Lengel F. Gonito
BSED III-A1 Mathematics
Mathematical
Investigation and Modeling
Mathematical
investigation
A
mathematical investigation is typically defined as an investigation that
involves a collection of mathematical and problem solving based issues. Such
issues generally have multiple uses and purposes. They are typically open-ended
in their content. There are generally more than one possible solution, and use
more than one method in solving the issue at hand. Mathematical investigations
may also consist of statistical data. Mathematical investigations are
furthermore centered on some sort of theme or central issue/hypothesis
(Marie, 2001).Mathematical investigation refers to the sustained exploration of
a mathematical situation. It distinguishes itself from problem solving because
it is open-ended. Mathematical investigation is more of a divergent activity. In
mathematical investigations, students are expected to pose their own problems
after initial exploration of the mathematical situation. Ronda (2010) said that
the exploration of the situation, the formulation of problems and its solution
give opportunity for the development of independent mathematical thinking
and in engaging in mathematical processes such as organizing and recording
data, pattern searching, conjecturing, inferring, justifying and explaining conjectures and
generalizations.
Mathematical
investigation refers to the sustained exploration of a mathematical situation.
It distinguishes itself from problem solving because it is open-ended. Problem
solving is a convergent activity. It has definite goal, the solution of the
problem. Mathematical investigation on the other hand is more of a divergent
activity. In mathematical investigations, students are expected to pose their
own problems after initial exploration of the mathematical situation. The
exploration of the situation, the formulation of problems and its solution give
opportunity for the development of independent mathematical thinking and in
engaging in mathematical processes such as organizing and recording data,
pattern searching, conjecturing, inferring, justifying and explaining conjectures
and generalizations. It is these thinking processes which enable an individual
to learn more mathematics, apply mathematics in other discipline and in
everyday situation and to solve mathematical (and non-mathematical) problems.
Teaching anchored on mathematical investigation allows for students to
learn about mathematics, especially the nature of mathematical activity and
thinking. It also make them realize that learning mathematics involves
intuition, systematic exploration, conjecturing and reasoning, etc and not
about memorizing and following existing procedures. The ultimate aim of
mathematical investigation is develop students’ mathematical habits
of mind.
Although students may do the same mathematical investigation it is not
expected that all of them will consider the same problem from a particular
starting point. The “open-endedness” of many investigations also means
that students may not completely cover the entire situation. However, at least
for a student’s own satisfaction, the achievement of some specific results for
an investigation is desirable. What is essential is that the students will
experience the following mathematical processes which are the emphasis of
mathematical investigation:
- systematic exploration of the given situation
- formulating problems and conjectures
- attempting to provide mathematical justifications for the conjectures.
Mathematical Modeling
Mathematical modeling is generally understood as the process of
applying mathematics to a real world problem with a view of understanding the
latter. One can argue that mathematical modeling is the same as applying mathematics where we also
start with a real world problem, we apply the necessary mathematics, but after
having found the solution we no longer think about the initial problem except
perhaps to check if our answer makes sense. This is not the case with
mathematical modeling where the use of mathematics is more for understanding the real world problem.
The modeling process may or may not result to solving the problem entirely
but it will shed light to the situation under investigation. The figure below
shows key steps in modeling process. Mathematical modeling approaches can be
categorized into four broad approaches: Empirical models, simulation models,
deterministic models, and stochastic models. The first three models can very
much is integrated in teaching high school mathematics. The last will need a
little stretching.
Empirical modeling involves examining data related to the problem with a view
of formulating or constructing a mathematical relationship between the
variables in the problem using the available data.
Simulation modeling involves the use of a computer program or some
technological tool to generate a scenario based on a set of rules. These rules
arise from an interpretation of how a certain process is supposed to evolve or
progress.
Deterministic modeling in general involves the use of
equation or set of equations to model or predicts the outcome of an event or
the value of a quantity.
Stochastic modeling takes deterministic modeling one further step. In
stochastic models, randomness and probabilities of events happening are taken
into account when the equations are formulated. The reason behind this is the
fact that events take place with some probability rather than with certainty.
This kind of modeling is very popular in business and marketing.
My Reflection
Mathematical
investigation and modeling it involves
a collection of mathematical and problem solving based issues and the
process of applying mathematics to a real world problem with a view of
understanding the latter.
Mathematical
investigation and modeling is one of those subjects I have taken that presents
challenges or the most challenging not only for me but also for my classmates. This
subject is a kind of subject that is not as simple as what we are thinking of.
When I enrolled at first semester of my 3rd year level in college
one of the subjects that I saw as I fill up the enrollment form is MIM which
stands for Mathematical Investigation and Modeling as a part of my chosen major
which is mathematics. It really caught my attention. I think for the possibilities
or ideas of what this subject is all about. By simply reading the word
investigation and modeling, the word immediately come in into my mind is that
maybe this subject is difficult and challenging. But instead of thinking
negative things I used to prepare myself to face the goodness, difficulties or
challenges at the back of this unfamiliar major subject.
The
most awaited moment finally come, the day when I first encounter Mathematical
Investigation and Modeling. Our professor defines it as a process of collection of mathematical and
problem solving based issues and the process of applying mathematics to
a real world problem with a view of understanding the important concepts and
problems. Through further discussion I learned lots of things about it, yes
indeed MIM is really challenging. Critical thinking, analytic thinking and
deeper understanding are the requirements for you to engage yourself to this
field.
Time
goes by as we go through studying MIM I have encountered challenges, I started
to experience solving problems which have higher level of difficulty compare to
the common mathematical problem, those problems under this subject really
arouse our mental abilities. But the good thing is that I am no alone as I
faced those challenges I or in solving those mathematical problems, our
professor allows us to have our groups to solve problems together and with the
help of every group member. For this instance although the complexity is still
there the tasks within solving is now more exciting because you are working
with you group, the difficulty of conducting investigation in solving such
situation is lessen because there are multiple numbers of ideas coming from
different minds that united or converged. Little by little I have started to cope
with the lesson and with this subject. Many
weeks and months had passed I started to enjoyed taking this subject, maybe
because through this I have meet my progress in engaging myself to the new
level of solving problems. Another reason is that through this I started to
adopted new environment of studying mathematical problems which improved my
mental ability though I can’t avoid feeling confused and headache brought by
the tricky word problems. Another good thing I realized after taking this
subject is that it molded great cooperation and collaboration within everyone
of us, as we work together. It made us to realize the importance of sharing
ideas then mixing them together to come up to the better result.
Being
part of this is what we called modeling, which refers to application of the
mathematical problems or constructing word problems with its relation to the
real life set up or application. In here I learned how to connect mathematical
problem to the situation wherein I or we commonly observed in our daily lives.
Through this I have noticed and proven that mathematics is really relevant when
talking about its connection to the life of people. This subject is a factual
proof about the importance of mathematics in the life of every one of us, that
its existence is not just to have this for us to study about but it has an
important function not only in education but also in real life situation.
The
bloody experience in solving word problems finally ended up, when midterm part
is finally over as well. It was nice to think that we have passed that level of
difficulty and hardship but breaking news suddenly heard by my ears. Our
adviser told us that we have to conduct a demonstration teaching as a part of
our requirements on his subject MIM, and for us to apply the knowledge and
learning we have acquired. We are asked to select topics we are going to used
and construct an investigation and modeling regarding that topic. It is hard to
make that one because we are required to make investigation and modeling which
is new and unique to our professor. We
are all challenge to make this for the sake of our grades. We find difficulty
in this kind of task, what we have conducted first is making lesson plan which
was MIM approach lesson plan and then have its application by means of
demonstration. Demonstration began, nervous among my classmates can be observed
until the day of my turn finally arrived. It served as a big challenge to my
part because many of my classmates who are done in their demonstration got a
good and best result. The rats inside my chest started to run, I don’t even
know how this presentation happened but all I did is just built the trust to me
and potentials. This is really worth it, I thanked God because I got a good
result though my grade is not as high as what others got, I feel contented and
deserved to what I work for. This experience taught me more things. It served
as my good practice on how I will handle my future, especially when I reach the
moment of being a successful teacher.
To sum it up, I therefore conclude that having this subject
as part of my chosen course really helpful. It enhanced one’s ability
especially when it comes to mathematics. It serves as an training and practice
especially to those who are in the pasture of studying mathematics as their
field of specialization. Although the suffering is there but it’s just a tool
or part of challenges that awakened one’s mind to achieved their progress
regarding their capability, as a mathematician and great problem solver. It
made me to realize that learning enables us to do things we
were never able to do, change our
perception of the world and our relationship to it, and extend our
capacity to create.
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