Tuesday, November 09, 2021

The "Rigorous" Case Against Teaching More Students More Math

California is in the process of updating its statewide framework for Mathematics education, and unsurprisingly there is controversy -- controversy that will strike many observers as having striking similarities to the recent flare-ups about "ethnic studies" in the state. Monica Osborne in Newsweek has a concise summary of the position of opponents (largely summarizing the objections made in this open letter). There are the de rigueur drive-bys against the curriculum becoming too "political" and non-"neutral", to the point at which it is allegedly not even "mathematics" anymore -- I actually addressed some of those objections in a prior post. But the main objection is claiming that the new math framework abolishes (or at least sharply discourages) "accelerated" or "advanced" math classes, doing disservice to students who genuinely are "gifted" in a misguided nod towards "equity".

One thing I will say is that the open letter (perhaps learning from the Ethnic Studies debacle, where a refusal to do basic citation led to outright fabrications being bandied about by curriculum opponents) does everyone the helpful service of citing its sources so one can see where, exactly, the Mathematics Framework allegedly does say the things its critics contend it is saying. It isn't perfect -- for example, the letter claims that the Framework "[e]ncourages keeping all students together in the same math program until the 11th grade and argues that offering differentiated programs causes student 'fragility' and racial animosity (Ch. 1, p. 15)"; but I wasn't able to find either the "keeping all students together" claim or the "racial animosity" claim on that page (as for the "fragility" point, which was mentioned, it is interesting to note that -- perhaps contrary to one's intuition reading that word out of context -- the claimed "fragility" is that of the gifted students who come to "fear times of struggle in case they lose the label"). Nonetheless, it is tremendously helpful in assessing these claims that they are consciously linked to particular portions of the draft framework. Kudos for that!

Having said that, what do we make of how the new framework addresses issues like "tracking" and "giftedness" and other like practices? To some extent, there seems to be the usual "talking past one another" that one finds in controversies such as this; the State Board explicitly denies that the new framework removes programming to serve "gifted" students, or eliminates "accelerated" classes. But clearly they are attempting to make some alterations that are designed -- I think it is fair to say -- to de-emphasize the degree to which math education is sharply divided between the naturally-gifted "haves" and the hapless "have-nots". Are those modifications salutary? Do they underserve brilliant minds? Or are they increasing access to rigorous mathematical education for all students?

I want to digress for a moment to talk about my own journey through mathematics education. I am not, to be clear, a math teacher (though I am a teacher, and so to that extent have practitioner-level experience with pedagogy in general). I also never considered myself, and to this day do not consider myself, "good at math" (a label which to some extent makes this debate more interesting to me). This is so even though under any objective metric I was actually very good at math -- I got a 720 on my math SAT and a 4 on my AP Calculus exam. That's quite good, especially for someone who always considered himself "bad at math"!

My recollection of how math education occurred at my (high-performing, suburban public) school is that between sixth and seventh grade students made a single, fateful choice. They could enroll in advanced math -- that would put them on track to take advanced calculus (multi-variable calculus) by twelfth grade. They could enroll in honors math -- that would put them on track to take regular calculus in twelfth grade. Or they could enroll in some form of "basic" or "remedial" math, in which case they would not be on track to take calculus at all.

Notice that this single choice at age 12 basically laid out the entirety of one's mathematical education for the next six years. There was essentially no moving up the ladder once one made the choice (there was theoretically the possibility of falling down the ladder, although that was a terrifying prospect that kept many a "gifted" student awake cramming at night). For my part, I chose the middle ("honors") route -- I didn't think of myself as "good at math", so the top path wasn't for me, but I was a conscientious student, so I wasn't going to go the bottom route either. And from that point forward, my path was set; and to a large extent so was my self-identity vis-a-vis math. Despite the fact that I objectively was perfectly capable of doing well at math (again, check those SAT and AP scores), I always viewed myself as not a math guy -- the people who were taking the advanced classes; they were the ones good at math. As someone who tends to think very verbally, I never connected with math, never fully saw its usefulness to someone like me and with my interests.

This persisted even after I graduated high school. Carleton had a math/science distribution requirement, which I skirted as best I could with classes like "Science and Society" or "Conservation Biology" (which basically was a semester of taking pleasant hikes through nature). The political science department did require one to take a statistical methods sequence, which I satisfied by taking (1) the lowest level stats class the school offered and then (2) a "methods" class where I resolutely avoided doing any of the hard statistical analysis by deliberately choosing a research question where I found no correlation between my variables (so no need for robustness checks). When I got to graduate school, there was no way in hell I was going to touch the methods classes (which were not required for political theorists) -- that was way to advanced for little ol' anti-math me. I can't quite say I "regret" not taking them -- even now, the thought of it fills me with dread -- but I can say that lacking the ability to conduct independent empirical research is probably the biggest gap in my scholarly toolset dividing the sorts of scholarship I'm interested in doing from that which I'm capable of doing, and I do regret that.

Thinking back on it, it is simply insane that this entire mathematical-education arc was more or less set in stone via a single decision made when I was twelve. That's nuts! All the more nuts since it ended up becoming quite obvious that I was perfectly capable of learning advanced math; the choice made then did not map onto any "natural" capacity I did or did not have. I can fully accept that some other students at twelve might have had more fire in the belly for math than I did at the time, and that in turn could suggest different styles of teaching math to them than would have been appropriate for me. But to lock either of us into a particular rigid track at that age, telling us "you will go exactly this high, and no further" (and "you can go lower, but only if you cop to being a disgraceful failure") seems absolutely absurd.

It was remembering that personal trajectory that informed my read on what California is proposing. The core of their argument is, more or less, "all or nearly all students are capable of learning high-level math, and so making decisions in middle school that lock students in or out of taking high-level math classes at the end of high school is both unnecessary and foolish". Instead, we should restructure math education so that there are many different pathways that offer the opportunity for more students to end up enrolled in high-level math courses -- the choice you make at twelve shouldn't be your destiny. And likewise, we should recognize that "calculus" is not the only example of "high-level" math -- it is one, but not the only one. The current system is defined by the "race to calculus", where success is defined solely by whether and how fast one gets to the calculus class of the math sequence. This was certainly my experience -- the advanced students got there in 11th grade (then took an extra class of advanced calculus senior year), the middle group I was in get there in 12th grade, and the bottom group doesn't get there at all and is thus looked down upon. We actually had an AP Statistics class offered as well, but I honestly do not recall who even took it -- it was definitely seen as the "lesser" math class, even though in retrospect a strong knowledge of statistics would have been far more useful to me than the calculus class I did take. But it did not lie on the rigid trajectory of the math sequence, which meant it was implicitly downgraded as the off-ramp for the failures.

Math education, even more so than other disciplines, is addicted to the "cult of the genius", where we can identify relatively early on certain students as naturally gifted and others as hapless drones, and sort them into appropriate tracks based on those assessments. The California Framework suggests that this cult, like many cults, is not backed by empirical evidence -- one needn't be a gifted genius to learn high-level math, and working on the assumption that one does need to be such a genius means providing decidedly suboptimal math education to the median California student. We could be successfully teaching more students more math, but we don't because we somehow decided that we can sort the geniuses from the worker bees at the tweenage years and after that we just let nature take its course. It is unsurprising that these gut-check instincts about who is a "genius" track the usual lines of social hierarchy and stereotyping, but that is just the tip of the problematic iceberg.

The reformulation proposed by the California framework is not to prohibit strong, enthusiastic math students from being able to pursue their interests. For those students, it suggests that optimal math education is not defined by racing them to calculus as soon as possible, but may entail giving them more opportunities to explore more elements of math in greater depth or rigor. Moreover, by broadening the pathway that leads to high-level math, we reduce the sometimes overwhelming pressure that falls upon these student where if they struggle anywhere they are a failure and have sacrificed their future. It stands to reason that there will be plenty of "gifted" students for whom algebra clicks easily and for whom geometry is a nightmare. The status quo assumes that anyone who happens to be good at that type of math that is prevalent in grade seven will likewise sail through all the other math concepts at a similar pace. It would be better if we recognized that all students likely will have some parts of math they find easy or which naturally connects (or which they just enjoy more), and other areas they find difficult and need more time on (or just find profoundly uninteresting), and that acknowledging the latter does not come at the cost of giving up one's chance to take upper level courses. The rigidity of the current framework doesn't serve "advanced" students even on its own terms. It locks them into a rat race with only one destination and exceptionally high stakes for failure. It is not a good thing that strong math students at my high school essentially could not take AP Statistics even if that interested them more than AP Calculus.

But the bigger reform being pitched is that, for the average California student, the math sequence being offered isn't rigorous enough. It knocks too many students off the path of taking advanced math classes at far too early a point. Pluralizing the sequence of math classes and making it so that many students -- not just those identified as "stars" as tweens -- are in position to take advanced coursework in the field is a step towards stronger math education. It is step towards teaching more students more math at a higher level -- an unambiguously good thing, as far as I'm concerned.

In this, Osborne's reference to the opportunities wealthier families have to place their children in bespoke math "enrichment" programs actually proves the opposite of her point. It is unlikely, after all, that all or even most of the students who enroll in such programs are "geniuses". But that's the point -- they don't have to be: give them access to advanced programming, and they're perfectly capable of learning the material, because this education is not the province of geniuses but of ordinary students. The lesson California draws from those programs is that most of their students, "genius"-labeled or no, can and should have access to strong, rigorous mathematical education. So they should reorient their math education away from rigid, imprecise sorting at young ages, and instead think in terms of providing a plurality of pathways to get as many students as possible to advanced classes. Rich parents get this already; they can provide their kids with advanced instruction regardless of how they're labeled as tweens. This is powerful evidence that the rest of the public school system should follow suit.

What may be true is that California is trying to at least diminish the obvious hierarchy within math, where we can say the students who took multi-variable calculus their senior year are better than the those who only took AB calculus are better than those who didn't take calculus at all (recall my high school, where taking AP Statistics was a sure mark of being a lesser student). Reorienting the math curriculum so that many different end points (calculus, statistics, formal modeling, and so on) are all rigorous and robust means consciously trying to blur hierarchical lines that say one of these ends is "higher" than the others. If the most important part of math education is providing for status-differentiation -- being crystal clear about who are the elite students and who are the normies -- then this is a loss. It is, we should be clear, a very different loss from depriving "gifted" students of the opportunity to learn advanced math -- they're still learning it, they if anything have more choices on exactly what arena of advanced math they want to concentrate on, but they are not simply by virtue of staying on the top rung of the ladder they climbed at age twelve automatically viewed as superior math students to their peers. There is good ground to believe that it is that loss --the status loss, not the educational loss -- which is what's motivating much of the anger at the reform (how can I guarantee I'll get into Stanford unless everyone knows I'm better than my classmates who weren't imprecisely identified as math geniuses as a tween?). 

But California is making a conscious decision to trade having clear ordinal hierarchical ranking for giving more students more rigorous math instruction than they have now. That, to me, sounds like a good trade. And to the extent it isn't -- the problem isn't that the new curriculum lacks rigor or ambition, or tries to bring everyone down to the lowest common denominator. It is because it is sincerely committed to doing more for more students that the framework is generating this animosity.

No comments: