Note: This essay and those that following are based on posts I made on Internet usegroups
in 2000-2001.
At the time, I got precisely the pushback that I expected, which is
why I posted in the first place.
Let's see if I can present my thesis any clearer this time.
Note: The link to the next essay in this series is at the bottom of this page.
We need to memorize things to understand things. Once we memorize things, to understand
(or comprehend) how they relate to each other to form a whole concept, we need to construct
relationships between them. I have been preaching this simple message since the mid-1980s,
but it falls largely on the deaf ears of people who have come to hate rote memorization. But
this is how Einstein viewed the process of conceptualization:
Some readers may be puzzled: didn't we learn all about the foundations of physics when
we were still at school? The answer is "yes" or "no," depending on the interpretation.
We have become acquainted with concepts and general relations that enable us to comprehend
an immense range of experiences and make them accessible to mathematical treatment.
"On the Generalized Theory of Gravitation," A. Einstein, Ideas and Opinions,
(first published in 1950 in Scientific American) Three Rivers Press,
p. 341, [1954, 1982].
On the other hand, you can skip my defense of the thesis that memorization comes
before understanding, and just read what ChatGPT thinks about it by using the link
ChatGPT defends the thesis that
memorization comes before understanding (presented in both English and French)..
I even asked the Bing chatbot to comment on my thesis and this link will
take you to its reply and my extensive rebuttals.
Bing's chat bot defends and attacks the thesis that
memorization comes before understanding..
Abstract: Throughout this series of essays on memorization, I will pursue two themes at the same time. The first theme is to define memorization and to advocate for why we need to do it. The second is to define understanding and to advocate for why we need to do that, too. On the controversy of the memorization vs. understanding debate in education, I will not take sides, except to debunk them both. At every level of one's formal educational completion, one should both know more (have memorized more) and understand more than one had at the beginning of it. We need to stop framing this debate in the form of an either-or fallacy and recognize that we need both. By the way, my definitions of memorization, understanding, and their related terms are meant to be strictly utilitarian.
Section 1: We're all born with a self-defeating 'entitled' attitude of wanting an easy road to knowledge.
The following is a quote from the book Zen and the Art of Motorcycle Maintenance. Background: The story tells of the adventures of its author Robert Pirsig and his son Chris (who was about 12 at the time), as they traveled the Northwestern United States by motorcycle, while the adult Pirsig regularly interjects his recounting to the reader of his philosophical and psychological adventures that followed him after he struggled to provide a rigorous metaphysical justification for the existence of objective Quality. Chris starts off this short dialogue with his father:
"Can I have a motorcycle when I get old enough?"
"If you take care of it."
"What do you have to do?"
"Lot's of things. You've been watching me."
"Will you show me all of them?"
"Sure."
"Is it hard?"
"Not if you have the right attitudes. It's having the right attitudes that's hard."
---- Zen and the Art of Motorcycle Maintenance
So, we probably all see in Chris an image of ourselves, who have to face the daunting prospects of learning a new and difficult subject, and feel inadequate to the task, and to succumb to self-doubt and resentment. And it's here that the elder Pirsig gives us some very good advice: If we want to learn something well, we must be all-in on it. We must have the right attitude on it. If we want to excel at something new, we need to get over our natural resentment to putting in the hard work to get there. In the midst of this present culture that denigrates memorization, we need to make a separate peace with the obligation to memorize a lot of things. We must be ready to do whatever it takes to accomplish that hard intellectual goal -- and that includes casting off our natural dislike of memorization, be it by rote or otherwise. As for me, I overcame my own natural resentment of rote memorization decades ago. Until you overcome your hatred of memorization, you are wasting precious time.
Now, I want to describe a bit of my own formal educational experiences. In fifth grade, I noticed that I had a little talent for math, but math in fifth grade wasn't very hard. In high school, I got pretty good grades, but I was never the best kid in the class at anything. I didn't at that time particularly like math, but I loved science, and I knew that I'd have to do well in math to do well in science, so that was my primary motivation to study math at that time. (Since then, I have developed a love for math, as well.) What defeated me the most in my high school math classes was algebra word problems. At that time, I never understood how to solve for mixed-rate formulas on my own. Since then, I have figured out how to do this with a general technique I call Scheme (and I'll return to this topic later). And because I couldn't at that time comprehend how to derive the formulas for mixed-rate problems on my own, I would characterize my knowledge of the subject at that time as 'fragile'.
But what do I mean by 'fragile'? I define a mixed-rate problem in the abstract as the situation where two or more 'machines' work together, usually at different rates, to accomplish one task or job. If the rate at which the nth machine works to complete the job on its own is Rn in units of 1 job / hour (to give it a convenient time unit), then we can solve for the time it will take the nth machine working by itself to complete the job as given by
1 job = Rn T.On solving for T, we get
T = 1 / Rn,where I have suppressed the units. Now, here's where it gets interesting. Say we now have two machines working together at rates R1 and R2. If the two machines start and stop at the same time, how long will it take them to complete the job? Well, we'll begin by defining this common work time as T. Then, the equation to write down is this
1 [job] = R1 T + R2 T = (R1 + R2) T.On solving for T, we get
T = 1 / (R1 + R2).But what if the rates for each machine is instead given to us in inverted units, so that R1 and R2 are replaced by r1-1 and r2-1, respectively, then this last equation would instead be expressed as
T = 1 / (1/r1 + 1/r2),and this result is more commonly expressed as
T = r1r2 / (r1+r2).This last formula is probably familiar to most of my readers, and it's okay to memorize it, but what happens if the problem is changed even slightly? Then the formula breaks because it's fragile. For instance, if the two machines do not start and stop at the same time, they may not work the same total time, as has been assumed to get this formula. Then what do we do? Simple. We assign to each machine the total time they each work on the job, T1 and T2, respectively.
This is what I mean when I claimed that my knowledge by rote learning alone was fragile. The problem does not lie in rote memorizing this formula. The problem lies in not understanding how to produce it on one's own, so as to correct it if the problem changes.
I'll return to the theory of solving word problems in the next installment.
Section 2: Advice from the experts.
In a video-taped interview, physicist Richard Feynman was asked about how an 'ordinary person' can achieve understanding of the difficult subjects he mastered. His answer was this:
You ask me if an ordinary person, by studying hard, will get to be able to imagine these things like I imagine. Of course! I was an ordinary person -- who studied hard. There's no miracle people. It just happens that they got interested in this thing and they learned all this stuff. They're just people. There's no talent, a special miracle ability to understand quantum mechanics or a miracle ability to imagine electromagnetic fields, that comes without practice and reading and learning and study. So, if you say, you take an ordinary person, who's willing to devote a great deal of time and study and work and thinking and mathematics and so on (?), then he's become a scientist.I have to disagree with Feynman to the extent that I have seen brilliant people who learn difficult subjects quickly, that I can bearly comprehend at all; though I agree with him that even a dummy like me can make some headway after a lot of patience and hard work.
Now, I'll let the crafty Lt. Columbo weigh-in on the matter. Columbo is speaking to the brilliant murder suspect he has been mentally sparring with:
All my life I kept running into smart people. I don't just mean smart like you and the people in this house. You know what I mean. In school there were lots of smarter kids. And when I first joined the force, sir, they had some very clever people there. And I could tell right away that it wasn't going to be easy making detective as long as they were around. What I figured, that if I worked harder than they did, put in more time, read the books, kept my eyes open, maybe I could make it happen. And I did! And I really love my work, sir.So, let me make the obvious observation. That the end result of all his putting in more time, reading the books, keeping his eyes open resulted in Columbo having a lot more information in his brain than when he first started -- and that is the process of memorization.
-- "The Bye Bye Sky High IQ Murder Case"
Section 3: Examples from my own life --'no pain, no gain'.
When I was in high school, although I got better than average grades, I was never that guy to whom difficult subjects came easy. It frustrated me a lot in those days. Actually, it still does. I graduated in 1983 with a BS in math, but that was many years longer than I was supposed to have graduated. Everything I have learned, has been because of great effort on my part. Nothing hard ever came easy to me.
In the mid-eighties, I started to tutor mathematics, which lasted for three or four years. I tutored A, B, and C level students, mostly. And at the end of that time, I noticed something profound about my students: When I showed up for the tutoring sessions, my A students were always more familiar with the prerequisite material and the section definitions than were the lower-scoring students. The point is that the better students were more prepared at tutoring time simply because (I presume that) they had put in more practice time and more memorization -- accomplishments that do not require superior intelligence or a knack or a superior talent. It just requires putting in the time and doing the hard work. Maybe the A students were a bit more motivated to learn and maybe they had a bit more natural ability (but I don't know that for sure), but I wouldn't have called any of them brilliant. After all, they too needed tutoring to guarantee their A in the class.
So, let's have an example: Say I have and A-student and a C-student at the same place in the same text, just at the start of learning trigonometry. So, at the point that my A-student has memorized the meanings of sine, cosine, and tangent of the interior angles of a right triangle, my C-student is still having trouble remembering the standard named parts of a right triangle. The A-student was used to memorization for the sake of good grades. He did this either out of long habit, so that it had become second nature to him, or because he thought it was expected of him (those were the 'bad old days', I'm told), or because in his or her moments of clear, rational thought they made a separate peace with the obligation to do a lot of memorization to succeed in difficult subjects in school. It's still just a case of 'no pain, no gain' -- at least for most of us.
Thus, I came up with a simple line to encapsulate this wisdom
Memorization comes before understanding.Now, I know that the modern knee-jerk reaction to this claim is to put it down totally and immediately. But this is ludicrous to me. How is one ever to piece together the parts of a concept into a meaningful complete conceptual whole until one has all the pieces in one's mind to mentally manipulate? For example, if you don't know the parts of a right triangle, you have no business trying to tackle trigonometric functions until you do.
Section 4: The fragility of only rote learning.
Now, for another quote from Feynman:
I don't know what's the matter with people: they don't learn by understanding; they learn by some other way--by rote, or something. Their knowledge is so fragile!I used the word 'fragile' before, when I was explaining my trouble with word problems, because I knew that I would place this Feynman quote here and I wanted to give it a context. I can't tell from this quote whether Feynman is putting down rote learning, per se, here, or whether he's putting down an over-emphasis on just rote learning. In the case of my own forced reliance on rote memorization, to solve word problems while taking algebra, it was not because I was lazy or because I didn't want to put in the time to learn. Rather, it was because apparently neither my teacher nor my textbook made clear to me how to solve such problems in general. In other words, what are the general heuristics of solving algebra word problems, especially of the mixed-rate type? It would take me about fifteen years after graduating high school that I would, by much effort, finally figure out how to approach such problems successfully. In fact, I have written up the heuristics and they are published on this website.
Algebra Word Problems (Scheme). A general heuristic for solving algebra word problems. Many solved problems presented and the techniques extended to stoichiometry. (This is word problems for chemistry).
Anyway, like I said, I disagree with Feynman that there are no miracle people, that is to say, brilliant people. I've seen a few of them myself. I was even once falsely accused of being a brilliant person! So, I'll explain, because it goes to show that people have some very messed up ideas about who is able to learn what, and how one should go about learning difficult stuff.
Section 5: Apparently, I'm hopelessly old-fashioned, or, Have 3 x 5 Cards, Will Travel.
If my memory serves me correctly, in the mid-2000s I had a dubious flirtation with the idea of going into the craft of computer-fixer technology. I had been studying the subject on my own for about ten years at that time and I thought that the best way to proceed would be to obtain standard credentials, such as A+ and Net+ certifications. So, to pass the A+ exam the first time, I worked even harder, put in more time, read the books, kept my eyes open -- all the stuff Columbo recommended. But I learned from my online queries that I could also take a class in A+ from a local community college -- so I did. Anyway, about two-thirds the way through the semester, shortly before class one day, I was walking around the parking lot, studying A+ from notes I had placed on 3 x 5 cards. Soon, one of my fellow classmates drove by me and said hello and then asked what in the world I was doing. I replied that I was studying for class from note cards -- yes, ugly rote memorization!! Well, the guy was clearly flabbergasted and said, "But you're the best student in the class!!" (Wow, I guess there's a first time for everything!) His obvious misconception was that good students don't need to do rote memorization. I blame this misconception on the misguided academic war against memorization that has been going on for the last fifty years or so.
I distinctly remember that on one of those cards was the voltages of the four colored pins of the Four-Pin Molex Power Supply Connector. Now, I know that one could argue, "That's just dumb busy work. You could look that up whenever you need to!" Although I agree that no one can set every aspect of computer-troubleshooting technology to long-term memory, you can't use that as an excuse to not memorize the important things of your subject area. If you're going to go out into the field as an expert, by which I mean go to someone's home or office to competently fix their computer, you're going to have to carry with you in your memory a lot of specialized computer information in both hardware and software to aid in hardware problem-solving. For the record, I passed the A+ test the first time. Yay!
Anyway, like I said, I made a separate peace with learning by rote in the mid-1990s, when I started to make plenty of 3 x 5 note cards and to carry them around with me and to study seriously from them, whether I'm in formal education or not. For example, recently I began to study category theory. One of my sources is a YouTube video series. When the instructor mentioned the phrase 'thin category', I didn't know what it was, so I looked it up and wrote it down on a note card to study from.
Section 6: Conceptual Understanding: Identifying relationships among the Parts = Understanding the Whole.
What we have here is the age-old problem of the pendulum swinging hard in the opposite direction. I am absolutely not recommending that formal education go back to emphasizing rote memorization at the expense of understanding. But, honestly now, how can any rational person maintain that it is possible that the parts of a complicated concept can be mentally put together to form the concinnity of a whole, if one hasn't bothered to memorize the parts first? And what I have described is called conceptual understanding. Let's formalize this:
Conceptual Understanding is any process by which relationships can beAnd, yes, I'm leaving the term 'comprehension' as undefined, so as not to make the mistake that Pirsig made in trying to define 'Quality'.
assigned between/among the parts of a whole to provide a comprehension that is greater
than the comprehension of the sum of the parts, considered individually.
For example, for a person to claim to understand what a right triangle is, he or she should know what a right angle is, what the interior angles are, what the sides and hypotenuse are, and, in my opinion, what an altitude from a vertex to its opposite side is. So, only when one has memorized these parts and memorized the relationships among them, can one honestly claim to 'know what a right triangle is'.
Okay, so this is my definition of Memorization:
Memorization is any process by which information has been stored
in the brain so that the conscious mind can retrieve it both instantly
and accurately. Any piece of information so stored is said to be memorized.
"Oh, just one more thing, sir." Why Memorize, Part 2 .