Mara Alagic: What Change Is Technology Bringing to Conceptual Understandings of Mathematical Ideas?
What Change Is Technology Bringing to
Conceptual Understandings of Mathematical Ideas?
(WORK IN PROGRESS)
Mara Alagic
Wichita State University, Wichita, USA
mara@math.twsu.edu
Through every rift of discovery some seeming anomaly drops out of the
darkness, and falls, as a golden link into the great chain of order.
- Ed Hubbel Chapin (1814-880).
Abstract
Technology integration is bringing new lenses to our understanding of key mathematical
ideas. This paper addresses some of the issues relevant to teaching mathematics for conceptual
understanding in the technology-based environment. By reflecting on some of the existing
knowledge in this area and teachers deliberation as they learn to integrate technology in their
mathematics teaching, it strives to bring more light to the scene and capture the “struggle”
Paper presented at the IMECT3 - Third International Conference on Mathematics Enrichment
with Communication Technology at Robinson College, University of Cambridge, Cambridge, England July 2002
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Mara Alagic: What Change Is Technology Bringing to Conceptual Understandings of Mathematical Ideas?
Introduction
... the defining characteristic of knowledge workers is that they
are themselves changed by the information they process
Kidd (1994, p. 186)
Current research suggests that the use of technology integrated into the curriculum is a powerful
learning tool, bringing new lenses to our understanding of key mathematical ideas. Specifically in the
domain of mathematical knowledge development, technology (1) empowers teachers and students to deal
1with multiple representations, (2) enhances ability to visualize, (3) increases opportunities for development
of conceptual understanding, and (4) enhances opportunity for individualized learning.
For many teachers understandings of key mathematical ideas are grounded in the ways they have
learned them before this technology-induced paradigm shift was so powerful. These same teachers are
teaching new generations of pupils born and being educated surrounded with the explosion of emerging
technologies. Bridging this kind of “digital divide” requires revisiting roles that representations, and
translations among representations, play in mathematical learning and problem solving. This paper, in its
first part, brings together teaching mathematics for conceptual understanding in the technology-based
environment, by considering mathematics teaching and technology in general terms, exploring conceptual
understanding via multiple representations and addressing some effects of the cognitive tools to teaching
and learning mathematics.
“Technology in the Mathematics Classroom K-12” is a course that pre-service and practicing
teachers take to advance their knowledge of technology integration. Learners’ interactions (instructorteachers, teacher-teacher) played a significant role in designing this course. The qualitative features of these
interactions as they relate to learning to teach mathematics for conceptual understanding in the technologybased environment will be explored. Other results have been reported elsewhere (Alagic, 2002; Alagic &
Langrall, 2002). Data collected is comprised of online interactions, questionnaires, assignments, and
interviews of the teachers (students in the course). The second part of this paper is a reflection on teachers’
deliberations related to teaching mathematics for conceptual understanding in the technology-based
environment. The following four questions are in the center of these deliberations:
How are technology-based representations of mathematical concepts different from standard
representations?
What do you think is the influence of technology-based representations on YOUR conceptual
understanding of specific mathematical ideas and on the development of your pedagogical content
knowledge?
Paper presented at the IMECT3 - Third International Conference on Mathematics Enrichment
with Communication Technology at Robinson College, University of Cambridge, Cambridge, England July 2002
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Mara Alagic: What Change Is Technology Bringing to Conceptual Understandings of Mathematical Ideas?
What criteria are you as a teacher going to use to balance between technology-based and "other"
representations?
Mathematics Teaching and Technology
... what theoretical reflection we need if we want to really help
teachers to adequately use technological tools...
(Lagrange, 2002, p. 15)
Technology Principle. The National Council of Teachers of Mathematics (NCTM, 2000)
identified the "Technology Principle" as one of six principles of high quality mathematics
education. The principle states: "Technology is essential in teaching and learning mathematics; it
influences the mathematics that is taught and enhances students' learning" (p. 24). Preparing
teachers to integrate technology appropriately requires professional development that focuses on
both conceptual and pedagogical issues, and ongoing support. This is a complex task for teacher
educators and its success significantly depends on teachers’ beliefs and openness to changes in the
classroom practice (e.g.,Waits & Demana, 2000).
The research on use of technology reveals the complexities of the interplay of technology
and teaching in the learning of mathematics. One thing remains constant. It is ultimately the
mathematics teachers, not the technological tools that remain the key to the success of the
mathematical learning environment. Their own perspective on the nature of mathematics, on the
potential of the technology, and the training that they receive determines their effectiveness in
integration of the technology in mathematics learning. (Garofalo, Drier, Harper, Timmerman, &
Shockey, 2000; Kaput, 1992; NCTM, 1991, 2000). The external world is interpreted according to
ones own experiences, beliefs, and knowledge and therefore each person visualizes the external
world at least slightly differently. As learners, teachers are able to comprehend a variety of
interpretations and use them. But they cannot map their own interpretations of the world directly
onto their students, because they do not share a set of common experiences and understandings.
Yet another key piece of teacher knowledge for building a technology-based learning environment
is how to teach for transfer. Teaching practices congruent with a metacognitive approach to
learning include those that focus on sense-making, self-assessment, and reflection on what worked
and what needs improving. These practices have been shown to increase the degree to which
students transfer their learning to new settings and events (Schoenfeld, 1991).
Paper presented at the IMECT3 - Third International Conference on Mathematics Enrichment
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Mara Alagic: What Change Is Technology Bringing to Conceptual Understandings of Mathematical Ideas?
Technology in Context. The use of technology in mathematics teaching should support and
facilitate conceptual development, exploration, reasoning and problem solving, as described by the
NCTM (1989, 1991, and 2000). Technology enables users to explore topics in more depth and in
more interactive ways. It makes accessible the study of mathematics topics that were previously
impractical, such as recursion and regression, by removing computational constraints. Technologyaugmented activities should take advantage of these capabilities of technology, and hence should
extend beyond or significantly enhance what could be done without technology. These activities
can facilitate mathematical connections in a variety of ways: (a) interconnect and integrate
mathematics topics, (b) connect mathematics to real-world phenomena, (c) multiple
representations.
What factors determine the success of the use of technology in learning mathematics?
The following guidelines provide the essential ideas for strengthening mathematics instruction
while integrating technology (Garofalo, Shockey, Harper, & Drier, 1999; Flick & Bell, 2000).
1. Technology should be introduced in the context of mathematics content.
2. Technology should address worthwhile mathematics with appropriate pedagogy.
3. Technology instruction in mathematics should take advantage of the unique features of
technology.
4. Technology should make scientific views more accessible.
5. Technology instruction should develop students' understanding of the relationship between
technology and mathematics.
These guidelines are interconnected and our classroom experiences always addressed more
than one.
Teaching Mathematics for Conceptual Understanding
“Students do not necessarily interpret results in the manner that is
“obvious” (to the mathematics teacher).
(Dreyfus, 2002, p. 26)
When students attain understanding, what have they achieved? What students do in
response to the questions that put understanding into action show their level of understanding.
Students might be able to solve an equation, but if there is no understanding of where the equation
is coming from or where and how to use it, they may just be using a memorized skill that is going
Paper presented at the IMECT3 - Third International Conference on Mathematics Enrichment
with Communication Technology at Robinson College, University of Cambridge, Cambridge, England July 2002
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Mara Alagic: What Change Is Technology Bringing to Conceptual Understandings of Mathematical Ideas?
to be useful only for that type of equation, nothing more. Understanding a concept or a topic of
study is being able to carry out a variety of actions or performances with the topic by the ways of
critical thinking: explain, apply, transfer, generalize, represent in a new way, make analogies and
metaphors, and so on. It is being able to take knowledge and use it in new ways (Perkins, 1993).
Teachers set the classroom atmosphere for the kinds of inquiry in which students engage,
whether with the teacher or among themselves (Cohen, 1990; Porter, 1989; Thompson &
Thompson, 1994). The images and beliefs that they have about the nature of mathematics
influence the way they are teaching (Bauersfeld, 1980; Cooney, 1985). These images reveal
themselves in two main orientations: calculational and conceptual (Thompson, Philipp,
Thompson, & Boyd, 1994). Students also have varying degrees of conceptual or calculational
orientations to mathematics. Those who have adapted to calculationally-oriented instruction will
expect that the classroom discussions will be about getting answers (Nicholls, Cobb, Yackel,
Wood, & Wheatley, 1990). They will not only have difficulty focusing on their and others'
reasoning, they may also consider such a focus as being irrelevant to their images of what
mathematics is about. On the other hand, students who have adopted a conceptual orientation will
likely engage in longer, more meaningful discussions (Cobb, Wood, & Yackel, 1991). A
conceptual approach aims for students to solve problems by working from their own
understandings (Thompson, Philipp, Thompson, & Boyd, 1994).
For teachers, it is not sufficient to know how to solve the problem with which the students
may be grappling, nor is it sufficient to know several solution methods (McDiarmid, Ball, &
Anderson, 1989). To be able to facilitate students' thinking in productive ways, teachers need to
have an image of students' thinking as they develop these ideas. Any teacher can begin building
this image by encouraging students to reason and express him or herself accordingly, by listening
to their reasoning, respecting it, and asking students to do likewise.
The interplay of factual knowledge, procedural proficiency, and conceptual understanding
makes all three components usable in powerful ways. Students who memorize facts or procedures
without understanding often are not sure when or how to use what they know, and such learning is
often quite fragile (Bransford, Brown, & Cocking, 1999). Mathematics makes more sense and is
easier to remember and to apply when students connect new knowledge to existing knowledge in
meaningful ways (Schoenfeld, 1988). Learning with understanding also makes subsequent learning
easier.
Paper presented at the IMECT3 - Third International Conference on Mathematics Enrichment
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Mara Alagic: What Change Is Technology Bringing to Conceptual Understandings of Mathematical Ideas?
Studies in complex domains such as solving science problems (Bromage & Mayer, 1981;
Heller & Reif, 1984; Robertson, 1986) have suggested that conceptual understanding is associated
with connections -- connections between science concepts and everyday life and connections
among the different concepts in a discipline. Someone who is good at solving transfer problems
does not randomly connect concepts (which might occur when using memorized algorithms to
solve problems) but rather integrates the concepts into a well-structured knowledge base.
Conceptual Understanding via Multiple Representations
Things before words, concrete before abstract.
– Johann Heinrich Pestalozzi (1803)
How is the information processed once it has been perceived and has entered the cognitive
system? The answer to this question depends on the way information is represented in the system.
Understanding this constitutes an important piece of teacher knowledge for designing an effective
technology-enhanced learning environment. Some types of knowledge representation preserve
much of the structure of the original perceptual experience. Those are called perception-based
representations. Our minds have an ability to best remember what is most important. Meaningbased representations are quite abstracted from the perceptual details and incorporate the meaning
of the experience (Anderson, 2000). Representations can be a process and a concept; a tool for
thinking and a finished product. They are observable both externally and internally (NCTM, 2000).
There are many ways of knowing through representations: examples, models, demonstrations,
simulations, analogies, and metaphors.
MANIPULATIVES
= CONCRETE
MANIPULATIVES
= CONCRETE
PICTURES
TRANSLATION
BETWEEN MODES OF
REPRESENTATION
REAL-WORLD
EXPERIENCE
REAL-WORLD
EXPERIENCE
PICTURES
SPOKEN
SYMBOLS
SPOKEN
SYMBOLS
WRITTEN
SYMBOLS
WRITTEN
SYMBOLS
Paper presented at the IMECT3 - Third International Conference on Mathematics Enrichment
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Mara Alagic: What Change Is Technology Bringing to Conceptual Understandings of Mathematical Ideas?
Teachers who can represent a concept in a variety of ways provide a vehicle for all students
to grasp the concept and make connections to previous and future understandings. Teachers' use of
representations can supply a rich repertoire of access points for accommodating the different ways
students have been found to learn (Fischer, 1980, Bidell & Fischer, 1992, as cited in Fink, 1993),
provided such representations are already familiar to students (Janvier, 1987, p. 102-103; DufourJanvier, Bednarz, & Belanger, 1987).
Multiple representations for certain concepts have been linked with greater flexibility in
student thinking (Ohlsson, 1987, as cited in Leinhardt, et. al. 1991). Such flexibility, in turn, has
also been associated with better transfer of learning into the ill-structured domains typical of the
real world (Spiro, Vispoel, Schmitz, Samarapungavan, Boerger, 1987). As an arena in which
teachers' creativity can come to the fore, instructional representations provide a temporary context
for incubating student understanding. By blending familiarity and challenge to stimulate
development, they are akin to Papert's "microworlds" (1980), Schoenfeld's "reference worlds"
(1986, as cited in Leinhardt, et. al., 1991) and Kegan's (1982) "holding environments."
Representing knowledge. Langer (1989) and others (cf. Salomon & Globerson, 1987)
emphasize the importance of mindfulness in learning. Students learn and retain the most from
thinking in critical and creative ways. Some of the best thinking results when students try to
represent what they know. Representing knowledge require students to think in meaningful ways
to represent what they know, actively engage in creating knowledge that reflects their
understanding of mathematical ideas rather than absorbing predetermined presentations of
knowledge.
To use a calculator, a real-world problem must be restated in symbolic form. Students
starting to use calculators, especially if they are using them with problems involving more than one
operation, usually need some help (Wiebe, 1989). The fundamental problem here, as in other
examples of learning to use intellectual tools, is understanding and becoming proficient in abstract
representational systems that convey concepts. When using a calculator, students have to
understand both mathematical representations and how to translate information depicted in
question format to the final solution. The step from natural language to symbolic notation is
probably too large. Therefore, an intermediate representation would be useful in order to promote
understanding. It is this aspect which is crucial and is therefore the principal cognitive model
within the environment.
Paper presented at the IMECT3 - Third International Conference on Mathematics Enrichment
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Mara Alagic: What Change Is Technology Bringing to Conceptual Understandings of Mathematical Ideas?
MENTALMODELS/
INTERNAL REPRESENTATIONS
EXTERNAL
REPRESENTATIONS
& TRANSLATIONS
Problem solving and “word problems” are about restating the problem in different format,
symbolic form. The necessary cognitive process requires the mapping of mental models to
representations. Since there are multiple representations, translation between them is essential in
order to assist learning. Teachers in a technology-based learning environment facilitate the
understanding of mathematical concepts through the use of multiple representations and its
cognitive modeling (i.e. environment plus embedded model), (a) the construction and use of
mental models; (b) the mapping of mental models to representations (i.e. cognitive modeling); and
(c) the translation (i.e. re-representation).
Representation standard (NCTM, 2000) recognizes representations as essential elements in
building students’ conceptual understanding of mathematical concepts and relationships and in
communicating mathematical arguments and understandings to one’s self and to others. The term
representation, in the same standard, refers both to the process of “capturing” representation and
to the product - the form itself. Furthermore, the term applies to processes and products that are
observable externally (external representations) as well as to those that occur “internally” (internal
representations, mental models).
Teachers can gain valuable insight into students’ ways of interpreting and thinking
about mathematics by looking at their representations and building bridges from students’
personal representation to more conventional ones. Different representations often clarify
Paper presented at the IMECT3 - Third International Conference on Mathematics Enrichment
with Communication Technology at Robinson College, University of Cambridge, Cambridge, England July 2002
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Mara Alagic: What Change Is Technology Bringing to Conceptual Understandings of Mathematical Ideas?
different aspects of a complex concept or relationship. Students need opportunities to construct,
refine, and use their own representations as tools to support their own learning and doing
mathematics for conceptual understanding.
Students demonstrate conceptual understanding in mathematics when they provide
evidence that they can recognize, label, and generate examples of concepts; use and interrelate
models, diagrams, manipulatives, and varied representations of concepts; identify and apply
principles; know and apply facts and definitions; compare, contrast, and integrate related concepts
and principles; recognize, interpret, and apply the signs, symbols, and terms used to represent
concepts. Conceptual understanding reflects a student's ability to reason in settings involving the
careful application of concept definitions, relations, or representations. (NCTM, 2000).
Since every transfer can be considered as a new representation, the principle that people
learn by using what they know to establish their new understandings can be reformulated into
“learning occurs through representations” where representations are considered either as concepts
(objects, fixed representations) or as transfer-processes between different media. Processes of
learning and transfer are central to understanding how people develop important competencies. It
is especially important to understand the kinds of learning experiences that lead to transfer,
defined as ability to extend what has been learned in one context to new contexts (NRC, 2000). It
goes to understanding defined as flexible performance (Perkins, 1993), and more specifically to
conceptual understanding.
Problem Representation. The main reason that experts are better problem solvers than
novices is that they construct richer, more integrated mental representations of problems than do
novices. Their representations integrate domain knowledge with problem types, so they are able to
better classify problem types (Chi, Feltovich, & Glaser, 1981; Larkin, 1983). Whether problems
are represented as production rules (Anderson, 1983) or as schema-like forms (Chi & Bassock,
1989; Larkin, 1983) it is generally accepted that problem solvers need some kind of internal
representation (mental model) of a problem in order to solve a problem. Problem representations
can guide further interpretation/simulation of the problem, and/or associate with and trigger a
particular solution schema (Savelsbergh, de Jong, & Ferguson-Hessler, 1998).
Problem representation is the key to problem solving among novice learners as well as
experts. Instruction must help learners to construct problem representations that integrate their
problem representations with their knowledge. What characterizes good problem representations?
Paper presented at the IMECT3 - Third International Conference on Mathematics Enrichment
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Mara Alagic: What Change Is Technology Bringing to Conceptual Understandings of Mathematical Ideas?
The quality of internal problem representations is a function of the coherence (internal structure)
and the integration of the different representations (qualitative and quantitative, abstract-concrete,
visual verbal). What make experienced problems solvers more effective is their richer, more
coherent and interconnected representations of problems.
Multiple Representations. Cox and Brna (1995) have shown that when people are learning
complicated new ideas it helps to interact with various representations like diagrams, graphs and
animations. If the learner can integrate information from representations with different formats
then they often acquire a deeper understanding of the concept. On the other hand, if the learner
fails to make the connection between the different kinds of information, then many of the benefits
that multiple representations provide will not occur (e.g.Tabachneck, Leonardo & Simon, 1994)
and it can even inhibit learning (Ainsworth, Bibby & Wood, 2002). What then can be done to
ensure that learners can translate information between representations in the technology –based
environment?
Activities should incorporate multiple representations of mathematical topics. Research
shows that many students have difficulty connecting the verbal, graphical, numerical and algebraic
representations of mathematical functions (Goldenberg, 1988; Leinhardt et al., 1990). Appropriate
use of technology can be effective in helping students make such connections (e.g., connecting
tabulated data to graphs and curves of best fit). "We, as mathematics educators, should make the
best use of multiple representations, especially those enhanced by the use of technology, encourage
and help our students to apply multiple approaches to mathematical problem solving and engage
them in creative thinking" (Jiang & McClintock, 2000, p.19).
Teaching Mathematics for Conceptual Understanding
in the Technology-based Environment
Students do not necessarily do what seems “natural”
(to the instructional designer).
(Dreyfus, 2002, p. 27)
Consideration for representations as a tool for meaningful learning has been given by
anumber of researchers (e.g., Kaput, 1987; Greeno & Hall, 1997; Schultz & Waters, 2000). Greeno
and Hall emphasize importance of students’ exploration in selecting representations when building
their conceptual understanding of mathematical ideas. In the presence of technology, “the ability of
Paper presented at the IMECT3 - Third International Conference on Mathematics Enrichment
with Communication Technology at Robinson College, University of Cambridge, Cambridge, England July 2002
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Mara Alagic: What Change Is Technology Bringing to Conceptual Understandings of Mathematical Ideas?
students to operate within and between different representations of the same concept or problem
setting is fundamental in effectively applying technology to enhance mathematics learning”
(Demana & Waits, 1990, p.218). Schultz & Waters (2000), while using a variety of representations
for a given mathematical situation, explore some criteria for selecting representations that would
facilitate students learning. For example, which representations best (a) promotes conceptual
understanding, (b) generalizes to higher-level mathematics, (c) applies to finding approximate
solutions, (d) applies to finding exact solutions, (e) suits the learning style and comfort level of the
student? Which representation is best for a given type of technology? What representations do
your student prefer? Which one do you prefer?
With multiple contexts, students are more likely to abstract (yet another representation) the
relevant features of the concepts and develop a more flexible representation of knowledge.
Research has also shown that developing a suite of representations enables learners to think
flexibly about complex domains and develop their conceptual understanding orientation.
Technology-augmented activities should facilitate mathematical connections in two ways:
(a) interconnect mathematics topics and (b) connect mathematics to real-world phenomena.
Technology "blurs some of the artificial separations among some topics in algebra, geometry and
data analysis by allowing students to use ideas from one area of mathematics to better understand
another area of mathematics" (NCTM, 2000, p. 26). Many school mathematics topics can be used
to model and resolve situations arising in the physical, biological, environmental, social, and
managerial sciences. Appropriate use of technology can facilitate such applications by providing
ready access to real data and information, by making the inclusion of mathematics topics useful for
applications more practical, and by making it easier for teachers and students to bring together
multiple representations of mathematics topics.
A potentially abstract problem statement or an abstract arithmetic expression may be
illustrated using the visualization tools to encourage inquiry: i.e. three external graphical
equivalent and linked representational styles. In order to reach a reflective abstraction, scaffolding
through multiple representations should encourage inquiry. Instructional design in a technologybased environment has to take into consideration opportunities that multiple representations and
translations among them in order to encourage a learner’s inquiry.
Representations are essential components of a learning environment in which learners are
required to think harder about the topic being studied and to generate thinking that would be
Paper presented at the IMECT3 - Third International Conference on Mathematics Enrichment
with Communication Technology at Robinson College, University of Cambridge, Cambridge, England July 2002
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Mara Alagic: What Change Is Technology Bringing to Conceptual Understandings of Mathematical Ideas?
impossible without these representations. This should lead them to enhancing their conceptual
understanding. But it is not an orientation that can be created easily, and once created, easily
maintained (Romberg & Price, 1981; von Glasersfeld, 1988; Wood, Cobb, & Yackel, 1991). In
short, the real power of technologies to improve education will only be realized when students
actively use them as cognitive tools for building their own representations and translations among
them.
Cognitive tools. Derry (1990) defines cognitive tools as both mental and computational
devices that support, guide, and extend the thinking processes of their users. Jonassen (1992)
describes them as: "generalisable tools that can facilitate cognitive processing " (p.2). Cognitive
tools can make it easier for learners to process information, but their main "goal is to make
effective use of the mental efforts of the learner" (Jonassen, 1996, p.10). These are tools that are
used to engage learners in meaningful cognitive processing of information. They are knowledge
construction and facilitation tools that can be applied to a variety of subject matter domains. These
cognitive tools include specially designed knowledge construction tools, such as semantic
networking tools and micro worlds for mediating learning.
Jonassen and Reeves (1996) assert that well designed cognitive tools should: represent
knowledge (how someone depicts content or personal knowledge); be generalisable (can represent
knowledge in different content areas); engage the learner in critical thinking about the subject;
assist learners to acquire skills that are generalisable and transferable to other contexts; be simple
but powerful in order to encourage deeper thinking and processing of information; be easy to learn
- therefore the mental effort needed to learn the software should not exceed the benefits.
The primary distinction between traditional learning applications of technologies and their
use as cognitive tools is best expressed by Salomon, Perkins, and Globerson (1991) as the effects
OF technology versus the effects WITH computer technology. When students work WITH
computer technology, instead of being controlled by it, they enhance the capabilities of the
computer, and the computer enhances their thinking and learning. The result of an “intellectual
partnership” with the computer is that "the appropriate role for a computer system is not that of a
teacher /expert, but rather, that of a mind-extension cognitive tool" (Derry & LaJoie, 1993, p.5).
Technology should not be used to carry out procedures without appropriate mathematical
and technological understanding (e.g., inserting rote formulas into spreadsheets). Nor should it be
used in ways that can distract from the underlying mathematics. Another way to prevent
Paper presented at the IMECT3 - Third International Conference on Mathematics Enrichment
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Mara Alagic: What Change Is Technology Bringing to Conceptual Understandings of Mathematical Ideas?
technology use from compromising mathematics is to encourage users to connect their experiential
findings to more formal aspects of mathematics. Technology should not influence students to take
things at face value or to become what Schoenfeld (1985) referred to as "naive empiricists."
An early stage of knowledge development for designing a technology-enhanced learning
environment might entail an understanding that some representations are better than others for
portraying particular aspects of a situation, knowledge of a range of representations, as well as a
developmental approach to the ways teachers and students can use them (Vergnaud, 1987; Alagic
& Langrall, 2002). With the increased availability of IC technology to provide students with easy
access to a range of representational formats, explicit instruction in crossing between symbol and
referent, as well as how certain representations convey mathematical content more efficiently than
others is now being seen as a crucial aspect of mathematics education (Kaput, 1987).
Another early stage of knowledge development for designing a technology-enhanced
learning environment might involve the awareness that students, faced with multiple
representations for the same concept often learn how to make one correspond with another using
the syntactic rules of math, but without developing a sense of the underlying concept being
represented. Premature or inappropriate use of representations can cause frustration and
misconceptions in children and place undue focus on the representation at the expense of the target
concept; thus, effective representations for the younger student must be based on students' own
drawings and codes (Dufour-Janvier, et. al., 1987, Alagic & Langrall, 2002).
Yet another stage of development might include the recognition that with today’s emerging
technologies the very nature of the problems that can be solved and methods used in the process
are changing: performing calculations; collecting, analyzing, and representing numeric
information; creating and using models and simulations; representational scaffolding higher levels
of abstraction, solving problems with mathematical premises. The hands-on, minds-on learning
experiences fostered through today’s interactive technology applications empower students with a
level of mathematical power they cannot achieve without technology. (Potential for stimulating
higher order thinking when freed from the mechanics of calculating.)
A more advanced stage could involve the understanding of the dialectic between perception
and conceptualization. For example, accessing geometrical knowledge is more often presented as
resulting from the ability to rely efficiently both on spatial and geometrical competencies, as
opposed to resulting from rejection of some perceptive apprehension of geometrical objects.
Paper presented at the IMECT3 - Third International Conference on Mathematics Enrichment
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Mara Alagic: What Change Is Technology Bringing to Conceptual Understandings of Mathematical Ideas?
Fostering the dialectic interplay between these differing competencies through more emphasis on
the relevant traits within problems and situations lead to the development of geometrical expertise
(Hoyles & Keith 1998, Laborde 1998, as reported in Lagrange, 2001).
Spreadsheets as cognitive tools for “producing” representations
Spreadsheets have become popular tools for exploring mathematical phenomena and
building conceptual understanding of mathematical ideas. Spreadsheets have three primary
functions: storing, calculating, and presenting information. A spreadsheet program can file
information, usually numerical, into a particular location (the cell). This enables information to be
accessed and retrieved efficiently. Most importantly, spreadsheets support calculation functions.
The numerical contents of any combination of cells can be mathematically related in just about any
way the user wishes. Cells can be added, multiplied, and factored in any combinations of ways.
Most spreadsheets provide mathematical functions such as logarithms and trigonometric functions.
It also includes sophisticated tools for generating tables and graphs.
Spreadsheets are rule-using tools that require users to become rulemakers (Vockell & van
Deusen, 1989). Calculating values in a spreadsheet requires that the user identify relationships and
patterns among the data that he or she wants to represent in the spreadsheet. Next, those
relationships must be modeled mathematically, using rules to describe the relationships in the
model. Building spreadsheets requires abstract reasoning by the user, thereby matching one of the
important goals of cognitive tools. The combined calculational and graphical capabilities of a
spreadsheet provide a context to engage students in analyzing and connecting multiple
representations.
Spreadsheets also support problem-solving activities. Given a problem situation with
complex quantitative relationships, spreadsheets can be used to represent those relationships. The
"what if?" thinking that is supported by spreadsheets is essential to decision analysis. Such
reasoning requires learners to consider implications of conditions or options, thereby engaging
higher order thinking (Sounderpandian, 1989). Identifying values and developing formulas to
interrelate them in spreadsheets enhance learners' understanding of the algorithms used to compare
them and also the mathematical models used to describe content domains. It is sometimes useful to
provide guided activities and problems to structure the use of spreadsheets. For example, to
support higher-level thinking skills such as collecting, describing, and interpreting data, Niess
(1992) provided students with a spreadsheet with wind data from various towns. Wind directions
Paper presented at the IMECT3 - Third International Conference on Mathematics Enrichment
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Mara Alagic: What Change Is Technology Bringing to Conceptual Understandings of Mathematical Ideas?
described rows of data, with the percentage of days for each month of the year representing
columns. She then asked students to use the spreadsheet to answer queries, such as: Are the winds
more predominant from one direction during certain months? Why do you think this is the case? In
which months is the wind the calmest? Which wind direction is the most stable during the year?
What Research Supports the Use of Spreadsheets as Cognitive Tools? A few studies have
examined the effects of different instructional treatments on learning to use spreadsheets (Charney,
Reder & Kusbit, 1990; Kerr & Payne, 1994; Tiemann & Markle, 1990). These studies were not
investigating the cognitive requirements or effects of using spreadsheets. Rather they were
interested in the effects of different computer-based tutorial treatments, and spreadsheets happened
to be the content or skill being learned. Baxter and Oatley (1991) compared the effectiveness of
two different spreadsheet packages. Not surprisingly, the users' prior experience level with
spreadsheets was far more important to learning than the usability of the software package. These
studies provide few insights about the effectiveness of spreadsheets as cognitive tools.
In one of the rare studies investigating spreadsheets as cognitive tools, Sutherland and
Rojano (1993) were interested in how prealgebra students could use spreadsheets to represent and
solve algebra problems. This study was conducted simultaneously in Britain and Mexico and took
place over a 5-month period. During that time, students moved from a strict cause-effect local
numerical notion of algebraic relationships to general rule-governed relationships that could be
symbolized both in the spreadsheet and in algebraic notation. Another study used spreadsheets in
community college math classes to help students solve linear and nonlinear equations problems
(Hulse, 1992). Non significant increases in mathematics achievement and decreases in numerical
computation anxiety were reported; however, this study was so methodologically flawed by short
treatment times and the use of inappropriate measures of achievement that it would be difficult to
generalize the results.
Teachers Deliberations: Capturing the “strugle”
"Technology in the Mathematics Classroom K-12” is a three-week summer course
(http://education.wichita.edu/alagic/summer2002/752r/752r_first_page.htm) which teachers take
either as a part of their requirements for graduate coursework or from a desire to advance their
knowledge of technology integration. The underlying themes are:
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with Communication Technology at Robinson College, University of Cambridge, Cambridge, England July 2002
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Mara Alagic: What Change Is Technology Bringing to Conceptual Understandings of Mathematical Ideas?
1. Experiencing and doing mathematics as problem solving, reasoning, connecting and
communicating through a variety of representations in the technology-based learning
environment” (NCTM 2000).
2. Recognizing how conceptual understanding and procedural knowledge are developed
together and that their mutual development is enhanced (reinforced) through technology.
3. Reinforcing awareness of changes in the teaching of school mathematics brought both by
current school reform for standards-based teaching that supports integration of technology
and by the development of IC technology.
4. Evaluating a variety of computer programs and web resources for learning and doing
mathematics.
5. Organizing a class for differentiated instruction.
Responses to four of the questions posed at the end of the course are reported below.
Question 1.
How are technology-based representations of mathematical concepts different from standard
representations? Explain in details an example appropriate for the level you are teaching (or will
be teaching).
Teachers mainly supported their answers with examples from this classroom experience.
These examples of mathematical concepts varied from interactive web-sites to the most
sophisticated software available to them (web-quests, interactive work sheets, concept mapping
software, computer algebra systems, and dynamic geometry). Teachers perceived technology
based-representations to be richer, hands-on and more interesting than standard representations,
visual, and dynamic. The main benefit that they report on is more flexibility for teachers. They
recognized variety in methods of presentation, levels of engagement for the learner, opportunities
for feedback and the ability for the teacher to connect many related topics at once. Specifically,
according to these teachers, advantages of the technology-based representations are that they

Appeal to more of the student's senses and help students with their visualization.

Add dimension to the solution of problems.

Are interactive.

Can be manipulated or changed by the students.

Allow the students to get a different picture of what the mathematical concept actually is.

Help make connections between what the students see through the visual representation
and the abstract concept the teacher is trying to teach.
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Mara Alagic: What Change Is Technology Bringing to Conceptual Understandings of Mathematical Ideas?

Are largely visual and hands on which appeal to many learners.

Are more dynamic, the concept can be animated or edited to “come alive”.

Can involve more of a student ownership/control learning situation.
Based on these answers, reflections and classroom discussions, this group of teachers
focused on advantages of well-designed tools. They seemed to neither have a critical stance in
making their choices nor recognize intrinsic changes that technology-based representations are
bringing to their teaching. Maybe the tone of the class sessions was set too strongly in the direction
of finding appropriate technology based representations that enrich our mathematical ideas. We
speculate that reflecting during and after classroom experiences in their own schools will trigger
some questions.
In the responses to the written question above, all but one teacher strongly recognized a
variety of differences between technology-based and “standard” representation. She writes,
In actuality, I do not believe that technology-based representations are all that different from the
standard representations. Tech-based reps are just in a different format. One enhancement for
tech-based representation was with the proof of the Pythagorean theorem. Using the Web, an
animated version of the proof could lead to a more concrete example of why and how this theorem
works... the worksheet maker for the math facts.
Some of the teachers related to their previous experiences with learning mathematics. One
of them writes,
I do not have the trigonometry background or as much algebra background as some of the
teachers have. The lessons they shared helped me to begin to understand what they were talking
about. As an example, one concept is hyperbola and graphing it. When they could use technology
to show me the actual path or lines, it helped me to see it in my mind. My math teachers from years
ago didn’t help me make the connection between algebra and geometry to the real world. We just
followed the formulas, did the work.
Question 2.
What do you think is the influence of technology-based representations on YOUR
conceptual understanding of specific mathematical ideas and on the development of your
pedagogical content knowledge? Give an example first. Then, elaborate.
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Mara Alagic: What Change Is Technology Bringing to Conceptual Understandings of Mathematical Ideas?
Mathematics related pedagogical content knowledge term was briefly mentioned in this
class, not really explored. The researcher’s uncertainty about these teachers understanding of
pedagogical content knowledge in terms of Shulman (1986) appeared to be warranted. In an
expectation that teachers would select a mathematical concept and elaborate on their specific PCK
related to that example, the question asked for an example to start with. But most of the teachers
kept their answers quite general. For example,

Technology-based representations will take a concept that is not thoroughly understood and
direct you to many other ways of looking at that concept, therefore elevating your level of
conceptual knowledge.

If there is some theory or rule in math that I do not fully understand, the Internet can be very
helpful in teaching me it. Also, by searching for various ways to teach something I can in turn
learn different ways that students may understand something and see various ways to teach it
to students.

.. a visual and hands on learner....Technology can help cement the concepts in my own brain
and help me gain a higher level of understanding so that I can teach the concepts to students
and have the ability to break down the concept in many different ways to serve the different
learning styles of the students.

... the visual representation ...teachers that struggle with specific concepts can use technology
to help them better understand and better teach that concept to students.

... foresee some of the misconceptions and prior knowledge....I should be able to improve my
pedagogical content knowledge
Examples are followed by general comments. Not much about technology-based
representation as support for scaffolding and differentiating instruction.

... tessellation ... My own pedagogical knowledge increased each time I visited a different site
because it would review the definition of the topic and also show other ways of understanding
and teaching the concept to students

I experience immediate feedback and am able to extend and explore ideas that occur to me
quickly, Green Globs.... Gauge just how much altering I need to make in order to accomplish...

... spreadsheet on probability by doing a word problem....This example, teach me that by using
spreadsheet, I can practice my knowledge of probability with a hand on experience, which was
wrote functions to do the calculation.
Paper presented at the IMECT3 - Third International Conference on Mathematics Enrichment
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Mara Alagic: What Change Is Technology Bringing to Conceptual Understandings of Mathematical Ideas?

... Pythagorean theorem...it has helped further my understanding of mathematical concepts and
has brought to light some other details I missed in other mathematical concepts.

.... compound interest. I have had the concept explained to me before, but when I applied the
information to the spreadsheet, I had a clearer understanding of the concept

...multiplication can be taught by using manipulative...multiplication spreadsheet where the
appropriate cells would fill in with color giving a visual picture of the math fact.
And, some additional reflections:

Since I was brought up before the computer age, I have a great respect for what technology
can do for the discovery of new mathematical concepts. For example, in calculus, the concepts
of a secant line “moving” closer to becoming a tangent line. To see that under animation
made me feel warm all over. It made the once difficult- to -understand concept, rather easy.

The use of technology adds the third dimension to any subject matter.

I plan to go through my curriculum with new insight. I will question myself as I plan, if
technology can be beneficial.

When I become a teacher, show them how to use technology-based representations to solve
real world problem. Then let them practice with the technology. I think this is one of the best
ways of learning.
Question 3.
What criteria are you as a teacher going to use to balance between technology-based and
"other" representations?
Teachers’ answers fell into the following categories:

Availability of the technology:
-
I can usually get in the lab about 3 times a week for about 45 minutes.
-
I will use what technology is available to me...take students to the computer lab to use
software to reinforce my teaching.

Improving students’ understanding and achievement (learning styles of students, reinforcing
teaching, students needs),
-
If the technology is going to improve student achievement.
-
One that can benefit most of students in the classroom.
-
If technology can help me to demonstrate some of the concepts that I am teaching,...
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Mara Alagic: What Change Is Technology Bringing to Conceptual Understandings of Mathematical Ideas?


Balance and variety
-
A balance between traditional and technological instruction.
-
By mixing your teaching up, your students are more likely to stay interested.
-
Varity is crucial for the students, and me.
Teacher’s knowledge and confidence with use of technology
-
When I am very comfortable with (technology), that I know how to use.
Although most of the teachers identify specific criteria, some of them are cautious to
explicate any specific ground for decision making. They are describing the process they think they
will go through when making choices. The following reflective answers capture the essence of
some teachers’ dilemmas

I won’t forgo all other forms of representations and go all out technology. The balance will
come when I get a “feel” for where my students are in their use of current technology. The
overuse of a certain technology tool will need to be avoided.... The initial concepts of the
subject need to be addressed initially in traditional ways. Problems solved on the blackboard
help the student see the steps needed to come up with the solution. Once the student is given the
background of the subject, the use of technology should be introduced to strengthen its
application or practice.

If I can put together a standard representation that will work just as well in a shorter amount
of time, I might opt for the standard.

I will experiment with a variety of technological representations and maybe even use more
than one in the same concept as needed by the individuals I am teaching. I do not believe that
all classical representations should be eliminated nor do I think a classroom without any
technology is the best either.
One teacher rank-ordered his criteria in the following response:
First, availability of technology...Secondly, learning styles of the teacher and how
effectively they understand the technology available. Lastly, which representation might be most
effective in the instruction of that particular concept?
Question 4.
How can technology help in differentiating instruction for individualized learning? Be
specific.
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with Communication Technology at Robinson College, University of Cambridge, Cambridge, England July 2002
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Mara Alagic: What Change Is Technology Bringing to Conceptual Understandings of Mathematical Ideas?
The attitudes toward differentiating instruction through the use of technology were
overwhelmingly positive. We see that as a major difference from some of our earlier findings, with
different cohort of teachers (Alagic & Langrall, 2001). Teachers quoted number of examples
explored during the instruction and explained how that could help them differentiate instruction.
Here are a few of those examples.

Web sites included step-by-step detailed instructions on multiplying which could be used as an
introduction for new students or as a review for students having difficulty, resource for whole
classroom activities, a computer center in the classroom, or to use for students ahead, or to use
as a tutor for students who are need extra help.

... multiplication....each student to work at their own pace and see immediate results. Providing
altered repetition for those students needing more instruction and giving the advanced student
an opportunity to explore on their own with guidance from the teacher.

Worksheet maker web sites are available to work with students’ individual abilities. A teacher
could easily tell if students needed further instruction and practice or if they were ready to
move on. Students could move ahead as they were ready and not be held back from further
learning because the teacher needed to work with students that hadn’t mastered the concept
yet.

Cooperative learning experiences could be used through WebQuests or other group projects
that are technology based.

It is important to try different ways to approach a concept, for example, learning to find area
and perimeter. By using the Geometer’s Sketch Pad, students can see the process appear.
Students can vary the method in which they receive the information. ... Also, a student can
repeat steps as many times as necessary to reinforce understanding.

If a teacher was using Web Quest or a PowerPoint software students could work at their own
pace. ... Students take charge of their own learning and move forward at their own pace.
Some teachers preferred to keep their answers more general and reflect on their
experiences:

A regular classroom ... multiple levels of abilities. Using technology can help close the gap.
Just like in our class some students just are starting out ... Other students can learn more
complex technology and add it to their presentation.
Paper presented at the IMECT3 - Third International Conference on Mathematics Enrichment
with Communication Technology at Robinson College, University of Cambridge, Cambridge, England July 2002
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Mara Alagic: What Change Is Technology Bringing to Conceptual Understandings of Mathematical Ideas?

... hands on learners who have been struggling to understand the abstract concepts of
mathematics ... reach the advanced learner ... an accelerated pace while the other learners
can continue at a pace that is comfortable for them ... those independent learners who are too
proud or too intimidated to ask for help.
Future Inquiry
... the defining characteristic of knowledge workers is that
they are themselves changed by the information they process
Kidd (1994, p. 186)
Shifts in the philosophy and theory of learning as well as emerging technologies support
the view that a paradigm shift in teaching and learning mathematics with the use of information
and computing technologies is taking place. The existence of increasingly efficient cognitive tools
lends support to the view that the learning environment in school mathematics is changing into a
more technological one. Teachers are aware of current changes and are involved in the processes
of these changes in their schools.
Technology integration is bringing new lenses to our understanding of key mathematical
ideas. For many teachers, understandings of these ideas are grounded in the ways they have
learned them before this paradigm shift that technology is providing was so powerful. These same
teachers are teaching new generations of pupils born and being educated surrounded with the
explosion of emerging technologies.
Many teachers are disillusioned by their experience with technology integration so far.
Marcinkiewicz (1991) points out that teachers are often not sure that the skills and experiences
they acquire in available technology training will be easily transferable to classroom instruction.
High-quality training, sufficient resources and awareness of necessary change are some of the
critical factors necessary to regain the trust (Cafolla & Knee, 1995). To build confidence, teachers
need successful experiences and ongoing pedagogical and technological support when integrating
technology into their curriculum (Byrom, 1997).
Mathematics teachers need opportunities to experience and do mathematics in
environments supported by diverse technologies (Dreyfus & Eisenberg, 1996). Understanding,
using, and appreciating mathematics are essential components of the development of mathematical
power. Empowering teachers through the use of technology in mathematics exploration, openPaper presented at the IMECT3 - Third International Conference on Mathematics Enrichment
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Mara Alagic: What Change Is Technology Bringing to Conceptual Understandings of Mathematical Ideas?
ended problem solving, interpreting mathematics, developing conceptual understandings and
communicating about mathematics is in the heart of professional development and teacher
education (Bransford, et al, 1996; Schoenfeld, 1982, 1992). Teachers need to experience and learn
in depth how conceptual understanding emerges in technologically based environment to better
understand

“the conditions under which their students will be able to see on the screen what is
evident to the software designer” and, we expect, to the teacher,

“do in activity what seems natural to the instructional designer,

conclude from the data what is obvious to the teacher and” hopefully

“think in a way that is logical to the mathematician” (Dreyfus, 2002, p.30).
Many other questions of interest remain open. For example:
 Because technology-based representations can make conventional representations
dynamic and interactive, do they provide a more immediate way to map students'
developing understandings? If so, how could such “maps” provide valuable insights
into students’ thinking to help new teachers develop their mathematics related
pedagogical content understanding more efficiently? (Alagic & Langrall, 2002).
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