What does math mean? And what should it mean for my students?
The answer to both these questions is up for grabs and not initially apparent. It requires an interpretation – the technical question is, what hermeneutic do we apply to math? How do we “read” math in a way that makes sense for us?
Math developed over time, and we can say things about who developed it, how they developed it, and can condense and summarize their work in retrospect. It is also utilized now for a variety of reasons. The interpretative question implies that neither set of facts – how math developed, or how it is used now – offer a definitive answer to a question of meaning and norm: what does math mean and what should it mean for us now?
Math, perhaps more than any other traditional secondary school subject, elicits reactions among people that emphasize its universality, objectivity, and invariance – and that these qualities are by far the most important thing about it. Ruminations on math in this key end up sounding grand and cosmic: math is a universal language, math is the underlying structure of natural laws, etc.
In this reading, the context in which math takes place is less important that the content that math delivers. Human disposition and purpose does not impact math’s impartiality. The only important context for math is the one in which it was developed – before now. The work has been done and we are responsible to the edifice that our ancestors have left.
In a million implicit or explicit ways, you can hear the argument that the math we receive now would be the same math no matter who came up with it or when they came up with it. In this frame of thought, reforming math education is unnecessary at best, and at worst obscures math’s universality and objectivity in the name of some lesser purpose (often, politics, or worse, political correctness). This explains much of the resistance to math reform. It is also a sensibility prominent among students. “Why don’t you just teach us the math,” frustrated students of mine have asked (they present themselves as experts of knowing math when they see it – but perhaps not yet hermeneutics).
In other fields of study, an emphasis on the message that takes no account of the messenger or the community receiving the message has been called “orthodoxy,” for better or worse, but that’s not quite specific enough. We’re talking about the interpretation of a text – mathematical statements, objects, and conjectures – so the term I’ll use has more to do with a lens, or a reading stance than a whole worldview.
I’ll call what I’ve described here: Mathematical Literalism. In this reading, the primary meaning of math is its universality and invariance, and students should approach it with humility and with an intention to take in the received wisdom. Furthermore, those students who by talent and effort can approach it are honored as having some special capacity for abstraction, which is often a cipher for general intelligence.
This is one interpretation of math. Another is functional and gives more attention to what math can accomplish when put into real-world applications (I mean this colloquially, as in, talking about math in close relation to stuff, as opposed to the category of math textbook word problems that are not-so-affectionately labeled, ”pseudocontexts”). Here, math is most often seen in service of the sciences, engineering, economics, and finance. Its universality is acknowledged, but mostly in that it delivers reliable quantification and abstraction that makes valuable tasks accessible. The context of mathematics matters, but not the context of its historical development – only the context where math finds its expression in the field. This is often measured in material terms.
Those who speak of math with this frame in mind are attentive to the demands of a modern economy and a computerized, data-driven society, one which the digital revolution is constantly transforming. There are also aware that education, especially in applied and technical fields, opens up career opportunities for students that are selective and profitable. The important context, again, is in the future – and is measured by material gain.
This kind of reading of math I call Mathematical Materialism. The primary meaning of math is how we may use it, and it should be approached by students with a practical and ends-driven spirit. Those who find success in this way are valued for the material benefit they have created – no small distinction in our materialist, pragmatic culture.
Clearly, these two interpretations are not mutually exclusive. Most people do not think too much about math, and if pressed on the matter will say some combination of: math is math, and if you can do it you’re pretty smart; you will probably be able to find a good job, and the social value of that work will allow give you the means to live a happy life. The broader American values of practicality, hard work, individualism, and material success all participate in promoting these two readings of math.
There is another reading of math that acknowledges the data that Literalism and Materialism start with, but goes in a different direction. We can find a helpful middle route between these two hermeneutics by noting where they locate the most important context for mathematical work.
Literalism puts the important context in the past: the work has been done, and now we have math and we have to absorb it. Materialism puts the important context in the future: it’ll be worthwhile math when it’s put in service of some project with results.
In between these two, we have the context of the present: the important work to be done for students of mathematics is the work that is done right now. Its meaning is in its actualization, and not in its majestic existence or its material expression.
What would we call a reading that makes its present context a priority? It would focus our intention not on retracing past steps or the preparation for future steps, but instead on the mathematical process as it is pursued in your math classroom, or your child’s math classroom, right now.
I propose Mathematical Praxis as a middle-way interpretation of math, between Literalism and Materialism. I think this implies what I intend – that our proper reading of math requires us to engage in mathematical activity, and the activity is more important than the content or the product. The meaning of math is found in its process – the only meaning is the meaning you create in doing it, right now.
This interpretation of math would not dismiss its historical development, or its potential products – but it would approach them in right relation with math as a present and creative intellectual activity, an activity that requires practice, a space for which needs to be cleared in a young person’s developing mind. In fact, without engaging in creative mathematical activity, students can neither appreciate the work and ingenuity of mathematicians past (and locate them in a variety of locations, not just in the ancient Greeks and early modern Europeans), nor can they be expected to skillfully (and compassionately) apply mathematics to its practical ends. Surely some do, but a privileged few. If math is powerful and valuable, it must be shared equitably and made accessible to all.
Math is a verb, something that students do. It’s meaning is not prescribed or premeditated. It has meaning only in as much as it’s free act of creative intellect. Experiencing this, students can start to answer a question they are rarely asked: what does math mean?