Some thoughts about why American math education stinks

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 lets 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  youre willing to hear me out, Im 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 couldnt 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.  Its 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 dont 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 youd 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 wasnt 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 Bs 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 didnt 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 theyve mastered concepts they should have mastered last year but didnt, failing to define terms precisely and  introducing new, more sophisticated concepts before the old ones have been fully assimilated, theres pretty much no way for kids to figure things out on their own. Forget deep understanding; they wont 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. Whats 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 doesnt 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 isnt 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 isnt sensitive to that context, explanations can easily end up being more superficial or more complex than what a student actually requires. Thats a big part of what makes teaching an art as well as a science. More often than not, students wont  come out and tell you when theyre confused; teachers must  be  attuned to  facial expressions and body language. If they  miss those cues and blithely keeps on going†¦ well, youve 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 isnt sequenced coherently, students end up with gaps and eventually hit a wall.   Likewise, if a teacher doesnt 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 its fair to assume that plenty of elementary school and even plenty of high school math teachers dont have a particularly strong grounding in the subject as a whole. It then stands to reason that they cant 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 cant imagine the subject from the perspective of a novice or figure out how to explain things that seem so obvious to them (or worse, dont 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 its fair to say that its often applied in a manner that leaves much to be desired. Ive 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.  Ive 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 Ive seen grammar get taught, I strongly suspect this isnt  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 theyre supposed to be figuring out; or 2) the teacher does not actually step in at some point and explain things clearly. (For the record, Im talking about a run-of-the-mill public high school math class, not a seminar at Exeter.) Ive 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.