INTEGRATING MATHEMATICAL MODELING AND
TECHNOLOGY IN TEACHING AND LEARNING
MATHEMATICS
Bülent Çetinkaya
Aysel Şen
Sinem Baş
bcetinka@metu.edu.tr
sen.aysel@gmail.com
bashsinem@gmail.com
Middle East Technical University, Faculty of Education
Department of Secondary Science and Mathematics Education, Ankara / Turkey
ABSTRACT
Models and modelling perspective as a new educational approach has promising implications for teaching, learning, and research and may
have important suggestions for the development and revisions of the new national mathematics curriculum in Turkey. According to the models
and modeling perspective, students construct, describe, represent, interpret and evaluate real life problems or situations through “modeleliciting activities.” Modeling is a cyclic process which includes simplification, mathematization, transformation and interpretation, and
verification. In this modeling process, students need to use technological tools, such as educational software, graphing calculators and
spreadsheets to be able mathematize and explain a given problem situation. Thus, the main purpose of this paper is to describe the models
and modeling perspective, present the importance of this perspective in teaching and learning mathematics, and discuss opportunities this
perspective offers to integrate technology into teaching and learning process. We also share an example of model-eliciting activities to
illustrate how to integrate technology in developing a mathematical concept.
Keywords: Model and Modeling Perspective, Teaching and Learning Mathematics, Use of Technology.
INTRODUCTION
Teaching mathematics in a collaborative environment, which
is important for students’ affective and social development,
is another objective emphasized in the new curriculum and
again is fundamental part of the modeling activities. “Models
and modeling perspectives suggest that zones of proximal
development are multi-dimensional regions where
interactions include not only teachers who are more capable
but also learners interacting with peers, with themselves,
and with powerful representational tools and media” (Lesh &
Doerr, 2003a, pp. 544). Group works provide opportunities
for students to develop their communication skills and make
them partially independent from the teacher and thus
provide environment to learn the mathematics themselves
and in a meaningful way.
In today’s world many of the most important inventions in
fields ranging from engineering to economics are created
through modeling some complex systems that exist in
nature or that are created by human. Education is one of the
relatively recent fields where modeling, especially
mathematical modeling, is used to understand some
phenomena in the educational system and to make
productive inferences about them.
Modeling as a perspective of teaching and learning
emphasize the importance of constructing, representing,
interpreting and assessing real life problems or situations in
a collaborative environment. The development of problem
solving skills (e.g., simplifying a problem; analyzing subproblems and data by using tables, figures or graphs;
exploring patterns; making predictions about the result of
the problem and testing them; developing equations from
data and using them; making selections among strategies)
is one of the major tenets of this perspective. According to
this perspective, through modeling activities, students are
expected to construct models, and express their
mathematical thinking and reasoning skills with associating
verbal and mathematical expressions in their modeling
process. These activities aim to develop students’ inquiry
and investigation skills including realization of the problem,
planning about how to solve the problems, develop ideas,
making decisions about the results whether they need
revisions or extensions, and whether they satisfy the
conditions and assumptions given in the problem (Lesh &
Doerr, 2003a).
Having students work on model-eliciting activities may help
them develop aforementioned skills that are emphasized in
the new mathematics curriculum. Thus, in the next section,
we explicate the models and modeling perspective, present
the role and the importance of this perspective in teaching
and learning mathematics, and discuss opportunities this
perspective offers to integrate technology into teaching and
learning process.
MODELS AND MODELING PERSPECTIVE
Theoretical Foundations
Models and modeling perspective is a new educational
approach that characterizes mathematics problem solving,
learning, and teaching in a new way (Lesh & Doerr, 2003a).
Although its assumptions are basically originated from
constructivism, they are different in several respects. Social
cognitive aspects in its theoretical foundation and several
other considerations move this perspective beyond
constructivism (Lesh, Doerr, Carmona, & Hjalmarson,
2003).
Similar emphasis can also be seen the new mathematics
curricula. Although this emphasis may be less explicit in
elementary level, it is overtly stressed in the secondary
school mathematics curriculum. The new curricula underline
the role of comprehension of mathematical systems and
concepts, making connections between them, improving
students’ reasoning and problem solving skills and applying
them in real-life problems (Bulut, 2007).
Models and modeling perspective is mainly based on three
basic assumptions. One of them is that people interpret their
experiences using internal conceptual systems (Lesh,
Doerr, Carmona, & Hjalmarson, 2003). These conceptual
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systems, like schemata or situated cognitive structures that
cognitive scientists refer to, are a kind of sense-making
systems. According to the models and modeling perspective
these systems are called “models” and described as
“conceptual systems consisting of elements, relations,
operations, and rules governing interactions that are
expressed using external notation systems, and that are
used to construct, describe, or explain the behaviors of
other systems- perhaps so that other system can be
manipulated or predicted intelligently” (Lesh & Doerr, 2003b,
p.10). Doerr and Tripp (1999) claim that it does not mean
that all conceptual systems are models. A conceptual
system can be considered as a model only if it is created for
specific purposes such as interpreting some complex
systems (Doerr & Tripp, 1999).
word problems regardless of their reasoning processes. For
this reason desired model development rarely occurs while
students solve these types of word problems.
Furthermore, in a traditional problem solving approach,
students are required to learn some mathematical concepts
prior to being able to solve a problem. However, Lesh and
Doerr (2003b) contend that “problem solving should be a
critical step on the way to learning; it is not simply an activity
that should occur after a concept has been learned…” (p.
30). Model-eliciting activities are based on this assumption,
and problem solving forms a context where students engage
in a learning process.
Considering shortcomings of the traditionally accepted
problem-solving activities in terms of supporting students’
mathematical thinking, researchers developed six principles
that are intended to guide the design of model-eliciting
activities (Lesh, Hole, Hoover, Kelly, & Post, 2000). One of
them is “the model construction principle” according to
which the task should enable students to develop a
description, explanation, or justified predictions of the given
problem to elicit their conceptual systems (p.12). The
second one is “the reality principle” that requires the task to
enable students to use their own existing knowledge and
experience and to reflect on significant real-life situation
(p.18). The third one is “the self-assessment principle” which
means that the task is designed in a way so that students
can assess if their interpretations or ideas are good enough
or need improvement based on the usefulness of their
responses (p. 21). The fourth principle is “the constructdocumentation principle” according to which the task should
require the students to produce documentation that would
explicitly reveal their thoughts and solution ways taken into
account throughout the activity (p.22). The fifth principle is
called “the construct shareability and reusability principle”
that requires the task to challenge students to develop
model to be applicable in other situations and be shareable
with other students or other interested groups (p 24). Finally,
according to “the effective prototype principle”, the task
should encourage students to develop a solution that have
explanatory power to provide a useful prototype for
interpreting other structurally similar situations (p.25).
The other assumption of this theory is that these models do
not completely reside within the minds of humans (i.e., they
are not totally internal) and they are generally expressed
using a variety of interacting representational media that
ranges from written symbols to technological tools such as
educational
software,
graphing
calculators,
and
spreadsheets (Lesh, Doerr, Carmona, & Hjalmarson, 2003).
The central point to highlight is that because different media
represents different aspects of models, multiple media can
be used to externalize these models at the same time (Lesh
& Doerr, 2003b). Thus, the term “model” encompasses
more than internal conceptual systems in the language of
models and modeling theory. Specific purpose for which
they are created and representational media through which
they are externalized are the other main components of
models (Lesh & Lehrer, 2003).
The final assumption is that models are a kind of
interpretation systems and they are frequently developed for
reinterpretation of experienced complex systems as well as
reinterpretation of models themselves (Lesh, Doerr,
Carmona, & Hjalmarson, 2003).
Model-Eliciting Activities
The literature suggests that modeling or mathematical
modeling can be understood differently. According to
Galbraith (1999), there are two types of mathematical
modeling: (a) structured modeling, and (b) open modeling.
In structured modeling tasks, question statement ask
students to find a particular unknown, which makes
mathematics and context inseparable (Brown, 2002). Open
modeling tasks, on the other hand, requires students to
“investigate the particular phenomena” (p.68) in the problem
statement which is given in real-world terms and have no
apparent links to mathematics (Brown, 2002). Thus, student
is expected to formulate the problem in mathematical terms.
The nature of the structured modeling, thus, brings about
the utilization of technology as an answer-checking tool
whereas open modeling tasks improve the potential and
complete use of technology (Brown, 2002). In the present
study the example of the latter one is presented.
The Modeling Process
In model-eliciting activities, students go through a modeling
process at the end of which they are expected to construct
shareable and reusable mathematical models. This process
has been described by many researchers in a slightly
different ways. Nonetheless, there is a consensus that the
process is cyclic and iterative, and students make
extensions, revisions, refinement or rejections of their initialprimitive models (Dossey, McCrone, Giordano, & Weir,
2002; Zbiek & Conner, 2006). This cyclic process continues
until students create productive and powerful models for the
predetermined purposes (Hodgson, 1999; Lesh & Doerr,
2003b).
In the mathematics classroom where traditional teaching
methods are used, problem-solving activities generally entail
linear problem solving process and this problem solving
process does not strengthen the mathematical thinking skills
of students too much and improve students’ problem solving
versatility. Lesh and Doerr (2003a) document that models
and modeling perspective is different from problem solving
approach in several different dimensions. In the models and
modeling perspective, students’ descriptions, explanations,
and interpretation processes about real life problem
situations are valued and the importance of development of
powerful mathematical models in mathematics education is
emphasized (Doerr & Tripp, 1999). It is known, however,
that traditional textbooks and standardized achievement
tests ask students to give short numeric answers to the
Figure 1 portrays the modeling process described in the
Standards (NCTM, 1989, p. 138). It shows that
mathematical modeling is a non-linear process including
interrelated steps. According to the author of NCTM, there
are five basic steps in the mathematical modeling process
(NCTM, 1989):
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1.
Identify and simplify the real world problem
situation
2.
Build a mathematical model
3.
Transform and solve the model
4.
Interpret the model
5.
Validate and use the model
REAL
Real-world
Problem
situation
Simplification
step requires students test if predictions and conclusions
reached through the model are meaningful and valid in the
real world situation (p.17). Thus, the model is evaluated
concerning its consistency with its predetermined specific
purpose (Zbiek & Conner, 2006).
ABSTRACT
Interpretation
If the constructed model passes mentioned tests in the
validation process, it can be considered as a powerful model
with its “sharable” and “reusable” attributes (Lesh & Doerr,
2003b, p.14). However, sometimes the solution to the realworld situation or problem is not accounted for by the
mathematical model. If this is the case, then the students go
back to earlier steps and repeat the entire process or part of
it several times. This iterative nature of modeling process is
explicated by Dossey et al. (2002). They asserted that there
is a dynamic interaction among the steps of a modeling
process. For example, in the event of that students can not
construct a model (in the second step) or con not reach
mathematically significant solution to the identified problem
within the model (in the third step), they should go back to
the first step and revise their conditions and assumptions.
Likewise, in the case of that the conclusions and predictions
made through the model do not make sense for both
identified problem and initial real world situation (in the
fourth and fifth steps), students are expected to revise prior
steps (Dossey et al., 2002). Zbiek and Conner (2006) use
the term of “aligning” to express these enduring
metacognitive control mechanisms of students and
emphasize their importance for a successful modeling
process (p.104).
Transformation
Validation
Problem
Formulation
Solution
within the model
Mathematization
Mathematical modele.g. Equation(s),
Graph(s)
Figure 1. The Standards’ model of the modeling process.
In the first step, students identify a problem to be solved in
the real world situation, and state it in its most precise form
as possible. By mathematically observing, questioning and
discussing, they think about what information in the given
situation is important or not. Thus, they simplify the situation
by ignoring less important components at first. Sometimes,
this is an easy step, while other times this may be the most
difficult step of the entire modeling process. Listing the key
features and relationship among those features may help
students to simplify the situation. This process also includes
“the action of specifying” because students specify
conditions and assumptions related to the situation in order
to consider and use them in the next step, build a
mathematical model (Zbiek & Conner, 2006, p.99).
In the second step, students create mathematical
representations of specified components of the problem and
the relationships among them. In this step, students define
variables, establish notation, and explicitly identify some
form of mathematical relationship and structure, make
graphs, and write equations. All of these mathematization
attempts eventually encourage students’ building of a
mathematical model. In their description of modeling
process, Zbiek and Conner (2006) explain this
mathematization process as finding “mathematical
properties and parameters” related to “the conditions and
assumptions” that were identified before (p.99). Lesh and
Doerr (2003b) combine these two steps, simplification and
mathematization, and call it as “description” (p.17).
The Role and Importance of the M&M Perspective in
Mathematics Education
Modeling process can be considered as a significant
learning process (Lesh & Doerr, 2003a). When going
through iterative modeling cycles, students have various
opportunities to learn mathematics, which is true for each
sub-process of modeling (Zbiek & Conner, 2006). For
example, in the modeling activities, students are confronted
with a real life problem situation in which some important
mathematical constructs are embedded. For dealing with
this problematic situation, they are encouraged to participate
actively in reorganizing and developing their existing
conceptual systems through modeling cycles. Once their
developed conceptual systems (i.e., models) are structurally
similar to those that underlie current mathematical
constructs embedded in this situation, learning of this
mathematical construct occurs (Doerr & Tripp, 1999).
According to models and modeling perspective, this learning
of mathematical constructs with modeling can be considered
as a local conceptual development that is similar to Piaget’s
general stages of development related to these constructs
(Lesh & Doerr, 2003b).
In the transformation step, students analyze and manipulate
the model to find mathematically significant solutions to the
identified problem. This step is usually familiar to students.
The model from second step is solved, and the answer is
understood in the context of the original problem. The
students may need to further simplify the model if it cannot
be solved. Many times the solution procedure involves
analytic, numeric, and graphic techniques.
In the interpretation step, students carry their mathematical
solution that is reached in the context of mathematical
model back to the specified (or formulated) real world
problem situation. Then, they test and evaluate whether the
solution is meaningful for this problem situation (Hodgson,
1999). In other words, they test if their solution created
through model makes sense or not in the problem context.
In a sense, this step is similar to mathematization in that
students are challenged to establish a link between the
model world and real world (Zbiek & Conner, 2006). This
sub-process is included as “translation” in Lesh and Doerr’s
diagram of modeling process (p.17).
Mathematical modeling addresses affective, cognitive, and
social aspects of learning. It considerably promotes
students’ motivation to learn in many respects. Firstly, when
students notice that they are able to deal with a situation
that they can encounter in daily life by using mathematics,
they recognize the benefit of mathematics to people and
motivate themselves to learn it (Hodgson, 1995). Secondly,
as they progress towards iterative modeling cycles with a
predetermined purpose, they may sometimes face the
weakness of their current ways of thinking and recognize
that their existing mathematical knowledge is not enough to
proceed. That can be another source of motivation to learn
new mathematics (Zbiek & Conner, 2006).
In the final step, students think about the validity and
usefulness of the created model in terms of initial real world
situation besides addressing the identified problem
(Hodgson, 1999). As mentioned before, models are created
for specific purpose in specific situations. Like Lesh and
Doerr’s (2003b) description of “verification” process, this
Modeling theory emphasizes that for productive use of
created conceptual systems (or models) in dealing with
complex problem situations students should externalize
them with representational media (Lesh & Lehrer, 2003).
Furthermore, according to this theory,
in order for
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2003) in students’ minds and these reorganizations
gradually improve their conceptual development. This is why
models and modeling perspective emphasize the multiple
representational media. Consequently, technology plays a
significant
role
in
coordinating
mathematizing,
transformation, interpretation steps in the modeling process.
significant forms of learning to occur during the modeling
process students’ externalized models should be “sharable
with other people” and “reusable for other purposes” (Lesh
& Doerr, 2003b, p.14). These underlying assumptions point
out consideration of learning as contextual and situated.
In their study, Doerr and Tripp (1999) attempt to portray
students’ mathematics learning from models and modeling
perspective. They reveal that the interactions of students
with their externalized models, current problem situation,
and other learners’ internal (mental) models are the most
important source of learning. As mentioned before, modeleliciting activities provide students with such an environment
in which students are challenged to experience these
interactions. Particularly, interactions among students are
considerable. In enduring discussion processes during
these activities, one student’s questions or conjectures
might challenge the others’ way of thinking and cause
development of it in a productive way (Doerr & Tripp, 1999).
Therefore, social aspect of learning is another important
component of modeling perspective.
Furthermore, technology can allow students to integrate
different branches of mathematics (i.e., algebra, geometry,
statistics and probability) or to relate mathematics to other
subject areas (i.e., physics, biology, and economics). In this
sense, by providing multiple connections, technology
supports students’ complete development of understanding
in the subject area (Pead, Ralph, Muller, 2002) which is an
important goal of the model-eliciting activities.
Another benefit of the technological environment in
modeling approach is that it enables students to deal with
problems despite they may not know some requisite
knowledge or have difficulties with computation. For
students, to be unacquainted with some arithmetic
operations should not unable them to solve some
meaningful problems. Moreover, difficult calculations will
consume students’ time and thus limit their work. However,
the ease of calculations and exploration of the solution with
the help of computers or other technological tools will help
students in making conjectures, testing assumptions, and
making sense of their findings.
The Use of Technology in Modeling Process
Technology plays an important role in teaching and learning
mathematics through modeling activities. The nature of
model-eliciting activities offers that the appropriate use of
technology enhances the potential of these activities to
provide enduring learning opportunities for students. Kendal
and Stacey (2004) reported that “problem solving abilities of
students
is
enhanced
through
linking
different
representations” (p.341) and technology has the power to
incorporate and generate representations for explaining and
transforming an indeterminate to a determinate situation
(Confrey & Maloney, 2002). Because they project on a
screen different dynamic, and graphic representations and
thus presents different ways of thinking, technological tools
such as spreadsheets, dynamic geometry software and
graphing calculators can be seen as an alternative to
enhance students’ problem solving abilities. It is known that
these technological tools help students access to graphical,
numeric and symbolic representations and additionally
move between them.
Use of technology based tools through model eliciting
activities could also help teachers to monitor and assess
progress of students and notice their conceptual strengths
and weaknesses. As stated in previous section, students’
understanding and thinking in model-eliciting activities can
be more complex because of the progressive nature of
activities and multiplicity of the solutions and answers. Since
these tools help students to express their current ways of
thinking, this testing will assess students’ work with more
diagnostic information. In this sense, use of technology with
modeling activities can help students develop mathematical
skills such as symbolization of the data, mathematization of
the problem situation, and generalization of the results.
To sum up, although the integration of technology into
mathematics instruction in our country is still in progress,
eventually, mathematical investigations that make use of
computers and calculators will be more effective and will be
more beneficial for students and teachers. Therefore,
instructional technologies, such as graphing calculators and
spreadsheets should be strongly linked to the content of
mathematics instruction.
Johnson and Lesh (2003) stated that the meaning of
underlying conceptual systems often distributed among the
number of interacting representational media. For modeleliciting activities when students pass through a series of
modeling cycles the results that students generate are some
combinations of lists, graphs or tables formed by these
media. When initial types of these representations are
created in the mathematization step, in transformation step
these linked representations enable students to modify one
system (e.g. tables) and at the same time to monitor the
effects of these activities in other systems (e.g. graphs).
Thus, by the help of technology, students can get feedback
readily to their conjectures about the problem and see
whether results are consistent with the theoretically
predicted ones. Thus, students can make sense the
relationships between changing graphs and links between
them.
Example of Model-Eliciting Activities
Bobcat Problem
Most species of wild cats are endangered including bobcat. This
project explores the behavior of a bobcat population using growth
rate data from the state of Florida (from Cox et. al, 1994). We
know annual growth rates for bobcats under best (r = 0.01676),
medium (r = 0.00549), and worst (r = -0.04500) environmental
conditions. For this project, assume that growth rates are constant
from year to year. For now consider each growth rate as
representing a different region in Florida.
As different ways of thinking and different media facilitates
or restricts the different aspects of the problem situations,
they may cause interpretations or inferences that are
incompatible with each other (Johnson & Lesh, 2003). For
instance, in some cases a graph produced by a spreadsheet
may be interpreted differently by group members. Thus, in
the interpretation step, this discrepancy will lead students to
integrate some other representational media such as written
symbols, equations or drawings. Thus, by examining their
interpretations, students try to understand the complex
patterns underlying problem situation and try to simplify
them. Hence, “reorganizations” are made (Johnson & Lesh,
1. Construct a spreadsheet (or other simulation) tracking three
bobcat populations, each initially consisting of 100 individuals,
over the 25 years under the three types of environmental
conditions. Plot all three simulations on a single graph and explain
the behavior of the bobcat population in these three types of
conditions.
2. When this bobcat population is growing under the best
conditions in an uncontrolled way, this growth may disturb the
ecological balance of animals. Several management plans have
been discussed for dealing with this situation. The first is to allow
one bobcat per year to be hunted. The second is to allow five
bobcats per year to be hunted. The third is to allow one percent of
the animals to be hunted. The last is to let five percent of the
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increased rapidly under the best conditions, increased
slightly under medium conditions, and declined significantly
under the worst conditions (the interpretation step). In the
task, it is expected to keep the growth of the population
under control. As it was assumed in the first step, therefore,
hunting a specific number or a specific percentage of
bobcats that enables the population to be stable at one point
under the best conditions can be a productive strategy (the
validation process).
animals be hunted. Construct a simulation which compares these
strategies. Explain which strategies are suitable for dealing with
uncontrolled growth of bobcat population.
3. Continuing with the theme of part 2, experiment to find
strategies which enable controlling the growth of bobcat
population under the best condition. Consider that 200 animals
can be acceptable for keeping the ecological balance.
Note. Adapted from Mooney & Swift (1999).
Examining the modeling process
At this point, how hunting a specific number or a specific
percentage of bobcats can change the population is
considered. Can these strategies increase, decrease or
stabilize the population under the best condition (making
assumption process)? In the text, there are four suggestions
towards this purpose. Again through using Excel, it can be
observed whether these management plans are suitable to
keep ecological balance or not under the best conditions. It
is expected that, these observations of the suggested
management plans through mathematization prompts the
invention of the new strategies.
As mentioned in the Standards’ model of the modeling
process, firstly, the mathematical problem that is embedded
in real world situation is identified through clarifying the
conditions and determining some assumptions (the
simplification step). The conditions are obvious in this realworld situation. There are, for example, three annual growth
rates for the bobcats populations which are referred to as
“best” (r = 0.01676), “medium” (r = 0.00549), and “worst” (r
= -0.04500). Moreover, initial number of the population is
100 and this number might increase and decrease in a way
that impacts the ecological balance. The ultimate purpose is
to find the way in which the undesired growth of the
population can be controlled and the balance is maintained.
Again, the behaviors of the bobcat population after the
implementation of the management plans can be
represented through the equations Pt+1 = Pt + r*Pt -1 (to
allow one bobcat per year to be hunted), Pt+1 = Pt + r*Pt -5
(to allow five bobcats per year to be hunted), Pt+1 = Pt + r*Pt
- (0.01)Pt (to allow one percent of the population to be
hunted), and Pt+1 = Pt + r*Pt - (0.05)Pt (to allow five percent
of the population to be hunted), respectively (the
mathematization step). The effectiveness of these
management plans can be examined in Figure 3.
Some assumptions are also specified. First one has already
been given in the text that the growth rates do not change
from year to year. Secondly, when examining the behavior
according to these rates over a time period, the numbers of
animals will become fractional. Because presence of
fractional bobcats is not meaningful, a round down is
required in the results. Another assumption can be related
to the already given and newly invented management plans.
If the 100 animals increase rapidly with approximately 1.6%
growth rate under the best condition, it may be the case that
other animal population would be endangered. In this case,
attempting to stabilize the population at one point by hunting
might be an efficient way of keeping the balance (the
simplification process).
When this mathematical model is analyzed, the
mathematical meanings of the graphs of the functions give
information about the behaviors of population after the
implementation
of
the
management
plans
(the
transformation and the interpretation steps). Namely, this
model enables some informative real-world conclusions
such as allowing hunting of 1% of the population and
allowing one bobcat per year to be hunted were almost the
same. In fact, the population was still increasing even after
these two cases of hunting were allowed. On the other
hand, allowing 5% or 5 bobcats to be hunted caused the
bobcat population to decrease and eventually approach
extinction. Indeed, hunting 5 bobcats per year seemed to
have the most adverse effect on the bobcat population (i.e.,
leads to extinction after only 25 years). Consequently, these
strategies could not enable the desired stabilization and
they are not suitable for keeping the ecological balance (the
validation process).
In the third step, based on the initial assumptions and
directions of the question, the behaviors of the bobcat
population in these three types of conditions can be
mathematically represented through the equation P t+1 = Pt +
r*Pt (note, P: population for a given year, t: time in years, r:
respective growth rate). Next, the behaviors of bobcat
populations in three environmental conditions (with different
growth rates) can be expressed numerically and graphically
through an Excel spreadsheet (See Figure 2).
Figure 2. A graph produced in solution to bobcat problem
By analyzing this graph (i.e., a simple model of the
situation), some mathematical conclusions can be made
(the transformation process). For example, the increase in
the first graph of function may be considered as linear and
this increase is more rapid than the second one, whereas
the third graph of function decreases curvilinearly. When
these mathematical conclusions are interpreted in simplified
problem context, it can be said that the bobcat population
Figure 3. Graphs of bobcat populations after implementation
of the four suggested management plans
At the end of the task, to stabilize the bobcat population at
200, new assumptions are introduced and the model is
refined. One assumption is to add certain number of
bobcats into population so the population rises to 200 and
then stabilize at 200. Additionally, although the text does not
mention the time limitation, the priority of the strategy that
245
enables the desired stabilization over a reasonable time
period can be considered.
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In light of these assumptions, two different strategies are
developed by manipulating the third suggested
management plan (to allow one percent of the population to
be hunted). First strategy is to allow one percent of the
population to be hunted until 200 and then to subtract a fix
number of bobcats that is approximate to the growth rate
(making assumption). This formulation can be represented
in a numeric table by using a spreadsheet (mathematization
step). Using this spreadsheet, the amount of bobcats to be
hunted and frequency of this process are determined: 10
bobcats per 3-year periods (transformation step). The graph
of this strategy can be seen in the Figure 4.
Figure 4. The graphs of management strategies
According to this graph, although the desired stabilization
occurs, this strategy (i.e., model) needs to be revised
because the time necessary for population to be stable at
200 is too long, approximately 110 years. This period of time
is not reasonable with respect to real world context (the
validation process). It is then assumed that the time period
can be shortened by allowing the routine annual growth
from 100 to 200 animals without any interruption. Once the
numbers of animals reach 200, 10 bobcats every three
years can be subtracted to stabilize the population at this
point. The mathematization of these assumptions can be
seen from the graph of the spreadsheet.
This graph indicates that, the last acknowledged strategy is
the most powerful and valid one because it is consistent
with all of the assumptions and conditions determined
throughout this modeling process. More importantly, it
addresses the initially defined purpose. Moreover, it makes
sense in the context of real world because the time period
required to implementation is practical (validation step).
In conclusion, this example shows that the modeling
process requires several modeling cycles. The primitive
models that are created at the beginning are iteratively
revised through these cycles and improved towards the
more useful and more powerful models. There are dynamic
transitions among the steps and these transitions do not
have to be linear. These dynamic interactions among these
sub-processes continue until the creation of productive and
powerful model(s) for the predetermined purpose(s) with the
help of technology.
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