Monday, May 13, 2013

Better learning through abstraction..…


There are many barriers to learning and teaching mathematics, some of which include:  anxiety, comprehension, and focus.  How do we get past these?  First, let’s look at anxiety. Why does the mere mention of mathematics strike fear and confusion in students?   It seems there are so many different causes of this to look at, that we are unable to pin point a single method to fix the problem.  In fact to find the solution, we need use the solution.   You will say   “well that sounds impossible”.   If we don’t know the solution, how can we use the solution to find it? There is a simple answer, abstraction.
I touched upon the idea of abstraction in Algebra: the true heart of mathematics.  So rather than just focusing on a solution to all of the different barriers to learning mathematics, we instead use abstraction to find a general solution.  We can then begin teaching abstraction rather than just mathematics.   If we find ways to take the classic “math” out of the picture we avoid the instant fear that many students face.  The statement “Today, class we are going to invent a new dictionary” is much less daunting than “Today, class we are going to learn algebra”.   The first statement will immediately spark a curiosity. The students will begin asking questions, and wanting to know more. 
What does inventing a new dictionary have to do with algebra?  To answer this, we first need to think what is algebra?  Algebra is basically a “dictionary” of formulas, theorems, and proofs which apply to a wide range of problems. 
How do these relate to each other?  Let’s look at the commutative property of addition. In algebra this is “defined” as   (a + b = b + a) .    To look at this in words rather than algebraic symbols, you get 

 Commutative property of addition: When adding two numbers together the order of the numbers does not change the result of the addition.


Where do we go from here?  This seems no different than how algebra is introduced normally.  What I am proposing is taking the math out of it in the beginning, and then slowly adding the math in after the concept of abstraction is understood.  We ask the students to look at some common problems they may face in their daily lives.  We will then list out some of the top suggestions. The students will then be asked to find the commonalities between the problems to group them together. Then the students are asked to make a name for each of the problem groups.  They can then begin defining the general solution to those problem types. Keep it simple, only spend about a week introducing the concept of abstraction with simple problems.   After the idea of abstraction seems clear then you could start adding the Math back into the process. You will begin with asking the students about math related problems. You could start with something simple like what is an even number. Many of the students by this point are familiar with the idea of what an even number is.  However, this time you can approach the same way you approached your dictionary idea. You ask the students to create their own definition of an even number. Allowing them to create their own mathematical rules to use empowers them with the ability to not only recognize the steps of problem solving, but to understand it rather than just memorize it.

               Teaching abstraction leaves a wide open doorway to many different possibilities. The hardest part of learning math for many students is the daunting amount of formulas and rules to remember. A student who knows how and why these rules work, and can look at a problem and devise their own algorithms for solving it, will be way ahead of the curve when they are faced with solving any problem they encounter. When you think about it you use this concept every day. As adults we face many different types of problems, and we solve them by first identifying the problem then abstracting a set of procedures for solving them. We sometimes have similar problems that we have solved before, and we encounter new problems that need new solutions.  However, being successful in life is dependent on the solidarity and efficiency of our ability to solve problems. When I say problems I don’t mean just mathematical, it is the seemingly mundane parts of life for some and the more perplexing or dire situations we encounter that I call problems.         Abstraction, logic, deductive and inductive reasoning are so important to a successful life that part of the curriculum of each grade level should be dedicated to them.

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