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More Methodical Than Guessing

Date: 06/12/2017 at 13:46:48
From: Egan
Subject: Guessing a solution vs Concrete Steps.

It seems like guessing is involved in many operations that we do in
maths.

For example, in finding square roots, we make guesses as to what they
could be. In factorisation, we think of numbers that can multiply to be
another, in a guessing kind of way.

Another example is in solving the general cubic equation. One important
step is to simply rearrange an algebraic expression to another, more
complex one -- just because it works. (How do we know we should
re-arrange it that way and not in any other way?)

While I only have a high school level of mathematical experience, I love
learning about more advanced concepts. But all this guessing is
difficult for me to master, especially in complex problems. In middle
school, I was very good in geometry, but bad in algebra where, for
example, we factorised expressions in a guessing kind of way.

All of which leads me to two questions. Are there better options for
these mathematical operations? Can we classify mathematical operations
into the categories of "guess it first" on the one hand; and on the
other hand, "Do steps 1, 2, 3, ... and then you will get the solution
directly"?



Date: 06/12/2017 at 18:51:52
From: Doctor Peterson
Subject: Re: Guessing a solution vs Concrete Steps.

Hi, Egan.

There are a number of different things that can be said about this.

My first thought is that we need to distinguish between an "operation"
(in the sense of what we want to accomplish; for example, find the
square root) and an "algorithm" or process or method to perform the
operation or otherwise reach the goal (do this, then that, ...).

There are usually multiple ways to accomplish a task. Some may involve
guessing, while others may not; some may be more efficient than others,
while others may be easier to explain. Sometimes guessing turns out to
be the most efficient method, even though non-guessing methods are
available.

Guessing is not inherent to the operation, itself -- just a feature of a
particular method you may choose to use. So we can't sort "operations"
according to whether they involve guessing.

Second, "guessing" can take several very different forms. You are
familiar with algorithms of the "guess and check" variety, where you
make a guess, see whether it works, and then make a new guess based on
the outcome. Sometimes the "guess" will almost always work, and the
"check" is mostly a matter of continuing on with your work. (I am
thinking of long division; longhand square roots are similar, once you
have the necessary experience.)

Other processes, such as some factoring techniques, can be done by a
quick "guess" (insight) that turns out to be correct, or by
systematically trying all possible answers (e.g., all pairs of factors
of a given number), with the conclusion being either the answer or the
knowledge that there is no answer. (Pure "guess and check" can never
determine whether you have just missed an answer.) Such a procedure may
be done blindly (just plodding through the list) or intelligently (using
clues to know what to skip).

Still other "guess-like" processes are not really guesses at all, but
successive approximations to the correct answer, which may not be
possible to find exactly. (Here I am thinking of the "divide and
average" method for finding a square root.)

Let's consider some examples.

Square root I've mentioned two different methods for finding a square
root, and there are a couple others. This illustrates my first point.
One method ("longhand") involves guessing for each digit (which should
rarely be far off). Another ("divide and average") starts with a
"guess," but goes on from there to repeatedly improve the guess, with no
more guessing needed:

  Square/cube roots without a calculator
    http://mathforum.org/dr.math/faq/faq.sqrt.by.hand.html 

Division
Long division as normally done involves guessing (I'd rather call it
estimating) quotient digits. But you could do it with no guessing at
all: Just make a list of multiples of the divisor, and you can look up
the closest multiple to the dividend. (We guess to save time, not
because it is needed!) Or, you could do what a computer or calculator
does: convert to binary, divide in binary (which requires no guessing),
and convert back. The method with guessing is easier and faster for
humans, but not the only way. See this page:

  Long Division, Egyptian Division, Guessing
    http://mathforum.org/library/drmath/view/58858.html 

Factoring a polynomial
Factoring can be done using methods that require various amounts of
guessing. Looking in our archives for examples that discuss guessing, I
found one that shows a traditional trial and error method with lots of
guessing:

  Factoring Trinomials: 9x^2 - 42x + 49
    http://mathforum.org/library/drmath/view/61570.html 

There, Doctor Ian refers to another page in which he shows how to factor
by completing the square, or using the quadratic formula, each of which
entirely bypasses guessing (and, unlike the other method, even works
when the factors do not involve integers):

  Factoring Quadratics Without Guessing
    http://mathforum.org/library/drmath/view/60700.html 

There is also a commonly-taught method of "ac-grouping," which uses a
simpler kind of guessing:

  Factorization by Decomposition and the Distributive
    http://mathforum.org/library/drmath/view/77809.html 

But when I saw the title of the first of these, I immediately saw that
9x^2 - 42x + 49 could be factored using only a single "guess": recognizing
that the first and last terms are both squares, we can guess that it MIGHT
be a perfect square, of the form (a - b)^2 = a^2 - 2ab + b^2. In that
case, the factorization has to be (3x - 7)^2. Then all I have to do is
check that by multiplication, and I can determine that it's correct. (If
the guess had been wrong, I'd have switched to one of the other methods.)

Do you see my point here? There are MANY ways to factor a quadratic,
which can range from no guesses, to one guess, to potentially hundreds
of guesses. The optimal strategy is to first "guess" what might be the
most efficient method for a particular problem, and then do it. The
methods with fewer guesses involve various levels of complexity and
risk, as a trade-off to the challenge of guessing.

Your other example was the cubic equation. I suppose you have read
something like this:

  Cubic and quartic equations
    http://mathforum.org/dr.math/faq/faq.cubic.equations.html 

There is actually a formula for this that requires no guessing. Far
beyond the scope of high-school algebra, it takes up half a page, at
least. I have never tried to master it. Alternatively, you can follow
the procedure discussed on our page, which does not involve guessing,
but is an orderly process.

You also allude to the role of guessing in not just the process, itself,
but rather in how that process was devised. You may be right that this
involved some guessing (somewhat like my perfect square guess above,
where I tried something that seemed useful, and it worked). That's a
very different kind of guessing, which arises in any non-routine
problem-solving. I compare this sort of thinking to finding your way to
a goal through a forest you have never seen before. You have to develop
an intuition about what might work, based on experience using the
mathematical "tools" you have learned.

There is often no way around that; that's why we call such a problem
"non-routine." And that kind of math may be the most useful in real
life: when you learn how to solve problems you have never seen before,
you are ready for the real world, where nothing is ever quite what you
have been taught! Willingness to try and fail and try something 
else -- to persevere -- is the most important thing you can learn.

As far as learning to do the guessing part, all I can say (apart from
the fact that you can often work around it) is that as you gain
experience in any field, you develop a feel for how things work; and
that sense helps you make the right guesses much of the time. How that
is done depends very much on the particular field you are trying to
master.

Perhaps we could discuss that in detail for a particular problem, or
kind of problem. Show us one that interests you and how you currently try
to solve it, and we can suggest ways to do it better.

- Doctor Peterson, The Math Forum at NCTM
  http://mathforum.org/dr.math/ 
  


Date: 06/12/2017 at 22:35:17
From: Egan
Subject: Thank you (Guessing a solution vs Concrete Steps.)

Thank you very much, sir, for your detailed answer! 

I have been using this site for about eight years, and I always
recommend it to my friends (especially those with more of a mathematical
background), as this is so much better than Quora or Mathematics Stack 
Exchange. :)
Associated Topics:
High School About Math
High School Square & Cube Roots
Middle School About Math
Middle School Factoring Expressions
Middle School Square Roots

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