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Mathematics in nature
From rainbows, river meanders and shadows to spider
webs, honeycombs and the markings on animal coats,
the visible world is full of patterns that can be
described mathematically. A closer observation of
such readily observable phenomena reveals
beautifully how nature uses Mathematics. Knowledge
of Mathematics enables a deeper appreciation of such
natural phenomena as cloud formation, halos, tree
heights and leaf patterns, butterfly and moth wings,
and even puddles and mud cracks. A beautiful
symbiosis exists between the basic scientific
principles at work in nature and their mathematical
expressions.The universe is made up of galaxies, mountains,
vehicles, living creatures and all manner of other
things, each seemingly unique. Moreover it is a
chaotic affair in which these interact with one
another in different ways, often violently, but
sometimes with great subtlety. But, thanks to
Mathematics, we are able to think about the world of
objects and events, and to communicate those
thoughts in ways that reveal unity and order.
The
numbers, lines, angles, shapes, dimensions,
averages, ratios, probabilities, operations, cycles,
correlations, etc that make up the world of
Mathematics enable us to make a sense of the
universe that otherwise seem to be hopelessly
complicated. Mathematical patterns and relationships
have been developed and refined over centuries, and
the process is as vigorous and productive now, as at
any other time in history. This is perhaps because,
today, Mathematics is applied in more fields of
endeavor than ever before, and has also become more
essential in everyday life.Mathematics can be found
in some surprising and beautiful places.
Mathematical concepts reveal themselves in natural
forms, in art, and in decorative patterns that are
traditional in many cultures. Perfect symmetry and
pattern are to be found in starfish, snow flakes and
Islamic tiles; ratios and mathematical sequences
describe the structure of sunflowers and of
classical Greek temples. Fractal geometry is
revealed in the formation of coastlines and
mountains. The Chaos Theory that can be simulated
with a bouncing ball can describe complex natural
phenomena such as weather systems.
Mathematics has been used to model plant growth. The
model treats leaves, stem and root as different
compartments, and then assumptions are made as to
how different environmental factors affect the rate
of change of biomass or nutrients in the different
compartments. These models are typically framed as
systems of differential equations. Population models
are used to determine the effects of different
assumptions about the age, size or the spatial
structure of a population on the dynamics of the
population. Models of graph theory describe the
topological structure of food webs. Natural resource
management models help set harvest quotas for fish
and game. Conservation ecology uses mathematical
models to help determine the relative effects of
alternative recovery plans for endangered species as
well as aid in the design of nature preserves. The
very idea of mathematical modeling of natural
phenomena must have been born from the realization
that Nature is the Mathematician Supreme.
Math Vs
Other Subjects…(Math Is Special)
• Math is best learnt by doing. The homework
problems help you learn the formulas and techniques
you need and improve your problem-solving skills.
• Each Math session builds up on the previous ones.
You must keep pace with the Instructor. Losing some
classes in between will not only sever the
continuity of thought but may also bring
down your morale.
• Since each session builds on the previous ones,
you're always reviewing previous material as you do
new material. Many small ideas together make a
bigger one.
• Focus on identifying the concepts and absorbing
them, rather than on memorization.
Math
At Different Levels - College Math Vs High School
Math
A College math class may meet less often and cover
more material in every class than a High School
class does. You are expected to absorb all new
material quickly. The instructor may not follow-up
your homework and the tests may be less frequent,
but will cover more material. Therefore, more time
has to be devoted to studies each day.
How Much To Study
Generally it is accepted that a one-hour session
calls for double the time, that is two hours of
careful study. This has to be kept in mind while
planning the day and the week. The last session of 2
hours each week can be a group study session, where
you may meet your friend and both of you go over the
difficult questions that you did during the week.
This helps in better retention of the concepts.
Also, chalk out the plan to meet your instructor
with questions that both of you could not solve.
How To Solve Problems
• The higher the math class, the more problem types.
In lower classes, problems require just one or at
the most two steps to solve. But now, problems will
require several steps to solve them. The trick is to
break each problem down into smaller pieces and
solve each piece - divide and rule!
• You will typically encounter the following problem
types:
o Problems that test your memory.
o Problems that test your skills.
o Problems that call for applying your skills to
handle practical situations.
o Problems that call for applying your skills to
handle “not so practical” situations.
• Lower classes generally involve problem types 1, 2
and 3. In higher classes you will deal mostly with
problems of types 2 and 3 and sometimes type 4. Type
4 problems usually need a multi-step approach, and
an amalgamation of ideas and techniques.
• Pólya's four-step approach to problem-solving:
1. The first step in solving a problem is to read
carefully and understand the problem. Identify
exactly what is the problem asking for.
2. Decide on an approach. Draw a picture if possible
or make a table. Recollect similar cases that you
may have handled before. Identify which skills and
techniques you have learnt are applicable to the
problem at hand.
3. Implement.
4. Verify. Does your answer appear reasonable in the
context of the problem? If not, start all over again
from Step 1.
• Every homework or assignment must be taken as a
test, with all the seriousness. Teachers have always
found that students who were regular with their
homework invariably did well in the tests. Preserve
all your work carefully.
Studying For Tests
• You are already halfway through your test
preparation if you have been doing your homework
seriously. Still, go over each topic; review your
notes or handouts. Memorize the formulas.
• Work out the homework questions again. If you have
forgotten something, refer back and find out how you
had done it earlier.
• Test yourself. You may know the subject matter but
yet you may not be confident about taking the test.
Work out difficult problems from the Exercises and
those that your teacher had drawn your special
attention to.
• Get enough rest. Your mind needs to be fresh and
alert for the test.
Action Plan For Taking Tests
• First look over the entire test to get a sense of
its length. Identify problems you can do right away
and those you expect to have to think about.
• Take the easy problems first and build up
confidence. This way, you will also not miss out on
sure points just because you run out of time. Then
try the problems you think you can figure out; then
finally try the ones you are least sure about.
• Work by the clock. Figure out how much time you
can spend on each question. Work as quickly and
continuously as you can. If you get stuck in any
problem, move on without wasting any more time on
that.
• Show all your work: In Math, steps do carry value.
Clearly show as many steps as required and no more.
• Do not waste time on erasing incorrect things.
Just scratch out and move on.
• In a multiple-step problem outline the steps
before actually working the problem.
• In a multi-part question, every part carries
points separately. Do whatever is possible.
• Verify your answers wherever possible.
• If you finish early, check every solution.
MATH HOMEWORK TIPS FOR PARENTS:
· Encourage your child to use a daily math assignment book.
· Follow the progress your child is making in math. Check with your child daily about his homework.
· If you don't understand your child's math assignments, engage in frequent communication with his or her teacher.
· If your child is experiencing problems in math, contact the teacher to learn whether he or she is working at grade level and what can be done at home to help improve academic progress.
· Request that your child's teacher schedule after-school math tutoring sessions if your child really needs help.
· Advocate with the principal for the use of research-based peer tutoring programs for math. These tutoring programs have proven results, and students really enjoy them.
· Use household chores as opportunities for reinforcing math learning such as cooking and repair activities.
· Try to be aware of how your child is being taught math, and don't teach strategies and shortcuts that conflict with the approach the teacher is using. Check in with the teacher and ask what you can do to help. Ask the teacher about online resources that you can use with your child at home.
· At the beginning of the year, ask your child's teacher for a list of suggestions that will enable you to help your child with math homework.
How to solve problems?...
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