Making mathematics teaching and learning meaningful
THERE is little contention that the teaching and learning of mathematics are equally challenging processes. The struggle to improve on both aspects must yield to the realisation that these processes are inherently messy.
As it relates to teaching, this messiness often involves re-teaching, un-teaching, over-teaching, and under-teaching. It is a complicated judgement that the teacher must make in order to become facilitators of understanding. But even though a lot of emphasis is rightly placed on the credentials of the mathematics teacher, another important variable in the calculus of the classroom is the relevance of the content being distilled. What is the point? Why should students learn algebra? Why should they learn trigonometry? Why should they even learn mathematics?
The argument being proffered here is that making knowledge relevant is the precursor to making learning successful. This has a deep ontological basis: Human being is strongly driven by the need for meaning. The argument posits that the drive to learn, the motivation or the desire to learn, is sparked and sponsored by an acute understanding of why learning is meaningful, or why the thing that is being taught matter.
This relevance is rooted in a deep connection with one's lived reality. It urges that students study and succeed in learning algebra, for example, because they perceive how it interacts with their social contexts. While it is true that this is a real concern for some students -- why should I learn this mathematics -- the truth is often less palatable: Some students do not really care about the mathematics enough to worry about its real-world relevance; most teachers were never taught nor care deeply about the real-world relevance; and, even more discouraging, a lot of the mathematics that students are taught they will never consciously use in any real-world context.
Success outside of relational meaning
Many students succeed in mathematics independent of a meaningful engagement with the real-world relevance of mathematics. The real question is: Why?
What was their source of motivation or inspiration? The important thing to recognise is that meaning, and therefore motivation, is relational in nature. If the meaning does not come from the relationship between the subject and the lived reality, it must come from the relationship between the person and the subject.
Students persevere with the subject because they realise what it can do for them in an instrumental way: It can be a ticket to their dreams. And as they say in Jamaica, if you don't have a ticket...
This, of course, is pedagogically dissatisfying. A lot of the mathematics taught at the primary or secondary level is simply irrelevant to student's lived contexts, beyond helping to meet the goal of making citizens numerate.
Mathematics that is embedded in the structures of our social interactions -- the mathematics of cellular use and data transmission, or the mathematics that makes Internet transactions safe is mathematics that is often beyond the scope of the standard high school curriculum. This curriculum, therefore, provides a little mathematics to make one curious, but not enough to make one meaningfully informed. For those who continue on the path of mathematical learning, the meaning between mathematics and the social reality usually comes later, rather than earlier. Telling students, however, they will understand in the sweet by and by is a tough sell.
State of play
So what is the state of play? First, we know that some mathematics is easy to derive meaning from because of the quick superficial connections to numerate citizenship: accounting; book-keeping; counting; measuring, and comparing. Life, at its most basic, requires a strong grasp of numeric competency; otherwise, we quickly find that the lack of money sense will make us centless.
Second, we know that for more substantial connections between mathematics and the world, we need to consume more mathematics. And the appeal is that mathematics is relevant in a nebulous way at the moment, but that the sun will blaze through the cognitive clouds once we persevere with the mathematics long enough.
Third, we know that mathematics serves a functional role in rationing scarce resources: Those who do well in the subject have a ticket in the lottery; and those who do not do well, don't. This can often incentivise students to learn: They simply need mathematics for the next stage in their education or career. And this is enough motivation.
The way forward
I think, however, that as teachers of mathematics the most important meaning that can be strived for in mathematics education is not in the relationship of mathematics to the social context, nor to the learner. Rather, it is in providing meaning that makes mathematics internally consistent and deeply engaging by appealing to the hand, the eyes, the heart; and the mind. It is making explicit the linkages in the development of mathematical ideas, and providing students the tools to explore the multi-dimensionality of the problem-solving approaches: the numeric, the graphical, the algebraic and, the statistical. Students who grasp the subject from its enactive to its iconic, through the symbolic stages, can enjoy a facility and appreciation for the power of the subject in economising complex relationships. And when students can find meaning inside the mathematics, they often do not need to seek it outside.
Ryan Palmer is a mathematics specialist with broad-based international teaching experience and content development, who writes from Kent, England. Send comments to Observer or email@example.com