In theory, at least, I should be the very last person to weigh in on this topic. As I usually joke, most of my students could tutor me in math. My mathematical education was mediocre in every way imaginable, and let’s not even talk about that 200+ point score gap between my SAT and Math and Verbal scores. With competent instruction, I probably could have become an excellent — or at least a decent — math student, but alas, that ship sailed many years ago.

So why on earth should anyone listen to me spout off about what ails math education? Well, because by this point, I know a fair amount about the functions and dysfunctions of the American educational system, about pedagogical trends, and about just how difficult good teaching really is. If you’re willing to hear me out, I’m going to start with an anecdote.

I occasionally receive emails from prospective test-prep authors (mostly math/science, incidentally) seeking advice. A few months ago, I got a message from someone looking to self-publish an ACT science book. In the course of his message, he mentioned that he couldn’t begin to understand my work because he trafficked in the world of logic and objectivity. Although it was undoubtedly unintentional, the implication was that my work was frilly and subjective, the academic equivalent of a pink cupcake.

It probably would have surprised him to learn that traditionally, grammar and rhetoric were grouped together with logic as areas of study. Until the nineteenth century, the boundary between the humanities and the sciences was quite fluid, the sciences being considered a form of “natural philosophy.” Although there are still some of areas where the two overlap in overt ways — music theory and math, for example, or philosophy and physics — they tend to be relatively esoteric. Current discussions typically pit humanities and sciences against one another or worse, contend that the humanities only have value insofar as they can be made to serve more pragmatic pursuits. (For an exceptionally heavy-handed example of this mentality, see Passage 4, Test 1 in the new SAT Official Guide.)

The reality, however, is that grammar and math actually have quite a bit in common, and the way I teach grammar probably has a lot more to with with what goes on (or should go on) in math class than what goes on English class. It’s not an exact analogy, of course, but consider that math and grammar share some essential characteristics. Both are formal, symbolic systems whose real-life applications are not always immediately obvious. Both are sequential and cumulative — if you don’t master basic terms and formulas and understand their applications, you are not really prepared to move on to the next level. And both can become very creative at a high level, but not without a thorough mastery of the basics.

If you’d asked me a year ago, I would have very naively said that training people to teach reading would be harder than training them to teach grammar. Reading, after all, is fairly subjective, and there are almost infinite ways for a student to misunderstand. As it turned out, I had things backwards. Because there are a fairly limited number of formal techniques that can be used to teach reading (focusing on the introduction and conclusion to determine the main point, using context clues, identifying transitions), there wasn’t a huge amount of wiggle room in terms of training people to teach it.

Grammar was a different story.

First, let me explain that I learned pretty much all of my grammar in foreign language class. Years of foreign language class, starting from when I was about seven through well after I graduated from college. Almost all of it was pretty traditional — pages and pages of exercises, progressing from the present tense to the imperfect subjunctive, from direct and indirect objects through relative pronouns. Although I have an excellent ear for languages, grammar did not come totally naturally to me. In fact, I got B’s in French for most of high school (albeit in an extremely accelerated class). But after covering the same concepts in more or less the same order in multiple classes, in multiple languages, over multiple years, there was pretty much no way I could not master them.

The way I teach, and the way I write my grammar books, very much reflects that experience. When I started teaching grammar, I simply mimicked what my teachers had taught me — teachers who were at worst merely competent and at best outstanding. Because I came from a foreign language background, my starting assumption was always that my students knew nothing, that every term had to be defined, and that I could not leave any step to be inferred. Since very few of my students had learned any grammar in school — and in the rare cases they had studied grammar, they usually had only the most fragmentary understanding of what they had learned — this approach proved highly effective.

When I started interviewing and training tutors, however, I was struck by a few things. First most tutors had a noticeable tendency to overcomplicate their explanations. They often attempted to cover multiple concepts simultaneously, using very fairly sophisticated terminology — and they didn’t stop to make sure the student truly understood all of the terminology they were using. They simply took for granted that the student had not only been exposed to but had also mastered the terminology they were using, even if that was not at all the case.

Now, in English, kids can still muddle along because, well, they speak the language (even if some of their writing is pretty hair-raising), but in math I suspect those types of oversights can be deadly. If a teacher is talking past their students, assuming that they’ve mastered concepts they should have mastered last year but didn’t, failing to define terms precisely and introducing new, more sophisticated concepts before the old ones have been fully assimilated, there’s pretty much no way for kids to figure things out on their own. Forget “deep understanding;” they won’t even get the basics.

That brings me to my next point, namely the false dichotomy between “rote learning” and “deep understanding.” I think most people would consider it common sense that lessons need to be calibrated to the level of the particular students, and that beginners usually need to have things explained in pretty simple ways. What’s somewhat less intuitive, and what often gets overlooked in debates about pedagogy, is that aiming for “deep understanding” too early on can be counterproductive because it often involves more unfamiliar terminology and concepts than students are prepared to handle. The strain on working memory is simply too great.

The initial goal, at least from my perspective, should be to give students tools that are simple to remember and that can actually be used. If an explanation of the underlying logic behind a rule happens to help students better grasp a rule, in such a way that they can apply it more effectively, then by all means the explanation should be provided. But if explanations are too confusing, they can do more harm than good. It doesn’t happen often, but sometimes straight-up memorization is actually the best approach at first. Then, when the student is ready, progressively more nuanced versions of the concept can be introduced.

Usually, though, the issue isn’t explanation vs. no explanation but rather how in-depth the explanation should be. There are countless gradations between pure “rote” memorization and in-depth conceptual learning, and there is a very fine line between explaining a concept thoroughly and explaining it in a way that brings in extraneous, potentially confusing information. A good deal of teaching involves walking that line. Sometimes a little bit of the theoretical underpinnings can be introduced, and sometimes it makes sense to go more in-depth. It all depends on where students are starting from and what they hope to accomplish. If a teacher isn’t sensitive to that context, explanations can easily end up being more superficial or more complex than what a student actually requires. That’s a big part of what makes teaching an art as well as a science. More often than not, students won’t come out and tell you when they’re confused; teachers must be attuned to facial expressions and body language. If they miss those cues and blithely keeps on going… well, you’ve probably had that experience.

Moreover, concepts being taught must be considered in context of the subject as a whole: what (if anything) has been taught before, and what must the student absolutely master at this point in order to move to the next level somewhere down the line? If a curriculum isn’t sequenced coherently, students end up with gaps and eventually hit a wall. 

Likewise, if a teacher doesn’t know enough about the subject to understand where the particular concept they are teaching fits in, they are unlikely to be capable of fully preparing students for the next level. I think it’s fair to assume that plenty of elementary school — and even plenty of high school — math teachers don’t have a particularly strong grounding in the subject as a whole. It then stands to reason that they can’t teach with an eye toward what might be required a year down the line, never mind five years down the line.

On the flip side, of course, some teachers are so naturally gifted in a subject, or take so much of their knowledge for granted, that they simply can’t imagine the subject from the perspective of a novice or figure out how to explain things that seem so obvious to them (or worse, don’t even realize that things need explaining). That was my 10th grade math class, and thinking about it still makes me shudder.

The other, related, issue I see has to do with the way in which both traditional and progressive forms of teaching are misapplied.

In traditional teaching, a general concept is presented, after which students work through a number examples to see it in action. This model has taken a lot of flack over the last century, some of it merited and most of it based on various types of distortion, but I think it’s fair to say that it’s often applied in a manner that leaves much to be desired. I’ve noticed that American teachers tend to overestimate students’ ability to infer the application of rules to complex/sophisticated situations after those rules are presented in a relatively superficial way. For example, subject-verb agreement can theoretically be covered in about five seconds: singular subjects take singular verbs, while plural subjects take plural verbs. Easy, right? In theory, perhaps.

In reality, many students must learn about gerunds, prepositions and prepositional phrases, non-essential clauses, compound subjects, etc. in order to answer the full range of SAT subject-verb agreement questions. You cannot skip parts — even seemingly obvious ones — and leave beginning students to figure out the rest; every step must be mapped out. Concepts must be continually reinforced and slowly built upon so that new concepts, as well as their relationships to other concepts, are gradually introduced and then explored in progressively more complex ways.

Furthermore, each concept must be drilled until it has been mastered; simply reiterating the logic behind a concept is not enough. I’ve been told that this is how math gets taught in most Asian countries, which not coincidentally tend to have the highest math scores. Based on the way I’ve seen grammar get taught, I strongly suspect this isn’t happening in American classrooms. (Also, American pedagogy is addicted to incoherence, confusing it with freedom and creativity; explicit, clearly sequenced lessons would be anathema, even if teachers were given leeway in implementing the specifics.)

An equal if not bigger problem results from the other extreme. In a progressive model, students are given a series of problems or examples and asked to figure out the general concept. While this approach has the potential to be useful, if done in a moderate and controlled way, it can lead to serious confusion if 1) students have insufficient background knowledge to figure out whatever it is that they’re supposed to be figuring out; or 2) the teacher does not actually step in at some point and explain things clearly. (For the record, I’m talking about a run-of-the-mill public high school math class, not a seminar at Exeter.)

I’ve seen tutors try to build lessons around students’ prior knowledge or intuitive understanding of a concept, when in fact those things were so spotty they provided virtually no basis for understanding. What they clearly perceived as guiding intended to “empower” the student was actually going nowhere. Far from realizing that, though, they seized on any scrap of understanding as evidence that their approach was working. Never mind that there was very little the student could apply in any meaningful way. Once again, that approach creates enough problems in English, but native speakers will still be able to utter more or less grammatically acceptable utterances regardless of whether they can distinguish between the present perfect and the past perfect. In math, the consequences are likely to be a lot more dire.