When studying mathematics at any
level, one might find themselves asking where did this stuff come from? , or why
is this useful?. The answer to these
questions can be summed up in one word , Algebra. Why algebra? Why is it so important? Well, let’s look at what algebra actually
means. Algebra throughout history has
developed many rules, relations, and symbols to answer some of the everyday problems
we encounter. The history of algebra and
how it was formed, in my opinion, has little to do with what it actually means.
I will instead explain algebra by use of philosophy. We can call it the
philosophical theory of algebraic reasoning.

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The Theory of forms and Algebra

The theory of Forms
is basically that real-world objects have forms. For example when you look at
your table, how do you know it’s a table?
In fact your table could be extremely different from all other tables,
but you know it’s a table because it has the form of a table. The form of a
table is that it has a flat surface at the top, and a support structure at the
bottom. The Form of the table can be
restated as the abstraction of a table. By having the abstract model or form of a table,
we can then manipulate or customize our table to look how we want so long as it
stays within the basic bounds of the form. We can also now take our base form and use it
to define specific types of tables.
Algebra works in the
same way as the theory of forms. In algebra we use letters to symbolize or
initialize the form of a number. We call the letters variables because they can
change their value. Let’s look at the statement 1+1=2. Even though we have
numbers and not variables this is still an abstraction. This is a perfect
abstraction for general purposes it gives us a specific form unity. The 1 is
the form or abstraction of a real world object. It can be any object as long as
it is a single object. So to translate this statement we would say that a
single object and a single object together has the form of two objects. For most the meaning of this statement is
obvious, however; the statement a + a = b is not as obvious. Also, this
statement does not mean the same as 1+1=2.
Table + Table = 2 tables this
statement is equivalent to 1+1=2 only because we are assuming the Knowledge of
what the symbols 1, 2, + , =, represent. So we build our philosophical theory
of algebra off of the assumption that we don’t need to define the symbols for
numbers and what they mean.

The symbols 1 and
2 have a very specific meaning, but their properties, and interactions have a
more complicated philosophical meaning. So we have a gap in our ability to
abstract numbers and their properties, which is why we use the letters or variables.
The variables are in fact a form of a
form. Sounds redundant, but it is necessary to provide abstraction from
something that is already an abstraction. This concept is where it gets a
little tricky. We can say a variable is
the form of a number but not all numbers only numbers that have the specific
properties we provide with the rest of the statement. Take our a + a = b statement.
This statement gives us a relationship between any two numbers that can be
represented in this form a + a = b which has many numerical equivalences. It can mean 1+1 =2,
2+2 = 4, 3+3 =6, 4+4 = 8…etc. In fact this statement provides an abstract definition
of even numbers. Statements like this
should then be called definitions of forms.
We shall go back to our table example again. The basic form of a table
is flat surface at the top and support structure at the bottom. What if we said
table + wood? We are giving an abstract definition of tables that are made out
of wood. So table + wood is the same as a + a =b they are both abstract
definitions of forms. The only difference is abstract definitions of
numerical forms take a little more interpretation. However both assume an
understanding of something else. The table + wood example assumes an
understanding of the form of wood and the form of table. The a+a=b example
assumes an understanding that 2,4,6,8,10…etc are called even numbers. a+a=b
just gives us a definition of even numbers without trying to list all of the
even numbers up to infinity. So abstraction is actually necessary to define
specific numbers and number relationships, because we cannot realistically list
every even number to infinity.

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Why is algebra the heart of mathematics?

The
abstraction of algebra gives us strong foundation by which to build many more
complex definitions with a wide range of uses in everyday life. A lot of these
concepts we use every day without even realizing it. In fact nearly every
branch of mathematics would not exist without the foundation laid down by
algebra. For example statistics, uses many definitions that would not exist
without the framework of algebra. We even use algebraic concepts in many other
subjects. For example in composition classes we use abstraction as a tool to write
papers we start with an outline, which is simply an abstraction. Our outline
gives us a simple abstract definition of our paper which makes writing our
paper much easier. Language itself is an
abstract definition of our thoughts and feelings.

Thank you, I have been following several articles about mathematics education. I saw a couple where some people actually suggested removing algebra from high school. I was shocked by this hearing educators make this suggestion, which was my original motivation for putting this together.

ReplyDeleteThis is a very comprehensive and informative article. As a math tutor I am particularly interested in others views on algebra and agree that it is shocking that some want to remove this from school.

ReplyDeleteThank you, It is very shocking that some educators have actually suggested the removal of such an essential subject. The problem solving aspects of it alone are worth keeping it around. One of the most important things to learn from algebra is how to analyze a problem and find a logical series of steps to approach solving it. These methods can be applied to all sorts of problems we face whether they are purely mathematical or not.

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