Thursday, October 22, 2009

I have come for division!

One of the sillier "education" notions these days - I mentioned it recently - is the idea that children ought to be taught "problem-solving skills" rather than the ordinary traditional subjects which they will need if they are ever to solve problems. No teacher, no administrator, no "doctor" of education (I use it in quotes since they are rarely PhDs and almost never actually teach, which is what the Latin word means!) - none of these educators actually specifies what skills these are, since the few which I know of (e.g. math induction & recursion, automata, graph theory, force diagrams, dimensional analysis, and so forth) all require the basic tools of learning. Ah, well - you think I mean these things ought to be taught in grade school? Certainly, but not necessarily. I think kids are lots smarter than any educator - or any Media Personage - can grasp, and could be getting far more meat far sooner, if there was discipline (which means living & acting as a student does!) and lots less trash and distractions. But this posting is not on the "Thursday" method of education, as interesting as that might be. Besides there is a better way of showing my point on this topic, and it is quite funny. For at the same time these dear ones are pushing for "problem solving skills" they are rejecting traditional skills - like (don't get scared now) - like "LONG DIVISION".

Now, the funny thing is that long division is a problem-solving skill! It solves the question of how many times some given number (say the number of stars) can be split up ("divided") among another given number (say the number of students in a classroom). If there are 5000 stars, and there are 40 students, then each person gets 125 stars. I know that was an easy one, but I am not lecturing about the technique today. (I can, if you want, but it will have to be on my own blogg. But see below for GKC's comments on the topic.)

These dear educatists will say that we use calculators for chores like long division - but that is like saying we can use bicycles to go around the bases during a baseball game! Sure, we could, and get Home lots faster than running! But as worthy a tool as the bicycle is, it is not admitted to be fair part of the game of baseball.

Nor is the calculator a fair part of the game of long division.

That is because long division is a skill which is necesary for other tasks than getting the result of dividing one number by another. It is a SKILL, writ large as FAther Jaki likes to write, and comes up in a whole range of places in mathematics, computing, and such disciplines. But there is another reason for it, which completely escapes the understanding of these educators.

That is Long Division is a means of teaching something much harder to describe than the very simple idea of getting the quotient. In fact, it exemplifies the First Problem Solving Skill one ought to have.

Oh, Doc! Really?

Yes, my child. Really. It is simply stated, and something I would guess you've heard from your mother, especially if you've ever helped her in the kitchen. It is simply this:
Follow the Directions.
Yes. You see, Long Division is a bunch of Directions - it is a - well, since I am a computer scientist, I should use the word "algorithm" - but I am often a baker (and even occasional cook) so I should use the word "recipe", and I am also a scientist (yes, I have a white lab coat!) and so can use the term "lab protocol"; I have been a musician so I could suggest the term "score", and I have read GKC's plays, so I could call it a "script", and I am also a Catholic so I might use the term "rite" (though that is a bit of a stretch).

Long Division is a lot more interesting than "Long Addition" or even "Long Multiplication" because it contains a "step" we computer people call a "conditional" - that is something with an "IF". That is nothing new to any of the above fields of human activity: recipes often have "if" statements, and everything from lab protocols to liturgical rites contain such things.

Does this division business connect to Chesterton? Sure, and in a startling way. I am sure you know the Gospel lines "Think ye, that I am come to give peace on earth? I tell you, no; but separation." (Luke 12:51) or "Do not think that I came to send peace upon earth: I came not to send peace, but the sword." (Matthew 10:34) which GKC relied on when he writes:
Christianity suddenly stepped in and offered a singular answer, which the world eventually accepted as answer. It was the answer then, and I think it is the answer now.
This answer was like the slash of a sword; it sundered; it did not in any sense sentimentally unite. Briefly, it divided God from the cosmos. ... And the root phrase for all Christian theism was this, that God was a creator, as an artist is a creator. A poet is so separate from his poem that he himself speaks of it as a little thing he has "thrown off." Even in giving it forth he has flung it away. This principle that all creation and procreation is a breaking off is at least as consistent through the cosmos as the evolutionary principle that all growth is a branching out. A woman loses a child even in having a child. All creation is separation. Birth is as solemn a parting as death.
[GKC Orthodoxy CW1:281]
Father Jaki elaborates on this idea in several places, notably in his Genesis 1 Through the Ages where he discusses the Hebrew bara which means "create" but also "divide, hack".

Not that I suggest the learning of Long Division is somehow a part of theological training - but of course it is. Theologians, like Philosophers and Historians and all the Students of Words, no less than the Students of Numbers, need to FIRST learn to think according to simple, easy, formulated rules - in order that they can proceed to examine issues for which there might not be such rules! Yes, Long Division is as important to the most esoteric branches of literature and philosophy as good grammar is to the most esoteric branches of engineering and science and mathematics.

Besides, and you may find this most surprising to learn: there are lovely problems in mathematics that calculators (and even computers) cannot solve, and that is one good reason why we need to learn Long Division. I've seen such things at work, and it was not something esoteric either. But as much fun as it is I cannot go into the math of such things here.

To conclude, I'll let you enjoy the only four excellent insights which I found where GKC uses the term "Long Division". The hilarious thing is that one of them says almost the same thing I've been trying to say - but it's nearly 100 years old. Odd that the modern up-to-date educators are still trying such failed and fusty old methods...

It is unfortunate that common-sense has come to mean almost the contrary of the sense that is common. Indeed, we might say that when men boast of common-sense, it generally means a contempt for common people. A man who will not listen to any evidence in favour of ghosts or witches may (especially in his own opinion) possess sense; but what exactly he does not possess is common-sense. He has no realisation of the common bond of human instinct and experience which binds him to the very varied memories and lives of his fellows. He may be right in saying that he has no nonsense about him; a very lamentable gap in any man's character. But the general impression of a borderland of abnormal experiences is not nonsense. It IS sense, even if to some it seems like the suggestion of a sixth sense. It is not nonsense either in the bad or in the good sense. It is not a confusion of thought or a contradiction in terms. It is not a fantastic form of art or a grotesque form of beauty. Spirit-rapping does not introduce us to the Mad Hatter or the Pobble Who Had No Toes; would that it ever introduced us to anybody so entertaining! On the other hand, it is not nonsense to say that a man's soul went out of his own body, as it is nonsense to say that he jumped down his own throat. It is simply an assertion, true or false, about certain conditions on another plane, which are different from the laws of our planet, but not different from the laws of our reason. It is certainly unknown; it may be unknowable; but it is not unthinkable. It is not like saying that long division is green, or that Wednesday is oblong, or that thought is a molecular movement.
[GKC ILN Jan 12 1929 CW35:21-22]

Shelley invented half a hundred goddesses, but he could not pray to them, not even as well as the old atheist Lucretius could pray to Venus, Mother of Rome. All Shelley's deities were abstractions; they were Beauty or Liberty or Love; but they might as well have been Algebra and Long Division, so far as inviting the gesture of worship goes. In this, as in everything else, what is the matter with the new pagan is that he is not a pagan; he has not any of the customs or consolations of a pagan.
[GKC Jul 5 1930 CW35:339]

A peasant who merely says, "I have five pigs; if I kill one I shall have four pigs," is thinking in an extremely simple and elementary way; but he is thinking as clearly and correctly as Aristotle or Euclid. But suppose he reads or half-reads newspapers and books of popular science. Suppose he starts to call one pig the Land and another pig Capital and a third pig Exports, and finally brings out the result that the more pigs he kills the more he possesses; or that every sow that litters decreases the number of pigs in the world. He has learnt economic terminology, merely as a means of becoming entangled in economic fallacy. It is a fallacy he could never have fallen into while he was grounded in the divine dogma that Pigs is Pigs. Now for that sort of intellectual instruction and advancement we have no use at all; and in that sense only it is true that we prefer the ignorant peasant to the instructed pedant. But that is not because we think ignorance better than instruction or barbarism better than culture. It is merely that we think a short length of the untangled logical chain is better than an interminable length of it that is interminably tangled. It is merely that we prefer a man to do a sum of simple addition right than a sum of long division wrong.
[GKC The Thing CW3:165]

Education is only truth in a state of transmission; and how can we pass on truth if it has never come into our hand? Thus we find that education is of all the cases the clearest for our general purpose. It is vain to save children; for they cannot remain children. By hypothesis we are teaching them to be men; and how can it be so simple to teach an ideal manhood to others if it is so vain and hopeless to find one for ourselves?
I know that certain crazy pedants have attempted to counter this difficulty by maintaining that education is not instruction at all, does not teach by authority at all. They present the process as coming, not from the outside, from the teacher, but entirely from inside the boy. Education, they say, is the Latin for leading out or drawing out the dormant faculties of each person. Somewhere far down in the dim boyish soul is a primordial yearning to learn Greek accents or to wear clean collars; and the schoolmaster only gently and tenderly liberates this imprisoned purpose. Sealed up in the newborn babe are the intrinsic secrets of how to eat asparagus and what was the date of Bannockburn. The educator only draws out the child's own unapparent love of long division; only leads out the child's slightly veiled preference for milk pudding to tarts. I am not sure that I believe in the derivation; I have heard the disgraceful suggestion that "educator," if applied to a Roman schoolmaster, did not mean leading our young functions into freedom; but only meant taking out little boys for a walk. But I am much more certain that I do not agree with the doctrine; I think it would be about as sane to say that the baby's milk comes from the baby as to say that the baby's educational merits do. There is, indeed, in each living creature a collection of forces and functions; but education means producing these in particular shapes and training them to particular purposes, or it means nothing at all. Speaking is the most practical instance of the whole situation. You may indeed "draw out" squeals and grunts from the child by simply poking him and pulling him about, a pleasant but cruel pastime to which many psychologists are addicted. But you will wait and watch very patiently indeed before you draw the English language out of him. That you have got to put into him; and there is an end of the matter.
[GKC What's Wrong With the World CW4:64-5]


  1. I've been reading your blog for about 2 weeks, very funny that you should talk about this. I was teaching CCD last night to 7th graders who for the most part couldn't divide the number number of Catholics by the number of priests in the world on the chalk board. They set up the problem all right, but when it got down to actually doing the division, they were completely baffled.

    I've always been opposed to the calculator though; evil tool of the communists. I've said so since 'I' was in 7th grade when I crazily hacked a calculator to bits during a speech running for class president. Something that would most definitely get on of the 7th graders in my class expelled from school forever nowadays. And it's only been 12 years since I was in 7th grade!

  2. Dr. Thursday, I agree with all the philosophical and pedagogical points in your post, and (of course) in your quotations from Chesterton. But I still hate long division and harbour violent impulses towards all the teachers who ever inflicted it on me. And I'm sure none of the maths I suffered through did me the slightest bit of good. Still, I think the whole "teach them problem solving techniques, draw their intrinsic creativity out" approach dates back at least to Bertrand Russell's unfortunate school.

  3. Being a math teacher, I very much enjoyed reading the post. The prevalent philosophy is definitely that teachers should never tell kids anything. They should figure everything out themselves by doing activities.

    In my class, I only lecture for 5 minutes while the other 75 min the kids all work on problems in pairs while I go around and help those in need.

    The very best way to teach someone math is to have them do it, because it requires them to figure it out themselves. Thus, forcing them to be "problem-solvers." But you don't do that by saying do 1533/3 and not tell them how to do it. You give examples and then have them mimic you and apply the procedure to their problems.

    Although, one of the most challenging aspects of teaching is having them set up the problem. If you try introducing real-world implementations of using equations they are very much challenged and have a hard time grasping how an equation or formula applies in a particular instance.

  4. It probably says something about me that I love long division.

    And as a homeschooling mother, I prohibit calculators until the appropriate time during advanced algebra.

    Even if one thinks math will never used in a person' future life, the brain skills of learning to do math do come in handy. Math skills expand the thinking powers of the brain, help one to reason logically, and exercise brain matter in ways that are helpful in all of life.

    Eventually, there will be a real problem to solve in life, and after doing the difficult convolutions of long division, the person's brain will solve the new problem in life.

    And I agree with DavyMax: you have to DO math, you have to solve the problems with your own pencil and paper.

    Math: there's an answer, and it's a real answer. There's something very satisfying in that, which the brain enjoys. At least, my brain does.

  5. I recall a newspaper headline from the pontificate of John Paul II: "POPE SEEKS END OF LONG DIVISION." The reference was to the division between the Roman and Orthodox communions (or maybe between Catholics and Anglicans), but as a mathematics teacher I got a chuckle out of it.

    I find that my students are usually at sea when it comes to finding not the quotient, but the remainder, when one integer is divided by another, because calculators typically convert the remainder into a decimal fraction. (And yes, there are times when the remnainder is more pertinent than the quotient.)

  6. Nancy, I have to disagree with you about maths being satisfying, and even about maths problems having an answer. I think I must be a genius because even before my voice broke I realised what it took Wittgenstein a long time to cotton to, viz., that maths is a system of tautologies, and decided there and then to have nothing more to do with it. That five and five is ten (I just checked it on my calculator) is equivalent to saying that ten is ten. There is no new knowledge, so how is there an answer?

    Also, computers and calculators can do sums much better than humans and that proves it's not worth doing in the first place, just like playing chess.

    (By the way, I'm sure you're right and I'm wrong. But if I want to justify the fact that I can't do the simplest mental arithmetic because I daydreamed through all my maths classes, why shouldn't I?)

  7. Are you being serious Maolscheachlann?

    If so, I think you're missing out on a huge concept in life. Just because a computer does something better than us doesn't mean it's not "worth doing in the first place." Also, computers still don't play better than the best humans in chess. Also, computers can't do math at all. The only thing they can do is compute (duh!). A computer will never solve a math problem. A computer will never prove that there are an infinite number of primes (Go ahead give it a try). It will never prove the Reimann hypothesis or the Axiom of Choice (even humans can't do that). You might think that these things are obscure and unimportant, but if you do anything on the computer requiring security you owe it to math. The computer itself would be impossible without the mathematical principles first being discovered, let alone the internet and online security. How about physics? Is that useless too? Without the non-euclidean geometries first investigated by mathematicians in the early to late 19th century, the theory of relativity and thus all modern physics would be absolutely impossible (not to mention all the math that goes into quantum physics and string theory). I could go on, but I got to stop eventually...

  8. I was pretty sure the computers had overtaken the humans in chess by now....and I know that drafts (or checkers) has been solved. The perfect game ends in a draw.

    But I'm glad to hear the Reimann hypothesis and the Axiom of Choice (which I've never heard of) won't be solved by machines...

    I think you learn what you're wired to learn. When I was a kid, I never forgot a new word once I heard it, and it was like getting a shiny new toy. I still remember the first time I heard many "big" words (like "opaque" or "capitalism"). But maths...maths was torture every day. And I get by just fine without it. I think it should be an optional subject.

  9. Well, maybe not quite that drastic....

  10. Maolsheachlann - you're a kindred spirit. I home educate my children. Of the ones who do math LESSONS (I'd say that they are all doing math, when they are playing piano or cooking recipes), math comes fairly naturally to two of them and it is pure torture to a third child. I sympathize with the girl, not only because it is like watching myself all over again, when I was her age, but also because I cannot look her straight in the eye and tell her that she's ever going to do most of what is in the book in her everyday life. The most complex it gets for me is figuring out how much paint I need to do a room or how much mulch I need for the garden.

    Davymax3 - not everyone wants to do those things. I'm thankful someone else has figured out how I can securely pay bills online and that someone else has built a series of dams to make my corner of the world habitable -- I'm just as thankful that God wired some brains to write poetry, compose music or coreograph ballets, all of which are mathematical in a very different way. Just not in a you-are-obligated-to-sit-through-trigonometry-in-high-school sort of way.

    Being exposed to a variety of subjects is important, I believe. However, I disagree with the way our current state laws will require my non-mathematical child to take three years of advanced math in high school. If you know that you are going to enter a cloistered convent, if you know that you want nothing more than to breed golden retrievers on the family farm, if you want to become a pastry chef, why are you obligated (either by law or by lofty moral ideals) to suffer (and have your GPA suffer) through most of high school mathematics?

  11. Well, my initial comment was more or less tongue-in-cheek, but I'm glad I made it, since I learned something that interested me and reassured me (Davymax's comment that computers "can't do math at all") and also it prompted Colleen's interesting contribution! I just hope I haven't dragged down the level of the discussion on Dr. Thursday's post.

    Colleeen....I sympathise with your daughter! (And I admire you so much for home-schooling, which must be incredibly difficult.) I don't want to come over all "poor me", but I remember maths being quite traumatic. The maths lesson was a Sword of Damocles hanging over the rest of the school day, and even when it was over, there was the evening's homework to suffer through.

    I do understand that we neeed standards and syllabuses (syllabi?) in school. But I think that, without falling into the "drawing out" fallacy that Chesterton (and Dr. Thursday) were complaining about, education should recognise that people have different intelligences. Just as some people are dyslexic, other people are mathematically challenged and just will NOT benefit from any amount of tuition beyond the purest basics. How to tailor an education policy to these differences? Well, I don't know.

    Again, I'm sorry I dragged us a little off topic.

  12. Actually, I think that many Doctors of Education today would agree (as I do) with a lot of what Dr. T. and Davymax3 say above--that students learn skills best when they are taught along with content, that instructors should teach principles and processes before asking students to apply those principles or engage in those processes, and even that long division is a problem-solving skill (as long as students learn how to use it to solve problems whose significance they can understand). Davymax3's pedagogical advice--"You give examples and then have them mimic you and apply the procedure to their problems"--sounds positively trendy (as well as positively right); just search for "scaffolding" on an education-related database like ERIC, and you'll find Ed.D's saying something close to the same thing. It sounds like Davy could probably point out a bunch of administrators who are not so enlightened; I get that, and I think it's partly a problem of a disconnect or lag between theory and practice. For a long time, education needed to move away from too much lecture and rote memorization towards more application and discovery; I think people who think seriously about education now realize the problems that arise when the pendulum swings too far in the other direction, but, in practice, it may take awhile for the pendulum to noticeably move towards the middle.

    I'm not saying that education studies as a field has somehow gone totally Chestertonian! Many Ed.D.s are social constructionists who would not agree that "education is ONLY truth in the state of being transmitted" (they would probably say that education is also truth in the process of being constructed or discovered). But there's enough common ground for dialogue, and it's possible to "divide" different point of view without polarizing.

  13. Brian. Rote memorization. That puts me in mind of one of my biggest pet peeves. Oh how I wish rote memorization back! Then I would not have editors of drama journals with PhDs in drama ask me to footnote Aristotle because they don't understand what "poetics" means.

    Then we wouldn't have college freshman who don't know the difference between a subject and a predicate.

    What good is memorization without understanding, some might ask. I ask what good is understanding when you can't remember what it is that you were supposed to understand in the first place.

  14. Sure--I agree that there's a place for memorization. I'd like more of my freshmen to have memorized what a subject is and what a predicate is.

    On the other hand, I'm thinking of an old composition placement exam that assessed whether students knew grammatical terms but did not ask them to do any substantial writing. And I remember that some of my own English teachers spent much more time on gramamtical terminology and sentence diagramming than on how to convincingly develop and communicate an idea in an essay.

    I think that the movement away from an overemphasis on memorization was a good thing, but I agree with you that ignoring memorization altogether is not good at all.

  15. Speaking of rote memorization, one thing that I would really like to see is more rote memorization of poetry. I personally think that, rather than all the comparing and contrasting and exploring, schoolkids should be required to learn fifty poems off by heart (over their entire schooling, I mean). Can you imagine how it would enrich a country's cultural life to have fifty classic poems that EVERYBODY recognised, and could quote at will? Can you imagine what a base that would be for future reading of poetry, at the age when people begin to appreciate having had to squeeze Ozymandias into their brains? I seriously think you could dispense with virtually ALL analysis of poetry if you only had a certain corpus committed to memory.

    Which is even further from problem-solving skills...sorry....

  16. Yes--I STILL remember the poems I had to memorize in sixth grade. Walt Whitman, Robert Frost, Shakespeare. I hear you Brian--I didn't mean to imply that you were damning all rote memorizing--only that it brought to mind my mourning of the gorgeous baby that went out with that bathwater.

    Memorization is a skill that needs to be exercised in order to strengthen. It's like mental gymnastics. Not to mention that memorized information operates differently at a cognitive level than information only abstractly conceptualized.

    As a theater department professor let me tell you--the student actor's inability to memorize simple text is nightmarish. It's because they've simply not done it, and have not developed the technique for memorizing it.

    It's the same with performing music. There is only so much one can do with sight-reading. To truly interpret a piece it must be memorized.

  17. I agree about the value of memorizing poetry. My dominant memories of memorization from primary school have to do with the multiplication tables--and those memories aren't pretty! I would definitely have warmer and fuzzier feelings about memorization if I'd been told to memorize poetry as a child. I might even have minded the multiplication tables less if I'd been flexing my memory muscles on poetry, which (I'm pretty sure) would have come easier for me. And as Maolsheachlann said (and as doctors of education also say) "education should recognise that people have different intelligences."

  18. I agree with the last, but I'd also say that you can embrace the contrapositive there. Education should also recognise that people have different stupidities.

    And that is where teacher meets student. :)

  19. Absolutely! I might use that "different stupidities" line with the Writing Center peer tutors I supervise; every now and then a tutor worries about "feeling stupid" when trying to help s student writer who knows a lot about some subject matter that baffles the tutors--but those writers often don't know as much about writing as the tutors do.

  20. I agree with the theory of different stupidities. In fact, I consider myself something of a polymoron.

  21. Maolsheachlann: polymoron. Okay. I'm stealing that.

    B: yes. You know I think it is funny this "different intelligences" I was trained to use this theory to design a curriculum "in which all could succeed" and my issue was--shouldn't this knowledge also be used to identify where students are FAILING and meet them there? As opposed to jerry-rigging the curriculum so that everyone feels good about their intelligences?

    It's a slight difference in understanding of education. But e duce lead out of . Stupidities are latent abilities that need to be led out. Intelligences are things that are already out there and then where does the need for "e duce" come from?

    Fun convo, as usual, hosted by the charming Dr. T. Thanks!

  22. Dr. Thursday, I am in complete agreement with what you have to say. I would humbly add that it would likewise be helpful if along the way we taught the sister to mathematics, Aristotlean logic as well.


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