CLARITY
OF DEFINITIONS
Methodical Approach to Education [click]
Clarity of Definitions [click]
The Age Factor [click]
Systemic and Incremental Approach [click]
Fundamental Skills [click]
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Ask your student a question if he or she
can give you the correct definition of the height
of a triangle or the a definition of a monomial.
In most cases the answer will be incomplete or entirely
incorrect.
The answer will
probably
not start with a noun, as it should when the
question is "what is it?". Clarity and
precision
of fundamental definitions are the driving force
behind true success in mathematics and science.
Indeed,
what is the height of a triangle - any triangle? Hardly
any student will be able to provide a complete
answer and this is where the first failure occurs:
we believe that students will be able to arrive at
their own definition. It is absolutely not true! Fundamental
knowledge is called fundamental
for
a reason: it
is a starting point for everything. Compare your student's
answer to the accurate definition:
THE HEIGHT
OF A TRIANGLE IS A PERPENDICULAR LINE DRAWN FROM
ONE
OF THE VERTICES OF THE TRIANGLE TO THE LINE CONTAINING
THE SIDE OPPOSITE TO THE VERTEX.
Sounds simple,
right? Perhaps, but not
until the definition is illustrated with examples
allowing the student to grasp the physical essense of the definition. Consider
the following diagram:
It is clear that the lines AD, BE and CF represent
the heights of the triangle ABC.
Everything seems complete and within bounds of conventional
logic. But what if we deal with an obtuse
triangle? Will
you be able to draw its heights without the definition
provided
above?
In most cases students will be unable to do so without
first recalling the correct definition. Try this with
your student and ask him or her to draw all three heights
for an obtuse triangle. Here's an example:
Do you understand this picture? What do
you suppose lines L1 and L2 represent?
These lines contain the opposite sides of the triangle:
the side AC is opposite to the vertex B and the side
AB is opposite to
the vertex
C. Thus, following the definition, now you will recognize
segments BE and CF as the other two heights of the triangle.
These heights are perpendicular to the lines containing
the opposite sides and, unlike the height AD, these heights
are located outside of the triangle. Yet, in both triangles
depicted above the main property of a height of any
triangle is fulfilled: either the
heights themselves or their continuations intersect at
the same point, in our case point O.
However, if this example was not enough
to convince you of the crucial role of an accurate definition,
we have another example for you: ask your student to define a
monomial. You will see that
once again he or she will stick to a
definition along the lines of "it is a polynomial of one
term..", or that he or she will
not be able to say anything at all. In reality
the definition is extraordinarily simple and immediately
reveals all the necessary
associations used in factoring:
A MONOMIAL
IS THE PRODUCT OF THE NUMERICAL COEFFICIENT AND THE
VARIABLES
The most important point is that the monomial is a product. Indeed, monomials such as
are nothing more than
products of some numbers (in this case the number is
-1) and variables.
Why is it so important to understand that monomials are products?
Because later on, when issues of factoring arise,
understanding the nature of a monomial as the product
will make it so much easier to follow the complex changes
of signs, finding algebraic common denominators, and
doing other operations.
And this is only the beginning.
How would you define a function (will you find anywhere
a definition describing a function as a dependency?);
or the likeness of geometric figures; what about coefficient
of proportionality? Will you learn the precise definition
of the Fermat's Theorem
in your calculus course; or will you learn that the
slope of a linear function is a tangent of the angle
under which the function crosses the X-axis? There can
be no serious mathematics without the knowledge and understanding
of precise definitions, and only this knowledge
enables the student to speak in the professional
and fascinating language of mathematics.
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