Transcript Mathematics as a Second Language

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Mathematics
as a
Second Language
Arithmetic Revisited
© 2010 Herb I. Gross
Developed by
Herb Gross and Richard A. Medeiros
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Prelude to Mathematics
as a Second Language, Part 1
There is a “tongue-in-cheek” piece of advice
to the effect that “if someone doesn’t
understand your explanation of something,
just explain it again; only louder”.
Many students have experienced this sort
of explanation too many times in their
study of arithmetic; that is, listening to
the same explanation being given
over and over again.
© 2010 Herb I. Gross
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To break this chain of events, we have
embarked on what we believe is
a fresh and productive approach.
More specifically, our innovative approach
to teaching basic mathematics, which we
call “Mathematics as a Second Language”,
is to introduce numbers in the same way
that people from all walks of life use them;
namely as adjectives that modify nouns.
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More specifically
numbers can be
viewed either as
nouns or adjectives.
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P
0
1
2
3
For example, in this case, 2 is a
noun that names the point P.
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2
Q
0
P
1
2
3
In this case, however, 2 is an
adjective that modifies
(measures) the distance between
points Q and P.
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However, most of us see
numbers as adjectives.
That is, we’ve seen…
3 people
1
2
3
3 apples
1
2
3
3 tally
marks
1
2
3
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But never
“threeness”
by itself.
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The theme of this
presentation begins with
the underlying principle
that numbers are adjectives
that modify nouns
(or other adjectives)
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As an application of how we can use
the “numbers as adjectives” principle,
consider the following situation.
Many students often fail to grasp the
relative size of a billion dollars with respect
to a million dollars in the sense that they
view both amounts as being “very
© 2010 Herb I. Gross
big”.
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However, suppose we now let the
adjectives “million” and “billion” both
modify “seconds”.
Some very elementary calculations show
us that a million seconds is a “little less”
than 12 days while a billion seconds is a
“little more” than 31 years. This
gives new life to the idea that although a
million seconds is quite “big”, it is still a
small fractional part of a billion seconds.
© 2010 Herb I. Gross
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It also gives students (and others as well)
an easy to understand ratio; namely, a
million is to a billion as
12 days is to 31 years.
While it may be easy to confuse
a million with a billion;
no one ever confuses
12 days with 31 years!
© 2010 Herb I. Gross
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However, size is relative.
A quantity that is small relative to one
quantity might be very large relative to
another quantity.
For example, while a million seconds is
small compared to a billion seconds, a
billion seconds is small when compared
to a million days. More specifically…
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Practice Problem #1
If there had been 400 days in a
year, how old would you have
been when you had lived
1 million days?
Answer: 2,500 years old
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Solution to Practice
Problem #1
The answer is found by dividing
1 million (days) by 400 (our estimated
number of days in a year).
1,000,000 ÷ 400 = 2,500
In other words, if there had been
400 days in a year, you would have been
2,500 years old when you had lived for
1 million days.
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Notes on
Practice
Problem #1
The fact that there are less
than 400 days in a year means
that our estimate is less than
the correct answer.
Thus, without having to do any further
computation, we are safe in saying
that you will not have lived a million
days until you were more than
2,500 years old.
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Notes on
Practice
Problem #1
Taking into account leap years,
there are 365 ¼ days in a year.
With a calculator it takes no
longer to divide by 365.25 than
to divide by 400. Doing this, we
would find that to the nearest
whole number of years…
1,000,000 ÷ 365.25 = 2,738
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A Note on
Using a
Calculator
The calculator is a great aid in
helping us to perform
cumbersome computations
quickly. It is not a replacement
for logic and number sense.
The user has to have enough knowledge
to be able to tell the calculator what to
compute. For example, the calculator will
not divide 1,000,000 by 365.25 unless it is
told to do so!
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A Note on
Using a
Calculator
Moreover, even with a
calculator we can enter data
incorrectly. To guard against
this, it helps to have a
number sense.
For example, in this discussion, our
number sense told us that the millionth
day since your birth cannot occur prior to
your 2,500th birthday. Hence, the answer
we got using a calculator (2,738) is at least
reasonable.
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Notes on
Practice
Problem #2
The point we wish to make
here is that to know the size of
a quantity, we must know both
the adjective and the noun it
modifies.
For example, even though a billion is more
than a million, the fact remains that
1 billion seconds (approximately 31 years)
is less time than 1 million days
(approximately 2,700 years)
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Key Point
In an era when education perhaps seems to
be overly concerned with the use of
manipulatives and high-tech visual aids,
there is a tendency to forget that innovative
use of language may in itself be both the
greatest manipulative and the best visual
aid that can ever exist; and of at least equal
importance, it is available to be used by
everyone.
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In this course, we shall discuss how by
treating mathematics as a language we can
“translate” many complicated
mathematical concepts into simpler but
equivalent mathematical concepts.
More specifically, we shall demonstrate
how by properly choosing the nouns that
numbers modify, we can make many
mathematical concepts easier for students
to comprehend, no matter what other
modes of instruction we may be using.
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In the next part of this
Lesson, we will discuss in
more detail the properties of
numbers as they are used in
our course.
3+2=5
© 2010 Herb I. Gross