I happen to agree that students who are not on accelerated math tracks are often shortchanged by poor instruction and - even more importantly - ridiculously low standards. I also happen to agree that every student - barring a severe LD/ID - should be able to master basic algebra, geometry, and statistics at the very least
if they are taught correctly. But the
Virginia Mathematics Pathways Initiative, which proposes a universal course sequence for all students in the Commonwealth in grades K-10, appears to be animated by wrong-headed ideas -- at least if the supporting documentation the VDOE provides is any indication.
Batty Theme 1: If You're High Achieving, You Need to Check Your Privilege
Despite the VDOE's recent denials, it does appear that their intention was - and perhaps still is - to eliminate acceleration and force high achieving students into mixed-ability classrooms until the last two years of high school. The links at the bottom of the page all display a hostility to explicit ability grouping, decrying tracking as "inequitable". This, from the NCTM, is representative:
"The practices of tracking students into course pathways that do not prepare them for continued study of mathematics and tracking teachers in ways that deny some students access to high-quality instruction are longstanding barriers to offering each and every student access to a high-quality mathematics education. These practices are not just, and they contribute to unjust differential student learning outcomes. These insidious practices must be dismantled if we are to achieve the goal of supporting each and every student in reaching his or her potential in mathematics."
But to call tracking “insidious” is biased, loaded language. Granted, in the past, high ability students from underserved groups were often excluded from “high” tracks due to bias. But the way to address this is not to get rid of tracking. It is, instead, to remove the subjective from tracking decisions. A focus on universal objective measures of mathematics achievement and talent will expand opportunity when coupled with “second look” programs that actively seek out low-SES students who may fall through the cracks.
In reality, tracking happens for practical reasons. Nobody in 2021 is intentionally trying to hold students back. It's simply a fact that in a typical middle/high school, you will encounter students who have strong ability/strong foundational development and students who do not. This variety arises because 1.) students are not all equally intelligent, sad to say, and 2.) even if they were equally capable in a theoretical sense, by the time students have entered the upper grades, it’s already too late to correct what’s been missed in previous classes without pulling out students who need extra help.
In fairness, the NCTM does mention "differentiation," which is basically a covert way to ability group students. But "differentiation" is very difficult to pull off fairly and effectively and, oh by the way, also dramatically increases a teacher's workload. So if we were to switch to mixed-ability classrooms, how would we bring the students who are behind up to “grade level” while also adequately serving talented, highly prepared students at the same time? How would we teach the student who still can’t divide fractions and the student who’s flying through basic algebra? I'll tell you: we wouldn't. We'd ignore the high achieving kid and teach to the lowest common denominator. I've seen it happen out here in the wild; I even catch myself devoting more time to struggling students even though I'm well aware, due to personal experience, that gifted children need special attention too.
It's also true that disadvantaged students often don’t have access to the most experienced, most effective teachers. But there are reasons for this too beyond some “insidious” conspiracy to screw certain kids. For one thing, the most talented, most experienced teachers are often the only ones who can teach AP/IB/honors level math. That's certainly the case at my own workplace. As a sixteen year vet, I'm the only instructor who knows our curriculum backwards and forwards and can confidently teach AP-level math and science. Thus, the highest achievers are funneled to my desk because I am the only one with the requisite content knowledge. Secondly, in many neighborhood schools, social dysfunction wholly unrelated to mathematics instruction often makes the environment an exceedingly unpleasant one in which to work. To be blunt, the highly talented instructors quite rightly want to work where they will be appreciated and treated well -- not where they risk constant disrespect and even physical assault from students we can no longer remove from our classrooms because “muh equity.” And lastly, talented teachers are more likely to want autonomy to teach as they see fit -- something they may not be given in low-ranking schools, where faddish nonsense often reigns supreme. Strengthen discipline in such schools. Protect them from the dangers of the surrounding community. And return to modes of instruction that are proven to succeed. Only then will you attract the best and the brightest.
Batty Theme 2: The Best Way to Teach Math Is Via Discovery & Endless Talk
This has been ed school orthodoxy since forever, and it shows no sign of dying out any time soon. (Which is one reason why I refuse to get my masters in math education even though it would almost certainly open up my job prospects). The problem? It's wrong. You can't learn math the same way you learn your native language because, in evolutionary time, we've only just yesterday started to use symbols and abstraction to represent both low and high level mathematical concepts. (Calculus was invented in the 17th century -- two seconds ago as far as our lagging wetware is concerned.)
The NCTM document describes a classroom scenario in which a teacher opens up a class discussion regarding how to correctly interpret a position-time graph -- a discussion that continues at some length until the students finally realize for themselves that a horizontal line on a position-time graph means that the object is standing in place. The theory behind this is that this floundering conveys some sort of "deep understanding" that explicit explanation does not. But there's no good evidence that discovery actually works -- at least not for our novice students, who simply don't have the grounding to conduct a fruitful investigation and reach the correct conclusion in a reasonable time frame. On the contrary, studies from Project Follow Through onward have concluded that well-designed direct, teacher-led instruction is best -- not just for procedural fluency but also for this vaunted "understanding" that is the education elite's central concern.
How do you best prepare students to tackle novel problems? You fill their mental bank with examples, algorithms, and formulae and give them scaffolded exercises that start with the routine and then gradually step up the challenge by adding more and more novel elements. You don't demand they behave like professional mathematicians at the outset in the name of "productive struggle". No scientist launches an experiment without first reviewing all the literature that's relevant to his research question. No pianist starts playing concertos without first learning scales. No basketball player skips his basic drills. Similarly, no student of mathematics should be thrown into the proverbial deep end without floaties -- i.e., the received knowledge generated by mathematicians of previous ages.
Which brings me to...
Batty Theme 3: Procedural fluency/correctness is unnecessary -- or at least not as important as "understanding."
From the NCTM again:
Or imagine a classroom in which Jacobi is told directly that his reasoning is incorrect and Charles is told directly that his is correct. In such a case, both Jacobi and Charles might base their mathematical identities on correctness—getting the right answer—rather than on reasoning and sense making.
And also:
Historically, much of school algebra has been concerned with the rewriting of algebraic expressions and the solving of equations and inequalities. Today, these tasks can be performed by technology that is available in a variety of forms, and this development should shift the focus of school algebra from learning how to perform algebraic manipulations “by hand” to learning how to recognize which techniques will produce a desired outcome, how to interpret the outcome mathematically, and how to use the outcome to move forward in analyzing a situation or solving a problem.
Beneath quotes like these lurk the assumption that procedural fluency and understanding are independent -- but that is not the case. Procedural fluency is, in fact, what enables understanding over time. Math is a linear subject. Topics build on each other. The lower leads to the higher.
In my own experience, students who have not thoroughly mastered adding, subtracting, multiplying and dividing fractions struggle with Algebra 1. That's because the skills required to competently manipulate fractions are intimately linked with the skills you need for entry-level high school courses. To cite one example: to efficiently reduce fractions requires strong knowledge of divisibility -- something that also happens to be required to efficiently factor quadratic trinomials. The fractions score on my workplace’s diagnostic exam, in short, turns out to be a fantastic proxy for estimating a student’s general number sense and problem-solving ability and is a strong predictor of success (or struggle) in math overall. And when we focus on addressing a student's lack of proficiency in manipulating fractions through direct, focused instruction - when we, in other words, do exactly the opposite of what the NCTM suggests - that student's algebra grade, in the vast majority of cases, goes up. True: this is the experience of one particular learning center. But my personal observations do line up with what we're seeing in our abysmal math performance stats nationwide after years of "reform" math and "discovery learning".
So no: forgoing the rich understanding that comes through doing lower level math with pencil and paper is not going to enable students to access higher level concepts more quickly. The reason for this is easily explained by something known as cognitive load theory. According to cognitive load theory, we can only hold roughly four items in our working memory at a time. But we can circumvent that by storing things in long-term memory - like the multiplication facts or the algorithm for finding a percent - thereby opening up the working memory slots for other things - like breaking down a multistep word problem. Thus, as it turns out, it is actively detrimental to force students - through instruction that deemphasizes the importance of procedural fluency - to consciously attend to lower level skills while they're attempting to understand something more advanced.
And as for these "reformers'" love affair with "technology": Number one, technology is not reliable; I see my calculator-dependent students miss-key things all the time without realizing that their magical box is giving them garbage output in response to their garbage input. Number two, using technology is often less efficient than simply knowing something; it takes much longer to type "8 X 7" into a calculator than it does to retrieve "56" from memory. But most importantly, the way you process information that's delivered to you on a computer is different from the way you'll process that same information in writing. When looking at a screen, we are less attentive. We scan. We become passive bystanders instead of actively grappling with the topic at hand. That's why smart researchers are now suggesting students eschew technology entirely when they're trying to absorb a lecture or otherwise assimilate new material.
Batty Theme 4: We Must Never Tell a Child He's Wrong
Instead, we must cultivate their "identities" as "doers of mathematics."
Yeah: it's basically the self-esteem movement all over again; it's just been re-dressed in social justice/critical theory clothes.
Of course, we should never tell a child that he's constitutionally bad at math and should just resign himself to that fact. As I stated at the start, I think even middling students are capable of mastering algebra, geometry, and basic statistics with the right interventions. And I also think that there should be flexibility in our sorting systems -- that there should be ample opportunity to jump tracks through sustained hard work and demonstrated achievement.
But while we should encourage our students to strive to achieve no matter their starting point, we shouldn't lie to them. Despite what appears to be popular ed school belief, kids are not stupid. They're going to know who usually gets the answers right and who often gets the answers wrong no matter how many times you insist that everyone in your class is "thinking mathematically" (whatever that means). And they're certainly going to notice all attempts to blow smoke up their butts.
The best way to build a child's so-called "identity" as a "doer of mathematics" is to actually teach them mathematics to mastery. Competence leads to self-esteem and self-efficacy. It doesn't work the other way around.
Sigh. Can we please drop all this BS and go back to the legitimate cognitive science? Kthx.
