doing.

Using technology in practice also means that the teacher role could change. This is also an aspect that can be studied throught the construct of teacher role, e.g. Counselor and Technical Assistant (Zbiek & Hollebrands, 2007).

The use of technology changes this role. But: this could very well clash with teacher's expectations. Beaudin and Bowes (1997), and later Zbiek and Hollebrands introduced the PURIA model for CAS implementation:

personal Play,
personal Use,
Recommendation,
Implementation,
and Assessment.

This could also be applied to the implementation of DME² use in the Netherlands.

Technology and curriculum

There are several reasons why technology is adapted:

Representational fluency: technology makes it possible to move easily between several representations. This also belongs to good design principles for technological environments (Underwood et al., 2005). For example, the applet Algebra Arrows has a multirepresentational aspect. One could think of the distinction: context-table-graph-formula.

Mathematical concordance is another construct looking at the level intended and real knowledge building are the same or not. In analyzing the way that teachers and students interact with cognitive tools, it is helpful to consider the mathematics of the tool, the mathematics of the teacher, the mathematics the teacher intended through particular technology-based activities, the mathematics that the student engaged in as a result of the technology-based activity arranged by the teacher, and the mathematics that is learned.

Amplifiers and Reorganizers can be used to describe curricular roles of technology. (Pea 1985) Amplifiers accept the goals of the current curriculum and work to achieve goals better. Reorganizers change the goal of the curriculum or the way the goals are obtained by replacing, adding and reordering parts. I would say the WELP³ project would be an amplifier.

“The French school”

In the early 90s the use of Computer Algebra Systems was seen as a possible means to get rid of manipulations (routine skills) and focus more on concepts and complex problem solving.
Artigue, in “the French school”, noticed that actual tool use should be scrutinized, as to discover obstacles and difficulties in the classroom. Her thoughts in the 90s on this become clear in this quotation:
“... we needed other frameworks in order to overcome some research traps that we were

more and more sensitive to, the first one being what we called the “technical-conceptual

cut”. Indeed, theoretical approaches used at that time ... tended to use this reference to

constructivism in order to caution in some sense the technical-conceptual cut, and we

felt the need to take some distance from these. Artigue (2002, p.247)”

 

Two approaches were combined to overcome these traps. On the one hand Vérillon and Rabardels work on instrumentation ((1995), the ergonomic approach) and on the other hand Chevallards anthropological approach (Task, Techniques used to solve Tasks, Technology or Talk used to explain and justify Techniques, and Theory).

Instrumental approach

The instrumental approach of tool use is easily summed up In this research the instrumental approach of tool use is more important:
Instrument = artefact + instrumentation scheme.
Verillon and Rabardel (1995) distinguish an artifact and an instrument. An artefact is only the tool. An instrument is the psychological notion: the relationship between a person and an artifact. Only when this relationship is established one can call this a "user agent". The mental processes that come with this are called schemes.
Instrumental genesis is the process of an artifact becoming an instrument. In this process both conceptual and technical knowledge play a role (again, "use to learn" and "learn to use"). In instrumental genesis three aspects come together: task, theory, technology. They are closely connected.
Trouche (2003) distinguishes a tool component with instrumentation (how the tool shapes the tool-use) and instrumentalisation (the way the user shapes a tool), and also a psychological component with schemes (Piaget & Inhelder, 1969) . According to several studies (Artigue, (2002); (Guin, Ruthven, & Trouche, 2005) genesis for computer algebra systems is a timeconsuming and lengthy process.
When focusing on particular aspects of instrumental genesis, for example instrumentation, instrumentalization and technique (Guin & Trouche, 1999), it becomes more clear how students can use tools more effectively and what obstacles hinder conceptual and technical understanding of a tool. Trouche sees three functions: a pragmatic one (it allows an agent to do something), heuristic (it allows the agent to anticipate and plan actions) and epistemic (it allows the agent to understand something).
Also instrumental orchestration concerns the external steering of students’ instrumental genesis (Guin & Trouche, 1999)
So there are also conceptual aspects within the cognitive instrumentation schemes. The instrumental approach provides a good framework for looking at the relation between tool use and learning from an individual perspective. Yackel and Cobb (1996) argued that coordinating both perspectives is expected to explain a lot on the advent an use of computer tools. Tool use and learning is especially apparent in mathematics education. Integration in the classroom is essential, and to understand this we need to observe instrumentisation.

Anthropological approach

As tools are used in practice, a context, in activity one’s view on practice becomes important. In the anthropological approach we discern task, techniques, technology and theory:
Technology and theory can be defined as knowledge per se

Task and technique: know-how relevant to a particular theory and technology

So, Artigue and Lagrange focus primarily on Task and Technique. Also, the distinction between, for example, pragmatic (efficient) techniques for doing tasks, and epistemic techniques (more focused, as I see it, on concepts on real symbol sense)
As Artigue says: 

“Professional worlds as well as society at large have a pragmatic relationship with computational tools: their legitimacy is mainly linked to their efficiency. But what schools aim for ... is much more than developing an effective instrumented mathematical practice. The educational legitimacy of tools for mathematical work has thus both epistemic and pragmatic sources: tools must be helpful for producing results but their use must also support and promote mathematical learning and understanding. (Artigue, 2005)”

 

So use to task, theory and technique go hand in hand. Again citing Lagrange (1999):

“The argument for this is essentially in five parts:

1. 
   Technical work does not disappear when doing mathematics with CAS, it is transformed.
   
2. 
   Within a theory, every topic has an accompanying set of tasks and techniques. Novices progressively become skilled in techniques by doing, talking about, and seeing the limits of techniques. This eventually leads to a theoretical understanding of the topic.
   
3. 
   Although rote repetition of a specific technique for a specific task is a mathematically impoverishing experience, this is not a reason to jettison techniques per se.
   
4. 
   Techniques and schemes are linked. Students need time to develop rich schemes by using techniques.
   
5. 
   The empirical observation that diminishing the role of techniques encourages teachers to avoid talking about them (Chevallard’s “technology”).”
   

 

Monaghan (Monaghan & Ozmantar, 2006) points out several issues in “the French school”, for example the tension in the “technique”. He wonders whether the two approaches could be integrated. 

Two missing aspects in French theory are teachers and affect. Concerning the first: one could say that orchestration has to do with teachers. This however seems more concerned with “what can be done” and not so much a description of teachers’ practices. Perhaps it would be good if studies would focus more on developing accounts of teachers coming to instrumental genesis. Concerning affect: as it surely has a large impact on tool use (beliefs, attitudes, motivations, emotions) it should have more attention.

 

Monaghan (2005) also has argued that de notion of schemes could be cut by looking towards “activity theory”:

“Activity theory considers the actions of a person towards an objective (affective motives are thus essential). Activity is mediated through artifacts, social procedures, and language, and Trouche may find the ideas of mediation relevant to his considerations of orchestration.”

So here we see common ground between two frameworks: mediation can take place through artifacts, procedures and language, corresponding well with the notion of orchestration.

C. ICT tool use in this research

We claim that tools can facilitate in the learning of algebraic skills. Therefore we want to study how instrumental genesis takes place when using a prototype.