Triangles

Waverly Station Scaffolding, CC-BY-SA

  • When my son asks “when will I ever have to know how to calculate the area of a triangle?” what do I say?

You could tell him, for one, that this might be the wrong question to be asking about his math homework, that while there is a good chance he won’t have a job where he has to do this precise calculation very often, that it’s nonetheless probably worth practicing it anyway, that it is worth getting a handle on how the calculation works, by manipulating a bunch of examples. The true value lies in practicing the application of calculative methods to abstract concepts.

Or if you wanted to be more specific, you might tell him that by breaking down a problem into its parts, identifying the right methods to solve those parts, and then fitting the solutions together, he is honing a skill set which will be necessary not only if he chooses a career in engineering, but also across a wide swath of both hands-on and math-heavy vocations; this description of the skill doubles as the basic task of those working computer science. If you were feeling sassy, you might mention that, beyond informing career options, identifying sub-problems and trying to reconcile solutions with one another also resonates with the analytical thrust necessary to the critique of political and philosophical arguments, therefore promises to develop more skeptical engagement with what is said in the media, not to mention that it spins cognitive wheels needed for the development and discovery of new perspectives on and forms of appreciation of art, film, dance, and music, and ultimately to creating a thoughtful approach to our own emotional lives, our relationships and our life choices.

If he was still listening, you might ask him to contemplate the common occurrence, in which one starts with a task that is somehow simultaneously boring and challenging, and becomes so familiar with its exercise that both the form of the problem and the nature of the solution becomes second nature. Attending to a task that is not inherently pleasant, fighting off distraction, overcoming the frustration that inevitably attends early efforts: all these are hard, but they come with a payoff. Taking a low-level skill from heartache to habit sets down a foundation for the pursuit of tasks that are in many cases innately more interesting (e.g. only by learning how to write in complete, grammatically correct sentences can you become capable of writing moving love letters). Measure twice, cut once may seem like a boring rule you have to force yourself to remember, until it simply becomes automatic in the way that the carpenter does her or his job—and an essential part of doing the job well. It turns out moreover, that the payoff is double, because while overcoming frustration and boredom are hard, one becomes better at them through practice. Even for the rarest of geniuses, the pleasures of virtuosity are built on a scaffolding of boredom, self-doubt, and frustration not only in their own field but across a variety of repetitive, basic tasks.

As for the triangles themselves, you might strongly put forward that building up a vocabulary of mathematical fundamentals offers an opportunity for your son to become a person, and to live a life, that he might not otherwise consider or have access to. Knowing, becoming familiar with, the universal relationships that govern the area and height, the angles, width and side lengths of every triangle means that each one takes on the gleam of an otherwise hidden meaning. Perhaps he’ll become the kind of person for whom those meanings (and the broader network of meanings of which they are a part) form a useful or interesting moment of his every day. Perhaps he won’t. But becoming familiar with those meanings—not just by plugging numbers into a formula, mind, but by learning how to manipulate the underlying ideas in ways that offers answers to questions—means he will have a choice, later on, about whether that’s a kind of person he wants to become, rather than being forced to choose a career and identity in which triangles, along with calculus, probability, and statistics remain mysteries whose secrets are the sole possession of others. There is beauty and power and magic in the abstract worlds of mathematics in which triangles are only one of the most basic inhabitants. You might ask if he really wants to decide so early on that he will never have reason to visit them.

Learning and Meta-Learning

Over at Tomorrow’s Professor, an excerpt from a book on ePortfolios (for the unnaturally curious, the book is Documenting Learning with ePortfolios: A Guide for College Instructors):

ePortfolios…allow learners to make connections among varied learning experiences and transfer knowledge and skills to new contexts and situations. This approach, particularly when it capitalizes on the features of ePortfolios together with a culture of folio thinking, can promote deep and integrative learning. For students, however, the value of ePortfolios and folio thinking may be unclear. Students may initially assume that the use of ePortfolios in a course or program is simply a new and faddish approach to teaching and learning. Indeed, without effectively communicating the purpose of ePortfolios and the benefits that ePortfolios are intended to produce for them, students may resist the approach, thereby making it challenging for them to really capitalize on those benefits.

This is a challenging issue. In my experience of the university setting, students often come to learning experiences with preconceptions both about what they are supposed to be learning, and about how they should best be taught those things. The solution presented here is to show your cards: make pedagogical methods explicit.

The difficulty of framing is that an entire level of learning gets lost. It may be true that students who are told how something will add to their knowledge-base or skill-set will overcome their “resistance” and allow them to “capitalize” on a learning technique. Yet being so explicit allows them to be smug in their presumption that they know how learning works, and how teachers should teach, informed, more often than not, by what Paul Freire called the banking model of education.”

Freire’s point, in his critique of this model, was partially that one should not view the teacher and the student as polar opposites, with the student as an empty vessel and the teacher as a the holder of knowledge with gets ‘desposited’ in the learners. On a substantive level, his argument implied that both teacher and students are learners, that both have knowledge to share, that education should aim to combine that knowledge in a mutual learning process. Fine: but if I want to learn Portoguese, then its likely that I am going to find a teacher who has more relevant knowledge than I do.

His criticism also has an implication about the process of learning. Education is not a mechanical process; I cannot, in fact, put my knowledge directly into your brain, techno-utopian fantasy notwithstanding. Rather, learning is necessarily active. I can tell you something – say, the definition of GDP – but your ability to remember it will depend on what you do when I tell you; on whether you are writing it down when I am talking; on what you are using to write it down; on how soon you return to it after first hearing it. My sense is that the best way to really learn the definition of GDP is to be forced to use it in practice, or to reflect on its meaning: why is it defined this way? Why does the result of this calculation matter? What would be wrong with other calculations? How else might we have tried to capture this information? How do we measure this aggregate in practice? I would argue, even further, that the definition of GDP only becomes useful once a person can provide answers to these questions. Memorizing the definition might get you marks on a test; only your ability to think about it in context will make you a better economist.

Telling someone how a process or technique is supposed to aid their learning treats becoming a better learner (“meta-learning”) as a passive, rather than an active process. Learning itself is a skill, and like all skills, it is only sharpened and refined through practice. Telling students what contribution ePortfolios might make to learning therefore ignores both elements of Freire’s insight: first, it assumes that the teacher knows exactly what contribution the process might make to the student’s competence as a learner and that this knowledge is simply transferred to the student; second, it does not require students to use this knowledge, and is almost sure to be ineffective at making them better learners. In other words, it may convince students to use ePortfolios, but it will not make them better learners.

The reality is, the best way to increase student learning competence is for them to be reflectively engaged in the learning process; to constantly push them to think about how they learn best, to consider what they might learn from a given experience, to adopt practices which maximize their own learning, to experiment with alternatives, to ask better questions. In other words, it requires departing from a simple image of education as a service that universities provide to students, and recognize that education is work which requires creativity, thought, engagement and participation by students.