So
You Think You Don't Know Algebra?
Zoë is in second grade and, while she might not realize it, she is already on the
road to learning algebra. In time, the box she is filling will give way to a letter (usually x) and the instruction
"Fill in the blank" will be replaced by the instruction "Find the value of x for which ... " Viewed in
a somewhat different way, consider the following two questions:
Question #1:
Are you comfortable answering a question of the form:
“Find the value of x for which 4 + x = 6” ?
Question
#2:
Are you comfortable answering a question
of the form:
“Fill in the blank: 4 + __ = 6”?
If you answered “No”
to Question #1 but “Yes" to Question #2, you do not
have a math problem. Rather you have a language problem. More
specifically, our arithmetic course uses language to center on
our “adjective/noun” theme; and our algebra course
uses language to talk about the ability to paraphrase. Very often,
rephrasing a problem into an equivalent problem make sit easier
for us to solve the original problem.
As an illustration,
let's use the “fill-in-the-blank” idea to highlight our procedure:
Suppose you are given a "fill-in-the-blank" question to see whether
you know that 4 + 2 = 6. Let's further assume that you do not understand arithmetic but you do know how to use a calculator.
You would be lucky if the question was worded in the form 4 + 2 = ___ since, in that case, all you would have to do is use
your calculator and enter the sequence of key strokes:
4 + 2 =
and 6 would appear as the answer.
Now suppose that the question had instead been
worded in the form 4 + __ = 6. In that case your luck would have run out! Namely, you could still begin by entering
the sequence of keystrokes:
4
+
but, unfortunately the calculator does not have a "blank key". 1
Finally,
suppose the question had been worded in the equivalent form 6 – 4 = __. In that case, you would have again been
very fortunate because you could obtain the correct anwer simply by using your calculator and entering the sequence of keys
strokes:
6
–
4 =
In this
context we may view algebra as the subject that allows us to paraphrase questions that the calculator does NOT "understand"
(for example, 4 + __ = 6) into equivalent questions that it DOES "understand" (for example, 6 – 4 =
__ ).
Of course if all mathematical fill-in-the-blank
questions were that easy to paraphrase it would not take much tie to complete the study of algebra. However, most of the time
the desired paraphrasing is not as easy to obtain. For example, consider this question:What is the number that when you: multiply
it by 3, add 4, divide the result by 2, subtract 3, add the original number to the previous result, you get the
answer 9?
In this context, algebra
is the subject that shows us how to use the rules of logic and arithmetic to show that the solution to the question
is 3. 2
1As
similar situation exists in other courses as well. For example if a fill-in-the-blank question is given to test whether you
know that Sacramento is the capital of California, your chances for success are much greater if the question is worded in
the form "Sacramento is the capital of ____” rather than in the form “_____ is the capital of California”.
In essence it is more likely that you will think of California when you hear the word “Sacramento” than it is
that you will think of Sacrament when you hear the word “California”.
2This is the reason that we have decided to talk about the “game” of algebra.
Namely, in any game there are rules and definitions and the objective is to use these rules and definitions to arrive at what
we call a winning situation. In the “game” of algebra the rules are the rules of arithmetic and our objective
is to use these rules to transform problems we can't solve into equivalent problems that we can solve.
