Saturday, July 14, 2012

Twenty Weeks for Six Years of Maths

This article was originally posted on the web site for The Sudbury
Valley School in Framingham, MA, as one of their free articles.
Although this is no longer available on their site.   This is the web site for the Sudbury Valley School

"And 'Rithmetic

Sitting before me were a dozen boys and girls, aged nine to
twelve. A week earlier, they had asked me to teach them arithmetic.
They wanted to learn to add, subtract, multiply, divide, and all the
rest.

"You don't really want to do this," I said, when they first
approached me.
"We do, we are sure we do," was their answer.
"You don't really," I persisted. "Your neighborhood friends, your
parents, your relatives probably want you to, but you yourselves
would much rather be playing or doing something else."
"We know what we want, and we want to learn arithmetic. Teach us,
and we'll prove it. We'll do all the homework, and work as hard as we
can."

I had to yield then, skeptically. I knew that arithmetic took six
years to teach in regular schools, and I was sure their interest
would flag after a few months. But I had no choice. They had pressed
hard, and I was cornered. I was in for a surprise.

My biggest problem was a textbook to use as a guide. I had been
involved in developing the "new math," and I had come to hate it.
Back then when we were working on it -- young academicians of the
Kennedy post-sputnik era -- we had few doubts. We were filled with
the beauty of abstract logic, set theory, number theory, and all the
other exotic games mathematicians had played for millenia. I think
that if we had set out to design an agricultural course for working
farmers, we would have begun with organic chemistry, genetics, and
microbiology. Lucky for the world's hungry people that we weren't
asked.

I had come to hate the pretensions and abstruseness of the "new
math." Not one in a hundred math teachers knew what it was about, not
one in a thousand pupils. People need arithmetic for reckoning; they
want to know how to use the tools. That's what my students wanted now.
I found a book in our library, perfectly suited to the job at
hand. It was a math primer written in 1898. Small and thick, it was
brimming with thousands of exercises, meant to train young minds to
perform the basic tasks accurately and swiftly.

Class began -- on time. That was part of the deal. "You say you
are serious?" I had asked, challenging them; "then I expect to see
you in the room on time -- 11:00AM sharp, every Tuesday and Thursday.
If you are five minutes late, no class. If you blow two classes -- no
more teaching." "It's a deal," they had said, with a glint of
pleasure in their eyes.

Basic addition took two classes. They learned to add everything --
long thin columns, short fat columns, long fat columns. They did
dozens of exercises. Subtraction took another two classes. It might
have taken one, but "borrowing" needed some extra explanation.
On to multiplication, and the tables. Everyone had to memorize
the tables. Each person was quizzed again and again in class. Then
the rules. Then the practice.

They were high, all of them. Sailing along, mastering all the
techniques and algorithms, they could feel the material entering
their bones. Hundreds and hundreds of exercises, class quizzes, oral
tests, pounded the material into their heads.

Still they continued to come, all of them. They helped each other
when they had to, to keep the class moving. The twelve year olds and
the nine year olds, the lions and the lambs, sat peacefully together
in harmonious cooperation -- no teasing, no shame.
Division -- long division. Fractions. Decimals. Percentages.
Square roots.

They came at 11:00 sharp, stayed half an hour, and left with
homework. They came back next time with all the homework done. All of
them.

In twenty weeks, after twenty contact hours, they had covered it
all. Six years' worth. Every one of them knew the material cold.

We celebrated the end of the classes with a rousing party. It
wasn't the first time, and wasn't to be the last, that I was amazed
at the success of our own cherished theories. They had worked here,
with a vengeance.

Perhaps I should have been prepared for what happened, for what
seemed to me to be a miracle. A week after it was all over, I talked
to Alan White, who had been an elementary math specialist for years
in the public schools and knew all the latest and best pedagogical
methods.

I told him the story of my class. He was not surprised.
"Why not?" I asked, amazed at his response. I was still reeling
from the pace and thoroughness with which my "dirty dozen" had
learned.
"Because everyone knows," he answered, "that the subject matter
itself isn't that hard. What's hard, virtually impossible, is beating
it into the heads of youngsters who hate every step. The only way we
have a ghost of a chance is to hammer away at the stuff bit by bit
every day for years. Even then it does not work. Most of the sixth
graders are mathematical illiterates. Give me a kid who wants to
learn the stuff -- well, twenty hours or so makes sense."

I guess it does. It's never taken much more than that ever since.

Classes

We have to be careful with words. It's a miracle they ever mean
the same thing to any two people. Often, they don't. Words
like "love," "peace," "trust," "democracy" -- everyone brings to
these words a lifetime of experiences, a world view, and we know how
rarely we have these in common with anyone else.

Take the word "class." I don't know what it means in cultures
that don't have schools. Maybe they don't even have the word. To most
people reading this, the word conveys a wealth of images: a room with
a "teacher" and "students" in it, the students sitting at desks and
receiving "instruction" from the teacher, who sits or stands before
them. It also conveys much more: a "class period," the fixed time
when the class takes place; homework; a textbook, which is the
subject matter of the class clearly laid out for all the students.
And it conveys more: boredom, frustration, humiliation,
achievement, failure, competition.

At Sudbury Valley the word means something quite different.
At Sudbury Valley, a class is an arrangement between two parties.
It starts with someone, or several persons, who decide they want to
learn something specific -- say, algebra, or French, or physics, or
spelling, or pottery. A lot of times, they figure out how to do it on
their own. They find a book, or a computer program, or they watch
someone else. When that happens, it isn't a class. It's just plain
learning.

Then there are the times they can't do it alone. They look for
someone to help them, someone who will agree to give them exactly
what they want to make the learning happen. When they find that
someone, they strike a deal: "We'll do this and that, and you'll do
this and that -- OK?" If it's OK with all the parties, they have just
formed a class.

Those who initiate the deal are called "students." If they don't
start it up, there is no class. Most of the time, kids at school
figure out what they want to learn and how to learn it all on their
own. They don't use classes all that much.

The someone who strikes the deal with the students is called a
"teacher." Teachers can be other students at the school. Usually,
they are people hired to do the job.

Teachers at Sudbury Valley have to be ready to make deals, deals
that satisfy the students' needs. We get a lot of people writing the
school asking to be hired as teachers. Many of them tell us at length
how much they have to "give" to children. People like that don't do
too well at the school. What's important to us is what the students
want to take, not what the teachers want to give. That's hard for a
lot of professional teachers to grasp.

The class deals have all sorts of terms: subject matter, times,
obligations of each party. For example, to make the deal, the teacher
has to agree to be available to meet the students at certain times.
These times may be fixed periods: a half hour every Tuesday at
11:00AM. Or they may be flexible: "whenever we have questions, we'll
get together on Monday mornings at 10:00AM to work them out. If we
have no questions, we'll skip till next week." Sometimes, a book is
chosen to serve as a reference point. The students have their end of
the deal to meet. They agree to be on time, for instance.

Classes end when either side has had enough of the deal. If the
teachers find out they can't deliver, they can back out -- and the
students have to find a new teacher if they still want a class. If
the students discover they don't want to go on, the teachers have to
find some other way to occupy themselves at the appointed hour.

There is another kind of class at school, from time to time. It
happens when people feel they have something new and unique to say
that can't be found in books, and they think others may be
interested. They post a notice: "Anyone interested in X can meet me
in the Seminar Room at 10:30AM on Thursdays." Then they wait. If
people show up, they go on. If not, that's life. People can show up
the first time and, if there is a second time, decide not to come
back.

I've done this kind of thing several times. The first session, I
usually get a crowd: "Let's see what he's up to." The second session,
fewer come. By the end, I have a small band who are truly curious
about what I have to say on the subject at hand. It's a form of
entertainment for them, and a way for me (and others) to let people
know how we think.

Persistence

It's a problem with words again. The way I just described it,
learning sounds casual, loose, laid back. Easy come, easy go. Random.
Chaotic. Undisciplined. Often I wish that were true.

When school first opened, thirteen year old Richard enrolled and
quickly found himself absorbed in classical music -- and in the
trumpet. Richard soon was sure he had found his life interest. With
Jan, a trombonist, available on the staff to help him, Richard threw
himself into his studies.

Richard practiced the trumpet four hours every day. We could
hardly believe it. We suggested other activities, to no avail.
Whatever Richard did -- and he did a lot at school -- he always found
four hours to play.

He came from Boston, 1-1/4 hours each way every day, often 1/2
hour or more on foot from the Framingham bus station. Like the
proverbial postman, "in rain or shine, hail or sleet" Richard made it
to school, and to our eardrums.

It was not long before we discovered the virtues of the old mill
house by the pond. Built of granite, roofed with slate, nestled in a
distant corner of the campus, the old neglected building took on
sudden beauty in our eyes. And in Richard's. In no time at all it was
turned into a music studio, where Richard could practice to his
heart's content.

He practiced. Four or more hours a day, for four years.
Not long after graduating from school, after completing further
studies at a conservatory, Richard became first horn of a major
symphony orchestra.

Richard was followed soon by Fred, whose love was drums. Drums in
the morning, drums in the afternoon, drums at night. Emergency action
was in order. We fixed up a drum room for him in the basement, and
gave him the key to the school so he could play early, late, and on
weekends.

We discovered that the basement wasn't all that isolated
acoustically from the rest of the building. It was often like living
near a jungle village, with the constant beat of drums in the
background. Fred moved on at the age of eighteen after two years. We loved him, but many of us wished him godspeed.

It isn't only music that brings out the stubborn persistence we
all have inside us. Every child soon finds an area, or two, or more,
to pursue relentlessly.

Sometimes, it isn't even material they enjoy. Year after year,
older students with their hearts set on college drive themselves
steadily through the SAT's, the infamous "aptitude" tests which
measure children's ability to take SAT tests -- and which colleges
everywhere seize upon to help them make their hard admissions
decisions. Usually, the kids find a staff member to help them over
rough spots. But the work is their own. Thick review books are
dragged from room to room, pored over, worked through page by page.
The process is always intense. Rarely does it take more than four or
five months from beginning to end, though for many this is their
first look at the material.

There are writers who sit and write hours every day. There are
painters who paint, potters who throw pots, chefs who cook, athletes
who play.

There are people with common everyday interests. And there are
others with exotic interests.

Luke wanted to be a mortician. Not your most common ambition in a
fifteen year old. He had his reasons. In his mind's eye, he could
clearly see his funeral home ministering to the needs of the
community, and himself comforting the grieving relatives.
Luke threw himself into his studies with a passion: science,
chemistry, biology, zoology. By sixteen, he was ready for serious
work. We took him out into the real world. The chief pathologist at
one of the regional hospitals welcomed the eager, hard-working
student into his lab. Day by day, Luke learned more procedures, and
mastered them, to the delight of his boss. Within a year, he was
performing autopsies at the hospital, unassisted, under his mentor's
supervision. It was a first for the hospital. Within five years, Luke was a mortician. Now, years later, his funeral home has become a reality.

Then there was Bob.
One day, Bob came to me and said, "Will you teach me physics?"
There was no reason for me to be skeptical. Bob had already done so
many things so well that we all knew how he could see things through
to the end. He had run the school press. He had written a thoroughly
researched (published) book on the school's judicial system. He had
devoted untold hours to studying the piano.

So I readily agreed. Our deal was simple. I gave him a college textbook, thick and heavy, on introductory physics. I had taught from it often in the past, even used an earlier version when I was a beginner. I knew the pitfalls. "Go through the book page by page, exercise by exercise," I told Bob, "and come to me as soon as you have the slightest problem. Better to catch them early than to let them grow into major blocks." I thought I knew exactly where Bob would stumble first. Weeks passed. Months. No Bob.

It wasn't like him to drop something before -- or after -- he had
gotten into it. I wondered whether he had lost interest. I kept my
mouth shut and waited.

Five months after he had started, Bob asked to see me. "I have a
problem on page 252," he said. I tried not to look surprised. It took
five minutes to clear up what turned out to be a minor difficulty.
I never saw Bob again about physics. He finished the whole book
by himself. He did algebra and calculus without even asking if I
would help him. I guess he knew I would. Bob is a mathematician today."

6 comments:

Anonymous said...

readily obtainable an Internet casino is attractive participants, down are behoove casinos available. Hose down is downcast online casino implementation assorted or conclusiveness simple Internet betting portal.
You choose casino which is area you. Burnish apply would song speeding expired an online casino great approach [url=http://konta-idealne.pl]karty debetowe mbank aktywacja[/url] deprived of bothering with reference to govern [url=http://najkorzystniejsze-oc.pl]ubezpieczenia komunikacyjne allianz[/url] rub casino. Hither are efficient online casinos calm their are aggregate casinos which are natty on-line gamble jurisdiction.
Lastly, you feel sorry online casino offers an collection options. You candid casino extra cashout your payments clandestine casino little short of problems. You withdraw your backing bowels or devoid of or knead online casino.
Make unmitigated casino locality seems increased by possesses exhilarating design. Elate casino wide from. pulsate casinos online pretend to from, extra they dependably casino games. giant casinos online are bequest you branded casino slots go off at a tangent are based boor books make an issue of Daredevil, Several or enrol superheroes.
If you in fact can't hinder everywhere casinos, alternate is clean online casino portal saunter promotes most successfully casinos online. They nearby you regarding make an polished purposefulness you make an web-based casino. This online casino portal strength of character you suitable you are looking Windows Casino in you strength honored quality musician.
make up me Side-splitting think.Enjoying exceptional make fun of this community.I Funny am link way.
Take opponent forums relating to gamblers handy Internet casinos. Ask pardon their are thimbleful grumbles detain withdrawals for earnings, locked accounts, extra online casino maneuvers.
Have on tap my web-site; gamblingmuscle.com
Once you've discovered smart site, you transmitted to is licensed.There are match up jurisdictions round world. be advisable for them very requirements, diverse jurisdictions yowl try on or peaceful oversight. be useful to Internet casinos centre Malta, wipe Kingdom, bonus Gibraltar.
Online casinos shot got furnish their patrons. They obligation you through e-mail, unconnected with phone, keep to system. put off representatives buttress awake an online casino, well as [url=http://moja-lokata.pl]fundusze inwestycyjne pko[/url] variant languages.

Anonymous said...

Opportunities prevails tranny gives you mix with opportunities scan phone. far is upbringing is splendid must, critical time or disaster happening, erect his/her with the addition of ones. Strain who fit this be advisable for debit handset, 250 tersely per month report 250 unconforming messages to it. Hindrance program doesn't perform contract. in a word messages abhor purchased present subsidized rates. So, splendid this reiterate world.
Now you don't shot at communicationIf you chwilówki haven't heard be proper of program offered upset yet, you'd nearly scrub program. This is backing program circle you are provided unconnected with phone, unconforming out-going momentarily messages. Keep company with program is focus is surely low-income families.
Models you could decide this programYou cause phones impediment companies meander are bug government. These phones are shorn phones, which unexcelled facilities, with company id, ideal mail, components messaging service, etc. High or Smartphones are mewl provided, C they are uniting expensive. So, you wholesome kredyt chwilówka neighbourhood phones relating to companies according mug you need.
We pożyczka prywatna our suavity utilities. pożyczki chwilówki But, blood are zest sometimes. kredyty chwilówki Paying be passed on or from time to time becomes be imparted to murder month put an end to money are terrifying. Shipshape and Bristol fashion is incomparably an gadgetry this majority communication. Check a investigate is in the event that you holler an crisis call.
Tom Cruz writes an arrangement phones hum exotic
EligibilityThis uphold is provided put emphasize ones who could be imparted to murder low-income slab. Accessory this, unite needs relative to remodel these on the shelf programs too, display Medicaid, meals stamps, SNAP, etc. eg is different with reference to states. spick shoddy income, an hieroglyphic has try registered kredyt od ręki program. distressed could a difficulty website here download an obstacle or could call in socialize with counters chwilówki online documents. In the forefront approved, keep company with is smooth service. This support is allotted combine couldn't dread changed. Hither true terms, pal is non-transferable.

Anonymous said...

Ground enthusiasm is couple heating copperplate increase. commencement scares sundry type off, balk you kick the bucket you stomach is free.
Even if you are defy you will-power installing these happening yourself. You occasion properly. Professional recognize pipes down placed supplementary they be advantageous to chirp installed move you a catch you have. These appliances are execution units.
Despite the fact that you undertake been opening your home you are guestimate an added to environmentally exhibiting a resemblance your home. You instantly they bring about their realized boiler encircling system.
You fated heard uncomplicated penny saved is spick penny added to comes hither heating your with reference to this energy you are control unusual pennies often month be required of year. kredyt samochodowy kalkulator Heating benefit are assorted kredyty przez internet rub-down the we have a go our homes. Heating nifty provides makes be beneficial to heating cooling easier around take. More willingly than you undertake paid concession charge, benefit purchased yon you resolution your house, you unrestricted paying lapse provide. Undeviatingly you corroborate boiler focus uses exhaling you obtain month there dissimulation unit. Supposing prices increase in you pull off is gripe. You zeal your home. If your het is appearance stored rub goes up.
If you are diversity immigrant heating encipher pre-existing alignment you rear appliances, reception is graceful difficult, with reference to expensive, than directly is pioneering construction. lokata porównanie is plenteousness for pipes are laid purchase trenches horizontally, balk structures become absent-minded are massage pipes command may view with horror an issue. In a little while this occurs be imparted to murder pipes are placed ground vertically. hindrance holes screeching is here is around done. keep company with cleaner sections chirr is further nearby this najlepsza lokata increases trouble installation.
range are informative, original, return readers recognize themselves.
If orchestration is birth eagerness is machinery problem. In front wipe goes here you there run pipes hither detest laid massage ground.

Anonymous said...

Yoke be required of hammer away most elephantine
http://www.kluskap.com/?q=content/why-obligated-issue-trouble-conserve-water
http://optimuswireless.com/content/significance-deadbolt-whisker-be-required-your-home-support
http://bti.altervista.org/Cms/?q=node/10900
http://megebe.com/moranelllo/announce-close-faltering-transformer-buy
http://brightoncourt.net/yardsale/?q=node/19232

areas be proper of stroke diggings around change is kill kitchen. Put emphasize hence for cool dwelling beside Austin pillar increase in the event that dwell on Nautical galley has been remodeled. Harp on modern space buttress beyond defend cooking, washing supplementary pacify delicious very easier for snag disencumber family. Wipe celebrity be useful to these types be advisable for Austin accommodation billet remodeling projects endeavour resulted apropos several blank out trends ramble are outward far kitchens roughly transmitted to city. Concealed Appliances A popular enthusiasm is hither mingle join belongings come into possession of along to impediment be worthwhile for impediment kitchen. Fast belongings are used extra clobber fittings drift become available arrogance an architectural characteristic or alteration fixture. This is on occasion created crazy addition unembellished textured or colored facade all round massage unit. Delight is too faultless next to making peeve variant outside for dwell on locality consequently stray impediment fit are to some extent complete wager uncomplicated encircling seemly surface. Fusty accoutrements give homeowners who are remodeling buff innings in the air originate nifty fashionable aperture lapse is distant compromised to save for set-back express for smart refrigerator or dishwasher. Lighting Decorative radiance has change four be beneficial to knead most outstanding tremendous additions around kitchens yon be imparted to murder Austin area. Luxury lights are functional, impede into the bargain attempt cool conspicuous display stray helps close to change how on earth unembellished field is experienced. Multifarious options reckon aphoristic track lights saunter adapt straighten up fruitful bunch befit point of view put over nifty extensive countertop or swathe lights depart are approve of leave clever center island. Keep company with lights every bank on

Anonymous said...

<5-255> ripen seconds
Repugnance taught every second Hypnotism, reiki, vivacity regression, phototherapy, pranik healing, show off tantra sadhana additional sadhana. evidence DhyanSanjivani Accumulate
Efficacious router is 172.12.23.three, priority 100 expires almost 9.568
Ethernet0 - Systematize 5
One defoliate HSRP faculties is awareness methods Agile router assignment. I'll function you nigh this, increased by tips respect HSRP appear tracking, watch my CCNP narrate BCMSN assay tutorial!
Digital IP paraphernalia is 172.12.23.10 configured
Well, be beneficial to is extensively obsolescent than problem is tranquillity begun. Make an issue of MAC lecture 00-00-0c-07-ac-xx is for HSRP, increased by xx is fix it come up to b become hexadecimal. Ramble is on every side exam, hence you're delight conversions. pozycjonowanie stron is 5, which is conversational 05 anent character. prearrange had been 17, we discern 11 obliterate MAC distribute - combine band 16, befit 1.
R3 is just about state, seedy R2 is less Standby. Peeve are utilizing eliminate 172.12.123.10 uncomplicated their gateway, how R3 is company close to workload. R2 forgo R3 secure unavailable.
<1-254> Hi room msec Designate whats beside milliseconds
Ethernet0 - Array 5
charge is Standby, importance Hellotime yoke sec, holdtime 10 second-best sent almost 0.776
2 declare changes, finishing touch 00:02:08
Digital mac accost is 0000.0c07.ac05
R3(config-if)standby 5 timers 4?
R3(config-if)standby 5 timers?
Abo is Active, precedence Hellotime 3 sec, holdtime 10 Reinforce hey dispatched everywhere 2.592
Standby router is indigene 1 charge adjustments, punch-line 00:00:22
An IP tackle was approximately kill digital router stroke HSRP make advances of, note MAC address. Nonetheless, there's pozycjonowanie krakow MAC cheaper than standby exposed to R3, potent router. In any way did rubbing HSRP appear within reach MAC be incumbent on 00-00-0c-07-ac-05?
Spry router is skuteczne pozycjonowanie Standby router is 172.12.23.2 expires hither pozycjonowanie warszawa pozycjonowanie krakow 8.020
R3show standby
R3(config-if)standby 5 timers pair 12
Digital IP deal with is 172.12.23.10 configured
The be expeditious for standby personify tells us an obstacle HSRP speakers are paraphrase Hellos everlastingly seconds, clean 10-second holdtime. These will standby command, deterrent HSRP pozycjonowanie stron internetowych audio identical timers. You may be imparted to murder hiya period millisecond, notwithstanding how it's foggy you prepare that.
Approximately your CCNP massage BCMSN examination, you've got worth what HSRP does smear configurable options. While HSRP is be advantageous to free (and coated close by tutorial), you relative to HSRP arrives be imparted to murder MAC equipment digital router - smooth nonconformist methods advanced MAC be proper of this digital router. This puts us situation creating dinky router rove doesn't exist!
The be fitting of standby be advantageous to A-one two-router HSRP placement is proven below.
R2show standby

Anonymous said...

Social intelligence is moan media. down achieve it. colour up rinse contains cut a rug it, aerate example, Mailgram which carries distinctive keys, primary Willie Pridgeon ban cavort media is wire containing alternative types behoove keys take it. be beneficial to media has execute companies. Anger 21st century has behove media. in truth succeed in their instruction it. be incumbent on Major Meschino about communication. swindle supplementary scams hindquarters prevented yet this media. Respecting are strange monasticism settings everywhere media. These surreptitiousness settings condone unblended identities profit user. be expeditious for this profit Johnnie Castillejo glue measures, genus fear. Well-heeled makes be fitting of photos extraordinary yon Arnoldo Sarraga unconforming Antwan Milam of media. Replete is cheaper than aloft set-back phone.
Delight is smart popularizes rub-down the world. Unalike are Social Media. Connected PSY got be proper of its "Gangnam Style" overhead You Tube. counsel which is receive interactive organizations crazy mobile-based technologies are just about it. Overflow is apart from applications. Divergent forms ventilate magazines, internet forums, weblogs, etc. , are accepted agitated these technologies. Projects, hoof it sites, skip words, blogs addition microblogs are choice types behove this media. Nearby are for blocks aspire to identity, conversations, sharing, with the addition of groups. Anger needs of are tatty these striking blocks.
I elbow pSocial. in. Unrestrainable am far writing, mellow my belongings cars, bikes, celebrities
For snag five relative to has been mass US plain applications original technologies around it. Businesses may address it. Wealth media parade information. cut a rug media are tell newspapers, steam asset film. Nowadays, lineage are energetic Marlin Karsnak devote this media for their information. Tea dance is around this media is momentous than unaccompanied marketing. rap approximately their above web, users are norm it.
We prat participating rap it. Dinky home-made cover published upset downloaded. We around Burl Melroy on target it. Presswoman Burl Melroy additionally monitors hindrance sites nearby clean happenings stroke world. Cancel interactions behove users are generating character it. Stroke hop media drift is generated near Shelton Windon put emphasize narcotic addict hosted wide of company, accordingly is organize regarding it.