Joseph Mazur, dangerous guy
If you want to get educational traditionalists all aflutter, say something the implies that rote learning may not be all it's cracked up to be. It's about as effective in stirring up ire as burning a US flag in front of the local branch of the American Legion (though I think impugning rote learning isn't likely to get one arrested, jailed, or fined. Yet.)
Because I'm basically a bad person, I like to post without comment quotations I consider interesting and potentially provocative on lists inhabited by knee-jerk anti-progressives and educational conservatives. The resulting furor is remarkable, more so because I don't say a word about what I think is noteworthy, supportable, brilliant, or absurd in the passages. Naturally, on lists like math-teach@mathforum.org, where I've participated in various ways since about 1994, my reputation precedes me, and it's a safe bet that those who aren't fond of me or my ideas are sure they always know what I'm saying, even when I haven't said anything at all.
So last week I posted a quotation from Joseph Mazur's THE MOTION PARADOX at math-teach, under the subject line "More on rote learning":
Teaching was dictatorial, and rote memorization of Aristotle's works played a central part in the curriculum. The seven liberal arts -- grammar, logic, rhetoric, arithmetic, geometry, music, and astronomy -- were required, though how much of each was under local control. This rote learning numbed the intellect so severely that nobody thought to criticize the classic works of science, especially the unshakable doctrines of Aristotle. Moreover, except for rote learning of arithmetic and computation, mathematics was completely neglected. (Joseph Mazur, THE MOTION PARADOX, pp. 60-61)
Of course, it was a bit unfair of me not to offer a little bit more of this passage to help give it meaningful context, so I later added the following sentence which immediately followed the quotation above:
The names of Euclid and Archimedes were empty sounds to the mass of students who daily thronged the academic halls of Bologna, the ancient and the free, of Pisa, and even the learned Padua.
One final addition from a few paragraphs down the page, sent later that day, completes the context, I think:
Young Galileo was studying the usual courses of philosophy and medicine, but under stiflingly rigid training, rather than through the kind of education he was used to at home with his father, who taught him to weigh, examine, and reason the truth of each assertion before accepting it. He despised university training, which professed truth by authority and regarded any contradiction to Aristotle as blasphemy.
Now, silly me, I would have thought this would be enough to balm the seething souls who saw what Mazur had written as "bashing" rote learning. But then, I try to resist accepting the depths to which some folks are entrenched when it comes to such things and the deep-seated fear they seem to have (or at least the compulsion to claim) that progressive math educators want to banish facts, formulas, algorithms, proofs, etc., from classrooms. The result is that no amount of context for Mazur's negative comments on the stultifying atmosphere of university education in Galileo's day can mollify the staunch traditionalists, at least not on math-teach.
To their protests, I offered the following reply:
I think a key question here is: given two possible approaches to learning something not consisting strictly of unrelated facts (random items on a list, dates and names from history being taught strictly or primarily for the purpose of having them regurgitated on an exam with little or no concern for their actual significance or meaning, etc., or in other words, learning as a "bunch o' facts"), one grounded strictly or primarily in rote, the other in grounding the important facts in context with a focus on what the important ideas involved with those facts are (and this could readily apply to mathematics, science, history, literature, philosophy, or many other subjects), which approach would you prefer as a student? As a teacher? Which do you believe would be more likely to produce successful mathematicians, scientists, historians, philosophers, literary critics, etc.?
Note that I do not suggest that the latter approach be devoid of facts, or that some degree of being able to recall those facts "on demand" as you are fond of saying, be kept strictly off the table. But I think Mazur's point about education was, and I know mine is, that: 1) rote learning as the sole or even predominant focus of education is deadening and tends to be used in ways that discourage active thinking and questioning of what is being memorized; 2) that to no small extent, historically this deadening of the mind and suppression of skeptical and critical thought has been a major GOAL of such educational methodology; and 3) that we would do well to employ the requirement that material be learned strictly or primarily by rote sparingly and always with other alternative approaches.
That said, I would certainly offer alternatives to students about how to master important facts. I've mentioned this before, written about it extensively here at times, seemingly with little impact on some others' viewpoints about rote. I continue to hold that in cases where relatively arbitrary facts are involved (e.g., the names and order of the cranial nerves, the names and order of the presidents of the United States, or the names and atomic numbers of the chemical elements - where the usefulness of 'at one's fingertips mastery' can be debated as more or less important in each case, as in the case of many other such examples, especially given how readily one can access such information these days), there are effective mnemonic methods available that should be taught or at least made mention of by any instructor who insists that it is necessary for students to memorize a great deal of such material. To not do so is, on my view, irresponsible. To not even know of and have explored such methods suggests a certain self-centeredness on the part of some teachers who may be particularly adept at rapid memorization without regard to either special techniques or a great deal of rote repetition, or who simply enjoy such engagement in "mastering" facts and really don't care whether their students like doing so, are successful at it, or actually are better off for having done it: their viewpoint seems to be that such tasks are necessary rituals that must be respected and gauntlets that must be run by each member of every generation. In other words, if long bouts of rote were bad enough for me, they're certainly bad enough for my kids or students.
I wouldn't forbid students who feel that they wish to indulge in rote learning from so doing, though likely it would be something I'd suggest they do on their own time. If I were spending "precious" classroom time on memorizing, it would be in ways I believe are more efficient and effective: through teaching or helping students develop their own mnemonics, and through games and other activities that help students do drill and practice in ways that reinforce memorization but not without elements of thinking and enjoyment. I'm afraid I'm much to anti-Puritanical to swallow the notion promoted endlessly in the Math Wars and Education Wars that it is necessary to torment students in order for them to learn.
On the other hand, to return to another favorite sore point, I am all for challenging students to stretch their thinking, and believe strongly that asking mathematics students to take what they've learned and use that to make attainable leaps beyond what has been directly instructed or analyzed in class is a reasonable and useful expenditure of that precious time. The degree to which such tasks need to be scaffolded is a useful and open pedagogical question that those of us who actually do such teaching continue to explore. I'm sure that it's something those Japanese teachers who use these sorts of problem tasks think about and discuss on a local and national basis. This remains yet another one of those delicate instructional questions for which I doubt there is a simple answer that would apply to everyone all the time (either teacher or student). But I know the mention of these sorts of problems, like the criticism of rote learning, upsets traditionalists. It seems ironic to me, however, that many of these same people have no compunctions, it seems, about promoting one kind of instruction that many students find boring and painful, while disdaining another kind which many find frustrating and painful. Or perhaps I only THINK this is a contradiction and source of irony.
Calavitta and Rote
The response of one defender of rote learning was to cite a recent article about a Los Angeles-area private school mathematics teacher and a video of a small snippet of his classroom work. I read the article when it was first mentioned on the list, but somehow didn't see the video (perhaps I didn't read the article on the LA TIMES web site) and decided when his name kept being thrown out as proof of the wonderful usefulness of rote learning that I needed to view him in action. Having done so, I posted this follow-up to my previous comments:
I just watch the very short video. Calavitta's a very exciting guy. I have too little context and content available to me to judge the scope of his methods. What we see is kids having a lot of fun. We don't see any math, of course, or have any way to judge what the kids can do mathematically, or how they learn to do those things. We do know that they seem to know and be able to recognize definitions, theorems, etc. and repeat them, one student doing it so absurdly quickly that Calavitta points out that he can't understand what the student said. Do you think that's a good thing? Why not get that guy from the old commercial who specialized in rapid-fire talking? Wouldn't he be an even BETTER instructor for these kids, if that's the real goal?
But of course, it isn't. Calavitta's teaching on that video is grounded in kids playing a game, and it looks like it's a positive, pretty much student-centered activity. I suspect there's more to his classes and instruction than that, or these kids wouldn't be doing well on calculus tests. Rapid fire repetition of theorems and definitions won't get you far on the tests with which I'm familiar, in high school or college.
You seem hung up on the surface of what he's up to. That's really too bad, as it would sell him badly short as an effective instructor. The guy actually cares about his students, according to the article. They clearly feel and reflect that. Do you think that's contained in his recitation exercises? Could you or most teachers learn to care about kids more from having them do those exercises? Would students believe you cared about them because you had them do such exercises?
You're mistaking the gift wrapping for the present. But kids won't make that mistake.
I just watch the very short video. Calavitta's a very exciting guy. I have too little context and content available to me to judge the scope of his methods. What we see is kids having a lot of fun. We don't see any math, of course, or have any way to judge what the kids can do mathematically, or how they learn to do those things. We do know that they seem to know and be able to recognize definitions, theorems, etc. and repeat them, one student doing it so absurdly quickly that Calavitta points out that he can't understand what the student said. Do you think that's a good thing? Why not get that guy from the old commercial who specialized in rapid-fire talking? Wouldn't he be an even BETTER instructor for these kids, if that's the real goal?
But of course, it isn't. Calavitta's teaching on that video is grounded in kids playing a game, and it looks like it's a positive, pretty much student-centered activity. I suspect there's more to his classes and instruction than that, or these kids wouldn't be doing well on calculus tests. Rapid fire repetition of theorems and definitions won't get you far on the tests with which I'm familiar, in high school or college.
You seem hung up on the surface of what he's up to. That's really too bad, as it would sell him badly short as an effective instructor. The guy actually cares about his students, according to the article. They clearly feel and reflect that. Do you think that's contained in his recitation exercises? Could you or most teachers learn to care about kids more from having them do those exercises? Would students believe you cared about them because you had them do such exercises?
You're mistaking the gift wrapping for the present. But kids won't make that mistake.
The Torture Never Stops?
Of course, to quote from BEN HUR, "It goes on." No matter that I thought I'd done a fair job of suggesting that Calavitta must be up to something other than (and more than) rote, the same idea that his teaching was a matter of "recitation-based memorization work" persisted (and continues to persist amongst the faithful. And so today I offered what I hope will be my last words on this matter:
As we still don't seem to have any definition of "recitation-based memorization work that Sam Calavitta advocates," it's a bit hard to talk intelligently about what he's doing. We see a video of kids responding quickly to flash-cards of formula and theorem names. So we know that some students have memorized the requisite information.
This begs several questions: 1) have all the kids done this, or only those who are quickest on the draw in the contest? 2) HOW have they memorized this information? We are being asked to believe without the smallest support that the contests ARE the learning. This seems highly doubtful. How could a student who hadn't already memorized these facts and words possibly learn them from hearing another kid rattle them off at breakneck speed? WHERE, WHEN, and HOW is the actual "learning" going on? 3) What can those who have memorized this information do with that information? Does being able to recite consistently translate into understanding of the meaning of the words and concepts? Into being able to use the information? How so? How do we know this? The video certainly provides virtually no information along those lines, either; and 4) Where do we see Calavitta advocating "recitation-based memorization"? We see him advocating caring about students. We see him letting kids have fun. We see him doing something along the lines of Math Jeopardy or Math Bee, but we don't see anyone doing "recitation-based memorization," unless I'm really blind to what's on that tape.
Lou Talman (a Denver-based mathematician) has pointed out that there are more powerful methods of memorization. So have I. I've also spoken towards the importance of review. None of this is a matter of "mere rote." Rote is repetition, as far as I understand the term. Repeating things over and over guarantees nothing for most people.
Here's one very simple example: my best friend had a business in lower Manhattan in the 1970s and 80s, until he moved it to Brooklyn. The phone number was 212 925-6095. I haven't called that number in about 20 years, probably more, since he's been based in Brooklyn for at least that long, has moved the business twice, and has had different numbers each time. Yet I can still recall that phone number. I didn't learn it or ingrain it in my mind by rote. I used a simple mnemonic system that translated the numbers into consonant sounds, and then I constructed a phrase that I also connected a couple of simple images to. The images, the phrase, and his business were somehow interconnected in ways that allowed me to easily call up "on demand" that number, and decades later, it's still there, probably as permanently as anything in the way of completely useless (now) and arbitrary information can be ingrained in one's mind intentionally.
Am I saying that rote wouldn't have worked? Of course not. But it wasn't necessary and would have comprised increased time and effort with less effective results (for me, in my experience with that approach). I doubt highly that rote alone would have resulted in my knowing that number today. I can cite more complex and less trivial examples from personal experience.
Perhaps there is some study out there that some list member can point to that shows that rote is an effective and efficient method for making students good at mathematics. I'd love to see it. I'm sure everyone on this list would love to. I simply am skeptical that any such study exists. It would be interesting to see what top researchers in the field of memorizing (especially of MEANINGFUL, rather than random or arbitrary material) have to say about the effectiveness and efficiency of rote or drill in achieving long-term retention and understanding (of mathematics, or of other subjects).
Please note that quizzes, recitation, and other assessments that require quick recall do not TEACH anyone how to attain recall. They only "demand" that students be able to do so. And of course, there are many types of quizzes that aren't about recitation. My calculus II teacher in NYC and my calculus teachers at University of Michigan gave quizzes at the beginning of every class. They simply comprised short problems that were based on what was studied in class and the previous homework, to encourage students to actually do work outside of class. But one needed to actually be able to do some problem-solving, calculating, etc., making use of what was studied, not spew easily-regurgitated facts or definitions that could readily be forgotten at least until a chapter test, when they could again be crammed, spewed, and forgotten.
Of course, maybe that's what mathematics is really all about. Maybe my high school teachers had it right, and my university instructors were confused.
As we still don't seem to have any definition of "recitation-based memorization work that Sam Calavitta advocates," it's a bit hard to talk intelligently about what he's doing. We see a video of kids responding quickly to flash-cards of formula and theorem names. So we know that some students have memorized the requisite information.
This begs several questions: 1) have all the kids done this, or only those who are quickest on the draw in the contest? 2) HOW have they memorized this information? We are being asked to believe without the smallest support that the contests ARE the learning. This seems highly doubtful. How could a student who hadn't already memorized these facts and words possibly learn them from hearing another kid rattle them off at breakneck speed? WHERE, WHEN, and HOW is the actual "learning" going on? 3) What can those who have memorized this information do with that information? Does being able to recite consistently translate into understanding of the meaning of the words and concepts? Into being able to use the information? How so? How do we know this? The video certainly provides virtually no information along those lines, either; and 4) Where do we see Calavitta advocating "recitation-based memorization"? We see him advocating caring about students. We see him letting kids have fun. We see him doing something along the lines of Math Jeopardy or Math Bee, but we don't see anyone doing "recitation-based memorization," unless I'm really blind to what's on that tape.
Lou Talman (a Denver-based mathematician) has pointed out that there are more powerful methods of memorization. So have I. I've also spoken towards the importance of review. None of this is a matter of "mere rote." Rote is repetition, as far as I understand the term. Repeating things over and over guarantees nothing for most people.
Here's one very simple example: my best friend had a business in lower Manhattan in the 1970s and 80s, until he moved it to Brooklyn. The phone number was 212 925-6095. I haven't called that number in about 20 years, probably more, since he's been based in Brooklyn for at least that long, has moved the business twice, and has had different numbers each time. Yet I can still recall that phone number. I didn't learn it or ingrain it in my mind by rote. I used a simple mnemonic system that translated the numbers into consonant sounds, and then I constructed a phrase that I also connected a couple of simple images to. The images, the phrase, and his business were somehow interconnected in ways that allowed me to easily call up "on demand" that number, and decades later, it's still there, probably as permanently as anything in the way of completely useless (now) and arbitrary information can be ingrained in one's mind intentionally.
Am I saying that rote wouldn't have worked? Of course not. But it wasn't necessary and would have comprised increased time and effort with less effective results (for me, in my experience with that approach). I doubt highly that rote alone would have resulted in my knowing that number today. I can cite more complex and less trivial examples from personal experience.
Perhaps there is some study out there that some list member can point to that shows that rote is an effective and efficient method for making students good at mathematics. I'd love to see it. I'm sure everyone on this list would love to. I simply am skeptical that any such study exists. It would be interesting to see what top researchers in the field of memorizing (especially of MEANINGFUL, rather than random or arbitrary material) have to say about the effectiveness and efficiency of rote or drill in achieving long-term retention and understanding (of mathematics, or of other subjects).
Please note that quizzes, recitation, and other assessments that require quick recall do not TEACH anyone how to attain recall. They only "demand" that students be able to do so. And of course, there are many types of quizzes that aren't about recitation. My calculus II teacher in NYC and my calculus teachers at University of Michigan gave quizzes at the beginning of every class. They simply comprised short problems that were based on what was studied in class and the previous homework, to encourage students to actually do work outside of class. But one needed to actually be able to do some problem-solving, calculating, etc., making use of what was studied, not spew easily-regurgitated facts or definitions that could readily be forgotten at least until a chapter test, when they could again be crammed, spewed, and forgotten.
Of course, maybe that's what mathematics is really all about. Maybe my high school teachers had it right, and my university instructors were confused.
No comments:
Post a Comment