Handal, B., & Herrington, A. (2003). Re-examining categories of computer-based learning in mathematics education. Contemporary Issues in Technology and Teacher
Education [Online serial], 3(3). Available:
Frameworks for Classifying Computer Use in Schools
In 2002, the online journal Contemporary Issues in Technology
and Teacher Education republished the seminal paper by Arthur
Luehrmann discussing approaches to using computers in schools in the 1970s
(Luehrmann, 2002a). Juxtaposed with this republished paper was a reflection
on these approaches, 30 years later (Luehrmann, 2002b). His papers
considered the categories of computer use in schools identified by Taylor (1980);
that is, as a tutor, tool, and tutee.
In the 70s and 80s most computer software was developed as a way to
tutor or guide individual learners. As a
tool students use computer applications such as spreadsheets, word processors, and graphing software to help
them complete inquiry-based tasks, such as problem solving and
mathematical modelling. Not surprisingly, in 1980, Taylor saw this application
occurring more frequently outside of schools. Students were seen to use the
computer as a tutee when they programmed the computer to solve problems, such as
in a LOGO environment (Papert, 1980).
Taylor indicated that his framework should not be followed slavishly
but only as long as it provides useful insights. Interestingly, he considered
toys as a fourth category, including simulations and games, but felt that they
were subsumed under the original three categories. With today's
school-age generation and their preoccupation with game consoles, this category
may well be the dominant one to apply to the computer
out of the school.
In 1985, Alessi and Trollip provided another framework for
conceptualising the role of computers in education. They suggested the following
categories for computer-assisted instruction (CAI): drill, tutorials, games,
simulations, and tests. Alessi and Trollip's original categories were aligned to a
traditional, expository model of instruction that followed four phases:
presenting information; guiding the student; practicing by the student; and
assessing student learning (p. 6). Each category of CAI was seen to emphasise one
or more of these instructional phases.
In comparing both frameworks, it appears that Alessi and Trollip's
categories of drill, tutorial, game, and simulation can be considered as
subcategories of tutor, as described in Taylor's scheme. However, electronic
tests appear to shift the nature of computer use from the learner to teacher and
are prescient of the administrative software available to teachers that now
for planning, monitoring, and evaluating learning environments. Over
the last 20 to 30 years, major changes in technology and conceptions of
teaching and learning necessitate a re-examination of these earlier taxonomies.
Changes in Technology
Technologies such as CDs, mobile phones, digital cameras, and
personal digital assistants are common accessories in the digital home and
workplace. The World Wide Web is increasingly becoming part of most schools
and classrooms. Online education has a number of benefits over
traditional computer-based technologies. Clearly, greater access is provided to
those students studying at a distance or unable to mainstream into regular
classrooms, as well as those students who wish to learn at their own
pace (Santoro, 1995). This type of flexible teaching can enable students
to assume greater responsibility for their own learning (Schwier &
Misanchuk, 1993; Winn, 1997).
In addition, such an approach can be used to reach a large number
of students in a classroom accommodating a broad range of learning
styles (Vargo, 1997; Winn, 1997). Nunan (1996) added that flexible
delivery through online methods fosters a culture of self-learning, problem
solving, and activity-based learning.
Through such tools as discussion boards and chat rooms, online
educational resources enable greater access to synchronous or asynchronous
collaborative learning (Mayadas, 1997). Asynchronized learning occurs when
a virtual classroom environment can be accessed anywhere at
anytime, whereas synchronized learning takes place when learners are connected
to an online environment at the same time (Bourne, McMaster, Rieger,
& Campbell, 1997). Collaboration is further enhanced because online
technologies are also effective in allowing computers with different platforms
and browsers to work together in a learning environment, in contrast to
other computer technologies that are platform specific (Hosie & Schibeci,
2001). Changes in the way teaching and learning are conceptualized have
paralleled changes in technology.
Changes in Pedagogy
Traditional teaching based on behaviorist views of learning is being
replaced by inquiry-based teaching, reflecting a constructivist view of
learning. According to Elliot, Kratochwill, and Travers (1996),
behaviorism focuses on the manipulation of external conditions to modify behaviors
that will lead eventually to learning. A behaviorist teaching style in
mathematics education tends to stress practices that emphasize rote learning and
memorization of formulas, single solutions, and adherence to procedures and
drill. Teaching is seen as a matter of enunciating objectives, providing the
means to reach those objectives and using constant repetition in class for
skill acquisition (Leder, 1994). Wood, Cobb, and Yackel (1991) argued that
such approaches lead to passive modes of learning.
In contrast, constructivism claims that knowledge must be actively
constructed by learners as they are already "knowing beings" who
bring previous knowledge and experience to any learning event. Learning
depends on the way the learners interact with situations, beliefs, attitudes,
and previous experiences (Biggs & Moore, 1993). For advocates of
constructivism, learning is an adaptive and experiential process. Learners tend to
look for similarities and differences within their own cognitive schema as
they encounter new situations. These contrasts are the end-product of
conflictive knowledge looking to be resolved through reorganizing schemes of
knowledge (Phillip, 1995). Constructivist teaching strategies include
more reflective learning activities, such as problem-solving and
inquiry-based learning (Murphy, 1997; Wood et al., 1991).
Given the changes to technology and pedagogy, how relevant and useful
for today are the original categories of computer use developed by Taylor
and Alessi and Trollip? In Luehrmann's original paper he saw the need
for students to learn to use computers as a tool and as a tutee rather than
as acting only as the subject of computer tutoring. In his reflection 30
years later, Luehrmann concluded that the
"teaching tool use is just about the
only impact that computers have had on schools" (p.1) but more often
than not, learning how to use the tool takes the focus, as opposed to
learning with the tool.
Fortunately, he argues, the tutor role has come and gone leaving
some residual pockets of drill and practice; but he bemoans the fact that the
role of the computer as a tutee has also disappeared except, perhaps, for
those studying computer science. His hope is still that computers will be used
as tools to understand learning across all subject areas.
Just as Luehrmann re-examined his framework in 2002, Alessi and
Trollip have similarly reflected on their original premise in later editions of
their textbook. They now propose an expanded list for the role of
computers: tutorials, hypermedia, drills, simulations, games, tools and
open-ended learning environments, tests, and web-based
learning. This second, expanded list results not only from new technological developments (in
particular the Web) but also from new paradigms of teaching and learning
Constructivist approaches clearly emphasise a different set of
instructional phases and learning outcomes than envisaged in this earlier work.
Hypermedia and the use of tools and open ended learning environments are seen
as categories that align better with constructivist approaches. It is
somewhat curious, however, that Alessi and Trollip suggested web-based learning
as another approach, because in their own words, "Web based learning can
be combined with any of these other methodologies (for the web is essentially
a delivery medium)" (Alessi & Trollip, 2001, p. 12).
These categories are not mutually exclusive and at times overlap each
other. For example, a tutorial program can be organized through nodes of
information, thus showing some hypermedia-based instruction (HBI)
characteristics. Similarly, tools and open-ended learning environments may include
some simulation features, while a drill-and-practice activity could take the form
of an instructional game. Selecting the appropriate category will depend on
the nature and planned outcomes for the mathematics lesson to be taught.
The following discussion looks at describing and giving online examples
within the context of mathematics education reflecting Alessi and Trollip's
Drill-and-practice activities, because of their repetitive nature, still reflect
a traditional, behaviorist approach that focuses on mastering basic skills
reviewing material that has been previously learned. A typical
drill-and-practice exercise presents learners with a question, followed by
response entry and corresponding evaluation of the question and feedback.
According to Schwier and Misanchik (1993), for a drill-and-practice activity to
be effective, there should be a gradual increase on the "types, amount,
and layers of stimuli and feedback presented" (p. 20).
An enriching drill and practice activity should provide opportunities
to increase understanding of a mathematical concept as the learner
progresses through the activity. Examples of drill-and-practice online sites
include Percentage Estimation, Lessons on Order of
Operations, Problem Solving, Ratios, Fractions, Percentages,
Shape, and Interactive Arithmetic
(Editor's note: See the
Resources section at the end of this article for all
Tutorials are one step ahead of drill-and-practice activity, because they
not only present information but also guide students through their
learning processes. A tutorial usually follows a structure and sequence. The
tutorial starts with an introduction to the lesson and information is presented.
Next, the learner answers a series of questions and the program evaluates
them. Typical responses are "sorry," "very good," "try again," and "right
answer is," among others. In contrast to drill-and-practice approaches, the
tutorial will give feedback on the procedure to get the correct answer. The
cycle closes when the lesson is terminated, either by the learner or by the
program. A summary appears at the close of the lesson.
Tutorials have potential in online interactive learning, because they
provide many possibilities to motivate students through multimedia
capabilities. Also, a tutorial allows learners to work at their own pace in an
individualized mode of instruction and provides many opportunities for
reinforcement, correction of mistakes, and elucidation of misunderstandings (Schwier
& Misanchuk, 1993). According to Alessi and Trollip (1991) tutorials
are effective for "presenting factual information, for learning rules and
principles, or for learning problem-solving strategies" (p.17).
One of the advantages of a tutorial is its potential to teach students who
do not have a qualified teacher in a particular topic, a problem that occurs
many isolated small schools or when the small number of students in
a particular class does not justify the hiring of a specialist teacher (Alessi
& Trollip, 1991). Tutorials are also useful in providing individualized
instruction but are less effective for collaboration.
There are, however, a number of different ways in which tutorial
software can be used. For instance, a tutorial may be employed to support
and reinforce classroom instruction, to teach a selected topic, to activate
prior knowledge in an area before going to the main topic, or to generate
classroom discussion and group work. Tutorials can also provide instruction
to students who have missed classes, to review previously encountered
topics, or for remediation (Merrill et al., 1992). Tutorials can be combined
with other computer devices, such as print, still video, full-motion video,
CD-quality audio, computer-generated graphics, animation, and textual
overlays. Selected examples of online tutorial sites are
Algebra Tutorial, Calculus Tutorial, and
What Is the Point?
The next two categories of games and simulations are both
goal-oriented activities that provide a multimedia simplification of reality. While
using simulators and games, learners encounter a dynamic situation to which
they must respond (Linser, Naidu, & Ip, 1999). In a game situation the
learner engages in a lose/win situation that requires the practice of skills assumed
to be known or in the process of development. Practice is said to
facilitate knowledge acquisition (Biggs & Moore, 1993). The artificial
environment provided by the software, therefore, motivates the learner through
an amusing activity that indirectly provides pedagogical benefits. Examples
of instructional games on the WWW include Multiplication
Matho, The Hanoi Tower, MathCar
Racing, The Number Machine, Worm
Hunt, Can You Add Them Up to 24? and
The idea behind simulations is to encourage learning within
artificial situations. One of the greatest advantages of simulators is their capacity
represent and connect huge amounts of information through
multimedia (Alessi & Trollip, 1991; Gibbons & Fairweather, 1998).
Simulations are similar to games in that they are goal-oriented. They
differ from games in that they are not explicitly governed by rules, and there is
no competition among the players or against the computer. The major
advantage of simulations is their capacity to represent the real-world in
circumstances when learning cannot be enacted in real terms; for example,
the manipulation of hazardous substances. Computational difficulties,
financial constraints in mimicking an activity, timeframe needed to replicate
the whole process, or magnitude of the equipment necessary for a
certain experiment, each creates unfeasible learning experience.
Teaching mathematics through online simulators requires the teacher to
keep a wise balance between personalized scaffolding and autonomous
learning. As with all the categories of computer based learning, it is helpful to
engage students in teacher and student led discussion before, during, and after
the activity. The following links take the reader to online simulation sites:
The Broken Calculator, Solve an
Equation, Finance Calculator, and
Hypermedia-based instruction (HBI) is a more complex form of
CAI (Ayersman & von Minden, 1995). The basic difference between HBI
and CAI is in the organisation of information. While most of the CAI
approaches present information on a relatively structured and linear sequence,
HBI organises information through a node-and-link structure.
Hypermedia approaches combine hypertext and multimedia. Multimedia delivers
content using several formats, such as text, sound, graphics, and video that work
to reinforce each other (Hall, 2000). Hypertexts are learning environments
in which knowledge is represented through a network of nodes of
information. Nodes of information are connected through clickable buttons to
other nodes, and users control navigation through the nodes. A hypertext has
been defined as "a database that has active cross-references and allows the
reader to jump to other parts of the database as desired" (Schneiderman &
Kearsley 1989, p. 3). The association of nodes on such a nonlinear structure permits
a learner to associate a variety of content within an exploratory context.
The non-linear dynamics of HBI empowers students, giving them
more autonomy, responsibility, and interactivity with the software (Hall,
2000). Such technical capabilities over traditional CAI approaches permit
the learner to build more meaningful connections among texts and
information. Ayersman and von Minden (1995) have argued that HBI allows students
to acquire more holistic understanding, participate actively in
explorative learning, and construct quality knowledge. It has also been claimed that
HBI proved successful in reaching a variety of learning styles given its
diverse use of media as compared to other traditional forms of instruction
(Hall, 2000; Liu & Reed, 1994; Melara, 1996; Summerville, 1999;
Weller, Repman, & Rooze, 1994). Some examples in mathematics education
occur at Maths Thesaurus, The Symmetry
Project, A Maths Dictionary and Matrix: The Virtual Maths
Tools and Open-Ended Learning Environments
Tools are electronic processes that assist learners in carrying out tasks,
such as planning, writing, calculating, drawing, composing, and
communicating (Alessi & Trollip, 2001). In mathematics education, tools such as
spreadsheets, databases, and graphics packages provide situations for
problem solving while also offering open-ended learning environments in
which students can investigate such concepts as geometrical and algebraic
patterns and relationships. Teachers can use these tools to assist their students
in learning mathematics through higher order thinking processes rather
than simply learning about the tool. As Luehrmann (2002a, 2002b)
indicated, such technological tools may be the predominant approach in many
schools. Although many multimedia packages are familiar to mathematics
teachers, such as Tesselmania (MECC, 1995) and the
Geometer's Sketchpad (Key Curriculum Press, 1995), their presence on the web is in its early
stage. Some of the following examples can be found:
Rotating Triangles, Function
Plotter, and Create a Graph, JAVA Gallery of Interactive On-Line
Geometry, and JavaSketchpad.
The previous discussion suggests that Alessi and Trollip's (2001)
recent categories of computer use in schools is a helpful framework for
classifying web-based mathematics learning activities. As a challenge it would
be interesting to see whether these software categories apply equally well
for other areas of the curriculum. Although each of these categories
facilitates different learning outcomes, simulations, hypermedia and tool-use
activities constitute the closest approach to a constructivist view of teaching
and learning mathematics. In fact, it could be argued that the sequence
of categories from drill, tutorial, game, simulation, and hypermedia to
tools and open learning environments mirrors a progression from behaviorist
to constructivist pedagogy.
Unfortunately, the frequency of online mathematical tasks reflects an
inverse relationship with a great deal less software available at the constructivist
end of the continuum. With an ever-increasing reliance on the web, both in
the classroom and at home, it is hoped that software developers and
educators will become aware of this apparent pattern and consolidate their
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A Maths Dictionary - http://www.amathsdictionaryforkids.com/
Algebra Tutorial -
Calculus Tutorial - http://www.karlscalculus.org/calc2_1.html
Can You Add Them Up to 24? -
Create a Graph -
Finance Calculator -
Function Plotter -
Interactive Arithmetic -
JAVA Gallery of Interactive On-Line Geometry -
Lessons on Order of Operations -
MathCar Racing -
Maths Thesaurus - http://thesaurus.maths.org/index.html
Matrix: The Virtual Maths Museum -
Multiplication Matho -
Percentage Estimation -
Problem Solving, Ratios, Fractions, Percentages, Shape
Rotating Triangles -
Solve an Equation -
Spacey Math - http://www.learningplanet.com/sam/sm/index.asp
The Broken Calculator -
The Hanoi Tower -
The Number Machine -
The Symmetry Project - http://www.adrianbruce.com/Symmetry/
What Is the Point? - http://www5.funbrain.com/cgi-bin/co.cgi
Worm Hunt - http://www.counton.org/index.html
Cumberland High School
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