So I’ve been catching up on some reading these past few days and one of the recent Stone articles (I do make sure to check those out always, though they’re not always interesting) spoke about what they termed as the “Math Wars”. I’d heard of this before briefly from a co-worker, but never termed as a “war” or anything like that.
What I took from the history the authors sketch of this conflict is between folks who advocate teaching and designing for “numerical reasoning,” which entails that learners navigate numbers and techniques in ways that work best for them (the reformers), and those who advocate for teaching the common methods used most universally throughout the world (the traditionalists). The article makes some claims that I agree and disagree with, and I’d like to address a few of them in relation to the things I’ve picked up about education in the past few years.
There’s certainly a great deal to which I sympathize with math traditionalists. There’s a great degree of utility to common language – thinking of the nervous attempts at German I’ve mustered throughout the years with native speakers brings to mind the comfort English speaking affords me in my day-to-day. This presumably extends to math as well, where I can conjure up two individualistic math-speakers who just can’t wrap their minds around how the other came to their numerical conclusions. Additionally, I agree with the authors point that teaching common algorithms leads to mechanical thinking. What one might attribute mechanical thinking comes from a weird perspective (so it seems to me) that on one hand hypothesizes the brain and the way humans think as analogous to computer processing while on the other hand believing that this sort of computer-like thinking is wrong and detrimental to some sort of true creativity. It must be hard to sleep at night for folks who hold this view, as they’re trapped into believing that the way humans think is fundamentally wrong. Getting back to the matter, what appears to be mechanical thinking is really just mastery, which is a beautiful thing I believe. It avails us a level of “yes, I got this!” while freeing us up to do other rad things with our time.
Beyond this, I’m reminded of the notion of power (I’m going to throw this term around quite a bit, bear with me?) and how it relates to all (I think I can be broad here) forms of knowledge. We ought to afford learners the chance to understand and interrogate X structure from the empowered position, whatever its’ form, with the terms and logic that that power engages with X. Again, this is to say that we should empower learners with the knowledge to critically evaluate things, most definitely including mathematics, in the ways that the current power does (critical anything is super important yet difficult to define, so just go with your favorite understanding of critical). This doesn’t mean that learners all have to truly, I don’t know, buy into those terms and logics, no, rather they should have the opportunity to understand them and be free to conjure alternatives.
This being said, the would-be imposed methods of teaching and learning math of the reformers certainly carries merit. With mastery comes a blindness to ways of determining solutions outside of those already known. Yes, traditional mathematics isn’t “creative” and neither are other potential ways as advocated or permitted by reformers. What exactly constitutes creative I couldn’t say, but at the very least, calling out traditional mathematics by opening up the discussion to new mathematical processes is less likely to make the learner “blind”. The authors put forth that learners “need to master bodies of facts,” in mathematics as opposed to other content areas, but positioning facts in that way is problematic. I say who gives a damn about facts in any subject, whether it’s math or history. Facts should become apparent to learners as they master useful ways of thinking about things. Knowledge of these mathematical facts have to be contextualized through the processes that use them so that the learners don’t, or don’t have to, take the facts for granted, but see them as tools for conversing in a particular way around a subject. The ways of thinking should be emphasized over the facts, which will be picked up as one gains a handle on the field.
On one hand, I feel as if I’m trying to refute the authors by simply stating the side their arguing against more loudly (and I’m trying to avoid their weak point about math books and rote learning), but by stating that the traditional methods of learning mathematics are (most elegant and powerful), you end up defining mathematics by it’s past rather than it’s potential. I don’t believe that reformers are actively against teaching traditional methods, but I would argue that it’s a matter of attempting to unhinge math from its’ roots with the hope that learners see math-as-we-do-it as a culturally mediated practice instead of universal and fundamental fact. In math and elsewhere, learners are often (in my experience) acquainted with the ambiguity and malleability of knowledge after hearing for years that “this is the way the world is”. This is bullshit. Christianity isn’t just the Holy Trinity, Islam isn’t just the Five Pillars, math isn’t just FOIL, computer science isn’t just java and so on.
We absolutely should avail learners of the traditional methods of numerical reasoning because it provides access to ways of understanding the world as it has been constructed. Without this way of thinking, it’s hard to imagine anyone being able to critique or creatively outdo traditions when they ought to be. But like hell we should limit them to that, and the only way that we’re going to expand the horizons of mathematical knowledge is if we open up our curriculum to reasoning that removes traditions from the gravitational centerpoint.
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