Note: This was intended as a Preface to a curriculum based on teaching Discrete Mathematics and a conceptualization of mathematics as a language for those people who were alienated, phobic, or otherwise performing poorly following a traditional calculation-oriented coursework.
The Associative Property
For some children, learning math in school isn't particularly difficult. They need to do the work, but once they do, they're able to understand the algorithm that is long division. These same children get to algebra, and again, some are able to continue methodically applying the instructions the teachers convey, and understand the basics of how to take a word problem and turn it into formula. For some of these students, they're still able to manage trigonometric rules, using algebraic techniques in their chemistry and physics classes, and all is well.
But for a lot of students, beginning in elementary school, math ranges from their least favorite class to a scene of daily embarrassment or even fear.
Why? And what can be done?
I believe that alienation with mathematics, rather than being a pedagogical issue or an issue with cognitive capacity, is due to a psychologically-oriented developmental process where children identify themselves as being at a certain place along a spectrum of sophistication in interpersonal communication.
I'd like some context with my information, please.
There's a way of defining that spectrum. Everyone is familiar with the concept of a word's connotation, the idea that although the strict definition of the word is one thing, maybe there's a sense of something else there.
Some people might object to being called "nurturing," because they feel it connotes weakness. Conversely, some people might object to being called "competitive," because they feel it connotes a lack of empathy.
In addition, there are non-verbal messages that can change the
information of the explicit verbal one. If I say, "Get your coat!" do
I mean, "Hurry! We're about to embark on fun adventure!" or am I
furious and want to leave immediately? I convey some messages with
tone of voice, others with my physical action, and still others can be
inferred by the context in which I give the message.
This is what is meant when people talk about "high context" and "low context" language. The idea is that our words can have additional meaning based on the situation in which they're used.
In a “high context” language, people can say a single word or give a significant look, and everyone understands not only what is meant, but how they are expected to react. The husband merely arches his eyebrow, and his wife knows that it’s time to let the hosts know they must leave the party and take the baby sitter home.
In a “low context” language, the person communicating conveys all the relevant information with their primary message in order to ensure what they’re saying is clearly understood and that the understanding leads to the reaction they’d prefer. Many people naturally speak in a very explicit, low context way to small children because children aren't fluent in either the verbal or non-verbal language:
TODDLER [Takes Christmas ornament off tree.]
MOMMY [occupied, realizing the toddler can't replace the ornament]: Sweetheart, you need to put that down.
TODDLER [Crouches down in place and releases object onto floor.]
MOM: No, not on the floor; put it on the table…
TODDLER [Picks up object, looks at Mommy, glances around the room, trying to identify "the" table out of several surfaces.]
MOMMY: [Notices the confusion] Right here, this table… the table next to Mommy [pats table].
TODDLER: [Puts the ornament on the table and lets go. Looks at Mommy expectantly.]
MOMMY: Very good!
Math is nearly a zero-context language: with the exception of carrying forward previously defined terms, it conveys all relevant information within each communication.
Smells like Junior High
So, there are two attributes of mathematics instruction that work against some students as they learn math. One is that for some children, they're more inclined towards higher context communication. For those students, math is a set of facts that need to be memorized; a set of chores, not in any way a coherent language. Since it happens at a developmental time when other learning is far more exploratory than codified, math often slips into last place in the race for determining these children's favorite subject to study. The child then feels relatively unenthusiastic during math which can lead to feeling unconfident doing homework and anxious during tests, so that the child would generally rather spend time reading, writing, or even doing conceptually-oriented scientific exploration. Typically these are "verbal" children, or those interested in the ways ideas interact, frequently girls.
The other difficulty arises as the students get a bit older. As students enter adolescence, each group, each clique in a school has its own context. To be part of a group, the student has to share the context of that group. We all watch students identify themselves with brands, with bands, with clothing styles; demonstrating that you understand the non-verbal messaging of your peer group is critically important to assimilation.
The first thing that happens developmentally is that a student starts to realize who they're unlike, or at least who they don't want to be like. It takes a while for the student to define themselves, but if a group of young friends can all agree that those children over there are in some fundamental way unlike them, they often feel they've begun to identify themselves.
Unfortunately, it's often in this first stage when students encounter math algorithms that get a bit more abstract. So while they are operating in a relatively high context social milieu, math becomes the subject that epitomizes context-free communication. They don't know much about themselves yet, but they do understand that they're fascinated with high context communication: they experiment with inventing slang and nicknames, and evaluate how others walk, talk, and dress.
These students might initially reject mathematics because the students who persistently try to lower the context of his or her environment by being highly specific and sometimes finding humor in the very ambiguity of high-context communication is anathema.
By the time they develop the self-confidence to determine their own interest more objectively, their performance may have begun to lag to greater or lesser degree.
Just the facts, ma'am.
So the best way to tap into the true interests of children who either naturally gravitate towards high context communication or who later differentiate themselves out of an aversion to being identified with the kids who excel in math is to teach mathematics by following its evolution as a philosophy.
As Keith Devlin says in The Language of Mathematics: Making the invisible visible, mathematics is the study of patterns. Every day, ordinary people think they see a pattern in real life, then they make statements to describe that pattern. But that's what mathematics really is! When we learn arithmetic, we’re actually learning specific examples of one particular set of statements; the one we encounter so often that schoolchildren need to learn it.
After the ideas about the pattern are said in Math Language, then the next step is to prove that these ideas make sense. How to prove things began being discussed as part of philosophy; as people, we know when we've been convinced or not. However, a mathematical proof is very careful about what it allows to be convincing. For example, a mathematical proof can’t contradict itself – it can’t allow something to be both true and a lie at the same time.
Unfortunately, people naturally have a higher context concept of life. We might allow someone to persuade us because, you know, we want to be persuaded! Maybe the person trying to convince us is a close friend, or maybe what they're saying would be a nice thing to imagine to be true. Maybe we just don’t see the contradiction in what the persuasive person is saying: they "bury the medicine in the cake," and smile warmly at just the time they see we're beginning to notice contradictions; they might convey through their actions all kinds of other information that distract us from the factual one on which we’re basing our decisions.
However, mathematics is not subject to persuasion, just to logic.
"This Puzzle Contains Over 5000 Die-cut Pieces!"
Yet, even if we understand the philosophical aspects of mathematics, we’re not actually born knowing that specific shapes represent a quantity any more than we’re born knowing that the shape K represents the sound “kuh.”
We have to learn all that. We have to learn about numbers, about counting, and about how all those numbers behave if we want to be able to tell time, make change, figure out how much money we’ll save buying the Economy Size package (assuming we use it all), how much over the speed limit we’re driving, and all kinds of other things that we think might be useful.
But underlying all that is the language of math. For the students who are alienated by the mechanics of math, understanding the philosophy will make sense out of the series of rote computations. We encourage children to read, knowing that by understanding and connecting with literature at an early age will help them with spelling and vocabulary; in the same way, we need to introduce children to the logic and philosophy of mathematics rather than concentrate on rote learning when there's a beautiful language just a step away.