The Concept Drives The Computation
I say this to my students all the time. I know my students understand what I'm talking about when I say it. But what do I mean?
Well I only say this to students who show either some proficiency in arithmetic or demonstrate that if they tried they would be able to become proficient. For other students, no beating around the bush here,
it's just a battle trying to get them to put forth effort. In other words I only say it (the concept drives the computation) if and when I think it might be well received and could result in giving the student a new perspective, a perspective that
includes putting forth effort.
So what does "The Concept Drives The Computation" mean? It means an understanding of the concept(s) involved is what tells you what to do in order to arrive at a solution. Math is a little like following
your nose. But you need to learn a lot of stuff before you can do it by simply following your nose. Following your nose means understanding the applicable concept and then doing the math by applying the
concept. A student needs a tremendous amount of experience before they are ready to do math by simply following their nose. Put another way - one of my professors explained it to me like this;
Understanding Math is a lot like bacteria growing in a petri dish. Knowledge of math comes in splotches and as those splotches grow they begin to bridge gaps from one splotch to another splotch. As these
gaps are bridged your understanding of math becomes stronger and stronger. Just like the colony of bacteria becomes stronger and stronger as the splotches of bacteria bridge to one another.
As a student progresses in math, there really isn't much concept to be had until they reach pre-algebra and/or algebra I. There is some but not much. When I say not much what I mean is that there's not much
concept when you look at the concepts in comparison to the body of mathematics. There's actually plenty of concept in just the number line and counting by 1's, 2's, 3's, and so on. There's concept in
fractions - they represent less than a whole. There's concept all over the place if we're willing to think about it. There's concept in understanding that multiplication is fast addition and division is fast
subtraction. There's lots and lots of arithmetic operations we're asked to become familiar with. Before pre-algebra it's really about becoming proficient in arithmetic. When we encounter pre-algebra / algebra
I it's not really about concept it's more about an idea. That idea is that we don't really need to know the value of something. We just need a way of expressing it. A way of expressing something that we do not
know the value of. We just know it exists and we may know some of its properties. Our first encounter with expressing an unknown quantity is usually the variable X. Up until the point when X is introduced all of
our learning has been to become proficient at adding, subtracting, multiplying, dividing, understanding fractions, understanding percents, decimals, and how both percents and decimals can be expressed as fractions.
We're also asked to understand some fundamental principles of arithmetic - additive identity, additive inverse, commutative property of addition, associative property of addition, why subtraction is neither
commutative nor associative, multiplicative identity, multiplicative inverse, commutative property of multiplication, associative property of multiplication, distributive law of multiplication, why division is
neither commutative nor associative, common denominators, least common denominators (this is actually an application of the multiplicative identity (from a conceptual point of view) believe it or not), least common
multiples, order of operations, what square roots are, etc., etc., etc., etc., ..., etc. Just a myriad of "junk". We must become proficient at all this "junk" (it's not really junk but it's a lot lot lot of stuff)
and most students think it's JUNK until they start understanding concepts. I actually think that thinking it's junk is an appropriate reaction to all the stuff they have to know. OK, I think I'm getting off topic
a little. I'll try to stay on course - now that students have learned about this new idea, the idea that X is a number - an unknown number, students must learn how to apply all the "stuff" they've learned how to
do previously to the unknown, X. This introduces new challenges. They must learn how to treat X as a number without knowing what X is. All the arithmetic computation they've previously mastered must now be
applied to X, a number that is unknown, but is still a number. This is definitely a challenge.
A student really has to go on faith that all this "junk" is going to have meaning at some point in their future. But believe me - all this "junk" has huge implications to ones ability to compete for a job with
all the other folks who are competing for the same job. This can't be emphasized enough.
After a student has become proficient at all of the arithmetic operations (computations) they practice from inception / introduction to mathematics, they are ready to start thinking about math from a conceptual
point of view. But the battle hasn't ended. The campaign to understand has just begun and if we've done a good job up until this point, we're well equipped to engage and evolve our understanding to include concepts.
All the computations they've learned up until that point puts them in a position where they are capable of understanding and applying concepts - the computations support the concepts they've learned and applying
the concepts using computation solidifies the computations. Applying concepts means modeling situations with mathematics.
There's so much more that can be said. Too much! I can't do it justice - I can't even try. Suffice it to say the rabbit hole goes deep, VERY VERY VERY VERY DEEP, and branches out in all directions. Branches
have branches which have branches. It doesn't stop! And I don't think it will stop - EVER!
There is some more I'd like to say about mathematics though - Math trains your mind in the art of thinking. It trains you in the ability to think about things in terms of clarity and relevance. It trains us in
the ability to think about things without letting our intuition and/or perception get in the way of what's real and true. Performing math correctly forces us to see things for what they really are and not
seeing them for what we think they are (what our intuition suggests to us they are). Math can't be done correctly unless the person doing the math can see things for what they are.
It allows you to see things for what they really are - and this transcends into all facets of our lives. Being good at math prepares us to be good at life (this doesn't mean you can't be good at life without a
solid understanding of math but math gives you a serious leg up!). We are all inundated with companies
that are marketing their products to us. To some extent math will make us immune to manipulation. As an example how often do you hear insurance companies say something like "we take care of you like no other"?
What does that actually mean? It means nothing - think about it. I'd like to finish this essay with two thoughts:
1) from Judge Judy - "If it doesn't make sense, it's not true",
(This doesn't mean the truth of a statement will always be apparent. It does mean that upon scrutinizing the premise/hypothesis and conclusion of
a statemtnt, the inherent truth/false of the statemtnt will emerge. Intuition is, many times, misleading.)
2) from Arthur Conan Doyle, Sr. (creator of the detective, Sherlock Holmes) - "Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth".
The Concept of Equality
An Effective Tutoring Session